Boundary-Layer Thickness at Stagnation Points and Attachment Lines
The attached viscous flow, i. e., the boundary layer, has its origin at the forward stagnation point of a flight vehicle configuration .
The primary attachment point, depending on the angle of attack, can lie away from the nose-point at the lower side of the configuration, as we have seen for the Blunt Delta Wing, Fig. 3.16. Also the primary attachment lines on a hypersonic flight vehicle with the typical large leading-edge sweep of the wing, can lie away from the blunt-wing leading edge at the lower side of the wing. Secondary and tertiary attachment lines can be present. We have shown this for the BDW in Figs. 3.16 and 3.17 (see also Figs. 3.19 and 3.20). In such cases we can have an infinite swept wing flow situation, i. e., zero or only weak changes of flow parameters along the attachment line, like on the (two) primary and the tertiary attachment line of the BDW. The same situation can be present at separation lines. This all is typical for RV’s and CAV’s.
The boundary layer at the forward stagnation point has a finite thickness despite the fact that there both the external inviscid velocity ue and the tangential boundary-layer velocity u(y) are zero . The situation is similar at a three-dimensional attachment line, however there exists a finite velocity along it.
In the following discussions and also for the consideration of the wall shear stress and the thermal state of the surface we idealize the flow situation. We study the situation at the stagnation point of a sphere, and of a circular cylinder (2-D case), and at the attachment line of an infinite swept circular cylinder. The reason is that the velocity gradient due/dxx=0, which governs the flow there (at the swept cylinder it is the gradient across the attachment line), can be introduced explicitly as function of the radius R, Sub-Section 6.7.2. This may be a rather crude approximation of the situation found in reality on a RV, but it fits the situation more or less exactly for a CAV. Nevertheless, it permits us to gain insight into the basic parameter dependencies of the boundary-layer thickness, and later also of the wall-shear stress, SubSection 7.2.4, and of the thermal state of the surface, Sub-Section 7.2.6, at an attachment point and at a primary attachment line.
The classical approach to describe the boundary layer at a stagnation point is to replace in the boundary-layer equation, explicitly or implicitly, for instance the velocity ux=o by (du/dx)x=0, which, like (due/dx)x=o, is finite. This operation can be made, for instance, by differentiating the momentum equation in question with respect to the corresponding tangential coordinate .
We consider now the flow along an attachment line. At the attachment line we have a finite inviscid velocity component along it, and hence a boundary layer of finite thickness. Both may not or only weakly change in the direction of the attachment line.
In order to obtain the basic dependencies of the boundary-layer thickness, we assume that the attachment-line flow locally can be represented by the flow at the attachment line of an infinite swept circular cylinder, Fig. 6.37 b). There we have constant flow properties at the attachment line in the z-direction.
We only replace the velocity gradient for the stagnation point in eq. (7.133) by that for the swept infinite cylinder, eq. (6.167), and find for the laminar boundary layer:
For a subsonic leading edge, i. e., Mcos p < 1, exact theory shows that the dependence of S indeed is oc l/^cosy, but for a supersonic leading edge 5 is somewhat larger .
Summary. We summarize the dependencies in Table 7.4. We chose, like in Sub-Section 7.2.1, ш = ш^ = 0.65 in the viscosity law, Section 4.2. In all cases S is proportional to /Д, i. e., the larger the nose radius or the leading-edge radius, the thicker is the boundary layer. Also in all cases 5 increases with increasing reference-temperature ratio, this means in particular also with increasing wall temperature.
In the case of the infinite swept circular cylinder 5 increases ж (cos p)-0 5, at least for small sweep angles p. This result will hold also for turbulent flow. For p — 90° we get 5 —— ж. This is consistent with the situation on an infinitely long cylinder aligned with the free-stream direction.
Table 7.4. Sphere/circular cylinder and infinite swept circular cylinder: dependence of the boundary-layer thicknesses (laminar flow) on the radius R, the sweep angle y, and the reference-temperature ratio T*/Tref (ш = = 0.65).