Numerical Method for the Solution of Boundary Layer Equations

A Cartesian mesh system is defined, with constant steps Ax and Ay and indices i and j in the x – and y-directions, respectively. This is an initial/boundary value problem. Before the obstacle, a uniform flow is specified ui, j = U, vi, j = 0, j = 1, jx. The pressure gradient or source term is specified from the inviscid flow solution. The continuity equation is used to solve for v and the momentum equation for u. Consider the x-momentum equation:

Подпись:_du _du dui d2 u

dx + p – y = >m~du + >‘dy

where u and v are evaluated from previous iteration (for initial guess, extrapolate from previous line). Two boundary conditions are specified for u, i. e.

Подпись: (8.39)У1 = 0, ui,1 = 0, yjx = h, ui, jx = U, i = 2,…

h is the ordinate of the upper boundary, h > 5(l), large enough to include the boundary layer.

Central finite differences are used for the first and second partial derivatives of u in y

Подпись:Numerical Method for the Solution of Boundary Layer EquationsПодпись:Numerical Method for the Solution of Boundary Layer Equations

Подпись: ui, j - ui-1, j Ax Подпись: i = 2,..., j = 2,..., jx — 1 Подпись: (8.41)
Numerical Method for the Solution of Boundary Layer Equations

and i

A tridiagonal solver is used to obtain ui, j for line i = 2,…

Note that, as long as ui, j > 0, the tridiagonal matrix is diagonally dominant, which insures stability with respect to round-off errors.

Consider next the continuity equation to solve for v. One boundary condition is specified

Подпись: (8.42)У1 = 0, v;,1 = 0, i = 2,…

Numerical Method for the Solution of Boundary Layer EquationsПодпись: i,jПодпись: (

Numerical Method for the Solution of Boundary Layer Equations
Подпись: Ax

The following scheme is used

Подпись: (8.43)i = 2,…, j = 1,…, jx — 1

This equation is marched upward along the vertical line, solving for vi, j+1.

The process is repeated, i. e. one iterates on u and v on line i until convergence (3 or 4 iterations). Then, one marches to the next line i + 1. The domain is swept once in the flow direction.

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