Prandtl Meyer Shock problem

Shock Wave:-

Shockwave is a type of propagating disturbance that moves faster than the local speed of sound in the medium which causes the change in pressure, temperature, density of the medium. The changes in the flow properties are irreversible and the entropy of the entire system increases. Because a shock wave does no work, and there is no heat addition, the total enthalpy and the total temperature are constant But because the flow is non-isentropic, the total pressure downstream of the shock is always less than the total pressure upstream of the shock.

Oblique shock:-

This types of waves occurs when supersonic flow encounters a corner that effectively turns the flow into itself and compresses also When a shock wave is inclined to the flow direction it is called an oblique shock. The oblique shock produced at atmosphere(in flow geometries) the outlet mach number can be supersonic (weak shock wave) and when produced to an confined geometries the outlet mach number can be subsonic (strong shock waves).

In detached shock waves, supersonic speed across the shock and changes to subsonic.

Normal shock:-

If the shock wave is perpendicular to the flow direction it is called a normal shock. Across the normal shock the flow changes from supersonic to subsonic conditions.

Normal shock generally is used to simulate the high heating environment of spacecraft.

Boundary conditions:-

A Boundary value problem is a system of ordinary differential equations with solution and derivative values specified at more than one point. Boundary conditions ar dependent on the variables which is time-independent. There are two main types used in the shock problem one is Dirichlet and Neumann boundary condition.

Dirichlet boundary condition:-

Dirichlet boundary conditions are applied in either ordinary differremtial equation or partial derivatives where the value of an particular variable is applied and that value taken throughout the boundary of the domain.

Neumann boundary condition:-

In Neumann boundary condition same as in Dirichlet boundary condition can be applied in both ordinary differential equation and partial derivatives but the condition specifies the values in which the derivative of a solution is applied within the boundary of the domin.

Boundary condition inside the Prandtl Meryer:-

• In Inlet the velocity is specified as 680m/s using Dirichlet boundary condition and pressure as zero normal gradient(Neumann).
• In Outlet the velocity and pressure as zero normal radient(Neumann).
• Above boundary setup tell that the Outlet which is depend on Inlet i.e the solver extrapolates the required information from interior.

Prandtl Meryer project:-

Generally, Prandtl Meryer expansion fan formed due to the supersonic flow encounters the convex corner. The expansion fan consists of infinite number of mach waves. Across the expansion waves the flow velocity increases and mach number increases as well as the flow properties changes such as static pressure, temperture and density decrease. Since the process is isentropic, the stagnation properties remain constant across the fan.

Geomentry:-

Geomentry scaled to required size which shown in below image.

Case setup:-

• Material - Air - default values specified in Gas simulation, Global transport parameters.
• Run Parameters - density based solver - compressible.
• Simulation time - Start time as 0.0cyc to End time as 25000cyc.
• Turbulence Modelling - Rng k-epsilon.

Boundary flagging:-

Region and initialization:-

Region 0 is added as well as te stream ID and velocity, temperature, pressure, species is specified.

Base grid:-

Base grid set as 0.8m in dx, dy, dz.

Adaptive Mesh Refinement:-

• Sub-grid criterion is specified as 0.05k.

• The image shows the Adaptive Mesh Refinement of 0.05k as well as the base grid size of 0.8m.
• The two level of embed is shown.

• `Grid Size = (Base Grid)/2^(Embed-Level)`.

• The above equation gives grid size for each level of embed.

This image shows the Sub-grid criterion of 0.03k.

This image shows the Sub-grid criterion of 0.01k.

On comparing the above three images the sub-grid of 0.01k gives better results than others.

Temperature Distribution :-

The above animation is for AMR = 0.05k, Velocity = 680 m/s.

The above animation is for AMR = 0.05k, Velocity = 100 m/s.

The above animation is for AMR = 0.03k, Velocity = 680 m/s.

The above animation is for AMR = 0.01k, Velocity = 680 m/s.

Conclusion :-

• By this simulation learnt how to use Adaptive Mesh Refinement for different sub-grid values.
• In this geomentry the sub-grid value of 0.01k gives better results.
• Applied Boundary conditions which gives the output values taken from input.