Verification of Prandtl Meyer shock wave by CONVERGE CFD

Title: Verification of Prandtl Meyer shock wave by CONVERGE CFD

Objective: 1.Importance of prandalt meyer expansion

                  2. Effect of shock wave on Velocity, Temprature , Pressure & Cell count.

                 3. Effect of Sub grid criteria (SGS) on cell count & shock location

Theory:

1.What is Prandalt Meyer expansion?

A supersonic expansion fan, technically known as Prandtl–Meyer expansion fan, a two-dimensional simple wave, 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 waves, diverging from a sharp corner. When a flow turns around a smooth and circular corner, these waves can be extended backwards to meet at a point         

 

                      

 

Each wave in the expansion fan turns the flow gradually (in small steps). It is physically impossible for the flow to turn through a single "shock" wave because this would violate the second law of thermodynamic.

                         

Across the expansion fan, the flow accelerates (velocity increases) and the Mach Number increases, while the static pressure, Temprature and density decrease. Since the process is isentropic, the stagnation properties (e.g. the total pressure and total temperature) remain constant across the fan.

 

2. Boundry condition for Shock Wave problem

 To get nearest answer for any problem boundry condition require. for solve the steay & transient state flow the initial & boundry condition is require.

Type of boundry condition

1. Drichlet boundry condition

2. Neumann boundry condition

3. Robin boundry condition

4. Mixed boundry condition

5. Cauche boundry condition

     As this shock wave problem is based on transient flow , we need to provide the initial condition. generally the Drichlet BC its used for the boundry in the form of value like pressure, velocity at the boundry.

        Neumann BC its imposed in the form of ODE & PDE. The condition specifies the values in which the derivative of a solution is applied within the boundry of the solution.

         If we wan to give the pressure outlet BC , its vey important to give the static pressure at outlet. static pressure is valid for subsonic flow, but the flow is supersonic in that case static pressure is depend on the initial condition of pressure. So, by the Neumann BC outlet condition pressure is depend on initial condition pressure. this will derive the preesure value form initial.

2.What is Shock Wave

In physics, a shock wave it is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a medium but is characterized by an abrupt, nearly discontinuous, change in prssure ,temprature and density of the medium.

For the purpose of comparison, in supersonic flows, additional increased expansion may be achieved through an expansion fan, also known as a prandalt-meyer expansion fan. The accompanying expansion wave may approach and eventually collide and recombine with the shock wave, creating a process of destructive interference.

3. Physics behind Shock Wave

we can see below the Mach no<0.8 there is no any shock wave generation , but from the Mach no>0.85 shock wave is start to generate.

                

Case setup 

1. Fluid used -Air

2. Solver- Steady state & Density  based, Full Hydrodynamic 

3. end time -25000cycle  &  Start Time: 0 Cyc

4. Intial & minimum step =1e-7

5. Maximum time step= 1

6. Inlet pressure= Zero normal gradient

7.outlet pressure=Zero normal gradient

7. Base Grid size = 0.8 m

8. Turbulence model: RNG K-E

Boundry condition

Inflow:

Velocity =680 m/s in x direction

Temperature= 286.1 K

Outflow:

Pressure- Zero normal gradient

Velocity- Zero normal gradient

Front and Back – 2D

Top and Bottom Wall – Slip Wall

 

     

 

Region & Initialization

1.Initial velocity =680m/s

2. Initial pressure=50000pa

3. Fluid material=Air

 

Geometery 

for crete the supersonic flow inlet velocity 680m/s is consider , while pressure is dont know so, consider the zero nomrmal gradient i.e neumann BC. as Supersonic flow is pass from corner there is create diffrent Mach no. profile.

 Boundry name & ID

 

Run parameter

Steaydy state monitor is ON for early conversion process

Adaptive mesh Refinment

 Grid Size= (base grid size)/2^n                      n=embed level

                        = 0.8/(2)^2

                        =0.2m

Base grid=0.8m

Level 1 Grid Size= 0.4m

Level 2 Grid size= 0.2m

Adaptive Mesh Refinment

1.Embed type is sub grid scale 

2. Max. emebdding level=2 

3. Sub-grid criteria=0.05k once temprature is more than SGC the embedding level is active

Case 1: Inlet velocity=680m/s ,SGC=0.05K

for create the supersonic flow inlet velocity consider the 680 m/s & once the temprature is go beyond the 0.05k the embed level 2 will active.

SGC=0.04K 

in case of 0.04k the embedd level is more active. we can see that half portion of shock wave there is refined mesh is present.

SGC=0.03K

in case of 0.03k the embedd level is more active than the 0.04 & 0.05k. we can see that full portion of shock wave there is refined mesh is present. adaptive mesh refinment is help for capture the accurate shock wave, pressure , velocity contour

Pressure contour

from pressure contour we can see that at inlet higher pressure. in the shock wave region there is medium pressure level, while on outlet there is lower pressure is present.

Temprature contour

from Temprature contour we can see that at inlet higherTemprature. in the shock wave region there is medium Temprature level, while on outlet there is lower Temprature is present.

Velocity contour

from  Velocity contour we can see that at inlet lower Velocity. in the shock wave region there is medium Velocity level, while on outlet there is higher Velocity is present. its means that at the outlet supersonic flow is present

 

Pressure animation

Temprature Animation

Velocity Animation

Case 1 Graph : Inlet velocity 680m/s 

from the below plot we can see that at the inlet Mach No.2 is provided, but as the inlet velocity is increases the Mach no is increases whic is more than the Mach= 2.15 

Mach No= Velocity of Flow /Speed of Sound 

Static Pressure

At the inlet initial pressure is provided 50000pa. as flow is proceed at the inlet around 1.4pa is reached. but at the outlet static pressure is lower than the inlet pressure, so that at outlet velocity is increases.

Avg. Velocity

at the inlet velocity is provide 680m/s. but as the flow is proceed the outlet velocity is around 700m/s

Avg. Temprature

At the inlet initial temp. is provided 286k. but as the flow is proceed the temprature is decreases up to 270 k due to decrease in pressure at the outlet

Total cell

As the SGC Temprature decreases from 0.05 to 0.03k the more refined mesh. so, that total cell count is also increases.

Case 2: Inlet velocity=100m/s 

To analyze the effect of inlet velocity on Mach no., shock wave, pressure ,temprature & velocity simulation is performed. so, its obsereved for 100m/s velocity very low shock wave is generated which negligable & so that its called subsonic flow.

Pressure contour

from the pressure contour we can see that there is not generate shock wave. but there is generate three diffrent pressure zone.

Temprature contour

from theTemprature contour we can see that there is not generate shock wave. but there is generate three diffrent Temprature zone

Velocity contour

we can see that at the expansion corner maximum velocity is 110m/s is generated , but its sufficient to genrate the supersonic flow 

Case 2 Graph : inlet velocity:100m/s 

 

from Avg. mach plot we can see that there is maximum Mach no. 0.3 is generated which tell us that, flow is subsonic.

Static Pressure

 

Overall conclusion

1. Adaptive mesh refinement help for capture the accurate simulation. For SGS 0.03k the higher mesh refinment & which covers the entire shock wave profile.

2. As the SGC Temprature decreases from 0.05 to 0.03k the more refined mesh. so, that total cell count is also increases.

3. As flow is proceed to outlet Mach no. is increases but temprature, pressure is decreases.

4.it can be observed that the expansion fan tends to become less narrow and distinct as the SGS parameter increase

5. For velocity 100m/s there is no generation of shock wave. & the flow is called subsonic flow