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During supersonic flight over a airfoil, both compression and expansion occur because an airfoil is both convex and concave. When an airflow encounters a concave corner it compresses, but Prandtl-Meyer expansion is the result of a supersonic flow that goes over a convex corner and as a result of that the flow expands. Expansion happens due to changes in the direction of the flow: the flow tries to stay parallel to the wall but since it bends away from the flow, the flow properties change. This supersonic expansion is named after Ludwig Prandtl and his student Theodor Meyer because they were the first ones to work out a theory for a centered expansion wave. The expansion relation that was defined by this theory is a result of a number of assumptions.

The most important assumption is that this flow phenomenon is only two dimensional. Furthermore it is assumed that all the stream lines are parallel to the surface and straight before the corner, which rules out a turbulent flow. Turbulent flows require a time dependency, which clutters the solution. It is believed that for this physical model, the gas is calorically perfect and as such is a perfect gas which allows the usage of the isentropic relations. The final assumptions are that the expansion is continuous and centered: there is only one expansion fan and within that fan all the expansion occurs.

## Prandtl-Meyer flow governing equation Edit

In order for the flow to stay parallel to the wall, the flow has has to change directions. As can be seen from the first figure, the speed parallel to the expansion wave does not change, while the speed perpendicular to the expansion wave does. There is no acceleration in the direction of the expansion wave since the flow cannot travel in that direction due to the wall. This fact is used in the Prandtl-Meyer flow differential equation.

Using the law of sines, it can be seen from the figure that

- $ \frac{V+dV}{V} = \frac {\sin \left(\frac{\pi}{2}+\mu\right)}{\sin\left(\frac{\pi}{2}-\mu d\theta\right)} $
**(1)**

Taking the limit of $ d\theta $ and inserting the Mach angle will yield the governing differential equation for a Prandtl-Meyer flow, which is

- $ d\theta = \sqrt{M^2-1}\frac{dV}{V} $
**(2)**

Using the isentropic relations for a perfect gas it becomes possible to write the differential equation into a single variable differential equation. The Mach number is the ratio of speed of the object and the speed of sound. This can be written as $ M = \frac{V}{a} = \frac {V}{\sqrt{\gamma R T}} $ and for a steady isentropic flow, the stagnation temperature is

- $ \frac{T_1}{T} = 1+\frac{\gamma-1}{2}M^2 $

Combining these equations, the squared speed becomes

- $ V^2 = \frac{\gamma RT_1 M^2}{ 1+\frac{\gamma-1}{2}M^2}=\frac{a^2 M^2}{ 1+\frac{\gamma-1}{2}M^2} $
**(3)**

Differentiating this equation logarithmically yields $ \frac{dV}{V} = \frac{1}{ 1+\frac{\gamma-1}{2}M^2}\frac{dM}{M} $ which fits perfectly into equation 2. Integrating this newly obtained equation will yield the Prandtl-Meyer function.

- $ \theta = \nu(M_2) - \nu(M_1) $
**(4)**

where

- $ \nu (M) = \sqrt{\frac{\gamma+1}{\gamma-1}}\arctan\sqrt{\frac{\gamma+1}{\gamma-1}\left(M^2-1\right)}-\arctan\sqrt{M^2-1} $

Because the calculation of $ \nu $ is cumbersome, the values for the given Mach angle and Mach number are usually tabulated. Knowing the angle of the corner and the initial Mach number, equation (4) yields the Mach number behind the expansion fan. The final static properties are a function of the final flow Mach number (M2) and can be related to the initial flow conditions as follows,

- $ \frac{T_2}{T_1} = \left(\frac{1+\frac{\gamma-1}{2}M_1^2}{1+\frac{\gamma-1}{2}M_2^2} \right) $

- $ \frac{p_2}{p_1} = \left(\frac{1+\frac{\gamma-1}{2}M_1^2}{1+\frac{\gamma-1}{2}M_2^2} \right)^{\frac{\gamma}{\gamma-1}} $
**(5)**

- $ \frac{\rho_2}{\rho_1} = \left(\frac{1+\frac{\gamma-1}{2}M_1^2}{1+\frac{\gamma-1}{2}M_2^2} \right)^{\frac{1}{\gamma-1}} $

Since the angle $ \theta $ is known from the geometry, it is possible with the use of the table to obtain the Mach number after the expansion fan. In order to determine the angles which the fan makes with the wall, the Mach angle has to be determined. These angles have to be determined, otherwise the angle is not possible. The Mach angles are tabulated with the Mach number and the Prandtl-Meyer function.

## Mach angle Edit

A normal shock waves is straight, but most shock waves in real life are oblique. This is due to the fact that if an object goes beyond the Speed of sound, or Mach number Mach = 1, the projectile moves faster than the Shock wave which travels at the speed of sound. This time delay is depicted in figure 'Propagation of a supersonic flow'. Assuming that a shock wave is continuously created during supersonic flight, it is allowed to say that every point of the supersonic flight path behaves like a sound source. Since shock wave occur only upon first contact of the projectile surface with the air, a cone of sound is usually formed around the nose of the projectile. Designers try to make sure that the rest of the projectile stays within this cone in order to reduce further shock waves over the rest projectile as shock waves produce a lot of Drag. Inside this cone the effects of compressibility occur, while the outside of this cone is called the zone of silence since there the air is not disturbed.

From this discussion we can conclude that the cone angle, which is better known as the Mach angle, is only a ratio between the speed of sound and the speed of the projectile which travels beyond the speed of sound. Because sound propagates equally in all direction, it can safely be assumed that the speed of sound moves perpendicular to the direction of flight. Now a speed triangle is formed,which is represented in the lower figure. From the speed comparison it can be seen that the angle $ sin \mu $ equals the speed of sound, **a**, divided by the forward speed, **v**. Rewriting this into an equation, the following relation holds.
$ \mu = \arcsin \frac {a}{v} = \frac{1}{M} $

Where the Mach number is defined as $ M = \frac{V}{a} $

In an expansion fan, the Mach angle is determined twice and it determines the size of the fan. In figure 1 an example is given for two angles are determined in the case of a very thin expansion fan.

## Flow property changes Edit

Knowing the Mach number after the expansion allows for the calculation of the Pressure, temperature and the density of the flow behind the expansion fan using the isentropic relations for pressure, temperature and density. Using the relationships of the three states in combination with the Mach number, equations (5), the following observations can be made.

During expansion the speed will increase. An increase in speed will reduce the static temperature, static pressure and static density. However, the total pressure, temperature and density will stay the same since the expansion is isentropic.

## References Edit

- M.J. Zucrow, Gas Dynamics, Volume I, 1976, John Wiley & Sons
- R.M. Rotty, Introduction to gas dynamics, 1962, John Wiley & Sons
- J.D. Anderson Jr., Fundamentals of Aerodynamics, Fourth edition, 2007, McGraw-Hill
- E.L. Houghton and P.W. Carpenter, Aerodynamics for engineering students, Fourth edition, 1993, John Wiley & Sons
- C.E. Dole and J.E. Lewis, Flight theory and aerodynamics, A practical guide for the operational safety, second edition, 2000, John Wiley & Sons

Author: Philip Sondervan