Coupling the Numerical Solution of the Partial Differential Equations of the Inviscid Flow with Integral Equations of Boundary Layers

9.1.2.1 Inviscid Potential Flow Field Calculations and Integral Momentum Equations of Boundary Layer

For small disturbance potential flows, the nonlinear governing equation is given by the conservation form

d F д2ф dx + dy2

where

This equation is valid for transonic flows with shocks as well as pure sub – and super-sonic flows, including the incompressible case (M0 = 0).

The body is augmented with the displacement thickness and the boundary condi­tion becomes, for the upper surface

Together with the integral momentum equation, the discrete equations can be solved using standard relaxation methods. To help the convergence, artificial time and viscosity terms are added in terms of the displacement thickness

The parameters a and e may be chosen guided by numerical analysis. Horizontal line relaxation with block pentadiagonal solver for the potential and S* at the lines above and below the airfoil (at у = 0), will allow tighter coupling for the case of boundary layer separation.

Moses et al. [17], solved the integral momentum equation and the integral mechan­ical energy equation simultaneously with the inviscid potential flow equation to sim­ulate separated flow in a diffuser.

In their work, the velocity profile is assumed to be a function of two parameters

where S is the thickness of the boundary layer and в is a profile shape parameter.

The integral equations can be written as a system of two ordinary differential equations

d S d в due

an + a2 = bn + b2

dx dx dx

d S d в due

a21 + a22 = bt2 + b22 (9.46)

dx dx dx

For attached flows, with a given velocity ue, these two equations can be solved numerically with a stepwise procedure in the downstream direction.

For separated flows, the equations are singular at the separation point; the determi­nant of the coefficient matrix, A = (at, j), is zero. Solutions exist only for a specific

pressure gradient and in this case, the solution is not unique. Moreover, the calcula­tion is unstable for reverse flows. To avoid the difficulties associated with separation, the edge velocity, ue, is treated as unknown and the discrete inviscid equations are solved, augmented with the two equations for S and в, using vertical line relaxation, sweeping in the main flow direction.

Moses et al. [17] represented the inviscid flow in terms of stream function, hence the governing equation is

V2 ф = 0 (9.47)

and the boundary condition is

ф = const. (9.48)

on the displaced wall, say on the upper surface, y = yu+S*. The coupling between the inviscid flow and the boundary layer is imposed through the relation at the displaced wall

To start the calculations, quasi-one dimensional problem is solved to provide a good initial guess. The development of boundary layer through separation and attachment is successfully simulated and the edge velocity, ue, the displacement thickness, S*, and the skin friction coefficient, Cf, are accurately calculated.