Obtain an explicit expression for the vertical velocity component using a finite difference scheme prescribed in Appendix 10.

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The unsteady potential flow solution gives us the time dependent value of the surface vortex sheet strength. The velocity component tangent to the airfoil surface can be obtained from the surface vortex sheet strength. This tangent velocity is nothing but the boundary layer edge velocity which is to be used as a boundary condition for the vorticity transport equation. At the edge of the boundary layer the vorticity value becomes zero. The boundary layer equations, Eqs. 8.4-8.7, can be solved with marching in the main flow direction as follows. If we discretize time with At, space with Ax and Ay, then Eq. 8.5 becomes an algebraic equation with superscript n showing the time step, and i, j indicating the discrete locations in x, y directions, as follows "-1 -ui.j 00-00-1, Ar +-- 00-1 2Ay Re 1+-200+ (Ay) (A10.1) Organizing Eq. A10.1 for the unknown values of coin j at a station i gives A+B+C D. J-2,J (A10.2) Here, co, is the unknown wall vorticity value and +1 = 0 is the vorticity at the edge of the boundary layer. This makes the number of unknowns, J, one more than the number of equations given by A10.2. If we find one more equations we can close the problem, i.e., have equal number of equations with unknowns. If we show the free stream velocity with U the velocity at the upper surface of the profile becomes Vu(x, y, t)=U+u' = U+(x, y, t)/2 and at the lower surface V4(x, y, t) = U-u' = U - Y(x, y, t)/2 (A10.3) (A10.4) Integrating the vorticity values normal to the surface as shown in Fig. A10.1 gives yj ^ Fig. A10.1. The boundary layer velocity profile -001/2002-003 - 01 = V/An V(x,y,t) (A10.5) Hence, from the simultaneous solution of Eqs. A10.2 and A10.5 we obtain the vorticity values. Once we know the vorticity profile at a station we can obtain the tangential velocity components at a point i, j by numerical integration as follows ui.j 1ij-1+ (10/2+ ) k=1 (A10.6) The vertical velocity components, on the other hand, are obtained with the proper discretezation of the continuity equation as follows. Vij = Vij-1 Ax (uijuij-1-ui-1j-ui-1,-1) (A10.7) The continuity equation is discretized involving the points shown in the mole- cule below. i-1.j Ay Ax ij i-1j-1 ij-1 Now, writing Eq. A10.5 as the first line and the open form of Eq. A10.2 as the rest of the lines, the matrix form of those become [1/2 1 1 1 000 -V/An A2 B C3 002 D. . A3 B3 C3 103 D3 (A10.8) @J-1 DJ-1 DJ Aj Bj 00J