Prandtl Meyer Shock problem
Aim:
To simulate Prandtl–Meyer Shock problem in CONVERGE.
Objective:
1.Literature Review of shock wave.
2.To understand diferent types of boundary conditions.
3.Effect of SGS parameter on shock wave,temperature and cell count.
Introduction:
Shock wave:
Shock wave 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 preassure, temperature and density of the medium. The energy and speed of a shock wave alone dissipates relatively quickly with distance. When a shock wave passes through matter, energy is preserved but entropy increases.
Prandtl–Meyer expansion fan:
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 Thermodynamics. 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. the total pressure and total temperature) remain constant across the fan.
Fig: Prandtl-Meyer expansion wave
As shown in Fig. The leading edge of the expansion fan makes an angle μ1 with respect to the upstream flow direction, and the trailing edge of the wave makes an angle μ2 with respect to the downstream flow direction. The angles μ1 and μ2 are Mach angles, defined by
Where M1 and M2 are the upstream and downstream Mach numbers, respectively.
The final static properties are a function of the final flow Mach number (M2) and can be related to the initial flow conditions as follows,
Boundary condition:
There are three main Boundary condition:
1.Dirichlet boundary condition
2.Neumann boundary condition
3.Robin boundary condition:
Dirichlet boundary condition:
Dirichlet boundary condition specifies the value of a variable at a boundary. It may also be called a fixed condition. Therefore, it is used when a value needs to be imposed at a boundary.
For example, a no-slip condition in fluid mechanics is a Dirichlet condition because it sets the value of the velocity to zero.
Neumann boundary condition:
Neumann boundary condition specifies the derivative of a variable at a boundary. It is used rather than the actual value when a certain rate of change in value needs to be imposed for a variable.
For example in fluid mechanics is the fully developed condition at an outlet where the gradient of flow variables is set to zero.
Robin boundary condition:
Robin boundary condition specifies a combination of the value and the derivative of a variable at a boundary. Therefore, these conditions are suited for more complex behaviour.
For example, in heat transfer, Robin conditions are used to model Newton’s law of cooling, where heat flux is proportional to temperature.
Mach Number:
Mach number is ration of local flow velocity with respect to the boundaries to the local speed of sound in the medium.
M = u/c
Where:
M is the local Mach number,
u is the local flow velocity with respect to the boundaries
c is the speed of sound in the medium,
Subsonic conditions occur for Mach numbers less than one, M < 1
Sonic condition occur for Mach numbers is equal to one, M = 1
Supersonic conditions occur for Mach numbers greater than one, M > 1
Praoblem Discription:
Geometry:
Geometry is imported in the CONVERGE studio.
Material: Air
Boundary Conditions:
Inlet: Boundary Type:INFLOW
Preassure : Zero Normal Gradient
Velocity : 680 m/s (X direction)
Temperature:286.1 K
Outlet: Boundary Type: OUTFLOW
Preassure : Zero Normal Gradient
Velocity : Zero Normal Gradient
Temperature:300 K
Upper and Lower Wall : Boundary Type:Two-D
Front and Back: Boundary Type: WALL
slip boundary condition
Run Parameter:
Solver: Steady state solver
Simulation time parameter:
Start Time = 0
End Time = 25000 cyc.
Initial Time Step = 1e-7
minimum Time Step = 1e-7
Maximum Time step = 1 S
Turbulence Modeling:
Turbulence Model: RNG k-ε">ε
ε">Initial Condition:
ε">Velocity: 680m/s and 100 m/s (in both the cases x direction)
ε">Temperature : 300k
ε">Grid Control:
ε">Base Grid = 0.8 m (in all X,Y and Z direction)
ε">Adaptive Mesh Refinement(AMR):
ε">Cycle steady = 100
ε">Minimum cells = 1
ε">Maximum Cells = 200000
ε">Maximum embedding level = 2
ε">Sub grid criterion(SGS):
ε">Here SGS is taken as variablle to study the Prandtl meyer shock problem.
ε">Sub grid criterion = 0.05 K
ε">Sub grid criterion = 0.01 K
ε">refined grid size = base grid/2^n
ε">n = max embedding level
ε">Therefor,for base grid size of 0.8 m,
ε">refined grid size =0.2 m
ε">Result:
ε">Case 1: Velocity = 680 m/s and SGS = 0.05 K
ε">Mesh:
ε">
ε">Staticε"> Preassure:
ε"> ε">
Average Density:
Mach Number:
Average mach number at inlet = velocity of air/ velocity of sound = 680/340 = 2
Average temperature:
Total Cell Count:
Velocity Contour:
Preassure Contour:
Temperature contour:
Case 2: Velocity =680 m/s ,SGS=0.01K
Mesh:
Total Preassure:
Static Preassure:
Average Density:
Mach Number:
Average mach number at inlet = velocity of air/ velocity of sound = 680/340 = 2
Average Temperature:
Total cell count:
Velocity Contour:
Preassure Contour:
Temperature contour:
Case 3: Velocity = 100 m/s & SGS = 0.05 K
Mesh:
Mach Number:
Total cell count:
Velocity contour:
Preassure Contour:
Temperature Contour:
Result:
1.Total cell count is as follows:
at velocity =680 m/s & SGS =0.05k : Total cell count~7700
at velocity =680 m/s & SGS =0.01k : Total cell count~15900
at velocity =100 m/s & SGS =0.05k : Total cell count~4300
2.As the fluid flow with velocity of 680 m/s,the is mach no. for given condition becomes 2and hence the flow is supersonic.As the supersonic flow is expanded at sharp corner infinite no.of expansion waves are formed and as the fluid flow across the expansion waves mach number increase and hence preassure,temperature and density decreases at outlet.
3.As SGS value decreses finer mesh is produced.Physical meaning of sub grid temperature is whenever difference between actual temperature and resolved temperature becomes SGS value the CONVERGE will refine the Mesh.
4.At the velocity = 100 m/s,the mach number is around 0.3 so flow is subsonic and hence flow doensn't get expanded at sharp corner and hence no expansion fan is observed.