Prandtl Meyerr Shock Problem

Introduction

In this project we will simulate a Prandtl Meyer expansion flow and take a deeper look at boundary conditions. we will try to answer the question of what boundary conditions are used in a shock flow problem and why? what is a shock wave? and we will look at the effect of SGS temperature value on cell count and shock location.

Pressure Boundary Condition

The ability to specify a pressure condition at one or more boundaries of a computational region is an important and useful computational tool. Pressure boundaries represent such things as confined reservoirs of fluid, ambient laboratory conditions, and applied pressures arising from mechanical devices.
Generally, a pressure condition cannot be used at a boundary where velocities are also specified, because velocities are influenced by pressure gradients. The only exception is when pressures are necessary to specify the fluid properties, e.g., density crossing a boundary through an equation of state.

Types of Pressure Boundary Conditions

There are typically two types of pressure boundary conditions, referred to as static (Dirichlet)or stagnation(Neumann) pressure conditions. In a static/Dirichlet condition the pressure is more or less continuous across the boundary, and the velocity at the boundary is assigned a value based on a zero normal-derivative condition across the boundary.
In contrast, a stagnation/Neumann pressure condition assumes stagnation conditions outside the boundary so that the velocity at the boundary is zero. This assumption requires a pressure drop across the boundary for flow to enter the computational region.
Since the static pressure condition assumes only that the normal fluid velocity at the boundary has a zero gradient, it is less specific than the stagnation pressure condition. In this sense the stagnation pressure condition is generally more physical and is recommended for most applications.

Shock Wave 

Consider an object in a fluid flow. As the fluid molecule approches the object it slows down due to the opposing atomic charges(exchange of photons), inturn increasing the local pressure, since we conceder the concept of continuum in fluid mechanics the information of this obstruction is passed on to the later molecule by this pressure/accoustic wave travelling at the speed of sound. When the flow speed reaches the speed of sound the wave generated by the wave source (object) interactes with the later waves and this causes a constructive interference which gives rise to a shock wave.The collision of fluid particle with the object creates a sonic boom(a high level of vibration). At atomic level the physics behined this is still in study as it is considered nothing touches anything and photon exchange stops the interaction of object, which keeps us in the mystery of sound prduction. untill then we will consider that the shock wave is a wave produced by wave front meeting at the wave source travelling at the speed of sound. and the coliision of object with the fluid particle due to the flow speed exceeding the speed of pressure/accoustic wave produces the sonic boom magically.

Mesh

basic mesh 

mesh with adaptive mesh refinement at:

0.01 SGC

0.05 SGC

0.09 SGC

Effect of SGS on shock location and cell count can be seen clearly from above images. as we increase the SGC the cell count decreases. the shock location gets bigger with decrease in SGC.

Total cell count at 0.09 SGC value

Total cell count at 0.05 SGC value

Total cell count at 0.01 SGC value

effect of change in velocity of initial condition

at the initial condition (in regions) the velocity is set to two values:

1. 100 m/s - subsonic

2. 680 m/s - supersonic

The plots for these are shown below.

Mass Flow

Total Pressure

Static Pressure

Dnesity

Velocity

Mach Number

from above velocity plots we can see that the initial velocity of 100 m/s changes to the final value and it does not have any effect on the final value.

Values for mass flow, pressure and density increases due to the initial state but as this simulation will go on without convergence, we can say that initially the value varies and continues to follow the curve pattern.

the video of temperature contour are shown below:

1. for Sub-Grid criterion 0.01

2. for Sub-Grid criterrion 0.05

3. for Sub-Grid criterion 0.09