Category When Is A Flow Compressible?

The Reference Temperature Method

In this section we discuss an approximate engineering method for predicting skin friction and heat transfer for laminar compressible flow. It is based on the simple idea of utilizing the formulas obtained from incompressible flow theory, wherein the ther­modynamic and transport properties in these formulas are evaluated at some reference temperature indicative of the temperature somewhere inside the boundary layer. This idea was first advanced by Rubesin and Johnson in Reference 80 and was modified by Eckert (Reference 81) to include a reference enthalpy. In this fashion, in some sense the classical incompressible formulas were “corrected” for compressibility effects. Reference temperature (or reference enthalpy) methods have enjoyed frequent appli­cation in engineering-oriented analyses, because of their simplicity. For this reason, we briefly describe the approach here.

Consider the incompressible laminar flow over a flat plate, as discussed in Section

18.2. The local skin friction coefficient is given by Equation (18.20), repeated below:

_ 0.664

Cf ~

For the compressible laminar flow over a flat plate, we write the analogous expression

Evaluating Equation (18.54) at the reference temperature, we have

£-** ______ ^ w______

p* u*(haw hw)

Example 1 8.2 | Use the reference temperature method to calculate the friction drag on the same flat plate at the same flow conditions as described in Example 8.1b. Compare the reference temperature results with that obtained in Example 8.1b, which reflected the “exact” laminar boundary layer theory.


The reference temperature is calculated from Equation (18.53), where we need the ratio Tm/Te. For the present case, the flat plate is at the adiabatic wall temperature, hence we need the ratio Taw/Te. To obtain this, we use the recovery factor, which for a flat plate laminar boundary layer is given by Equation (18.47):

Also, the value of д* that corresponds to T* is obtained from Sutherland’s law, given by Equation (15.3)

_Д _ /rV,/2 T0 + 110 M о UJ T + 110

Recall: In Equation (15.3), до is the reference viscosity coefficient at the reference temperature T0. In Equation (15.3) T0 denotes the reference temperature, not the total temperature. Here we have a case of the same notation for two different quantities, but the meaning of T0 in Equation (15.3) is clear from its context. We will use the standard sea level conditions for the values of T0 and д0, that is,

до = 1-7894 x КГ5 kg/(m)(s) and T0 = 288 К

Hence, from Equation (15.3)

д* /Г*у1/2 T0+ 110 /612.7Л3/2 288 + 110

/У VW T* + 110 ~ V 288 / 612.7 + 110


д* = 1.709ДО = (1.709)(1.7894 x КГ5) = 3.058 x КГ5 kg/(m)(s)

From Equation (18.52) integrated over the entire chord of the plate, we have the same form as Equation (18.22), namely,

_ K328

7 У*?

Hence, the friction drag on one side of the plate is

Df = p*VfSC*f = 5(0.574)(1000)2(40)(2.167 x 10“4) = 2844 N

The total friction drag taking into account both the top and bottom surfaces of the plate is

D = 2(2488) =

The result obtained from classical compressible boundary layer theory in Example 18.1b is D = 5026 N. The result from the reference temperature method used here is within one percent of the “exact” value found in Example 18.1b, a stunning example of the accuracy of the reference temperature method, at least for the case treated here.

CFD Applications: Transonic Airfoils and Wings

The analysis of subsonic compressible flow over airfoils discussed in this chapter, re­sulting in classic compressibility corrections such as the Prandtl-Glauert mle (Section 11.4), fits into the category of “closed-form” theory as discussed in Section 2.17.1. Although this theory is elegant and useful, it is restricted to:

1. Thin airfoils at small angles of attack

2. Subsonic numbers that do not approach too close to one, that is, Mach numbers typically below 0.7

3. Inviscid, irrotational flow

However, modern subsonic transports (Boeing 747, 111, etc.) cruise at freestream Mach numbers on the order of 0.85, and high-performance military combat airplanes spend time at high subsonic speeds near Mach one. These airplanes are in the transonic flight regime, as discussed in Section 1.10.4 and noted in Figure 1.37. The closed – form theory discussed in this chapter does not apply in this flight regime. The only approach that allows the accurate calculation of airfoil and wing characteristics at transonic speeds is to use computational fluid dynamics; the basic philosophy of CFD is discussed in Section 2.17.2, which should be reviewed before you progress further.

The need to calculate accurately the transonic flow over airfoils and wings was one of the two areas that drove advances in CFD in the early days of its development, the other area being hypersonic flow. The growing importance of high-speed jet civil transports during the 1960s and 1970s made the accurate calculation of transonic flow imperative, and CFD was (and still is) the only way of making such calculations. In this section we will give only the flavor of such calculations; see Chapter 14 of Reference 21 for more details, as well as the modern aerodynamic literature for the latest developments.

Beginning in the 1960s, transonic CFD calculations historically evolved through four distinct steps, as follows:

1. The earliest calculations numerically solved the nonlinear small-perturbation potential equation for transonic flow, obtained from Equation (11.6) by dropping all terms on the right-hand side except for the leading term, which is not small near M„о = 1. This yields

8u 8v

(l – Ml;)— + ~ = Ml
^ 8x 8v

which in terms of the perturbation velocity potential is

Equation (11.69) is the transonic small perturbation potential equation; it is non­linear due to the term on the right-hand side, which involves a product of deriva­tives of the dependent variable <j>. This necessitated a numerical CFD solution. However, the results were limited to the assumptions embodied in this equation, namely, small perturbations and hence thin airfoils at small angles of attack.

2. The next step was numerical solutions of the full potential equation, Equation

(11.12) . This allowed applications to airfoils of any shape at any angle of attack. However, the flow was still assumed to be isentropic, and even though shock waves appeared in the results, the properties of these shocks were not always accurately predicted.

3. As CFD algorithms became more sophisticated, numerical solutions of the Eu­ler equations (the full continuity, momentum, and energy equations for inviscid flow, such as Equations (7.40), (7.42), and (7.44)) were obtained. The advantage of these Euler solutions was that shock waves were properly treated. However,

none of the approaches discussed in steps 1-3 accounted for the effects of viscous flow, the importance of which in transonic flows soon became more appreciated because of the interaction of the shock wave with the boundary layer. This inter­action, with the attendant flow separation is dominant in the prediction of drag.

4. This led to the CFD solution of the viscous flow equations (the Navier-Stokes equations, such as Equations (2.43), (2.61), and (2.87) with the viscous terms in­cluded) for transonic flow. The Navier-Stokes equations are developed in detail in Chapter 15. Such CFD solutions of the Navier-Stokes equations are currently the state of the art in transonic flow calculations. These solutions contain all of the realistic physics of such flows, with the exception that some type of turbu­lence model must be included to deal with turbulent boundary layers, and such turbulent models are frequently the Archilles heel of these calculations.

An example of a CFD calculation for the transonic flow over an NACA 0012 airfoil at 2° angle of attack with = 0.8 is shown in Figure 11.21. The contour lines shown here are lines of constant Mach number, and the bunching of these lines together clearly shows the nearly normal shock wave occurring on the top surface. In reference to our calculation in Example 11.3 showing that the critical Mach number for the NACA 0012 airfoil at zero angle of attack is 0.74, and the experimental confirmation of this shown in Figure 11.1 Ofo, clearly the flow over the same airfoil shown in Figure 11.21 is well beyond the critical Mach number. Indeed, the boundary layer downstream of the shock wave in Figure 11.21 is separated, and the airfoil is squarely in the drag-divergence region. The CFD calculations predict this separated flow because a version of the Navier-Stokes equations (called the thin shear layer approximation) is being numerically solved, taking into account the viscous flow effects. The results shown in Figure 11.21 are from the work of Nakahashi and Deiwert at the NASA Ames Research Center (Reference 74); these results are a graphic illustration of the power of CFD applied to transonic flow. For details on these types of CFD calculations, see the definitive books by Hirsch (Reference 75).

Today, CFD is an integral part of modem transonic airfoil and wing design. A recent example of how CFD is combined with modem optimization design techniques for the design of complete wings for transonic aircraft is shown in Figures 11.22 and 11.23, taken from the survey paper by Jameson (Reference 76). On the left side of Figure 11.22a the airfoil shape distribution along the semispan of a baseline, initial wing shape at Мх = 0.83 is given, with the computed pressure coefficient distributions shown at the right. The abrupt drop in Cp in these distributions is due to a relatively strong shock wave along the wing. After repeated iterations, the optimized design at the same = 0.83 is shown in Figure 11.22b. Again, the new airfoil shape distribution is shown on the left, and the Cp distribution is given on the right. The new, optimized wing design shown in Figure 11.22b is virtually shock free, as indicated by the smooth Cp distributions, with a consequent reduction in drag of 7.6 percent. The optimization shown in Figure 11.22 was subject to the constraint of keeping the wing thickness the same. Another but similar case of wing design optimization is shown in Figure 11.23. Flere, the final optimized wing planform shape is shown for Mqq = 0.86, with the final computed pressure contour lines shown on

Figure 1 1.21 Mach number contours in the transonic flow over an NACA 001 2 airfoil at Mtx; = 0.8 and at 2° angle of attack. (Source: Nakahasi and Deiwert, Reference 74.1

the planform. Straddling the wing planform on both the left and right of Figure 11.23 are the pressure coefficient plots at six spanwise stations. The dashed curves show the Cp variations for the initial baseline wing, with the tell-tale oscillations indicating a shock wave, whereas the solid curves are the final Cp variations for the optimized wing, showing smoother variations that are almost shock-free. At the time of writing, the results shown in Figures 11.22 and 11.23 are reflective of the best combination of multidisciplinary design optimization using CFD for transonic wings. For more details on this and other design applications, see the special issue of the Journal of Aircraft, vol. 36, no. 1, Jan./Feb. 1999, devoted to aspects of multidisciplinary design optimization.


Figure I 1.22 The use of CFD for optimized transonic wing design. Moo = 0.83. (a) Baseline wing with a shock wave, (b) Optimized wing, virtually shock free. Source: Jameson, Reference 76.

Incompressible (Constant Property) Couette Flow

In the study of viscous flows, a flow field in which p, p, and к are treated as constants is sometimes labeled as “constant property” flow. This assumption is made in the present section. On a physical basis, this means that we are dealing with an incompressible flow, where p is constant. Also, since /і and к are functions of temperature (see Section 15.3), constant property flow implies that T is constant also. (We will relax this assumption slightly at the end of this section.)

The governing equations for Couette flow were derived in Section 16.2. In partic­ular, the у-momentum equation, Equation (16.2), along with the geometrical property that Hp/Hx = 0, states that the pressure is constant throughout the flow. Consequently, all the information about the velocity field comes from the x-momentum equation, Equation (16.1), repeated below:


For constant д, this becomes


Integrating with respect to у twice, we obtain

и = ay + b

where a and b are constants of integration. These constant can be obtained from the boundary conditions illustrated in Figure 16.2, as follows:

At у = 0, и = 0; hence, b — 0.

At у = D, и = ue; hence, a = ue/D.

Thus, the variation of velocity for incompressible Couette flow is given by Equa­tion (16.5) as


Note the important result that the velocity varies linearly across the flow. This result is sketched in Figure 16.3.

Once the velocity profile is obtained, we can obtain the shear stress at any point in the flow from Equation (15.1), repeated below (the subscript yx is dropped here because we know the only shear stress acting in this problem is that in the x direction):


From Equation (16.6),

3 и ue

з7 ~ z>

Hence, from Equations (16.7) and (16.8), we have


Note that the shear stress is constant throughout the flow. Moreover, the straight­forward result given by Equation (16.9) illustrates two important physical trends— trends that we will find to be almost universally present in all viscous flows:

1. As ue increases, the shear stress increases. From Equation (16.9), r increases linearly with ue this is a specific result germane to Couette flow. For other problems, the increase is not necessarily linear.

2. As D increases, the shear stress decreases; that is, as the thickness of the viscous shear layer increases, all other things being equal, the shear stress becomes smaller. From Equation (16.9), r is inversely proportional to D—again a result germane to Couette flow. For other problems, the decrease in r is not necessarily in direct inverse proportion to the shear-layer thickness.

With the above results in mind, reflect for a moment on the quotation from Isaac Newton’s Principia given at the beginning of this chapter. Here, the “want of lubric­ity” is, in modem terms, interpreted as the shear stress. This want of lubricity is, according to Newton, “proportional to the velocity with which the parts of the fluid are separated from one another,” that is, in the context of the present problem propor­tional to ue/D. This is precisely the statement contained in Equation (16.9). In more recent times, Newton’s statement is generalized to the form given by Equation (16.7), and even more generalized by Equation (15.1). For this reason, Equations (15.1) and

(16.7) are frequently called the newtonian shear stress law, and fluids which obey this law are called newtonian fluids. [There are some specialized fluids which do not obey Equation (15.1) or (16.7); they are called non-newtonian fluids—some polymers and blood are two such examples.] By far, the vast majority of aeronautical appli­cations deal with air or other gases, which are newtonian fluids. In hydrodynamics, water is the primary medium, and it is a newtonian fluid. Therefore, we will deal exclusively with newtonian fluids in this book. Consequently, the quote given at the beginning of this chapter is one of Newton’s most important contributions to fluid mechanics—it represents the first time in history where shear stress is recognized as being proportional to a velocity gradient.

Let us now turn our attention to heat transfer in a Couette flow. Here, we continue our assumptions of constant p, p. and k. but at the same time, we will allow a temperature gradient to exist in the flow. In an exact sense, this is inconsistent; if T varies throughout the flow, then p, p, and к also vary. However, for this application, we assume that the temperature variations are small—indeed, small enough such that p, p, and к are approximately constant—and treat them so in the equations. On the other hand, the temperature changes, although small on an absolute basis, are sufficient to result in meaningful heat flux through the fluid. The results obtained will reflect some of the important trends in aerodynamic heating associated with high-speed flows, to be discussed in subsequent chapters.

For Couette flow with heat transfer, return to Figure 16.2. Here, the temperature of the upper plate is Te and that of the lower plate is Tw. Hence, we have as boundary conditions for the temperature of the fluid:

At у = 0: At у = D:

The temperature profile in the flow is governed by the energy equation, Equation

(16.3) . For constant p and k, this equation is written as


Also, since p is assumed to be constant, Equations (16.10) and (16.1) are totally uncoupled. That is, for the constant property flow considered here, the solution of the momentum equation [Equation (16.1)] is totally separate from the solution of the energy equation [Equation (16.10)]. Therefore, in this problem, although the temperature is allowed to vary, the velocity field is still given by Equation (16.6), as sketched in Figure 16.3.

In dealing with flows where energy concepts are important, the enthalpy h is fre­quently a more fundamental variable than temperature; we have seen much evidence of this in Part 3, where energy changes were a vital consideration. In the present problem, where the temperature changes are small enough to justify the assumptions of constant p, p, and k, this is not quite the same situation. However, because we will need to solve Equation (16.10), which is an energy equation for a flow (no matter how small the energy changes), and because we are using Couette flow as an example to set the stage for more complex problems, it is instructional (but by no means necessary) to couch Equation (16.10) in terms of enthalpy. Assuming constant specific heat, we have


Equation (16.11) is valid for the Couette flow of any fluid with constant heat capacity; here, the germane specific heat is that at constant pressure cp because the entire flow field is at constant pressure. In this sense, Equation (16.11) is a result of applying the first law of thermodynamics to a constant pressure process and recalling the funda­mental definition of heat capacity as the heat added per unit change in temperature, Sq/dT. Of course, if the fluid is a calorically perfect gas, then Equation (16.11) is

a basic thermodynamic property of such a gas quite independent of what the pro­cess may

Note that h varies parabolically with y/D across the flow. Since T = h/cp, then the temperature profile across the flow is also parabolic. The precise shape of the parabolic curve depends on hw (or Tw), he (or Te), and Pr. Also note that, as expected from our discussion of the viscous flow similarity parameters in Section 15.6, the

Prandtl number is clearly a strong player in the results; Equation (16.16) is one such example.

Once the enthalpy (or temperature) profile is obtained, we can obtain the heat flux at any point in the flow from Equation (15.2), repeated below (the subscript у is dropped here because we know the only direction of heat transfer is in the у direction for this problem):

д T

q = – k — [16.17]


Equation (16.17) can be written as

k dh r,

g =——— [16.18]

cP dy

In Equation (16.18), the enthalpy gradient is obtained by differentiating Equation (16.16) as follows:

Inserting Equation (16.19) into Equation (16.18), and writing k/cp as /і/ Pr, we have

From Equation (16.20), note that q is not constant across the flow, unlike the shear stress discussed earlier. Rather, q varies linearly with y. The physical reason for the variation of q is viscous dissipation which takes place within the flow, and which is associated with the shear stress in the flow. Indeed, the last term in Equation (16.20), in light of Equations (16.6) and (16.9), can be written as

= ти

Hence, Equation (16.20) becomes

The variation of q across the flow is due to the last term in Equation (16.21), and this term involves shear stress multiplied by velocity. The term ти is viscous dissipation; it is the time rate of heat generated at a point in the flow by one streamline at a given velocity “rubbing” against an adjacent streamline at a slightly different velocity— analogous to the heat you feel when rubbing your hands together vigorously. Note that, if ue is negligibly small, then the viscous dissipation is small and can be ne­glected; that is, in Equation (16.20) the last term can be neglected (ue is small), and in Equation (16.21) the last term can be neglected (r is small if ue is small). In this case, the heat flux becomes constant across the flow, simply equal to

In this case, the “driving potential” for heat transfer across the flow is simply the enthalpy difference (he — h „ ) or, in other words, the temperature difference (Te — 7 „) across the flow. However, as we have emphasized, if ue is not negligible, then viscous dissipation becomes another factor that drives the heat transfer across the flow.

Of particular practical interest is the heat flux at the walls—the aerodynamic heating as we label it here. We denote the heat transfer at a wall as qw. Moreover, it is conventional to quote aerodynamic heating at a wall without any sign convention. For example, if the heat transfer from the fluid to the wall is 10 W/cm2, or, if in reverse the heat transfer from the wall to the fluid is 10 W/cm2, it is simply quoted as such; in both cases, qw is given as 10 W/cm2 without any sign convention. In this sense, we write Equation (16.18) as




qw = —




where the subscript w implies conditions at the wall. The direction of the net heat transfer at the wall, whether it is from the fluid to the wall or from the wall to the fluid, is easily seen from the temperature gradient at the wall; if the wall is cooler than the adjacent fluid, heat is transferred into the wall, and if the wall is hotter than the adjacent fluid, heat is transferred into the fluid. Another criterion is to compare the wall temperature with the adiabatic wall temperature, to be defined shortly.

Return to the picture of Couette flow in Figure 16.2. To calculate the heat transfer at the lower wall, use Equation (16.23) with the enthalpy gradient given by Equation (16.19) evaluated at у = 0:

To calculate the heat transfer at the upper wall, use Equation (16.23) with the enthalpy gradient given by Equation (16.19) evaluated at у = D. In this case, Equation (16.19) yields

dh he – hw + Pr и2 Pr u] _ he – hw – Pr и2 By D D D

In turn, from Equation (16.23)

Let us examine the above results for three different scenarios, namely, (1) negligi­ble viscous dissipation, (2) equal wall temperature, and (3) adiabatic wall conditions (no heat transfer to the wall). In the process, we define three important concepts in the analysis of aerodynamic heating: (1) adiabatic wall temperatue, (2) recovery factor, and (3) Reynolds analogy.

Flow over an Airfoil

The viscous compressible flow over an airfoil was studied in Reference 56. For the treatment of this problem, a nonrectangular finite-difference grid is wrapped around the airfoil, as shown in Figure 20.3. Equations (20.1) to (20.5) have to be transformed into the new curvilinear coordinate system in Figure 20.3. The details are beyond the scope of this book; see Reference 56 for a complete discussion. Some results for the streamline patterns are shown in Figure 20.4a and b. Here, the flow over a Wortmann airfoil at zero angle of attack is shown. The freestream Mach number is 0.5, and the Reynolds number based on chord is relatively low, Re = 100,000. The completely laminar flow over this airfoil is shown in Figure 20.4a. Because of the peculiar aerodynamic properties of some low Reynolds number flows over airfoils (see References 51 and 56), we note that the laminar flow separated over both the top and bottom surfaces of the airfoil. However, in Figure 20.4Й, the turbulence model is turned on for the calculation; note that the flow is now completely attached. The differences in Figure 20.4a and b vividly demonstrate the basic trend that turbulent flow resists flow separation much more strongly than laminar flow.