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Supersonic Potential Flow • For supersonic flow, we write the small perturbation potential equation as: 2xx yy 0 M 2 1 • 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: ellipse: ax2 by2 0 • hyperbola: ax2 by2 0 However, the differences between the equations is more fundamental then appearances. AE 401 Advanced Aerodynamics 233 7/16/2015 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 234 7/16/2015 Supersonic Wavy Wall • Once again, consider the wavy wall problem: y yw h cos2x / l x l • x h The flow tangency boundary condition is the same as before: dyw 2hV 2x y ( x,0) vw V dx l sin l • However, this time we will be solving: • Since this equation is hyperbolic, we will not be using separation of variable to solve this. 2xx yy 0 AE 401 Advanced Aerodynamics 235 7/16/2015 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: x, y f x y g x y f x y f ' x y x 2 f x y f x y 2 x f x y f ' x y y g x y g ' x y x 2 g x y g x y 2 x g x y g ' x y y 2 2 f x y f x y 2 y 2 2 g x y g x y 2 y AE 401 Advanced Aerodynamics 236 7/16/2015 Supersonic Wavy Wall [3] • Combining these yields: 2 f 2 f 2 2 f f 0 2 2 x y 2 2 g 2 g 2 2 g g 0 2 2 x y 2 • 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 237 7/16/2015 Supersonic Wavy Wall [4] Left running waves f = constant x y constant M Right running waves • • g = constant x y constant Note that the angle shown, , is given by: 1 1 dy atan atan asin M 2 1 dx M Thus, these lines represent Mach waves. AE 401 Advanced Aerodynamics 238 7/16/2015 dy 1 dx dy 1 dx Supersonic Wavy Wall [5] • For our solution, we are obviously only interested in the left running waves. Thus: x, y f x y • With the velocity components: u x f • • v y f To satisfy our flow tangency boundary condition, lets use surface slopes: dy v f tan dx V V Thus, the u perturbation velocity, assuming small angles, is: V tan V u AE 401 Advanced Aerodynamics 239 M 2 1 7/16/2015 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. 2u 2 Cp V M 2 1 AE 401 Advanced Aerodynamics 240 7/16/2015 Supersonic Wavy Wall [7] • The pressures predicted by our new relation are 90o out of phase with the wall oscillations: x x • 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 241 7/16/2015 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 242 7/16/2015 Supersonic Similarity • Our solution for the supersonic pressure coefficient also yields a supersonic similarity rule: C p1 M 12 1 1 C p 2 M 22 1 2 • 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 243 7/16/2015