Category When Is A Flow Compressible?

Laminar Boundary Layers

Lamina—A thin scale or sheet. A layer or coat lying over another.

18.1 Introduction

Within the panoply of boundary-layer analyses, the solution of laminar boundary layers is well in hand compared to the status for turbulent boundary layers. This chapter is exclusively devoted to laminar boundary layers; turbulent boundary layers is the subject of Chapter 19. The basic definitions of laminar and turbulent flows are discussed in Section 15.1, and some characteristics of these flows are illustrated in Figures 15.5 and 15.6; it is recommended that you review that material before progressing further.

The roadmap for this chapter is given in Figure 18.1. We will first deal with some well-established classical solutions that come under the heading of self-similar solutions, a term that is defined in Section 18.2. In this regard, we will deal with both incompressible and compressible flows over a flat plate, as noted on the left side of our roadmap in Figure 18.1. We will also discuss the boundary-layer solution in the region surrounding the stagnation point on a blunt-nosed body, because this solution gives us important information on aerodynamic heating at the stagnation point—vital information for high-speed flight vehicles. As part of the classical solution of com­pressible boundary layers, we will discuss the reference temperature method—a very useful engineering calculation that makes use of classical incompressible boundary – layer results to predict skin friction and aerodynamic heating for a compressible flow. Then we will move to the right side of the roadmap in Figure 18.1 and discuss some

more modern computational fluid dynamic solutions to laminar boundary layers. Un­like the classical self-similar solutions, which are limited to a few (albeit important) applications such as flat plates and the stagnation region, these CFD numerical solu­tions deal with the laminar boundary layer over bodies of arbitrary shapes.

Note: As we progress through this chapter, we will encounter ideas and results that are already familiar to us from our discussion of Couette flow in Chapter 16. Indeed, this is one of the primary reasons for Chapter 16—to introduce these concepts within the context of a relatively straightforward flow problem before dealing with the more intricate boundary-layer solutions.

Critical Mach Number

Return to the road map given in Figure 11.1. We have now finished our discussion of linearized flow and the associated compressibility corrections. Keep in mind that such linearized theory does not apply to the transonic flow regime, 0.8 < Мж <

1.2. Transonic flow is highly nonlinear, and theoretical transonic aerodynamics is a challenging and sophisticated subject. For the remainder of this chapter, we deal with several aspects of transonic flow from a qualitative point of view. The theory of transonic aerodynamics is beyond the scope of this book.

Consider an airfoil in a low-speed flow, say, with = 0.3, as sketched in Figure 11.5a. In the expansion over the top surface of the airfoil, the local flow Mach number M increases. Let point A represent the location on the airfoil surface

t Local MA

where the pressure is a minimum, hence where M is a maximum. In Figure 11.5a, let us say this maximum is MA = 0.435. Now assume that we gradually increase the freestream Mach number. As M^ increases, MA also increases. For example, if is increased to M = 0.5, the maximum local value of M will be 0.772, as shown in Figure 11.5b. Let us continue to increase Moo until we achieve just the right value such that the local Mach number at the minimum pressure point equals 1, that is, such that MA = 1.0, as shown in Figure 11.5c. When this happens, the freestream Mach number Mx is called the critical Mach number, denoted by Mcr. By definition, the critical Mach number is that freestream Mach number at which sonic flow is first achieved on the airfoil surface. In Figure 11.5c, Ma = 0.61.

One of the most important problems in high-speed aerodynamics is the determi­nation of the critical Mach number of a given airfoil, because at values of Мж slightly above Mcr, the airfoil experiences a dramatic increase in drag coefficient (discussed in Section 11.7). The purpose of the present section is to give a rather straightforward method for estimating Mcr.

Let poo and pA represent the static pressures in the freestream and at point A, respectively, in Figure 11.5. For isentropic flow, where the total pressure po is constant, these static pressures are related through Equation (8.42) as follows:

Pa_ = Pa/Po = /1 + [(у-1)/2]Л^у/(у~П [11 56]

Рос Poc/Po l + [(y -)/2]Mj) *

The pressure coefficient at point A is given by Equation (11.22) as

Combining Equations (11.56) and (11.57), we have

Equation (11.58) is useful in its own right; for a given freestream Mach number, it relates the local value of Cp to the local Mach number. [Note that Equation (11.58) is the compressible flow analogue of Bernoulli’s equation, Equation (3.13), which for incompressible flow with a given freestream velocity and pressure relates the local pressure at a point in the flow to the local velocity at that point.] However, for our purposes here, we ask the question, What is the value of the local Cp when the local Mach number is unity? By definition, this value of the pressure coefficient is called the critical pressure coefficient, denoted by Cp. cr. For a given freestream Mach number Moo, the value of Cp. cr can be obtained by inserting MA = 1 into Equation (11.58):

Equation (11.59) allows us to calculate the pressure coefficient at any point in the flow where the local Mach number is 1, for a given freestream Mach number M^. For example, if Mx is slightly greater than Mcr, say, Mx = 0.65 as shown in Figure

11.5d, then a finite region of supersonic flow will exist above the airfoil; Equation

(11.59) allows us to calculate the pressure coefficient at only those points where M = 1, that is, at only those points that fall on the sonic line in Figure 11,5d. Now, returning to Figure 11.5c, when the freestream Mach number is precisely equal to the critical Mach number, there is only one point where M = 1, namely, point A. The pressure coefficient at point A will be Cp<cr, which is obtained from Equation (11.59). In this case, Moo in Equation (11.59) is precisely Mcr. Hence,

[1 1.60]

Equation (11.60) shows that Cpa is a unique function of Mcr; this variation is plotted as curve C in Figure 11.6. Note that Equation (11.60) is simply an aerodynamic relation for isentropic flow—it has no connection with the shape of a given airfoil. In this sense, Equation (11.60), and hence curve C in Figure 11.6, is a type of “universal relation” which can be used for all airfoils.

Equation (11.60), in conjunction with any one of the compressibility corrections given by Equations (11.51), (11.54), or (11.55), allows us to estimate the critical Mach number for a given airfoil as follows:

1. By some means, either experimental or theoretical, obtain the low-speed incom­pressible value of the pressure coefficient Cpq at the minimum pressure point on the given airfoil.

2. Using any of the compressibility corrections, Equation (11.51), (11.54), or

(11.55) , plot the variation of Cp with Moo■ This is represented by curve В in Figure 11.6.

Eq. (11.51), (11.54), or (11.55)

cr

3. Somewhere on curve B, there will be a single point where the pressure coefficient corresponds to locally sonic flow. Indeed, this point must coincide with Equa­tion (11.60), represented by curve C in Figure 11.6. Hence, the intersection of curves В and C represents the point corresponding to sonic flow at the minimum pressure location on the airfoil. In turn, the value of M^ at this intersection is, by definition, the critical Mach number, as shown in Figure 11.6.

The graphical construction in Figure 11.6 is not an exact determination of MCT. Al­though curve C is exact, curve В is approximate because it represents the approximate compressibility correction. Hence, Figure 11.6 gives only an estimation of MCT. How­ever, such an estimation is quite useful for preliminary design, and the results from Figure 11.6 are accurate enough for most applications.

Consider two airfoils, one thin and the other thick, as sketched in Figure 11.7. First consider the low-speed incompressible flow over these airfoils. The flow over the thin airfoil is only slightly perturbed from the freestream. Hence, the expansion over the top surface is mild, and Cp о at the minimum pressure point is a negative number of only small absolute magnitude, as shown in Figure 11.7. [Recall from Equation

(11.32) that Cp ос й; hence, the smaller the perturbation, the smaller is the absolute magnitude of Cp.] In contrast, the flow over the thick airfoil experiences a large perturbation from the freestream. The expansion over the top surface is strong, and Cp (t at the minimum pressure point is a negative number of large magnitude, as shown in Figure 11.7. If we now perform the same construction for each airfoil as given in Figure 11.6, we see that the thick airfoil will have a lower critical Mach number than

the thin airfoil. This is clearly illustrated in Figure 11.7. For high-speed airplanes, it is desirable to have Mcr as high as possible. Hence, modern high-speed subsonic airplanes are usually designed with relatively thin airfoils. (The development of the supercritical airfoil has somewhat loosened this criterion, as discussed in Section

11.8. ) For example, the Gates Lear jet high-speed jet executive transport utilizes a 9 percent thick airfoil; contrast this with the low-speed Piper Aztec, a twin-engine propeller-driven general aviation aircraft designed with a 14 percent thick airfoil.

Example 1 1 .3 I In this example, we illustrate the estimation of the critical Mach number for an airfoil using (a) the graphical solution discussed in this section, and (b) an analytical solution using a closed – form equation obtained from a combination of Equations (11.51) and (11.60). Consider the NACA 0012 airfoil at zero angle of attack shown at the top of Figure 11.8. The pressure coefficient distribution over this airfoil, measured in a wind tunnel at low speed, is given at the

c

Figure 1 1.8 Low-speed pressure coefficient distribution

over the surface of an NACA 0012 airfoil at zero angle of attack. Re = 3.65 x 106. (Source; R. J. Freuler and G. M. Gregorek, "An Evaluation of Four Single Element Airfoil Analytical Methods," in Advanced Technology Airfoil Research, NASA CP 2045, 1978, pp. 133-162.)

Following step three, these values are plotted as Curve В in Figure 11.9. The intersection of curves В and C is at point D. The freestream Mach number associated with point D is the critical Mach number for the NACA 0012 airfoil. From Figure 11.9, we have

(.b) Analytical Solution. In Figure 11.9, curve В is given by Equation (11.61)

At the intersection point D, (Cp)mill in Equation (11.61) is the critical pressure coefficient and Moo is the critical Mach number

Cp a = 0,43 (at point D) [11.62]

Vі – Mcr

Also, at point D the value of Cp, cr is given by Equation (11.60). Hence, at point D we can equate the right-hand sides of Equations (11.62) and (11.60),

-0.43 _ _j?_ [( +[(y – 1)/2]МС2ГУ/У ‘

yi^Mj У Ml [ 1 + (y – l)/2 J

Equation (11.63) is one equation with one unknown, namely, Mcr. The solution of Equation (11.63) gives the value of Mcr associated with the intersection point D in Figure 11.9, that is, the solution of Equation (11.63) is the critical Mach number for the NACA 0012 airfoil. Since Mcr appears in a complicated fashion on both sides of Equation (11.63), we solve the equation by trial-and-error by assuming different values of Mcr, calculating the values of both sides of Equation (11.63), and iterating until we find a value of Ma that results in both the right and left sides being the same value.

м -0.43 2 ГП+[(у-])/2]МІу/г-‘ _ ~

СГ 1 + (к — l)/2 ) _

0.72

-0.6196

-0.6996

0.73

-0.6292

-0.6621

0.74

-0.6393

-0.6260

0.738

-0.6372

-0.6331

0.737

-0.6362

-0.6367

0.7371

-0.6363

-0.6363

To four-place accuracy, when Mcr = 0.7371, both the left and right sides of Equation (11.63) have the same value. Therefore, the analytical solution yields

Note: Within the two-place accuracy of the graphical solution in part (a), both the graphical and analytical solutions give the same value of Mcr.

Question: How accurate is the estimate of the critical Mach number in this example? To answer this question, we examine some experimental pressure coefficient distributions for the NACA 0012 airfoil obtained at higher freestream Mach numbers. Wind tunnel measurements of the surface pressure distributions for this airfoil at zero angle of attack in a high-speed flow are shown in Figure 11.10; for Figure 11.10a, Mx = 0.575, and for Figure ll. lOfc,

(a)

(h)

Figure 11.10 Wind tunnel measurements of surface pressure coefficient distribution for the NACA 0012 airfoil at zero angle of attack. Experimental data of Frueler and Gregorek, NASA CP 2045.

(a) Мх = 0.575, (b) Moo = 0.725.

Mx = 0.725. In Figure 11.10a, the value of Cpxr = -1.465 at = 0.575 is shown as the dashed horizontal line. From the definition of critical pressure coefficient, any local value of Cp above this horizontal line corresponds to locally supersonic flow, and any local value below the horizontal line corresponds to locally subsonic flow. Clearly from the measured surface pressure coefficient distribution at Mx = 0.575 shown in Figure 11.10a, the flow is

locally subsonic at every point on the surface. Hence, Mж = 0.575 is below the critical Mach number. In Figure 1 l. lOh, which is for a higher Mach number, the value of C pa = —0.681 at Moo = 0.725 is shown as the dashed horizontal line. Here, the local pressure coefficient on the airfoil is higher than Cpxr at every point on the surface except at the point of minimum pressure, where (Cp)mi„ is essentially equal to Cp, cl. This means that for = 0.725, the flow is locally subsonic at every point on the surface except the point of minimum pressure, where the flow is essentially sonic. Hence, these experimental measurements indicate that the critical Mach number of the NACA 0012 airfoil at zero angle of attack is approximately 0.73. Comparing this experimental result with the calculated value of Mcr = 0.74 in this example, we see that our calculations are amazingly accurate, to within about one percent.

Similarity Parameters

In Section 1.7, we introduced the concept of dimensional analysis, from which sprung the similarity parameters necessary to ensure the dynamic similarity between two or more different flows (see Section 1.8). In the present section, we revisit the governing similarity parameters, but cast them in a slightly different light.

Consider a steady two-dimensional, viscous, compressible flow. The x-momen­tum equation for such a flow is given by Equation (15.19a), which for the present case reduces to

In Equation (15.27), p, u, p, etc., are the actual dimensional variables, say, p = kg/m3, etc. Let us introduce the following dimensionless variables:

/ _ p

, u

, _ V

U = ——

Poo

Voo

V ~ Voo

/ p

, _ X

, У

/X = ——

У = –

P’00

c

с

where Poo, Vqo, poo, and рьж are reference values (say, e. g., freestream values) and c is a reference length (say, the chord of an airfoil). In terms of these dimensionless

variables, Equation (15.27) becomes

where M0о and Reoo are the freestream Mach and Reynolds numbers, respectively, Equation (15.28) becomes

Equation (15.29) tells us something important. Consider two different flows over two bodies of different shapes. In one flow, the ratio of specific heats, Mach number, and Reynolds number are у і, Мж, and Re^j, respectively; in the other flow, these parameters have different values, y2, M^, and Re^. Equation (15.29) is valid for both flows. It can, in principle, be solved to obtain u’ as a function of x’ and y’. However, since y, Moo, and Reoo are different for the two cases, the coefficients of the derivatives in Equation (15.29) will be different. This will ensure, if

f (x y’)

represents the solution for one flow and

и = h(x y’)

represents the solution for the other flow, that

However, consider now the case where the two different flows have the same values of y, Moo, and Reoo – Now the coefficients of the derivatives in Equation (15.29) will be the same for both flows; that is, Equation (15.29) is numerically identical for the two flows. In addition, assume the two bodies are geometrically similar, so that the boundary conditions in terms of the nondimensional variables are the same. Then, the solutions of Equation (15.29) for the two flows in terms of u’ — f(x’, v’) and и’ = /г(х’, у’) must be identical; that is,

f{(x’,y’) = f2(x, y) [15.30]

Recall the definition of dynamically similar flows given in Section 1.8. There, we stated in part that two flows are dynamically similar if the distributions of V/Vx, p! P<x>, etc., are the same throughout the flow field when plotted against common nondimensional coordinates. This is precisely what Equation (15.30) is saying—that

u’ as a function of x’ and v’ is the same for the two flows. That is, the variation of the nondimensional velocity as a function of the nondimensional coordinates is the same for the two flows. How did we obtain Equation (15.30)? Simply by saying that y, Mqo, and Reoo are the same for the two flows and that the two bodies are geometrically similar. These are precisely the criteria for two flows to be dynamically similar, as originally stated in Section 1.8.

What we have seen in the above derivation is a formal mechanism to identify governing similarity parameters for a flow. By couching the governing flow equations in terms of nondimensional variables, we find that the coefficients of the derivatives in these equations are dimensionless similarity parameters or combinations thereof.

To see this more clearly, and to extend our analysis further, consider the en­ergy equation for a steady, two-dimensional, viscous, compressible flow, which from Equation (15.26) can be written as (assuming no volumetric heating and neglecting the normal stresses)

Turbulence Modeling

The simple equations given in Section 19.2 for boundary-layer thickness and skin friction coefficient for a turbulent flow over a flat plate are simplified results that are heavily empirically based. Modern calculations of turbulent flows over arbitrarily – shaped bodies involve the solution of the continuity, momentum, and energy equations along with some model of the turbulence. The calculations are carried out by means of computational fluid dynamic techniques. Here we will discuss only one model of tubulence, the Baldwin-Lomax turbulence model, which has become popular over the past two decades. We emphasize that the following discussion is intended only to give you the flavor of what is meant by a turbulence model.

19.3.1 The Baldwin-Lomax Model

In order to include the effects of turbulence in any analysis or computation, it is first necessary to have a model for the turbulence itself. Turbulence modeling is a state – of-the-art subject, and a recent survey of such modeling as applied to computations is given in Reference 85. Again, it is beyond the scope of the present book to give a detailed presentation of various turbulence models; the reader is referred to the literature for such matters. Instead, we choose to discuss only one such model here, because: (a) it is a typical example of an engineering-oriented turbulence model, (b) it is the model used in the majority of modern applications in turbulent, sub­sonic, supersonic, and hypersonic flows, and (c) we will discuss in the next chapter several applications which use this model. The model is called the Baldwin-Lomax turbulence model, first proposed in Reference 86. It is in the class of what is called an “eddy viscosity” model, where the effects of turbulence in the governing viscous

flow equations (such as the boundary-layer equations or the Navier-Stokes equations) are included simply by adding an additional term to the transport coefficients. For example, in all our previous viscous flow equations, /л is replaced by (ц. + t±r) and к by (к + kj) where jij and kT are the eddy viscosity and eddy thermal conductivity, respectively—both due to turbulence. In these expressions, ji and к are denoted as the “molecular” viscosity and thermal conductivity, respectively. For example, the x momentum boundary-layer equation for turbulent flow is written as

Moreover, the Baldwin-Lomax model is also in the class of “algebraic,” or “zero – equation,” models meaning that the formulation of the turbulence model utilizes just algebraic relations involving the flow properties. This is in contrast to one – and two-equation models which involve partial differential equations for the convection, creation, and dissipation of the turbulent kinetic energy and (frequently) the local vorticity. (See Reference 87 for a concise description of such one – and two-equation turbulence models.)

The Baldwin-Lomax turbulence model is described below. We give just a “cook­book” prescription for the model; the motivation and justification for the model are described at length in Reference 86. This, like all other turbulence models, is highly empirical. The final justification for its use is that it yields reasonable results across a wide range of Mach numbers, from subsonic to hypersonic. The model assumes that the turbulent-boundary layer is split into two layers, an inner and an outer layer, with different expressions for jiT in each layer:

{

(Mr) inner У — ycrossover r 1

, , . [19.6]

‘ M ‘/’ Jouter У _ ycrossover

where у is the local normal distance from the wall, and the crossover point from the inner to the outer layer is denoted by ycrossover- By definition, ycrossover is that point in the turbulent boundary where (мг)outer becomes less than (mt)inner – F°r the inner region:

and к and А+ are two dimensionless constants, specified later. In Equation (19.7), m is the local vorticity, defined for a two dimensional flow as

du dv З у dx

For the outer region:

where К and Ccp are two additional constants, and Fwake and /’кіеь are related to the function

Equation (19.12) will have a maximum value along a given normal distance y; this maximum value and the location where it occurs are denoted by Fmax and ymax, respec­tively. ІП Equation (19.11), Fwake is taken to be either Углах Fmax ОҐ (2wk Углах Fj. j –

whichever is smaller, where Cwk is constant, and

t/dif = у/ U2 + V2

Also, in Equation (19.11), FKieb is the Klebanoff intermittency factor, given by

The six dimensionless constants which appear in the above equations are: A+ = 26.0, Ccp = 1.6, Скіеь = 0-3, Cwk = 0.25, к = 0.4, and К = 0.0168. These constants are taken directly from Reference 86 with the understanding that, while they are not precisely the correct constants for most flows in general, they have been used successfully for a number of different applications. Note that, unlike many algebraic eddy viscosity models which are based on a characteristic length, the Baldwin-Lomax model is based on the local vorticity a>. This is a distinct advantage for the analysis of flows without an obvious mixing length, such as separated flows. Note that, like all eddy-viscosity turbulent models, the value of iij obtained above is dependent on the flowfield properties themselves (for example ш and p); this is in contrast to the molecular viscosity ц. which is solely a property of the gas itself.

The molecular values of viscosity coefficient and thermal conductivity are related through the Prandtl number

[19.15]

In lieu of developing a detailed turbulence model for the turbulent thermal conductivity kT, the usual procedure is to define a “turbulent” Prandtl number as Pr7 = цтср/кт. Thus, analogous to Equation (19.15), we have

where the usual assumption is that Pr7 = 1. Therefore, /і 7 is obtained from a given eddy-viscosity model (such as the Baldwin-Lomax model), and the corresponding kT is obtained from Equation (19.16).

Turbulence itself is a flowfield; it is not a simple property of the gas. This is why, as mentioned above, in an algebraic eddy viscosity model the values of /x7 and kT depend on the solution of the flow field—they are not pure properties of the gas as are /г and k. This is clearly seen in the Baldwin-Lomax model via Equation (19.7),

where fi t is a function of the local vorticity in the flow, oj—a flow field variable which comes out as part of the solution for the particular case at hand.