Prandlt-Glauert Similarity

1. Prandlt-Glauert Similarity • Consider the linearized potential equation for subsonic flow: • where: • Consider a coordinate transformation in the x direction • No changes occur in the y direction, but the x derivative become: AE 401 Advanced Aerodynamics

2. Prandlt-Glauert Similarity [2] • With this transformation, the governing equation becomes: • Thus, the solution in this transformed space is simply a solution for incompressible flow. • To proceed from this point, we must consider how to apply the boundary conditions. • If two solutions are obtained for the same geometry but using the two equations above: • The solutions will differ by: AE 401 Advanced Aerodynamics

3. Prandlt-Glauert Similarity [3] • The pressure coefficients, found from the x derivatives, will be related by the same factor: • Similarly, the forces found by integrating pressures will differ by the same factor. • This idea of correcting incompressible solutions for compressibility is know as Prandlt-Glauert similarity. • Generally, this approach works to speed of around Mach = 0.6 – but breaks down soon after that. AE 401 Advanced Aerodynamics

4. Prandlt-Glauert Similarity [4] • Another approach is to find the changes in geometry to achieve the same potential solution. • Since the solution is dictated by the boundary condition, this requirement, in 2-D, becomes: • We can handle this at least two ways: • fix the y axis and stretch the x axis, as we did when we started. • or, in order to keep the planform the same, fix the x axis and stretch the y axis: AE 401 Advanced Aerodynamics

5. Prandlt-Glauert Similarity [5] • Thus, to have the same solution (including lift), the compressible flow geometry would be thinner then the incompressible geometry by the factor . • This agrees with what you may have already observed for aircraft. • To capture both these effects, a generalization of the Prandlt-Glauert similarity rule is: • Which relates the thickness, pressure coefficient and Mach number at one condition to that at another. AE 401 Advanced Aerodynamics

6. y x x h l Subsonic Wavy Wall • The wavy wall problem is a classical solution which demonstrates differences between flow types. • Consider the infinite wall given by: • The flow along this wall will satisfy our small perturbation potential: • With the flow tangency boundary condition: AE 401 Advanced Aerodynamics

7. Subsonic Wavy Wall [2] • However, rather than apply this BC on the wall, it is consistent with our small perturbation assumptions to apply it on the y=0 axis: • To solve this system, use the classical separation of variables technique. • First assume that the solution can be expressed by the product of functions, each of which are dependant upon only one variable: • The derivatives of our potential function are then: AE 401 Advanced Aerodynamics

8. Subsonic Wavy Wall [3] • Thus, our governing equation becomes: • Or: • This last equation states that the some relation which depends only on x must be equal to another relation which depends only on y. • The only way for this to be true for all x and y is if the two relations equal a common constant, k2: AE 401 Advanced Aerodynamics

9. Subsonic Wavy Wall [4] • Thus, our assumption of separation of variables leads to two independent 2nd order ODE’s: • Assuming that k is real, then these two ODE’s have solutions: • The first BC that we can apply is the fact that in subsonic flow, the disturbances created by the wall should not propagate to infinity. • This BC can only be satisfied if: AE 401 Advanced Aerodynamics

10. Subsonic Wavy Wall [5] • The remaining terms, combined, become: • And the perturbation vertical velocity: • Or, on the axis: • Comparing with our flow tangency, BC: • shows that: AE 401 Advanced Aerodynamics

11. Subsonic Wavy Wall [6] • Our solution to this problem is thus: • To get the pressure coefficient, we need the x velocity perturbation: • And then: AE 401 Advanced Aerodynamics

12. x x Subsonic Wavy Wall [7] • The first thing we note about this solution is that the pressure distribution is 180o out of phase with the wall. • Thus high pressures occur a the low spots, low pressures on the peaks. • More importantly, the pressure distribution is symmetric about the peaks (or troughs) • Thus, an integration of the pressures would produce no net force in the x-direction - no drag! • A closer evaluation would show that the pressures are related to the 2nd derivative – or curvature – of the wall. AE 401 Advanced Aerodynamics

13. Subsonic Wavy Wall [8] • Next, the rate of attenuation of the disturbance away from the wall decreases with the Mach number. • Thus, the higher the Mach number, the disturbances are felt further away from the wall – or stronger at the same location: M = 0.6 M = 0 AE 401 Advanced Aerodynamics

14. Improved Compressibility Corrections • In the wavy wall solution, the Prandlt-Glauert factor once again appears as a similarity parameter. • An improvement on the Prandlt-Glauert rule was suggested by Laitone using the local Mach number rather than the freestream. • Bu using issentropic relations to relate the local mach and pressure coefficient, Laitone obtained: AE 401 Advanced Aerodynamics

15. Improved Compressibility Corrections [2] • Laitone’s correction is an improvement over Prandlt-Glauert, although with less mathematical basis. • However, Laitone’s correction does tend to overpredict the compressible effect. • Another correction based upon some simplified solution methods was suggested by Karman-Tsien: • This correction, while of comparable accuracy to Laitone’s, has greater acceptance. AE 401 Advanced Aerodynamics