Week 5: Prandtl Meyer Shock problem

Prandt-Meyer Expansion Wave

In a Prandt Meyer Expansion Wave, a supersonic flow is expanded over a sharp expansion corner. An expansion wave that consists of infinite number of weak mach waves fans out from the corner. The leading edge makes an angle `mu1` given by the following formula

`mu1=sin^-1(1/(M1))`

whereas, the lagging edge makes an angle `mu2` given by the following formula

`mu2=sin^-1(1/(M2))`

M1 and M2 are upstream and downstream mach numbers.

Following are properties of this flow:

1. The flow is isentropic.
2. As the flow passes through the expansion corner, the mach number increases but temperature, pressure and density decreases.
3. The properties changes smoothly in the expansion wave itself, except for the corner where properties are discontinous.

Shock Wave:

A wave in which pressure changes abruptly in an elastic medium i.e. air, water e.t.c. as is the case in above illustration of Prandtl Meyer Shock Problem.

Boundary Conditions:

Neuman Bounday Condition:

This specify the value of derivative of a function on boundary.

Drichlet Boundary Condition:

This specify the value of function on the boundary.

Mixed Boundary Condition:

This specify a combination of Neuman,Drichlet and Robin boundary conditions on the boundaries. e.g. Neuman on one boundary and Drichlet on the other.

Robin Boundary Condition:

This specify a linear combination of a value of function and an its derivative on a particular boundary.

Simulation of a Prandtl Meyer Shock Problem in Converge:

Geometry:

For this simulation Density based Steady State Solver is used, with RNG K-epsilon as turbulance model and base mesh of dx=dy=dz=0.8m . Moreover , simulations are done in supersonic (680 m/s) and subsonic (100 m/s) domains with adaptive mesh refinement using temperature and Sub Grid Criteria(SGS) equal to 0.1 and 0.01

Mesh:

Case 1: 680 m/s and SGS 0.01:

Inlet Boundary Condition :

AMR Settings:

Pressure Contour:

Temperature Contour:

Velocity Contour:

Cell Count:

Case 2: 680 m/s and SGS 0.1:

Pressure Contour:

Temperature Contour:

Velocity Contour:

Cell Count:

Case 3: 100 m/s and SGS 0.01:

Pressure Contour:

Temperature Contour:

Velocity Contour:

Case 4: 100 m/s and SGS 0.1:

Pressure Contour:

Temperature Contour:

Velocity Contour:

Youtube Link:

Case1: 680 m/s and SGS 0.01

Pressure:

Temprature:

Velocity:

Case2: 680 m/s and SGS 0.1

Pressure:

Temperature:

Velocity:

Observations:

• Its noted that as the SGS(Sub Grid Criteria) is decreased the total Cell Count increases.
• As the SGS parameter decreases the expansion fan's adaptive mesh refinement length from the sharp corner to the outlet of the nozzle increases.
• Also the Expansion Fan is only observed in the supersonic flow and not in the subsonic flow.