Week 5: Prandtl Meyer Shock problem

                       LITERATURE REVIEW ON SHOCK FLOW AND SIMULATION PF PRANDTL MEYER EXPANSION FAN OVER AN EXPANSION CORNER.

 

l. OBJECTIVES

1. Perform a literature review on shock flow and associated boundary conditions.

2. Analyse the effect of the Prandtl Meyer expansion fan on the flow properties by simulating the supersonic flow of fluid through a geometry having an expansion corner.

3. Simulate and analyze the effect of the Sub-Grid Scale (SGS) parameter on the shock location.

4. Analyse the flow behavior of fluids by simulating the subsonic flow through the given geometry having an expansion corner.

 

ll. SHOCKS FLOWS AND ASSOCIATED BOUNDARY CONDITIONS

1. Shock Waves

A shock wave is propagating disturbance that moves faster than the local speed of sound in the medium. Like any ordinary wave, it carries energy and can propagate through a medium (solid, liquid, gas, or plasma) or in some cases in the absence of a material medium, through a field such as the electromagnetic field.

A shock wave is a very thin region in a flow where the supersonic flow decelerates to subsonic flow. The process is adiabatic but non-isentropic.

The shock waves are characterized by an abrupt change in the characteristic of the medium. For example, across the shock, there is always a rapid rise in pressure, temperature, and density of the flow.

2. Types of Shock waves

2.1. Normal Shock Waves

If the wave is perpendicular to the flow direction, it is called a normal shock wave. A normal shock occurs in front of a supersonic object if the flow is turned by a large amount and the shock cannot remain attached to the body.

2.2. Oblique Shock Wave

An oblique shock wave is not perpendicular to the direction of fluid flow. Such a shock wave arises when a fluid stream flowing at a supersonic move along a convergent or divergent boundary.

2.3. Curved Shock Wave

When an object moves faster than the speed of sound, and there is an abrupt decrease in the flow area, shock waves are generated. However, if the flow area increases, a different flow phenomenon is observed. If the rise is abrupt, we encounter a centered expansion fan called Prandtl Meyer expansion fan.

The fan consists of an infinite number of Mach waves, diverging from a sharp corner. Across the expansion fan, the flow accelerates, and the Mach number increases, while the static pressure, temperature, and density decrease. Since the process is isentropic, the stagnation properties (e.g., total pressure and total temperature) remain constant across the fan.

4. Difference Between Shock Wave and Expansion Fans

There are some marked differences between shock waves and expansion fans.

Figure 4.1 - Shock Waves

The Mach number decreases across a shock wave, the static pressure increases, and there is a loss of total pressure because the process is irreversible.

Figure 4.2 - Expansion Fans

The Mach number increases through an expansion fan, the static pressure decreases, and the total pressure remains constant. Expansion fans are isentropic.

 

5. Boundary Conditions

Boundary conditions are constraints necessary for the solution of the boundary value problem. A boundary value problem is a differential equation ( or system of differential equations ) to be solved in a domain on whose boundary a set of conditions is known.

Boundary value problems are extremely important as they model a vast amount of phenomena and applications, from solid mechanics to heat transfer, from fluid mechanics to acoustic diffusion.

Type of Boundary Conditions

Both ordinary and partial differential equations need boundary conditions to be solved. Different types of boundary conditions can be imposed on the boundary of the domain. The choice of boundary condition is very important as a bad imposition of boundary condition may lead to the divergence of the solution or may also lead to the convergence of an incorrect solution.

A. Dirichlet Boundary Condition

This condition specifies the value that the unknown function needs t take on along the boundary of the domain.

Example:

In CFD, the classical Dirichlet boundary condition consists of the value of velocity or pressure to be taken by a certain set of nodes. It is common to refer to some sets of boundary conditions according to the following terminology:

• Slip boundary condition: The velocity normal to the boundary is set to zero, while the velocity parallel to the boundary is let free.
• No-slip boundary condition: Both the velocity normal to the boundary and the velocity parallel to the boundary are set equal to zero.

At least one homogeneous boundary condition on the pressure (i.e. p = 0) has to be imposed as a reference for open domains, for instance, in the highest boundary of the air domain.

B. Neumann Boundary Condition

When imposed on an ordinary or a partial differential equation, the Neumann boundary condition specifies the values that the derivative of solution is going to take on the boundary of the domain.

Example:

Constraints on the derivative of velocity can be seen in the application of a symmetry plane - 

• `(delu)/(deln)=0`
• Since this condition is always applied in addition to a Dirichlet boundary condition on the velocity normal to the boundary, it is naturally satisfied.

C. Robin Boundary Condition

The Robin boundary condition consists of a linear combination of the values of the field and its derivatives on the boundary. Thus, it can also be said to be the linear combination of the Dirichlet and Neumann boundary conditions.

D. Mixed Boundary Condition

It consists of applying different types of boundary conditions in different parts of the domain. The mixed boundary condition differs from the Robin condition because the latter consists of different types of boundary conditions applied to the same region of the boundary, while the mixed condition implies different types of boundary conditions applied to different parts of the boundary.

E. Cauchy Boundary Condition

The Cauchy boundary condition is the condition on both the unknown field and its derivatives. It differs from the Robin condition because the Cauchy condition implies the imposition of two constraints (1 Dirichlety + 1 Neumann), while the Robin condition implies only one constraint on the linear combination of the unknown function and its derivatives.

 

6. Shock Flow Boundary Conditions

The boundary conditions need to be defined for the surfaces bounding the entire domain. In a shock wave flow, the flow fields get separated into two regions due to the generation of an expansion fan, the flow field before the expansion fan is called the upstream region, and the flow field after the expansion fan is called the downstream region

A. Inlet Boundary Conditions

Initially, in the upstream region, the flow needs to be supersonic to generate an expansion fan. This is done by specifying Dirichlet boundary conditions at the inlet.

In the given projects, we take - 

• Velocity of Fluid at Inlet = 680 m/s
• Temperature = 286.1 K

The speed of sound depends on many factors like altitude, medium, and temperature, but for the given calculation, we can assume it to be 340 m/s.

Therefore, Mach Number (M) = Velocity of Fluid / Speed of Sound = 2

Since M> 1, the fluid travels at supersonic speed.

B. Outlet Boundary Conditions

Once the flow field passes the expansion fan, the fluid properties changes and hence are unknown across the expansion fan. Since the fluid properties are unknown, we define the Neumann boundary condition at the outlet.

 

ll. SUPERSONIC FLOW OF FLUID OVER AN EXPANSION CORNER

A. GEOMETRY

1. Geometry Setup

The geometry is setup in converge studio as shown in the figure below -

2. Diagnosis Check

A diagnosis check is performed to check for errors - 

3. Boundary Flagging

The boundaries are flagged into five components as shown in the figure below -

 

B. CASE SETUP

1. Fluid: Air

• Mass fraction of Nitrogen: 0.77
• Mass fraction of Oxygen: 0.23

2.Slover:

A steady-state density-based solver is used to simulate the flow.

3. Simulation Cycles:

• Total Number of Cycles: 25,000 cycles
• Minimum number of cycles to be executed by the steady-state solver: 25,000 cycles
• Initial time-step: 1e-9 s
• Minimum time-step: 1e-9 s
• Maximum time-step: 1 s

4. Boundary Conditions:

Boundary	Boundary Type	Dirichlet Boundary Condition	Neumann Boundary Condition
Inlet	INFLOW	Velocity = 680 m/s  Temperature = 286.1 k  Pressure = 101000 pa	-
Outlet	OUTFLOW	-	Velocity            Pressure           Temperature (Zero gradient to all the parameters )
Front 2-D	TWO - D	-	-
Back 2-D	TWO - D	-	-
Top and Bottom Wall	WALL	Velocity: Slip	Temperature

 

5. Regions and Initialization:

• Flow Region: Volumetric Flow Region
• Velocity: 680 m/s(x-direction)
• Total Pressure: 50000 Pa
• Temperature: 300 K
• Species: Air

6. Turbulence Model: In our simulation, we don't have to use Turbulence modeling since there is no Reynolds stress because the flow is inviscid.

7. Grid Control:

• Base Grid:
  • dx = dy = dz = 0.8 m
• Adaptive Mesh Refinement: AMR Group 1
  • Active Region: Volumetric Flow Region
  • Temperature:
    • Embedded Type: Sub-Grid Scale (SGS)
    • Maximum Embedding Level: 2
    • Sub-grid criterion: 0.05 K
    • Timing Control Types: Sequential
    • Start: 5000 cycles
    • End: 999999999 cycles

8. Output Files:

• Time interval for writing 3-D output files: 1000 cycles
• Time interval for writing restarting files: 1000 cycles

 

C. OUTPUTS

1. Mesh

The mesh is kept constant for the first 5000 cycles. Converge monitors the curvature of the fluid property provides in the AMR and refines the mesh when the property curvature variation between the consecutive grid is more than the defined SGS value. This is done to capture the flow properties in the expansion fan region accurately.

Figure 1.1 - Mesh (At the start of the simulation)

Figure 1.2 - Mesh (At the end of the simulation)

2. Flow Properties Contours

Animation 2.1 - Temperature Contour

Animation 2.2 - Temperature Contour with AMR

Animation 2.3 - Velocity Contour

Animation 2.4 - Velocity Contour with AMR

 3. Temperature, Velocity, Pressure, and Density Contours

we can see that when the supersonic fluid flows past the expansion corner, it is accompanied by an increase in the velocity and a decrease in the temperature, pressure, and density of the fluid.

4. Prandtl Meyer Shock Representation in Temperature Contour

 

D. VARIATION OF FLOW PROPERTY AT INLET AND OUTLET

1. Variation of Mach number w.r.t. Number of Cycles

2. Variation of Temperature w.r.t. Number of Cycles

3. Variation of Velocity w.r.t. Number of Cycles

4. Variation of Mass Flow Rate w.r.t. Number of Cycles

5. Variation of Cell Count w.r.t. Number of Cycles

E. RESULTS

1. The Prandtl-Meyer expansion fans are isentropic processes that generate continuous and smooth changes in the flow, causing the flows to total properties to be conserved.

2. The total values are the values that the flow properties produce if brought to stagnation, while static values represent the actual values of the flow properties ate a certain speed.

3. There is a decrease in static temperature, pressure, and density as well as an increase in the Mach number and velocity of flow from the inlet to the outlet due to the presence of the expansion fan.

4. The Mach number of supersonic flow increases through an expansion fan. The amount of the increase depends on the incoming Mach number and the angle of the expansion.

5. The mass flow rate at the outlet is equal to the negative of the mass flow rate at the outlet at the end of the simulation. This shows that the mass flow is conserved.

6. The cells are unevenly distributed among the four processors to achieve a good load balance. Also, since the grid undergoes adaptive mesh refinement, the number of cells changes continuously after a fixed number of cycles.

 

lll. EFFECTS OF SUB-GRID SCALE (SGS) ON THE FLOW PROPERTIES

A. GEOMETRY

The geometry is the same as used in the previous simulation.

B. CASE SETUP

The case setup is similar to the one used in the previous simulation. Since we are observing the effects of the SGS parameter on the flow properties, we will simulate four cases having the following SGS values -

• Case 1: SGS= 0.01K
• Case 2: SGS= 0.05K
• Case 3: SGS= 0.1K
• Case 4: SGS= 0.3K

C. OUTPUTS AND RESULTS

Figure - Temperature Contours for different SGS

From the figures and animations, it can be observed that as the SGS parameter increases, the expansion fan becomes less distinct, and the mesh is unable to capture the effects of the expansion fan accurately. This shows that the adaptive mesh refinement is dependent on the sub-grid scale value. Decreasing the SGS parameter helps to capture the expansion fan accurately.

lV. SUBSONIC FLOW OF FLUID OVER AN EXPANSION CORNER

A. GEOMETRY

The geometry is the same as used in the previous simulations.

B. CASE SETUP

The case setup is similar to that for supersonic flow. Only the velocity at the inlet and the initial velocity conditions of the inlet are changed to obtain subsonic flow.

 

1. Boundary Conditions:

Boundary	Boundary Type	Dirichlet Boundary Condition	Neumann Boundary Condition
Inlet	INFLOW	Velocity = 680 m/s  Temperature = 286.1 k  Pressure = 101000 pa	-
Outlet	OUTFLOW	-	Velocity            Pressure           Temperature (Zero gradient to all the parameters )
Front 2-D	TWO - D	-	-
Back 2-D	TWO - D	-	-
Top and Bottom Wall	WALL	Velocity: Slip	Temperature

 

2. Regions and Initialization:

• Flow Region: Volumetric Flow Region
• Velocity: 680 m/s(x-direction)
• Total Pressure: 50000 Pa
• Temperature: 300 K
• Species: Air

C. OUTPUTS AND RESULTS:

Figure 2 - Flow Parameters Contour

The contours obtained vastly differ from those obtained for supersonic flow. There is no expansion fan-generated and there is no abrupt change in the flow parameters. The flow parameters passing through the given geometry can be described by conserving momentum and energy. 

A slight increase in velocity and decrease in pressure is observed at the expansion corner, which can be developed into expansion fans if the flow becomes supersonic.

 

V. CONCLUSION

We were able to successfully design and simulate the flow of air over an expansion corner and analyze the effects of the SGS parameter. We were also able to simulate the subsonic flow of air over an expansion corner and analyze the differences in the obtained results.

This is useful for understanding the flow behavior in models subjected to supersonic speed and leading to the generation of an expansion fan. This can help us to design aircraft models traveling at a supersonic speed such that the expansion wave does not affect the airflow around the aircraft.