Laval Nozzle Flow Calculations

6LAVAL NOZZLE MEKHANIKA ZHIDKOSTI I GAZA FLOW CALCULATION U. G. P i r u m o v Izv. AN SSSR. M e k h a n i k a Z h i d k o s t i i G a z a , Vol. 2, No. 5, pp~ 1 0 - 2 2 1 i 9 6 7 The inverse problem of the theory of the Laval nozzle is considered, which leads to the Cauchy problem for the gasdynamic equations; the streamlines and the flow parameters are found from the known velocity distribution on the axis of symmetry. The inverse problem of LavaI nozzle theory was considered in 1908 by Meyer [1], who expanded the velocity potential into a series in powers of the Cartesian coordinates and constructed the subsonic and supersonic solutions in the vicinity of the center of the nozzle. Taylor [2] used a similar method to construct a flowfield which is subsonic but has local supersonic zones in the vicinity of the minimal section. Frankl [3] and Fal'kovich [4] studied the flow in the vicinity of the nozzle center in the hodograph plane. Their solution, just as the Meyer solution, made it possible to obtain an idea of the structure of the transonic flow in the vicinity of the center of the nozzle. A large number of studies on transonic flow in the vicinity of the center of the nozzle have been made using the method of small perturbations. The approximate equation for the transonic velocity potential in the physical plane, obtained in [3-6], has been studied in detail for the plane and axisymmetric cases. In [7] Ryzhov used this equation to study the question of the formation of shock waves in the vicinity of the center of the nozzle, and conditions were formulated for the plane and axisymmetric cases under which the flow will not contain shock waves. However, none of the solutions listed above for the inverse problem of Laval nozzle theory makes it possible to calculate the flow in the subsonic and transonic parts of the nozzles with Iarge gradients of the gasdynamic parameters along the normal to the axis of symmetry. Among the studies devoted to the numerical calculation of the flow in the subsonic portion of the Laval nozzle we should note the study of Alikhashkin et al., and the work of Favorskii [9], in which the method of integral relations was used to solve the direct problem for the plane and axisymmetric cases. The present paper provides a numerical solution of the inverse problem of Lava1 nozzIe theory. A stable difference scheme is presented which permits analysis with a high degree of accuracy of the subsonic, transonic, and supersonic flow regions. The result of the calculations is a series of nozzles with rectilinear and curvilinear transition surfaces in which the flow is significantly different from the one-dimensional flow. The flowfield in the subsonic and transonic portions of the nozzIes is studied. Several asymptotic solutions are obtained and a comparison is made of these solutions with the numerical solution. w1. P r o b l e m f o r m u l a t i o n a n d c a l c u l a t i o n m e t h o d . This paper considers the inverse problem of Laval n o z z l e t h e o r y . An a n a l y t i c v e l o c i t y d i s t r i b u t i o n i s g i v e n o n t h e a x i s of s y m m e t r y i n t h e s u b s o n i c , t r a n s o n i c , a n d s u p e r s o n i c r e g i o n s . A c c o r d i n g to t h e Cauchy-Kovalevskii theorem, for analytic initial conditions for the equations of gasdynarnies there is a unique solution of the Cauehy problem in some vicinity of t h e a x i s of s y m m e t r y . The formulation of the inverse problem, which is in essence a Cauchy problem, does not involve any fundamental difficulties. We note that in this regard the inverse problem differs favorably from the direct problem, for which up till now no unique solution has been obtained and the conditions f o r s h o c k - f r e e f l o w in t h e t r a n s o n i c r e g i o n h a v e n o t b e e n f o r m u l a t e d evefl f o r n o Z z l e s w h o s e w a i l s a r e m a d e i n t h e f o r m of a n a l y t i c c u r v e s [7]. H o w e v e r , d e f i n i t e p r o b l e m s a r i s e in t h e n u m e r i c a l s o l u t i o n of t h e C a n c h y p r o b l e m . In t h e g e n e r a l c a s e t h e C a n c h y p r o b l e m in t h e e l l i p t i c r e g i o n i s i n c o r r e c t , a l t h o u g h , if t h e c l a s s of a n a l y t i c f u n c t i o n s i s c o n s i d e r e d , t h e n in a l i m i t e d r e g i o n t h e p r o b l e m b e c o m e s c o r r e c t [10]. N e v e r t h e l e s s , e v e n f o r a n a l y t i c i n i t i a l c o n d i t i o n s i n t h e s u b s o n i c p o r t i o n of t h e n o z z l e , w h e r e the gasdynamic equations are elliptic, with a poorly selected difference scheme the roundoff errors, which inevitably arise in the numerical solution, increase e x t r e m e l y r a p i d l y i n t h e s o l u t i o n of t h e C a n c h y p r o b l e m . T h e r e f o r e , to o b t a i n a s t a b l e s o l u t i o n w e m u s t select a difference scheme such that its use will not l e a d to t h e r o u n d o f f e r r o r s e x c e e d i n g t h e a p p r o x i m a tion errors significantly. On t h e o t h e r h a n d , i n t h e g e n e r a l c a s e t h e r e l a t i o n s h i p of t h e s t e p s i n t h e d i f f e r e n c e s c h e m e i n t h e h y p e r b o l i c r e g i o n s h o u l d b e s u c h t h a t t h e r e g i o n of i n f l u e n c e of t h e a p p r o x i m a t i n g s y s t e m d o e s n o t e x t e n d b e y o n d t h e r e g i o n of i n f l u e n c e of t h e b a s i c s y s t e m of d i f f e r e n t i a l e q u a t i o n s [11]. H o w e v e r , i n t h e c l a s s of a n a l y t i c f u n c t i o n s t h e r a t i o of t h e s t e p s i n t h e d i f f e r e n c e scheme cannot be arbitrary, s i n c e i n v i e w of t h e a n a l y t i c i t y of t h e i n i t i a l c o n d i t i o n s w e c a n n o t a l t e r t h e m in any great segment without altering them throughout t h e e n t i r e r e g i o n of a n a l y t i c i t y [12]. The system of gasdynamic equations which des c r i b e s t h e i r r o t a t i o n a l , i s e n t r o p i c flow of a n i d e a l gas with a constant adiabatic exponent in the variables (r i s t h e s t r e a m f u n c t i o n ) a n d x h a s t h e f o r m [13] Ogz~ 2J Og v Op k Ov O~ pu' 8x u O~ gJ Ox' p = p,.4~, u=- k-- I k-- I H e r e u a n d v a r e t h e p r o j e c t i o n s of t h e v e l o c i t y v e c t o r w o n t h e x a n d y a x e s of t h e C a r t e s i a n c o o r d i n a t e s y s t e m , r e f e r r e d to a . - - t h e c r i t i c a l s p e e d of s o u n d ; p, p a r e p r e s s u r e a n d d e n s i t y , r e f e r r e d to t h e pressure and density for w = a.; k is the specific heat ratio; and j = 0 and 1 for the plane or axisymmetric cases, respectively. Let us write the difference scheme, corresponding to (1~ which is used in the present study. Let all t h e f l o w p a r a m e t e r s o n t h e n - t h l a y e r ~n = c o n s t b e known at i points, which in the general case are not e q u a l l y d i s t a n t f r o m o n e a n o t h e r (i = 0.1 . . . . . M)o T h e n t h e p a r a m e t e r s o n t h e (n + 1 ) - t h l a y e r r = = const are defined by the formulas [ (~) ,~ yi(n+l)J" = ~]'in23 "@ 2 J - ' h ~ [ ~ 1 t ], + (pu)~.,:,,l),/}~ (1.2) S p e c i a l c a l c u l a t i o n s m a d e in t h e p r e s e n t s t u d y on t h e s e l e c t i o n of t h e n u m b e r of p o i n t s f o r c a l c u l a t i n g t h e d e r i v a t i v e s showed that the three-point scheme is most stable.4 M( / y~ M=/ Fig. the variation of v is slight.(xi+~ + xi-l) +9i (x~ . s i n c e as a r e s u l t of t h e i o w g r a d i e n t s in t h i s r e g i o n t h e a p p r o x i m a t i o n e r r o r s w i l l b e s m a l l . b u t w i t h s o m e s t r e a m l i n e n e a r the a x i s of s y m m e t r y . conversely.3) h a s a r e m o v a b l e s i n g u l a r i t y on t h e a x i s of s y m m e t r y ..z i + d z~) ( x i . Figure i shows a typical variation of v with nozzle length. the approximation errors in regions 2 and 4 became so large that.(xi + xi+Q + ~ + 1 (x~+. r e s p e c t i v e l y . when calculating the flow in the elliptic region it is advantageous to use a difference grid with a variable step. in t h e c a l c u l a t i o n w i t h a f i v e .. Special calculations were made to verify these facts.2) and (1. t h e d e r i v a t i v e s w i t h r e s p e c t to x were calculated using the formula ( &P ) -~x 2x--(x~+' + xd ~= ~-~ (x~_.~) [k+l u~(. In t h e r e g i o n of t h e m a x i m u m .p o i n t s c h e m e . In this connection it is natural when replacing the derivative by a difference relation in region 1 to select larger difference g r i d s t e p s in t h e d i r e c t i o n of t h e x a x i s . and A r i s t h e i n t e g r a t i o n s t e p a l o n g t h e n o r m a l to t h e s t r e a m l i n e s . in the vicinity of the maximum.. the calculation rapidly becomes unseable.) t- 2x -. in t h e a x i s y m m e t r i e e a s e t h e c a l c u l a t i o n b e g i n s not w i t h t h e a x i s of s y m m e t r y a s in t h e p l a n e e a s e . we u s e (1. in which a different arrangement of the points on the layer was used. The use of large difference grid steps in regions with small gradients leads to a situation in which the growth of the roundoff errors in the numerical solution of the Cauchy problem for the elliptic equations is practica!ly not noticeable and has no effect on the calculation stability. h e r e t h e q u a n t i t i e s w i t h s u p e r s c r i p t (o) a r e t a k e n e q u a l to t h e c o r r e s p o n d i n g q u a n t i t i e s on t h e n . V a r i o u s d i f f e r e n c e s c h e m e s m a y b e u s e d to c a l c u l a t e t h e d e r i v a t i v e s 8 v / 0 x and 0 y / ~ . . ) (xi . in r e g i o n s 2 and 4 . T h e p a r a m e t e r s in all t h e f o l l o w i n g iterations are calculated similarly.t }- 1 = IpZI+.( 1 . 2 Thus.t h l a y e r .FLUID DYNAMICS 7 ('~) t h e d i f f e r e n c e g r i d s t e p m u s t be s u f f i c i e n t l y s m a l l so t h a t t h e a p p r o x i m a t i o n e r r o r s a r e s m a l l .3) to c a l c u l a t e t h e q u a n t i t i e s y(1)i(n+ D and p(1)i(n+l) at a l l p o i n t s of the ( n + 1 ) .8 g x Z Fig. 1 Let us turn to the difference form of the derivatives 3v/0x and 3y/Sx.5) to d e t e r m i n e t h e v a l u e s of p and u. corresponding to subsonic flow with low velocities.. If we use a difference grid with constant but small step. 5 ) .t h l a y e r . j u s t a s in r e g i o n 1. such that the growth of the roundoff error in region I became practically imperceptible.. Only with the use of a difference grid with small steps in the regions 2 and 4 and large steps in regions 1 and 3 was it possible to obtain a stable solution with high accuracy in the entire flow region right up to the singular point in the transonic region.~.y+ [(@70. c a l c u l a t i o n i n s t a b i l i t y s h o w e d up f o r a c o n s i d e r a b l y s m a l l e r n u m b e r of s t e p s a l o n g t h e n o r m a l to t h e streamlines than for the calculation using the threep o i n t s c h e m e .- xd' w h e r e r is e i t h e r of t h e f u n c t i o n s v and y. while in region 3.4) t o d e t e r m i n e v(1)i(n+l).)1~ L k. T h u s . We see from this figure that in region i. On t h e o t h e r h a n d . 2 ) .>. regions 2 and 4 are regions of large gradients of the function v.[d~)]2] '1=" (1. In c a l c u l a t i o n of t h e d e r i v a t i v e s at t h e e x t r e m e p o i n t s of t h e l a y e r . in t h i s e a s e t h e r o u n d o f f e r r o r s in t h i s r e g i o n a r e a l s o smM1 b e c a u s e of t h e l a r g e v a l u e s of the d e r i v a t i v e s .x i . T h e a p p r o x i m a t i o n e r r o r in t h i s d i r e c t i o n is (Ar 2. In t h e a x i s y m m e t r i e e a s e (1. In t h i s c o n n e c tion. If we use a difference grid with a constant but large step. In t h e f i r s t a p p r o x i m a t i o n we u s e (1. T h e n we u s e t h e r e s u l t i n g v a l u e s of y to d e t e r m i n e (Oy/0x)~})+l) (the m e t h o d f o r c a l c u l a t i n g 0 y / 0 x and Ov/0x is d e s c r i b e d l a t e r ) and w e u s e (1. T h u s . the roundoff error growth in region 1 leads to the calculation becoming unstable after a small number of steps in the direction along the normal to the streamline. At t h e r e m a i n i n g p o i n t s of t h e l a y e r x = x i. 3 0. (.p o i n t s c h e m e u s i n g t h e s a m e a r r a n g e m e n t of p o i n t s on t h e l a y e r a s u s e d in the c a l c u l a t i o n w i t h t h e t h r e e . Let us examine the iteration method for calculating t h e p a r a m e t e r s u s i n g ( 1 . .5) 1 -- k -- 1 H e r e v is t h e i t e r a t i o n n u m b e r .+~) = k -- 2 [P'(:'~)t)](k--')/k. F i n a I l y . as before.x~_~) (x~+~ . x is s e t e q u a l to x i _ 1 o r xi+ ~ f o r t h e l e f t and r i g h t e n d s . the variation of the function v is not large and its derivatives are small.. t h e d i f f e r ence grid step should be selected sufficiently large for analogous reasons. - 2x -. Pi(n+i) -'~ (~) 12i(n+f) (v) ~.)(2. - xi+. and t h e r o u n d o f f e r r o r s b e c a u s e of t h e l a r g e d i f f e r e n c e g r i d s t e p s w i l l a l s o b e s m a l l .. 1) dyo ' ] + : w h e r e f is any of the functions p. It was foun&that with accuracy to 0.1 9 10 -z a n d 0 . 0.1 ~ Expansion into a series in @in the vicinity of the symmetry axis.~. For the approximate solution we use the equation for absence of vorticity n~O pOUO1]O 2 ~ (duo/dx) 2 ~--Uo~(k--i)(k+i) and. Then g~ C ]. This fact also means that the roundoff errors have practically no effect on the calculation results. it is natural to seek the solution of (2.I ]h/(a-t) 2 u~ I 2 ( k+ f u~- k -.i -- _ rk+t + 2uv-'~x -[- k. solutions Below we present which will later be com- solution.~ Fig. dx LV2 . although it is not difficult to obtain analogous results for the plane case as well. 0 . The system for determining the first and second coefficients of the series (2. A detailed study of the flow in the vicinity of the rectilinear sonic line has been made by Ovsyannikov [14]. the region of convergence of the series (2. (2. ul(y ) ~.1) is bounded form f = ~ ] f ~ ( x ) $ . w Some asymptotic some asymptotic pared with the numerical makes it p o s s i b l e convergence Solutions. so that the pivotal method was not suitable for calculating the flow in the subsonic region. v2(y) ~. as comparison with the numerical solution shows (see the following section). This comparison not only to establish the region of o f t h e a s y m p t o t i c s o l u t i o n s b u t a l s o to e v a l - uate the accuracy of the numerical a definite range.5) and (2. we represent the sought parameters in the /tO ~ dx ~ and the continuity equation [k.3) In solving the inverse problem we set u = u0(x) for ~ = 0 on the symmetry axis.(y)x+u~(y)x2§ vdy)x+v2(y)xZ+v~(~)x~+ +v~(g)x~-}-.7) After substituting (2.5 9 10-~agreed with one another. in solution. Vn' Un. J d~o { P'=--2[i--Uo~(k'i)/(k+t)] fg \V(~-k)Th u3Z) 2 l--uo2(k--i)/(k-t-1)-~z vo: Xl 3.4) were used to calculate the streamline lying close to the axis. we obtain a system of equations for determining the functions Pn. _ U2 k--i " 2 lt2 "lay . should coincide with the approximate solution which. Let us construct the solution of the system (1. Here the gasdynamic equations are considered in cylindrical coordinates for axisymmetric flow. 2 .+ (u2 + vz) dyo dy~ ui---Jc'uo ~ dx dx Yo(p~uo+poa~)-]-4y~poao~O.. However.01% the results of the calculation with steps 0. equal to 0. . th(x) similarly from (2. Yn. 10 -s.1 9 10 -~.0.= -. v / ' ~ . 5 .05% dot k . To do this we calculated the difference in the momentum of a fixed section at the left end and several running sections. Therefore this solution cannot be used for the calculation of regions quite far from the symmetry axis. Analytic expressions were obtained for these quantities.- j 0.1) in the vicinity of the symmetry axis we can calculate the flow in both the supersonic and subsonic regions. p.2 9 10 "z.0. In the vicinity of a rectilinear sonic line. Formulas (2.0. 0.1) and equate coefficients of like powers of in the left and right sides of the equations.. (2. In view of the flow symmetry relative to the axis. An integral verification was also made of the calculation accuracy. The momentum difference was compared with the integral calculated along the streamline of the pressure forces projected on the x axis.7) into (2.6) and equating coefficients of zero and first powers of x in the left and right sides of these equations. Expansion into a series in x in the vicinity of the rectilinear sonic line.1 2 2 uo z -.1 " 10 "2. a series of calculations was made with variable steps A~.5) and (2. In this subsection we consider only axisymmetric flow.1) was initiated. y / ~ ' .+ "' = 2. The results presented in the present subsection repeat certain results of Ovsyannikov. 2. on which u = 1 and v = 0. it is not difficult to find that vi(y) ~. but are not presented here because of the length of the formulas..4:-v~(g)x ~ + . 0. from which the numerical integration of the system (1.4) 2.1) in the vicinity of the symmetry axis in the form of series. with the aid of series (2. From (2. (2. In the process of the calculations it was found that in order to ensure stability of the pivotal method in the subsonic region it is necessary to increase the difference grid step so much that the approximation errors become excessively large. 0. The difference of these quantities in the two calculations was no more than 0. u. If we substitute these expansions into (1.6) l+u. dvo k--~--yop~.oL o---..4 9 9 10 -z.. On.(YtP~ -[-'2goP~). 10 -2 .1) is the following: ' We can find vl(x). yl(x). i~ is particularly small in the transonic flow region.= Po= [ k-~2t k--. and 0.6) in the form v= ~3~ (2.2) (2.8 MEKHANIKA ZHIDKOSTI I GAZA In order to determine the maximal step A~Oin the three-point scheme with variable step on the layer.I k x[ L 2. just as for y2(x)..2". (2. vi. In a certain sense this solution will be a generalization of the Meyer solution.2 9 10 "~. 3 An attempt was made to calculate the flow in the subsonic portion using the matrix pivotal method [11].0 l/k po:Po-.3).2) we find go /I (2. dx =0.5) Ou l Og = Ov / Ox u= 2. The system of equations ( 2 . i .12) with the solution which is obtained in the n u m e r i c a l integration of (2. .8) and equating coefficients of l i k e powers of y . for y = O . vs(y). o In the a x i s y m m e t r i c case.(y . . In the v i c i n i t y of the point y = Y0. . b e c o m i n g positive.10) etc.t. A= 10. t do d~-~-----.11) and (2. 1 0 ) cannot be i n t e g r a t e d in closed form for a2 e 0 in the a x i s y m m e t r i c case. In the plane case.5 Let the v e l o c i t y distribution on the s y m m e t r y axis in the v i c i n i t y of the r e c t i l i n e a r sonic line on the subsonic side for x > 0 be g i v e n in the form a = i + a~x~ + a~x ~ + a~x~ + .~. aa~aa 2. k+l d~az 5yo (2.Y o ) + d~. 12-~105 u 1 02a --~+ Ox 2 t Ou +]----=0.9) 5o5. for the system (2.yo) v=Re: ['X ---- u0(x--iysdnt) sintdt JI (y . = Ro-. For an arbitrary v e l o c i t y distribution along the axis.yo) 4 l n ( y ~ Yo)+ C~(y -. dy 2 y dy du~ -~ d 0 ~ . . when the second of Eqs.. 2).12) that the point x = 0.O ! 0.( 2 . we can find the coefficients d i.x '~ + . the solution of the inverse problem has the form u = Re [~o (~ + iy)].~ 0.! 0. the point x = 0.. Then in the r e a l p l a n e the solution of the inverse p r o b l e m has the form Fig. dg dZa~ l da~ ----+------20(k+t)uauz~O. and at this point there is a Prandtl-Meyer flow [14].d~ -~(~+t). u 5 In the l o c a l supersonic zone in the v i c i n i t y of the r e c t i l i n e a r sonic l i n e there arises a c o m p r e s s i o n . Ov Ou Oy -- Ox 02a . b e g i n n i n g from the point y = y0 of the r e c t i l i n e a r sonic l i n e and e x t e n d i n g u p s t r e a m from the r e c t i l i n e a r sonic line toward the subsonic zone [14]. . e .Y0. dr5 --=6(k+l)uau~+3(k+t)(2k--l)u~+6v~----..14) .+-~-=0. 8 ) . The flow of the i n c o m p r e s s i b l e fluid was c a l c u l a t e d for the a x i s y m m e t r i c case and a v e l o c i t y distribution along the axis g i v e n in the form ao = a ~ + ( t .13) to the Darboux equation.12) ] (2.8).8 ~ Solution of the inverse problem of Laval n o z z l e theory for an incompressible fluid. It follows from (2. . For this v e l o c i t y distribution in the p l a n e x = 0 for y = ~ the v e l o c i t y u goes to infinity.a ~ ) / (t + Ax2). O* v = Re [iao(z + iy)]. For low v e l o c i t i e s we can consider the flow in the subsonic portion of the n o z z l e as the flow of an incompressible fluid. 4 For some v a l u e y : y0 the function 89 vanishes and then changes sign.FLUID DYNAMICS 9 These conditions are the known Gortler conditions [15].d~(y . the solution of the second e q u a t i o n of (2.3 y U. The c h a n g e of sign of the function 89 m e a n s that in addition to the r e c t i l i n e a r sonic l i n e at the point where uz(y ) = 0 there arises stilI another sonic Iine. and for some y = Y0 the function 89 goes to infinity.yo) z "q-'ds(y . .tl) 02 Then the boundary conditions wilI be: for the system (2. whose solution after s i m p l e transformations m a y be reduced to the form [16] t~ I u 0. 7] do (y .+ \ F<+-+y +i~.1 and A = 10 we h a v e Y0 = 0. the corner point for the flow of an i n c o m p r e s s i b l e fluid arises at the poles of . Equating coefficients of the other powers of x. For the distribution of the v e l o c i t y along the axis u = uo~ + (1 .~ where F(x .d6(y . the functions u with odd numbers and the functioh~ v with e v e n numbers are e q u a l to zero.d G. Here the fl0w b e t w e e n the r e c t i l i n e a r and c u r v i l i n e a r sonic lines is supersonic (Fig..a.8) d~t2/dy=O..u ~ ) / ( 1 + Ax z) for u ~ = 0. 2) and t e r m i n a t e s with the r e c t i l i n e a r characteristic OA.w a v e fan w h i c h begins from the chara c t e r i s t i c OB (Fig. (2./+.yo) 3 "1- (2.yo) ~ + .13) Oy For the inverse problem the boundary conditions are v = 0 and u = = u0(x ) for y = 0.13) is the Laplace equation.1 and A = 10. . y = ~ w i l l be a corner. Therefore for the v e l o c i t y distribution on the axis.8) is sought in the form [14.yo) -b dz -}.5 0 0 0. g i v e n in the form u = u ~ + (1 .. In particular. we obtain du2 t d~z --=3~.s) (2.9) du~/dg~O.~+--~.12) into (2. } u~ = 0. for u~ = a~ g = O. The flow of an incompressible fluid is described by the equations for the system (%10) du~ / dg = O. where the function 89 goes to infinity.2 = a2 < Fig. ~ y (2.u ~ ) / ( 1 + Ax z) with u ~ = 0. this system was i n t e g r a t e d n u m e r i c a 1 i y and the values of 89 u~(y).. c o i n c i d i n g with the sonic line. .2 l. d~ -t. 0 for g = O . y = Y0 is a corner point. and vs(y).725. by the c h a n g e Yi = iy we reduce the second of (2. udy). Q. x-0. which are necessary and sufficient for the sonic l i n e to be r e c t i l i n e a r . vT(y) were c a l c u l a t e d . (2.iy sin t) is an arbitrary function. . r/. @ Substituting (2.00! . vs dy y The values of Y0 and Cz were d e t e r m i n e d from the condition of m a t c h i n g the a s y m p t o t i c solution (2. duz --=4c. (2.. This nature of the distribution has been observed experimentally and is obtained as a result of the calculations made in the present article.. For the flow of a compressible fluid it is not possible to establish the coordinates of the corner singular point so simply.uz + ut2).+ -. obtained in the numerical solution of the inverse problem with the use of a three-point scheme with a variable . . p. P2 + -. The boundary conditions for the inverse problem have the form UI y=v=O. Z2 6 F r o m (2. the pressure variation has the same nature as in the inverse problem. :C57. It was shown that the exact and approximate values of the coordinates of the sonic line and the line O = 0 differ by about 10-20%.. For a given 0 k we find from the boundary conditions and (2.e..5". Asymptotic solution in the vicinity of an infinitely remote point in the subsonic region. but 0 must not exceed the criticaI value for which the flow in the subsonic region does not depend on the backpressure. The approximate equation for the transonic flow potential was obtained in [3-6]. 7 w Calculation results. cp) T-" g' = ------ I.17) it follows... We note that the asymptotic behavior of the subsonic flow at infinity with application to the external problems was considered in [18]. y and v. Vn(0).~ (~ = o). ~(a~) P~-- n(i) ' u /o+ M~ z . Similar results are obtained in the present paper (see Section 3. Therefore the approximate transonic solution [3-6] is essentially not suitable for calculating flows in nozzles in cases of practical interest. It was also shown that the exact and approximate values of the flow parameters... The exact solution obtained with the Frankl' solution [9] and the approximate transonic solution were compared in [17] for the plane case in the vicinity of the nozzle center.r Fig.. etc.0 . and supersonic regions. ) This fact is quite obvious physically and indicates that we can vary only the single parameter Ok in the numerical solution of the direct problem with the use of the asymptotic expansion in the vicinity of an infinitely remote point. Substituting the series (2.1) i n the axisymmetric case for the direct and inverse problems. transonic.15) f = ~. pn(0). if Pl = 0. which may be solved in closed form.18)) are uniquely determined by the single parameter Ok. the inclination of the velocity at the sonic line and the Math number On the line 0 = 0. (2.+ Z (2. 2 X2 X3 (~ = ~ ) . 2. which is also the first term of the series constructed by Meyer near the center of the nozzle. The present section describes the results of the calculation of an axisymmetric flow in the subsonic.' po ~ const~ (2. lll~ Ig. q ( ~ ) = ~ [V~(k + t ) .t I l l u f_L+I__2+ This equation implies that i f f 1 < 0 (i. ~n+l In the direct problem ~0k is the value of the discharge on the nozzle contour and must be given as a function of the flow regime. This means that a positive pressure gradient at the point of joining of the cylindrical and converging . Let us construct the asymptotic solution in the vicinity of an infinitely remote point for the system (1. Pz = - 1 pi ~ Pi 0 "q# EO Fig... .~O For the inverse problem the unknown constants Po.tt. In particular. L e.kp~tt~ut.10 MEKHANIKA ZHIDKOSTI I GAZA the initial function continued into the complex plane. p = p~ kp| k p~ 2 (9. Pi.1) and equating coefficients of the same powers of x. We write the sought parameters in the form (2.% ( k .16) and (2..2) for the axisymmetric case.n u=w:-l--~'l--~'t-'. we obtain the system of equations for determining the functions yn(~).segments in the subsonic part of the nozzle must also be accompanied by a minimum pressure segment. in the asymptotic expansion of the function p do not depend on 0 (the dependence on 0 shows up in the following terms of the expansion. the pressure on all the streamlines can only diminish (Pl < 0. and the angle between the velocity direction and the x axis is s m a l l We derived the exact solution for this equation. e. as was assumed in deriving the approximate equation for the transonic flow potential. i. Solution of the approximate equations for the transonic flow potential.. then P1 < 0 on all the streamlines and the pressure on all the streamlines in the vicinity of the infinitely remote point diminishes in the direction of the gas motion. X A -- h Xn nl.. X U 2p~u~ p~hkM~'i/ P' ~ For the direct problem the boundary conditions have the form u=v=O '~.1~) 08 /~F 2.17) . if the subsonic part is converging).15) into (1. in the direction of motion of the gas in the vicinity of an infinitely remote point. In deriving this equation it was assumed that the magnitude of the gas speed is close to the speed of sound. as the results of the calculations show. we have vo = vi = O. n~0 where f is any of the functions p. and ih are determined with the aid of the boundary conditions from the known uoo and u~. (This statement is still valid if we consider all the subsequent expansion terms as well.16) t)~ = q ( ~ ) = 2% 1/o. ~/Z5 ~ u. ( * p~ = const.18) 'll. ).. In the direct problem the constants P0 and Pi are determined as follows. . second. for * = 0 .@-~-"l-. although. l.t ) ~ p / ( ~ pi = const. such a point always exists.005 Pt = -. P2 < 0. differ significantiy even for small distances from the axis. .. W e have 0. first.~ /. Pn~).4 ~.. Un(0). Only in nozzles in which the flow in the vicinity of the critical section differs little from onerdimensinnal flow can this solution be used for calculation.. that with reduction of x.(~)z-~. that the first three terms 2. which may be obtained by similar methods) and. u. We note that in the direct problem with a given equation for the nozzle contour the flow parameters (see 2. for example. Flows with rectilinear and curviIinear transition surfaces are analyzed separately. for example.75.3. Each of these streamlines may be selected as a nozzle contour. We see from this figure that the sonic line and the line 0 = 0 are inclined upstream from the nozzle center so that the sonic point is located downstream of the minimal nozzle section. To ensure a rectilinear sonic line it is necessary and sufficient that dw/dx -.06 (curve 1). the lines w = const. t .6 that for small values of x ~ 0. which shows the family of streamlines and the lines w = const. particularly in the velocity range from 1. 7). 3 . This same figure shows the values of w calculated from one-dimensional theory (curve 2) and the variation of w with length along the axis (curve 3).FLUID DYNAMICS 11 step on the layer. 1~Ax line shows that for x = 0.0 for w = 1 [15].w~)e-~/b + ( ~ -.t) The results of the calculation for the distribution (3. For the first velocity distribution numerical calculations were made of the flowfield and.2) (t -. 1. Figure 7 shows the values of w calculated from one-dimensional theory (curve 2) and the variation of w with distance along the axis (curve 3). a = 2. the flow parameters of the compressible and incompressible fluids differ little. 2. which leads to the appearance in this region of a positive pressure gradient.67. in addition. w~ ~ 0 .2 to 0.1) account was taken of three terms. 10 -z.08 (curve 1).5) with • = 0.05). We also see from Figs. 7. Although for low supersonic speeds the velocity w may with small error be calculated from one-dimensional theory.arctg(e :x -. Figure 7 also shows the variation of the velocity w with distance along the streamline with ~ = 0. 2. 2). on the streamlines which are remote from the axis of symmetry ~ e flow differs markedly from one-dimensionality. We see from this figure that the calculated flow differs markedly from the onedimensional flow. We see from this figure that the flow differs markedly from one-dimensional.6 with the solution obtained as a result of the expansion in a series in x in the vicinity of the rectilinear sonic -~ -2 -r 0 Fig. the sonic line will be eurvilinear. which. This means that a sharp variation of the contour form in the transonic region may lead to the formation of positive pressure gradients in this region. For large vaiues of x the radius of convergence of series (2. 3. For x ~ 2 the series (2.25. For this velocity distribution along the axis the flow in the supersonic region for w > 1.16 (h = 0. in the transonic region behind the transition line the variation of the velocity is not monotonic. and for x > 1 the results of the exact and approximate solutions practically coincide. for y > 0. Again in this case a local supersonic zone arises in the flowfleld. and the velocity components for the flow of the incompressible fluid were calculated using (2.88.. This figure shows the family of streamlines. When the velocity gradient in the center of the nozzle is nonzero. Thus. 4 .2-0. the occurrence of positive pressure gradients in this region (Fig. 1.1) with Woo = = 0. particularly for high velocities. Calculations of the flowfield were made for the following velocity distributions along the axis: w= i + . and the line 0 = 0 (light circles).00.2 is nearly one-dimensional. and on each layer -. the solution of the inverse problem for the incompressible fluid cannot be used for describing the flow of the compressible fluid. 2. in the region of low subsonic velocities. S. 1. Second.00. 1. The flow in this zone will be shoekfree. For large values of x.05 the expansion in a series in ~ diverges even for small values of y (this fact may also be established analytically).const the nodes of the difference grid were located at the points x = 0-0. the second sonic line has a common point with the rectilinear sonic line at infinity. We see from this figure that the vicinity of the critical section must be gently sloping to ensure a rectilinear sonic l i n e .5.25 a local supersonic zone and a second sonic line appear.50. locared in the plane x = 0. Z i 8 3.1.w~)/(1 + Ax a) along the axis. which may lead to separation of the boundary layer. 4 . w = i ~- (3. The calculation results are shown in Fig.70.76. and with the solution of the inverse problem for the incompressible fluid (curve 4).6 that for small values of x. however. because of the analyticity of the velocity distribution along file axis. 0. Laval nozzle with eurvilinear transition surface. the approximate solution obtained as a resutt of expanding the parameters in a series in r in the vicinity of the symmetry axis may be used for calculating the flow only for sufficiently low subsonic velocities. b = 0. In the vicinity of the rectilinear sonic line.7) diverge.02). where the nozzle contour is practically rectilinear and parallel to the axis.2)-(1. particularly for high velocity values. and the calculation of the subsonic and supersonic regions must be carried out jointly. there also arises a zone with a positive pressure gradient. the streamlines in the vicinity of the rectilinear sonic line are parallel to the axis over a segment which is still longer than in the preceding case.2. First.02). Here. 4 . We see from this figure that while the flow is nearly one-dimensional on the streamlines sufficiently close to the nozzle axis (~ = 0. for which the effects of compressibility are significant. In connection with this the following velocity distributions aiong the nozzle axis were given: w w~-t t~Ax 2 .00. For low velocities the flow is also almost one-dimensional.14). We see from Figs. the approximate solutions described in the preceding section were obtained. in expansions (2. The comparison shown in Figs. the calculation region on the side of the supersonic portion will not necessarily be bounded by a limiting characteristic of the second family. 0. 7 has two singularities. For the velocity distribution w = woo + (1 . 2. Between the second sonic line and the rectilinear sonic line the gas is initially accelerated and then decelerated (Fig. 0.2". with the solution obtained as a result of expanding the solution in a series in x in the vicinity of the rectilinear sonic line (curve 3). A comparison is made with the approximate solutions described in the preceding section.e-~=).2 are shown in Fig. the one-dimensional theory naturally does not m a r e it possible to study the fine details of the flow. I A = 10. Figures 4 . the sonic line (dark circles). The velocity distribution on the streamline ~ = 0. In the vicinity of the rectilinear sonic line the streamlines are practically parallel to the axis. in accordance . 0.40. when the gas motion velocity is not 1argo (w < 0. Calculations of the flowfield were made for various values of the adiabatic exponent.08 shown in Fig.1) (3.3).05 the agreement between the exact and approximate solutions is very good.15. Figure 3 shows also the variation of the velocity w = with length on the streamline ~ = 0. 3.1 ~ L~val n o z z l e with rectilinear transition surface.6 (h = 0. The calculation results show that flow parameters in the subsonic region depend little on k. The calculations were made using (1.6 show the comparison of the numerical solution (curve !) obtained in the present study with the approximate solution obtained as a result of expanding the flow parameters in a series in {0 in the vicinity of the symmetry axis (curve 2). The velocity distribution along the axis is given in the subsonic and supersonic regions.11) increases. Frankl.3. matem~ vol. 1947. k~--~-t vi Po21 After r e p l a c i n g the p a r a m e t e r s with index 1 on the l i n e ~ = ~0p by the p a r a m e t e r s with index 2. About the same values of (d)v/dx'y' were obtained as a result of the c a l culations. in contrast with the p r e c e d i n g case. The results of the c a l c u l a t i o n for the distribution (3. Figure 8 also shows the v a r i a t i o n of the v e l o c i t y with distance on the s t r e a m l i n e with @ = 0. 9. in this case the iota1 temperatures. fiz.5).06 are very close to the coordinates of the contour of the subsonic portion of the n o z z l e for which the v i c i n i t y of the c r i t i c a l section toward the subsonic part of the n o z z l e is m a d e in the form of a circular arc with radius R2 = 1. " Zh.I k'-. " J. i matem. Set. and a d i a b a t i c indices in the layers m a y be different. vychisl. It should be p a r t i c u l a r l y e m p h a s i z e d that as a result of the c a l c u l a t i o n s of the nozzles with a r e c t i l i n e a r sonic line. Uber Zweidimensionale Bewegung vorgSnge in einem Gas das mit Ubershallgesohwindigkeit str~mt.~ 0. 4. on the contour with ~ = 0./h2. a positive pressure g r a d i e n t arises in the c y l i n d r i c a l portion of the subsonic part of the nozzle.5. 182-190. Meyer. Th. Thus.poo)/(poov~) with distance along the nozzle. there is a positive pressure gradient whose m a g nitude increases with increase of the Iength of the c y l i n d r i c a l portion. "On the theory of Laval nozzles. a shock w a v e e m a n a t e s from this point. precedes a region in which the pressure diminishes. Forsehungs Heft.1) the pressure decreases m o n o t o n i c a l l y along the e n t i r e n o z z l e length. Fal'kovich. Lavrent'ev. vol. AN SSSR. Thus. Von Karman.. In a l l cases the c a l c u l a t i o n breaks down on some s t r e a m l i n e in c o n n e c t i o n with the f o r m a t i o n of corner points in the transonic region. vol. G. and Phys. w h i l e on the contour with ~ = 0.2. and also the v e l o c i t y c o m p o n e n t on the l i n e 0 = 0 and the v e l o c i t y c o m p o n e n t v on the l i n e w = 1. the flow differs s i g n i f i c a n t l y from o n e . i0.01. 5. We note that the coordinates of the n o z z l e contour with ~ = 0. whose m a x i m a l vaIue for the considered case is 0. The author wishes to thank G. i matem. no.. As a result of the c a l c u l a tions it was found that for y > 0.5 [17]. M.2 hours. fiz. vychisl.5 are shown in Fig. 5. Such a c a l c u l a t i o n may be m a d e within a framework of the i d e a l fluid without account for m i x i n g of the layers. 1945. The Flow of Air at High Speed past Curved Surfaces. "On the theory of Laval nozzles. the Iine 0 = 0. 6 . t h e r e a r e n o s h o c k . K. REFERENCES i. G. Even for dX/dx = 0 a corner point is formed in the flow (Section 2. A. 1965. 6. Ser. while b e g i n n i n g with this s t r e a m l i n e the a d i a b a t i c index k = k z and the t o t a l pressure is P0z. 6. 1381. the c a l c u l a t i o n using the proposed s c h e m e of the entire flowfield occupies only 1 0 . while the c a l c u l a t i o n of the supersonic part of the n o z z l e using the method of c h a r a c t e r i s t i c s requires 1 .1 5 minutes on the M-20 computer. 8. Ryzhov. According to these data the values of (dk/dx~ ~ are e q u a l to about 0 . 1956. 4. We see from these figures and from the c a l c u l a t i o n results that on a l l the streamlines. matem. VTs AN SSSR. The proposed method may be used for calculating equilibrium and nonequilibrium flows. In the supersonic region. it was found that for c e r t a i n forms of the transonic portion of the n o z z l e contour the occurrence in this area of regions with a positive pressure gradient is possible.06 (curve 1). I.05 for the v e l o c i t y distribution (3. corresponding to the v e l o c i t y distribution (3. Thus. t o t a l pressures. 1946. 3. 2. M. 1965. no. O.6r. P.1 the a p p r o x i m a t e solution gives a considerable error in c a l c u l a t i n g the v e l o c i t y components. F. to which is joined a c o n i c a l portion with angle 0in = 35 ~ which joins the c y l i n d r i c a l portion with radius R1 = r ~ (Fig.f r e e flows with (dk/dx-) > (dh/dx-):'. V. The coordinates of the line 0 = 0 and of the sonic Iine were c a l c u l a t e d suff i c i e n t l y a c c u r a t e l y to a v a l u e of y . " PMM.f r e e f l o w a corner point c a n n o t exist on a fluid s t r e a m line.5 the streamlines are p r a c t i c a l l y r e c t i l i n e a r and para!1e1 to the axis. Ya. were c a l c u l a t e d with the aid of the a p p r o x i m a t e transonic solution. In the plane case this value is about 0.. Favorskii. The coordinates of the sonic line. The m e t h o d proposed in the present paper permits c a l c u l a t i n g the flow in both the subsonic and supersonic portions of the n o z z l e with a high d e g r e e of a c c u r a c y . matem. on the s t r e a m l i n e ~ = ~p the foIIowing r e l a tions must hold: = (k2 + J ~J(h'--l)(i (j pz \ va ~ 2 / k2-. 7. I~ Taylor. 3. " Tr. 5. No. I./dx (x is the ratio of the x c o o r d i n a t e to the throat radius) exceeds some l i m i t i n g v a l u e (d)v/dx-)*. AN SSSR. i0.1.. The positive pressure g r a d ient in this region m a y be e l i m i n a t e d if we use contours which suffic i e n t l y g r a d u a l l y approach the g i v e n radius of the entry to the subsonic part of the nozzle. Th. Assume that up to some s t r e a m l i n e ~ = ~p the gas has the a d i a b a t i c index k = k 1 and the t o t a l pressure P01. 9. matem. S. "Calculating Laval nozzles. 26. "Study of transonic flows in Laval nozzles. The method suggested here is an i m p r o v e m e n t on the method of c h a r a c t e r istics in that a single s c h e m e m a y be used to c a l c u l a t e s i m u l t a n e o u s l y the subsonic and supersonic regions of the nozzle. We see from the results of the c a l c u l a t i o n s presented in this section that. w m = 0. = pz. 2 ) - (1. 8. beginning with the s t r e a m I i n e with ~ = 0. inostr. Chushkin. 7 for the a x i s y m m e t r i c case. Great Britain Aeronautieal Research Committee Reports and Memoranda. " Izv. no. Transonic Flow Theory [Russian translation]. I. " Zh. The v a l u e of (dk/dx)* was d e t e r m i n e d from the results of an exp e r i m e n t a l study m a d e by Sergienko of the v e l o c i t y distribution along MEKHANIKA ZHIDKOSTI I GAZA the n o z z l e axis which arises with flow about corner points. 6. just as for the nozzles with a curvilinear sonic line. In this connection.5. P.08 the m a x i m a l v a l u e of Cp = 0. Math. 20. vol. 1 / b = 3. shock waves were formed on those s t r e a m l i n e s on w_hieh the d e r i v a t i v e of the v e l o c i t y at the c e n t e r of the n o z z l e d>. 1908...1).0 . "The similarity law of transonic flow. On the s t r e a m l i n e # = e p both flows must have the same directions of the v e l o c i t i e s and the same values of the static pressure. and P. As a rule.12 with the analysis in Section 2. Figure 7 shows the v a r i a t i o n of the pressure coefficient Cp = 2(p .2) with woo = = 1. at the b e g i n n i n g of the c y l i n d r i c a l portion of the n o z z l e contour.9. "On the Cauchy problem for the Laplace equation. 72.iw. A. as a rule. S. " Izv. The supersonic part of the n o z z l e m a y also be c a l c u l a t e d by the m e t h o d of c h a r a c t e r i s t i c s i f the flow p a r a m e t e r s are known on some i n i t i a l characteristics in the transonic region. while for (dk/dx-) < (dk/d~)* the e x i s t e n c e of both shock-free flows and flows with shock waves is possible. it is c o n v e n i e n t to m a k e the c a l c u l a t i o n s of m u i t i l a y e r flows with different physical properties. "On the calculation of flow in a plane Laval nozzle. 5. . In the coordinates ~. 2 )hl/(kl--l) p.t k - k2-. lit. vol. In addition. 1965. Alikhashkin. Since in s h o c k . Favorskii. on the n o z z l e contour corresponding to the s t r e a m l i n e with ~ = 0. the c a l c u l a t i o n proceeds using ( 1 . no.d i m e n s i o n a l flow. Vladimirov for compiling the large number of programs and carrying out the calculations on the M-20 computer.08 it is 0. 8). Izd. 03.2). 1960. e x t e n d i n g d o w m t r e a m . and a positive pressure g r a d i e n t does not arise in either the transonic or subsonic regions. 1930. D. C h a r a c t e r i s t i c for this v e l o c i t y distribution will b e the fact that in the subsonic portion for x -> 1. x used in the present paper. Guderley. 19 November 1966 Moscow . M a t h e m a t i c a l Questions of Subsonic and T r a n s o n i c G a s d y n a m i c s [Russian t r a n s l a t i o n ] . B e r e z i n and N. F i z m a t g i z . Izd. 1938. 1959. L. no. 13 16. l i t . und Mech. 1961. S. 33. 1954. LKVVIA. N. G S r t t e r . Courant and D. Ye. L. Bers. 6. Methods of Mathe m a t i c a l P h y s i c s . 19. Kibel. 18. Ovsyannikov. Vol.. "Computational study of a s u p e r s o n i c j e t issuing f r o m an opening with plane walls. I. vol. A. 1950. Hilbert. V. "Study of gas flows with a n o r m a l shock line. P. Z. 17." P M T F [Journal of Applied Mechanics and Technical P h y s i c s ] . 13. F i z m a t g i z . Kamzolov and U.F LUID DYNAMICS 11. Math. N. 2. G . " T r . P a r t 2 [in Russian]. Gostekhizdat. Dahlquist. 1963. Math. Zhidkov. 2 [Russian translation]. V. and N. G. no. Roze. R. 1967. Zum ~bergang yon U n t e r s c h a l l zu 0 b e r s c h a l l g e s e h w i n d i g k e i t e n in Diisen. "Convergence and s t a b i l i t y f o r a hyperbolic d i f f e r e n c e equation with analytic values. 12. H . Pirumov. G. 1951. angew. 2. I. 14. V. T h e o r e t i c a l H y d r o m e c h a n i c s . i n o s t r . Kochin. 1 5 . no. Computational Methods [in Russian]. . Scand. 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