Study of the Prandtl Meyer Shock problem and effect of subgrid mesh

Aim

Study of the Prandtl Meyer Shock problem and effect of subgrid mesh.

Shock wave

The shock wave, constructive interference of sound created by an object moving faster than sound.

Shock wave boundary interactions occur when a shock wave and a boundary layer converge and, since both can be found in almost every supersonic flow, these interactions are commonplace. The most obvious way for them to arise is for an externally generated shock wave to impinge onto a surface on which there is a boundary layer. However, these interactions also can be produced if the slope of the body surface changes in such a way as to produce if the slope of the body surface changes in such a way  as to produce a sharp compression of the flow near the surface-  as occurs, for example, at the beginning of a ramp or a flare, or in front of an isolated object attached to a surface such as a vertical fin. If the flow is supersonic, compression of this sort usually produces a shock wave that has its origin within the boundary layer. This has the same effect on the various flow as an impinging wave coming from an external source.

The shock wave, strong pressure wave in an elastic medium such as air, water, or a solid substance, produced by supersonic aircraft, explosions, lightning, or other phenomena that create violent changes in pressure. Shock waves differ from sound waves in that the wavefront, in which compression takes place, is a region of sudden and violent change in stress, density, and temperature. Because of this, shock waves propagate in a manner different from that of ordinary acoustic waves. In particular, shock waves travel faster than sound, and their speed increases as the amplitude are raised; but the intensity of a shock wave also decreases faster than does that of a sound wave, because some of the energy of the shock wave is expended to heat the medium in which it travels. The amplitude of a strong shock wave, as created in the air by an explosion, decreases almost like the inverse square of the distance until the wave has become so weak that it obeys the laws of acoustic waves. Shock waves alter the mechanical, electrical, and thermal properties of solids and, thus, can be used to study the equation of state (a relation between pressure, temperature, and volume) of any material.

Sound waves are caused by infinitesimally small pressure disturbances and travel through a medium at the speed of sound. Under certain flow conditions, abrupt changes in fluid properties occur across a very thin section: shock wave. Shock waves are characteristic of supersonic flows, that is when the fluid velocity is greater than the speed of sound. Flow across a shock is highly irreversible and cannot be approximated as isentropic.

• Sound waves are caused by infinitesimally small pressure disturbances and travel through a medium at the speed of sound.
• Under certain flow conditions, abrupt changes in fluid properties occur across a very thin section: shock wave.
• Shock waves are characteristic of supersonic flows, that is when the fluid velocity is greater than the speed of sound.
• Flow across a shock is highly irreversible and cannot be approximated as isentropic.

In physics, a 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 the energy and can propagate through a medium but is characterized by an abrupt, nearly discontinuous, change in pressure, temperature, and density of the medium.

Normal Shocks

Shock waves that occur in a plane normal to the direction of flow: Normal shocks. A supersonic flow across a normal shock becomes subsonic. Conservation of energy principle requires that the enthalpy remains constant across the shock. h01 = h02

For an ideal gas,

h = h(T)

and thus

T01 = T02  

Across the normal shock we apply the governing equations of fluid motion:

Mass:           ρ1A1V1= ρ2A2V2

Energy:             h01 = h02

Momentum: A (P1 –P2) = (V2 –V1)

Entropy:       s2 – s1 ≥ 0

If we combine mass and energy equations and plot them on the h-s diagram: Fanno line

Similarly combining mass and momentum gives: Rayleigh line

Shock waves that are inclined to the flow at an angle: oblique shocks. In a supersonic flow, information about obstacles cannot flow upstream and the flow takes an abrupt turn when it hits the obstacle. This abrupt turning takes place through shock waves. The angle through which the fluid turns: deflection angle or turning angle, θ. The inclination of the shock: shock angle or wave angle, β.

Oblique shocks are possible only in supersonic flows. However, the flow downstream of the shock can be either supersonic, sonic or subsonic, depending upon the upstream Mach number and the turning angle. To analyze an oblique shock, we decompose the velocity vectors upstream and downstream of the shock into normal and tangential components.

There are two possible values of β for θ<θmax. θ=θmax line: Weak oblique shocks occur to the left of this line, while strong oblique shocks are to the right of this line. M=1 line: Supersonic flow to the left and subsonic flow to the right of this line. For a given value of upstream Mach number, there are two shock angles. β=βmin represents the weakest possible oblique shock at that Mach number, which is called a Mach wave.

Prandtl-Meyer Expansion waves

An expanding supersonic flow, for example, on a two-dimensional wedge, does not result in a shock wave. There are infinite Mach waves forming an expansion fan. These waves are called Prandtl-Meyer expansion waves. The Mach number downstream of the expansion increases (M2>M1), while pressure, density, and temperature decreases.

Prandtl-Meyer expansion waves are inclined at the local Mach angle µ. The Mach angle of the first expansion wave

µ1 = sin-1(1/M1)

µ2 = sin-1(1/M2)

Turning angle across an expansion fan is

Θ = v(M2) – v(M1)

V(M) is called the Prandtl-Meyer function

V(M) =  `√((γ+1)/(γ-1) ) 〖 tan〗^(-1) [√((γ+1)/(γ-1)) (M^2-1)]-〖tan〗^(-1) (√(M^2-1))`

Shock Boundary Conditions

For solving the steady state flow equation as derived above, appropriate boundary conditions are needed. It is one of the required components of the mathematical model. On the other hand, for solving the transient flow equation, the appropriate initial condition is also required. Boundary conditions are generally three types. They are Dirichlet boundary condition, Neumann boundary condition, and mixed boundary condition.

1. Dirichlet Boundary Conditions

Dirichlet boundary condition, prescribe the value of the variable h(x,y,z,t) is specified at the boundary of the problem domain. This is also known as type I boundary condition. The head may be constant or may vary in space or in time.

2. Neumann Boundary Conditions

Neumann boundary condition, the gradient of the variable is specified at the boundary of the problem domain. Here, n is the direction, x, y, and z. One of the most frequently use Neumann boundary condition is the no-flow boundary condition, i.e.  = 0 at the boundary.

As discussed, in the case of Neumann boundary condition, we have

`(dh)/(dn)=c_1`

Where C₁ is constant

`-K_n (dh)/(dn)= -K_nC1=q_n`

Where q_n is Darcy\'s flux in the n^th direction. As such in case of Neumann boundary condition, we can also specify Darcy\'s flux at the boundary instead of the gradient of the variable. The Neumann boundary condition is also knowing as Type II boundary condition.

“Dirichlet boundary conditions describe an absorbing phenomenon on the boundary and Neumann boundary conditions describe a reflecting phenomenon on the boundary.” Dirichlet boundary condition, the value of the variable is prescribed at the boundary. Neumann boundary Conditions, When Neumann boundary Conditions imposed on an ordinary or a PDE, the condition specifies the values in which the derivative of a solution is applied within the boundary of the domain.

Pressure outlet boundary conditions require the specification of a static (gauge) pressure at the outlet boundary. The value of the specified static pressure is used only while the flow is subsonic. If the flow is supersonic, the specified pressure will no longer be used, the pressure will be extrapolated from the flow in the interior. This means that Neumann Boundary Conditions which say that the outlet condition is dependent on the initial conditions of the flow need to be used in this case, that will derive the value of pressure from the initial conditions.

3.Mixed Boundary Conditions

We can also specify a mixed boundary condition in the form given below.

This is also known as Type III boundary condition and is the linear combination of Type I and Type II boundary condition.

`ah+b(dh)/(dn)=Constant`

4. Robin Boundary Condition

When imposed on an ordinary, or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the boundary of the domain. Robin boundary conditions are a weighted combination of Dirichlet boundary conditions and Neumann boundary conditions. This contrasts to mixed boundary conditions, which are boundary conditions of different types specified on different subsets of the boundary.

5. Cauchhe Boundary Condition

Cauchy boundary conditions augment an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so to ensure that a unique solution exists. A Cauchy boundary condition specifies both the function value and normal derivative on the boundary of the domain.

Computational Parameter

1. Density Based Steady State Solver
2. Boundary Condition

• Inlet – Velocity – 680 m/s (Temp - 286.1K)
• Outlet – Zero gradient value

1. Simulation Parameter

• Start time 0
• End time 25000
• Initial time step 1e-7
• Minimum time step 1e-7
• Maximum time step 1

1. Base grid 0.8
2. Adaptive Mesh Refinement

• Sub Grid – 0.05
• Grid Size = Base Grid / (2) ^ (embed -level)

Supersonic Condition

(of speed) greater than the speed of sound in a given medium (especially air)

from the above fig, we observe that at 5000 cycle cell is constant and after that higher increase in the total cell that is the effect of Adaptive mesh refinement.

From the above fig of mass flow, it is easily showing the converse condition of inlet and outlet mass flow curve nature symmetrical. 

From the above figure of pressure, we easily observed that an increase no of the cycle with respect static and total pressure is increasing. Total pressure slop is increasing high compared to static pressure.

As the temp plot, Inlet temp is constant for all the cycle. Outlet temp plot it is observed a decrease in temp and then constant temp behavior.

Mach number is constant for the inlet. Increment in Mach number for outlet profile.

When using a Dirichlet boundary condition, the value of the variable is prescribed at the boundary. Neumann boundary Conditions, When Neumann boundary Conditions imposed on an ordinary or a PDE, the condition specifies the values in which the derivative of a solution is applied within the boundary of the domain.

Pressure outlet boundary conditions require the specification of a static (gauge) pressure at the outlet boundary. The value of the specified static pressure is used only while the flow is subsonic. If the flow is supersonic, the specified pressure will no longer be used, the pressure will be extrapolated from the flow in the interior. This means that Neumann Boundary Conditions which say that the outlet condition is dependent on the initial conditions of the flow need to be used in this case, that will derive the value of pressure from the initial conditions.

SubSonic Condition

Subsonic is below the speed of sound, where air behaves non-compressibly, similar to water in many ways, but many times less dense. Drag is minimized by using teardrop-shaped vehicles and projectiles.

• from the compare, the figure of pressure, velocity, and the temp is showing non-compressive behavior of the fluid.
• Using adaptive mesh refinement, the total cells in the supersonic waves dynamically increase. However, in the case of subsonic flow, it remains constant as there is no abrupt change in the properties of the fluid.
•  

Boundary Condition

• Inlet – Velocity – 100 m/s (Temp - 286.1K)
• Outlet – Zero gradient value

Effect of SGS parameter on shock location

From the above figure, As Sub grid size decreases the adaptive mesh refinement observed.

Conclusion

• From SGS parameter it is noticed that lower the SGS will result in good refinement of cells as the SGS was increased it resulted in lowering the refinement cells.
• It is clearly seen that the speed changes once the enlargement wave and it\'s been inflated.
• The Pressure, Temperature is additionally been decreased because of the flow past the wave.
• The process is a physical property, stagnation properties stay constant.
• The subsonic flow, there\'ll no shock wave generated as a result of shock wave only generated if the fluid speed is greater than the speed of sound.
• Super-sonic flow, shock wave started because space suddenly will increase as a result of in supersonic flow speed is greater than the speed of sound.