Prandtl meyer Shock wave Challenge
Introduction:
A boundary condition is assigned to any model or geometry which helps in solving an ODE or PDE along with the initial conditions depending on whether the flow is steady or transient. There are three types of boundary conditions, Dirichlet boundary condition, the Neumann boundary condition, and mixed boundary condition also known as Robin boundary conditions which are applied to inlet and outlet. When a value of unknown at the boundary is known or has a fixed value that boundary condition is known as a Dirichlet boundary condition. When a boundary condition is specified in terms of a derivative of a function as a constant then we call it as a Neumann boundary condition. In Robin or mixed boundary condition, a constant value or derivative are used but that derivative is used to find the unknown value which we want to find.
A shock wave is defined by the property of Mach no which is greater than 1 or 340m/s and can occur in any medium gas or air. Shock waves generally occur in Transonic, hypersonic and supersonic sound waves. The shockwaves property is defined by the temperature and pressure of the particle moving with greater than the speed of sound and as this speed increases, this can cause abrupt changes in pressure and temperature in a gas medium. If a shock wave hits an object it creates an increase in pressure and temperature in the downstream of the body creating a sharp parabolic profile hence causing a decrease in velocity. As the shock wave moves through the medium this can cause a density change along the length hence changing the density gradient.
Prandtl Meyer shock problem, the speed of the object as it approaches the speed of sound gets unstable and doesn\'t remain constant anymore. In such cases the gas surrounding the object gets compressed and flow area gets decreased so the compressibility factor of the gas needs to be considered. When the flow is expanded it is termed as center expansion fan. In shock waves the static pressure increases which cause a loss in total pressure. In expansion waves, the Mach number decreases due to which the static pressure decreases and the total pressure remain constant. the process is isentropic for this reason.
The Prandtl Meyer function nu is defined by Mach number and the real gas constant. The function is derived from mass momentum and energy conservation for small differential defelction
The simulation was carried out for subsonic and supersonic initial conditions. Also, a new concept of adaptive mesh refinement was introduced in this lecture. Changes have been studied on how the sub-grid criterion can affect the velocity pressure and temperature contours. A box-shaped geometry has been considered with a slight negative slope in the bottom layer to see how the expansion fan is created at two different initial conditions. A density-based solver is used in this since this case has some tremendous density variation along with the profile of the system.
For Subgrid Criterion-0.05
1. Initial conditions V= 100m/s T=300K and Pa= 50000. Boundary conditions at inlet, V= 680 m/s at 286.1 K i.e Mach 2 with Air and at the outlet Neumann boundary conditions have been applied for velocity and temperature.
The grid generation for the following specification is shown above.
The Mach no is indicated as per the boundary conditions and initial conditions that are applied. The sub-sonic velocity transcends to supersonic velocity after a few cycles and then remains steady.
The Total pressure rises tremendously as compared to the static pressure as the velocity changes by a huge amount in the region.
Pressure and Velocity magnitude is shown above at the last time step when the solution converges.
1. Initial conditions V= 680m/s T=300K and Pa= 50000. Boundary conditions at inlet, V= 680 m/s at 286.1 K i.e Mach 2 with Air and at the outlet Neumann boundary conditions have been applied for velocity and temperature.
There is slight variation in the grid at a few locations if observed carefully.
The velocity has a slight variation for the Mach no since it doesn\'t have have much difference in the initial and boundary conditions.
The velocity and Pressure contours show that how the static and total pressure varies by such a huge scale for supersonic when compared with the subsonic scale.
In order to check the effect of subgrid criterion on the number of cells and the post-processing results, few changes were made with 0.1 and 0.02 as the value.
Subgrid 0.1
3. Initial conditions V= 100m/s T=300K and Pa= 50000. Boundary conditions at inlet, V= 680 m/s at 286.1 K i.e Mach 2 with Air and at the outlet Neumann boundary conditions have been applied for velocity and temperature.
The grid generation is less as compared to 0.05 due to which the expansion fan details cannot be identified in the contour effectively as shown below. The results haven\'t shown much difference as compared to the above conditions except for the grid size and cell count.
Subgrid 0.02
3. Initial conditions V= 100m/s T=300K and Pa= 50000. Boundary conditions at inlet, V= 680 m/s at 286.1 K i.e Mach 2 with Air and at the outlet Neumann boundary conditions have been applied for velocity and temperature.
The grid generation is more as compared to 0.05 due to which the expansion fan details are very prominent in the contour as shown below. The results haven\'t shown much difference as compared to the above conditions except for the grid size and cell count.
Animations have been shown below and the following links have been provided for each subgrid values.
For 0.05: Supersonic Pressure condition for 680m/s
For 0.02: Supersonic Velocity contour for 680m/s
https://www.youtube.com/watch?v=LnhH5mkcFk4
For 0.1: Supersonic Velocity contour for 680m/s
https://www.youtube.com/watch?v=ot7HW5ggCIc&feature=youtu.be
Conclusion:
Hence the shock wave properties with two initial condition have been analyzed and discussed accordingly. And also how the subgrid effects the flow contours and helps us in analyzing the data more precisely and effectively.