Hello Folks, I've been trying to write a finite difference code that will solve for transonic flow over a lifting wing. The scheme I am using is an adaptation of the transonic small disturbance solver proposed by Murman & Cole in 1971. I am trying to reproduce the work of Krupp and Murman (1973 ish) as a steping stone to go on to further TSD work.

I am having a GREAT deal of difficulty in reproducing their results although I have duplicated the method as presented in their papers and in Krupp's PhD thesis. I cannot get the Kutta condition satisfied exactly at the trailing edge of my airfoil.

I am fairly sure that the far-field boundary condition is not to blame for this problem, I am using a finer mesh than Krupp & Murman did but this shouldn't cause the problem. The boundary conditions upon the central slit seem to be causing the problem. The flow must be tangential to the surfaces of the airfoil and this is satisfied in my results. At the trailing edge of the airfoil, however, the flow velocity values are not exactly the same on the upper & lower surfaces.

A plot of the pressure coefficient shows that the vaues upon the upper and lower surfaces equalise approaching the trailing edge but at the actual trailing edge there is a rather sudden (but small) deviation between the 2 values. Behind the airfoil the pressure become equal within a few mesh points.

If anyone could offer some advice I'd be very grateful - I've been trying to solve this problem for months now and am starting to go insane )

Bren

You have to make sure that you do not have a grid point at the trailing edge since the potential function is double valued there. Downstream of the trailing edge a jump in potential has to be introduced for a lifting airfoil. At each iteration this jump in potential has to be obtained by extrapolating two values on the upper and the lower surface of the airfoil all the way to the trailing edge. This jump in potential can then be used at the next iteration for the points in the wake.

A discussion can be found in Lecture Notes in Physics Vol 41 Progress in Numerical Fluid Dynamics edited by H.J. Wirz, which is a von Karman Institute lecture series held in February 1974. There is a paper by F.R. Bailey which discusses relaxation methods for steady transonic flows.

Thanks very much for pointing me in the right direction. I looked up the Bailey paper and it was a big help.

In the end I found that the problem was caused by the way in which I was calculating the potential values on the upper & lower airfoil surfaces. The work presented by Krupp indicates that the airfoil lies upon a gridline of the cartesian mesh with the nodes that make up the airfoil (which is represented as a slit on the y=0 axis) corresponding to the lower surface of the airfoil. The potential upon the upper surface is found via linear extrapolation using neighbouring nodes.

My problem was caused by the linear extrapolation; the extraplation is only used for the upper airfoil surface. The lower airfoil surface is dealt with as part of the iterative scheme. The discrepancy between the potential values on the upper and lower surfaces at the trailing edge caused the Kutta condition to be violated.

There are 2 techniques I found to avoid the problem: 1. Solve the problem normally then after calculations are completed re-calculate the potential values upon upper AND lower surfaces using linear extrapolation. 2. The airfoil lies between gridlines & thus no need to use different methods for upper & lower surfaces. This also makes the tangential flow condition more accurate.