Prandtl-Meyer Shock Problem - Converge Studio

This project objective is to simulate the Prandtl Meyer shock wave phenomena using Converge Studio. First, a quick literature review about shock waves and their boundary conditions is provided. Then , the problem is set up and solved, focusing in how the different paramenters in the simulation affect the shock wave location.

Theory

A shock wave is a propagating disturbance that can appear when a medium moves faster than the local speed sound. As any wave, a shock wave carries energy and can propagate through the medium, but it can be caracterized by an abrupt change in speed, temperature and pressure. This discontinous change in the fluid characteristics is what makes the analysis of shock waves problems so important, as it can suppose important changes in the operating conditions or properties of the system. For example, shock waves appear on external flows when an object is moving faster than M=1, compressing the fluid/medium in front of it. In the case of internal flows, such as the nozzle of an gas turbine engine, the shock waves appear to accomodate the flow expansion to the outlet boundary conditions (usually atmospheric pressure).

Boundary Conditions:

There are 3 main boundary conditions in any kind of fluid dynamics problem: Dirichlet, Neumann and Combined boundary conditions. A Dirichlet boundary condition specifies the values of the desired variables at the boundaries of the domain, while the Neumann boundary condition specifies the derivative of those variables. The third part are just combinations of these two, such as the Cauchy or Robin boundary conditions.

So, why is it important to always determine Neumann boundary conditions in shock waves problems? Well, shock waves problems are asssociated to supersonic problems. In any medium, the information travels aways through pressure waves, which can only move as fast as the speed of sound. This is, in a supersonic problem, the properties of the fluid inside the domain can't be altered fast enought to try to match the flow at the outlet. Therefore the information at the boundary condition cannotbe carried to the flow inside the domain, and therefore this fuid is completely dependent on the initial conditions. That is why Dirichlet boundary conditions make no sense at the outlet of a supersonic problem.

Prandtl-Meyer Expansion Problem

The Prandtl-Meyer expansion fan is a centered expansion process that occurs when a supersonic flow turns around a convex corner. The fan consists of an infinite number of Mach waver, diverging from the sharp corner. Each wave in the expansion fan turns the flow gradually in small steps, as according to the second lay of thermodynamics it is physically impossible for the flow to turn through a single expansion wave. Across this expansion fan, the flow accelerates (supersonic expansion), while the static pressure, temperature and density decrease. This process is in theory isentropic, which means that the stagnation properties (total pressure and total temperature) remain constant across the fan.

Geometry, Mesh  and Problem Setup:

For this problem, the simple following channel is created with a convex corner. The flow simulated is a non-reacting species, air (23% O2 and 77% of N2). This air is made to flow at Mach number grater than 1 through a channel with supersonic initial speed. 

The base grid of the problem is 0.8m, but an adaptative mesh refinement is neccesary for this type of problems. This option tracks the change in the varaibles desired, and employs a thinner mesh when it detects a sudden change in the variable (which is going to happen at the expansion waves). In this case, a temperature dependent adaptative mesh refinement is given, with a maximum embedding level of n=2 (this means, the smallest mesh size is d=0.8/(2^n)=0.02 m), and the value of sub-grid scale criteria was initially set to 0.05 K. One of the objectives of the simulation is to analyze the impact of this sgs parameter into the solution of the problem, and therefore it will be varied accordingly.

A density-based steady solver with a RNG k-eps turbulence model is used, as it is a highly compressible problem. To simplify, the back and the bottom boundaries are specified as 2D. The top and bottom boundaries are specified as slip-walls, with zero normal gradient (Neumann type). At the inlet, the flow speed is specified as 680 m/s and the temperature as 286.1 K (this is M=2). Now, remember that the oulet boundary conditions cannot be Dirichlet, this is a pressure or velocity value cannot be given, and is specified agains as zero normal gradient for pressure and velocity. 

Results and Post-processing:

SGS = 0.05 K

Static Pressure Profile:

Velocity Profile:

The results show what was already expected. When the flow starts the supersonic expansion, expansion waves appear and there is a sudden drop in pressure (also in Temperature and density) while the fluid accelerates. It is clear that the change does not happen instantaneously, but there are several expansion waves. Note also that the sub-scale criteria worked, as the mesh size is thinner when the shock wave appears, but it comes back to the previous value at the exit of the fan.

SGS = 0.01 K

Static Pressure Profile:

Velocity Profile:

SGS = 0.08 K

Static Pressure Profile:

Velocity Profile:

In the previous images, the effect of changing the SGS is shown. This criteria evaluates the minimum value of change in temperature that can happen in one cell, which means that the 0.01 value makes the mesh thinner than the 0.05 value. Although there is not much of a difference between the point where the expansion waves appear, it is clear that the slope of this waves changes with the accuracy of the mesh. Also, the thinner mesh allows to appreciate better the entire problem, and defines accurately the expansion zone, from beggining to end. In both the other cases, the change during the expansion seem to be more sudden, while in the thinner mesh the entire fan can be identified. 

Also, note that the velocity values does not change, but the pressure values does. This is also shown in the plots below. The reason is that 25,000 cycles has not been enought for the pressure profile or the outlet mass flow to reach an steady solution. The velocity and temperatures profiles have.

Conclusion

- The Prandtl-Meyer expansion problem can be accurately simulated using Converge Studio.

- The location of the start of the shock phenomena hasn't changed with the mesh resolution, but the resolution of the expansion fan was improved when the sgs was set to 0.01.

- Although the mean temperature and velocity reached similar values, thinner refinement of the mesh lead to different static pressure and outlet mass flow values.


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