Supersonic Potential Flow

1. Supersonic Potential Flow • For supersonic flow, we write the small perturbation potential equation as: • Writing the equation in this form highlights the difference from the subsonic equation. • A PDE in this form is said to be hyperbolic - while the subsonic equation is said to be elliptical. • The names come from the form of the equations for ellipses and hyperbolas: • However, the differences between the equations is more fundamental then appearances. AE 401 Advanced Aerodynamics

2. Supersonic Potential Flow [2] • Elliptical equations are characterized by the following: • Smooth and continuous interior solutions; maximum and minimum on boundaries. • All points on the interior depend upon all points on the boundary (if only very slightly!). • In contrast, hyperbolic equations are characterized by: • The possible existence of discontinuities in interior. • Wavelike propagation of information from boundaries into interior. • Regions of influence and regions of “silence”. • You might recognize some of these from the theories for subsonic and supersonic flow you have learned. • These characteristics dictate how to approach solutions! AE 401 Advanced Aerodynamics

3. y x x h l Supersonic Wavy Wall • Once again, consider the wavy wall problem: • The flow tangency boundary condition is the same as before: • However, this time we will be solving: • Since this equation is hyperbolic, we will not be using separation of variable to solve this. AE 401 Advanced Aerodynamics

4. Supersonic Wavy Wall [2] • Instead, assume a solution in the form of: • Either of these two functions are solutions to the governing equation as can be demonstrated by: AE 401 Advanced Aerodynamics

5. Supersonic Wavy Wall [3] • Combining these yields: • However, these two functions represent two different types of solutions. • Since f is a function of x-y, then it will have constant values along lines described by equation. • Similarly, g is a function of x+y, then it will have constant values along these lines. • These lines thus form waves as seen below. AE 401 Advanced Aerodynamics

6. Supersonic Wavy Wall [4] • Note that the angle shown, , is given by: • Thus, these lines represent Mach waves. Left running waves f = constant Right running waves g = constant AE 401 Advanced Aerodynamics

7. Supersonic Wavy Wall [5] • For our solution, we are obviously only interested in the left running waves. Thus: • With the velocity components: • To satisfy our flow tangency boundary condition, lets use surface slopes: • Thus, the u perturbation velocity, assuming small angles, is: AE 401 Advanced Aerodynamics

8. Supersonic Wavy Wall [6] • So, finally, our pressure distribution along this wall is: • You may recall deriving the same equation as the limit for weak shocks on a thin airfoil. • From this solution, we see that the supersonic pressure coefficient varies directly with slope - rather than curvature like the subsonic solution. • Also, we never applied a far field boundary condition since these waves, in 2-D, will propagate to infinity. • Note that these waves don’t coalesce or fan-out like shock or expansion waves - a result of our linearization. AE 401 Advanced Aerodynamics

9. x x Supersonic Wavy Wall [7] • The pressures predicted by our new relation are 90o out of phase with the wall oscillations: • High pressures occur on the front face of each wave, low pressures on the back face. • As a result, an integration of pressures results in a drag force in the x direction - wave drag. • In application, there is also a slight drag due to the total pressure loss in shock waves. But this is usually small unless it is a normal shock - I.e. transonic flow! AE 401 Advanced Aerodynamics

10. Supersonic airfoils • A nice thing about the supersonic solution is due to the hyperbolic nature of the problem. • Due to the limited regions of influence, the solution applies equally to a isolated segment of the wall as to the whole wall. • Thus, for a parabolic arc airfoil (constant curvature, so linear slope variation), the pressure distribution looks like: Cp - 1 x/c + AE 401 Advanced Aerodynamics

11. Supersonic Similarity • Our solution for the supersonic pressure coefficient also yields a supersonic similarity rule: • Note the close similarity to the subsonic Prandlt-Glauert rule. • Paradoxically, this rule states that, for the same geometry, as the Mach number increases, the pressure coefficient (and thus lift coefficient) decreases. • However, since dynamic pressure also increases with Mach number, the actual pressures and forces increase. AE 401 Advanced Aerodynamics