# The Governing Equations and the Computational Algorithm

The governing equations are the compressible Navier-Stokes equations in two dimensions. In dimensionless form with D(D = 3.3 mm is the thickness of the cavity overhang) as length scale, U, the free-stream velocity (from left to right), as velocity scale, D/U as the time scale, p0, the ambient gas density, as the density scale and p0U2 as the pressure scale, they are

 dp 9Uj dp + p + и j = 0 91 H dXj 1 dXj (12.37) 9и 9и _ dp dtij “I – U — “I­dt 19 Xj 9 xt 9 Xj (12.38) 9 p 9 p 9 Uj It + и1 Ц + Y p dXj = 0 (12.39) T = _jl + дииЛ 4 Rd dxj dxi (12.40)

where Rd is the Reynolds number based on D.

These equations are solved in time by the multi-size-mesh multi-time-step DRP algorithm. In each subdomain of Figure 12.15, the equations are discretized by the DRP scheme. At the mesh-size-change boundaries, special stencils as given in

Section 12.1 are used. The time steps of adjacent subdomains differ by a factor of 2 just as the mesh size does. With the use of the multi-size-mesh multi-time-step algorithm, most of the computation effort and time are spent in the opening region of the cavity where the resolution of the unsteady viscous layers is of paramount importance.

12.4.2.1 Numerical Boundary Conditions

Along the solid surfaces of the cavity and the outside wall, the no-slip boundary con­dition is enforced by the ghost point method. Along the vertical external boundary regions (3 mesh points adjacent to the boundary), the flow variables are split into a mean flow and a time-dependent component. The mean flow, with a given boundary layer thickness, is provided by the Blasius solution. The time-dependent part of the solution is the only portion of the solution that is computed by the time marching scheme. The boundary conditions used for the computation are as follows. Along the top and left external boundaries, the asymptotic radiation boundary conditions of Section 6.1 are imposed. Along the right boundary, the outflow boundary conditions of Section 9.3 are used.