Challenge

Prandtl Meyer Shock problem

EEssha Shinde1 Sept 2020

Shock Wave

All flow processes aim to change the state of the medium through which they traverse.

the medium can undergo state change by 3 ways

isentropic porcess: effects of heat transfer and friction is neglected. change of state is achieved by change n cross sectional area.

Fanno flow: change of state is caused by friction. croos sectional area remains constant. heat transfer is neglected

rayleigh flow: the cross sectional area remains constant. no friction takes place. change of state is caused by change in stagnation temperature.

every type of application pertaining to fluid mechanics observes 4 different types of waves.

Mach waves: these are isentropic waves which are represented in grid. they cause negligible changes in flow properties. these are present in supersonic and subsonic flow.

isentropic compression waves: these are weak compression fronts accross which flow property changes are finite but small(<5%). the changes in entropy is 0

expansion waves: type of isentropic waves where entropy change is 0 however property changes can be neglected. strength of waves is finite.

non isentropic compression waves : Also known as shock waves are compression fronts accross which flow properties such as pressure, temperature, density, entropy tend to jump at a higher level, and velocity accross the front decreases.

they have very thin region of order of mean free path (10^-6 cm)

shock waves are of 3 types

normal shock waves: when the wave is normal to flow direction

oblique shock waves: these waves are at an angle to flow direction

the static properties such as pressure, temperature, density accross the shock wave increases whereas the upstream stagnation pressure remains greater than the downstream stagnation pressure. the stagnation temperature remains the same

Shock waves are Adiabatic and Non isentropic in nature.

Shock waves are observed only in supersonic and hypersonic flow regimes.(M>1)

when the flow is subsonic, and it encounters any obstacle or change in flow area , the fluid element has sufficient time to adjust themselves according to downstream obstacle or area change.
The acoustic signal which carries the information regarding any obstacle or change in flow area downstream travels at the speed of sound.
however, in supersonic case the information coded in the acoustic signal doesnt get sufficient time to pass this information upstream.

Flow accross shock wave are adiabatic and irreversible in nature

the shock wave is very thin compression wavefront of thickness comparable to mean free path of gas molecules in flow field.
the flow accorss the shockwave is of high velocity.
owning to the above factors, the fluid crosses the wave in infinitesimal time.
thus thus the fluid particles are unable to exchange energy with the surrounding.

Natural and external shock waves are predominently oblique in nature

Shock waves are classifies as follows:

Normal shock waves
Oblique shock waves
Expansion shock waves

Boundary conditions:

boundary conditions are constraints for solution of boundary value problem
there are 3 types of boundary conditions

Dirichlet boundary condition: determines value on the domain boundary of unknown function

eg: u(x,t)displacement of vibrating string with end fixed at x=0 and x=h

u(0,t)=0 and u(h,t)=0 are the dirichlet b.c

neumann boundary condition: determines value of derivatve of solution at domain boundary

robin bc: consists of linear combination of values of field variable and derivatives at boundary

boundary conditions used in case set up

inlet: a dirichlet boundary condition used for velocity specifying value 680m/s that gives mach 1.6 at temperature boundary condition maintained at 286.1k at which speed of sound is 417.57m/s. Pressure boundary condition set to zsero normal gradient which is neumann boundary condition.

outlet: the pressure and velocity bc are set to zero normal gradient. Supersonic outlets are set to neumann bc as as boundary pressure and velocity is extrapolated from the interior field.

top-botton wall: top bottom wall set at slip condition. the temperature boundary condition set at zero normal gradient

The steady state simulation ran for 25000 cycles. A density based solver is used , since in supersonic compressible flow the density increases on downstream side.

a temperature based adaptive mesh refinement is used with a base grid of 0.8m. Maximum embedd level is set at 2. the results are observed for 3 sub grid criterion; 0.01,0.1,0.05

AMR is a technique which is used to refine accuracy of specific areas within a domain where quantification of precision is required, while maintaining the precision and resolution of other areas at lower level. In this simulation the region where shock occurs will have higher order of precision therefore finer cells than the surrounding region which has the base grid size of 0.8m.

Temperature based mesh refinement computes the rate of temperature change. I.e. the temperature curvature within the shock domain. If the curvature exceeds user defined value the mesh will be refined. Maximum embedding level is 2(n)

Thus, the refined mesh will have cell size:

Grid size= Base grid (0.8m)/2^2(n)=0.2m

SGS 0.01