Compressible Flow over a Flat Plate

The properties of the incompressible, laminar, flat-plate boundary layer were devel­oped in Section 18.2. These results hold at low Mach numbers where the density is essentially constant through the boundary layer. However, what happens to these properties at high Mach numbers where the density becomes a variable; that is, what are the compressibility effects? The purpose of the present section is to outline briefly the effects of compressibility on both the derivations and the final results for laminar flow over a flat plate. We do not intend to present much detail; rather, we exam­ine some of the salient aspects which distinguish compressible from incompressible boundary layers.

The compressible boundary-layer equations were derived in Section 17.3, and were presented as Equations (17.28) to (17.31). For flow over a flat plate, where dpeldx = 0, these equations become

Compare these equations with those for the incompressible case given by Equations (18.1) to (18.3). Note that, for a compressible boundary layer, (1) the energy equation must be included, (2) the density is treated as a variable, and (3) in general, p and к are functions of temperature and hence also must be treated as variables. As a result, the system of equations for the compressible case, Equations (18.31) to (18.34), is more complex than for the incompressible case, Equations (18.1) to (18.3).

It is sometimes convenient to deal with total enthalpy, ho = h + V2/2, as the dependent variable in the energy equation, rather than the static enthalpy as given in Equation (18.34). Note that, consistent with the boundary-layer approximation, where v is small, ho — h + V2/2 = h + (n + v2)/2 ~ h + и2/2. To obtain the energy equation in terms of ho, multiply Equation (18.32) by u, and add to Equation (18.34), as follows. From Equation (18.32) multiplied by u,

Adding Equation (18.35) to (18.34), we obtain

Recall that for a calorically perfect gas, dh = cpdT hence,

Substituting Equations (18.39) and (18.40) into (18.38), we obtain

dh0 dh0 d /X dh0

pu——- h pv—- = —————–

dx dy dy Pr dy

which is an alternate form of the boundary-layer energy equation. In this equation, Pr is the local Prandtl number, which, in general, is a function of T and hence varies throughout the boundary layer.

For the laminar, compressible flow over a flat plate, the system of governing equations can now be considered to be Equations (18.31) to (18.33) and (18.41). These are nonlinear partial differential equations. As in the incompressible case, let us seek a self-similar solution; however, the transformed independent variables must be defined differently.

£ — PeP^e^eX

The dependent variables are transformed as follows:

и._

/ = — (which is consistent with defining stream function = v2|/)

ue

_ hp
8 ~ iho)e

The mechanics of the transformation using the chain rule are similar to that described in Section 18.2. Hence, without detailing the precise steps (which are left for your

solution to Equations (18.42) and (18.43) is the shooting technique described in Section 16.4. The approach here is directly analogous to that used for the solution of compressible Couette flow discussed in Section 16.4. Since Equation (18.42) is third order, we need three boundary conditions at r] = 0. We have only two, namely, / = /’ = 0. Therefore, assume a value for /"(0), and iterate until the boundary condition at the boundary-layer edge, /’ = 1, is matched. Similarly, Equation (18.43) is a second-order equation. It requires two boundary conditions at the wall in order to integrate numerically across the boundary layer; we have only one, namely, g(0) = gw. Thus, assume g'(0), and integrate Equation (18.43). Iterate until the outer boundary condition is satisfied; that is, g = 1. Since Equation (18.42) is coupled to Equation (18.43), that is, since pji in Equation (18.42) requires a knowledge of the enthalpy (or temperature) profile across the boundary layer, the entire process must be repeated again. This is directly analogous to the two minor iterations nested within the major iteration that was described in the discussion of the shooting method in Section 16.4. The approach here is virtually the same philosophy as described in Section 16.4, which should be reviewed at this stage. Therefore, no further details will be given here.

Typical solutions of Equations (18.42) and (18.43) for the velocity and temper­ature profiles through a compressible boundary layer on a flat plate are shown in Figures 18.4-18.7, obtained from van Driest (Reference 79). Figures 18.4 and 18.5 contain results for an insulated flat plate (zero-heat transfer) using Sutherland’s law for [jl, and assuming a constant Pr = 0.75. The velocity profiles are shown in Fig­ure 18.4 for different Mach numbers ranging from 0 (incompressible flow) to the large hypersonic value of 20. Note that at a given x station at a given Re*, the boundary – layer thickness increases markedly as Me is increased to hypersonic values. This clearly demonstrates one of the most important aspects of compressible boundary layers, namely, that the boundary-layer thickness becomes large at large Mach num­bers. Figure 18.5 illustrates the temperature profiles for the same case as Figure 18.4. Note the obvious physical trend that, as Me increases to large hypersonic values, the temperatures increase markedly. Also note in Figure 18.5 that at the wall (у = 0), (ЗT/3y)w = 0, as it should be for an insulated surface (qw = 0). Figures 18.6 and 18.7 also contain results by van Driest, but now for the case of heat transfer to the wall. Such a case is called a “cold wall” case, because Tw < Taw. (The opposite case would be a “hot wall,” where heat is transferred from the wall into the flow; in this case, Tw > Taw.) For the results shown in Figures 18.6 and 18.7, Tw/Te = 0.25 and Pr = 0.75 = constant. Figure 18.6 shows velocity profiles for various different values of Me, again demonstrating the rapid growth in boundary layer thickness with increasing Me. In addition, the effect of a cold wall on the boundary layer thickness can be seen by comparing Figures 18.4 and 18.6. For example, consider the case of Me = 20 in both figures. For the insulated wall at Mach 20 (Figure 18.4), the bound­ary layer thickness reaches out beyond a value of (y/x)^Rex = 60, whereas for the cold wall at Mach 20 (Figure 18.6), the boundary-layer thickness is slightly above (y/x)«/Re^ = 30. This illustrates the general fact that the effect of a cold wall is to reduce the boundary-layer thickness. This trend is easily explainable on a physical basis when we examine Figure 18.7, which illustrates the temperature profiles through

0 0.2 0.4 0.6 0.8 1.0 «/« Figure 1 8.4 Velocity profiles in a compressible laminar boundary layer over an insulated flat plate (Source: van Driest, Reference 79.)

the boundary layer for the cold-wall case. Comparing Figures 18.5 and 18.7, we note that, as expected, the temperature levels in the cold-wall case are considerably lower than in the insulated case. In turn, because the pressure is the same in both cases, we have from the equation of state p = pRT, that the density in the cold-wall case is

T/Te Figure 18.5 Temperature profiles in a compressible laminar boundary layer over an insulated flat plate. (Source: van Driest Reference 79.)

much higher. If the density is higher, the mass flow within the boundary layer can be accommodated within a smaller boundary-layer thickness; hence, the effect of a cold wall is to thin the boundary layer. Also note in Figure 18.7 that, starting at the outer edge of the boundary layer and going toward the wall, the temperature first increases, reaches a peak somewhere within the boundary layer, and then decreases to its pre­scribed cold-wall value of Tw. The peak temperature inside the boundary layer is an indication of the amount of viscous dissipation occurring within the boundary layer. Figure 18.7 clearly demonstrates the rapidly growing effect of this viscous dissipation as Me increases—yet another basic aspect of compressible boundary layers.

Carefully study the boundary-layer profiles shown in Figures 18.4-18.7. They are an example of the detailed results which emerge from a solution of Equations (18.42) and (18.43); indeed, these figures are graphical representations of Equations (18.43) and (18.42), with the results cast in the physical (x, y) space (rather than in terms of the transformed variable r). In turn, the surface values Cf and Сц can be obtained from the velocity and temperature gradients respectively at the wall as given by the velocity and temperature profiles evaluated at the wall. Recall from Equations (16.51) and (16.55) that Cf and Сц are defined as

rw

PeU2e

and where (3u/dy)w and (dT/dy)w are the values obtained from the velocity and temperature profiles, respectively, evaluated at the wall. In turn, the overall flat plate skin friction drag coefficient C/ can be obtained by integrating c/ over the plate via Equation (18.21).

Return to Equation (18.22) for the friction drag coefficient for incompressible flow. The analogous compressible result can be written as

In Equation (18.44), the function F is determined from the numerical solution. Sam­ple results are given in Figure 18.8, which shows that the product C/VRec decreases as Me increases. Moreover, the adiabatic wall is warmer than the wall in the case of Tw/Te = 1.0. Hence, Figure 18.8 demonstrates that a hot wall also reduces C/ VRec.

Return to Equation (18.23) for the thickness of the incompressible flat-plate boundary layer. The analogous result for compressible flow is

In Equation (18.45), the function G is obtained from the numerical solution. Sample results are given in Figure 18.9, which shows that the product (&^/ШГх/х) increases as Me increases. Everything else being equal, boundary layers are thicker at higher

Figure 1 8.8 Friction drag coefficient for laminar, compressible flow over a flat plate, illustrating the effect of Mach number and wall temperature. Pr = 0.75. (Calculations by E. R. van Driest, N АСА Tech. Note 2597.)

Figure 1 8-9 Boundary-layer thickness for laminar, compressible flow over a flat plate, illustrating the effect of Mach number and wall temperature. Pr = 0.75. (Calculations by E. R. van Driest, NАСА Tech. Note 2597.)

Mach numbers. This fact was stated earlier, as shown in Figures 18.4 and 18.6. Note also from Figure 18.9 that a hot wall thickens the boundary layer, as discussed earlier.

Recall our discussion of Couette flow in Chapter 16. There, we introduced the concept of the recovery factor r where

haw=he + r-^ [18.46]

This is a general concept, and can be applied to the boundary-layer solutions here. If we assume a constant Prandtl number for the compressible flat-plate flow, the numerical solution shows that

r = VPr

for the flat plate. Note that Equation (18.47) is analogous to the result given for Couette flow in that the recovery factor is a function of the Prandtl number only. However, for the flat plate, r = л/Рг, whereas for Couette flow, r = Pr.

Aerodynamic heating for the flat plate can be treated via Reynolds analogy. The Stanton number and skin friction coefficients are defined respectively as

(See our discussion of these coefficients in Chapter 16.) Our results for Couette flow proved that a relation existed between C# and c/—namely, Reynolds analogy, given by Equation (16.59) for Couette flow. Moreover, in this relation, the ratio Сн/cf was a function of the Prandtl number only. A directly analogous result holds for the compressible flat-plate flow. If we assume that the Prandtl number is constant, then for a flat plate, Reynolds analogy is, from the numerical solution,

In Equation (18.50), the local skin friction coefficient Cf which is given by Equa­tion (18.20) for the incompressible flat-plate case, becomes the following form for the compressible flat-plate flow:

In Equation (18.51), F is the same function as appears in Equation (18.44), and its variation with Me and Tw/Te is the same as shown in Figure 18.8.

The speed of sound is

«oo = yJyRTx = л/(1-4)(287)(288) = 340.2 m/s

The Mach number is Mx = 100/340.2 = 0.29. Hence, Mx is low enough to assume incompressible flow, and we can use Equation (18.22),

1.328

v’ReZ

Please note that for the flow over a flat plate at zero angle of attack, the freestream velocity and density, Vx and p0c, are the same as the velocity and density at the outer edge of the boundary layer, ue and pe. Hence, these quantities can be used interchangeably. Thus,

The total drag due to friction is generated by the shear stress acting on both the top and bottom of the plate. Since D f above is the friction drag on only one surface, we have

Total friction drag = D = 2D/ = 2(87.8) = (b) For Poo = 1000 m/s, we have

Clearly, the flow is compressible, and we have to use Equation (18.44), or more directly, Figure 18.8. From Figure 18.8, we have for Mx = Me = 2.94 and an adiabatic wall,

С/л/ReZ = 1.2

The friction drag on one surface is

Df = {pooVlSCf = і(1.22)(1 ООО)2(40)(1.03 x IQ-4) = 2513 N

Taking into account both the top and bottom surfaces,

Total friction drag = D = 2Df = 2(2513) =