Navier-Stokes Equation in Moving Reference Frame (MRF)

Most of the cfd applications consist of stationary objects around/inside which fluid will be flowing. So meshes are stationary. If the flow is incompressible, Incompressible Steady Navier-Stokes equations are solved. Very common applications are – Flow around Aerofoil, Flow inside Pipe. In both cases aerofoil and pipe are stationary hence meshes are stationary & solved using Incompressible Steady Navier-Stokes equations.

But there are certain applications involving motion of the geometry for example turbo machines. In that case mesh motion is required. It could be rotation or translation. Now our problem is unsteady because of the motion of the mesh. At each and every time step mesh is being moved. Now we will be solving Incompressible Unsteady Navier-Stokes equations which are more computationally expensive to solve ofcourse.

The Motivation

Problem comes when we have to deal with very high speed rotation or translation. We need to keep “time step size” very small such that any flow variable does not jump more than one/two mesh cell in one time step.

For example if there is a turbine rotating at 36000 rpm and one would want to solve for 10 revolutions. So in this case 600 revolutions are happening in 1 seconds. 1 revolution will happen in 0.0016 second. For 10 revolutions, We have to solve totally for 0.0167 seconds. This is our total solution time, 0.0167 second.

Here we have focus on this – 1 revolution is happening in 0.0016 seconds. In order to capture full transient flow physics we have to keep time step extremely small. For example let say it is only allowed to mesh move 10 degree per time step based on the accuracy requirements. So in this case time step will be 0.000044 seconds which is extremely small. So we can see how computational expensive situations can become.

What about initial condition. If the given initial conditions are poor. We have to further increase total solution time so that error in initial solution can be eliminated. Some time poor initial condition can lead to divergence also.

Other issue is when there is mesh in motion. Interface between moving and stationary region will generally going to be non-conformal. In order to transfer solution from moving region to the stationary region, CFD solver have to do some extra interpolation which makes computation more expensive and unstable.

What about initial condition. If the given initial conditions are poor. We have to further increase total solution time so that error in initial solution can be eliminated. Some time poor initial condition can lead to divergence also.

Other issue is when there is mesh in motion. Interface between moving and stationary region will generally going to be non-conformal. In order to transfer solution from moving region to the stationary region, CFD solver have to do some extra interpolation which makes computation more expensive and unstable.

Alternative Approach – Moving Reference Frame (MRF)

This is the concept of MRF, Governing equations are solved in a reference frame that is rotating or translating with the same speed of the rotating/translating geometry. Physically it means we are sitting on the moving body and seeing the flow field around it. This makes the flow field steady relative to the geometry.

If there is a turbine rotating and we are standing on the floor then flow field around turbine would be  transient from our perspective.

Instead of watching rotating turbine from a distance, If we sit on turbine blade and rotate with the blade then the flow field around us or turbine would be steady from our perspective.

Another well known example is River & Boat. Boat is moving with fixed velocity. There is a bridge above river. If you stand on bridge and observe flow of river water, First water below bridge is stand still then boat comes and create disturbance. Then boat goes and disturbance of water goes down. Observer standing on the bridge, flow appears to him will be unsteady because at a given location below the bridge, river water speed in changing.

If a person sitting in boat, moving with the boat itself. Now with respect to that person, River water will not change and He will see river water as steady flow field.

Steady state problems in the moving frame are easier to solve then transient problem with moving mesh.

This approach significantly reduce computational cost.

Now, Equation of Fluid Dynamics are defined with respect to Moving Reference Frame (MRF). We must account for additional accelerations terms which models the affect of fluid motion in the moving frame.

Fixed Vector in a Rotating Frame

What if vector itself is moving in the Rotating Frame?

Then the total dA/dT is the dA/dT as we see from the stationary frame + due to the change in the direction of vector A which is Omega X vector(A).

For Momentum Conservation (Navier-Stokes Equation), Our interest is Acceleration, Which is double derivative of displacement.

Momentum Conservation

Coriolis Acceleration – If we are translating in rotating frame, we will experience a lateral force.

Centrifugal Acceleration – Experience when sitting on rotating frame.

Translation Acceleration –  If we are sitting in translating reference frame.

Rotational Acceleration/Euler Force – This is due to angular acceleration of reference frame.

Strength of source term centrifugal acceleration will increase as we go away from the origin of MRF

Navier Stokes in Moving Reference Frame

Let’s look at the left-hand side of the momentum equation of Eqn , by taking into account Eqn  for the acceleration term:

First term on RHS can be expanded and written as :-

Finally :-

Incompressible Navier-Stokes equations in the rotating frame, in terms of relative velocities

We want to solve single set of navier stokes equation in entire domain but with extra source term in rotating region only. As we can see image above we define separate region and assign it as MRF. By applying vector identities and doing rearrangements we can arrive to the Navier-Stokes equations in the relative frame with absolute velocity.

This “source term” is only applied to the region of MRF only.

In the convection term we have absolute as well as relative velocity.

In CFD codes of Finite volume method. We discretize the Convection term in usual way which is integrating the terms and applying Gauss’s divergence theorem & we arrive to face volume fluxes which is shown below.

Now we substitute this expression shown below into the face volume fluxes.

Finally we arrive at :-

First term on RHS is same as we would have as in general NV-Stokes Equation.

Second term “Flux  Correction”, Physically when we are jumping from absolute frame to rotating frame we have to correct the fluxes in order to account for relative velocity as we would see if we were in rotating with the MRF.

Summary

References: ANSYS Officials, openFOAM Foundation