TREATMENT OF PHYSICAL AND NUMERICAL DIFFUSION IN FLUID DYNAMIC SIMULATIONS byKang Yul Huh and Michael W. Golay Energy Laboratory Report No. MIT-EL-83-011 August 1983 TREATMENT OF PHYSICAL AND NUMERICAL DIFFUSION IN FLUID DYNAMIC SIMULATIONS by Kang Yul and Huh Michael W. Golav Energy Laboratory and Department of Nuclear Engineering Massachusetts Institute of Technology Cambridge, MA 02139 MIT-EL-83-011 August 1983 Sponsored by: Boston Edison Co. Duke Power Co. Northeast Utilities Service Corp. Public Service Electric and Gas Co. of New Jersey -2- ABSTRACT TREATMENT OF PHYSICAL AND NUMERICAL DIFFUSION IN FLUID DYNAMIC SIMULATIONS by Kang Y. Huh and Michael W. Golay A computer code is developed to predict the behavior of the hydrogen gas in the containment after a loss-ofThe conservation equations for the four coolant accident. components, i.e., air, hydrogen, steam and water, are set up and solved numerically by decoupling the continuity and momentum equations from the energy, mass diffusion and turThe homogeneous mixture form is used bulence equations. for the momentum and energy equations and the steam and liquid droplets are assumed to be in the saturation state. There are two diffusion processes, molecular and turMolebulent, which should be modelled in different ways. cular diffusion is modelled by Wilke's formula for the multi-component gas diffusion, where the diffusion constants are dependent on the relative concentrations. Turbulent diffusion is basically modelled by the k-e model with modifications for low Reynolds number flow effects. Numerical diffusion is eliminated by a corrective scheme which is based on accurate prediction of cross-flow diffusion. The corrective scheme in a fully explicit treatment is both conservative and stable, therefore can be used in long transient calculations. The corrective scheme allows relatively large mesh sizes without introducing the false diffusion and the time step size of the same order of magnitude as the Courant limit may be used. A -3- ACKNOWLEDGEMENTS This research was conducted under the sponsorship of Boston Edison Co., Duke Power Co., Northeast Utilities Service Corp. and Public Service Electric and Gas Co. of New Jersey. support. The authors gratefully acknowledge this thank all those who assisted them The authors during this research especially Dr. Vincent P. Manno, Mr. B.K. Riggs, Ms. Rachel Morton and Prof. Andrei Schor. work was expertly typed by Mr. Ted Brand and Ms. Hakala. Eva The 2 Governing Equations 2.-4TABLE OF CONTENTS Page ABSTRACT ACKNOWLEDGEMENTS TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES NOMENCLATURE CHAPTER 1: CHAPTER 2: INTRODUCTION PROBLEM STATEMENT AND SOLUTION SCHEME 2 3 4 6 12 13 16 21 21 23 25 27 28 29 29 30 30 37 37 42 43 44 45 48 48 53 57 59 62 76 79 81 82 82 83 84 86 87 88 89 91 2.3 Truncation Error Diffusion in a Two Dimensional Problem 4.2 Comparison of Explicit.3 Discussion of Skew Differencing and Corrective Schemes Raithby's Scheme 4.3.2 Truncation Error Diffusion 4.1 Truncation Error Diffusion in a One Dimensional Problem Another Approach for a One Dimensional 4.2 Turbuletit Diffusion 3.3.3 Low Reynolds Number Flow CHAPTER 4: NUMERICAL DIFFUSION Truncation Error Diffusion 4.4.5 SMAC Scheme and Compressibility CHAPTER 3: DIFFUSION MODELLING 3.2.2 Cross-flow Diffusion 4.3.3.1.3.3.2.2.2.2.4.1 Derivation of Turbulence Equations 3.4.1 Mesh Point Implementation Mesh Interface Implementation 4.2 k-e Model 3.4 Solution Scheme 2.3 Basic Assumptions and Limitations 2.1 4.4.2 Limitations 2.1 Continuity and Mass Diffusion Equations 2.1 Skew Differencing by Huh 4.1 Problem Statement 2.2 Energy Equation 2.1.1 Molecular Diffusion 3.2 Physical Constraint 4.4 4.1.2 .1 4.3.3.4.4.1 Assumptions 2.1 Diffusion Convection 4. Implicit and 4.2.3 Tensor Viscosity Method Corrective Scheme by Huh 4.2 4.4 ADI Schemes Conservation 4. -5- Page 4.4.3 Accuracy 4.4.4 Stability 4.4.4.1 Explicit Scheme 4.4.4.1.1 One Dimensional Central Differencing of Convection Term 4.4.4.1.2 'One Dimensional Donor Cell Differencing of Convection Term 4.4.4.1.3 Two and Three Dimensional Donor Cell Differencing of Convection Term 4.4.4.1.4 Stability Condition in Terms of Cell Reynolds Number 4.4.4.2 Implicit Scheme 4.4.4.3 ADI Scheme CHAPTER 5: RESULTS . . . . . . . . . . . . . . . . . 91 93 94 94 98 101 106 113 117 119 119 119 123 123 125 126 126 132 132 138 158 164 164 164 . . . . 168 168 168 .. . . . 170 176 181 184 204 5.1 Natural Convection Results 5.1.1 Updating of the Reference State 5.1.2 Energy Convection 5.1.3 Heat Transfer Modelling 5.1.4 Turbulence Modelling 5.2 Numerical Diffusion 5.2.1 Truncation Error and Cross-flow Diffusion 5.2.2 Validation of Huh's Correction Formula 5.2.3 Skew Differencing Scheme 5.2.4 Corrective Scheme 5.3 ADI Scheme CHAPTER 6: 6.1 6.2 CONCLUSION . . . . . . . . . . . . . . . Physical Models Numerical Schemes RECOMMENDATIONS FOR FUTURE WORK . CHAPTER 7: 7.1 7.2 Physical Models Numerical Schemes . . . . . . . . . . . . . . . . REFERENCES APPENDIX A: APPENDIX B: APPENDIX C: APPENDIX D: ANALYTICAL SOLUTION FOR THE PROBLEM IN FIGURE 5.3(B) . ............ ADI SCHEME.FOR MOMENTUM EQUATION . . . . . . . . . . . . . . . COMPUTER PROGRAMS TYPICAL INPUT FILES ........... -6- LIST OF FIGURES Number 4.1 4.2 Illustration of the cross-flow diffusion in a single mesh with pure convection ... ... Illustration of the cross-flow diffusion in a row of meshes aligned in the xdirection with pure convection . ........ Geometry for explanation of the cross-flow diffusion in a two dimensional . . . Cartesian coordinate ...... . Page 63 65 4.3 ... . . 66 4.4 Geometry for explanation of the cross-flow diffusion in a rotated two dimensional Cartesian coordinate . . . . . . . ....... Comparison of the prediction formulas for cross-flow diffusion constant by De Vahl Davis and Mallinson and Huh . . . . . . . . . . . 68 4.5 . . 71 4.6 Geometry for explanation of the cross-flow diffusion in a three dimensional Cartesian .. coordinate . ......... ...... A segment of the calculation domain showing grid lines, control volume and notations in Raithby's scheme . ........ Three possible flow configurations for mesh interface implementation of the corrective scheme in a two dimensional ....... .. ..... . . ... problem ... . 72 4.7 . 80 4.8 . 85 92 4.9 4.10 Profile assumptions of 0 in space and time for various differencing schemes . ....... Three distinct shapes of the parabola, f(t), according to the coefficient of second order term ................. Stability condition in the plane (C xd x ) for an explicit scheme with central differencing of convection term in a one .......... dimensional problem . ... Stability condition in the plane (C ,d) for an explicit sc'heme with donor cell differencing of convection term in a one dimensional problem .............. . . 97 4.11 . . 99 4.12 .102 -7- 4.13 The region in the complex plane where (al+a2 ) should exist for the stability of an explicit scheme with donor cell differencing of convection term in a general two dimensional problem .............. The region in the complex plane where the amplification factor, r, should exist for the stability of an explicit scheme with donor cell differencing of convection . term in a general two dimensional probl Stability condition in the planes (C ,d ) and (C ,d ) for an explicit scheme wfthx donor elY differencing of convection term in a general two dimensional problem . .104 4.14 . . . 105 4.15 . . .107 4.16 Stability condition in the plane (C ,d ) etc. for an explicit scheme with doiorx cell differencing of convection term in a general three dimensional problem . ..... . .108 Stability condition in the plane (Rx,C ) and (Ry,Cy ) for an explicit scheme with donor cell differencing of convection term in a general two dimensional problem . .. .111 4.17 4.18 Stability condition in the plane (Rx Cx) for an explicit scheme with donor cell differencing of convection term in a two dimensional problem with f=v/ u/x.... .. 112 4.19 Stability condition in the plane (Cx , d x ) for an implicit scheme with donor cell differencing of convection term in a one dimensional problem .............. Stability condition in the plane (R xCx) for an implicit scheme with donor cell differencing of convection term in a one ........... dimensional problem . ... Natural convection flow field due to heat transfer between the obstacle and air in the containment at t=10.8 sec . . .115 4.20 .116 5.1 . . . . 120 -8- 5.2 Natural convection flow field due to heat transfer between the obstacle and air in the containment at t=62.5 sec . ..... Problem geometry for the two cases, (A) parallel flow and (B) cross flow, for evaluation of numerical diffusion . . . Comparison of the analytic solution and numerical solution with donor cell differencing of convection term in 10x10 meshes along the line CD in Fig. 5.3(A) . Comparison of the analytic solution and numerical solution with donor cell differencing of convection term in 6x6 meshes along the line CD in Fig. 5.3(A) . Comparison of the analytic solution, analytic solution with increased diffusion constant and numerical solution with donor cell differencing of convection term in 10xlO meshes along . the line CA in Fig. 5.3(B) ........ Comparison of the analytic solution, analytic solution with increased diffusion constant and numerical solution with donor cell differencing of convection term in 6x6 meshes along the line CA in Fig. 5.3(B) . . . . . . . . .... Problem geometry with arbitrary angles of e and 81 to compare De Vahl Davis and Mallinson's and Huh's formulas for prediction of cross-flow diffusion constant . . . . . . . . . . . . . . . . . Comparison of the numerical solution with donor cell differencing of convection term and analytic solution with increased diffusion constant, (D+Dcf), by Huh's formula along the line BD'in Fig. 5.8 ... . . . . . . . . . . . . 121 5.3 127 5.4 128 5.5 .129 5.6 . .. 130 5.7 . 131 5.8 . . . 133 5.9 134 .430 . .14 140 5. 137 5..-9- 5. .12 .13 139 5..3(B) . skew differencing scheme by Raithby and skew differencing scheme by Huh in a pure convection problem with 6 = 63.. .convection problem with 6 = 450 Comparison of the true solution and solutions by donor cell scheme.16 143 5. Comparison of the analytic solution and numerical solutions by donor cell scheme and skew differencing scheme by Huh for 6x6 meshes along the line .3(B) . . 5. . skew differencing scheme by Raithby and skew differencing scheme by Huh in a pure . by De Vahl Davis and Mallinson's formula along the line BD in Fig. . . 144 . .. (D+DDM). . . ...10 Comparison of the numerical solution with donor cell differencing of convection term and analytic solution with increased diffusion constant. .43* .. .8 . CA in Fig. . . 142 5. 135 5.15 . 5. True solution and donor cell solutions with and without diffusion correction for a pure convection .17 Comparison of the analytic solution. problem with 6 = 63. 5. 5. . . . . .3(B) . Donor cell solutions with various diffusion corrections for a pure convection problem with 8 = 450 . 136 5. . . . .. Comparison of the analytic solution and numerical solutions by donor cell scheme and skew differencing scheme by Huh for 10x10 meshes along the line CA in Fig. .11 Comparison of the true solution and solutions by donor cell scheme. .. .. . . . . numerical solution by donor cell scheme and numerical solution by corrective scheme for 10x10 meshes along the line CA in Fig. 19 . .19 . ..147 5. .21 S.. .-10- 5.. .18 Comparison of the analytic solution.. .22 150 5. 151 5. . . .. 5. . . . . .. donor cell solution and solutions by the mesh point and mesh interface implementations of the corrective scheme for the recirculating flow field given in pig.20 . Simple recirculating flow field for test of the implementation strategies of the corrective scheme . . 146 5.. . . . . . 152 5.. numerical solution by donor cell scheme and numerical solution by corrective scheme for 6x6 meshes along the line CA in Fig.. ... . . .. . . . . 153 . .3(B) ... .. . . .25 Diffusion constant corrections of the mesh point implementation of the corrective scheme for calculation of the steam concentration distribution in the two dimensional containment .. ..23 . . .. 5. . .. 148 5.. . ..24 . 145 5.. Recirculating flow field for calculation of the steam concentration distribution in the two dimensional containment to test the mesh point and mesh interface implementations of the corrective scheme Detailed flow field for calculation of the steam concentration distribution in the two dimensional containment to test the mesh point and riesh interface implementations of the corrective scheme Initial steam concentration distribution in the two dimensional containment to test the mesh point and mesh interface implementations of the corrective scheme . Diffusion constant corrections at each interface for the mesh point and mesh interface implementations of the corrective scheme for recirculating flow field given in Fig. . Comparison of the true solution. .. 5.....19 ..... cx T where 4 is given by. . . . . . 157 159 5. 5.. 5. . . . . 0.0 sec .32 161 5. . Comparison of the explicit and ADI solutions for the profile of 4 along the line AB in Fig. Steam concentration distribution after one time step. . .3 sec.3(B) = e.. different time step sizes Boundary conditions in terms of 4 for the problem in Fig. . . . . 156 5.. 0.0 sec and the Courant limit of 1. 5.. . . . .33 Comparison of the ADI solutions for the profile of 4 at t = 10.. with mesh interface implementation of the .3 sec. corrective scheme . with donor cell differencing of convection term without any diffusion correction .29 Steam concentration distribution after one time step.26 Diffusion constant corrections of the mesh interface implementation of the corrective scheme for calculation of the steam concentration distribution in the two dimensional containment . .0 sec along the line AB in Fig. . .. .3 sec. . . .. . . 5.5 sec and the Courant limit of 1.. . 1 178 . 155 5. Comparison of the explicit and ADI solutions for the profile of 4 along the line AB in Fig.. with mesh point implementation of the corrective scheme . .27 .. .. .31 1i60 5. . . .-1I- 5. . . . Steam concentration distribution after one time step. . . 163 A.30 with .. Problem geometry for comparison of the explicit and ADI schemes . . .30 for At = 1. . 0..28 ..30 for At = 0. . ...0 sec . .30 5. . 154 5. 78 4. . . .1 List of the schemes for numerical representation of advection presented at the Third Meeting of the International Association for Hydraulic Research. . Page .-12- LIST OF TABLES Number 4. . .114 . 2-D. . Comparison of the Von Neumann analysis and numerical experiments for the stability condition in terms of the cell Reynolds number in an explicit. donor cell differencing of the convection term . . . . . . . . . . . 1981 .2 . . . . . . . . . . . D d ./Imaginary (1) Turbulent kinetic energy (2) Thermal conductivity (3) Ratio of specific heats.D y.y and z directions Diffusion numbers in x and y directions Internal energy Function of Body force vector per unit mass Drag force vector per unit mass Gravity Grashof number Enthalpy Heat transfer coefficient Imaginary unit.C x y D D ah D as Dhs D .-13- NOMENCLATURE Letter A c c Definition Area Specific heat at constant pressure Specific heat at constant volume Courant numbers in x and y directions (1) Dimension (2) Diffusion constant (3) Divergence (4) Discriminant of a quadratic equation Binary mixture diffusion constant between air and hydrogen Binary mixture diffusion constant between air and steam Binary mixture diffusion constant between hydrogen and steam Cross-flow diffusion constants in x.d xy e f Gr h I Im k g M Nu P Pe p y Az Pr Q Prandtl number Heat source . cp/c Length Mass Molecular weight Nusselt number Peclet number Effective Peclet number (1) Pressure u u (2) L* ax or Ax Ay Ax + p v C . 1 tan-- STime a Z Superscript (1) Vorticity (2) Amplification factor in Von Neunann analysis constant of heat transfer Traction tensor due to viscosity Summation Time step n .1 v u Mesh configuration.-14- Letter R Re Re Rt Rx S Sc T t At u v v w Ax y Ay Az Greek Definition Universal gas constant Reynolds number Real Turbulence Reynolds number Reynolds number in x direction Source Schmidt number Temperature (1) Time (2) Parameter of a quadratic equation in Von Neumann analysis Time step size x-direction velocity y-direction velocity Velocity vector z-direction velocity x-direction mesh spacing Mole fraction in multicomponent gas y-direction mesh spacing z-direction mesh spacing a ala2 8 6ij C p v 4(1) 6 61 Thermal diffusivity or diffusion constant Parameters used in Von Neumann analysis Thermal volumetric expansion coefficient Kronecker delta Turbulent dissipation rate Density Kinematic viscosity Vpiscosity Phase change (2) Any general conserved quantity Velocity direction. tan. -15- Subscript cell cf DM f g i=l 2 3 4 L t m mix new ob old R t total US Cell Cross-flow diffusion De Vahl Davis and Mallinson's Fluid Gas Air Hydrogen Steam Liquid Droplets Left hand side Liquid Molecular Multicomponent gas mixture New time step Obstacle Old time step Right hand side Turbulent Total calculation domain Upstream . ( 1~111111-_~. The governing conservation equations are decoupled in order to simplify the solution procedure. steam and liquid droplets are assumed to be in thermodynamic equilibrium and the relative humidity is assumed to be 100%. the fluid dynamic phenomena in the containment in order to justify those remedies. Some simplifying assumptions are made concerning the thermodynamic state in the slow mixing stage. air. The research work reported here is primarily con- cerned with the formulation and validation of the physical models and numerical schemes in the second slow mixing stage. The distinct feature of the second stage is a much longer time scale in comparison with the first blowdown stage. Some remedies have been proposed to deal with However. The error due to decoupling is negligible in a slow transient where the . it is necessary to understand such problems.-__ -16- CHAPTER 1 INTRODUCTION One of the major concerns in the Three Mile Island (TMI) accident was hydrogen gas accumulation in the containment.--I~~U1-i* 11*11044~1 -1~. although it may be less than 100% when there is no liquid component present. hydrogen. From the hydrogen transport point of view. the response of the containment during an accident can be divided into two stages. The four components. the fast blowdown stage and the slow mixing stage. The turbulent kinetic energy. molecular and turbulent. and the total diffusion constant is the sum of the diffusion constants of those two mechanisms. The turbulent diffusion constant is predicted by the k-c model [45). of each component in multi-component gas depends on the mole fraction of that component. In the second step the energy and mass diffusion equations without phase change and turbulence equations are solved using the flow field obtained in the first step. resulting in the same convection velocity for the four components. In the first step the continuity and momentum equations are decoupled from the energy and other scalar transport equations and solved by the Simplified Marker and Cell (SMAC) method in order to obtain the flow field. and turbulent dissipation rate. are determined by their own transport equations. The turbulent kinematic viscosity. Convection is assumed to occur as a homogeneous mixture. The diffusion constant. Diffusion occurs by two independent mechanisms. The molecular diffusion constant is predicted by Wilke's formula [73) for multi-component diffusion and ChapmanEnskog formula [5) for binary diffusion. the phase change is taken into consideration to maintain 100% relative humidity.-17- state change over one time step is small. Convection and diffusion are the central issues in physical modelling efforts of hydrogen transport. Finally. c. k. . however the latter turns out to be the dominant error source in most convection dominant problems. numerical diffusion. Therefore. There has been much debate on the numerical diffusion and many schemes have been suggested for the past two . truncation error diffusion and cross-flow diffusion. Since the error usually appears as an additional diffusion. The leading issue in numerical modelling of convection and diffusion is to minimize the error that occurs in the numerical solution procedures. Truncation error diffusion occurs in the flow direction while cross-flow diffusion occurs primarily in the direction normal to the flow.-18- which is the diffusion constant for momentum transport can be calculated directly from k and E. the major obstacle in accurate numerical modelling of convection and diffusion has been the cross-flow diffusion error which arises in donor cell treatment of the convection term in multi-dimensional problems. The numerical diffusion has two different sources. This is because the gradient of the scalar quantity under consideration is small in the flow direction in comparison with that in the direction normal to the flow. Truncation error diffusion is a one dimensional profile error and cross-flow diffusion is a multi-dimensional operator error [68) of the finite difference equation. the term. has been used to describe the numerical error in general. The turbulent Prandtl and Schmidt numbers are assumed to be equal to one. The effective diffusion constants of the two errors are of the same order of magnitude. Raithby's [53) and S. The mesh interface . However. This situation is partly due to misunder- standing of the sources of numerical diffusion and also partly due to use of inappropriate approaches for its elimination. The corrective scheme can be implemented with any of the explicit. although with different stability conditions. The conservative property is essential in a long transient problem like the hydrogen transport in the containment. The stability conditions for each of the aforementioned schemes can be obtained by a Von Neumann analysis and turns out to be consistent with numerical experiments. have been tested for recirculating flow problems. The corrective scheme is inherently better than the skew differencing scheme in that it is conservative and does not affect the simple solution procedure.-19- decades. ADI and implicit schemes. mesh point and mesh interface implementations. Chang's methods [16) are examples of skew differencing schemes and Huh's corrective scheme and tensor viscosity method [25] are examples of corrective schemes. Two implementation strategies for the corrective scheme. a recent review paper on this topic [65) shows that there is no scheme universally acceptable consistent with using reasonable mesh spacings and computation times. The schemes presently being used can be divided into two categories. skew differencing and corrective schemes. The suggested physical models and numerical schemes have been used to simulate the LOCA experiments performed in Battelle-Frankfurt [44) and HEDL (Hanford Engineering Ref.-20- implementation has always given physically reasonable solutions and may be used extensively for all diffusionconvection problems. [49) includes all the Development Laboratory) [6]. simulation results compared with experimental data. . . In order to justify the design of any mitigation system. a large amount of steam and water will come into the containment in the fast blowdown stage increasing the containment pressure. e. installation of ignition devices. containment inerting. After a while the slow mixing stage follows the initial blowdown stage and continues for an extended period of time.1 Problem Statement Once a loss-of-coolant accident (LOCA) occurs in a Light Water Reactor (LWR). The hydrogen is generated by radiolysis and chemical reaction between water and zirconium in the cladding and may react explosively with oxygen in the air. In addition to the pressure increase due to the primary coolant. The major safety concern is that of keeping the containment pressure below a certain level to prevent a large scale leakage of radioactive materials. hydrogen generation gave a serious concern about the integrity of the containment in the TMI-2 accident.-21- CHAPTER 2 PROBLEM STATEMENT AND SOLUTION SCHEME 2. Thereafter. it is essential to understand the fluid dynamic phenomena in the containment. . There have been some mitigation procedures suggested. use of flame suppressants and enhanced venting capability.. for dealing with this problem. the hydrogen has received much attention in the safety analysis of nuclear power plants.g. e.g. etc. thermal nonequilibrium. some simplifying assumptions should be made to use the numerical procedure without impairing the acceptable solution accuracy. phase change and heat transfer between the gas components and wall.. laminar and turbulent flows. ._ __ _ __ _ I_ ~ -22- Since a numerical method is suitable for this purpose. Therefore. The most difficult aspects of this analysis are the complicated geometry and chaotic post-LOCA conditions. a good computational tool has been required to predict the hydrogen concentration distribution. 1) Momentum: pgI + d + V-0 (2.3) Mas s diffusion: S+ V*vp i .T) (2. I+ 3x 1 .S + + t Bui P au k aui + ax~ ) x k X axk gk Pr t t 'T xk (2.xk k 2 v t BT -C2 + kg k Pr 3x 2k k t (2.7) .t = V-D.6) Equation of state: p = f p (ii . The physical implications of the conservation equations will be given with their basic assumptions and limitations. Continuity: V'v= 0 pl + V-vv ] = -Vp + -Vv = 0 (2.-23- 2.2 Governing Equations The governing conservation equations are set up to describe the post-LOCA fluid dynamics in the containment.Vp i + 4.2) Energy: p[e e + V-vh] + = V-kVT + e fg (2.4) Turbulence kinetic energy: Dk 1 t 3k 3 P 1 k ok x . (2.5) Turbulence dissipation rate: D 1 _ 1t StE xk Dt P Txk 0E ] + C t*( 1 p k ui xk ouk au. k + k t m Dim = fD(Pi'T) (2.14) m = fk (PiT) (2. Im + Dit it (2.T) (2. .11) k = . = f (Pi.8) f (v) (2.T) (2.16) kt = . = D.13) D.15) Dt = v t (2.-24- Constitutive equations: 'N = f (p .12) (2.19) Note that 1. The subscript i denotes the four components as .18) p = Epi (2.9) D.t (2.10) v = vm + Vt (2.17) t = C k 2/C E (2. . The turbulence equations are written in terms of the Cartesian components in tensor notation.20) i = 2 = 0 43 + 04 = 0 (2. (2. one continuity. air. liquid droplet. The mass diffusion equations are summed up for the four components. 2. Therefore.i (4). one energy. three momentum. there is one more-equation than is required. i=1-4.k C for a three dimensional case. w.-25- follows: i=l i=2 i=3 i=4 2. The phase change occurs between steam and liquid droplets and the diffusion constant of liquid droplets is equal to zero.21) D4 = 0 (2. four mass diffusion and two turbulence equations with ten primary unknowns. u. T. p.2. 3.22) There are eleven conservation equations.1 Continuity and Mass Diffusion Equations The continuity equation is redundant with the four mass diffusion equations and there is'an inconsistency between them. v. H2' steam. 27) Consequently Eq. 2. . Since the concern is only for the slow mixing stage after a LOCA. Therefore.25 and Eq.Vp. 2.-26- -. 2. it is reasonable to assume that Eq.4 are consistent only if the temporal and spatial variations of the total density are very small.27.24) Now it can be seen that the continuity equation is valid if and only if the following Eq. 2. 2. Eq.26 can be reduced to the following Eq.25 and Eq. 2. at p= 0 (2.27 remain valid throughout the transient.) + V-(Ep Jt i = (V'D.23) The sum of the phase change terms is equal to zero. 2. Vp = 0 (2.1 and Eq. 2.) i + E 1 (2.25) Vp i = 0 (2.26 are satisfied. ap at S+ V-p = E(V-DiVp i ) 1 (2.26) It is a reasonable assumption in a turbulent flow regime that the four components have the same diffusion constant.(Ep i ) a . LeaPi LeQ. E it p + e.-27- 2. 2.iei) + V'[Ep.29 to give the following. -(p.ih. ae.29) The second term on the left hand side of Eq.28) The temporal term can be divided as follows. .29 denotes the energy change due to the concentration change in a given control volume. 2.31 so that the internal energy and enthalpy can be written in terms of the temperature. [c pihi . (2.2. p + £e.ih i ] = V*kVT (2.3. It is clear that the major contribution to this term will come from the latent heat of the phase change. Be. 2. 2. I at t + V [vp.30 is substituted in Eq.30) Eq.31) The thermal equilibrium assumption is incorporated in Eq. Eq.2.i + V. is also of an approximate nature and the exact form is given in the following.i = V'kVT (2. = V-kVT (2.2 Energy Equation The energy conservation equation. 1 = c Vl. P (2 33) (2.3 Basic Assumptions and Limitations The governing equations are based on some simplifying assumptions about the physical phenomena in the containment after a LOCA. .32) where ei i = eg4g + e 4) Sefg g e.T h.34) 2. It is necessary to clarify these assumptions and their limitations for a safe use of the given physical models and solution scheme. = c p1 .T P P e Piei p ih p.33) h (2. 2.T 1 Eq.-28- (Picvi )~ +V"[vTZpic pi] + efg 9-kVT = (2.32 become identical by defining average internal energy and enthalpy as follows.3 and Eq.c . Pic vi T c p p. 2. 2 Limitations 1. The four components are treated as being nearly incompressible and the phase change rate is moderately small. with no slip.3. Dt and k t over one time step should be small because the turbulence equations are decoupled from the velocity field calculations. The temporal and spatial variations of the total density should be negligibly small because of the incompressibility assumption. The phase change rate should be moderately small because the solution scheme decouples the phase change term from the energy conservation equation. 3. 4. 2.3. 2. The change of the diffusion constants.-29- 2. The relative humidity may be less than 100% when there is no liquid component left. 5. The relative humidity in the containment is 100%.1 Assumptions 1. vt. The turbulent Prandtl and Schmidt numbers are equal to one. 2. 3. The four components all have the same convection The liquid droplets move with the gas components velocity. The four components are in a state of thermodynamic equilibrium. . k. When there is no liquid component left.-30- 2. explained in detail in section 2.5 SMAC Scheme and Compressibility The SMAC (Simplified Marker And Cell) scheme [21J has been used for the incompressible fluid flow with Boussinesq approximation for the buoyancy force.5. energy equation and two turbulence equations. 2.7 are a coupled set of equations with primary unknowns. T. The SMAC scheme is In the second step. v. is used to update the reference pressure and reference density which is important in calculating the buoyancy force. 2. The equation of state. In the third step phase change is taken into consideration to maintain 100% relative humidity. the relative humidity may be less than 100%.i Since it is too difficult to solve the whole system of the equations and obtain a consistent solution for all the unknowns.1-Eq. which are the four mass diffusion equations. . 2. Pi(4). Eq. 2. In the first step the SMAC scheme is used to get the convergent velocity field with zero divergence for every mesh. the conservation equations are decoupled into two sets. E. continuity/momentum equations set and other scalar transport equations set. p. The phase change is assumed to be zero in this step.7. the obtained velocity field is substituted in the scalar transport equations. It solves the . u.4 Solution Scheme Eq. . However. ay -(Eq. Continuity: au ax av ay 0 (2. The continuity and momentum equations are given in two dimensional Cartesian coordinates in the following.36) . it is possible to modify the SMAC scheme to accommodate a slight compressibility effect. Since the continuity equation used in the SMAC is an incompressible form.1_ _~ -31- continuity and momentum equations to get the velocity and pressure field in an iterative way.e.37)y (2.37): . 2.ax 2. the divergence of the velocity field vanishes everywhere. 2. the SMAC scheme can be used only for incompressible flow calculations.36 and Eq.36) y-direction momentum: av av ua av 1 + pg + a y x2 Nv 2 (2. which can arise with net inflow or outflow boundary conditions. i.37) Py The vorticity conservation equation is derived from Eq. -(Eq.35) x-direction momentum: au au + u-+v at ax au ay 1 p ax + Pgx + a 2u 2 ax 2 2au + 2 2 (2. 2.37 as follows.. 40) S (2.36 and Eq. 2.x ay ax 2 ay2 where au av (2.37 as follows. 2.39) where av au D= -.-32- a C 2 a2 + u+ V = v+ ) at . -(Eq. 2. ax 2 2. 2.36) + -~(Eq. ay 2. u n+l-u n -u At n+l At n n 1 Apn+l A + n 1 p Ax n+l + gn 1 Ap p by (2.+ -y ax ay The basic idea of the SMAC scheme is to separate each calculational cycle into two parts. In the tilde phase the approximate velocity field at time step (n+l) is obtained from Eq.38) =ay ax The Poisson equation for the pressure field is also derived from Eq.37 as follows.37): a2 = 2 ax 22 axay a 2 ay 2 (a2D + a2D aD +1 2 at Re ax 2 ay (2.41) where fn and gn are the explicit quantities at time step n. . the so-called tilde phase and pressure iteration.36 and Eq. 2.39 may be rewritten as A finite difference form of Eq.45) . au ay au av ax - (2. the following relations hold after the tilde phase. 2.40 and Eq.O 0 (2. 2. Eq. The equation of vorticity transport. 2. a guessed pressure field or the pressure field at time step n is used to start the iteration. 2. Therefore. follows.44) Eq.41 are still unknown.-33- Since the pressure field in Eq. 2.42) av . 2.41 have the right vorticity.v) -D (2. 2.39 that the pressure field is not right due to the fact that the divergence of the velocity field is not equal to zero.44 is given in the following.43) ax ay It can be seen from Eq. V2p = f(u. _ Pi+j-2P +P iPij+ pij f(uv) - Dn + l D n D D Ax 2y t (2. shows that the vorticity is independent of the pressure field. The impor- tant point is that un + l and vn+l calculated from Eq.38.40 and Eq. 49) . The slight compressibility means that the Navier-Stokes equations may be solved safely with the incompressibility assumption.-34- The pressure correction formula is obtained by conn+l only and putting D. sidering the cell (i. 2. Once the zero divergence velocity field is obtained. no more correction will be made.47) ) The pressure correction is given in terms of the residual divergence for every mesh.46) (2. equal to zero. (6u) L = - pAx (6u) R = t6p pAx (2.. S+ at V"(p) = 0 (2.37 so that it does not change the vorticity implemented in the tilde phase. while the continuity equation has the following compressible form. 2.j) -26p(-- 1 Abx 6p = -h + _) Ay 1 At( A + 1 Ay 1 Dt D (2. The velocity correction formula is derived from Eq.36 and Eq.48) The SMAC scheme can be extended to a slightly compressible fluid flow. 2. : in -n =Vp v - total net inflow of mass at the boundary during the time At. .51) v p where m.uniform over the whole containment.50 may be given from the previous time information and boundary conditions.p_ p at 1 min/V At n p n n (2. 2.51 and Eq. The associated physical assumption in Eq. 1 .52) (2.49 the nonzero divergence may be derived as follows.50) The two terms on the right hand side of Eq. D = V'v 1 p p at V p (2. 2. V: total volume of the containment. 2. Now the problem is how to converge to a predetermined nonzero divergence field.52 is that the pressure perturbation due to the inflow at the boundary propagates throughout the containment instantaneously and the pressure and density increase is.-35- From Eq. This can be done with the following modification of the pressure correction formula of the SMAC scheme. 50.(D-D 1 1 + -) 2At( Ax Ay (2.48. 2.53) where D O is given by Eq. The velocity correction formula remains the same as Eq. . 2.6p = . 1. As collisions occur more frequently. which should be modelled independently. hydrogen. steam and liquid components are transported by convection and diffusion in the containment. components. It gives the diffusion constant of each component in terms of . molecular and turbulent. 3. The thermal conductivity and viscosity are also diffusion constants in a broader sense for momentum and energy transport. in the containment after a LOCA. the diffusion constant of each component is calculated separately by Wi~k-'s formula [73] in Eq. Diffusion occurs by two different mechanisms. the process Since there may be three gas of diffusion is also increased. 3. Although turbulent diffusion is greater than molecular diffusion by a few orders of magnitude.1 Molecular Diffusion Molecular diffusion occurs by the collisions of gas molecules.-37- CHAPTER 3 DIFFUSION MODELLING The hydrogen. air. proper modelling of the latter is important because the molecular diffusion of hydrogen is significantly greater than those of other gases and also because there may be a laminar flow region in the containment. Convection occurs as a homogeneous mixture resulting in the same convection velocity for the four components. air and steam. 1) where D. stants are given as follows. 1-Y A B DAB 1-Y B yA DBA 1-Y C YA CA C YC CB B YC DBC C D = C YB DCB (3. T3 D = 0.2. The property of air will be replaced The resulting diffusion con- with that of nitrogen gas. The binary mixture diffusion constants are calculated by the Chapman-Enskog formula.-38- the binary mixture diffusion constants and mole fraction of that component..AB where DAB: AB Binary mixture diffusion constant in [cm /sec] .: Binary mixture diffusion constant between the i-th and j-th components.0018583 1 + M A MA MB (3. Eq. 3.2) poAB D.: yi: Diffusion constant of the i-th component Mole fraction of the i-th component D. OAB = (o A + oB (3. The error range of the Chapman-Enskog formula is reported to be about 6-10% [5).AB: Dimensionless function of the temperature and intermolecular potential field for one molecule of A and one of B. For non-polar gases OAB and ODAB are calcu- lated by Eq. 3. the following relations are required.----.3) CAB = AB For the pair of polar and non-polar gases a correction factor is introduced to account for the polarity.UIIII ~ -II-~PI-^ICI-L1I n~s^ - -^ -39- T: p: aAB: MAMB : Temperature in [OK] [atm) Total pressure in Lennard-Jones parameter in [A] Molecular weight QD.3.lll~*.5) 2 n . ap = np £ (o n + o p) E-E e (3. 3.4) np = np 2 where the correction factor 4 is given by.2. P n ]. In order to calculate the para- meters aAB and 0DAB in Eq. =n [ p n = [+ S n . (3. 5 are defined as follows. H2 a n = 7.9492 x 10 D ah D 1 T .6810 x 10 . 3. The containment pressure is assumed to be-1 atm. -5 = 4.9035 = 4. a * = a an/ : Reduced polarizability of the non-polar molecule ~ * = p p/ p co 3: a pp Reduced dipole moment of the polar molecule t * = p //8 p p For the steam component the following values will be used.65 A The polarizability is given for the hydrogen and nitrogen (air) components.6 x 10 The fitting functions for the binary mixture diffusion constants are obtained by the least-square-fit method. t* .9 x 10 -25 25 -25 [cm3 3 [cm ] : N2 : an = 17.6947 as .-40- The parameters in Eq.= 1.6 T1.2 E/K = 380 0 K a = 2. k. n mix = Yipi n i E j=l Yj ij (3.8916 x 10 where T: D: [OK1 [cm 2 /sec] -5 T 1.7) k.6) The viscosity and thermal conductivity can be considered as diffusion constants for momentum and energy transport and calculated similarly as follows [5). = n i=l y.67 x 10 where -: T: a: Svi : o2 2 (3.__Li~n_~PIIII ~.8) j=l where ij 8 Mj j ij M The viscosity of non-polar gases may be obtained by the following [5).lllI_ -41- Dhs = 2.lii-. i (3. = 2.8144 (3.10) Viscosity in [poise) Temperature in [K]) Lennard-Jones parameter in [A) Dimensionless nTumber . 009 cp at 1 atm.2 Turbulent Diffusion The diffusion process is greatly enhanced by the turbulence of fluid flow. E. 200 C There is another formula suggested for the viscosity of multicomponent gas in the following [5]. 3.11) for the Prandtl number of Pr = c (3. and turbulent dissipation rate.385 nn y.-42- The viscosity of steam is. There are several models suggested for Presently the best model calculating turbulence effects. Wsteam = 0. seems to be the k-c model originally developed by Launder and Spalding [45). The . 2 n Yi umix = C 2 i=l Yi + 1. The effective diffusion constant is therefore the sum of the molecular and turbulent diffusion constants.[5]. k. The k-s model sets up the transport equations with some empirical constants for the turbulent kinetic energy.12) where c P is the specific heat per mole at constant pressure. c (3.y RT k=l PMiDik k i There is a useful relationship non-polar polyatomic gases. 3.13) (3.u. momentum and energy equations by decomposing the variables into mean and fluctuating parts and averaging the resulting equations. These terms are treated by the eddy viscosity concept of Boussinesq and eddy diffusivity concept which are given in the following. .-- (3.14) .-43- turbulent viscosity is given directly in terms of k and C.= v ( Dt + ui j u P= Dt ax x 2 k6 ij 3x_) 3 i (3. u. The effect of turbulence appears as the additional terms due to the product of fluctuations which do not necessarily cancel out. + u. 29. etc.' j T'= a t x. u. 45] The transport equations for the turbulent kinetic energy and turbulent dissipation rate are derived from the continuity.' For example.2. 34. + Pi Pi = Pi + p. = u.15) .u.1 Derivation of Turbulence Equations [15.' etc. 3.2.' u is transformed to the turbulent kinetic energy equation by the contraction. The turbulent dissipation rate equation can also be derived in a similar way and some assumptions should be made in modelling various terms that appear in the derivation.16.c +g t Pr v BT az Tt (rue) + + 1 r ar a(w)-= z [) [ pr ar o ar + z a ( a az + C 1 aw [ 2 au 2 t + 2(-) 2 (-) p az ar p k 2 2 + 2 2 C + ( r + -) az 2 2 k + k Pr atT zt (3. j=i. A two dimensional cylindrical coordinate form of the k-E model is introduced in Eq.-44- The transport equation for u.16) .2 k-e Model The transport equations for the turbulent kinetic energy and turbulent dissipation rate form the basis of the k-E model. 3. -t + at r a (ruk) + (wk) = -[ p r (r oka r ) + (t ) az ok a au aw + -) r aw au 2t + 2( + -1[2(-) 3 3r p + 2- + u2 2 ] . 0 o 1.44 C2 1.metry.09 C1 1. C 0.17) The suggested values of the constants are given in the following table. = C pk 2/ (3. 3. 3.17. Since the k-E model is applicable to fully .3 Low Reynolds Number Flow While both turbulent and laminar flows are possible in the containment after a LOCA depending on the hydrogen generation rate and geo.-45- The turbulent viscosity pt is given in terms of k and E in Eq.92 ok 1. it is difficult to determine whether a region under consideration is in a turbulent or laminar flow. it ak t a +it m ok lit m a E C The turbulent Prandtl and Schmidt numbers are usually assumed to be equal to one.3 The molecular effects can be included in Eq. 3.16 as follows. A more refined model is introduced in this section because consistent treatment of laminar and turbulent flows and transition between them may be required in the future modelling efforts. The model presently being used is just to use the sum of the molecular and turbulent contributions for every region in the containment.3 exp(-R2t ) (3.19) = pk2/VE . while C1 and C2 are to vary with turbulence Reynolds number.5/(l+Rt/50)] C2 =2 where R 2 0 [1. D Dt p ax k [(t Dt+ ) e k + C k i + k x ) au axk C 2 E2 2 t 2.lit k + l ak aui i + --t (ax k + xk auk axi axk e .2vak 2(-) k 2 (3.0.0 .-46- turbulent flow. 3. 41).16.0 a ui ( 2 p ax ax 2 ) Dk Dt 1 p ak -[(. Rt. The following k-c model is a modified form by Jones and Launder [40. CP = CP0 exp[-2.18) In the above equations C 1 1 ok and a retain the values assigned in the ordinary k-c model. Eq. it is required to extend the model to laminar flow regime or to switch to laminar models according to some criterion. -47- The C O and C2. and C 2 0 are fully turbulent values of C This modified form of the k-s model was developed to cover all the laminar, transitional and fully turbulent regions. The constants were fitted to the data of a low Re flow in a round pipe to include the effect of laminar wall boundary layer. Although its applicability to a low Reynolds number flow in general is questionable, the model may be used with some confidence if it has the proper limit of laminar regime with the decay of turbulence. It may also be used in the transitional regime by assuming an adequate interpolation between the two extremes. Equation 3.18 shows that k and e have the same order of magnitude with the decay of turbulence. In other words, both k2/c and c2/k will go to zero as k and E go to zero separately. Therefore, the turbulent viscosity Vt which is proportional to k2/E will vanish with the decay of turbulence. In order to have the proper laminar limit, the total diffusion constant should be expressed as the sum of the turbulent and laminar diffusion constants. It may also be reasonable to assume that the turbulent Prandtl and Schmidt numbers are equal to one. -48- CHAPTER 4 NUMERICAL DIFFUSION The final numerical solution involves two types of errors which come from physical modelling and numerical solution procedure. The error in physical modelling is an intrinsic error which cannot be eliminated by the solution procedure. The error in numerical solution procedure is the difference between exact and numerical solutions of the governing equation, which usually appears as additional diffusion. It is clarified that there are two sources of numerical diffusion, truncation error diffusion and cross-flow diffusion. This chapter is primarily con- cerned with how to predict and eliminate these additional false diffusions to get an accurate solution. 4.1 Truncation Error Diffusion Truncation error diffusion occurs due to the approxi- mate nature of the finite difference formulations. The name of the truncation error originates from the Taylor series expansion where the second and higher order terms are truncated. Although the Taylor series expansibn is not a proper way of interpreting a finite difference equation, the name of the truncation error will be retained here. Since the truncation error exists in multidimensional problems in the same way as it exists in one dimensional problems, it can be analyzed in the following one dimensional _I_ I~___ _L~ ~ _ I conservation equation without any loss of generality. at where 4: S: -- U + ax a 2 ax + S (4.1) any general conserved quantity source term and diffusion constant a will be assumed The velocity u constant. [xi, xi+ Equation 4.1 is integrated in the domain ] and [tn , tn+l]. x Xi n+1 at dt dx + xi+ x tn+l t n aa 2 a2 ax x tn+1 u-- dt dx tn dt dx + S Ax At (4.2) where At = t n+l -t Ax n xi_ = Xi+ - When a function f(x) is continuous, there exists a point x=c in [a,b) such that, b b-a 1 f a f(x) dx = f(c) (4.3) Using Eq. 4.3, we may transform Eq. 4.2 as follows. -50- -t 1 [ n+l i+f n+f + Ax [ n+ 9 1+h n+g i- = -- S[(a4n+g Lx [ ax ) i (--x n+g ] + s + ax a+ i-h (4.4) (4.4) where <f < 0 < g-< 1 Equation 4.1 may also be reduced to Eq. 4.5. It is fully explicit and the convection term is finitely differenced by the donor scheme. h+1 At = n + Ax (Q i - 1(4.5) x2 Ilx n i+ - 24P + 4i-1 ) + S 1 From Eq. 4.4 and 4.5, obtained as follows. (4 +)EXACT and ( )FD can be n n Sn+l i FD n i + At [ -u i Ax bx 2 S n+g _n+g n i+1 - + + l 1-1 ) (4.6) + S] n+l i+f EXACT n i+f At i+h Ax i-1 + Ss + a( n+g ) i-6 + Cx x[( i+6 _ ( ax)n+g ax ] ] (4.7) n+g 4 i+ Ax n i Ax 1 xji+ n i+l .6 from Eq. It can be extended to a two dimensional form as follows.a(?) ) 2(.y+b].3.x+a) and [y.-51- Subtracing Eq.7.9) a + ( ) b AA B some B in the domainB ay whereinterior andpoints denote where A and B denote some interior points in the domain bounded by [x.a i- = ( nn+g n+g ax i+a - < a < n i-1 8a ( ) n i+b a 2 4n+g ax 2 i+c 2 ox n -1 < b < 0 n+g (a n+g ax i-. Eq. is for a one dimensional case. 4. 4. af f(x+a. we obtain the error at time step (n+l). .i+c T i+d ax where the following relations have already been used. n+l i+f EXACT n+l i FD n i+f - 4P n a n+g i + At [ -u( x ) i+a (4. -1 < c < n 2 4 + i Ax 2 1-1 4 2 -1 i+d < d < 1 The mean value theorem.y) + (f) af (4. 4.8) 3 @n + u(-)n + a( ax i+b ax 4 n+g _ n )n+g .y+b) = f(x. n . .tax .xi+ ] and [tn tn+).D +aAx(c-d)At--ax 2 +ag-t . 4.A -fAxAt axat ----------2 B -u-t-x(a+b)---- -------.8 to the following form. 4.-52- Now Eq. D and F are time derivative terms which become negligible for slow transients. 4. 4. 2 n+l (4i ) EXACT - (4+) FD = i n+1 f )2 2fx 1-f 2 ) (f 2 2 ------.F (4.10) The derivatives with respect to time and space in Eq. C and E are truncation error diffusion terms while the terms B.----- E ---------.C ax 22 +At g t ------------.9 reduces Eq. In a stable scheme the error introduced at a certain step decreases in its absolute magnitude in the following steps and most of the error occurs between neighboring time steps. The terms A.10 are at some appropriate points in the domain [xi .10 are based on the assumption of perfect information at time step n. Although the error terms in Eq. they may represent the errors over many time steps in a stable scheme. 11) ax Consider a case in which the domain is divided into N equal meshes and the mesh size is Ax. The results may be extended to a general one dimensional problem if the effects of the transient and source terms are not dominant in determining the profile of 4. 4. 4= Cecx + C2 where c = u/a . cL (4.11 is given as. The analytical solution of Eq.12) and C 1 =N cL -1 e C2 ecL -1 . u ax a2-2 (4. o 1 2 3 L i-1 1 i+l N 4(x=0) = 40 N Q(x=L) = NAx = L Two boundary values 0 and QN are given as constant.1 Truncation Error Diffusion in a One Dimensional Problem The truncation error diffusion in a one dimensional problem is analyzed for a steady-state case with no source.1.-53- 4. 16 is easily obtained from Eq.13 can also be solved with the given boundary conditions by reducing it to the following form.13) where the velocity u is positive. 4.16) .11 is finitely differenced using donor cell scheme for the convection term in the following. P(4).40) . i+ - i = (P+l)(4) - 4)i- (4.14 may be recast to Eq. 4. where P =-u a - l -i24 + i-i (4. Equation 4. .-54- Equation 4. 4.14) The parameter P in Eq. i u x i+l2. i+l i = (P+l)( 1 . (4.15 and solved for in subsequent procedures.14 is the cell Reynolds number or cell Peclet number according to whether the diffusion constant a is the kinematic viscosity or thermal diffusivity.15. Equation 4.15) Equation 4.i + 4i2 Ax = a (4. (1- 0 ) is ) P( N -¢0 (1+P) given as follows.'N-1 (1+P) N-1 - ) 0 N .11 and Eq.1 Inserting Eq.20) Now the analytical solution Eq. 4.I_.18) (l+P) + 1 Therefore.41 = N ) (1+P) [(1+P) N-1 - 2) (4. 4. we obtain the solution for 4.17) 01 can be expressed in terms of 40 and 4N N N by Eq. (1+P) 1 . as.20 have been obtained without any approximation for Eq.1 (1+P) N 1 (4.~-LI LIII~.LI II .17.19) (1 . 4. 1 (1+P) i .19 into Eq.13.(1+P) P (1+P) 0 + N . 4. 4. 4.16.12 and numerical solution Eq. i - 0 = 4. Both the analytical and numerical . 2 1 = (1+P) (4 0) 4' N 92 = N-2 (1+P) 2 ( - 0) = N.-~l*IIIIX1~~^ ~ ^I ^~Xi -55- Then. 4. 4. -24 j.25. Eq. 4.13.12 is substituted for 4i Eq.23) (4. and the effective diffusion constant for the numerical solution will be obtained.+1 i + ) i-l 0 (4.24) It is also possible to go through the similar procedures with a central differencing form of the convection term in Eq. in Eq.22 we can obtain the effective Peclet number Pe of the finite difference solution in terms of the real Peclet number P as follows. 4.25) .1 (4. 2 i-i P ( (1+P) (1+) N _ 1 N 4). i+l 2Ax i-l S S2ax 4) i+l -24i + i- 2 i- (4.1 e Therefore. Pe = in (1+P) (l+P) i-Ip2 N (1+P) .13. 4.20 is also substituted for 4.21 and Eq. 4. + -24 S i i+1- i-1 =e e l NP - 1 (e P -P 1) 2) ) ( (e 0 (421) The finite difference solution Eq. 4. 4.13. The in analytical solution Eq.-56- solutions will be reinserted to the finite difference equation.25) (4. 4. e (i-l)Pe (ePe-1)2 NPe .22) Comparing Eq. 4.25 can be derived as follows. Although this approach is approximate and overpredicts the truncation error diffusion.1. it helps understanding the origin of the truncation error diffusion. 4. The convection terms in .-57- The effective Peclet number for the finite difference solution of Eq. Pe = kn (2-P) 2+P (4.27) In general. As in the previous section the velocity u is assumed to be positive and constant.26 may be a good indication of the truncation error diffusion occurring in a general one dimensional problem.2 Another Approach For a One Dimensional Truncation Error Diffusion The truncation error diffusion in a one dimensional problem can be evaluated in a different approach.26) where -2 < P < 2 The effective diffusion constant De can be readily obtained from the following relation. 4.24 and Eq. the transient and source terms and the mixed type of boundary conditions will not affect the numerical diffusion appreciably if they are not dominant terms in determining the profile of 4. Eq. uLx D De Pe - (4. 4. Therefore. i. 4. ae -u( x where 0 <f < ae + u() ax i+ - u~x a2 ) -. the diffusion is dominant over the convection.30) Both Eq. 4. the convection is dominant over the diffusion. the profile of 4 will be linear and the truncation error will be reduced to zero.( i+f 2 (4.e. 4.5 may be approximated as follows. Pe also goes to zero resulting in no diffusion error in Eq. 4.28 and Eq. 4. If the Peclet number is small. 4.24 and Eq.29 from Eq..29 hold only when the Peclet number is large. 4.-58- Eq. -u i Ax i-1 -u i+ a4 -u(--) ax -u(a) ax (4.31) ND .29) i+ The truncation error diffusion can be quantified by subtracting Eq. Therefore.26. 4.28) Ax (4. the truncation error diffusion constant for a convection dominant problem may be given approximately as follows. i.4 and Eq. D uAX 2 (4.e. It can be shown that as P goes to zero.28. two dimensional. 2 u + v 2 + +( ) (4. conservation equation with no source term is given in the following.32) = x 3y Equation 4.1.-59- 4. u i 6x i-lj + v i by ij_a(i+1 2 4x i + i-l1 +ij+l -2i 2 + 0iji by (4. The formulas for the truncation error and cross-flow diffusion constants derived in this chapter reveal that they are approximatly of the same order of magnitude.3 Truncation Error Diffusion in a Two Dimensional Problem There are two sources of numerical diffusion in a two dimensional problem. A steady-state. Since the gradient of 4 in the flow direction is negligible in a convection dominant problem.33) . one is the truncation error diffusion and the other is the cross-flow diffusion. It will be shown that the truncation error diffusion in the direction normal to the flow cancels out and there is only a flow direction component left. the total diffusion quantity of 4 due to the truncation error is not significant in comparison with that due to the cross-flow diffusion.32 is finitely differenced using donor cell scheme for the convection term. 34) where tane = Ay/Ax tane1 = v/u The coordinate (x.v( ay ) ij .2 is applied to Eq.j+ + v(a) ly i+h. 4. ij Ax 1 1x Axhy 'i-1 i Ay ij-i 1 +AxAy fj+ j i+ i axx dt + Yj+ yj+ yj- xi+4 xi+ a (v-)dx dy ay Xiih + u(t-) x i+ a =-U(ax--) )3x .-60- The approach in section 4.j+ 2 2 ay 2 a2 2 A Ax U(a 2 ) + bay U( 924 )+ Ax 82 2 axay 2 ax2_-I = AX Ax A 2 axay va a yyaxa 2 ay 2 S[uAx-ax + (uAy+vbX) +v = .35) . transformed to the coordinate ( .33.y) is by rotation.n) ( = x cos6 + y sine I = -x sine + y cose (4.1. then the truncation error of the convection term is given as follows.x[cos + (cose tane 1 + sine)- + sine tane -] ax xY (4. dominant truncation error diffusion occurs When only in the flow direction.37 shows that the diffusion component in the n direction (normal to the flow direction) is zero and the cross differential term also vanishes when e is equal to e1 Therefore.37) Equation 4.35. e is equal to 0 or 7/2. 2 -2 2 + a = ( 2 ) (4.-61- where 5 is the coordinate in the flow direction.32 is reduced to the following form.34 will be transformed to the coordinate system (t.n) given by Eq. 4. the problem is basically one dimensional and the truncation . 2 (A) - [(sine tane1 + cose) 2 + a2 (-sine + cosetan6 1 ) a + (0) -4 an U" (4. y v e Sx Then Eq.36) where U = /u2v 2 Now Eq. 4. 4. 3.l1(A). The cross-flow diffusion comes from the multi-dimensionality of the problem.1. It is entirely different Recently this addi- from the truncation error diffusion. tional false diffusion was explained by Patankar [52] and Stubley et al. therefore it exists only in two or three dimensional problems when the flow direction is not aligned with the mesh configuration.-62- error diffusion occurs in the flow direction as in a one dimensional problem. 4. the hot and cold fluid enters the mesh from the left and bottom surfaces at 45* angle. In Fig. that will be named here as the cross-flow diffusion.2 Cross-flow Diffusion Most of the confusion about the nature of numerical diffusion comes from the fact that there exists an additional source of false diffusion. 4.4. In this section the origin of the cross-flow diffusion will be illustrated and the corresponding diffusion constant will be quantified so that they can be used in the corrective scheme of section 4. The hot fluid will come out of the top surface and the cold fluid out of the right surface. [67. 4. The origin of the cross-flow diffusion is illustrated in Fig. In the donor cell scheme Fig.1(A) is transformed into . 4. which shows a single mesh with pure convection.68) and it was clarified that this is the dominant source of the error in most multi-dimensional problems. -63- Hot Mledium Hot 7 Hot Cold (A) Cold (B) Cold (C) Fig.1. 4. Illustration of the cross-flow diffusion in a single mesh with pure convection . 1(B) are again interpreted as Fig. ] = (l-p)4- = where u . in The numerical results Fig. . The value of 0 at the point A should reappear at the point X because there is no physical diffusion.2 which shows a row of meshes aligned in the x-direction. A 3 . Another illustration of the cross-flow diffusion is given in Fig.&x u v ax Ay The effective diffusion constant for the cross-flow diffusion will be calculated in the two dimensional Cartesian coordinate in Fig. .1(C) by a Therefore. This is a pure convection problem Consider the fluid element The fluid with no physical diffusion. diffusion occurs in multidimensional donor cell differencing of the convection term. 4.3. CAB which looks like a long one-dimensional rod. of the value of 4 causes the cross-flow diffusion. 4. .1(B) where the velocity components normal to the surface are considered. an appreciable amount of false program user. and the sum of the values This distribution along the points A1 . 2 (l-p)[l+p+p + .-64- Fig. 4. at those points is exactly equal to 4. A 2 . .and intermediate temperature fluid will come out of the top and right surfaces. Then homogeneous mixing occurs in the mesh. 4. 4. In the donor cell scheme it is distributed all . -65- (1-p)4' x p (1-p) 2 p (l-p) p3 (l-p)4 p4 (1-p) A5 Ay A Ax 2 4 y p t-* U where p = Ax Ay Fig.2. 4. Illustration of the cross-flow diffusion in a row of meshes aligned in the x-direction with pure convection . V A2 e -A. 4. Geometry for explanation of the cross-flow diffusion in a two dimensional Cartesian coordinate .3.-66- 4. AY B Ax t^ t where t -t0 = tane = v/u u/Ax v P U = 3/U tane= Ly/Ax Ax by Fig. Diffusion A t1 . v. The travel of the fluid element is analyzed in Fig. and is translated The value 4A at the point A should reappear at the Point A'.-67- element was on the line CAB at time t in the velocity direction. 4. p4A and (l-p)4A and appears at the points Al and A 2 . right and left hand sides.t 0 PA £3 (Current) 2 1 constant ) where 1 = (Gradient) = U3 Lx/cose. Gradient) A /k 1 =A of 4 Current 2 2U of 4 Therefore. which shows that the value 4A diffuses out along the fluid element.4. The diffusion occurs both to the The diffusion to the right hand side is considered first in the following. sine cose cos= sx Uacos61 sin(O+e ) I DRHSS . the following right hand side diffusion constant results. In the donor cell scheme it is divided into two portions. k2 = Ax sine/sin(el+8) k3 = Ax sine 1 /sin(e 1 +8) After substituting appropriate expression for each term. . 4. Geometry for explanation of the cross-flow diffusion in a rotated two dimensional Cartesian coordinate .ct t2 t3 Fig.-68- ti . Ya.4. 38) Equation 4. There are two diffusion components in a two dimensional problem as follows.-69- Repeating the same procedure for the left hand side we can find that the right and left hand side diffusion constants are equal to the following. sine cose RHS LHS cos1 sin(+8) (4.38 is the effective diffusion constant when the diffusion is confined to the one dimensional fluid element. D D = uAx(l-p) = VAy p (4.39) Equation 4. The cross-flow diffusion constants are the ratios of the current and gradient of 4 in the x and y directions. sine cos6 D = UAx sin cos8 x sin(e+e 1 ) 1osel 2 sine cose sin 81 D = UAx y sin (8+8 ) cos1 1 (4. 82 x 9x 2 2 By 2 2 The current on the one dimensional fluid element can be factored into the x and y components.39 is identical to the following simple expressions.40) . 4. 4. 4. Eq. De Vahl Davis and Mallinson [24] also derived the effective cross-flow diffusion constant and their result is given in Eq. When both 6 and 61 are equal to .41) Equation 4. (4. In Fig.42 give the same result.42 underestimates the cross-flow diffusion as shown in Fig.41 shows that DRH S (or DLH S ) is maximum at an Tr7LHS angle of 61 < 6 e between 4 and e1.6. < 6 < 81 or < .38 should be differentiated with respect to 6 in order to find out the velocity direction with maximum cross-flow diffusion. 4.5. RHS D = It can be shown that.40 and Eq. 3 0 when tan 6 = tan6 1 . In the donor cell p' y scheme the value 4 is divided into three portions px 0 .-70- where p = u Ax u v By Equation 4. D DDM = UAlxy sin2e= 3 3 4(by sin36 + Ax cos36) UAx tan6 1 sin26 3 4(tan8 sin6 3 + cos3) (4. 4.42. 4.42) Equation 4. that is to say. Now the analysis will be extended to a three dimensional case in Fig.6 the value at the point 0 is 4 and that value should reappear at the point X where the velocity vector meets the plane ABC. 4.0 150 300 450 600 750 900 Fig. 0.0t Dcf UAx de Vahl Davis & Mallinson Huh 81=8C 1.5.-71- Cross flow Diffusion 2.5 I I 0.5 80 \ % % 4 1. Comparison of the prediction formulas for crossflow diffusion constant by De Vahl Davis and Mallinson and Huh .0 0 0 . Geometry for explanation of the cross-flow diffusion in a three dimensional Cartesian coordinate .-72- z C V Az X 0 o a X AY Ay B y x z C p . 4. p Fig.6. n C denote The coordinate of the Any point on the plane ABC should satisfy the following Eq. and C instead of at the point X. the following relation holds.44) 4 Therefore. t4 cos8. point X is given as follows. X(£ 4cosa.-73- z occurring at the points A. £4 cosy) nB . CX which are denoted as £i' z2' k3 can be obtained as follows.and nA unit vectors along those directions. S+ Ly Ax + Az = 1 plane ABC (4. menon. along the directions XA. B. This is the so-called cross-flow diffusion phenoThe currents due to the cross-flow diffusion are . cosa cosB cosy (4.43) Since the point X is on the plane ABC. XB. S= 4 + y z zAx 1 cos+ + cos Ay Ax + cosy Az (4. BX.45) The lengths of AX. XC.43. 4. £1 = AX =/(x4cosa)2 + (4cos)2 4cos) 2 = Ax 2 + 42 -2Ax£ 4 cosa . y. (Ay4 cosS).46) -+ The unit vectors nA .-74- k2 = BX = /( 4cos)) 2 + (Ay-_4cos) 2 + (k 4 cosy)2 = - y2 + + 42 L 4 £3 = CX = /( 4cosc) 2 + (£4cos8)2 '' + (Az-L 4 cosy) / -z + £42 -2Azi + 4 cosy (4.48) The currents in the directions nA . I (Current)xAi XAI = px Px C1 1 Lt .47) etc. nB. - n A = XA +1A - OA . Therefore. -Z 4 cos8. -4 cosy] 1 nB = [-i4cosa. n C can be obtained in terms of their components in x. nB' n C can simply be given as the products of the transported quantities and velocities.OX (4. z directions. 2 -o C -14cosy] = 1 -- £ 4 cosa. (Az- 4 cosy)] (4. (4.49) The gradients of 4 in the x. (Grad)x = 4. z directions are given as follows./ Ay 4/6x Q/by (Grad)z = 4/Azl (4.51) p Ax D x = lt (Ax-k 4 cosa) + p Ax -.(-4 pz x co s a ) ct 4 + at (- 4 cosa) 4 . y.50) Now the x direction cross-flow diffusion constant may be given as the ratio of the current to the gradient of 4 in the x direction./ Ax (Grad) = 4.-75- I(Current) XB = Py p2 t I(Current) XCI where At = k /U z '3 At (4. Px D (Ax-£ + Py D + -nC-(-4 (Current)x 4 cosa) c osa) At (- 4 (cosa) (Grad)x = 4/Ax Therefore. we can simplify the result as follows. .54) 4. finite element method.3 Discussion of Skew Differencing and Corrective Schemes The origins of the numerical diffusion and their effec- tive diffusion constants have been given in the previous sections.1 = uAx(l-px) (4. tensor viscosity method. indices will also give the same result.52) Inserting the expression for At given by At = £4 /U. etc. This section is a review of the schemes that The have been used to eliminate the numerical diffusion. Dy = vLy(l-py) Dz = wAz(l-pz ) The rotation of (4.-76- p D x Ax At A2 At t cosa(p +p +p) 4 x y z SxA At Ax L cosa at 4 (4. p Ax D = ux[ £4cos x . although none of them has been successful enough to be accepted widely. most important ones may be the skew differencing.53) The cross-flow diffusion constants in the y and z directions can be obtained in the same way. The donor cell treatment of the convection term results in a 5-point relationship in a two dimensional problem.3. However. In the skew differencing type schemes finite differencing of the convection term is done not in terms of the x and y directions. Since the upstream point in the velocity direc- tion does not necessarily fall on the mesh point where the quantity under consideration is defined.-II1~LI1I~~ ICII--1_1___~1___*_____11__ -77- Most of the schemes in Table 4. The corrective scheme is to reduce the diffusion constants in order to compensate for the additional false .. The cross-flow diffusion is not eliminated entirely in the skew differencing scheme.4 fall in the category of a corrective scheme. skew differencing and corrective schemes..~_ 1 i-_. Basically the inclusion of four additional corner points contributes tomore accurate numerical modelling of the convection.^ILli--. the interpolation should be done between neighboring points to obtain the value at the upstream point.. S..1 can be categorized into two types. but just appreciably lower than that of the donor cell scheme. but directly along the velocity direction. the interpolation between neighboring points may still give some cross-flow diffusion. In the skew differencing scheme the 5-point relationship becomes a 9-point relationship.. Chang's method 1161 and Raithby's scheme 1533 are of a skew differencing type while the tensor viscosity method [25] and Huh's corrective scheme that will be introduced in section 4. viscosity method Sel.aptjve me.j. List of the schemes for numerical representation of advection presented at the Third Meeting of the International Association for Hydraulic Research.thod of chaaceriics Fiite anuryic method Tenso. upwind f~iite di Teretn atl. Esposito Glas and Rodi Huffenus Lnd Khali.ordc.-78- Chss' iction Cen:ered . 1981 [65) .1.zky Chen Rue) ct &L Schonaucr Sykes Wad ct aL 41 X 21 21 x 1 40 x 40 100 x 50 81 x41 100 x 50 41 4l 3E 80 x x x x 21 21 19 40 . Grandotto Hickmott Firs.hod Lillinron Ozlandi Snd hod m u macsh? Yes Yes Yes Pc = 10' only Yes Yes Yes Pe = 10' only es mesh 23 x 11 40 x 40 65 41 21 41 x x x x 33 21 11 21 di 'frencc Gt:Iz:kin .-ad. £Ebzhau t aL/PATANKAR Lilinpion Orlandi Priddin Wada cit l.tnce Vuer:o: upwind Yes Yes Yes Yes 2) x II 21 x ]1 21 x 11 21 x 11 41 x 21 41 x 21 21 x 11 81 x 41 28 x 14 Se Tzb)t 2 Up'nd f-t(e elemmnt !.1QUICK t ETovh ci ./LLUE Huint/QUICK Wilkes/LUE Lillimpon/SUD LiiL-.e elements Clifft Dontcl t al. Wilkes Elbahu ct l.ronLUD Lillinion/VUDCC Nuana.thod LLx-W'en d-off - 21 x11 80 x 400 Table 4.Highe cdtr upwi:nd :"'ie Cdi't..hod Mesh *txLnsformation rr.inte Con-ribo:r/me. The corrective scheme is valid only if the effective numerical diffusion constant can be predicted accurately. and D z in Eq.53 and Eq. the corrective scheme can give a numerical solution which is almost free from numerical diffusion.-79- diffusion. 4. The correction formula The diffusion con- used here is Eq. although the validity of its correction formula is not clear. 4.54. 4. D. 4 = C1 + C2 n = C1 + C2 ( - (4. The finite difference equation for Raithby's scheme is derived on the assumption of a linear profile of 0 normal to the flow and a uniform profile in the flow direction.53 and Eq. where the balance equation is set up for a given control volume by considering convection and diffusion at the control surface.1 Raithby's Scheme Raithby's scheme [53] is an Eulerian type of approach. stants in x. y and z directions will be reduced by the amounts of D . The tensor viscosity method is one example of the corrective scheme.. 4. Since the cross-flow diffusion is dominant over the truncation error diffusion and the effective diffusion constants for the cross-flow diffusion can be predicted theoretically. It is therefore a conservative scheme. 4.55) where n is the distance normal to the flow and U = /u 2 +v 2 .54.3. A segment of. 4.-80- Fig. control volume and notations in Raithby's scheme .7. the calculation domain showing grid lines. -81- The convection quantity at the control surface is calculated from the upstream value in the flow direction and the upstream value is obtained by interpolating two neighboring points. not in terms of the convection that occurs at mesh interfaces.3. u 4 + 'v. 'ij- US (4. it is much simpler than Raithby's scheme and becomes conservative in a unidirectional flow. The inclusion of the four additional points also complicates the structure of the coefficient matrix in an implicit scheme.56) where U = /u is the coordinate in the velocity direction and 2 +v 2 . This should be repeated for the four control surfaces of every mesh in a two dimensional problem. The convection term is differ- enced in the velocity direction between mesh points.2 Skew Differencing By Huh Skew differencing scheme here treats the convection term in a Lagrangian way.= U ax dy SC (4.57) where the upstream value €US may be calculated by . 4. which may be too complicated and time consuming for a practical purpose. Although this scheme is nonconservative. Dx = uAx(l-px) D Yy = Ay (l-py ) The expressions for . The validity of this scheme is based on the fact that the cross-flow diffusion can be predicted accurately. D-D z in x.54.53 and Eq.2.4 Corrective Scheme By Huh The corrective scheme is to use the reduced diffusion constants. D-D y .58 for the numerical diffusion constant is apparently not consistent with the results in Eq. 4. 4. Some of the results of this skew differencing scheme is given in section 5.58) t2U= Eq.3 Tensor Viscosity Method The tensor viscosity method treats the numerical dif- fusion constant as a tensor quantity. D-Dx.3. 4. the cross-flow diffusion constants D in section 4. uv (4.3.-82- interpolating two neighboring points.3. 4. D are derived . 4. the tensor viscosity [25) T = Lt uu u2 The expression for is given in the following. z directions to compenx sate for the cross-flow diffusion effect. y.2. D . + -. Dy ._IIU__lpULI~~I___~l--Llt i~Lli--- ~-i ~iL~p-n-~sl.59) v u -. In a coarse mesh or recirculating flow field the latter implementation strategy is definitely preferred. Two implementation strategies have been tested so far. the mesh point implementation and mesh interface implementation.1 Mesh Point Implementation The mesh point implementation is to assign the crossflow diffusion constants Dx .. The velocities at two boundary faces are averaged to get the representative velocity. .These two become identical when the flow field is uniform or a very fine mesh is used so that the flow change over neighboring meshes is negligible.+ w Ay Lz x etc. After the cross-flow diffusion constants are assigned to every mesh point.4.3. they should be averaged again between neighboring mesh points to calculate the cross-flow diffusion current at mesh interfaces. 4. -83- D z = wAz(l-p z ) where px = u (4.Dz at the center of a mesh where all quantities are defined except the flow field. which is used with the cell dimensions to get the cross-flow diffusion constants. Dyl and Dy 2 . 4. There are three possible cases that should be treated independently in a two dimensional case. The right interface will have the cross-flow diffusion constant (Dx1+Dx2 ) and the top and bottom interfaces. The mesh interface implementation is always recommended although it has a more complex program logic than the mesh point implementation. .4. U1 and U2.59.V2) are used to give (DxlDyl) and(Dx 2 'Dy2 ). For the first two cases the cross- flow diffusion constants are zero because the velocity vector is aligned with the mesh configuration. and the resulting D x and D y will be assigned to each interface. can be directly used with cell dimensions in Eq. as shown in Fig.-84- 4. U and V. For the second case the two outward velocities. The velocity U is split into two components. so that they are proportional to V1 and V2.2 Mesh Interface Implementation The mesh interface implementation is to assign the cross-flow diffusion constants directly to the mesh interfaces where the cross-flow diffusion current should be calculated. Note that the cross-flow diffusion constant for the interface with an inward velocity does not have to be considered in that mesh because it will be considered in the neighboring mesh. The same logic can easily Appendix C includes be extended to a three dimensional case. The two velocity sets (Ul.Vl) and (U2.3. 4. The third case is expected to have a flow split as shown in the figure.8. 8. Three possible flow configurations for mesh interface implementation of the corrective scheme in a two dimensional problem ._~I~~ _~~~~_I__~__~~ __ -85- (1) (2) (3) Ul U2 V2 VI U U V14-V2 VI+V2 V2 U2 = UV2 U = Ul + U2 TV2 Fig. 4. 61) Implicit (l-D): n+1 i n 6t At i u -( Ax n+l i - n+l i-1 ) = D A 2 n+l i+-2 i+l n+l + i n+l i i-1 ) (4. 2. for the general conservation equation. 4. which is a parabolic partial differential equation. 4. Eq. 3.62) . Implicit and ADI (Alternate Direction Implicit). 4. The pros and cons of the three schemes will be compared in the following four viewpoints. Explicit. Implicit and ADP schemes There are three typical solution schemes. 1.) = i-1 n D Ax ( i+1 -2i l n n + n ) + iS i-1 (4.60.4 Comparison of Explicit. conservation physical constraint accuracy stability 2 ax 2 2 y 2 +ux +at + 7 + -T7) + S sz (4.-86- the fortran program for calculating cross-flow diffusion constants in an arbitrary flow field.60) Explicit (l-D): n+l I -n i At + u -(i i x n i. n+ n+ + 4n+ n+l + D ij+l - 2 n+1 + 4n+l i i-1 + Ax2 ADI (2-D) : * Ay 2 (B) n *i * 1 j + 1 1- i+l - At _ S +Ax 24. or affected only by the source term contribution (SO). when the net inflow or outflow at .63) 4.. + 4). 2 Ax n+l -2 * ij v n+l ) ij n+l Ay 13j-1 n+1 ij+l n+l 1j-1 At Ay2 + S (4.1 Conservation The conservation principle is that the total amount of 4 should be conserved (S=0).-87- ADI (2-D): n+ ij (A) _n ij In+ + u ij _ cn+ in + v n ij-1 At 2 Ax Ay )n+ i+lj 24 n+ Ax 2 +n+ n -l + D j+l n ij 2 Ay n+l ij n ij-i + S n+l ij n+ ij n+ ) ij n+ i-1 n+l ij-1 At 2 Ax Ay n+ =D i+i S2.4. -88- the domain boundary is equal to zero. A scheme may be both stable and nonconservative and also both unstable and conservative because the conservative property is independent of the stability. there are some nonconservative solution schemes like ICE (Implicit Continuous Eulerian).2 Physical Constraint The physical constraint is closely related with the stability in a practical sense. the conservative property may not be crucial.4. where the convection terms of momentum equations are linearized in a nonconservative way. implicit and ADI schemes can be made conservative by using appropriate differencing forms for the convection term. For a long time transient. the same exchange term should be used for neighboring meshes. Although a conservative form is preferred to a nonconservative one. When we are concerned with a steady state solution. the conservative property is as important as the stability because the error from nonconservation may accumulate over many time steps. however. Although they usually give . All the explicit. + CI 4. In order to guarantee conservation. 4.1 Diffusion There is a maximum net flow that can occur between neighboring meshes by diffusion. R C AxAy = CAxy + D R Ax Ax CAy C y At At ¢4AxAy = 1RAxAy . )C - 4R Ax 6y 4R C Ax Exchanged quantity of 4) by diffusion for time At = DR Ax CAy At After the exchange of the diffusion current. The laws of physics impose some restrictions on the numerical scheme to get physically reasonable solutions. 4.D . new values of 4 are given as follows. they are from entirely different origins.2.-89- similar constraints on the time step size and mesh spacing. The Courant condition and Hirt's stability condition [36] that the finite domain of influence should at least include the continuum domain of influence are good examples of the physical constraint condition. d 2D t < 1 (4. we have the following physical constraint.-90- Since the relationship. 4. R AxAy(4) C) > 2D (R Ax C-yAt 2 DAt X2 < 1 - (4. there will be a damping oscillation in the solution when the time step size is greater than that given by Eq. should still hold. When the physical constraint is ignored and the time step size greater than that given by Eq. however.64 or 4.65 may be a too conservative criterion in a real problem. .64 or 4.65 is used.65) Ax Equation 4. Therefore the physical constraint should be respected in addition to the stability condition to get a meaningful transient solution. t' > Q' .64 may be extended to a d-dimensional case as follows.63 is unconditionally stable. scheme given in Eq. Both of the oscillation and instaThe ADI bility should be avoided in a transient problem. the solution will be unstable or stable with damping oscillations.64) Eq. 4. 4.65. 4. Therefore. It is based on the assumption that 4A is the repreIf At is greater sentative value of the control volume A.4. Then uAttay is no more an appropriate expression for the convection quantity during At.2 Convection C A u B C A ). the damping oscillation or instability may occur as in the case of diffusion.-91- 4.B 4y S The explicit and donor cell treatment of the convection term gives the convection quantity at interface S as ulAAtAy. a portion of the control volume C will cross the interface S. At < 'x u (4. than the Courant limit.66) When a time step size greater than the Courant limit is used.3 Accuracy Realistic interpretation of a finite difference equa- tion comes from an integral form of the conservation . 4.2.4. -92- n+l n+l 4' I I. Profile assumptions of Q in space and time for various differencing schemes . ~~CIIL Is CIIII r r I I I I I! US Ds > I r I US DS Donor Cell Central Differencing Reverse of Donor Cell where US(Upstream) DS (Downstream) Fig.9. I I t i r _ I I [ n n+1 n n+l -_ 0 It Explicit Implicit CrankNicholson $i+. 4. However.-93- equation. The Von Neumann . 0(Ax 2 ) is The also not appropriate for finitely large At and Ax. whether the error introduced at a certain step will increase or decrease as the calculation goes over to the following steps. 4. Since the exact profile depends on the Peclet and Strouhal numbers of the governing equation. The accuracy expressed in terms of O(At). appropriate schemes should be chosen according to those numbers or weighting should be done between neighboring time steps and mesh points to increase the accuracy of the solution. The infinitesimal approximation of differential terms is not appropriate in most problems of our consideration. the error propagation is suppressed and the largest error contribution will be from the previous time step. In other words. the accuracy depends on the profiles 4 in time and space because the exact profiles assumed for are not known a priori. The stability can be checked by the theorems about matrix eigenvalues or Von Neumann analysis.4.4 Stability The stability is a mathematical problem. accuracy depends on how well we can compute the convection and diffusion quantities at control surfaces over the time At. It depends on the maximum eigenvalue of the iteration matrix. the accuracy is usually the least important among the given four considerations. If its absolute magnitude is less than one. lowing sections.4. 4..4.2 2i + D i+1 i n 2 -i i-1 (4. 1 4n = E~nei where I = . but its algebra may get complicated. is expanded in Fourier series as follows.-94- analysis is to expand the solution of a finite difference equation in Fourier series and check the decay of each mode separately. (4.68) . implicit and ADI schemes using the Von Neumann analysis.1 Explicit Scheme The Von Neumann stability analysis of a finite difference equation is simple in its idea.4. 4.4. stability criteria will be derived for the explicit.67) The solution 4. The stability is not a sufficient conIn the fol- dition for an acceptable numerical scheme. n+l i n €i+i I + u i+1 n i-i 2Ax i_1 lt n n i+l .1 One Dimensional Central Differencing of Convection Term The finite difference equation is given for a one dimensional general conservation equation with central differencing of the convection term.1. -95- The absolute value of should be less than one for every When Eq.68 is substituted results. 4.70 may be rewritten as follows by defining cos6 x as t.69) Dbt Dt 2 Tx. 4.2dx(-cosex) 2 + [Cxsinex 2 < 1 (4.2dx(-t) where t = cose -1 < t < 1 ] 2 +C 2(1-t 2) 1 (4. in Eq.67. f(t) = i 12 = (4d2_ 2 )t2 + (-Sd 2+4d )t + C 2+4d 2-4d (4. 2 11 = [1 . Therefore.71) The function f(t) will be defined as follows.72) In order to get the stability of Eq.70) Equation 4. mode to achieve stability.67. [i . the following expression for = l-Cxsin xI where cx d x uAt A tx - 2d (l-cos x) (4. the function . 4. f(t) < 0 (4. (1) 4d x 2 -Cx 2 > 0 x : concave upward : linear 2 (2) 4d 2 -C 2 = 0 (3) 4d x 2 -Cx 2 < 0 : concave downward. The parabola f(t) always meets the t-axis because the discriminant is always greater than or equal to zero.75) The parabola f(t) may have three distinct shapes according to the sign of the coefficient of the second order term.74) One of the two meeting points with the t-axis is the point t=l because f(l) is equal to zero.1]. D(Discriminant) = (C 2 .2d )2 > 0 (4.1] Equation 4. In case (1) f(t) is a parabola concave upward and f(-l) should be less than or equal to zero as shown in Fig.73 can be solved by a graphical method using the characteristics of a quadratic equation.-96- f(t) should be less than or equal to zero for any t in [-1. . 4.10(1).73) for any t in [-1. Therefore. f(1) = 0 (4. -97- f(t) f (t) f (t) 'I I -4. f(t). Three distinct shapes of the parabola.3 0 0 /1 i2 2 4d -2d x x 22 4d -c xx t / (1) 4d 2 -c 2 >0 x x (2) 4d 2 -c =0 x x 2 2 (3) 2 2 4d2-c2<0 x x Fig..10.- r I I t -i t -1 1 t 0. 4.. according to the sign of the coefficient of second order term . In case (3) the parabola f(t) is concave downward and the axis of symmetry should be on the right hand side of the point.-98- f(-l) = 8dx(2dx-1) < 0 0 < d < (4. 4. 4.C (4.78 is a finite difference equation with donor cell differencing of the convection term. . 4. we get the following stability conditions.4. 4.2d Axis of symmetry: t = 4dx dx > C 2 x > . 4.4.77) Summing up the results in Eq. 4d 2 .dx) is shown in Fig.2 One Dimensional Donor Cell Differencing of Convection Term The following Eq.76 and Eq. t=l as shown in Fig.10(3).1. 4d 2 -C 2 4d C > 0 < 0 : : 0 < d dx > < 2 Cx The stable region in the coordinate system (Cx.77. 4.76) In case (2) f(t) is a straight line and f(-l) again should be less than or equal to zero.11. 11. 4. Stability -ondition in the plane (Cx.-99- d x 1 -1 0 1 dx = C2 x 2x C Fig.d ) for an explicit scheme with central differencing of convection term in a one dimensional problem . -100- n+l i n At At i n + u i Ax n i-i n = D i+l n i n i-i Ax 2 2 (4.78) where the velocity u is positive. The Von Neumann analysis of Eq. 4.78 gives the following stability condition. 2 It 1-(C x +2d [ x )(1-cose x ) 2 + [C sine ) x 1 (4.79) (479) can be shown that the value of IC12 in Eq. 4.79 does not change when the velocity u is less than zero, if the Courant number, Cx, is defined such that it is always positive. The function f(t) is defined as follows to make it easier to solve Eq. 4.79. f(t) = 12 1- S[(C x+2d) 2-Cx ]2 t + [-2(C x+2d )2 -2(C +2d ) ] + 2(C x+2d )]t (4.80) + [(C +2d ) +C The function f(t) in Eq. 4.80 should be less that or equal to zero for any t in [-1,1] for the stability of Eq. 4.78. f(t) < 0 for any t in [-1,1] (4.81) The parabola f(t) meets the t-axis at the point, t=l. f(1) = .0 (4.82) -101- Equation 4.81 canbe solved by the graphical method as in the previous section. 2 2 (C +2d ) - C > 0 : 0 < Cx +2d X -< 1 - (4.83) (c x +2d x) 2 -C 2 x < 0 d >(C 2-C ) dx x (4.84) The results, Eq. 4.83 and Eq. 4.84, are shown graphically in Fig. 4.12. 4.4.4.1.3 Two and Three Dimensional Donor Cell Differencing of Convection Term The two dimensional extension of Eq. 4.78 is given in the following. n+l ij At n D . i+D 3 24 Lx n ii n + u ij Ax n n i-1 n n + vii by n ij-i n + 4 A2 1i-l3 n n 4 - 24 + D2+l ,_2 ay + i-i (4.85) The Von Neumann analysis is applied to the finite difference equation, Eq. 4.85, yielding the following result. - 1 = -C x (l-e lex) - C (l-e -I y (l-cos8 y) _ 2d (-csex ) x x (4.86) - 2dy (1-cosey) -102- d x 1 2 x- 1 2xx 1 d 1 2 1 C -Cx) Fig. 4.12. Stability condition in the plane (Cx,d x ) for an explicit scheme with donor cell differencing of convection term in a one dimensional problem -103- where C x - Ax C - vat d - DAt2 x2 x Y DAt Ay2 The Courant numbers, Cx and C y , are defined so that they are always positive, independent of the directions of the velocities u and v. Equation 4.86 can be simplified by defining al and a 2 as given in the following Eq. 4.87. = 1 + al1 where a= 1 -C x - 2d x + (Cx+dx)e x -I6 x + a2 (4.87) + de x le x S = 2 -C Y -2d + (C +d )e YY Y -I6 x + de Y Iy The stability condition is that the absolute magnitude of 5 should be less than or equal to one for any values of 8x e and By. From Fig. 4.13 and Fig. 4.14 it can be seen that indea1 and a2 should satisfy the following conditions pendently. a + 1 4 (4.88) (4.89) The region in the complex plane where (a +c ) should exist for the stability of an explicit conscheme with donor cell differencing of vection term in a general two dimensional problem .13.-104- Im \ .- .1 SRe -2 al +2 1+a +a 2 Fig. 4. Re Both al and a 2 should be in the shaded circle to ensure that Ial+a 2 +1I< i.. . Fig.-105- . The region in the complex plane where the amplificatior factor. should exist for the stability of an explicit scheme with donor convection term in cell differencing of a general two dimensional problem . 4.14. . where the points (Cx. 4. 4. Therefore. At.e Y) (4.4 Stability Condition in Terms of Cell Reynolds Number The stability conditions in the previous three sections are in terms of the dimensionless numbers. 4.dx) and (C y.4.e Rx . the stability However. When Ax and At are already given. and time step.88 and 4.88 and Eq. the stable range of Cx should be determined in terms of the cell Reynolds number Cx/dx'. which is independent of the time step size.90) a 2 = C [-1 y R R Y + R (4.89 can be solved by the graphical method as in the previous sections.1. Cx and d .d) should be in the shaded region for stability. 4. which depend on both the mesh spacing. -I 1 2 + (1+-)e Rx Rx 2 + (1+)e a1 = Cx [1 + Ie 1 3 .15. At. The results are given in Fig. The stability condition is again given by Eq. Ax. the stable time step size for a given mesh spacing.91) .4. The stable region shrinks on a linear scale with dimensionality.-106- Both Eq.16.89 and al and a2 may be rewritten as follows. A three dimensional case can also be solved through the same procedures and the results are given in Fig. 4. 4. our usual concern is can be checked directly. 15. . 4.dy) fcr an explicit scheme with donor cell differencing of convection term in a general two dimensional problem. Stability cc.idition in the planes (Cx.-107- d (d ) d -= x 2 1 x + -1 4 C x(C ) 16 d= C2_ C x 2x x Fig.dx) and (Cy. for an explicit scheme with donor cell differconvection term in a general three encing of dimensional problem . 4.-108- d xy(d or d z ) Cx d- dx + 1 1 Cx (C d! S3 2 .16.(C Cx x y or Cz) z d ) Fig. Stability condition in the planes (Cx.dx) etc. (1) Cx 2 K2 -1) > 0 concave upward .94 and Eq.88 and Eq. [ .__LI___IIII__LI*X__ ID I-- -109- The following condition for Cx and Rx can be derived from Eq.1] f(l) = 0 (4.93) ) where 2 K = 1 +.95.94 ) D(Discriminant) = C 2 (K-2C ) 2 > 0 (4.R and x t = cosex The stability condition is now that f(t) should be less than or equal to zero for any t in [-1. f(t) = C (1+K) 2 (l-t) .90.92) Equation 4.95 ) The function f(t) satisfies Eq. and has distinct shapes for the Following three cases.C K(l-t) + C ( l_1-t = (C 22 K-C 2 2 )t2 + (2C 2 22 22 K +C K)t + C 22 K - C K + C 2 < 0 (4. 4. 4. 4.92 can be simplified as follows.C (+2x .) RX (1-cose )) x + [C sine] x x < (4. 4. 17.17 is applicable to all general 2-D problems. 4.97) The results in Eq. therefore the subscript x may be replaced with the subscript y in Fig.97 are shown in Fig. f(-l) should be less than or equal For case (3) the axis of symmetry of the parabola The should be on the right hand side of the point. (4. although it can be relaxed if there is a constraint on the velocity direction and mesh spacings such that the following' relation is satisfied.17.yu < 1 (4. The stability condition in Fig. 4.98) Ax . 4.96) R -1 x < 0 < C x < (1+2-) Rx . t=l.96 and Eq. solution procedure is similar to those in the previous sections and the results are given in the following as. to zero. R > 0 x : 0 < C < x- 2 2 2(1+-) x and (4. 4. V f . 4.-110- (2) C 2(K 2-1) = 0 linear (3) 22 C 2(K 2_-1) < 0 concave downward For case (1) and (2). The same result will be obtained for the y-direction. Cx) and (Ry.Cy) for an explicit scheme with donor cell ditferencing of convection term in a general two dimensional problem .17. 4. Stability condition in the plane (Rx.-111- x (C y C x - 2 1 2 (1+ L) R x C = 2(1+ ) x -2 0 R (R x y Fig. Cx) for an explicit scheme with donor cell differencing of convection term in a two dimensional problem with f v u/Lx . Stability condition in the plane (Rx.18. 4.-112- 1 1 x CX x 1 (l+f) (1+ x ) 1+f -1- 1 f 0 x V where f =-y u Ax Fig. 4.99) (4.uLx. parison between the analytical result in Fig.2 shows the com- 1 l+f in Fig.17 and numerical experiments.-113- Then the number replaced with on the C -axis in Fig. d > - greater Cx (4. (4.C Y- y . 4.19 and Fig. and it can be seen that the implicit scheme is unconditionally stable if the diffusion constant is than . 4.99 is a fully implicit finite difference equation for a general one dimensional problem and the Von Neumann analysis will be applied in the same way as in the explicit scheme.99) The stability condition is shown in Fig. 4.2 Implicit Scheme The following Eq. the stability condition is found to be the following. 4.100) For a general two dimensional problem.Cx d > .20.18.101) (4.102) dx > .17 should be Table 4. n+l n t n+l n+l n+l n+l n+l S i + u i +xu i -i= D i+l D i Lx 2 i-i (4.4.4. 4. 4. 780 0. R x 0.380 0.833 8.488 0.260 0.019 Numerical Experiments 0.3 -8.333 833.167 -2.540 0.147 0.060 Table 4.120 0.660 0. Comparison of the stability conditions in terms of the cell Reynolds number by Von Neumann analysis and numerical experiments in a two dimensional explicit scheme with convection donor cell differencing of term .083 Von Neumann Analysi s 0.403 0.499 0.-114- Maximum Stable C.120 0.180 0.629 -2.333 -4.660 0.720 0.333 83.2. -115- -d = - x d =. Stability c-ndition in the plane (Cx.19. 4.dx) for an implicit scheme with donor cell differencing of convection term in a one dimensional problem .-C x 2x x 2x C x dx 1(C2+C 2xx ) Fig. 20. 4.Cx) for an implicit scheme with donor cell differencing convection term in a one dimensional of problem . Stability condition in the plane (Rx.-116- Cx=-1 - 2 Rx -2 -1 0 R x -R C x R +2 Fig. 63 gives the following stability condition.-117- 4. the following condition should be satisfied.+d )] C ) {[1+(l-cosex +dx)) +[fsinex }{[l+(1-cosy) (+dy) + Y[siney } (4.104) .jto be less than or equal to one. treating them implicitly one by one at each fractional step and the other formulation has only the x or y direction component treated implicitly in each step. C d x>2 (4.63. 4. 4.103) jsj <1 In order for I. 2 _C C 2 {[(-cos )(-+d X))] C [sine I2} {(1-cos6 C 2 -)(. One of the formulations has both the x and y direction components in each step. The Von Neumann analysis of the first formulation in Eq.3 ADI Scheme There are two possible formulations of the Alternate Direction Implicit (ADI) scheme given in Eq.4.4. 105) The stability condition of the second formulation in Eq.63 can also be given by the Von Neumann analysis as follows. 4.107) Therefore the second formulation of the ADI scheme in Eq.106) C d y - > - -~ 2 (4.63 has the same stability condition as the fully implicit scheme. . C d y> - 2 - 4 (4. C d x- > 2 2x (4. 4.-118- The same condition should hold for the y-direction. 5. The physical models are tested in a natural convection problem and the numerical schemes are tested in a simple geometry to compare numerical solutions with exact solutions. energy convection and heat transfer models are considered and the resulting flow fields are given in Fig.-119- CHAPTER 5 RESULTS Some calculations are performed to validate the physical models and numerical schemes in the previous chapters.1.1 Natural Convection Results The containment is modelled as a two dimensional rectangular compartment with an obstacle as a heat sink. density is determined by temperaSince the natural convection of air ture and pressure. 5. 5. Natural convection occurs due to the heat transfer between the obstacle and containment air.2.1 and Fig. occurs as a result of the buoyancy force due to density . For example.1 Updating of the Reference State The state of air is determined by any two independent properties. the ADI scheme is explored to -use a time step size larger than the Courant limit to save the overall computational time. Finally. Updating of the reference state. 5. 29 Ibm A= 2 ft2 T. 5.0 Btu/lbm 0F (Q ob = 208. 0.8 sec .1.-120- -p ). Natural convection flow field due to heat transfer between the obstacle and air in the containment at t = 10. = 677.4 sec AT = 100F T .5 ft/sec h = 1. = 680F air T = 580F Fig.8537x10-tu/ft20 F sec (Cp) = 1. Natural convection flow field due to heat transfer between the obstacle and air in the containment at t = 62.0 Btu/lbm OF (c )b = 208.4 sec 1 0 AT = 10 F T air = 680F .-121- > 0. 5.8537x10- tu/ft2 F sec p)ob = 1.29 ibm (K 2 A= 2 ft T.2. T ab = 580 F Fig. = 677. 5 ft/sec h = 1.5 sec . Told' Tnew: Temperature of the fluid at old and new time steps.total Pold Told new old Qcell cv Mf. depends primarily on the temperature. Pold' Pnew: .1) T Pnew = old + Ttotal c v M f.-122- gradients. temperature and pressure are equally important parameters in determining the flow field. The equation of state of an ideal gas is given in the following.cell where Pold' Pnew: Density of the fluid at old and new time steps. however. The density change of liquid. time Pressure of the fluid at old and new steps. new Pnew RT new (5. not on the pressure because of a high modulus of elasticity. c : Specific heat of the fluid at constant volume. 2 Energy Convection Another difference between liquid and gas is the ratio of the Specific heats.0 Monatomic. The energy convection will be underestimated by a factor of 1.3 Heat Transfer Modelling The heat transfer to the obstacle is modelled as a natural convection from a vertical surface.1.total : Mass of the fluid in one cell and in the total domain. The value of k is 1.1.2 in terms of the dimensionless numbers. An experimental correlation is given in Eq. The Nussett and Grashof numbers here are based on the axial length of one computational mesh.cell. 5.4 when internal energy is used for enthalpy in air flow. convection energy transfer should be in terms of enthalpy instead of internal energy. The VARR calculations 121] have shown that the magnitude of velocity can be affected by a factor two by updating the reference pressure at every time step.. k = c /C for liquid and 1. Mf.-123- Mf. 5. but they may also be based on the vertical height of the obstacle. . 5. Qcell total : Heat input to the fluid in one cell and in the total domain. .4 for air. diatomic and polyTherefore. the atomic gases have different values of k. 8537 x 10 .2 ft/sec' -5 V = 1.708 (air) Gr-Pr 105 .25 0. Gr = 9.3) Pr = 0. h = 1.923 x 10-3 Therefore. .4 lbm/ft-sec Btu/ft 2 F.-124- Nu = C(Gr-Pr) n ( 5.0763 ibm/ft 3 g = 32.2179 x 10 B = 1.109 > 109 C 0.4 The following data are used to calculate the heat transfer coefficient at the obstacle wall.sec .7134 x 108 Nu = 89.2) 3 2 Gr = gS(AT)L p 2 (5. This may be a typical value for the natural convection heat transfer in the containment without any phase change.876 The heat transfer coefficient is given as.021 n 0.555 0. L = 2 ft AT = 500 F p = 0. Heat transfer area between the obstacle and fluid over one computational mesh. can be obtained from heat balance between the obstacle and fluid as follows.4 Turbulence Modelling Laminar flow is assumed in obtaining the results in Fig. 5. Obstacle density. Therefore. Heat transfer coefficient between the fluid and obstacle. The time constant. 5. pf Pob: c h : Fluid density.2. CQ = Cpob (T-T) (5. the heat . Specific heat at constant pressure.-125- The VARR input [21) requires the time constant of the heat transfer instead of the heat transfer coefficient.4 sec. 5. (Mc )ob (MC) obf hApo b ob f (5.4) where -CQ is the source term in the energy equation of fluid.1. T = 677. T.1 and Fig.5) where M A : Mass of one computational mesh. If the flow is turbulent. The flow in Fig.2. The flow in Fig.3. 5. The corrective scheme is tested in recirculating flow problems for mesh point and mesh interface implementations. 5.to the obstacle will be enhanced and the diffusion transport of momentum and energy will also be enhanced in the containment air.4 and Fig.5 show that the numerical solutions is close to the analytical solution and the truncation error can be neglected. 5. 5. is compared with De Vahl Davis and Mallinson's in predicting the magnitude of cross-flow diffusion. steady-state problem in Fig.6 and Fig. 5.2 Numerical Diffusion The truncation and cross-flow diffusion errors are compared in a simple geometry where an analytical solution can be obtained.3(B) is skewed at 450 to the mesh orientation and both truncation error and cross-flow diffusions are included in the solution error.7 show that excessive numerical diffusion has occurred due to the . 5. Figure 5. 5.-126- transfer. 5. The skew differencing and corrective Huh's formula schemes are tested for various problems.3(A) is parallel with the mesh orientation so that no cross-flow diffusion occurs. The results in Fig.1 Truncation Error and Cross-flow Diffusion The truncation error diffusion is compared with the cross-flow diffusion in a two dimensional. 07 (B) =0.0 B T specified Fig.0 x BT - @x a( aT + 2 2 a2 ay ) Th= 10.0 u = v = 7. 5. for evaluation of numerical diffusion .0 u = 10.0 a= 1.-127- (A) .0 BT S=0. Problem geometry for the two cases.3.0 ay AB=BC=5. 0 ST By =0.0 B T_ -0. (A) parallel flow and (B) cross flow.0.0 (A) (B) AB=BC=5. spatial coordinate Fig. 5. Comparison of the analytic solution and numerical solution with donor cell differencing of convection term in 10x10 meshes along the line CD in Fig.-128- Analytic solution Numerical solution with Donor Cell T 10. 5.3(A) .4. -~l r.-__~ .~ I~. 5.~ -129- Analytic solution Numerical solution with Donor Cell C spatial coordinate Fig.5.3(A) .__LL l.^XI ill I~LX-_L-~LI. Comparison of the analytic solution and numerical solution with donor cell differencing of convection term in 6x6 meshes along the line CD in Fig. 5. 5.-130- Analytic solution Numerical solution with Donor Cell Analytic solution with D+Df cf T 10 spatial A coordinate Fig.3(B) .6. analytic solution with increased diffusion constant and numerical solution with donor cell differencing convection term in 10x10 meshes along the of line CA in Fig. 5. Comparison of the analytic solution. Comparison cT the analytic solution.J 0 I' C spatial A coordinate Fig. analytic solution with increased diffusion constant and numerical solution with donor cell differencing of convection term in 6x6 meshes along the line CA in Fig. . 1/ 5.3(B) s' / line CA in .7. 5..-131- Analytic solution Numerical solution with Donor Cell Analytic solution with D+Dcf T 10 / I / / . 5.3(B) ./ Fig. 10 x 10 meshes and Fig. Figure 5.11 and Fig. it is necessary to test them for various cases. 5. Eq.7 show that the numerical solution can be reproduced by the analytical solution with increased diffusion constants by the amount given by Huh's formula. 6 = 6 = 450. 5. Figure 5. 5.7 than in Fig.53 and Eq.2. Since the corrective scheme is based on the validity of those correction formulas.7 are for 6 x 6 It is seen that a finer mesh gives a better Since the cross-flow diffusion constant is pro- portional to the mesh spacing. De Vahl Davis and Mallinson's and Huh's formulas give the same cross-flow diffusion constants for the case. 5. 5.12 for a simple pure . It can be seen that the former result is much better than the latter. 5.9 is the result by Huh's formula and Fig.10 by De Vahl Davis and Mallinson's for the problem in Fig.6.8 has the arbitrary angles of 6 and 61 such that 6 = 600 and 61 76. 5. 5.54. 4. Raithby's and Huh's. 5. The problem geometry in Fig. 4.-132- cross-flow diffusion.6 and Fig. 5.2 Validation of Huh's Correction Formula The cross-flow diffusion can be predicted by Huh's formula. are tested in Fig. solution.810.5 and Fig.3 Skew Differencing Scheme Two skew differencing schemes.8. more diffusion has occurred in Fig. 5. Figure 5.6 are for 5.2. meshes.4 and Fig. 5. 335 Huh A e1 = 76. Problem geomctry with arbitrary angles of 6 and 81 to compare De Vahl Davis and Mallinson's and Huh's formulas for prediction of cross-flow diffusion constant .8.6603 DDM= 0.-133- Yo 2 T=0 D a = 1.810 8 = 600 8 Meshes 24 Meshes Fig.0 Ax = 0. 5.2406 by = 1.7674 DH u h = 3.0267 u = 5.0 v = 8. 5 0 spatial coordinate Fig. 5.-134- Analytic solution with D+Dcf Numerical solution with Donor Cell T 10. 5. by Huh's formula along the line BD in Fig.9. Comparision of the numerical solution with donor convection term and cell differencing of analytic solution with increased diffusion constant. (D+Dcf).8 . 5.-135- Analytic solution with D+DDM Numerical solution with Donor Cell T 101 spatial coordinate Fig. by De Vahl Davis and Mallinson's formula along the line BD in Fig. 5.8 . Comparision of the numerical solution donor cell convection term and analytic differencing of with increased diffusion constant.10. solution (D+DDM). 5.5 68. Comparison of the true solution and solutions by donor cell scheme.75 75 50 U Skew differencing by Raithby 50 100 100 100 100 50 100 100 100 50 0 100 100 50 0 0 100 50 0 0 0 50 0 0 0 0 50 34.75 25 U 12.-136- Donor cell 100 100 100 100 100 93.25 0 Skew differencing by Huh (True solution) 100 100 100 100 50 0 0 0.5 0 6. skew differencing scheme by Raithby and skew differencing scheme by Huh in a pure convection problem with e = 45* .25 18.25 65.63 87.38 50 31.11. 0 100 100 100 100 100 50 100 50 0 50 0 -0 V Fig.75 81. 12.71 20.-137- Donor cell 100 100 100 100 50 -. 62.70 4.01 55.3E 21. 5.9 33.E 21 0 12 True solution 3.1 6.22 -9.430 0 50 0 Comparison of the true solution and solutions by donor cell scheme.88 81.86 16.52 1.33 70.25 53.67 -5.25 -1.25 0 0 0 .11 3.430 .2! 90.96 17.85 -0.74 38.5 0 0 63.6 0 Skew differencing by Huh 100 90.6.3' 40.59.11 0 11.89-22.91 71.83 72.91 31.56 1.5E 25.3: 11.43 7.25 37. skew differencing scheme by Raithby and skew differencing scheme by Huh in a pure co-vection problem with e = 63.A Skew differencing by Raithby 100 100 100 100 96.26 80.99 10.50 2.47 -23.5C 100 100 50 Fig.13 0 0 100 100 100 100 100 100 50 25 0 50 25 0 0 0 0 0 0 0 0 0 0 0 0 6. It is indicated that Raithby's scheme may give unphysical results in problems with steep gradients of the quantity under consideration. Some cross-flow diffusion has occurred in Huh's scheme.11 has a flow given in sections 4.12 has a flow at a 63. method while Huh's scheme is Lagrangian. 5.2.4 Corrective Scheme The prediction formulas of the cross-flow diffusion constants are validated by showing that the numerical solution can be reproduced by the analytical solution with the appropriately increased diffusion constant. Raithby's scheme is an Eulerian Therefore. The details of those schemes are Figure 5.13 and Fig. The purpose of the corrective scheme is. but less than inthe donor cell scheme.14 show that the analytical solution can be reproduced by Huh's skew differencing scheme for the case.-138- convection problem. former is conservative while the latter is not. at a 450 angle with respect to the mesh orientation and shows that the donor cell scheme introduces appreciable cross-flow diffusion while Raithby's and Huh's skew differencing schemes give the true solution.2. Fig.1 and 4. Figure 5. the However. 5. both of them are conservative in a uni-directional flow. to obtain the true solution by subtracting the additional cross-flow diffusion . 5.3.3. 8 = 1e= 45*.430 angle and neither Raithby's nor Huh's scheme can reproduce the true solution. however. Comparison c f the analytic solution and numerical solutions by donor cell scheme and skew differencing scheme by Huh for 10x10 meshes along the line CA in Fig.-139- Analytic solution Numerical solution with Donor Cell Skew differencing scheme by Huh C A spatial coordinate Fig.13.3(B) . 5. 5. -140- Analytic solution Numerical solution with Donor Cell Skew differencing scheme by Huh 10 5 0 spatial A coordinate Fig. Comparison of the analytic solution and numerical solutions by donor cell scheme and skew differencing scheme by Huh for 6x6 meshes along the line CA in Fig. 5.3(B) . 5.14. Two implementation strategies of the corrective scheme.21 gives the true solution. Figure 5. donor cell solution and the solutions by mesh point aand mesh interface implementations. The corrective scheme gives a physically reasonable solution.18 show that the cross-flow diffusion can be eliminated by the corrective scheme and that a finer mesh spacing always gives a better solution. 5. 5. Fig. although not identical to the true solution. Figure 5. The corrective scheme is also tested in the problem geometry of Fig. Both of them are tested in a recirculating flow problem with pure convection. are introduced in Chapter 4.15 numerical solutions are given for a simple pure convection problem with various diffusion corrections. mesh point and mesh interface.3(B).430.16 has a flow with 8 = 63. In Fig. Figure.-141- constant from the total diffusion constant of the finite difference equation.20 shows the diffusion corrections of the mesh point and mesh interface implementations at each interface. Fig. 5. The full correction of the cross-flow diffusion gives the true solution without any numerical diffusion error. The mesh point implementation gives an unphysical solution . 5. 5.19 is a hypothetical 3 x 3 recirculating flow field where the inlet boundary values are specified on the left and bottom surfaces. 5.17 and Fig. The flow is at a 45* angle with respect to the mesh orientation. 92 5.88 5 4.99 D=-0.71 5.5 0 1.0 2.75 6.3 D=-0.59 7. Donor cell solutions with various diffusion corrections for a pure convection problem with e = 450 .29 0.0 8.0 2.5 - 3.88 10 5 0 2.0 1.13 7.-142- D=0.5 (True Solution) 9.25 0 5.1 10 8.5 x y Fig.01 7. 0 V ________ D=-0.71 0.29 5.0 10 7.0 2.15.41 u=v=l Ax=Ay=l D =D =0.08 5. 5.12 5. 56 - 2.07 2.07 0.07 10 3.91 0.16.73 10 7.37 3. 5.96 3.430 . 0 D-2 3 0.33 1.-143- D=0. v=2 x= y=l 2 D =D =-- x y 3 Fig.59 1.93 10 5. True solution and donor cell solutions with and without diffusion correction for a pure convection problem with e = 63.430 u=l.41 0.11 6.04 4.10 9.82 -0.11 True Solution V 63.11 0. 5. numerical solution by donor cell scheme and numerical solution by corrective scheme for 10x10 meshes along the line CA in Fig.17. Comparison of the analytic solution.3(B) . 5.-144- Analytic solution Numerical solution with donor cell Solution by corrective scheme 10 A spatial coordinate Fig. 18.-145- Analytic solution Numerical solution with donor Solution by corrective scheme 10 5 0 spatial coordinate Fig. 5. Comparisoin of the analytic solution. 5.3(B) . numerical solution by donor cell scheme and numerical solution by corrective scheme for 6x6 meshes along the line CA in Fig. 3 -)- 2 3 2 Lx = Ay = 1 Fig.-146- 0 Of-r 3 ~LC~' / 10 1 1 5 0 1 3 3.19. Simple recirculating flow field for test of the implementation strategies of the corrective scheme . 5. Diffusion coastant corrections at each interface for the mesh point and mesh interface implementations of the corrective for the recirculating flow field given in Fig. 5.625 0 .5 0.75 1.19 .2 0 0 0 mesh interface implementation Fig.25 1.75 1.125 0.25 mesh point implementation 0.75 0. 5.5 0 0 5 0 0 .75 1.2 0.20.75 0 0 0.625 0.-147- 0. 83 6.10 8.75 6.51 10 5 0 Fig.16 11.26 8.10 9. 5.21 7. donor cell solution and solutions by the mesh point and mesh interface implementations of the corrective scheme for the recirculating flow field given in Fig.82 8.-148- True solution Donor cell mesh point implementation -5.21 mesh interface implementation -0.21. Comparison of the true solution. 5.86 12.94 9.16 6.19 .18 7.08 -9. 5. 0. The true solution cannot be reproduced because some of the information about the flow field has already been lost in Fig. Figure 5.-149- while the mesh interface implementation gives a physically reasonable solution. After a while a flow field is set up in the This containment by the input mass and momentum of steam.28 show the steam concentration distributions after one time step.23. The result in Fig.27 and Fig.3 sec with the corrections in Fig.25 and Fig. Another test calculation is done for a two dimensional. Since the steam concentration distribution and flow field are given at the start of the calculation. 5. The steam concentration distribution at the beginning is given in Fig. There was initially air in the containment and steam is introduced in the source mesh at the rate of 0. respectively.28 is not a reasonable solution while the mesh interface implementation result in Fig. 5.2 kg/sec. 5.26 show the diffusion constant corrections of the mesh point and mesh interface implementations and Fig.19. 5. 5. 5.25 and Fig.26. 5.22 shows the flow field in the containment and more detailed information about the flow field is given in Fig. flow field is used here to test the mesh point and mesh interface implementations of the corrective scheme.27 is physically reasonable and even better . 5. rectangular containment with an arbitrary recirculating flow field. Figure 5.24 in which the source mesh has the highest steam concentration. 5. the steam concentration for the new time step can be calculated in an explicit way. Recirculating flow field for calculation of the steam concentration distribution in the two dimensional containment to test the mesh point and mesh interface implementations of the convective scheme .5m Ay = 1.22. 5.0m Fig.-150- Ax = 0. 3 095 1.2629 0.2190 0.34 0. 1317 0. 5955 SOURCE 0.3253 0.531 0. 0735 0.3074 0.5174 1.1139 0 .4870 0.23. 5.2159 0. 3563 0.456 0.0127 0.-151- 0.1276 0 .7126 0. Detailed flow field for calculation of the steam concentration distribution in the two dimensional containment to test the mesh point and mesh interface implementations of the corrective scheme .4920 0 .93 2.3957 0. 045 0. 0648 1.2216 0.1422 2148 0.0348 0.5893 0. 2214 velocity [m/sec] Ax = 0.5 m Ay = 1 m Fig.1229 0.2474 04 1379 0.0492 0 .2435 0.1385 0.2460 0.207 2. 861 1.0937 0 1. 5m by = Im Source: steam concentration in Ikg/m 3 0.1085 0.0095 0.1420 0.0179 0. 5.1178 0.0358 0.0620 0.0007 0.2 kg/sec of steam Fig.0136 0.0049 0.0346 0.1266 0.1244 0.24.-152- 0.0676 0.0031 Ax = 0.1710 0.0305 0.1592 0.0011 0.1712 0. Initial steam concentration distribution in the two dimensional containment to test the mesh point and mesh interface implementations of the corrective scheme .0002 0.1750 0.1859 SOURCE 0.0819 0.0298 0. 0 183 Ax = 0.1583 0. 0396 0.0937 0.1452 0.0731 116 0.0253 0.0 362 0.0311 0. Diffusion co.2740 0.. 5.0440 0.3262 0.1854 0.0 278 0.1178 0. 0152 0.0600 0.1701 0. 0227 0. 0.C209 0.0794 0.0193 0.1763 0.1 389 0.1451 0.1694 3106 0.25. 0086 0.0255 SOURCE 0.0300 0.0311 0.1279 0. 0488 0.0590 0.stant corrections of the mesh point implementation of the corrective scheme for calculation of the steam concentration distribution in the two dimensional containment . 0. 2422 0.0224 0.-153- 0.1014 258 0.0348 0.5m by = Im Diffusion constant in Im2/sec] Fig. 0. 0 923 0 0. 0579 0.4732 0.0479 0. 5.-154- 0 0 0.2134 0.0 76 0.0 257 0.0215 0. 0706 0.0860 0. 0240 0.0632 0.0534 0.0287 0.0192 0 0.1149 0 0 0.1026 0 SOURCE 0 0 0.3694 0.0 111 0 Lx = 0.0518 0 0 0.2529 0.5m by = lm Diffusion constant in Im2/sec] Fig.2317 0 0.26.9769 0 0 0.0443 0. Diffusion constant corrections of the mesh interface implementation of the corrective scheme for calculation of the steam concentration distribution in the two dimensional containment . 1623 0.1733 0.0465 0.0011 0. 5.0068 0.1243 0.0003 0.1321 0. with donor cell differencing convection term without any diffusion of correction .0107 0.0016 0.1711 0.27.0342 0.0947 0.0195 0.0342 0.1847 SOURCE 0.1706 0.0833 0.0735 0.2 kg/sec of steam Fig.0448 0.1475 0.-155- 0.1363 0.0233 0.0046 Lx = 0.3 sec.1135 0.5m Ly = Im steam concentration in [kg/m 3 ] Source: o. 0. Steam concentration distribution after one time step. 1334 0.-156- 0.0465 0.1646 0.0250 0.0815 0.0029 0. 5.0010 0. Steam concentration distribution after one time step.2 kg/sec of steam Source: 3 Fig.0042 Lx = 0.0092 0.1493 0.1797 0.0287 0.5m Ly = Im steam concentration in Ikg/m 0.1247 0.28.3 sec. 0.0433 0.1911 0.1240 0.2086 SOURCE 0.1729 0.1747 0.0005 -0.0937 0.0229 0.0709 0.0355 0. with mesh point implementation of the corrective scheme .0068 0. 0360 0.1847 SOURCE 0.1112 0.0815 0.0228 0.0058 0.0192 0.3 sec. 0.1243 0.0100 0.0947 0.1487 0.1756 0.-157- 0.2 kg/sec of steam Fig.5m by = Im steam concentration in [kg/m 3 Source: 0.0722 0.1794 0.1628 0.1738 0.1310 0. 5.0424 0.0295 0.0481 0.0010 0.0003 0.29.1342 0. with mesh interface implementation of the corrective scheme .0007 0. Steam concercration distribution after one time step.0039 ax = 0. 0 and the inlet boundary value of 4 is. 5. 5.-158- than the donor cell solution.32 shows the profiles of 4 at the same loca- tion for the time step size of 1 sec. it has the maximum time step size that can give a physically reasonable solution.31 shows the profiles of 4 along the dotted line for the time step size of 0. Figure 5. There is a source of 4 on the right boundary over two computational meshes and the profile of Q along the dotted line is affected by that source. In Fig. =10. 5. The results of the ADI and explicit schemes are compared in the problem geometry of Fig.30 there is an inflow from the bottom surface at the velocity of v=1.5 sec.3 ADI Solution The ADI scheme is tested to use a time step size larger than the Courant limit. which is exactly the . Although the Von Neumann analysis shows that the ADI scheme is unconditionally stable. It is found that a wise use of the ADI scheme can reduce the overall computation time in comparison with the explicit and implicit schemes. The ADI and explicit schemes give almost identical results during all the transient.0.30. In the meshes at the top of the source mesh. when the Courant limit is 1 sec. Figure 5. the steam concentration should decrease as the flow goes up to the ceiling because most of the steam introduced in the source mesh goes upward and gets distributed by convection and diffusion. 0 Fig. 5.Y-XI~^-.~ ~ILI_~_~L--I*-L~-l-*IIIU*XI -1 9- IB Source of ' 'A Profiles of 4 = 10.0 Ax = Ly = 1. Problem geometry for comparison of the explicit and ADI solution schemes .0 c= 1.1I -I _--~ 1_.30.0 on this line are compared. v = 1. 0 10. Comparison of the explicit and ADI solutions for along the line AB in Fig.-160- 12.5 sec 12.0 11.0 10. 5.0 coordinate 11.01 12.0 12. 0A Explicit -----ADI 10.0 11.0 0.0A 12.0 11.30 the profile of sec and the Courant limit of 1.0 10.0 sec for Lt=0.5 .0 A iatlai B 10.0 Fig.0 10.31.0 12.0 11.0 11. 5. 0 10.0 16 sec 20 sec 12.0 4 sec 11.0 11.0 sec . Comparison of the explicit and ADI solutions for the profile of 4 along the line AB in Fig.0 11. 12.0 10.-161- 1 sec 12.0 10.0 ~ spatial coordinate 12.0 11.0 10. 5.0 Explicit ADI 10. 5.0 Fig.0 sec and the Courant limit of 1.0 12.32. 11.0 12.30 for At=1.0 . 5. No cross-flow diffusion corrections are involved in the results of Fig.32 and Fig. . 5. Fig. 5. 5.33 shows that the ADI scheme can use a time step size up to five times the Courant limit.-162- the Courant limit. there is some deviation between the two results. Initially. Fig.31.33. The deviation for the time step size of 5 sec is expected to damp out as the calculation goes over to the next time steps because of the unconditional stability. but it damps out rapidly and the ADI scheme gives almost identical results with the explicit scheme. 5.0 sec along the line AB in Fig.-163- 12-.0 At =0. Comparison of the ADI solutions for the profile of at t=10.5 sec ec 11.30 with different time step sizes .33.0 t spatial coordinate B Fig. 5. (2) The models of laminar and turbulent diffusions in Chapter 3 may be adequate for predicting the hydrogen transport in the containment. or superheated steam as may be appropriate. The thermal equilibrium is assumed and the phase change occurs to maintain 100% relative humidity. The con- tinuity/momentum equations are decoupled from the scalar transport equations and solved by the SMAC scheme to obtain the flow field. The mass diffusion.-164- CHAPTER 6 CONCLUSION 6. The total diffusion con- stant is the sum of the laminar and turbulent diffusion constants.2 Numerical Schemes (1) There are two numerical diffusion sources. in the finite difference donor cell treatment of convection. 6. Cross-flow diffusion occurs due to the donor cell treatment of the convection term in a multi-dimensional problem. The .1 Physical Models (1) The solution scheme presented in Chapter 2 has been adequate for modelling the slow mixing stage in the containment after a loss-of-coolant accident. and turbulence equations are solved separately using that flow field. energy. truncation error and cross-flow diffusion. convection dominant. most of the numerical diffusion error in a multi-dimensional. It gives unphysical results for most recirculating flow and coarse mesh problems. recirculating flow problem is due to the cross-flow diffusion. It is conservative and an explicit scheme can be used without affecting the simple solution structure. but not always. The truncation error diffusion occurs in the flow direction while the cross-flow diffusion occurs in the diagonal direction of neighboring mesh points. The skew differencing scheme gives good results for some problems. The corrective scheme is based on the fact that the additional cross-flow diffusion can be predicted theoretically for every mesh at every time step. Therefore. The gradient of the scalar quantity under consideration is usually small in the flow direction in comparison with that in the direction normal to the flow. (2) Two types of approaches. Therefore. the corrective scheme is generally preferred to the skew differencing scheme. skew differencing and corrective schemes. ~unx.-165- effective diffusion constants of the truncation error and cross-flow diffusion are of the same order of magnitude. . have been tried in order to eliminate the numerical diffusion. The inclusion of corner points complicates the matrix structure and a fully implicit scheme should be used for maintenance of stability. 2 imposes a maximum time step size that can give a physically . (4) The Von Neumann stability analysis is applied to various finite difference forms of a general conservation equation. the physical constraint in section 4. using the characteristics of a quadratic equation. The results of numerical experiments are found to be consistent with the Von Neumann analysis. i. The graphical method is used to obtain the stability condition in terms of the Courant. Although the Von Neumann analysis shows that the ADI scheme is unconditionally stable. The mesh interface implementation is always preferred to the mesh point implementation in a recirculating flow problem. Cx.e. First. diffusion and cell Reynolds numbers. are tried for the corrective scheme. dx and Rx .. it has its own limitations due to the following factors. mesh point and They mesh interface. (5) The maximum time step size is limited by the Courant The Alternate Direction condition in an explicit scheme. are identical in a unidirectional flow.4. The results of various sample problems show that the mesh interface implementation of the corrective scheme always gives a physically reasonable solution with negligible numerical diffusion error.-166- (3) Two implementation strategies. Implicit (ADI) scheme can be used to increase the time step size and decrease the overall computational effort. asymmetry of the given problem may give different results according to the sweeping sequence.-167- reasonable solution. For example. a steadyThe state solution may never be reached due to the fact that one time step is composed of a few fractional steps. Second. the ADI scheme is a fractional step method. . The modified k-E model in a low Reynolds number flow may be a guide in this approach. distance from the wall). . The model should show proper limiting behaviors as the Reynolds number goes to infinity (turbulent) and zero (laminar).2 Numerical Schemes There have been two approaches.-168- CHAPTER 7 RECOMMENDATIONS FOR FUTURE WORK 7.. The flow regime criterion may be replaced with a model that covers all the turbulent. it is necessary to predict the flow regime in order to use appropriate physical models. skew-differencing and corrective schemes. previous time step information.g. velocity gradient. transitional and laminar flow regimes. 7.1 Physical Models Since both the laminar and turbulent flow regimes are possible in a complicated geometry and flow field. time step size. multi-dimensional recirculating flow problem. The important parameters may be the velocity. A criterion for the flow regime should be developed in such a form that it may be implemented into a computer code. geometry (e. to eliminate the cross-flow diffusion. They may still be improved further to get an accurate solution in a transient. etc. It may be possible to develop a reasonably simple form that may or may not include the corner points. 2. The solution in Appendix A may be a useful guide in this work.-169- Corrective scheme The corrective scheme has been successful by the mesh interface implementation strategy in the scope of this work. The conservative form of the skew differencing scheme is complicated and time consuming in comparison with the corrective scheme. The stability of the skew-differencing scheme is open to question in the explicit. ADI and implicit schemes although only the implicit scheme has been used up to now. . Skew differencing scheme 1. More tests and validation calculations are required in recirculating flow problems. Bradshaw.. Methods in Applied Mech. 6.. Cebeci." Journal of Comp...G. A. "Hydrogen Distribution in a Containment With a High-Velocity Hydrogen-Steam Source. Briley.V.H.R. (April 1980). Comp. 24. Phys.E." HEDL-SA-2682 (October 1982). John Wiley & Sons (1960)." Journal of Comp.. C. 54-73 (1980). thesis (1979). Briley." Numerical Heat Transfer 3. Transport Phenomena. Briggs. 6.H. W. A. 372-397 (1977). Baliga.. and Lightfoot. "The SMAC method: A numerical technique for calculating incompressible flows.. and Harlow.User's Manual.N. Stewart. Academic Press (1981). G. 393-409 (1980). 332-325 (1970).R." Journal of Comp.A. S. and Whitelaw. Phys.. and Harlow. Boyle. and McDonald.A." MIT Ph." Comp. Phys. Bird. "A Finite Difference Scheme for the Incompressible Advection-Diffusion Equation. 94-110 Amsden.R." LA-4370 (May 1970). P. H.. H. "Solution of the Multi-dimensional Compressible Navier-Stokes Equations by a Generalized Implicit Method. [2] [3) [4] [5) [6] [7] [8) [9) [10) [11) [12] .-170- REFERENCES [1] Amsden. J.R. A. E. and Harlow.. Amsden. "A Simplified MAC Technique for Incompressible Fluid Flow Calculations. Bloom. 3.B. and Patankar.R.. "On the Structure and Use of Linearized Block Implicit Schemes. F. "Transport of Turbulence in Numerical Fluid Dynamics. W.A.D. "BEACON/MOD 3: A Computer Program for Thermal-hydraulic Analysis of Nuclear Reactor Containments ." Journal of (1968). Phys. F.." NUREG/CR-1148 (EGG-208). 233-241 (1975). et al.. Engineering Calculation Methods for Turbulent Flow.R. T. and McDonald. et al.. W.H. R. D. and Eng.H. B. 34. F. "A New Finite Element Formulation for Convection-Diffusion Problems. "Transient Effects in Turbulence Modelling. Broadus. D. . [14] [15) [16] [17] 118] [19] [20) [21] [22] 123] [24] [25) . L. Hirt.. "Buoyancy Effects in Mathematical and Numerical Turbulence Modelling. G.. J. K. De Vahl Davis. N. S. "Studies of Numerical Methods for the Plane NavierStokes Equations. and Mallison. thesis (1977). Academic Press (1964). 32." MIT Ph.. "Tensor Viscosity Dukowicz. Jr.. 1. 95-107 (1982).. and Ramshaw. Methods in Applied Mech.. Chang. "VARR-2 . Covelli. Crowley. and Nakayama." The Phys. Clark.." CRBRPARD-0106. Phys.I." MIT Ph.K. H..D. Varadi." LA-6236 (July 1976). "Simulation of Containment Cooling with Outside Spray After a Core Meltdown. 1976). Hirsh.H. Phys. G. "Comparative Analysis of Numerical Methods for the Solution of Navier-STokes Equations.. "Study of Non-homogeneous Conditions in Reactor Containments Due to Separation Effects. thesis (1980). "An Evaluation of Upwind and Central Difference Approximations by a Study of Recircul ing Flow.H. Cebeci. Chang. Vol.. and Romero. and Eng. Chen.. Y. "SOLA-ICE: A Numerical Solution Algorithm for Transient Compressible Fluid Flows.L. G. S.I. and Design 69. Williams. 471-484 (1967). Cheng. Vol. M.P. and Design 69. 4.-171- [13) Carbajo.B.D.D. "Accuracy of Difference Formulation of Navier-Stokes Equations. 13-44 (1981)..G." Nuclear Eng." Computers and Fluids.D.S.F.A Computer Program for Calculating Time-dependent Turbulent Fluid Flows with Slight Density Variation. thesis (1981).S. Nielsen. Cloutman." Comp. 1 (Nov. P.C.." MIT M. "Coolant Mixing in the LMFBR Outlet Plenum. Cook. W. and Hansen. J. "Numerical Methods of Reactor Analysis.. L. B.." Journal of Comp.. and Lewis. 71-79 (1979). Keller. "Second-Order Numerical Advection. J. J.B. R." Journal of Comp.." Nuclear Eng. 27. P.J. of Fluids Supplement 2 (1969). C.W.. 29-43 (1976).. S. Method for Convection in Numerical Fluid Dynamics.D. M. 127-137 (1982). T. Hoffman. T.. 81-95 (1981). "Practical Investigation of Convective Difference Approximations of Reduced Dispersion. Vol. J. C. 5-28 (March 1979). Hinze. and Amsden.. 29. and Amsden...The 1978 Freeman Scholar Lecture. "A Numerical Fluid Dynamics Calculation Method for All Flow Speeds. A." The Phys. J. "A Comparison of Hybrid and Quadratic Upstream Differencing in High Reynolds Number Elliptic Flows.. G. A.F. Hirt.O..W. "Buoyancy Induced Fluid Motions Characteristic of Applications in Technology .O.. J.A. Wiley." Journal of Fluids Eng.H. "Improved Form of the Low Reynolds Number k-c Turbulence Model. 17. Han. "Numerical Calculation of Almost Incompressible Flow." Comp." The Phys..W. A. B. 18.. Hirschfelder... F. for Num. 8. 15..-172- [26] Fromm." Journal of Comp. and Launder. and Amsden.H.. "Behavior of Buoyant Moist Plumes in Turbulent Atmospheres. and Bird..." The Phys. R.E." Journal of Comp. 101. McGraw-Hill Book Company. Phys.. 2. J. of Fluids Supplement 2 (1969). 557-573 (1980). F. Gupta. and Marohar. 3. 309-312 (March 1975). "Heuristic Stability Theory for Finite Difference Equations. Harlow. of Fluids Supplement 2 (1969)." Int. Harlow. "On the Use of Central Difference Scheme for Navier-Stokes Equations. 2nd ed. [27] [28] [29) [30) [31] [32) [33) [34] [35] [36) [37) Hirt.H. R.A. "Computer Studies of Time-dependent Turbulent Flows. 197-213 (1971). 19-52 (1975).E.. J. Humphrey. B. M..W." Journal of Comp. Methods in Eng. R..H. Methods in Applied Mech. Vol. and Golay. C. New York (1954). 339-355 (1968). Phys. Gebhart. F." Journal of Comp.C. Turbulence.M. Vol.A. (1975). Molecular Theory of Gases and Liquids. C. of Fluids. M. Phys. and Eng.." MIT-EL 81-020 (June 1981). "Numerical Calculation of Multiphase Fluid Flow. Phys.. Hamza. Harlow.A.B. Curtiss. [38) .P. 80-93 (1968).. F.V... M.P.. 3. Vol. 1119-1130 (1973). C. 537-566 (1975). Methods in Applied Mech. Jones. J. Phys... Cl. and Rodi..D." Int.363-3. "Experimental Investigation of Hydrogen Distribution in a Containment of a Light Water Reactor Following a Coolant Loss Accident.E. Vol. Jones. Vol. "Progress in the Development of a Reynolds Stress Turbulence Closure." Comp. and Spalding. G. 269-289 (1974).-173- [39) Houston.. and Bremhorst. "The Numerical Comparison of Turbulent Flows." BF-63.. Heat Mass Transfer. et al. Manno...." MIT Sc. W.W. 456-460 (Sept. Fluid Mech.G.. Lam.H. W." Int. Methods in Applied Mech. 16. Vol." Trans.K. B.P. V." Report submitted to 2nd International Topical Meeting on Nuclear Reactor Thermalhydraulics (March 1982)..E. "Practical Evaluation of Three Finite Difference Schemes for the Computation of Steady-State Recirculating Flows. and De Braemaecker.E. and Launder. B. M. [40) [41) [42) [43] [44) [45] [46] [47] Leschziner.P. of the ASME. Reece. Battelle Institute e. John Wiley & Sons (1978). Keenan. J. 16." Journal of Comp. 103. and Keyes. D.E. "Analytical Modelling of Hydrogen Transport In Reactor Containments. J.A. 148) [49) . Launder. Frankfurt. "The Prediction of Laminarization with a Two-equation Model of Turbulence. Langer. B. 1981).B. "Analytical Modelling of Hydrogen Transport in Reactor Containments. K. translated (October 1980). B. 293-312 (1980).. G." J. 15.. "The Calculation of Low Reynolds Number Phenomena with a Two-equation Model of Turbulence. Launder. 301-314 (1972). W. and Eng." Comp. "ADI Solution of Free Convection in a Variable Viscosity Fluid.J. Jr. and Eng. Manno.. 68. Steam Tables. part 3. 221-239 (1974). Heat Mass Transfer.P..G. J.H. V. M. and Golay. "A Modified Form of the k-c Model for Predicting Wall Turbulence. thesis (1983). and Launder. 23. 15.. Computational Methods in Engineering and Science." The Phys. S... D. Phys.J.. (1977). Roberts..V. Hemisphere Publishing Corporation (1980). 169-184 (1972). Numerical Heat Transfer and Fluid Flow." Journal of Comp. "Convective Difference Schemes. "A Critical Evaluation of Upstream Differencing Applied to Problems Involving Fluid Flow. S. 10. Schonauer.E. "On Artificial Viscosity.. and Weiss. "Skew-Upstream Differencing Schemes for Problems Involving Fluid Flow. and Spalding.. and Torrance. Hermosa Publishers. McGraw-Hill.D. Pergamon Press (1982). 9. Roache.. S. P. G. W.D. 75-103 (1976). Methods in Eng. Heat Mass Transfer.K. W. Rodi. G. 1975). The Properties of Gases and Liquids. Rodi.C.. Turbulent Buoyant Jets and Plumes. Inc. J. J.K. Mass and Momentum Transfer in Three Dimensional Parabolic Flows. Methods in Applied Mech. 191-206 (1974). and Eng.B. Vol.V. T. P.. K.. of the ASME. 4. A. W." Mathematics of Computation." Trans. 541-550 (1972). Raithby.. "A Note on the Empirical Constant in the Kolmogorov-Prandtl Eddy-Viscosity Expression. "Upstream-weighted Differencing Schemes and Their Application to Elliptic Problems Involving Fluid Flow." Comp.J..." Int. New Mexico (1976). 9. Runchal. 153-164 (1976). New York (1958). Raithby.B. "Numerical Experiments with a Difference Model for the Navier-Stokes Equations (Turbulence Model). of Fluids Supplement 2 (1969). Patankar. and Sherwood." Int. and'Eng. "Convergence and Accuracy of Three Finite Difference Schemes for a Two Dimensional Conduction and Convection Problem. R. G. for Num.. John Wiley' & Sons. "A Calculational Procedure for Heat. N. 20 (1966). Patankar. K. Raithby..O.-174- [50] [51) Nakamura. 386-389 (Sept." Comp. Reid. 2." Computers and Fluids. Computational Fluid Dynamics.. Roache. [52) [53) [54) [55) [56) [57) [58) [59] [60) [61) [62] [63) . 1787-1806 (1972).. Methods in Applied Mech. Albuquerque.D. R. 551-559 (1972). and Woolner.. "Development and Validation of a Two-Variable Turbulence Model." Numerical Heat Transfer.D.. and Strong. H. "The Numerical Treatment of Advection: A Performance Comparison of Current Methods. 4. "TEMPEST: A Three-dimensional Time-dependent Computer Program [72] for Hydrothermal Analysis. K. 3.. "A Novel Finite Difference Formulation for Differential Expressions Involving Both First and Second Derivatives.-Dec." Comp." Chemical Engineering Progress. "Numerical Analysis of Fluid Hammer Waves by the Method of Characteristics.H. G.A.B. 20. A. 35. J. MIT Press (1970). R. A. Methods in Applied Mech. Phys. 2.. D. Vol. 4 (Oct. C..L.. 411-428 (1980). and Valentin. 72B. No.M. MatheResearch matical Sciences. Methods in Eng. 46. A First Course in Tur- [65] [66) [67] [68) [69) [70] [71] Torrance. 1974). 411-428 (1980).G." of the National Bureau of Science .J.. Wilke. "Proposal for a New Discrete Method Based on an Assessment of Discretization Errors. No. L. 153-168 (1982).W..B. J. "Comparison of Finite Difference Journal of Computations of Natural Convection." SAI-74-509-LJ.R. . Raithby. D.S. G. For Num. Stubley. and Eyler.D. G. and Eng." FATE-80-114 [73] (Jan.D. 220-237 (1976).-175- [64] Shin. A.. "Diffusional Properties of Multicomponent Gases.." Journal of Comp. Smith. "Simulation of Convection and Diffusion Processes by Standard Finite Difference Schemes and by Influence Schemes. Spalding. Tennekes. and Lumley. Raithby. Stubley. Y.A. K. Stuhmiller.. Budden.D. 1981)..E. Strong.B.. and Hutton.. 95-102 (1950). 1968)." Int.. 3. G. (Jan. Trent. J. M.L." Numerical Heat Transfer. bulence... Vol.B. 2 is simplified further by substitution of 4 for T as follows.1.a2T aT 2 + a2 2 2c-- ax ax 2 y ay (A. c. pc u aT = k(-a2T + 2aT 2 x2 p x ax ay (A.2) Equation A. A.3(B) and the constants can be grouped into one constant. Eq.1) In Eq.-176- APPENDIX A ANALYTICAL SOLUTION FOR THE PROBLEM IN FIGURE 5. Therefore.3) ax . The boundary conditions are specified in Fig. a=-- k PCp C =2 u 2A.1 is2 reduced to the following form.1 convection occurs in the x direction and diffusion occurs in both x and y directions.. as follows. A. 2 x 2 a2 ay = c 2 (A.3(B) A steady state two dimensional energy conservation equation without any source is given in the follbwing Eq. 5. Equation A.4) The boundary conditions should also be expressed in terms of 4 as given in Fig. A.5) Then Eq. ables. A.-177- where 4 = e-CXT (A.3 is reduced to the following form. tively. c 2 X cx 2 = (A.1.8) dy where a > c > 0 . A. I d2X + 1 d 2Y 1 = c2 C (A. which are functions of x and y only respec- (x. The variable 1 is separated into two vari- X and Y.y) = X(x)Y(y) (A.6) dx dy The two terms on the left hand side of Eq.7) d -Y c .a = - (A.3 can be solved by the method of separation of variables.6 should be equal to some constants because they are functions of x and y only respectively and the sum of them is equal to a constant. -178- . Boundary conditions in terms of 4 for the problem in Fig. 4 = ec T . 5.-= YO T 0 -ii ii i h YO + co = 0 4 = 0.1.3(B) where 0 is given by. A.0 0 = 0 x Fig. 14) (A.9) Y = C3sin8y + C4cosSy (A. A. A.10) The boundary conditions in Fig. The boundary con- dition at x=0 will be considered later. 13) In order to satisfy the boundary conditions given in Eq. A.8 can be easily obtained as follows.12) (d dx _cX) = 0 x=xO (A.11) dY dy=Y 0 = 0 (A. X = Cleax + C2 e-ax (A. A.12 and Eq. A.7 and Eq. dY dy y=O Y=O 0 (A. A.1 may be given in terms of the variables X and Y as follows.16) . C3 = 0 (A.15) ny 0 = n C2 _ n+c 2 e nX 1 n (a n 9 C) (A. A.11. the constants in Eq.9 and Eq.10 should satisfy the following relations. 13. Eq.-179- The solutions to Eq. 2. the final solution can be expressed as follows. . 17) T(x. . -ax = A0e + C ax Z A [e n=l n a +c n n 2a x 0 n -ea nx + ( a -c )e e ]cOSSnY n (A. Th 0 2. nw sin-2 nO0 (n=1.y) where nr (A.0 2 A- n Th h nr 1 n 1+( a -c )e n 1 a +c 2a x .18) n n Since the boundary condition at x=0 should also be satisfied. Therefore.19) .y) = e cx (x.1 are determined as follows.3 .) (A. the constants A.-180- where 2 =c 2 + n (O) n 2 yO =c 2 + 8 2 . Equations B. The ADI scheme is applied to a two dimeraional momentum equation in the following Eq.3 are the z and x direction sweeps of the x direction momentum . The most important sweep should be given the first priority and most updated information. the computational effort for one time step is greater than that of the explicit scheme. there are six sweeps in a three dimensional problem and four sweeps in a two dimensional problem.1-Eq. The sequence of the sweeps is arbitrary and depends on the problem under consideration. When the tilde phase of the SMAC scheme and other scalar transport equations are solved by the ADI scheme. B.4.i. They are formulated for a natural convection problem where the gravity acts in the z-direction. Since there are two steps involved in solving the momentum equation in each direction.-- -181- APPENDIX B ADI SCHEME FOR MOMENTUM EQUATION The ADI scheme can replace the explicit scheme in the momentum equation in order to eliminate the Courant condition for the time step size. Therefore. the time step size in the ADI should be at least two or three times the Courant limit in order to have a gain in the overall computational efforts required for a problem.1 and B. B. The time step size limitation in the ADI scheme comes from the semi-implicit treatment of the nonlinear convection term.iillaY~i~~l_ ^YII^^~II ( -. Rxi+ jui+ jui+ j RX * n+l (B. ar .-182- equation.1) * * n * * - ij+ At Wi+1 j+ ij+ + i+Ij+wij+ ui-j+Wi-lj+ 1 P- n~ . u n * u j -i+ Ui+D Dt j + * n i+ j+ Ui+ j .ui+ j At - n+l * u i+lj i+ .2 and B. sweep of the z direction momentum equation is treated as the last one in order to be given the most updated information.2) n+l i+j * .P z - n * RZij+ ij+ ij+ (B. 2 Az + Ui+ -1 P - n -- i+j i+ jui+4j (B.2ii+ Ar 2 + Wi-lj+ .u Lr * n+1 .Wi+ Az j Ui+ j 1 ui+ j+l * .2ui+. Equations B.4 are the x and z direction The z direction sweeps of the z-direction momentum equation.3) .u n+l ui+2 2 n+l Ar2 +2u j n+l ui j . w At n+l Wi.3.4 have given almost identical results with the explicit scheme for a time step size below the Courant limit. B.-183- n+l i+ * wij+ * i n+l +lWij+ * n+l zW.4) Equations B. .+ n+1 n+l Az -2wi + w _P z -P0 -zP -g z RZ n+l ij + wij+ 3wij+ (B. and B.2.1. B. 'difprg fortran'. Then four cross-flow diffusion constants at every interface are summed up to give a total cross-flow diffusion constant .4.-184- APPENDIX C COMPUTER PROGRAMS The files.3. 5. two different cases treated separately. The file. Fig. are the computer programs for calculating the numerical values of the analytical solution in the problem geometries. the calculational logic of the mesh interface implementation of the corrective scheme in section 4. 4. is the program for testing Fig. 5. the cross-flow diffusion constants can be calculated in a simple way as in the first part of the program. In the second part of the program the flow split is considered such that every outflow velocity is partitioned into four components in a three dimensional case in proportion to the velocities of the four neighboring surfaces in contact with the surface under consideration.8. There are When there are inflow and outflow. All the inflow velocities The component velocities are used to calculate the cross-flow diffusion constants for eight subspaces in a three dimensional coordinate space.3(B) and Fig. a flow split occurs as shown in Fig. 'progl fortran'.3(A). or all inflows in x. 'prog2 fortran' and 'prog3 fortran'. 5.2. When there are two outflows in any of the x. y and z direction mesh interfaces. y and z directions.8.. are assumed to be zero. A portion of the VARR program is introduced to show how the ADI scheme is implemented in the energy and momentum equations. The ADI scheme for the momentum equation is given in Appendix B. has four neighboring subspaces. The subroutine 'tdm' solves tridiagonal matrix problems by forward and backward sweeps. .-185- because one axis direction. It is used to solve a one dimensional implicit finite difference equation in the ADI scheme. the positive x direction. The total cross-flow diffusion constant can be calculated in the same way for every mesh interface. NX.1415926536 C = U/(2.3F7.5F11. AAN*2.OALNOEXP(ALN.ALP II FOIRMAT(F7.14 55 ATOT * THIODEXP(-ALN*XO)/2.26 II IF - IP - I GO TO 55 ALN = DSORT(CSO*II*II*YSO) (II. C*C XO " NXOX YO * PROOOO0 PROOOO20 PROOO30 PROO0040 PRO000OOOO P14000070 PRO000OGO P11000090 POOOOT100 PROOOO10 PIIOO0120 PRO00130 PROOO140 PROOO150 PRO00160 PROO000170 PRO00100 P1.5)/NY ADD * ATOTTDCOS(YY) SUM T(d) v SUM + * ADO * DEXP(C*XO) 20 CONTINUE SUM 10 CONTINUE WRITE(G.1 ) DX.000190 PI00O200 NY*OX YSO v PI*PI/(YO.O) ALNX .O-Z) T(tO) DIMENSION REAO(10.2) PI a 3.XO)/(ALN-C) GO TO .NY - 0.0*TII/PI PI/2.NY.NY) 100 FOnMAT(IOX.EO.04ALP) CSO .215.Tl.3) SlOP END PRO00210 PRO00220 PROO0230 P1ZOOO240 P1I000250 PROOO260 PROOO270 PROOO200 PROOO290 P11000300 PROO0310 PRO00320 PRO00330 PROOO00340 PRO00350 PROOO360 PROOO370 PROOO300 PROOO390 PROOO400 .IALNIC)*DEXP(2.0 00 20 IPsi.O*ALNXO)/(ALN-C) COF * CCOFF/lI AAN .0 44 YY I IIPI*(UJ-0.100) (T(I).-COF*DSIN(II*HPI)/(1.4ALNX) ATOT .YO) CCOrrF 2.I=l.0 IHPI DO SUM 10 JJ-I.-N FILE: PROG FORrRAN A VM/SP CONVERSATIONAL MONITOR SYSTEM PAGE 001 IMPLICIT REAL*0 (A-H.U.2. 0 SUM2 = 0.-COF*DSIN(II*HIPI)/(I.1.0 .2-1)*DX = X 4 YO/2.0 X + XO/2.+ALNX) BEN 11*PI/YO YOO .O) GO TO 55 ALNX .OCOS(OEN*Y4) Y4S DOSIN(3EN'Y4) ADO * AAN*(EXALO + ALNX*XALO)*YOO PROOO520 PR000530 P1OOO00540 PR000550 .3F7 .5+YO 00 20 IPs1.21 II = IP - I ALN = DSORT(CSO + IIII*YSQ) EXALO = DEXP(ALN*XOO. GL(10).215.0 NY2 XXX NY/2.0 SUMaX SUM4 XN vI Y2 Y3 Y4 • 0.DSIN(BEN+Y3) Y4C .0 = YO/2.p A FILE: PROG2 FORTRAN A VM/SP CONVERSATIONAL MONITOR SYSTEM PAGE 001 MPI. IC1l REALi0 (A-II.O*ALN*XO)/(ALN-C) COF = CCOFF/II AAN .5)*DX = YC = (JJ. U. TL(10).DCOS(BEN*YC) .2.OALP) CSO = C'C XO * PROOOO O PRO00OO20 P R000030 PROOOO40 PROO00SO POOG0050 PROOOO000 PROOOOOO PROO0090 PIR000 100 P11000110 PRO00 120 PROOO 1.0 SjU3 " = 0.OTH-IH/PI 11PI = PI/2.0 .0 = 0.XN = XN .2) PI .0 X = (JJ-O.0 SUM3X 0.FO.0-Z) DIMENSION TII(10).5) XALO = DEXP(-ALN*XO*O.O.X = 1. NX.5) EXAL = OEXP(ALNOX) XAL = DEXP(-ALN'X) EXALN D EXP(ALN*XN) XALN = DEXP(-ALNXN) IF(II. 1415926536 C .0 SUMO SUMI r 0.3.30 P1OOO 140 POOO00150 900440 PROO0170 PRO00180 PROOO 190 PROO0200 PROOO0210 PROO0220 PROOO230 PRO00O240 PRO00250 PR11000260 PRO00270 P1O00200 PR000290 PROOO300 PROO0310 PR000320 PROOO330 PROO0340 PRO00350 PROO03GO PROO0370 PROOO300 PR000390 P11000400 PROOO4 O PROOO420 PROO0430 PROO440 POO450 PI0OO0460 PR000O470 PROO0480 PR000490 PROOOSOO PR00510 DX'NX YO " DXNY VSO " rPI*P/(YOYO) CCOTF = 2.(ALN#C)*DEXP(2.0 - RTO DO = - OSORT(XXX) 10 JJ=1.5*YO .11) DX. 11 FORMAT(F7. TC(IO) TH14.NY2 0.U/(2.DCOS(BEN*YI) YI Y22 • DCOS(BENY2) OCOS(BEN*Y3) Y3C Y3S . ALP READ(10. NY. GH(10). 0 ADO3Y * 0.0 ADOvY * 0.100) (GH(I).5 A)DO * TIHl'XALO*O.AAN*(EXALALNX*XAL)*Y22 ADO3X v AAN*((C+ALN).tOO) (TC(I).5 A002 * TMI-IIXAL0.I WRITE(6.(" % FILE: PROG2 FORTRAN A VM/SP CONVERSATIONAL MONITOR SYSTEM PAGE 002 ADOI w AAN*(EXAL+ALNX*XAL)*YII AD02 .NY2) = I.SFII.NY2) WRITE(6.3) STOP ENO SUMO*DEXP(C*XO0O.NY2) WRITE(G6.DEXP(COXN)/RTO PROO000560 PROO0570 PROOOS80 PROOO590 PROOO600 PROO0610 PROOOG0620 PR000G30 PROOO640 PRO0650 PRIOOOG60 PROOG0670 PROOO680 PROOOG90 PROOO700 PROOO7 O PROO0720 PROOO730 PROO00740 PROOO750 PROOO7GO 1PI000770 PROOO700 P000790 PROO000000 PROOOIO1 PIOO000120 PROOO030 PROO0040 PROO000050 PROOO060 PROOO870 PROO000 PROOO090 PRO00900 PROO0910 PROOO920 PRO00930 PR000940 I0760 00 oo I . ALNX*XALN)*Y3S ADO3Y * ADD4X * AAN*((C+ALN)*EXALN + ALNX*(C-ALN)*XALN)*Y4C + ALNX*XALN)#Y4S AIDD4Y * -EN*AAN*(EXALN GO TO 77 55 AOOI = TIIH*XAL*O.00) (GL(I). 1OO) (TH(I).tOO) (TL(l).NY2) WRITE(.0 77 SUMI = SUMI + ADO SIIMO a SUMO + A000 SUM2 a SUM2 + A002 SUM3X = SUM3X + A00O3X SUM3Y w SUM3Y + ADD3Y SUM4X s SUM4X + AOO4X SUM4Y a SUM4Y + ADO4Y 20 CONTINUE C TC(dJ) THI(JJ) TL(J) G-i(JJ) = = GL(dJ) 10 CONTINUE C WI ITE(6.I-I.0 AOOX a 0.5) SUMI*DEXP(C*X) SUM2*DEXP(C'X) (SUM3X+SUM3Y)*DEXP(C*XN)/RTO (SUM4X+SUM4Y).EXALN + ALNX*(C-ALN)*XALN)*Y3C EN*AAN#(EXALN .I 1OO FORMAT(IOX.5 ADO3X 0.I-I.NY2) = t.II. S)#(XO-YO/(2.0 SUM4X = 0. I 1) DX.0 SUM3Y = 0.0) DO 10 00=1. ALP PROOOO10 GL( O).0 SUMI = 0.OtTIIIM/PI HIPI = PI/2.+ALNX) OEN = II'PI/YO YOO .N*X2)+ALNX')OEXP(-ALN*X2))*Y22 AD003XwAAN((C+ALN)*DEXP(ALN9X3)4ALNX*(C-ALN)*DEXP(-ALNtX3))Y3C PRO000410 PROO420 PR000430 1R'000440 P1100050 PR00O4GO PROO0470 PlOOO400 POO11000490 PROOOSOO 1100005 10 PROOO520 P0000520 P1OO0530 PROOO040 P1000550 .C P11000220 PROO0230 PROO0240 P1000250 PROOO2GO PROO0270 P11000200 PROO290 PR000300 PROO03 10 P 11000310 PROOO320 00 PRO00330 PR00OO340 PROO0350 00 20 IP=1. FILE: PROG3 FORIRAN A VM/SP CONVERSATIONAL MONITOR SYSTEM PAGE 001 IMPLICIT DIMENSION C 11 REALtO (A-H. G1(10).0.NY2 SUMO = 0.(dJ-0.0 PR00O170 PR000100 PROO0190 PR000200 PlO00210 SUM2 = 0.0 NY2 nRo P000O070 13100000 PROOOOOO PROOO0000 PRO00110 = NY/2.DCOS(iEN*tY2) YV3C = DCOS(BEN'Y3) Y3S = DSIN(BENtY3) YAC = OCOS(3EN*Y.*0RT3#NY2) Y4=YO/3.0 XC=Y0/(2.0*ALP) CSOQ C*C XO = DX'NX YO = DX*NY YSO = PI*PI/(YO*YO) CCOFF = 2.NX*DEXP(-ALNXI))*YI I ADD2-AAN*(DEXP(AI.RT3)+(dJ-O.2) PI = 3.5)*YO/(2.0+(dJ-O.5)*(XO-YO/1r3)/NY2 YC=(J-O.0)/NY2 X4-XO-YO/(2.0)/NY2 X2=(JJ-0.O*NY2) . TC(1tO) PROO0020 PROO0030 PROO0040 PnOOO050 PROOG000 READ( 10.0 .0-(JJ-0.5)*YO/(2. ( 10).O*RT3)+(JJ-O.DCOS(UENOYC) YIl DCOS(OEN'YI) Y22 .o) ALNX = (ALNwC)tDEXP(2.0*NY2) X3-YO/(2.EO. rORMAT(F7.5)YO/(2 .5)*(XO/RT3-YO/6.0 SUM4Y = 0.3F7.0*RT3#NY2) Y2=YO/2.5)YO/(2. U.0.O*ALN*XO)/(ALN-C) COF = CCOFF/II AAN = -COF*OSIN(II'HPI)/(I.-XO/RT3. NY.0-2) 111(10).S)*(XO/RT3-YO/6. NX.SO5T(2.0 SUM3X = 0.2.0*RT3)+(J-0.5) * (YO/3.5)*(XO-YO/(2.C-. TI-Il.O' XO/RT3)/NY2 XI=(dJ-0.22 II IP I PR00360 PRO00370 PROOO3UO P(1000390 P000400 ALN = DSORT(CSO + II*laIYSO) GO 10 55 Ir(II.0*RT3))/NY2 Y3=(JJ-O.0) PROO000120 PROO000130 PROO0140 PROOO150 PR000O160 RT3 = C SORT(3.215. TL(IO).0*RTJ))/NY2 YIYO/2.4) Y4S = DSIN(6ENlY4) ADDO=AAN*(DEXP(ALN*XC)4ALNX*DEXP(-ALN*XC))*YOO ADDI-AAN9(DEXP(ALN*XI)+AI.1415926536 C = U/(2. 5*RT3+SUM4YtO.SF11.0 ADO4Y = 0.100) (GL(I).5 ADO3X PROOOSGO PROO00570 PR1000500 PROO000590 PROOOGOO P1100000.0OEXP(CXI) PROO0670 PR000600 PROOOG90 PRO00700 PRO000710 PROO00720 PROO0730 PROO00740 PROO00750 PR 00760 PROO070 PRO00790 PRO00810 PRO00020 P'11000830 P1i000840 PROOO00OSO PRO006O PROO00070 PRO0000 PR000890 PRO00900 PROOO0910 TL(JJ)=SUM20+EXP(CX2) GL(JJ)=(SUM3X*O..NY2) WRITE(6.OEN*AAN(DEXP(ALN*X4)4ALNX*)DEXP(-ALN*X4))*Y4S GO TO 77 55 ADDO-TIII-I*DEXP( -ALN*XC)*O. 100) (GI(I).NY2) tOo rORMAT(IOX.I= .SUMI + P1I000620 PROO00630 PRO00640 SUMO w SUMO + SUM2 a SUM2 + SUM3X * SUM3X SUtM3Y = SUM3Y SUM4X SUM4Y * * ADO ADO A002 + ADD3X + ADO3Y SUM4X + ADD4X SUM4Y + AO04Y 20 CONTINUE TC(JJ)=SUMO*OEXP(C*XC) TtI(JJ)=SUMI.FILE: PROG3 FORTRAN A VM/SP CONVERSATIONAL MONITOR SYSTEM PAGE 002 AO3Y=BEN*AAN*(DEXP(ALN*X3)4ALNXODEXP(-ALN*X3))*Y3S ADI)4XAAN*((CALN)*DEXP(ALN.NY2) WRITE(6.100) (TH(I).100) (TC(I).5)*DEXP(C*X4) 10 CONTINUE WRITE(6.I=I.X4) ALNX*(C-ALN)*OEXP(-ALN*X4))*Y4C ADO4Y.5*RT3)*DEXP(COX3) GHi(JJ)-(SUM4X#O.I-1. PROOO6tIO .3) STOP ENO .NY2) WRITE(G.1-1.5+SUM3Y*0.5 ADD2TtII#DEXP(-ALN*X2)*O.I-=. = 0.5 ADDH11-H*DEXP(-ALN*XI)*0.1OO) (TL(I).0 ADDOX a 0.NY2) WRITE(6.0 77 SUM1 .0 ADO3Y = 0. 0 RFOT 2 0.VI + V2 WW = WI 4+ W2 PPX .IFZA(0).VP.WW4DELTII#( PPX + PPZ )/PSUM C RiFGII .IS .DIFII'IS2 rGR RfGTI .WM.0 n I FOO 130 OMTI m 0.GT.EO.0 ) C DIFR .O) IF(WP.0 OMT .DIrn*ISI .GT.O.EQ.0) IF(VP.) ISA .PPX $ PPZ + PPT GO TO 500 IF( PSUM.O.5*( VP WI * -0.0) IF(UP.OR.ISC.2.LT.0) IF(WM.5*( UP VI -0.LT.EO.t = 1 = i v 1 -* .0.WW/DELTII PSUM .5*( VM V2 0.0.VM.I 01o00200 DIF00210 DIF00220 DIFOO230 DIFOO240 DIF00250 DIF00260 DIFO0270 DIF00200 OlFOO290 DIFOO300 DI F003 10 I'-I .GT.O.UU*DELR*( PPZ + PPT )/PSUM DIFZ * VV*DELZ*( PPX + PPT )/PSUM DIFT .DIFRA(0)1.0 RFGTI .WP 1SI * 0 152 .OR.0 IF(UM.VELT(O).0.UM.UP.O 15S3 .DEL'TII.5.DIFZ'IS3 * DIFZ*IS4 RFGr GO TO 00 AOS(UM) + ABS(UP) DI F00170 DIFOO IGO DIFOO00190 DIFOOlOO ISI IS2 IS3 IS4 155 ISG .0) IF(VM.5+( UM U2 0.VELZ(0).0 RFGCR 0.2) UU * UI + U2 VV .( WM W2 0.AOS(VM) + AOS(VP) AUS(WM) + AOS(WP) DIF00320 DIFOO00330 DIF0340 DIF00350 01F00370 D)1F0030 D roo390 DIF00400 DIFOO4 IO 1FOO00420 I) FO0430 DIFOO440 DIFOO450 DIFOO4GO )DIFOO470 D r004flO DIFOO4O DIFOO500 DIFO0510 DIOO00520 ODFOO530 DIFOO540 DIF00550 bMtt .DIFTA(0) REA( 10O.L1.O.DELZ.O.O.4 -~ -FILE: O FI'IlG FOITIAN A VM/SP COINVERSATIONAL MONITOR SYSTEM PACE 001 DIMENSION VELX(0). bt ts .5) DELR.O 151 4 0 ISS1 O 156 O DIFOOOIO DIFOOO20 DIFOOO30 OIF0040 011:O0050 DIFOOOGO DIF00070 DIFOOOTO DIF00110 RFGRI = 0.ISt + IS2 ISO IS3 + IS4 ISC ISS5 + ISG C UI = -0.2.5*( WP C IF( ISA.EO.UU/DFLR PPZ * VV/OELZ PPT . 0-PFX)o( t.DELTH.PFX*PFT V2* I'F X'PF T V2.0 -0.0 30 CONTINUE RFGRI RFGR RFCTI - DIFO 10 tO 011:'01020 DIFO 1030 01 FOt040 DO 1050s DI FOtOGO DIFO 1070 01000 DIFO 01 FOtO0J0 OIFOi tOO DIFRA(5) v DIFfIA(t) aDIFZA(2) + DIFRA(6) + DIFRA(2) + OIFZA(3) DIFPA(7) + OIFRA(3) + DIFZA(6) + + + OIFRA(fl) OlfRA(4) + DIFZA(7) r % I .PFT u i.Z (3 ) VELZ(4) V EL ( 5 ) 011'00720 DIF00730o DI F00740 ui roo75o DI F007GO ui FO0llO DIF 00700 0D F00190 0117OO8OO OIFOOII10 o)1roos~o OI F001130 DI FOO 40 otrooaso 01 roooGo Dl FOOD 70 VELZ( 6) VELZ(7) V EL Z( 0) VrELT( I) VELT(2) VELT( 3) VELT(4) VELT(5) VELT (6) VELT (7) VELT (0) Dl FOODO DIFOOR90 0 fF00900 01 FO009tiO 01F-00920 DIFOO930 oDIF 00940 OlE 00950 Di FOO960 Dl F00970 0 F009-0 01 F00990 oiro0tooo W2$( t.0-PFX)*PFZ DO 30 1-1. oi rooG to Dl F00G20 D1F00630 01F00640 Ut 'PFZ .0-PFT) V t.o 'FX'( 1.0 oiETA(I) .VELX( I)/I)ELR PPZ . ronTRAN A FILE: CON4VERSATIONAL MONITOR SYSTEM DFPRG AVM/SI' FOTRAN DI F00560 DIF00570 PACE 002 PG 0 OMT GO a 0IFT'156 TO 999 00 PFX PF Z *0.0.0-PET) Vt .0 oirZA(l) . Z(2) vrI-.X(I)'DELR*( PPZ + PPT )0 'PSLJM DIFZA(1) m VELZ(I)*DELz#( rrX +.*( PPX + PPZ )/PSUM GO TO 30 600 OIFRA(I) .FILE: DIFPRC'.0-PET) Vl*( 1.0-PFZ) W20( 1.0-PFz) W2 &1EX'&( 1.0 Pr T Ir (UI+U2).0 (Vl*V2).NC.0-PFZ) DIFO00650 oDI roofGo oir00670 nir006s0 mrof0069 ot rooioo DIF00710o VEIX(S) VELX(6) VIELX(7) VELX(0) VEL7( I) V El.I-PFZ)s( 1.0 ) GO TO 600 DIFRA(I) .V(LT(I).0-PFT) U I PFZ (I 1.PrT )/PSUM DIFTA(I) .0-PFZ)'$( .0-9'FX)#(t1.O.PFT) WI#PI'X#PFZ WI'*ix( .VEU.0.( 1.0-PFX)'PFZ Wt'(I.0-PFX)*PrET V2*( 1.o-PF7Z)'PFT UIs .0.0-PFX)#PFT V2-( 1.o VELX( 1) VCL~X(7) VELX(3) VCLX(4 ) PFX i'rz PFT - - U1/(UI'U2) Vt/(VI14V2) Wl/(W14W2) r DoIooseo 01F00590 DirooGoo.) U2 4PF Z PET 1J2'( 1.VELT(I)/OELTH PSUM a PPX + PVZ + PPT IE( PSUM.VELZ(I)/DELZ PPT .0 Ir (W1#W2).NIE.0-PFX)#(t.0O-PF Z) W2 *P FX' IF Z WI'.( .0 PPX .0-PET) Vi*PFX*( 1..EQ.0.0-PFT.0.0.( i.0-PFT) U2*PFZ*( 1.0-PFZ)*PFT U20( 1.NE.O-PFX)O( I. FILE: OIFPRG FOR TRAN A VM/SP CONVERSATIONAL MONITOR SYSTEM PAGE 003 RFGT OMrt OMT = DIFZA(1) * DIFZA(4) a DIFTA(3) + OIFTA(4) DIFTA(1) + DIFTA(2) DIFZA(S) + DIFZA(0) 4 OIFA(D) 4 DIFTA(5) I DIFTA(6) DIFTA(7) D 999 WIllTE(GG) RFGRI .2X.20X.6F5.RFGT.I.RFGR.OMT GO TO 200 500 WRITE(G.OMT I.7) 5 FORMAT(9F5.RrGT I.511S ORRY) 200 STOP ENO DIFOi IfO DIFOI 120 DIFOI 130 oDIFO140 DIFO 150 DIFOI 160 DIFO 170 DIOt t0100oo oIro 190 DIFO1200 DIF01220 I I .2) 7 FORMAT( ItII .2) G FOIMAT( III . D.P(400).0t4O01 X( I ) I~) 13( ) 0(i1) NM I *N -I 00) 2 1-I.XN) DIMENSION A(N) .C .NMI 1+1 )/0( I) P( 1+I) -A( 141)#*C( I) -P( N(I+ I) 0(11-1) 3 X(N) *X(N)/0(N) DO 3 1-I.NMI .0.1O9:190 VA1109"IO() VAn0094 t VA11O0d1O '4A110943?0 VAIIW14 40 VAfl0O450 VAI*0946) VAI?94 70) VAR0O9400 VAR094 90 VAIR095C VAflfO!'j 10 VA110IJ520 .o(N).I K-N a (X(I() X(K) RE TURN END - X(K41)*COK))/O(K) VAR09Z350 VAflo9:160 VA1109370 VA11093001 VAI.SUarIouT INE TDM(A .O(N).C(N).X(N). K) z 0.E0.CR. DNF.TIONS SECT ION AD !ST ]F(ICLL-]. C C C C U AND v LDE VELOCITY EDU.F2(J.5-(V( K)*:(. 1CONV..L!S( -(]KZZ) ) .:X GO TO 40000 "0OO D.4 SwEEP IN X-DORECTION !KS'-= a DOR-KS VAR17980 VAR 7 990 • VAR i8000 V/R 160O0 VAR 8010 VA 18020 VAR1C030 VR180^0 vR 14BO1O VAR.G...KBP2 " (3.xx'.L.-VG-RD2/(.K2NC JKZ2 .S .S(0. iB10 . D2).O.I.0.E-1O 0005 FP" * IF(.LT.P2) GO TO 40002 (U(iK)*U(-KX))) LUG = LS(O.UG.KZ2))) .P.C . DA-'' G.)-K2NC " (K-i)-'NWPC .2D -LUG-RDX/(AUGOX +~D-G'RD)' ..K) • AVG.E230 .LE0.EO.FFX ( .PM...3.DFFUSI'O =.I.S( 4 .C "hT.K) a LUGDX. 1) GO TO 400 IF(K. I O50 VI8090 K1 K12 2 KEP CONSTA'NT CLLCULATION C CRO5S-FLD'.K t -KXX a jK . I vtR I5S0 rooOT 00G2 UC- '-S(U(]K) ) VLRi 1210 vAR1E230 VAR E20 U(3KXX) ) LUGc .G - V RIS150 VAR.EO.K) * 0.KS 2.JSP2 DO -O40000 DO 4000 K.E0.1) GO T0 49998 SOLUTiON FOR SIE.]K .) G0 TOrOOiC D'.UG.0 OOC COI~'T2]'UE C VAR P182 vLR a210 VAR E220 VAR-.0 D'rF2( I .-FX(3.ItD.NPC o1 ]F(K.LT. 5 -UODUMKS) 16D(1PK) * .S(Uft-H)-(UC-UQ) - -VL21S660..ET/TADD3 VAR IS420 VL-O IS440 3F~lDw. UD( 3K) UL UR UT Uo PK) UD( 3KP) U0( 3KMS) DCR DCL D!FFCD(IKL) D) F CO( KL. ( '(K ULH C CON'.0..5 )r0.I) -K2WC' + I-I)-K NC *K2NC "00Z-0100 K=2. .I DIFFCD(3KL2) D!~FFCD(IKL-3) *0(!. 3F(3~. -054 (K.R2CLI.PK-KNUO(IPK) ).1.]BR ( '.E.5 PC) ')-.'O('.1 SRKSr-I a * N BUOJY a LP0-E7.. a uJD(2K) )-0.KSPI I KL I lK z lK 1A 0.P( TJ"'. VLRIE6-t0 V4RI15650 VAR 15690 ULWM'(uLLIC) ( IR-'-(L'C-4UV) VARIE *7 DO.E003 DO E0100O ) K GO TO50205 222.I ) I KIPC # (K-I)'tN'PCL VAR16D VAR 11510 VARi18520 !PK a 31. KS I KLS a3)K I KP a IK - NP 2 I N1--PC VAR 185501 YLR 15560 VAR 1C570 VARias590 VAR 86201 VARi1Es30.C RD20 3KS-0 S3G'~ to VAR 8340: VARIS5360' RDZDRD2 VAR i !-:-0.0.2)GO a TO 29990VALR iS. --.) .o2m.Cw'))'A.o.=OStd7A 0506LdV - ~ ~ ~ ~ -Hm )CC-7r~! Sall*~ ZN07J ±.(iim1 (.'N-(t-X) + 06Md~7A 1 Et a 6-Nr -NZ At 2 SW SWA r r -).t) 2 td9N'Z-t oo0to oa 3flN tLNOD s0?_O3 M 6 dVA + NAcns + ZntO * (Nt)d-(dXt)d ).8~ VA 0899tMVA (oaQN 3NC647. (W (4I) 09O6t~u7A CA ~g0 *0 H IL~ Iun S:)o ( (dN I)an +(>(!M)Onl 'S0 06SLIA OEOG bVA 0L6O t'7A 0CS8~ 100 VA )O (dm)OoA1 0669'.onH.OSZ6 M7A OdI'N Al dN O7I~t 17A OdM -( A) ONZ Nt + 1IOZN-4(Vt) * + 3NZ)-(1-1) + t bi > >4 E A- c TA .(.+fl)-IdM Y(Qd...% At t x NdE ONCA + lA b + A t + OZ~N(L-t) OdAN 00gla aVA 06!. Qti 0*0 o3 a itflt NACAS dsm~rsN td3t*Z9t Oozos 00 00ZO9 00 0LL8Z!VA 0SL8dVA (Zito ZNOO YNOO0 -YO- + )- 00L3'ttVA 06L9tMVA 0 LStMVA (At)d-(Adt)d - 11 )*.( (nf-1l)-(Himlss7 -(In-3n)-(uLM)S9V ) ZNQ.(?n-On) i13-( fl-2f) unc.(on-in) M0'a(On-.)-10 6666a 01 0 MfN i NGO 07 6 "VA ZNOO NCO ZOM-( + (Xt)0M £ ) O~t6tVA (~ ~ .uQ + + t (Nt)0ftl x = XIt)f ( oo.7A 0060tMIA 0L6Sgv7A 0968LVA I 3a CA~ SimI) 0L96 t'7A O9S63tM7A j0 0 VA d061N-t-) AoflsaNx0As + Od.vt Z1t0 7 Z(J~Si0(o(n-sn)(a)a (1n. KBR DO 62100 1=2.( URT EAS(URT) ) .X.K2NC .C1.$ ]K .SIGMhARDZDODCB E1(J) = RDT * 0.IKBR) CALL TDM(A 1.W(1K)-RDT 82000 CONTINUE C CALL TDM(A .DIFFCO(IKL) DCL a D0FFCD(IKL+2) URT x 0.Ke1P ) VAR 19390 VAR19400 VAR 194 10 : VAR19420 VAR19430 VAR19440 VAR19450 VAR19460 VAR 19470 VAR19480 VAR19490 VAR19500 VR 19510 VAR19520 VAR 9530 VAR 19540 VAR19550 VAR19560 VAR19570 VAR19580 VARi9590 VAR19600 VAR19610 VAR19620 VAR19630 VAR19640 VAR 19650 VAR19660 VAR19670 VA2 1960O VAR 19690 VLR 19700 VAR19710O VAR19720 VARIC720 VAR197540 VARS19760 VR 197 70 VAR19780 VR 19790 VAR19800 .( DCT + DCS ) C1(J) a 0.IBR DO 81100 K-2.NWPC DCT 3 DIFFCO(IKL+1) DC.KBR) C DO 82100 Ka2.5 AI(J) a 0.SIGIArRDZD3DCT 01(J) a U(IK)-ROT 81000 CONTINUE C .ULT * ABS(ULT) 81(J) .8e1..IMKS IKt.5-RDX-( URT 1 + SIGMA-RDXD.X.DI.ABS(ULT) ) .'5 WRS (W(1KMS) * W(IPK-NWPC))-O.1SP J a (K-2)]IeR * 1 .5*RD2-( WRT .1 ]K £ + (1]-)-K2NC + (K-i)-NWPC ) VAR 19260 VAR 19270 VAR19 280 vLRlg2O VAR 19290 VAR19300.5.VRE + ABS(WRB) 1 * SIGMA*RDZD.5-RD02( -WRe .]K .SIGMA-RDXD*DCL ACBS(URT) . VAR19310 VAR19320 VAR19330 VLR193AO VAR19350 VAR19360 VAR1S370 VARie380 IKL a C 1 + (1-)'K2NCL + (K-1)-NWPCL IPK .AeS(WRB)) .RDX.KBR + K 81100 U(IK) a X(IX) C DO 82000 K=2. C 00 l8100 I=2.5-( U()]KS) * U(IMKS+NWPC) ) Ai(J) = 0.SIGt. = DIFFCD(IKL+3) WRT 3 (W(3K) + W(iPK))-O.( DCL * DCR ) C1(J) z 0.ABS(VRT) ) .KER D00 82000 1-2.KSPI ]K = 1 + (1-1)-K2NC + (K-I)-NWPC 1X a (1-2).C1 .5-RDZ*( VRT * ASS(wRT) .]K + K2NC IKP a 1MKS 1K + NWPC ]K K K2NC NWPC 1KIAS = C DCR .5*RDX-( -ULT .5-( U(IK) + U(]KP) ) ULT C a 0.16PI 1K a 1 * (]-i)-K2NC + (K-1)NWPC IX x (K-2)-ISR + I 1 E2100 W(]K) * x(Ix) 00 823000 K2.0.'-RDXDDCR D01(J) .ROT * 0. (1-) L >1 * (-~-~t A4d! zz IN I s)Z-t! + Afl9nszAofl3 (SCOP 001 0OZ17A 060ZZ2A OSOOZZ17~JA 1 OdIA'Na-)) * - -AC-47-0103t1)1E 0*0 a NAC(3 a9AzsA 007 OC Idgl*Zal OOOVS 00 OLOOZ!7A 0900ZM7A * ()t(z->q) :)NN-4-1) + I a as9tz:t 001C3 + Cxt)y 2 (x)Xfl( ( 00t ES A4t cc tdaA'Z-x4 001.ZC(3a-E1~S (d~t)d - 00oo' ((HIiA)SS7 I )-'ZQZ .' ).v%4!)iS +. 3)-3QO.~) Al NP 1=I000ca 00 .0xCd. )'ZQd-S50 )-Zc!-S* - a -' a - (HS.)ssv .'%)SS'7 + HSA . z (r 01)~S~7~-).L 0.(HL')S.~tSC(Hifl)ss b3(3 )..LG!1-CE)PA HIM.ycajSpo Hlfl-)bQS 00L661 dVA 096tdV'A C(S~wt)fl + (Ar4)fl )-s0o )'s0o Hlfl C(xid[)n + own)f - Htf 0OS66taVA * 0bVA I OS6J&1VA Z+1~~)41 2jE 1d)a O1'66t7A 01L61 J7A 3dI% (-) UMNN (-) >1I a SWN1 * D ).Htin 0~Q0~*/'.Q ( (S'feXtiM 094=J7A OSZO0t7A (>I).S'Q g* -2 S = CI)' 1 C + ># O oc OZMVA ~dN OdMN -)Z' 1I A W> = S AVI + A!t d>'[ 06t0MIA M17A 0910~1VA 0C10aVA +N *) 1:)dfMN'(-)) + IONZ)A-(L-l) OdAIY-( .fl.mNtAz + + ONA(-lt.40 C0~7A O CO0d7A NAoC1s (A(I)a .3 00 000O~d7A "L661 aVA (AI - (Ad 1) d !OSGOZi VA OG61 !VA 03661 SVA b33aXac~tjS ( 100 (Hll)SV H-~ln - - C(Hzn)sev - Ya - - ~ A +Laa Hfl )-xCi~s~o = x (r)La ()9 t ()1 2 (H.v4)tS -~m * io1. ).3nwNtJ. VAR20420.0/RC OR = CYL-0.12 J .NSEO) 33333 CONTINUE 15E0 a 0 DO 60000 K=Ki.EO.25*RRC CYRX = CYLO0.2.1 (1-2)-(KER--1) 84100 W(JK) a X(lX) C 89999 CONTINUE IF(ICONV.NWPC CFR = CF(1PK) CFL .O.IKBR) DO 84100 l=2.K2 DO 60000 13a1.SIE(IKP) S]EB * SIE(]KW.1CALI.1SP1 00 84100 K-2. VAR20430 VAR20440 VAR20450 VAR204GO0 VAR20470 VAR20480 VAR20490 VAR20500 VAR20510 VAR20520 VAR20530 VAR20540 VAR20550 VLR2050O VAR20570 VAR20580 VAR20590 VAR20600 VAR20610 VAR20620 VAR20630 VAR20640 VAR20650 VAR20660 VAR20670 VAR2060 VAR20690 VAR20700 i VAR20710 VAR20720 VAR20730 VAR20740 VAR20750 VAR20760 VAR20770 VAR20780 VAR20790 VAR20800 VAR208 10 VAR20820 VAR20830 VAR20840 VAR20850 VAR20860 VAR20870 VAR20880 VAR20890 VAP20S00 .EO.HDX RRC a I.I) GO TO 55000 DCR .5-(TS(IK)+TS(IPK)) .CF(]MKS) UC c U(IK) WC = W(!K) UL .0.S) SIECO .DIFFCO(1KL*i) DCL a D!FFCO(1KL+2) DCB * D]FFCO(JKL*3) ]PK )KP a a + K2NC * NWPC = 1K .KSR IK IX a I + (I-1)*K2NC + (K-i)*NWPC + K .CF(UK) IF (CFC.DIFFCC(IKL) DCT .S]EO(]K) TSRA TSLA .LE.SIE(IPK) S*EL . IBR-(K-2) + I ' (1-i)-K2NC IK = i + (K-1)-NWPC IKL a 1 + (1-1)-K2NCL + (K-1)-NWPCL RC a FLOAT(I-1)bDX .B.Ci.SIE(IMKS) SIET .5-RRC*RDX CFC .U(JMKS) vW = W(IKMS) AUC .1) GO TO 2040 ISEO = ISEO + 1 IF(ISEO.NE.5-(TS(]K)+TS(lMKS)) VAR203G0 VAR20370 VAR20380 VAR20390 VAR20400 VAR20410 .CALL TDM(A .0P.X.K2NC ]K IK 7]KS IKMS = IK .DASS(UC) AUL .DLeS(UL) ALC = DAES(WC) 1wS * DASS(WE) = SIE(1K) S!EC SIER .D1. x)j! 2(AI1)3rS oo0t 9 06CI ~1VA' 09ZI dVA 0LI."ai 0*0 000 09ML VA 3flNIINOD 00009 =(fil) 0O~VA 0Lt t aVA OOtZVA 06tt~7A XC-(QXC + dAO)-dQ-ZJlj + 01 0 00009 02.NOD (L03-e.1! Al z(IJ)L3 (NOfl'Ol V(ISJX0-1-7) Ot)t0dVA 0001ZlA N7 dlxiol ((L-1)x~JNG4(IV~~sNQ)ts0 -. 06C Zd7IA ZltrEzf 00 OOOL CO 0SCtLb7A NOI0t1D3JC-Z NE d33MS CNZ 0L~tZZ7A LCECE OZ :tZ7A 1 05 3 3lNTI. C0 r~t (r)LC 00~ttIdVA 060k ~wVA 0OLt dVA 00009 01 13t5.(r'.(N t0 *090~bVA 0O0I bVA 0O'0t MVA OSOI aVA 0OP0tZSVA (Q QtlY.Q ~ 000 fl co(3 S6 . t7) . -10 + ( 0S669 01 00 (V-03VIJfl S0.~t J -'~1f ((t0AX'1ZJANO ) C dt0* ) * N )- S 4~ I .OSrlth7A OV7 t!VA t.± 4:7) 21J3 1u L7S1L'q'l OS 60ZMVA (Slx () I s I)-SO I ((diE)l+ m1)l)'c = vzs I VIL .-+c a 21J10 to..I vo 0009 0:9>4 I )c t a.C*~ t )4* .L. t :) a .At~VA OSZI~7A clt'L fa ZAVtAZ'4 0=0i9 00 0=O9 00 -I . (nN*31. -N *:dM o zi Z)q4s)4 coo 0.OlI)03(r v+n . DIFFT-DCT-RDZD C2(J) a 0.1) GO TO 65500 D1(J) * 01(J) .C2(J)-SIE(IKP) C2(J) = 0.RDZ C2(J) = 0.0 69950 IF(CFT.(WC*AwC-WS*iWB).0 E2(J) .K2 Ix = KS.X.i) GO TO 70000 IF(CFT.IKBR) C DO 71000 ]=m1.NWPC C CFT = CF(IKP) CFB .NE.EO.0 70000 CONTINUE C CALL TDM.1.ND. 1 * 0.E0..01.0.K.5"(ws*Aw )'RD2 .LE..12 DO 71000 KKI.5 -0.1) GO TO 65000 .C2.0 C2(J) = 0.K)*DNFF2( .21730 VAR21740 A2(J) = B2(J) = VAR21750 VAR21760 VAR21770 VAR21780 VAR21790 VAR2800 VAR21810 VAR21820 VAR21830 VAR21840 VAR21850 VAR21860 VAR21870 VAR21880 VAR21890 VAR21900 VAR2190 VAR21920 VAR21030 VAR2190 VAR21950 VAR21960 VAZR21970 VAR21980 VAIR21990 VAR22000 .-(1-2) + K .0.5-(WC-WC).0.11.0 GO TO 70000 65500 C2(j) E82(J) .DIFFCO(IKL) = D]IFFCO(IKL+1) = DIFFCO(]KL+2) .5-(DNFFZ(I.(A2.TTSPLL . .5..DIFFCO(IKL+3) 1PK a 1K + K2NC IKP a !K + NWPC IMKS a 1K .5-(TS(]K)+TS(3KP)) TSeL .5-(rS(lK)+TS(lKMS)) C GAT DIFFT DIFFB C C * RDTSIE(]K) .X GtA.K2NC IKWS VAR21460 VAR21470 VAR21480 VA021490 VAR21500 VAR215 O VAR21520 VAR2rS30 VLR21540 VAR21550 VAR21560 = ]K .EO.0 DI(J) D(J) + DIFFT-DCT*RDZ GO 70 70000 65000 01( ) = SIE(]K) L2(j) a 0.IF C DCR OCT DCL DC8 C ( CFC.DIFFE-DC-RD2D RDT * (DIFFTDCT+D)FFS*DC6)*A02D .1) GO TO e9950 Dl(J) = 01() .B2.5-(DNFF2(3.NU) GAMT * RPRAN .M * * IF(TS(]K).k)+DNFFZ(1.OlFFT-DCT-RDZD IF(CFB EO.A2(J)-SIE(3KMS) A2(J) • 0.I DI(J) a TGA.CO(IK)-0.CF(IKt S) WS W(IKMS) WC a W(IK) VAR21570 VAR21580 VAR21590 VAR21600 VAR21610 AWB a DLeS(we) AWC DOASS(WC) TSTA a 0.K+1))-X.RDZ .NNOPT.I) )XMX GAMT-TSTA VAR21620 VAR21630 VAR21640 VAR21650 VAR216GO0 VAR21670 VAR21680 VAR21690 VAR2I700 VAR217 tO VAR21720 VA. NWPC DO 2.1) GO TO 33333 GO TO 33339 IF(ISEQ.NwPC 71000 SIE(]K) = X(IX) C IF(KIRK.E0.K2 1K 1 + (]-i)'K2NC + (K-I)*NWPC 33338 SIE(K) = 0.-S)=W(KK+) TO(KMS)=TO(KK+1) VAR22470 VAR22480 VAR22490 VLR22500 VAR22510 VAR 22520 VAR22530 VAR225ZO VAR22550 .K) = SIE(IK) 333335 SIE(IK) . TRANSFERS VELOCITIES TO STORAGE ARRAY( 2040 KI1 K2xKBP2 LWPC=I .5'( SIE(IK) + STOR(I. 12 KK=KK4K2NC KKL VAR22010 VAR22020 VAR22030 VAR22040 VAR2 2050 VAR22060 VAR?22070 VAR22080 VAR220s0O VAR22100 VAR22 110 VAR22120 VAR22130 VAR?22140 VAR22150 vLR22160 VAR22170 VAR22180 VAR22190 VAR22200 VAR222A0 VAR22220 VAR22230 VtR22240 VAR22250 VAR22260 VAR22270 VAR22280 VAR22290 VAR22300 VAR22310 VAR22320 VAR22330 VAR22340 2109 VAR22350 VAR22360 V R22370 VAR22380 VAR22390 VAR22A00 VAR22410 = KKL + K2NCL VAR22420 VAR22430 VLR2240 VAR22450 LWPCL = LWPC=I 1 VAR22460 SIE( 1)=S [(KMS) U(1)=U(!KMS) w( ) (IKVS) 0(l)-TO(IKMS) TS(i)=TS(IKMS) SIE(]K S)=SIE(KK4+ ) U(KtMS)U(KK+1) W(IKt.K2 IK v 1 + (1-I1)K2NC + (K-1)*NWPC STOR(I.K) ) KIRK a 0 33329 CONTINUE C C NOTE.SIEO(IK) KIRK a i GO TO 33334 33337 CONTINUE DO 33338 1=1112 DO 33338 K=KI.NE.IK r 1 + (I-I)*K2NC + (K-1).12 DO 33335 K=KI.K2 LWPC=LWPC+NWPC AT TIME-N ) .09 KzK1. IK=LWPC IKS212K2 4 1K SIE(IKS)=SIE(IK) U(IKS)=U(1K) W(IKS)uW( K) TO( IKS)TO( IK) TS(IKS)rTS( K) CONTINUE 12-]BPI K1=2 K2rKSP2 KK=0 KKL O0 DO 29F9 l.O) 00 33335 I11i.11. 6 and Fig.8. 4. The files. 'inputd 3' and 'inputd 7'. is for calculating the natural convection flow field in a Cartesian closed compartment. 5.1 and Fig. 5. 5. 5. results are in Fig. Fig. are for calculating the numerical solution by donor cell scheme in the problem geometry. The output The files. 'input mom'. are for comparing Huh's And De Vahl Davis and Mallinson's correction formulas in the problem geometry.APPENDIX D TYPICAL INPUT FILES The file.2. 'inputn dat'.3(B). 'inputd HUH' and 'inputd DM'. . 5. The output results are in Fig. is the input to test the stability of the ADI scheme for momentum equation.7. The file. Fig. 0 1.9 0.0 0.0 1.0 0.0 .7 0.-4 1.0 0.0 0.0 I 3 I 16G -1.0 60.0 0.0 1.0 0.0 2 5 10.01 0 1.2 10.0 0.0 1.0 1.0 10.5 6.0 0.0 1.013 0.-4 0.0 0.0 0.0 0.0 0.0 0.0 -1.0 0.0 60.44 0.0 4.0 0.0 2 l.0 0.0 0.-4 0.0 0 0.0 2.0 0 0 1.t20 1.0 4 0.5 0.045 0.01 0.09375 0.0 0.-4 5 1 1.0 0.120 1.rTh FILE: INPUT MOM VM/SP CONVERSATIONAL MONITOR SYSTEM PAGE 001 4 4 O STABILITY TEST FOR MOMENTUM 2.0 0.0 0.0 0.0 0.0 0.0 10.0 1.0 0 2 1 3 O0 2 2 O O ADI 10.0 0.0 1.0 0.0 0.75 10.0 0.0 10.0 1.0 60.0 1.0 1.0 0. -20 2 1.751.0 0.0 1.0 677.0 I 43 1.0 0.0 1.0 0.-5 1.0 1.0 1.0 0.0 0 1.0 8 2 I 3 1 3 0 3 0 1 1.2 1.FILE: INPUTN DAT A VM/SP CONVERSATIONAL MONITOR SYSTEM 'PAGE 001 5 10 0 1 0.0 6 6 50.0 -32.0 1.-20 II I 1.67 1 1I 2.4 0 0 0 0 O 0 0 6 2 0.0 2.7 1.0 0 0 2 1.0 0.90 0.0 0.-20 1.0 10.5 0.-20 1.2179-5 1. 20 1.0 0.0 1.0 1.44 0.-20 0.5 5 1.0 NATURAL CONVECTION IN A CARTESIAN CLOSED COMPARTMENT 1.0 0.0 0.0 4.0 68.0 .045 0.-20 030 1.0 1.0 1.0 0.0 0.0 60.01 1.01 0.+20 60. 000024 60.0 1.249 3 1.497 -0.-3 1 I -1.07107 111 3 1 1 2 5.0 1.01 0 0.0 7.0 0.207 0.0 8 10.0 9.0 5.0 0.0 0.0 0.0 1.0 0.01 0.0 5.07107 0.0 0.0 1.0 1.0 0.0 0.013 2.9 0.0 0.0 GO.0 9.0 0.0 0.999 1.497 2.0 0.457 -1.940 1.0 1.0 0.0 1.0 0.000 -0.07107 7.0 0.0 2 4 2 4 1.II I 11 2 2 1.0 1.O.511 1 1.000 10 1.751 2 1.204 -0.0 9.0 -1.0 0 0 NUMERICAL blFFUSION .0 12 11 2 121 t 5.-47.000 0 0 O O I.948 -1.0 0.009 4 1.75 0.0 0.0 9.0 0.409 1.0 4 I 36 -1.0 0.0 .000 9 1.0 9.0 0.0 0 2 5 3 0 7 6 1.09375 0.07107 1.001 0 1.0 60.991 6 1.0 0.0 0.CASE 3 2.000 1. 020 2.02 1.0 0.07107 0.0 1.026 5 1.05.0 1.063 0.07107 12 12 Ill 2 5.014 ).0 0.0 2 1t 1 113 5.70711 1.000 10 1.000 0 0 0 O .0 1.0 1.0 0.000024 0.OGI -2.0 0.+20 I 0.0 7.0 1.0 1.0 0.0 9 10.5 1.061 1 0.002 -0.0 0.997 7 1.0 1.0 7.0 1.0 0.5 0.974 5 1.009 G 1.0 0.0 0.0 0.0 1.014 -0.000 0.0 5.0 9.FILE: INPUTD 3 A VM/SP CONVERSATIONAL MONITOR SYSTEM PAGE 001 10 10 0 0.911 3 4 1.063 -( ).0 7.70711 0.003 7 1.07107 0.045 1.204 0.5 1.0 7.44 4.0 1.O 2 1.457 1.0 0.0 0.0 0.0 10.207 0.0 0.0 2 1.0 1 .0 0.0 0.002 0. 0 0.002 5 1.CASE 7 0.333327 1.000 6 1.0 0.0.07107 7.5 1.420 1.0 0.0 0.0 5.0 1.07 07 0.0 1.00.0 1.0 .0 2 2 0.0 2 9.014 0.0 0.4 8 2 1 3 0 0.0 -1.0 1.0 7.626 1.003 17051 1.013 4 1.0 0.0 4 2 4 1.0 0.0 10.0 0.0 0.75 2.0 0.0 5.633 1.0 4 I 36 1.907 3 1.143 -0.333327 60.0 0.457 2.374 1.07107 711 8 8 2 5.0 1.0 5.0 0.0 7.5 1.0 0.0 0.013 -2.0 0.000 0 0 0 1.0 1.0 1.0 0.0 3 0.0 2 0.045 1.0 7.143 0.0 0.071107 0.0 0.0 1 9.911 9.0 0.990 5 1.0 1.0 10.00.0 60.+20 2.0 0.0 7.01 2.001 -0.9 0.0 0.07107 I 113 7 2 1.01 0 0.0 1.17051 0.009 1.0 0.0 NUMER ICAL DIFFUSION .0 1.0 9.000 6 1.0 0 0 2 2 1 0.0 0.0 0.0 -3.0 4 1.0 I 1.0 -1.0 1.4 1.457 -2.000 0 O 0 0 0 7 1 2 7 2 5.07107 1 1 713 2 5.014 0.003 -0.0 1.07107 0.-3 1 1 1.633 -1.0 0.09375 .0 1.0 2 7 8 5.167 1.0 0.001 0.0 0.FILE: INPUTO 7 A VM/SP CONVERSATIONAL MONITOR SYSTEM PAGE 001 6 0.0 1.0 0.0 7.-47.0 0.01 0.0 0.0 0.0 1.44 4.0 60. 0 01l 0'00' C0'I C.L Z01 'L L'09C91C P09c91''? 1'Q9C9p 0'19C91' 1 iLO'O 1'LO*0 GL00O LLO'0 £00.u *0 Ul' I 6P I 0011* I 0.0 0.09 0. 6IP'0 O1'1 991'0 LOCO0 9LV'O 060*0 Cc£00 910'0 1 0 C Iz 0 0 0*1 OL+*t Ylz OZ4*1 zlo' IfIAH .0 0*1 i 0*0 099C00'LOOo 0*0 01 CI0'0 0*0 010 £LC6OO0 0*1 VL9'C) P99*0 OWO' LLC'O 1'CC*0 16L' 9L1 0 100 ~W31.0 9?LL '0 G'.09 W '09 0.0 GLO10 ZU10 SL9*0 109.z 10'o 0 1 0'l GL'O 1O1 '0 I 0*0 01l 0'l L910* 199*0 099. 1'0 0. 0 LG0 .0 09G"0 9Cs£0 661''0 6sv1'.*I riI 00 0*0 0*0 0i'00 0 10*0 6 v 0*0 0'0 top' .NOIS(fJJIG IV:)IUIW(IN too 39vd IVNOIIVSU3ANIX) dS/W~A v VIIi III)II OIMINI O~N :311A II . 0 1'0l '0 011 '0 sCl '0 LSI *0 6LI -0 G0Z 0 cpz *0 01 01 0 01 oz 61 LI 01 01 91 cI zI 01 6 0 Ia z 0 6ZC *0 OLC 0 &Pt'v O CZi.0 CGO.SAS 3~~~)Vd kOIINOW 0 0'0 0*0 0.1 1' 0*0 0 *1 D99C00 0'1 0IO' 0*0 0*0 0*0 0'0 0 *I0*0 9c 1 t.1bl b 690*6 £00*0 LC9 0 90c.. *0 PI 9.0 0 *I O'l 0' *1II1 01l 0*1 0 1 0'L 01 O'1 0. FILE: INPUTO IIU4U VM/SP CONVERSATIONAL MONITOR SYSTEM PAGE 002 2 0.0 8.6603 0.0 0.-20 9 1 1.-20 1 1.0 2 25 25 5.0 0.0 0.-20 2 1.0 2 25 5.0 0 2 1.-20 913 1.-20 10 1.-20 911 1.6603 0.0 .-20 5.-20 I13 1.0 26 26 5.-20 2 1.0 5.0 I 1 5.-20 1011 1.0 5.6603 0. 0 0.0 1.02 2.2406 1.5 1.0 1.0 1.003 0.174 1.013 2.420 0.0 0.0 5.0 0.01 0 0.260 -0.365 9.951 0. 4G1IG04 7.705 0.0 0.021 0.003 0.5 1.0 1.003660 60.4 4 1 36 1.496 0.002 0.0 1.099 -0.700 .GOG 2 1.0 1.0 0.0 1.09375 0.0 -1.0 60.0 0.5 1.0 1.002 -0.0 0.02.0 0.0036G0 0.0 0.20 1.026 -0.OGI 0..0 -1.463G04 2.767 0.96G 0.076 0.566 0.01 0.920 0.9 0.0 0.034 0.33G 6 1.OM 0.030 0.051 0.0 I 1.0 2 4 2 4 1.463604 2.0 1.0 0.009 0.rILE: INPUIO OM A VM/SP CONIVEIlSATIONAL MONI(fro SYSTEM PAGE 001 24 f( 0 I 0.-4 0.0 0.002 2..4GI6 0.0 0.302 0.0 1.463604 2.0 5.0 1.223 3 1.133 10.0 0.0 0 2 1 3 0 7 6 I -0.0 3 10 1.932 1.636 0.522 -0.0 1.0 0.0 0 0 NUMEtICAL IrFFUSION .0 1.0 1.0 1.0 1.02.0 0.0 0.005 0.093 0.567 9.011" 0.526 3 4 5 G 2-.30i9 1 0.007 0.0 0.0411 0.242 7 1.072 ' II 12 13 14 15 16 17 10 19 20 21 22 23 24 0 0 0 I '.0 1.0 0.0 1.0 1.0 0.0 5 1.197 0.0 1.0 0.030 0.0 0.0 60.403604 2.0 1.246 0.045 1.156 0.0 1.-3 1.0. 1.0 0.949 0.969 0.000 0.0 0.0 2 2 2 0.0 0.0 0.120 2.003 0.903 0.0 1.0 0.044 9.016 0.0 0.463704 9..75 1.029 0.0 -1.144 -1.0 0.0267 0.0 1.360 0.44 4.0 1.0 0.115 0.460 4 1.0 0.0 0. 10 OM A VM/SP CONVERSATIONAL MONITOnr SYSTEM PAGE 002 2 0.-20 1011 1. & .0 26 5.0 2 5.-20 1.0 5. 6603 0.6603 0.0 5.-0 26 2 1.0 I* - .0 0.0 0.-20 113 1. GG03 0.0 0 29 25 I 25 2 1.0 .-20 9 1 1.0 I 5.0 0.SILr.: INP.-20 911 I.-20 913 1.0 2 5.-20 I 2 1.-20 5.