Prandtl -Meyer Expansion Fan

Introduction:

Supersonic expansion fan also known as Prandtl Meyer expansion fan, is a two-dimensional simple wave expansion process centred at the convex corner. Too many jargons correct let’s take step by step.

Assuming a bullet moving through air at sea level too fast right, lets looks it in slow speed in the next pic.

The above figure shows differently shaped bullets, but there is one thing common in them i.e. the bow shaped thick black line. This is precisely what is called a shock wave. So long story short when something moves in a blanket of air or any other gas there are vibration or disturbance that are produced by the objects motion and these disturbances tend to travel through the blanket in all directions. This is shown below.

There are following possible cases:

• The object is stationary and produces some disturbance. V_obj = 0, Fig a.
• Object moves with some velocity but less than velocity of sound i.e. V_obj < V_Sound, Fig b. (SUBSONIC)
• Object moves with velocity around the velocity of sound i.e. V_obj = V_Sound, Fig c. (TRANSONIC)
• Object moves with velocity greater the velocity of sound i.e. V_obj > V_Sound, Fig d. (SUPERSONIC)

The blue colour represents the wave front, its circular because disturbance propagates in all direction. These wave fronts propagate at speed of sound. So, when the object itself starts moving as in second case the direction in which the object moves waves fronts come closer in that direction and are far apar in the opposite direction. When the speed of object is same as that of the sound in that medium the wave in the front of the object almost coincide as seen in figure c, and when object speed surpasses the speed of sound shock waves occur.

Shock Wave: It’s a type of propagating disturbance that moves faster than the local speed of sound in that medium. Shock wave also carries energy but is characterized by an abrupt, nearly discontinuous, change in pressure, temperature and density of medium.

Types of shock waves:

• Moving shocks:

1. Characterized by shock way propagating into a stationary medium.
2. Gas behind the shock can be supersonic.
3. Speed of the shock is a function of pressure difference between two gas bodies.
4. Usually generated by two gas bodies at different pressures, with shock wave propagating into higher pressure gas and expansion wave expanding into higher pressure gas.
5. Eg: shock tube, shock wave from explosion, balloon burst.

• Detonation Wave:

1. Characterized by shock wave supported by trailing exothermic reactions.
2. Shock waves generally travel through highly combustible or chemically unstable medium.
3. Chemical reactions take place following the shock wave and the wave is driven by the energy from these chemical reactions.
4. Its little different from ordinary shock waves, and travel at speeds governed by particular equation.
5. It’s usually accompanied by or give rise to moving shock waves.
6. Shock waves created from explosion of high explosives is of this type.

• Bow Shock or Bow wave or Detached shock wave

1. They are curved resembling bow; they form at distance from the object surface itself.
2. For a blunt body they may from in combination of a normal wave followed by a bow wave.
3. It occurs when the maximum deflection angle is exceeded. A detached shock is commonly seen on blunt bodies, but may also occur on sharp bodies at low Mach numbers.
4. Eg: Space return vehicle, bullets, etc.

• Attached Shock waves:

1. Shock appear to be attached to the tip of the sharp object at supersonic speeds.
2. There are of quite an interest in aerodynamics because for perfect gas and inviscid flow field a complete solution is available.
3. They may appear as strong shock with smaller shock angle or as weak shock with shocks appearing at 90 to the oncoming flow.
4. g. supersonic wedge and cone with smaller wedge angle.

And there are many other on which we can dwell on, but the point of interest here is the shock wave in presence of an expanding area of flow.

This drives us to Prandtl Meyer expansion fan concept.

Prandtl–Meyer expansion fan, a two-dimensional simple wave, is a centred expansion process that occurs when a supersonic flow turns around a convex corner. The fan consists of an infinite number of Mach waves, diverging from a sharp corner. When a flow turns around a smooth and circular corner, these waves can be extended backwards to meet at a point.

Each wave in the expansion fan turns the flow gradually (in small steps). It is physically impossible for the flow to turn through a single "shock" wave because this would violate the second law of thermodynamics.

Across the expansion fan, the flow accelerates (velocity increases) and the Mach number increases, while the static pressure, temperature and density decrease. Since the process is isentropic, the stagnation properties (e.g. the total pressure and total temperature) remain constant across the fan.

The expansion fan consists of an infinite number of expansion waves or Mach lines.

The first Mach line is at an angle mu1 with respect to the flow direction, and the last Mach line is at an angle mu2 with respect to final flow direction. Since the flow turns in small angles and the changes across each expansion wave are small, the whole process is isentropic.

The Mach number after the turn is related to the initial Mach number and the turn angle by,

q=n(M2)-n(M1) where, n ( or nu)is the Prandtl Meyer Function

There are some marked differences between shock waves and expansion fans. Across a shock wave, the Mach number decreases, the static pressure increases, and there is a loss of total pressure because the process is irreversible. Through an expansion fan, the Mach number increases, the static pressure decreases and the total pressure remains constant. Expansion fans are isentropic.

Boundary conditions:

• Inlet:

velocity:

type: Dirichlet

value: [680, 0, 0]

pressure:

type: Neumann

value: 0

temperature:

type: Dirichlet

value: 286.1

• Outlet:

velocity:

type: Neumann

value: [0, 0, 0]

pressure:

type: Neumann

value: 0

temperature:

type: Neumann

value: 0

• top-bottom:

type: wall

motion: STATIONARY

velocity:

type: Slip

value: [0, 0, 0]

pressure:

type: Neumann

temperature:

type: Neumann

value: 0

Why do we provide only zero normal Neumann at the outlet for pressure & velocity?

The reason is simple, the governing equation for the type of problem being solved here takes the hyperbolic behaviour and hence the information flow only in one direction. This is related to the concept of zone of dependence for the solution at a point in the domain. This is further clear form the figure below. As for the case of inviscid, supersonic (compressible), steady flow problem the behavioural characteristic of the governing equation takes the hyperbolic form, and hence we have to provide only zero normal Neumann boundary condition for pressure and velocity.

Domain under observation:

Cases:

1. Base case:
   • Grid size: 0.8m
   • AMR base: temperature
   • AMR max. embedding level: 2
   • AMR SGS: 0.05

2. Second case:
   • Grid size: 0.8m
   • AMR base: temperature
   • AMR max. embedding level: 2
   • AMR SGS: 0.01

Mesh:

AMR CELLS:

Results:

• Above figure shows the average total pressure which according to the characteristics of phenomenon should remain constant.
• Its observed that for all the cases the total pressure at inlet is equal to total pressure at the outlet which is expected as a characteristic of the Prandtl-Meyer expansion fan.
• There is definite difference in the results obtained from the two cases, but for individual cases as such the results are as expected.
• It turns out that the SGS criteria make the difference in the results for the two cases.
• A similar change in the static pressure is also evident form the figure below.

• This is quite clear from the graph that there is constant difference in the static pressure between the inlet and outlet for each case. This energy is used for maintaining the flow at the specified velocity.

• The average Mach number at inlet is defined at an inlet condition of 680m/s speed corresponding to Mach 2 while at outlet it should be greater as an expected characteristic of the phenomenon.
• Average Mach at outlet is at 2.13 while local maximum at 2.2 Mach.
• This allows us to have an idea of the angles at which the two extreme shock waves exist.

• The speed at the inlet is at 680m/s while at the outlet the average speed stand at 708m/s, which is in accordance to the trait of the phenomenon.

• The average temperature graph shows the same trait as the static pressure i.e. decrease across the expansion fan, and so is the case with density as seen below.

Calculations:

Mach at inlet: 2

Mach at outlet: 2.2

n(2) = 26.38°

n(2.2) = 31.73°

q(˚) = 31.73-26.38 = 5.35˚ is the turn angle.

See Fig. below

μ (M@2) = 30 ˚ w.r.t. flow direction at inlet

μ ([email protected]) = 27 ˚ w.r.t. to flow direction at outlet

See Fig. below

Check the Video here.

Reference:

1. Chapter 9, BOUNDARY CONDITIONS AND BOUNDARY TREATMENT, Computational Fluid Dynamics, Computational Fluid Dynamics, Audi Department of Vehicle Engineering, Széchenyi University.
2. An Introduction to Computational Fluid Dynamics, H K Versteeg and W Malalasekera.
3. Wikipedia search on Expansion fan.
4. NASA article on Expansion fan.