2 Different Transonic Small Disturbance Equations?
 

Hello Folks, I am currently looking at solutions of the transonic small disturbance. It seems, however, that there are two different forms of the equation and I'm unsure why this would be the case.

One form (as found in the famous Murman & Cole Paper of 1971 - the "Karman-Gudrely" equation) comes from a singular perturbation technique using the airfoil thickness ratio 'tr' as a small parameter. The equation is given as

(K -(gam +1)*P_x)*P_xx + P_yy =0

where K is a similarity constant including the mach number 'M' and thickness ratio 'tr': K=(1-M^2)/tr^(2/3). 'P' is the velocity potential 'gam' the thermodynamic constant and _x denotes differentiation w.r.t x.

I have seen another form of the equation which seems inconsistent with the KG equation, however: (1-M^2)*P_xx - (gam+1)*M^2*P_x*P_xx + P_yy =0;

By substituting the expression for K into the above expression the 2 equations are different. The two equations are both used to solve for flow around an airfoil with the same boundary conditions. I can't understand how two different equations can correspond to the same physical system. The second of the two equations doesn't even contain the airfoil thickness ratio!

Could someone please shed some light on this issue?

Cheers F

Re: 2 Different Transonic Small Disturbance Equati
 

In the second equation, the spatial coordinates have not been made non-dimensional. These equations are application for M very close to 1 with M>1. In the first form, the M² multiplying (γ+1) has been approximated as 1.