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This page explores quasi two-dimensional calculation of the supersonic flow over a wedge-shaped infinity-span wing

• Introduction
• History
• How To Work with Applet
• What's in the Result ?
• Two Pence of Theory
• Mathematical Model
  • General Simplifications
  • Shocks and Expansions
  • Aerodynamic Force
  • Streams Intersection
• How To Work Offline
• How To Contact the Author

Introduction

Whenever a wedge-shaped wing moves faster than the speed of sound (M>1) then the nice-looking picture of shock waves, series of supersonic expansion waves, and streams intersection appears.

• calculate and display the magical picture of supersonic flow over the wing
• find the exact values of flow thermo-gas-dynamic parameters in specific areas
• control wing geometry in "real-time" mode using a "point-and-click" interface
• calculate and display wave aerodynamic polar of your wing

History

In 1992, when I was working as an ordinary engineer at "Yakovlev" Aviation Company, my boss asked me to write the program that should estimate the supersonic performance of Yak-141 three-shock variable-geometry supersonic air inlet. It was not too complicated an assignment, and later I decided to implement this functionality in some kind of "real-time" software simulator. This simulator was designed for use by students at the Moscow Aviation Institute in addition to training me in the ways of the C++ Builder :-).

How To Work with Applet

1. You may drag and drop edge(s) of wing and/or drag and drop the whole wing itself.
   In case you drag the leading or trailing edge - the wing changes only its angle of attack and chord length. In other words, the wing's shape remains constant while the wing revolves around its center.
   If you drag top or bottom edge - the wing changes the geometry of its corresponding part but remain the same chord length and angle of attack.
2. In case you decide to calculate the exact point - (if you already know the geometry of your wing) - you may alter the appropriate fields on the right data panel and press Enter or click Flow button.

Also you may calculate and draw the Wave Aerodynamic Polar of the Wing. I wrote the word "Wave" to emphasize that viscosity is not taken into account and, therefore Drag Coefficient does include only pressure component and not the friction one. The Polar is a geometric illustration of dependences between Aerodynamic coefficients ( C_d - drag coefficient, C_l - lift coefficient, and L/D - lift_to_drag ratio ) and Angle of Attack.
If you move the mouse inside the plot area, then the program will show the current values of Angle of Attack and corresponded Aerodynamic coefficients.

What's in the Result ?

The graphic at the upper left shows the airfoil, shock and expansion waves, and streamline of the flow past the wing. The Applet draws only that part of flow where the waves do not intersect with each other.

Numerical output from the program is displayed in boxes at the lower left. Output parameters include the values of the following:

• corrected velocity (l)
• Mach number
• static pressure
• total pressure recovery coefficient (in shock waves)

Two Pence of Theory

The pattern of supersonic flow past the surface with a sharp turn highly depends on the direction of this turn.

In case of positive angle of turn (toward the flow interior) the direction of the stream and all its parameters change their values almost instantaneously producing so-called shock wave - flow compression. The depth of shock wave is supposed to be approximately equal to the free path of gas molecule. The static thermodynamic parameters of the flow (temperature, pressure, density) increase across a shock wave, while the velocity - decreases. Total pressure after the shock wave is always lower than that before. The process of supersonic flow compression is irreversible, the entropy increases and the adiabatic gas relations cannot be used. If the shock wave is inclined to the direction of stream it is called an oblique shock. In the other way if the shock is oriented normally to the stream it is called normal or detached shock.
These types of current take place in all inlets of supersonic air-breathing engines, before the airplane fuselage, wing and so on.

On the contrary, in case of the negative angle of surface turn, the direction of stream and all thermodynamic parameters change their values continuously in supersonic expansion process of flow turning. Theoretically, this process is isentropic and all stagnation (total) gas parameters remain their input values. The radius of "elementary" part of the stream (streamline) increases. The velocity of gas increases too, and all static parameters decrease. But if the turn angle is too large - then the velocity may reach its limit value and static parameters run down to zero! It means that all gas molecules move "in one order" and there is no stochastic submotion in their velocities. This effect is called over-expansion.
Supersonic expansion type of current was named after Prandtl and Meyer. It usually takes place after the rear edge of wing, in the flow issuing from the scarfed nozzle and turbine vane.

You think it is not enough? Try the Beginner's Guide to Aeronautics

Mathematical Model

• viscosity and boundary layer effects are not taken into account (this is, certainly, the main simplification)
• airflow is two-dimensional ( infinity span, no sweep angle, no tip effect )
• only Mach number (or corrected velocity) and the ratio of specific heats are taken into account in thermo-gas-dynamic calculations (no Reynolds, no Prandtl, no Frud number ... )
• static pressures in upper and lower parts of flow after the trailing edge are equal

Common Nomenclature
thermodynamics k = Cp / Cv - specific heats ratio U = (k-1)/(k+1) - auxiliary coefficient T - total temperature Subscripts: 0 - free stream; 1 - after turn; t - total parameter; n - normal projection	aerodynamics M = V / [ k·R·T ]½ - Mach number l = V / [ 2k/(k+1)·R·T ]½ - corrected velocity (velocity / critical sound speed) l2= [(k+1)M2]/[2+(k-1)M2] f - angle between the first and the last sonic waves
Input Parameters
l0 - Corrected velocity of free stream   (or Mach number) w - Turn Angle
Shock Wave   (w>0)	Supersonic Expansion   (w<0)
Find
a - Angle of oblique shock wave l1- Corrected velocity after shock wave s  - Total pressure recovery coefficient	a1 - Angle of the last sonic wave (characteristic) l1 - Corrected velocity after turn r1/r0 - Flow radius enlargement ratio
Pictures
if greek symbols are not supported, then: a - alfa;   l - lambda;   w - omega;   s - sigma;   f - fi
Base Equations
tg a = tg(a-w)·l02·sin2a /(1-U·l02·cos2a)   a may be found using Muller's method  l1 = l0 cos a / cos(a-w) l0n2 = tg a / tg(a-w) s = l0n2 [(1+U·l0n2) / (1-U/l0n2) ] 1/(k-1)	System of 7 Equations  lj2 = 1 + 2/(k+1) sin2( U½fj)  (w + f + a)j = p/2 (90°);   w1 = w + w0  sin aj = 1/Mj ,  where j = 0, 1 (subscript) is solved using quasi-Newton method    r1/r0 = [cos(U½f0) / cos(U½f1) ]1/U

Aerodynamic Force

Resultant of aerodynamic forces is equal to the vector Sum of 4 pressure forces that act on wing sides. Origin of Resultant has such coordinates that the summed moment of forces relative to this origin is zero.

where

• i number of wing side and force (i = 1 .. 4)
• L(i) - length of i side; L - length of chord
• X_mid(i) and Y_mid(i) - coordinates of the center of i side
• q = Pstatic₀·kM²/2 - dynamic pressure

Streams Intersection

Streams Intersection after the trailing edge of wing is calculated with general condition that static pressures in upper and lower parts of flow (above and below the boundary of streams) are equal. It means that velocities in correspondent areas are different. (That causes vortex generation after the wing, but the applet does not calculate thisrather complicated flow).

Input Parameters
t2 - above top rear surface lt2 Corrected velocity lb2 b2 - below bottom rear surface Pt2 total pressure Pb2 t3 - above streams boundary wr - rear angle of wing b3 - below streams boundary
Find
Turning angles of top (wt) and bottom (wb) parts of the flow All parameters of output Shocks and Expansions (according to the previous table)
Two possible Pictures
Base Equations
wt + wb = wr Pstatict3 = Pstaticb3 All equations about Shocks and Expansions from the previous table

How To Work Offline

Download wing.zip or wing_win.zip file. Unzip it, browse wing.htm or run wing.exe (the last one is Windows-based application) and enjoy !

How To Contact the Author

In case you decide to make any notices and / or suggest improvements - please feel free to write me - I'll try to take them into account. Especially I will be glad if you are able to make this applet work improperly !!

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