Category When Is A Flow Compressible?

The Boundary-Layer Equations

For the remainder of this chapter, we consider two-dimensional, steady flow. The nondimensionalized form of the x-momentum equation (one of the Navier-Stokes equations) was developed in Section 15.6 and was given by Equation (15.29);

Let us now reduce Equation (15.29) to an approximate form which holds reasonably well within a boundary layer.

Consider the boundary layer along a flat plate of length c as sketched in Fig­ure 17.7. The basic assumption of boundary-layer theory is that a boundary layer is

very thin in comparison with the scale of the body; that is,

Consider the continuity equation for a steady, two-dimensional flow,

d(pu) d(pv) __ dx dy

In terms of the nondimensional variables defined in Section 15.6, Equation (17.16) becomes

d(p’u’) Э(рУ) dx1 dy’

Because u’ varies from 0 at the wall to 1 at the edge of the boundary layer, let us say that u’ is of the order of magnitude equal to 1, symbolized by 0(1). Similarly, p’ = 0(1). Also, since x varies from 0 to c, x’ = 0(1). However, since у varies from 0 to <5, where 8 <£ c, then y’ is of the smaller order of magnitude, denoted by у’ = О(8/с). Without loss of generality, we can assume that c is a unit length. Therefore, y’ = 0(8). Putting these orders of magnitude in Equation (17.17), we have

!в + Ш=0 [17.18]

0(1) 0(8)

Hence, from Equation (17.18), clearly v’ must be of an order of magnitude equal to 8; that is, v’ = 0(8). Now examine the order of magnitude of the terms in Equation (15.29). We have

Let us now introduce another assumption of boundary-layer theory, namely, the Reynolds number is large, indeed, large enough such that

Then, Equation (17.19) becomes

From Equation (17.26), we see that dp’/dy’ = 0(8) or smaller, assuming that yM^ = 0(1). Since 8 is very small, this implies that dp’/dy’ is very small. There­fore, from the у-momentum equation specialized to a boundary layer, we have

Equation (17.26a) is important; it states that at a given x station, the pressure is constant through the boundary layer in a direction normal to the surface. This implies that the pressure distribution at the outer edge of the boundary layer is impressed directly to the surface without change. Hence, throughout the boundary layer, p — P(x) = pe (x).

It is interesting to note that if is very large, as in the case of large hypersonic Mach numbers, then from Equation (17.26) dp’/dy’ does not have to be small. For example, if Mqo were large enough such that 1/yM^ = 0(8), then dp’/dy’ could be as large as 0(1), and Equation (17.26) would still be satisfied. Thus, for very large hypersonic Mach numbers, the assumption that p is constant in the normal direction through a boundary layer is not always valid.

Consider the general energy equation given by Equation (15.26). The nondi­mensional form of this equation for two-dimensional, steady flow is given in Equa­tion (15.33). Inserting e = h — p/p into this equation, subtracting the momentum equation multiplied by velocity, and performing an order-of-magnitude analysis sim­ilar to those above, we can obtain the boundary-layer energy equation as

The details are left to you.

In summary, by making the combined assumptions of 8 <SC c and Re > 1/82, the complete Navier-Stokes equations derived in Chapter 15 can be reduced to simpler forms which apply to a boundary layer. These boundary-layer equations are



Note that, as in the case of the Navier-Stokes equations, the boundary-layer equations are nonlinear. However, the boundary-layer equations are simpler, and therefore are more readily solved. Also, since p = p,. (x), the pressure gradient expressed as dp/dx in Equations (17.23) and (17.27) is reexpressed as dpe/dx in Equations (17.29) and

(17.31) . In the above equations, the unknowns are u, v, p, and h p is known from p = pe (x), and д and к are properties of the fluid which vary with temperature. To complete the system, we have

and h=cpT [17.33]

Hence, Equations (17.28), (17.29), and (17.31) to (17.33) are five equations for the five unknowns, и, n, p, T, and h.

The boundary conditions for the above equations are as follows:

At the wall: у = 0 и = 0 v = О T — Tw

At the boundary-layer edge: у —»• oo и —*■ ue T —*■ Te

Note that since the boundary-layer thickness is not known a priori, the boundary con­dition at the edge of the boundary layer is given at large y, essentially у approaching infinity.

Prandtl-Glauert Compressibility orrection

The aerodynamic theory for incompressible flow over thin airfoils at small angles of attack was presented in Chapter 4. For aircraft of the period 1903-1940, such theory

was adequate for predicting airfoil properties. However, with the rapid evolution of high-power reciprocating engines spurred by World war II, the velocities of military fighter planes began to push close to 450 mi/h. Then, with the advent of the first operational jet-propelled airplanes in 1944 (the German Me 262), flight velocities took a sudden spurt into the 550 mi/h range and faster. As a result, the incompressible flow theory of Chapter 4 was no longer applicable to such aircraft; rather, high-speed airfoil theory had to deal with compressible flow. Because a vast bulk of data and experience had been collected over the years in low-speed aerodynamics, and because there was no desire to totally discard such data, the natural approach to high-speed subsonic aerodynamics was to search for methods that would allow relatively simple corrections to existing incompressible flow results which would approximately take into account the effects of compressibility. Such methods are called compressibility corrections. The first, and most widely known of these corrections is the Prandtl – Glauert compressibility correction, to be derived in this section. The Prandtl-Glauert method is based on the linearized perturbation velocity potential equation given by Equation (11.18). Therefore, it is limited to thin airfoils at small angles of attack. Moreover, it is purely a subsonic theory and begins to give inappropriate results at values of Moo = 0.7 and above.

Consider the subsonic, compressible, inviscid flow over the airfoil sketched in Figure 11.3. The shape of the airfoil is given by у = f(x). Assume that the airfoil is thin and that the angle of attack is small; in such a case, the flow is reasonably approximated by Equation (11.18). Define

P2 – і – мі

so that Equation (11.18) can be written as

2д2ф д2ф

Let us transform the independent variables x and у into a new space, £ and ij, such that



%=x [11.36a]

П = РУ [11.36b]

Moreover, in this transformed space, consider a new velocity potential ф such that

</>(£, rj) = Рф(х, у) [11.36c]

To recast Equation (11.35) in terms of the transformed variables, recall the chain rule of partial differentiation; that is,

Зф 3ф 3§ 3ф dt]

дх 3£ dx dr] dx

dip d0 3§ 30 dr]

dy Э§ 3у dr] dy

From Equations (11.36a and b), we have

Differentiating Equation (11.41) with respect to x (again using the chain rule), we obtain

З20 1 d2ф

3×2 “ W

Differentiating Equation (11.42) with respect to y, we hnd that the result is

Substitute Equations (11.43) and (11.44) into (11.35):

Examine Equation (11.45)—it should look familiar. Indeed, Equation (11.45) is Laplace’s equation. Recall from Chapter 3 that Laplace’s equation is the governing relation for incompressible flow. Hence, starting with a subsonic compressible flow in physical (x, у) space where the flow is represented by ф(х, у) obtained from Equation (11.35), we have related this flow to an incompressible flow in transformed (£, і)) space, where the flow is represented by ф(%, і) ‘) obtained from Equation (11.45). The relation between ф and ф is given by Equation (11.36c).

Consider again the shape of the airfoil given in physical space by у = f(x). The shape of the airfoil in the transformed space is expressed as r) — q(^). Let us compare the two shapes. First, apply the approximate boundary condition, Equation

(11.34) , in physical space, noting that df/dx = tan в. We obtain

df _дф _ 1 дф _ дф 00 dx ду p ду 3 г)

Similarly, apply the flow-tangency condition in transformed space, which from Equa­tion (11.34) is

dq_ _ дф

°°d$ ~ dr)

Examine Equations (11.46) and (11.47) closely. Note that the right-hand sides of these two equations are identical. Thus, from the left-hand sides, we obtain

df_ _dq_

dx </£

Equation (11.48) implies that the shape of the airfoil in the transformed space is the same as in the physical space. Hence, the above transformation relates the compress­ible flow over an airfoil in (x, y) space to the incompressible flow in (£, >)) space over the same airfoil.

The above theory leads to an immensely practical result, as follows. Recall Equa­tion (11.32) for the linearized pressure coefficient. Inserting the above transformation into Equation (11.32), we obtain

2Й 2 3ф 2 1 3ф 2 1 3ф ^ ^ 49]

Question: What is the significance of 3$/3£ in Equation (11.49)? Recall that ф is the perturbation velocity potential for an incompressible flow in transformed space. Hence, from the definition of velocity potential, дф/ді; = й, where й is a perturbation

velocity for the incompressible flow. Hence, Equation (11.49) can be written as

From Equation (11.32), the expression f—2u/V00) is simply the linearized pressure coefficient for the incompressible flow. Denote this incompressible pressure coeffi­cient by Орд. Hence, Equation (11.50) gives

or recalling that f = ,/F— M^, we have


Equation (11.51) is called the Prandtl-Glauert rule; it states that, if we know the incompressible pressure distribution over an airfoil, then the compressible pressure distribution over the same airfoil can be obtained from Equation (11.51). Therefore, Equation (11.51) is truly a compressibility correction to incompressible data.

Consider the lift and moment coefficients for the airfoil. For an inviscid flow, the aerodynamic lift and moment on a body are simply integrals of the pressure distribution over the body, as described in Section 1.5. (If this is somewhat foggy in your mind, review Section 1.5 before progressing further.) In turn, the lift and moment coefficients are obtained from the integral of the pressure coefficient via Equations (1.15) to (1.19). Since Equation (11.51) relates the compressible and incompressible pressure coefficients, the same relation must therefore hold for lift and moment coefficients:

The Prandtl-Glauert rule, embodied in Equations (11.51) to (11.53), was histor­ically the first compressibility correction to be obtained. As early as 1922, Prandtl was using this result in his lectures at Gottingen, although without written proof. The derivation of Equations (11.51) to (11.53) was first formally published by the British aerodynamicist, Hermann Glauert, in 1928. Hence, the rule is named after both men. The Prandtl-Glauert rule was used exclusively until 1939, when an improved com­pressibility correction was developed. Because of their simplicity, Equations (11.51) to (11.53) are still used today for initial estimates of compressibility effects.

Recall that the results of Chapters 3 and 4 proved that inviscid, incompressible flow over a closed, two-dimensional body theoretically produces zero drag—the well – known d’Alembert’s paradox. Does the same paradox hold for inviscid, subsonic,

compressible flow? The answer can be obtained by again noting that the only source of drag is the integral of the pressure distribution. If this integral is zero for an incompressible flow, and since the compressible pressure coefficient differs from the incompressible pressure coefficient by only a constant scale factor, p, then the integral must also be zero for a compressible flow. Hence, d’Alembert’s paradox also prevails for inviscid, subsonic, compressible flow. However, as soon as the freestream Mach number is high enough to produce locally supersonic flow on the body surface with attendant shock waves, as shown in Figure 137b, then a positive wave drag is produced, and d’Alembert’s paradox no longer prevails.

Example 11.1 | At a given point on the surface of an airfoil, the pressure coefficient is —0.3 at very low speeds. If the freestream Mach number is 0.6, calculate Cp at this point.


From Equation (11.51),

^ _ Cp, o _ ~~0-3

p ~~ Vl – M2 ~ y/ – (0.6)2

Example 1 1.2 | From Chapter 4, the theoretical lift coefficient for a thin, symmetric airfoil in an incompressible

flow is с/ = 2ла. Calculate the lift coefficient for Mx = 0.7.


From Equation (11.52),

C/,o 2na

C‘ ~ У1 – Ml ~ Vl – (0.7)2

Note: The effect of compressibility at Mach 0.7 is to increase the lift slope by the ratio 8.8/27Г = 1.4, or by 40 percent.

The Navier-Stokes Equations

In Chapter 2, Newton’s second law was applied to obtain the fluid-flow momentum equation in both integral and differential forms. In particular, recall Equations (2.13a to c), where the influence of viscous forces was expressed simply by the generic terms (•Tudviscous, (dd'(viscousi and (d%)viSCous – The purpose of this section is to obtain the analogous forms of Equations (2.13a to c) where the viscous forces are expressed explicitly in terms of the appropriate flow-field variables. The resulting equations are called the Navier-Stokes equations—probably the most pivotal equations in all of theoretical fluid dynamics.

In Section 2.3, we discussed the philosophy behind the derivation of the governing equations, namely, certain physical principles are applied to a suitable model of the fluid flow. Moreover, we saw that such a model could be either a finite control volume (moving or fixed in space) or an infinitesimally small element (moving or fixed in space). In Chapter 2, we chose the fixed, finite control volume for our model and obtained integral forms of the continuity, momentum, and energy equations directly from this model. Then, indirectly, we went on to extract partial differential equations from the integral forms. Before progressing further, it would be wise for you to review these matters from Chapter 2.

For the sake of variety, let us not use the fixed, finite control volume employed in Chapter 2; rather, in this section, let us adopt an infinitesimally small moving fluid element of fixed mass as our model of the flow, as sketched in Figure 15.11. To this model let us apply Newton’s second law in the form F = ma. Moreover, for the time being consider only the jc component of Newton’s second law:

Fx = max [15.14]

In Equation (15.14), Fx is the sum of all the body and surface forces acting on the fluid element in the x direction. Let us ignore body forces; hence, the net force acting on the element in Figure 15.11 is simply due to the pressure and viscous stress distributions over the surface of the element. For example, on face abed, the only force in the jc direction is that due to shear stress, rvv dx dz. Face efgh is a

Figure 1 5.1 1 Infinitesimally small, moving fluid element. Only the forces in the x direction are shown.

distance dy above face abed; hence, the shear force in the x direction on face efgh is [xyx + (дтух/ду) dy] dx dz. Note the directions of the shear stress on faces abed and efgh; on the bottom face, zyx is to the left (the negative x direction), whereas on the top face, zyx + (dzyx/dy) dy is to the right (the positive x direction). These directions are due to the convention that positive increases in all three components of velocity, u, v, and w, occur in the positive directions of the axes. For example, in Figure 15.11, и increases in the positive у direction. Therefore, concentrating on face efgh, и is higher just above the face than on the face; this causes a “tugging” action which tries to pull the fluid element in the positive x direction (to the right) as shown in Figure 15.11. In turn, concentrating on face abed, и is lower just beneath the face than on the face; this causes a retarding or dragging action on the fluid element, which acts in the negative x direction (to the left), as shown in Figure 15.11. The directions of all the other viscous stresses shown in Figure 15.11, including zxx, can be justified in a like fashion. Specifically, on face degh, zzx acts in the negative x direction, whereas on face abfe, zzx + (dzzx/8z)dz acts in the positive x direction. On face adhe, which is perpendicular to the x axis, the only forces in the x direction are the pressure force p dy dz, which always acts in the direction into the fluid element, and zxx dy dz, which is in the negative x direction. In Figure 15.11, the reason why zxx on face adhe is to the left hinges on the convention mentioned earlier for the direction of increasing velocity. Here, by convention, a positive increase in и takes place in the positive x direction. Hence, the value of и just to the left of face adhe is smaller than the value of и on the face itself. As a result, the viscous action of the normal stress

acts as a “suction” on face adhe: that is, there is a dragging action toward the left that wants to retard the motion of the fluid element. In contrast, on face bcgf, the pressure force [p + (dp/dx) dx] dy dz presses inward on the fluid element (in the negative x direction), and because the value of и just to the right of face bcgf is larger than the value of и on the face, there is a “suction” due to the viscous normal stress which tries to pull the element to the right (in the positive x direction) with a force equal to [txx + (dzxx/dx)dx]dydz.

Return to Equation (15.14). Examining Figure 15.11 in light of our previous discussion, we can write for the net force in the л direction acting on the fluid element:

і dP

Equation (15.15) represents the left-hand side of Equation (15.14). Considering the right-hand side of Equation (15.14), recall that the mass of the fluid element is fixed and is equal to

m = p dx dy dz

Also, recall that the acceleration of the fluid element is the time rate of change of its velocity. Hence, the component of acceleration in the x direction, denoted by ax, is simply the time rate of change of n; since we are following a moving fluid element, this time rate of change is given by the substantial derivative (see Section 2.9 for a review of the meaning of substantial derivative). Thus,

Du _ .

ax = — [15.17]


Combining Equations (15.14) to (15.17), we obtain


which is the x component of the momentum equation for a viscous flow. In a similar fashion, the у and z components can be obtained as

Dw dp drxz dzyz drzz

Dt dz dx dy dz

Equations (15.18a to c) are the momentum equations in the x, y, and z directions, respectively. They are scalar equations and are called the Navier-Stokes equations in

honor of two men—the Frenchman M. Navier and the Englishman G. Stokes—who independently obtained the equations in the first half of the nineteenth century.

With the expressions for rxy = xyx, xyi = xzy, rzx = rxz, rxx, xyy, and xzz from Equations (15.5) to (15.10), the Navier-Stokes equations, Equations (15.18a to c), can be written as

dи Эи Эи Зи dp Э / Эи

РЧ7 + риТ + pv^~ + pwV~ = UV • V + 2/х —

dt dx dy dz dx dx dx J

Equations (15.19a to c) represent the complete Navier-Stokes equations for an un­steady, compressible, three-dimensional viscous flow. To analyze incompressible vis­cous flow, Equations (15.19a to c) and the continuity equation [say, Equation (2.52)] are sufficient. However, for a compressible flow, we need an additional equation, namely, the energy equation to be discussed in the next section.

In the above form, the Navier-Stokes equations are suitable for the analysis of laminar flow. For a turbulent flow, the flow variables in Equations (15.19a to c) can be assumed to be time-mean values over the turbulent fluctuations, and p can be replaced by p + є, as discussed in Section 15.3. For more details, see References 42 and 43.

Results for Turbulent Boundary Layers on a Flat Plate

In this section, we discuss a few results for the turbulent boundary layer on a flat plate, both incompressible and compressible, simply to provide a basis of comparison with the laminar results described in the previous section. For considerably more detail on the subject of turbulent boundary layers, consult References 42 to 44.

For incompressible flow over a flat plate, the boundary-layer thickness is given approximately by

Note from Equation (19.1) that the turbulent boundary-layer thickness varies approx­imately as RejTl/5 in contrast to Re“I/2 for a laminar boundary layer. Also, turbulent values of <5 grow more rapidly with distance along the surface; 8 <x x4/5 for a turbulent flow in contrast to 8 oc x1 /2 for a laminar flow. With regard to skin friction drag, for incompressible turbulent flow over a flat plate, we have

Note that for turbulent flow, C f varies as Re~1//5 in comparison with the Re’1/2 vari­ation for laminar flow. Hence, Equation (19.2) yields larger friction drag coefficients for turbulent flow in comparison with Equation (18.22) for laminar flow.

The effects of compressibility on Equation (19.2) are shown in Figure 19.1, where C f is plotted versus ReTO with Мто as a parameter. The turbulent flow results are shown toward the right of Figure 19.1, at the higher values of Reynolds numbers where turbulent conditions are expected to occur, and laminar flow results are shown toward the left of the figure, at lower values of Reynolds numbers. This type of figure—friction drag coefficient for both laminar and turbulent flow as a function of Re on a log-log plot—is a classic picture, and it allows a ready contrast of the two types of flow. From this figure, we can see that, for the same Re^, turbulent skin

friction is higher than laminar; also, the slopes of the turbulent curves are smaller than the slopes of the laminar curves—a graphic comparison of the Re,/? variation in contrast to the laminar Re 1/2 variation. Note that the effect of increasing Мж is to reduce Cf at constant Re and that this effect is stronger on the turbulent flow results. Indeed, C/ for the turbulent results decreases by almost an order of magnitude (at the higher values of Re,*,) when M0c is increased from 0 to 10. For the laminar flow, the decrease in Су as Мж is increased though the same Mach number range is far less pronounced.