Wall Boundary Conditions

If the body in Figure 3.18 has a solid surface, then it is impossible for the flow to penetrate the surface. Instead, if the flow is viscous, the influence of friction between the fluid and the solid surface creates a zero velocity at the surface. Such viscous flows are discussed in Chapters 15 to 20. In contrast, for inviscid flows the velocity at the surface can be finite, but because the flow cannot penetrate the surface, the velocity vector must be tangent to the surface. This “wall tangency” condition is illustrated in Figure 3.18, which shows V tangent to the body surface. If the flow is tangent to the surface, then the component of velocity normal to the surface must be zero. Fet n be a unit vector normal to the surface as shown in Figure 3.18. The wall boundary condition can be written as

Equation (3.48a or b) gives the boundary condition for velocity at the wall; it is expressed in terms of ф. If we are dealing with ф rather than ф, then the wall boundary condition is

[3.48c]

where 5 is the distance measured along the body surface, as shown in Figure 3.18. Note that the body contour is a streamline of the flow, as also shown in Figure 3.18. Recall that ф = constant is the equation of a streamline. Thus, if the shape of the body in Figure 3.18 is given by уъ = fix), then

^surface — Ф у=}h — Const

is an alternative expression for the boundary condition given in Equation (3.48c).

If we are dealing with neither ф nor ф, but rather with the velocity components и and v themselves, then the wall boundary condition is obtained from the equation of a streamline, Equation (2.118), evaluated at the body surface; that is,

[3.48e]

Equation (3.48e) states simply that the body surface is a streamline of the flow. The form given in Equation (3.48e) for the flow tangency condition at the body surface is used for all inviscid flows, incompressible to hypersonic, and does not depend on the formulation of the problem in terms of ф or ф (or ф).