Level 2.Fluid mechanics CHAPTER 3. EXTERNAL INCOMPRESSIBLE FLOW In the previous chapter we examined internal flows and studied in detail the effects of wall friction on the behaviour of flow in pipes. This allowed us to quantify such things as the energy dissipation in a fluid and to derive, either theoretically or experimentally, the typical flow profiles in our pipes. In this chapter we shall adopt a similar approach but this time looking at external flows. External flow is the flow over a body which is submerged in an unbounded fluid. Think, for example, of the flow of air over an aerofoil. External flows share many similarities with the internal flows discussed in the last chapter. In our study of internal flow we noted that the fluid velocity at the pipe wall was always zero (the no slip condition). An examination of the velocity profiles derived previously, both for laminar and turbulent flow, show the velocity increasing from zero at the wall to a maximum value at the centre of the duct. This change in velocity is caused by the viscosity of the fluid. The arguments used when studying internal flows are applicable also to external flows but only in a region very close to the surface of the submerged body. Again the fluid sticks to the wall of the body but the velocity then gradually changes from zero at the wall to a free stream velocity U, which is the bulk velocity of the surrounding flow, not a centreline velocity. This region of velocity change (or gradient) is known as the boundary layer, and it has a thickness δ which is the distance from the wall (where the velocity is zero) to the free stream (where the velocity is U). Thus we could equally apply the concept of a boundary layer to 29 Level 2. Fluid mechanics both internal and external flows, however it is most commonly referred to in the study of external flows which we shall cover next. 3.1 The Boundary Layer Concept The concept of a boundary layer was first introduced by Prandtl in 1904. The concept was a significant breakthrough since, up until his work, theoretical ideas were still based on equations for inviscid flow. Of course inviscid flow predictions differed greatly from experimental measurements; the boundary layer concept provided the missing link between theory and practice. Prandtl showed that many viscous flow problems may be analysed by dividing the flow into two regions: a thin region adjacent to the solid boundary in which the effect of viscosity is important (the boundary layer) and an outer region in which the effect of viscosity is negligible and the fluid may be treated as inviscid. The boundary layer concept permitted the theoretical analysis of many viscous flow problems that previously had been considered impossible - and thus founded the modern era of fluid mechanics. The most straightforward example with which to introduce the concept of the boundary layer is the flow of a fluid over a stationary plate. U U U u(y) y u(y) x Laminar Transition 30 Turbulent that is the flow behaves as a series of ordered layers. slowing the fluid down near the wall. If the flat plate is stationary then the velocity of the fluid adjacent to the wall is zero. Eventually. However. There is no fixed distance at which a laminar boundary layer changes into a turbulent boundary layer.this is the no-slip condition. Under most conditions. in turn. The effect of viscosity sets up a velocity gradient in the fluid and hence a shear stress according to the equation τ =µ du dy As the flow moves along the plate the viscous effects in the fluid propagate further into the bulk fluid and the boundary layer thickness grows. Eventually the flow in the boundary layer becomes fully turbulent and is characterised by a random. Fluid mechanics The flow over the plate thus separates into three distinct regions: As the fluid impinges on the plate the fluid particles in direct contact with the wall assume the same velocity as the wall itself . a point is reached at which instabilities in the boundary layer cause a breakdown in the laminar structure of the flow and turbulent mixing begins to take hold. flowing over one another. retard the adjacent particles in the bulk fluid above . This stationary element of fluid will.this is viscosity in action. in fact under some conditions the laminar boundary layer may disappear extremely quickly and be virtually undetectable. chaotic mixing of the fluid. In this region of transition from fully laminar to fully turbulent flow.we may still assume that the outer flow behaves like an inviscid fluid. the boundary layer is initially laminar. The onset 31 . these turbulent effects are limited only to the region of the boundary layer . a mixture of both laminar and turbulent flow is present and the boundary layer thickness grows rapidly. or laminae.Level 2. Level 2. pressure gradients greatly affect the nature of the boundary layer. Generally speaking.2 The effect of a pressure gradient The arguments introduced in the previous section assumed a zero pressure gradient. To introduce the concept of a pressure gradient we shall look at a simple duct of varying cross-section: 32 . the Reynolds number is significant when characterising boundary layer flow. values of Reδ ≈ 3900 (the Reynolds number here is based on the boundary layer thickness δ) are common for the transition point. there is no unique value for Re at which transition from laminar to turbulent flow will occur . For experiments on flat plates. however for most practical engineering problems. Fluid mechanics of transition depends on the ratio between viscous and inertial forces in the boundary layer and so. However this is not normally the case. such as flow over a bluff body or an aerofoil. unsurprisingly. 3. So far we have looked at an ideal flat plate with a zero pressure gradient.among the factors that effect the onset of turbulence are: (i) Pressure gradient (ii) Surface roughness (iii) Heat transfer (iv) Body forces (v) Free stream disturbances One of the most important factors above is the effect of the pressure gradient. and under a zero pressure gradient. However this value should be treated only as a very rough guide. Therefore for a zero pressure gradient there is no danger of flow reversal and subsequent separation. ∂p ∂x < 0 : Favourable Pressure Gradient In this region. This is known as a favourable pressure gradient.Level 2.just like in the favourable pressure gradient above. Region 2. Under these circumstances it can be shown that the velocity of the fluid in the vicinity of the solid surface cannot be brought to zero . ∂p ∂x = 0 We have already examined a simple flat plate for which the pressure gradient is zero. i. the pressure behind a particle lying in the boundary layer is greater than the pressure opposing the motion. in effect.e. Fluid mechanics Region 1 Region 2 Region 3 ∂p ∂p ∂p <0 ∂x =0 >0 ∂x ∂x U U U U Backflow δ Region 1. 33 . "sliding down a pressure hill" without any danger of being slowed to zero velocity at a point away from the wall. the pressure gradient is aiding the movement of the particle and the particle is. "climb a pressure hill". Fluid mechanics Region 3. Separation occurs only when ∂p ∂x > 0 . the flow separates from the sphere and a viscous. It is this turbulent wake which contributes to the majority of the drag force on the sphere and so the point of separation is vitally important in determining the behaviour of the flow around the sphere. Just downstream from the point of separation the flow nearest to the wall is reversed. turbulent. the pressure gradient retards fluid flow in the boundary layer and the fluid must. wake is formed. Eventually. On bringing a fluid to rest we reach what is known as the point of separation. a favourable pressure gradient exists over the front section and so the boundary layer "sticks" to the wall of the sphere. Similar arguments may be applied to any body immersed in an external flow field and we shall look at this further in the study of drag forces towards the end of this chapter. the low energy fluid in the separated region is forced back upstream by the increased pressure downstream. However separation does not always necessarily occur if ∂p ∂x > 0 . We may now apply these concepts to a simple problem such as the flow over a sphere: Wake Boundary Layer Point of separation For the sphere. Here. effectively. The adverse pressure gradient may therefore bring to rest a fluid in the vicinity of the wall. 34 . ∂p ∂x > 0 : Adverse Pressure Gradient In region 3 we encounter what is known as an adverse pressure gradient. this depends on local flow conditions. As we progress over the top of the sphere the pressure gradient changes to an adverse pressure gradient.Level 2. Here. 35 .e.e. y =δ when u ( y ) = 0.99 times the free stream velocity.why choose 99%? . these are (i) The boundary layer disturbance thickness δ (ii) The boundary layer displacement thickness δ ∗ (iii) The boundary layer momentum thickness θ Boundary layer disturbance thickness δ The is also known simply as the boundary layer thickness (and was the thickness alluded to in the previous section) and is defined as the distance from the plate to a point in the fluid where the velocity is 0. Typically there are three different definitions for boundary layer thickness in common use. In this section we shall quantify exactly what we mean by the thickness of a boundary layer. Fluid mechanics 3.Level 2. the concept of the boundary layer displacement thickness was introduced. a defined beginning and end to the boundary layer was assumed (if you like. U u(y)=0.99U Obviously this definition is rather arbitrary .99U y δ u(y) x i. the viscous area of influence of the wall). i.also this boundary layer thickness is very difficult to measure accurately.3 Boundary Layer Thickness and Profile The previous discussion developed the concept of a boundary layer and. in the y direction. To remove these problems. and so ∞ 0 δ ∗ = ∫ 1 − δ u u dy ≈ ∫ 1 − dy 0 U U The displacement thickness represents the additional outward displacement of the streamlines caused by the viscous effects of the plate. that which would be present if the flow was inviscid.Level 2. 36 . Fluid mechanics Boundary layer displacement thickness δ ∗ This measure of boundary layer thickness seeks to compare the boundary layer profile with the profile of an equivalent flow but with no viscosity present. Normally this is chosen to be δ. one should truncate this integral at a suitable limit. what is the effect of the viscosity of the fluid. over and above. And so if we equate mass flow rates we find that ∞ ρδ ∗U = ∫ ρ (U − u )dy 0 For incompressible flow ∞ 0 δ = ∫ 1 − ∗ u dy U Practically. i. U U Equal area δ u(y) U-u(y) ∗ Inviscid (uniform) flow Boundary layer flow Thus the boundary layer displacement thickness is a balance between deficit in the mass flow rate of the actual boundary layer flow and that of an equivalent inviscid flow.e. when studying drag. They are however far easier to evaluate accurately from experimental data when compared to the boundary layer thickness δ. we shall focus on empirical methods. We shall look briefly in the next section at how we compute velocity profiles within the boundary layer region. the theoretical analysis of drag is however rather complex and beyond the scope of this course. 37 .3. Analysis may also be extended to turbulent boundary layers by use of the velocity power law discussed in Section 2. exact solutions exist for laminar boundary layer drag. Therefore δ ∗ and θ are far more commonly used when studying boundary layer properties. The concept of displacement and momentum thickness may be extended further. with the aid of the continuity and momentum equations. Thus the momentum thickness θ is defined as the thickness of a layer of fluid. Fluid mechanics Boundary layer momentum thickness θ The boundary layer momentum thickness is often used when determining the drag on an object. As in the case of internal flows. Flow retardation (caused by the object) within the boundary layer results in a reduction in momentum within the boundary layer when compared to inviscid flow. for which the momentum is equal to the deficit of momentum through the boundary layer. This gives ∞ ρu θ = ∫ ρu (U − u )dy 2 0 For incompressible flow u u 1 − dy 0U U ∞ θ =∫ The boundary layer displacement and momentum thickness may appear rather abstract. to the study of drag on an object.2. Instead.Level 2. of velocity U. For adverse pressure gradients. parabolic. similar to that seen for internal flow.1 Velocity profiles in the boundary layer In general. This assumes that the theory for two parallel plates. The equation above does of course predict a linear velocity profile however this is not always a good approximation for actual laminar boundary layer profiles. one of which is moving with a velocity U is valid here also. or even sinusoidal velocity profiles have been suggested as a better fit for an external laminar boundary layer. For turbulent boundary layers the power law velocity relation discussed in Chapter 2 provides an adequate correlation: 1 y n = U δ u and so this equation is equally applicable to external boundary layers. the velocity profile may take on many different forms. for a zero pressure gradient. A comparison of (non-dimensional) velocity profiles for a laminar and turbulent boundary layer are shown. the shape of a velocity profile will depend upon the nature of the pressure gradient present. Instead cubic.3.Level 2. the most obvious velocity profile to choose is u= Uy u or δ U = y δ where δ is the boundary layer thickness. However for a zero pressure gradient. as you would expect. If we look first at laminar flow. below: 38 . Fluid mechanics 3. the typical velocity profile for both laminar and turbulent flow is. then based on the analysis in Chapter 2. as seen in the previous chapter. if we consider the (gauge) pressure and the shear stress distribution over an aerofoil we find. It is evident here that. For example. a viscous fluid will exert a net force on that body. Fluid mechanics 1 Laminar (linear) Laminar (parabolic) y δ Turbulent n=7 0 0 1 uU Note here that both linear and parabolic profiles are shown for laminar flow (the parabolic profile generally provides a better fit for actual data).4 Drag and Lift On flowing over a solid body. 3.Level 2. the turbulent profile is much fuller (more blunt) than the laminar profile. This force is made up of the wall shear stresses τ w and the pressure force p (it is this pressure force that we indirectly discussed when looking at pressure gradients). typically. the following type of behaviour: p<0 Pressure U p>0 Shear Stress U 39 . even for very simple objects. The drag force for any body immersed in an incompressible flow is given by 1 FD = C D ρU 2 A 2 where CD is known as the drag coefficient and is obtained from experimental measurements.4. Of course we may write also that CD = FD 1 2 40 ρU 2 A . Fluid mechanics The total drag force acting on the aerofoil is a summation of the pressure force and the shear stress. Therefore. 3. Thus calculating pressure and shear forces theoretically. We have already seen that in the presence of an adverse pressure gradient (present here over most of the upper surface of the aerofoil) flow may separate and very complex flow patterns may develop. The magnitude of the drag force tends to depend strongly on the shape of the object. is usually prohibitive. Of course. for most shapes of interest we resort to experimental measurements and to do this we use lift and drag coefficients.1 Drag The drag force is the component of the force acting on a body which acts parallel to the direction of motion: U Drag Force FD The drag force consists of pressure and shear forces acting on the surface of the object.Level 2. to fully compute both sets of forces we should need to know in detail both the local pressure and the shear stress distribution at all points around the aerofoil. 17 Re > 103 4 π a/d ≤ 0.9 2.0 3.1 0.15 Re > 104 a D Drag Coefficient A = bD Sold Hemisphere A= Thin Disk A= π CD 1. This is because the separation point is fixed by the geometry of the sharp edge 41 .0 2.1 Re = 2 ×10 4 A = bD 2.0 4 Note here that the objects listed above have sharp edges and in most cases the drag coefficients may be listed as constant for a range of Reynolds numbers.15 1. Of course it should be remembered that the nature of the flow over the immersed body will depend heavily on the Reynolds number and so CD is not a constant.Level 2. Fluid mechanics The number "1 2 " is inserted to form the familiar dynamic pressure.2 1.5 2.65 1.42 Re > 104 D2 1. Instead we should be careful to identify the correct value for CD for a given Reynolds number.17 0. Thus to find the drag force acting on a body we need only identify CD for that particular body. Some drag coefficient for a few objects are given below: Shape Semicircular shell Semicircular cylinder D D Reference Area A (b = length) D D Rectangle Reynolds Number A = bD 2.3 Re = 105 D2 1.9 2.5 0.6 1.3 1. At a value of Re ≈ 2 ×105 a sudden reduction in drag occurs. but the relative contribution of the pressure drag increases until at Re ≈ 1000 the pressure drag contributes approximately 95% of the total drag. the wake is laminar and the drag is predominantly composed of friction drag in a laminar boundary layer.for examples objects such as a sphere or a cylinder. This is caused by the point of separation suddenly moving downstream of the sphere midsection and 42 . theoretically. This is not the case for blunt objects otherwise known as bluff bodies . Stokes was able to show.1 -1 10 1 3 10 10 5 10 Re At very low Reynolds numbers (Re<1) there is no fluid separation from the sphere. Fluid mechanics and so drag coefficients tend to be independent of the Reynolds number above values of Re of approximately 1000. The drag coefficient for flow over a sphere is shown below: 100 CD 10 1 0.Level 2. In the region 103 < Re < 2 ×105 the drag coefficient curve is relatively flat and the point of separation of the flow is just upstream of the sphere midsection. that in this region CD = 24 Re As Re is increased CD drops continuously. are more desirable on a bluff body because this delays the point of separation and thus reduces the pressure drag. A classic example of harnessing this effect is the "dimples" inserted on a golf ball. Of course one must be careful not to increase by too large a margin the overall surface area of the body since skin friction may begin to appear again and outweigh any reductions achieved for the pressure drag. Work in this area normally relies on delaying the onset of separation by streamlining the body shape. 43 .by virtue of their increased momentum when compared to laminar boundary layers . A turbulent boundary layer possess more momentum than does a laminar boundary layer and so is greater able to resist the adverse pressure gradient. The dimples are designed to speed up the onset of turbulence in the boundary layer. Thus the drag coefficient with a turbulent boundary layer may be up to 5 times less than that for a laminar boundary layer on either side of a critical Reynolds number (for this example Re critical ≈ 4 × 105 ). The turbulent boundary layer then resists separation and thus reduces the wake and minimises pressure drag.Level 2. Fluid mechanics so the size of the wake decreases. The objective of streamlining being to reduce the adverse pressure gradient that occurs behind the point of maximum thickness on the body. A detailed knowledge of the likely flow patterns over a body may therefore provide clues as how to best minimise drag. The latter point is an important one: turbulent boundary layers . The point of separation moves from just upstream to just downstream of the sphere midsection because the boundary layer changes abruptly from being laminar to being turbulent. An example of the coefficient of lift. and so the lift force changes with α. For an aerofoil the projected area Ap depends on the angle of attack α. plotted for an aerofoil. Fluid mechanics 3. 1 FL = C L ρU 2 Ap 2 or CL = FL 1 ρU 2 Ap 2 where Ap is the (planform) area projected at right angles to the flow.e. The lift force is defined in the same way as the drag force. one can clearly see where the wing stalls and the coefficient of lift drops off rapidly. 44 . i.4.2 Lift Lift is the component of force acting perpendicular to the fluid motion.Level 2. is given below: CL α Here. Fluid mechanics 3.83 N Torque T = 2 Fr = 2 × 71.2 Nm P = 2 Fu Power P = ×2 × 71. The mixer is rotated at 60 rpm in a large vessel containing a brine solution (specific gravity σ = 1.1).17 ] Now the mixer will be working against the drag force imparted by each circular disk. and (ii) the power required to drive the mixer. If the drag on the rods and the motion induced in the liquid may be neglected.6 m [Take the drag coefficient for each disk to be C D = 1.Level 2.77 = 541.77 × π × 2 4 FD = 71.83 × 3.12 FD = ×1.6 m 0.6 × 60 × 2 × π = 3.6 W 45 . ω=60 rpm d=100 mm 0.1×1000 × 3.83 × 0.6 = 86. We saw earlier that 1 FD = C D ρU 2 A 2 Now U = rω therefore U = 0. A rotating mixer is constructed from two circular disks.17 ×1.5 Example Problems Q1. estimate (i) The maximum torque.77 m/s 60 1 2 0. e. 46 . A dragster weighing 700 kg achieves a speed of 380 km/hr after travelling 0. Immediately after passing through the timing lights the driver opens the drag chute.Level 2. [Take ρ air = 1.5 km. the chute dominates the drag) find the time required for the machine to decelerate to 150 km/hr.e.2 kg/m 3 ] If we draw a free body diagram of the car: FD u y x ui = 380 km//hr u f = 150 km//hr Applying Newton's second law: − FD = ma = m du dt now 1 FD = C D ρu 2 A 2 and so 1 du − CD ρu 2 A = m 2 dt This expression must now be integrated. If the air and the rolling resistance of the car may be neglected (i.2 . which may be accomplished after first separating the variables. The area of the chute is 2.5 m2 and it has a constant drag coefficient of C D = 1. i. Fluid mechanics Q2. 2 × 2. Now for a linear laminar velocity profile we have u = U y δ and the momentum thickness was defined as u u θ = ∫ 1 − dy ∞ 0 U 47 U .Level 2.65 seconds Q3. The laminar and turbulent velocity profiles derived in Chapter 2 may be applied also to the study of the boundary layer profiles discussed here in Chapter 3.2 ×1.5 150. (i) Taking a simple linear velocity profile for laminar flow.000 2 × 700 t = 5.000 380. evaluate the ratio θ δ. Fluid mechanics uf T 1 1 − C D ρA∫ dt = m ∫ du 2 o ui u 2 thus uf 1 − C D ρAt = m − 2 u ui 1 − 1 C D ρAt = − 2m t= t= 1 + uf 1 ui 2m 1 1 − ρCD A u f ui 1 1 − × 3600 1. 097 δ A smaller value for θ δ suggests a "flatter" more uniform velocity profile.1667 δ (ii) Evaluate θ δ for a 1 7 power law profile (i.e. 48 .Level 2. n=7) used to represent turbulent flow: 1 u y 7 = U δ 1 2 δ 17 y 7 y 7 y y 7 θ = ∫ 1 − dy = ∫ 1 − 2 dy 0 δ 0 δ 7 δ 7 δ 1 δ δ 8 9 7 y 87 7 y 97 7 7 7δ 7 7δ 7 = θ = 1 − − = δ − 2 1 2 8 δ 7 9 δ 7 8δ 7 9δ 7 8 9 0 θ = 0. we have δ y2 y y y3 δ = θ = ∫ 1 − dy = − 0δ δ 2δ 2δ 2 0 6 δ thus θ = 0. Fluid mechanics terminating the integral at δ.