Prandtl-Meyer Expansion Wave

Simulation of flow across the Expansion waves using Converge_Studio: -

1. AIM OF THE PROJECT: -

1.) To analyse the properties of Expansion Waves.

2.) To study the flow over the Convex Corner or across the Expansion waves/Prandtl-Meyer Expansion waves.

3.)To study the influence of Sub-Grid Criterion on the results.

2. INTRODUCTION

Mach wave is infinitely weak oblique wave. When the body moves with supersonic speed, it creates the zone of disturbance behind it. The lines which work as an envelope for the disturbance zone is known as Mach Wave. When the disturbance is stronger, Oblique wave is created.

Figure 01 :- BC is the Mach Wave and µ is Mach angle.

Supersonic flow over a corner: -

There are two types of corner: - 1.) Concave Corner; 2.) Convex Corner. Concave corner leads to change the direction of upstream flow towards the direction of mainstream and This sudden change in flow direction leads to the formation of Oblique Shock Wave. Conversely, Convex wave changes the direction of upstream away from the direction of mainstream and it leads to the formation of Expansion waves or PRANDTL-MEYER EXPANSION WAVES.

Figure 02 :- It shows the supersonic flow over the two corners and change in the flow properties across the oblique shock/Expansion fan waves.

Prandtl-Meyer expansion waves are the infinite series of waves changing the flow properties smoothly over the complete range of the Expansion waves unlike the shock Waves. Flow forms the curved streamlines over the corner and finally flows parallel to the inclined surface.Prandtl-Meyer expansion waves are the infinite series of waves changing the flow properties smoothly over the complete range of the Expansion waves unlike the shock Waves. Flow forms the curved streamlines over the corner and finally flows parallel to the inclined surface. The two Mach waves which bounds the Expansion waves are the Forward Mach line and the Rearward Mach line. The forward Mach lines bounds the upstream flow and the rearward Mach line bounds the downstream side. Flow through the expansion wave is an isentropic process.

The following are the equations and formulae for 2-Dimensional, compressible, isentropic flow through the Expansion Waves.

dӨ = (M²-1)^0.5 *(dv/v) -> equation of flow motion through the Expansion Waves.

ᶹ(M) =((ϒ+1)/(ϒ-1))^0.5.tan^-1(((ϒ+1)/(ϒ-1)).(M²-1))^0.5- tan^-1(M²-1)^0.5 -> Prandtl-Meyer Function

Ө = ᶹ(M₂) - ᶹ(M₁) -> Equation for calculating M₂ using known Ө and ᶹ(M₁). ---->(Equation 01)

T₂/T₁= (1+[(ϒ-1)/2]*M₁²)/( 1+[(ϒ-1)/2]*M₂²) -----> (equation 02)

P₂/P₁= ((1+[(ϒ-1)/2]*M₁²)/( 1+[(ϒ-1)/2]*M₂²)) ^ϒ/ϒ-1 ------> (equation 03)

Here, Ө is the angle of deflection.

M is Mach Number.

ϒ is ratio of specific heat.

For other terms refer to figure 02(b).

Figure 03 :- Prandtl-Meyer Expansion

3. PROBLEM STATEMENT: -

The geometry used in the simulation is shown in figure 04. The fluid used for the simulation is Air (assumed to be consisted of Nitrogen and Oxygen only). The mole fraction of Oxygen present in air is 0.23 and that of Nitrogen is 0.77. The Temperature at which air flows is 286.1K and the speed of sound in air at 286.1K is 338.4m/s (Equation 01). The speed of air provided at the inlet or the upstream speed is 680m/s i.e. Mach Number =2.01. The flow is compressible, supersonic, isentropic and 2-Dimensional Flow.

(ϒ*R*T)0.5= a ---------(Equation 04)

a = (1.4*286*286.1)0.5m/s

a = 338.4m/s

ϒ=1.4 (Ratio of the specific heat)

R=286 m²/s²/K (Gas Constant)

T=286.1k (Temperature)

a =speed of sound.

Figure 04 Geometry set up of the problem

4. ANALYTICAL SOLUTION FOR THE GIVEN PROBLEM STATEMENT

For M₁ = 2.01, ᶹ(M₁) = 26.38⁰ using Prandtl-Meyer Function and Ө = 5⁰ for the case study so using equation 01 and Prandtl-Meyer Function M₂=2.186.ϒ=1.4 has been used for the calcultion.

Using Equation 02 :-

T₂/T₁= 0.920377

if T₁=286.1K then, T₂=263.319K

similarily using Equation 03:-

P₂/P₁=0.747963

5.) PROBLEM SET UP IN CONVERGE_Studio FOR SIMULATION

STEP 01: - Creating the geometry and Boundary Flagging

STEP 02: - Setting the case Setup Files: -

I have setup the case as per the problem statement provided and the additional information for setup is as follow: -

1.) Meshing -> Base grid of size 0.8m in all the three directions.For capturing the details around the Expansion waves,the base grid has further meshed around the shock for cells which temperture is greater than 0.05K (Case 01,Figure 06) and 0.01K(Case 02,Figure 07). This property of grading mesh is available in Adaptive Mesh Refinement and Sub-Grid Scale has been used in AMR.The property helps to save computational cost by not further meshing the entire domain but to mesh only around the particular domain of interest.

Figure 05 (A)-> Base Grid

Figure 05 (B)-> Grid size at the last step for case 01

Figure 05 (C)-> Grid size at the last step for case 02

2.) Turbulence Modeling -> K-epsilon model has been used.The turbulence coefficient have been taken the default values.

3.) Boundary Conditions:-> There are mainly three types of Boundary Conditions:- Dirichlet Boundary Condition,Neumann Boundary COndition and Mixed Boundary Condition. We have to provide only condition across the flow dimension of the geometry to initiate the flow.We can eiher use pressure difference or the velocity at the inlet.Now, in this case i have provided inlet velocity so it is not required to give pressure differnece otherwise it will be the redundent boundary condition.The pressure will be calculated after the simulation ends.So, we provide Neumann Boundary Condition for pressure that indicated the mass of gas will move across the inlet without any pressure change across it.For the outlet wall the pressure has been mentioned and Neumann condition has been applied to velocity.

6. RESULTS: - I have provided the velocity,density,Temperture and pressure profile for the given setup and especially focused on the change of the above parameters across the Expansion waves.I have also provided animation at the end for showing the time variation of the above mentioned properties.I have provided graph showing variation of the parameters quantatively across the Expansion waves. I have used paraview for the above purpose.Two case Study has been done.

Case 01: - SGC = 0.01K

Case 02: - SGC =0.05K

For Case 01 :- The temperature has decreased in the the expansion wave region continuously as it was expected to be (Figure 08 (A),09(A)).Similarly pressure (Figure 08 (B),09(B),Density (Figure 08 (C),09 (C)) have also decreased while velocity (Figure 08(D),Figure 09(D)) has increased.

PROFILES: -

Figure 08 (A) Temperture profile at the last step (2500 iteration)

Figure 08 (B) Pressure profile at the last step (2500 iteration)

Figure 08 (C) Density profile at the last step (2500 iteration)

Figure 08 (D) Velocity profile at the last step (2500 iteration).

GRAPHS: -

Figure 09 (A) Temperture Graph from bottom to top at the last step (2500 iteration)

Figure 09 (B) Pressure Graph from bottom to top at the last step (2500 iteration)

Figure 09 (C) Density Graph from bottom to top at the last step (2500 iteration)

Figure 09 (C)Velocity Graph from bottom to top at the last step (2500 iteration)

CASE 02: - The temperature has decreased in the the expansion wave region continuously as it was expected to be (Figure 10 (A),11(A)).Similarly pressure (Figure 10(B),11(B),Density (Figure 10 (C),11 (C)) have also decreased while velocity (Figure 10(D),Figure 11(D)) has increased.

PROFILES : -

Figure 10 (A) Temperture profile at the last step (2500 iteration)

Figure 10 (A) Pressure profile at the last step (2500 iteration)

Figure 10 (C) Density profile at the last step (2500 iteration)

Figure 10 (D)Velocity profile at the last step (2500 iteration)

Graphs: -

Figure 11 (A) Temperture Graph from top to bottom at the last step (2500 iteration)

Figure 11 (B) Pressure Graph from top to bottom at the last step (2500 iteration)

Figure 11 (C)Density Graph from top to bottom at the last step (2500 iteration)

Figure 11 (D)Velocity Graph from top to bottom at the last step (2500 iteration)

COMPARISION OF CELL NUMBERS FOR BOTH THE CASE: -

Figure 12: - Cell Numbers vs Cycles

7. CONCLUSIONS:- The results obtained for SGC=0.005 K should be more accurate than SGC=0.01 for given number of cycles.It is so because if SGC=0.005 K then the meshing will be refined for any cell temperature greater than 0.005 K and if SGC =0.01 K ,the meshing will be refined for any cell temperature greater than 0.01 k. Hence, the number of cells is larger for SGC =0.005 K than that of SGC =0.01 K.So, the result should be accurate more in case 01.Now, from figure 12 there is a difference of 2000 cells between both the cases.From figure 08,09,10,11; we can draw the conclusions that temperature of air after passing through the Expansion Waves is same for both the case, for pressure and density the final value is lesser for SGC =0.005 K and velocity graph is same for both the cases.P₂/P₁=0.7241 for SGC =0.01 K;P₂/P₁=0.7193 for SGC=0.005 K.

ANIMATIONS:-