Category When Is A Flow Compressible?

Corrector Step

Inserting the flow variables obtained above into the governing equations, Equations

(13.59) to (13.62), using rearward differences for the spatial derivatives, predicted

values of the time derivatives at t + At are obtained, for example, [(3p/30,j](t+A,). In turn, these are averaged with the time derivatives from the predictor step to obtain; for example,

Finally, the average time derivative obtained from Equation (13.64) is inserted into Equation (13.63) to yield the corrected value of density at time t + At. The same procedure is used for all the dependent variables, u, v, etc.

Starting from the assumed initial conditions at t — 0, the repeated application of Equation (13.63) along with the above predictor-corrector algorithm at each time step allows the calculation of the flow-field variables and shock shape and location as a function of time. As stated above, after a large number of time steps, the calculated flow-field variables approach a steady state, where [(3p/3r)(j]ave —>■ 0 in Equation

(13.63) . Once again, we emphasize that we are interested in the steady-state answer, and the time-dependent technique is simply a means to that end.

Note that the applications of MacCormack’s technique to both the steady flow calculations described in Section 13.4 and the time-dependent calculations described in the present section are analogous; in the former, we march forward in the spatial coordinate x, starting with known values along with a constant у line, whereas, in the latter, we march forward in time starting with a known flow field at t = 0.

Why do we bother with a time-dependent solution? Is it not an added complica­tion to deal with an extra independent variable t in addition to the spatial variables x and _y? The answers to these questions are as follows. The governing unsteady flow equations given by Equations (13.59) to (13.62) are hyperbolic with respect to time, independent of whether the flow is locally subsonic or supersonic. In Figure 13.9a, some of the grid points are in the subsonic region and others are in the supersonic re­gion. However, the time-dependent solution progresses in the same manner at all these points, independent of the local Mach number. Hence, the time-dependent technique is the only approach known today which allows the uniform calculation of a mixed subsonic-supersonic flow field of arbitrary extent. For this reason, the application of the time-dependent technique, although it adds one additional independent variable, allows the straightforward solution of a flow field which is extremely difficult to solve by a purely steady-state approach.

A much more detailed description of the time-dependent technique is given in Chapter 12 of Reference 21, which you should study before attempting to apply this technique to a specific problem. The intent of our description here has been to give you simply a “feeling” for the philosophy and general approach of the technique.

Some typical results for supersonic blunt-body flow fields are given in Figures 13.12 to 13.15. These results were obtained with a time-dependent solution described in Reference 35. Figures 13.12 and 13.13 illustrate the behavior of a time-dependent solution during its approach to the steady state. In Figure 13.12, the time-dependent motion of the shock wave is shown for a parabolic cylinder in a Mach 4 freestream. The shock labeled 0 At is the initially assumed shock wave at t = 0. At early times, the shock wave rapidly moves away from the body; however, after about

300 time steps, it has slowed considerably, and between 300 and 500 time steps, the shock wave is virtually motionless—it has reached its steady-state shape and location. The time variation of the stagnation point pressure is given in Figure 13.13. Note that the pressure shows strong timewise oscillations at early times, but then it asymptotically approaches a steady value at large times. Again, it is this asymptotic steady state that we want, and the intermediate transient results are just a means to that

end. Concentrating on just the steady-state results, Figure 13.14 gives the pressure distribution (nondimensionalized by stagnation point pressure) over the body surface for the cases of both Мж = 4 and 8. The time-dependent numerical results are shown as the solid curves, whereas the open symbols are from newtonian theory, to be discussed in Chapter 14. Note that the pressure is a maximum at the stagnation point and decreases as a function of distance away from the stagnation point—a variation that we most certainly would expect based on our previous aerodynamic experience. The steady shock shapes and sonic lines are shown in Figure 13.15 for

Figure 13.1 5 Shock shapes and sonic lines, parabolic cylinder.

Problems

Note: The purpose of the following problem is to provide an exercise in carrying out a unit process for the method of characteristics. A more extensive application to a complete flow field is left to your specific desires. Also, an extensive practical problem utilizing the finite-difference method requires a large number of arithmetic operations and is practical only on a digital computer. You are encouraged to set up such a problem at your leisure. The main purpose of the present chapter is to present the essence of several numerical methods, not to burden the reader with a lot of calculations or the requirement to write an extensive computer program.

1. Consider two points in a supersonic flow. These points are located in a cartesian coordinate system at {x, yi) = (0,0.0684) and (xi, у2) = (0.0121,0), where the units are meters. At point {x, yi): и і = 639 m/s, г 1 = 232.6 m/s, p = 1 atm, T = 288 K. At point {x3, Уі)’- R2 = 680 m/s, i>2 = 0, P2 = 1 atm, T2 = 288 K. Consider point 3 downstream of points 1 and 2 located by the intersection of the C+ characteristic through point 2 and the C_ characteristic through point 1. At point 3, calculate: m3, i>3, p3, and T3. Also, calculate the location of point 3, assuming the characteristics between these points are straight lines.

Boundary-Layer Properties

Consider the viscous flow over a flat plate as sketched in Figure 17.3. The viscous effects are contained within a thin layer adjacent to the surface; the thickness is exaggerated in Figure 17.3 for clarity. Immediately at the surface, the flow velocity is zero; this is the “no-slip” condition discussed in Section 15.2. In addition, the temperature of the fluid immediately at the surface is equal to the temperature of the surface; this is called the wall temperature Tw, as shown in Figure 17.3. Above the surface, the flow velocity increases in the у direction until, for all practical purposes, it equals the freestream velocity. This will occur at a height above the wall equal to S, as shown in Figure 17.3. More precisely, S is defined as that distance above the wall

Figure I 7.3 Boundary-layer properties.

where и = 0.99ue; here, ue is the velocity at the outer edge of the boundary layer. In Figure 17.3, which illustrates the flow over a flat plate, the velocity at the edge of the boundary layer will be V, x; that is, ue = Уж. For a body of general shape, ue is the velocity obtained from an inviscid flow solution evaluated at the body surface (or at the “effective body” surface, as discussed later). The quantity S is called the velocity boundary-layer thickness. At any given x station, the variation of и between у = 0 and у = S, that is, и = и (у), is defined as the velocity profile within the boundary layer, as sketched in Figure 17.3. This profile is different for different x stations. Similarly, the flow temperature will change above the wall, ranging from T = Tw at у = 0 to T — 0.997), at у = 8T. Here, St is defined as the thermal boundary-layer thickness. At any given x station, the variation of T between у = 0 and у = ST, that is, Г = T(y), is called the temperature profile within the boundary layer, as sketched in Figure 17.3. (In the above, Te is the temperature at the edge of the thermal boundary layer. For the flow over a flat plate, as sketched in Figure 17.3, Te = Too. For a general body, Te is obtained from an inviscid flow solution evaluated at the body surface, or at the “effective body” surface, to be discussed later.) Hence, two boundary layers can be defined: a velocity boundary layer with thickness S and a temperature boundary layer with thickness St. In general, Sj ф S. The relative thicknesses depend on the Prandtl number: it can be shown that if Pr = 1, then S = 5^;ifPr > l, then<5r < <5;ifPr < l, then<5r > S. For air at standard conditions, Pr = 0.71; hence, the thermal boundary layer is thicker than the velocity boundary layer, as shown in Figure 17.3. Note that both boundary-layer thicknesses increase with distance from the leading edge; that is, 8 = 8(x) and 8T = ST(x).

The consequence of the velocity gradient at the wall is the generation of shear stress at the wall,

where (du/dy)w is the velocity gradient evaluated at у = 0 (i. e., at the wall). Simi­larly, the temperature gradient at the wall generates heat transfer at the wall,

[17.6]

Equation (17.6) is identical to the definition of S* given in Equation (17.3). Hence, clearly <5* is a height proportional to the missing mass flow. If this missing mass flow was crammed into a streamtube where the flow properties were constant at pe and ue, then Equation (17.5) says that <5* is the height of this hypothetical streamtube.

2. The second physical interpretation of 8* is more practical than the one discussed above. Consider the flow over a flat surface as sketched in Figure 17.5. At the left is a picture of the hypothetical inviscid flow over the surface; a streamline through point yi is straight and parallel to the surface. The actual viscous flow is shown at the right of Figure 17.5; here, the retarded flow inside the boundary layer acts as a partial obstruction to the freestream flow. As a result, the streamline external to the boundary layer passing through point yi is deflected upward through a distance 8*. We now prove that this 5* is precisely the displacement thickness defined by Equation (17.3). At station 1 in Figure 17.5, the mass flow (per unit depth perpendicular to the page) between the surface and the external streamline is

[17.7]

At station 2, the mass flow between the surface and the external streamline is

m= I pudy + peue8* [17.8]

Jo

Since the surface and the external streamline form the boundaries of a streamtube, the mass flows across stations 1 and 2 are equal. Hence, equating Equations (17.7) and (17.8), we have

Streamline

1

1

ue

1

1

1

L_____

Pe

”щшшіишшшшшшшшгшшишіт.

Hypothetical flow with no boundary layer (inviscid case)

Figure 1 7.5 Displacement thickness is the distance by which an external flow streamline is displaced by the presence of the boundary layer.

Л’1 AVI

I peue dy — I pu dy + peue <5*

Jo Jo

or s*= Г’ (l-—)dy [17.9]

Jo PeM-e )

Hence, the height by which the streamline in Figure 17.5 is displaced upward by the presence of the boundary layer, namely, &*, is given by Equation (17.9). However, Equation (17.9) is precisely the definition of the displacement thickness given by Equation (17.3). Thus, the displacement thickness, first defined by Equation (17.3), is physically the distance through which the external inviscid flow is displaced by the presence of the boundary layer.

This second interpretation of 8* gives rise to the concept of an effective body. Consider the aerodynamic shape sketched in Figure 17.6. The actual contour of the body is given by curve ab. However, due to the displacement effect of the boundary layer, the shape of the body effectively seen by the freestream is not given by curve ab; rather, the freestream sees an effective body given by curve ac. In order to obtain the conditions ue, Te, etc., at the outer edge of the boundary layer on the actual body ab, an inviscid flow solution should be carried out for the effective body, and pe, u,_, Te, etc., are obtained from this inviscid solution evaluated along curve ac.

Note that in order to solve for S* from Equation (17.3), we need the profiles of и and p from a solution of the boundary-layer flow. In turn, to solve the boundary-layer flow, we need pe, ue, 7,. etc. However, pe, ue, Tc, etc., depend on <5*. This leads to an iterative solution. To calculate accurately the boundary-layer properties as well as the pressure distribution over the surface of the body in Figure 17.6, we proceed as follows:

1. Carry out an inviscid solution for the given body shape ab. Evaluate p, . ue. Te, etc., along curve ab.

2. Using these values of pe, ue, Te, etc., solve the boundary-layer equations (dis­cussed in Sections 17.3 to 17.6) for и — и {у), p = p( у), etc., at various stations along the body.

3. Obtain 8* at these stations from Equation (17.3). This will not be an accurate 5* because pe, ue, Te, etc., were evaluated on curve ab, not the proper effective body. Using this intermediate 8*, calculate an effective body, given by a curve ad (not shown in Figure 17.6).

4. Carry out an inviscid solution for the flow over the intermediate effective body ad, and evaluate new values of pe, ue, Te, etc., along ad.

5. Repeat steps 2 to 4 above until the solution at one iteration essentially does not deviate from the solution at the previous iteration. At this stage, a converged solution will be obtained, and the final results will pertain to the flow over the proper effective body ac shown in Figure 17.6.

In some cases, the boundary layers are so thin that the effective body can be ignored, and a boundary-layer solution can proceed directly from pe, ue, Te, etc., obtained from an inviscid solution evaluated on the actual body surface (ab in Fig­ure 17.6). However, for highly accurate solutions, and for cases where the boundary – layer thickness is relatively large (such as for hypersonic flow as discussed in Chap­ter 14), the iterative procedure described above should be carried out. Also, we note parenthetically that 5* is usually smaller than 5; typically, 8* «a 0.3 8.

Another boundary-layer property of importance is the momentum thickness в, defined as

[17.10]

To understand the physical interpretation of в, return again to Figure 17.4. Consider the mass flow across a segment dy, given by dm — pu dy. Then

A = momentum flow across dy = dm и = pu2 dy

If this same elemental mass flow were associated with the freestream, where the velocity is ue, then

{

momentum flow at freestream velocity associated with mass dm = dm ue = (pu dy)ue

Hence, decrement in momentum flow

(missing momentum flow) associated = pu(ue — u) dy with mass dm

The total decrement in momentum flow across the vertical line from у = 0 to у = y in Figure 17.4 is the integral of Equation (17.11),

[i7.i a]

Assume that the missing momentum flow is the product of peu2e and a height в. Then, Missing momentum flow = peu2e6 [ 17.13]

Equating Equations (17.12) and (17.13), we obtain

[17.14]

Equation (17.14) is precisely the definition of the momentum thickness given by Equation (17.10). Therefore, в is an index that is proportional to the decrement in momentum flow due to the presence of the boundary layer. It is the height of a hypothetical streamtube which is carrying the missing momentum flow at freestream conditions.

Note that в =9(x). In more detailed discussions of boundary-layer theory, it can be shown that в evaluated at a given station x = X is proportional to the integrated friction drag coefficient from the leading edge to x;; that is,

0(xi) a — f x Jo

where Cf is the local skin friction coefficient defined in Section 1.5 and C/ is the total skin friction drag coefficient for the length of surface from x = 0 to x = x. Hence, the concept of momentum thickness is useful in the prediction of drag coefficient.

All the boundary-layer properties discussed above are general concepts; they apply to compressible as well as incompressible flows, and to turbulent as well as laminar flows. The differences between turbulent and laminar flows were introduced in Section 15.2. Here, we extend that discussion by noting that the increased momen­tum and energy exchange that occur within a turbulent flow cause a turbulent boundary layer to be thicker than a laminar boundary layer. That is, for the same edge condi­tions, /)(., Uc. Te, etC., We have ^turbulent ^ ^laminar ttnd ($7) turbulent ■> (^7′)laminar – A hСП a boundary layer changes from laminar to turbulent flow, as sketched in Figure 15.8, the boundary-layer thickness markedly increases. Similarly, 8* and в are larger for turbulent flows.

The Linearized Velocity Potential Equation

Consider the two-dimensional, irrotational, isentropic flow over the body shown in Figure 11.2. The body is immersed in a uniform flow with velocity Voo oriented in the x direction. At an arbitrary point P in the flow field, the velocity is V with the x and у components given by и and v, respectively. Let us now visualize the velocity V as the sum of the uniform flow velocity plus some extra increments in velocity. For example, the x component of velocity и in Figure 11.2 can be visualized as Voo plus an increment in velocity (positive or negative). Similarly, the у component of velocity v can be visualized as a simple increment itself, because the uniform flow has a zero component in the у direction. These increments are called perturbations, and

и — Voo + й v = v

where u and v are called the perturbation velocities. These perturbation velocities are not necessarily small; indeed, they can be quite large in the stagnation region in front of the blunt nose of the body shown in Figure 11.2. In the same vein, because V = Уф, we can define a perturbation velocity potential ф such that

ф = Voo* + ф

дф

dx

дф

dy

Hence,

З 2ф д2ф

дх ду дх ду

Substituting the above definitions into Equation (11.12), and multiplying by a2, we obtain

[1 1.14]

Equation (11.14) is called the perturbation velocity potential equation. It is precisely the same equation as Equation (11.12) except that it is expressed in terms of ф instead of ф. It is still a nonlinear equation.

To obtain better physical insight in some of our subsequent discussion, let us recast Equation (11.14) in terms of the perturbation velocities. From the definition of ф given earlier, Equation (11.14) can be written as

r9 /Т7 Л, 7. 9 „лЗе лл3л

[a ~ (Voo + и)2]— + (a2 – v2)—– 2(Тоо + u)v— = 0

dx dy dy

From the energy equation in the form of Equation (8.32), we have

Кэс _ I (Too + n)2 + P2

2 “ у – 1 +

Substituting Equation (11.15) into (11.14a), and algebraically rearranging, we obtain

й у + 1 й2 у — їй2

tr + ,)^ + — vi + ~K

‘ й у + 1 v2

2 V^ +

Equation (11.16) is still exact for irrotational, isentropic flow. Note that the left-hand side of Equation (11.16) is linear but the right-hand side is nonlinear. Also, keep in mind that the size of the perturbations й and v can be large or small; Equation (11.16) holds for both cases.

Let us now limit our considerations to small perturbations; that is, assume that the body in Figure 11.2 is a slender body at small angle of attack. In such a case, й and v will be small in comparison with V-^. Therefore, we have

V2 ’ V2

Keep in mind that products of й and v with their derivatives are also very small. With this in mind, examine Equation (11.16). Compare like terms (coefficients of like derivatives) on the left – and right-hand sides of Equation (11.16). We find

1. For 0 < Moo £ 0.8 or Moo > 1.2, the magnitude of

or in terms of the perturbation velocity potential,

Examine Equation (11.18). It is a linear partial differential equation, and therefore is inherently simpler to solve than its parent equation, Equation (11.16). However, we have paid a price for this simplicity. Equation (11.18) is no longer exact. It is only an approximation to the physics of the flow. Due to the assumptions made in obtaining Equation (11.18), it is reasonably valid (but not exact) for the following combined situations:

1. Small perturbation, that is, thin bodies at small angles of attack

2. Subsonic and supersonic Mach numbers

In contrast, Equation (11.18) is not valid for thick bodies and for large angles of attack. Moreover, it cannot be used for transonic flow, where 0.8 < Мж < 1.2, or for hypersonic flow, where Мж > 5.

We are interested in solving Equation (11.18) in order to obtain the pressure distribution along the surface of a slender body. Since we are now dealing with approximate equations, it is consistent to obtain a linearized expression for the pres­sure coefficient—an expression which is approximate to the same degree as Equation (11.18), but which is extremely simple and convenient to use. The linearized pressure coefficient can be derived as follows.

First, recall the definition of the pressure coefficient Cp given in Section 1.5:

C„ = -—— [11.19]

QoC

where qoo = 5Pco^tx; = dynamic pressure. The dynamic pressure can be expressed in terms of M0c as follows:

Substituting Equation (11.21) into (11.19), we have

[11.22]

Equation (11.22) is simply an alternate form of the pressure coefficient expressed in terms of Moo. It is still an exact representation of the definition of C

To obtain a linearized form of the pressure coefficient, recall that we are dealing with an adiabatic flow of a calorically perfect gas; hence, from Equation (8.39),

V2 V2

T+- = roo + -^ [11.23]

2,Cp 2cp

Recalling from Equation (7.9) that cp = yR/(y — 1), Equation (11.23) can be written as

T – Tso = —— 25———–

°° 2yR/(y – 1)

Also, recalling that = *Jy Equation (11.24) becomes T x у – 1 – У 2 ^ у _ ! y2 _ V2

Too 2 yRToo 2 a^

In terms of the perturbation velocities

V2 = (Voo + m)2 + v2

Equation (11.25) can be written as

Since the flow is isentropic, p/poo = (T/TooyRy 1 ’, and Equation (11.26) becomes

2 ‘ 00 V V2

[2]oo ‘oo

Equation (11.27) is still an exact expression. However, let us now make the as­sumption that the perturbations are small, that is, й/V.^ 1, u2/ <SY 1, and

v2/ <gc 1. In this case, Equation (11.27) is of the form

— = (1 – e)^"1) [11.28]

Poo

where e is small. From the binomial expansion, neglecting higher-order terms, Equa­tion (11.28) becomes

p v

— = 1——– -—£ + ••• [11.29]

Poo Y – 1

Comparing Equation (11.27) to (11.29), we can express Equation (11.27) as

7- = 1 –

Poo 2


Substituting Equation (11.30) into the expression for the pressure coefficient, Equa­tion (11.22), we obtain

cP =

2

_i – ^

Ґ 2й | й2 + v2 ^

■ – 1

умі

VEoo+ ) +

or

cP =

2 й Й2 + v2

[1 1.31]

Since й2/and v2/V^ <ж. 1, Equation (11.31) becomes

Equation (11.32) is the linearized form for the pressure coefficient; it is valid only for small perturbations. Equation (11.32) is consistent with the linearized perturbation velocity potential equation, Equation (11.18). Note the simplicity of Equation (11.32); it depends only on the x component of the velocity perturbation, namely, й.

To round out our discussion on the basics of the linearized equations, we note that any solution to Equation (11.18) must satisfy the usual boundary conditions at infinity and at the body surface. At infinity, clearly ф = constant; that is, й = v — 0. At the body, the flow-tangency condition holds. Let в be the angle between the tangent to the surface and the freestream. Then, at the surface, the boundary condition is obtained from Equation (3.48c):

v v

tan# = – = ———– – [11.33]

и Ех, + и

which is an exact expression for the flow-tangency condition at the body surface. A simpler, approximate expression for Equation (11.33), consistent with linearized theory, can be obtained by noting that for small perturbations, й « Hence, Equation (11.33) becomes v

Equation (11.34) is an approximate expression for the flow-tangency condition at the body surface, with accuracy of the same order as Equations (11.18) and (11.32).

Viscosity and Thermal Conduction

The basic physical phenomena of viscosity and thermal conduction in a fluid are due to the transport of momentum and energy via random molecular motion. Each molecule in a fluid has momentum and energy, which it carries with it when it moves from one location to another in space before colliding with another molecule. The transport of molecular momentum gives rise to the macroscopic effect we call viscosity, and the transport of molecular energy gives rise to the macroscopic effect we call thermal conduction. This is why viscosity and thermal conduction are labeled as transport phenomena. A study of these transport phenomena at the molecular level is part of kinetic theory, which is beyond the scope of this book. Instead, in this section we simply state the macroscopic results of such molecular motion.

Consider the flow sketched in Figure 15.9. For simplicity, we consider a one­dimensional shear flow, that is, a flow with horizontal streamlines in the x direction but with gradients in the у direction of velocity, du/dy, and temperature, dT/dy.

conduction to velocity and temperature gradients, respectively.

Consider a plane ab perpendicular to the у axis, as shown in Figure 15.9. The shear stress exerted on plane ab by the flow is denoted by xyx and is proportional to the velocity gradient in the у direction, xyx ос 3u/dy. The constant of proportionality is defined as the viscosity coefficient ц. Hence,

Эи

tyx = Iі T~

The subscripts on xyx denote that the shear stress is acting in the x direction and is being exerted on a plane perpendicular to the у axis. The velocity gradient du/dy is also taken perpendicular to this plane (i. e., in the у direction). The dimensions of /і are mass/length x time, as originally stated in Section 1.7 and as can be seen from Equation (15.1). In addition, the time rate of heat conducted per unit area across plane ab in Figure 15.9 is denoted by qY and is proportional to the temperature gradient in the у direction, qy ос ЗT/3y. The constant of proportionality is defined as the thermal conductivity k. Hence,

[15.2]

where the minus sign accounts for the fact that the heat is transferred from a region of high temperature to a region of lower temperature; that is, qy is in the opposite direction of the temperature gradient. The dimensions of к are mass x length/(s2 • K), which can be obtained from Equation (15.2) keeping in mind that qY is energy per second per unit area.

Both д and к are physical properties of the fluid and, for most normal situa­tions, are functions of temperature only. A conventional relation for the temperature variation of ц for air is given by Sutherland’s law,

[15.3]

where T is in kelvin and до is a reference viscosity at a reference temperature, 7q. For example, if we choose reference conditions to be standard sea level values, then до = 1 -7894 x 10-5kg/(m • s) and 7b = 288.16 K. The temperature variation of к is analogous to Equation (15.3) because the results of elementary kinetic theory show that к ос jiCp; for air at standard conditions,

k = l.45 fiCp

where cp = 1000 J/(kg • K).

Equations (15.3) and (15.4) are only approximate and do not hold at high tem­peratures. They are given here as representative expressions which are handy to use. For any detailed viscous flow calculation, you should consult the published literature for more precise values of д and k.

In order to simplify our introduction of the relation between shear stress and viscosity, we considered the case of a one-dimensional shear flow in Figure 15.9. In this picture, the у and z components of velocity, v and w, respectively, are zero. However, in a general three-dimensional flow, u, v, and w are finite, and this requires a generalization of our treatment of stress in the fluid. Consider the fluid element sketched in Figure 15.10. In a three-dimensional flow, each face of the fluid element experiences both tangential and normal stresses. For example, on face abed, xxy and xxz are the tangential stresses, and xxx is the normal stress. As before, the nomenclature rtj denotes a stress in the j direction exerted on a plane perpendicular to the і axis. Similarly, on face abfe, we have the tangential stresses xyx and xyz, and the normal stress xyy. On face adge, we have the tangential stresses x7X and xzy, and the normal stress xzz. Now recall the discussion in the last part of Section 2.12 concerning the strain of a fluid element, that is, the change in the angle к shown in Figure 2.31. What is the force which causes this deformation shown in Figure 2.31? Returning to Figure 15.10, we have to say that the strain is caused by the tangential shear stress.

z

Figure 1 5*10 Shear and normal stresses caused

by viscous action on a fluid element.

However, in contrast to solid mechanics where stress is proportional to strain, in fluid mechanics the stress is proportional to the time rate of strain. The time rate of strain in the xy plane was given in Section 2.12 as Equation (2.135a):

Examining Figure 15.10, the strain in the xy plane must be carried out by тху and ryx. Moreover, we assume that moments on the fluid element in Figure 15.10 are zero; hence, xxy = xyx. Finally, from the above, we know that xxy – xyx ос єху. The proportionality constant is the viscosity coefficient д. Hence, from Equation (2.135a), we have

f dv du

tXy — tyx — Д ( ~dy ) ^ ®-®]

which is a generalization of Equation (15.1), extended to the case of multidimensional flow. For the shear stresses in the other planes, Equations (2.135i> and c) yield

The normal stresses xxx, xyy, and xzz shown in Figure 15.10 may at first seem strange. In our previous treatments of inviscid flow, the only force normal to a surface in a fluid is the pressure force. However, if the gradients in velocity Эм/dx, dv/dy, and dw/dz are extremely large on the faces of the fluid element, there can be a meaningful viscous-induced normal force on each face which acts in addition to the pressure. These normal stresses act to compress or expand the fluid element, hence changing its volume. Recall from Section 2.12 that the derivatives du/dx, dv/dy, and dw/dz are related to the dilatation of a fluid element, that is, to V • V. Hence, the normal stresses should in turn be related to these derivatives. Indeed, it can be shown that

du

= A(V • V) + 2д — dx

[15.8]

dv

= A(V-V) + 2 pt-

dy

[15.9]

dw

= A(V • V) + 2д—— 3 z

[15.10]

In Equations (15.8) to (15.10), A. is called the bulk viscosity coefficient. In 1845, the Englishman George Stokes hypothesized that

A = -| д [15.11]

To this day, the correct expression for the bulk viscosity is still somewhat controversial, and so we continue to use the above expression given by Stokes. Once again, the

normal stresses are important only where the derivatives du/dx, dv/dy, and 3w/dz are very large. For most practical flow problems, xxx, xyy, and rzz are small, and hence the uncertainty regarding X is essentially an academic question. An example where the normal stress is important is inside the internal structure of a shock wave. Recall that, in real life, shock waves have a finite but small thickness. If we consider a normal shock wave across which large changes in velocity occur over a small distance (typically 10-5 cm), then clearly du/dx will be very large, and rxx becomes important inside the shock wave.

To this point in our discussion, the transport coefficients /x and к have been considered molecular phenomena, involving the transport of momentum and energy by random molecular motion. This molecular picture prevails in a laminar flow. The values of /x and к are physical properties of the fluid; that is, their values for different gases can be found in standard reference sources, such as the Handbook of Chemistry and Physics (The Chemical Rubber Co.). In contrast, for a turbulent flow the transport of momentum and energy can also take place by random motion of large turbulent eddies, or globs of fluid. This turbulent transport gives rise to effective values of viscosity and thermal conductivity defined as eddy viscosity є and eddy thermal conductivity k, respectively. (Please do not confuse this use of the symbols £ and к with the time rate of strain and strain itself, as used earlier.) These turbulent transport coefficients є and к can be much larger (typically 10 to 100 times larger) than the respective molecular values /x and k. Moreover, є and к predominantly depend on characteristics of the flow field, such as velocity gradients; they are not just a molecular property of the fluid such as ji and k. The proper calculation of є and к for a given flow has remained a state-of-the-art research question for the past 80 years; indeed, the attempt to model the complexities of turbulence by defining an eddy viscosity and thermal conductivity is even questionable. The details and basic understanding of turbulence remain one of the greatest unsolved problems in physics today. For our purpose here, we simply adopt the ideas of eddy viscosity and eddy thermal conductivity, and for the transport of momentum and energy in a turbulent flow, we replace /г and k in Equations (15.1) to (15.10) by the combination /x + £ and k + к; that is,

An example of the calculation of є and к is as follows. In 1925, Prandtl suggested

that

for a flow where the dominant velocity gradient is in the у direction. In Equa­tion (15.12),/ is called the mixing length, which is different for different applications; it is an empirical constant which must be obtained from experiment. Indeed, all turbulence models require the input of empirical data; no self-contained purely the-

oretical turbulence model exists today. Prandtl’s mixing length theory, embodied in Equation (15.12), is a simple relation which appears to be adequate for a number of engineering problems. For these reasons, the mixing length model for є has been used extensively since 1925. In regard to к, a relation similar to Equation (15.4) can be assumed (using 1.0 for the constant); that is,

к = єср [15.13]

The comments on eddy viscosity and thermal conductivity are purely introduc­tory. The modern aerodynamicist has a whole stable of turbulence models to choose from, and before tackling the analysis of a turbulent flow, you should be familiar with the modern approaches described in such books as References 42 to 45.

Turbulent Boundary Layers

The one uncontroversial fact about turbulence is that it is the most complicated kind of fluid motion.

Peter Bradshaw Imperial College of Science and Technology, London 1978

Turbulence was, and still is, one of the great unsolved mysteries of science, and it intrigued some of the best scientific minds of the day. Arnold Sommerfeld, the noted German theoretical physicist of the 1920s, once told me, for instance, that before he died he would like to understand two phenomena—quantum mechanics and turbulence. Sommerfeld died in 1924. I believe he was somewhat nearer to an understanding of the quantum, the discovery that led to modern physics, but no closer to the meaning of turbulence.

Theodore von Karman, 1967

19.1 Introduction

The subject of turbulent flow is deep, extensively studied, but at the time of writing still imprecise. The basic nature of turbulence, and therefore our ability to predict its characteristics, is still an unsolved problem in classical physics. Many books have been written on turbulent flows, and many people have spent their professional lives working on the subject. As a result, it is presumptuous for us to try to carry out a thorough discussion of turbulent boundary layers in this chapter. Instead, the purpose of this chapter is simply to provide a contrast with our study of laminar boundary layers in Chapter 18. Here, we will only be able to provide a flavor of turbulent

boundary layers, but this is all that is necessary in the present book. Turbulence is a subject that we leave for you to study more extensively as a subject on its own.

Before proceeding further, return to Section 15.2 and review the basic discussion of the nature of turbulence that is given there. In the present chapter, we will pick up where Section 15.2 leaves off.

Also, we note that no pure theory of turbulent flow exists. Every analysis of turbulent flows requires some type of empirical data in order to obtain a practical answer. As we examine the calculation of turbulent boundary layers in the following sections, the impact of this statement will become blatantly obvious. Finally, because this chapter is short, there is no need for a roadmap to act as a guide.

Derivation of the Linearized Supersonic Pressure Coefficient Formula

For the case of supersonic flow, let us write Equation (11.18) as

л^-4=о

Эх2 dy2

where Л = J— 1. A solution to this equation is the functional relation

Ф — f(x-ky) [12.2]

We can demonstrate this by substituting Equation (12.2) into Equation (12.1) as follows. The partial derivative of Equation (12.2) with respect to x can be written as

З ф. d(x — ky)

= / (x — ky)——- —

dx dx

In Equation (12.3), the prime denotes differentiation of / with respect to its argument, x — ky. Differentiating Equation (12.3) again with respect to x, we obtain

d-±= f" dx2 1

Similarly,

Substituting Equations (12.4) and (12.6) into (12.1), we obtain the identity

Xі f" – Xі f" = 0

Hence, Equation (12.2) is indeed a solution of Equation (12.1).

Examine Equation (12.2) closely. This solution is not very specific, because / can be any function of x — Xy. However, Equation (12.2) tells us something specific about the flow, namely, that ф is constant along lines of л – Xy = constant. The slope of these lines is obtained from

x — Xy — const

Hence, ± = ! =_______ ‘___

dx x !Mlc -1

From Equation (9.31) and the accompanying Figure 9.25, we know that

tan/r = , : [12.8]

where p. is the Mach angle. Therefore, comparing Equations (12.7) and (12.8), we see that a line along which ф is constant is a Mach line. This result is sketched in Figure 12.1, which shows supersonic flow over a surface with a small hump in the middle, where 9 is the angle of the surface relative to the horizontal. According to Equations (12.1) to (12.8), all disturbances created at the wall (represented by the perturbation potential ф) propagate unchanged away from the wall along Mach waves. All the Mach waves have the same slope, namely, dy/dx — (M^. — 1)~1/2. Note that the Mach waves slope downstream above the wall. Hence, any disturbance at the wall cannot propagate upstream; its effect is limited to the region of the flow downstream of the Mach wave emanating from the point of the disturbance. This is a further substantiation of the major difference between subsonic and supersonic flows mentioned in previous chapters, namely, disturbances propagate everywhere throughout a subsonic flow, whereas they cannot propagate upstream in a steady supersonic flow.

Keep in mind that the above results, as well as the picture in Figure 12.1, pertain to linearized supersonic flow [because Equation (12.1) is a linear equation]. Hence, these results assume small perturbations; that is, the hump in Figure 12.1 is small,

and thus в is small. Of course, we know from Chapter 9 that in reality a shock wave will be induced by the forward part of the hump, and an expansion wave will emanate from the rearward part of the hump. These are waves of finite strength and are not a part of linearized theory. Linearized theory is approximate; one of the consequences of this approximation is that waves of finite strength (shock and expansion waves) are not admitted.

The above results allow us to obtain a simple expression for the pressure coeffi­cient in supersonic flow, as follows. From Equation (12.3),

and from Equation (12.5),

~ 9<^ і f’

v = — = – A./
dy

Eliminating /’ from Equations (12.9) and (12.10), we obtain

Figure 1 2.2 Variation of the linearized

pressure coefficient with Mach number (schematic).

portion. This is denoted by the (+) and (—) signs in front of and behind the hump shown in Figure 12.1. This is also somewhat consistent with our discussions in Chapter 9; in the real flow over the hump, a shock wave forms above the front portion where the flow is being turned into itself, and hence p > whereas an expansion wave occurs over the remainder of the hump, and the pressure decreases. Think about the picture shown in Figure 12.1; the pressure is higher on the front section of the hump, and lower on the rear section. As a result, a drag force exists on the hump. This drag is called wave drag and is a characteristic of supersonic flows. Wave drag was discussed in Section 9.7 in conjunction with shock-expansion theory applied to supersonic airfoils. It is interesting that linearized supersonic theory also predicts a finite wave drag, although shock waves themselves are not treated in such linearized theory.

Examining Equation (12.15), we note that Cp oc (Af£, — l)-l/2; hence, for su­personic flow, Cp decreases as M0c increases. This is in direct contrast with subsonic flow, where Equation (11.51) shows that Cp cx (1 — M^,)^1/2; hence, for subsonic flow, Cp increases as M^ increases. These trends are illustrated in Figure 12.2. Note that both results predict Cp —>■ oo as M —> 1 from either side. However, keep in mind that neither Equation (12.15) nor (11.51) is valid in the transonic range around Mach 1.

Recovery Factor

As a corollary to the above case for the adiabatic wall, we take this opportunity to define the recovery factor—a useful engineering parameter in the analysis of aerody­namic heating. The total enthalpy of the flow at the upper plate (which represents the

upper boundary on a viscous shear layer) is, by definition,

И2

ho = he + ~

(The significance and definition of total enthalpy are discussed in Section 7.5.) Com­pare Equation (16.45), which is a general definition, with Equation (16.39), repeated below, which is for the special case of Couette flow:

haw = he + Pr^ [16.36]

Note that haw is different from ho, the difference provided by the value of Pr as it appears in Equation (16.39). We now generalize Equation (16.39) to a form which holds for any viscous flow, as follows:

Similarly, Equation (16.40) can be generalized to

[16.46b]

In Equations (16.46a and b), r is defined as the recovery factor. It is the factor that tells us how close the adiabatic wall enthalpy is to the total enthalpy at the upper boundary of the viscous flow. If r = 1, then haw = ho – An alternate expression for the recovery factor can be obtained by combining Equations (16.46) and (16.45) as follows. From Equation (16.46),

haw he

Г = «2/2

From Equation (16.45),

и2

~f=ho~he

Inserting Equation (16.48) into (16.47), we have

where To is the total temperature. Equation (16.49) is frequently used as an alternate definition of the recovery factor.

In the special case of Couette flow, by comparing Equation (16.39) or (16.40) with Equation (16.46a) or (16.46b), we find that

[16.50]

For Couette flow, the recovery factor is simply the Prandtl number. Note that, if Pr < 1, then haw < h0; conversely, if Pr > 1, then haw > h0.

In more general viscous flow cases, the recovery factor is not simply the Prandtl number; however, in general, for incompressible viscous flows, we will find that the recovery factor is some function of Pr. Hence, the Prandtl number is playing its role as an important viscous flow parameter. As expected from Section 15.6, for a compressible viscous flow, the recovery factor is a function of Pr along with the Mach number and the ratio of specific heats.

The Issue of Accuracy for the Prediction of Skin Friction Drag

The aerodynamic drag on a body is the sum of pressure drag and skin friction drag. For attached flows, the prediction of pressure drag is obtained from inviscid flow analyses such as those presented in Parts 2 and 3 of this book. For separated flows, various approximate theories for pressure drag have been advanced over the last century, but today the only viable and general method of the analysis of pressure drag for such flows is a complete numerical Navier-Stokes solution.

The prediction of skin friction on the surface of a body in an attached flow is nicely accomplished by means of a boundary-layer solution coupled with an inviscid flow analyses to define the flow conditions at the edge of the boundary layer. Such an approach is well-developed, and the calculations can be rapidly carried out on

local computer workstations. Therefore, the use of boundary-layer solutions for skin friction and aerodynamic heating is the preferred engineering approach. However, as mentioned above, if regions of flow separation are present, this approach cannot be used. In its place, a full Navier-Stokes solution can be used to obtain local skin friction and heat transfer, but these Navier-Stokes solutions are still not in the category of “quick engineering calculations.”

Zoom view of protuberance grid along the bottom surface of the airfoil.

This leads us to the question of the accuracy of CFD Navier-Stokes solutions for skin friction drag and heat transfer. There are three aspects that tend to diminish the accuracy of such solutions for the prediction of tw and qw (or alternately, c/ and ChY

1. The need to have a very closely spaced grid in the vicinity of the wall in order to obtain an accurate numerical value of (du/dy)w and (ЗT/dy)w, from which rw and qw are obtained.

2. The uncertainty in the accuracy of turbulence models when a turbulent flow is being calculated.

3. The lack of ability of most turbulent models to predict transition from laminar to turbulent flow.

Computed velocity vector field around and downstream of the protuberance.

In spite of all the advances made in CFD to the present, and all the work that has gone into turbulence modeling, at the time of writing the ability of Navier-Stokes

solutions to predict skin friction in a turbulent flow seems to be no better than about 20 percent accuracy, on the average. A recent study by Lombardi et al. (Reference 92) has made this clear. They calculated the skin friction drag on an NACA 0012 airfoil at zero angle of attack in a low-speed flow using both a standard boundary-layer code and a state-of-the-art Navier-Stokes solver with three different state-of-the-art turbulence models. The results for friction drag from the boundary-layer code had been validated with experiment, and were considered the baseline for accuracy. The boundary-layer code also had a prediction for transition that was considered reliable. Some typical results reported in Reference 92 for the integrated friction drag coefficient C/ are as follows, where NS represents Navier-Stokes solver and with the turbulence model in parenthesis. The calculations were all for Re = 3 x 106.

Cf X 103

NS (Standard к — є)

7.486

NS (RNG к-є)

6.272

NS (Reynolds stress)

6.792

Boundary Layer Solution

5.340

Clearly, the accuracy of the various Navier-Stokes calculations ranged from 18 percent to 40 percent.

More insight can be gained from the spatial distribution of the local skin friction coefficient Cf along the surface of the airfoil, as shown in Figure 20.15. Again the three different Navier-Stokes calculations are compared with the results from the boundary layer code. All the Navier-Stokes calculations greatly overestimated the peak in c/ just downstream of the leading edge, and slightly underestimated c/ near the trailing edge.

For a completely different reason not having to do with our discussion of accuracy, but for purposes of showing and contrasting the physically different distribution of Cf along a flat plate compared with that along the surface of the airfoil, we show Figure 20.16. Here the heavy curve is the variation of с/ with distance from the leading edge for a flat plate; the monotonic decrease is expected from our previous discussions of flat plate boundary layers. In contrast, for the airfoil Cf rapidly increases from a value of zero at the stagnation point to a peak value shortly downstream of the leading edge. This rapid increase is due to the rapidly increasing velocity as the flow external to the boundary layer rapidly expands around the leading edge. Beyond the peak, c/ then monotonically decreases in the same qualitative manner as for a flat plate. It is simply interesting to note these different variations for c/ over an airfoil compared to that for a flat plate, especially since we devoted so much attention to flat plates in the previous chapters.

20.5 Summary

With this, we end our discussion of viscous flow. The purpose of all of Part 4 has been to introduce you to the basic aspects of viscous flow. The subject is so vast that it demands a book in itself—many of which have been written (see, e. g., References 41 through 45). Here, we have presented only enough material to give you a flavor for some of the basic ideas and results. This is a subject of great importance in aerody­namics, and if you wish to expand your knowledge and expertise of aerodynamics in general, we encourage you to read further on the subject.

We are also out of our allotted space for this book. Therefore, we hope that you have enjoyed and benefited from our presentation of the fundamentals of aerodynam­ics. However, before closing the cover, it might be useful to return once again to Figure 1.38, which is the block diagram categorizing the different general types of aerodynamic flows. Recall the curious, uninitiated thoughts you might have had when you first examined this figure during your study of Chapter 1, and compare these with the informed and mature thoughts that you now have—honed by the aerodynamic knowledge packed into the intervening pages. Hopefully, each block in Figure 1.38 has substantially more meaning for you now than when we first started. If this is true, then my efforts as an author have not gone in vain.

[1] у –

1 H—- ~r(Mj — 1)

Y + 1 .

From Equation (8.68), we see that the entropy change S2 — s 1 across the shock is a function of Mi only. The second law dictates that

S2 — S >0

In Equation (8.68), if Mi = l, s2 = v,, and if Mi > 1, then, v2 — .? 1 > 0, both of which

[2] = Voo tan в

[3] vx

Detached Shock Wave in Front of a Blunt Body

The curved bow shock which stands in front of a blunt body in a supersonic flow is sketched in Figure 8.1. We are now in a position to better understand the properties of this bow shock, as follows.

The flow in Figure 8.1 is sketched in more detail in Figure 9.21. Here, the shock wave stands a distance 8 in front of the nose of the blunt body; 8 is defined as the shock detachment distance. At point a, the shock wave is normal to the upstream flow; hence, point a corresponds to a normal shock wave. Away from point a, the shock wave gradually becomes curved and weaker, eventually evolving into a Mach wave at large distances from the body (illustrated by point e in Figure 9.21).

Figure 9.31 Flow over a supersonic blunt body.

A curved bow shock wave is one of the instances in nature when you can observe all possible oblique shock solutions at once for a given freestream Mach number M|. This takes place between points a and e. To see this more clearly, consider the в-Р-М diagram sketched in Figure 9.22 in conjunction with Figure 9.21. In Figure 9.22, point a corresponds to the normal shock, and point e corresponds to the Mach wave. Slightly above the centerline, at point b in Figure 9.21, the shock is oblique but pertains to the strong shock-wave solution in Figure 9.22. The flow is deflected slightly upward behind the shock at point b. As we move further along the shock, the wave angle becomes more oblique, and the flow deflection increases until we encounter point c. Point c on the bow shock corresponds to the maximum deflection angle shown in Figure 9.22. Above point c, from c to e, all points on the shock correspond to the weak shock solution. Slightly above point c, at point c’, the flow behind the shock becomes sonic. From a to c the flow is subsonic behind the bow shock; from c’ to e, it is supersonic. Hence, the flow field between the curved bow shock and the blunt body is a mixed region of both subsonic and supersonic flow. The dividing line between the subsonic and supersonic regions is called the sonic line, shown as the dashed line in Figure 9.21.

The shape of the detached shock wave, its detachment distance <5, and the com­plete flow field between the shock and the body depend on M and the size and shape

Figure 9.22 9-fi-hA diagram for the sketch shown in Figure 9.21.

of the body. The solution of this flow field is not trivial. Indeed, the supersonic blunt – body problem was a major focus for supersonic aerodynamicists during the 1950s and 1960s, spurred by the need to understand the high-speed flow over blunt-nosed missiles and reentry bodies. Indeed, it was not until the late 1960s that truly suffi­cient numerical techniques became available for satisfactory engineering solutions of supersonic blunt-body flows. These modem techniques are discussed in Chapter 13.

Elements of Hypersonic Flow

Almost everyone has their own definition of the term hypersonic. If we were to conduct something like a public opinion poll among those present, and asked everyone to name a Mach number above which the flow of a gas should properly be described as hypersonic there would be a majority of answers round about 5 or 6, but it would be quite possible for someone to advocate, and defend, numbers as small as 3, or as high as 12.

P. L. Roe, comment made in a lecture at the von Karman Institute, Belgium January 1970

14.1 Introduction

The history of aviation has always been driven by the philosophy of “faster and higher,” starting with the Wright brothers’ sea level flights at 35 mi/h in 1903, and progressing exponentially to the manned space flight missions of the 1960s and 1970s. The current altitude and speed records for manned flight are the moon and 36,000 ft/s—more than 36 times the speed of sound—set by the Apollo lunar capsule in 1969. Although most of the flight of the Apollo took place in space, outside the earth’s atmosphere, one of its most critical aspects was reentry into the atmosphere after completion of the lunar mission. The aerodynamic phenomena associated with very high-speed flight, such as encountered during atmospheric reentry, are classified as hypersonic aerodynamics—the subject of this chapter. In addition to reentry vehicles, both manned and unmanned, there are other hypersonic applications on the horizon, such as ramjet-powered hypersonic missiles now under consideration by the military and the concept of a hypersonic transport, the basic technology of which is now being studied by NASA. Therefore, although hypersonic aerodynamics is at one extreme

end of the whole flight spectrum (see Section 1.10), it is important enough to justify one small chapter in our presentation of the fundamentals of aerodynamics.

This chapter is short; its purpose is simply to introduce some basic considerations of hypersonic flow. Therefore, we have no need for a chapter road map or a summary at the end. Also, before progressing further, return to Chapter 1 and review the short discussion on hypersonic flow given in Section 1.10. For an in-depth study of hypersonic flow, see the author’s book listed as Reference 55.