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.