Navier-Stokes Equation in Moving Reference Frame (MRF)

Reference: ANSYS Officials

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 of course.

The Motivation

The problem comes when we have to deal with very high-speed rotation or translation. We need to keep the “time step size” very small such that any flow variable does not jump more than one/two mesh cells in a 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 second. and 1 revolution will happen in 0.0016 seconds. For 10 revolutions, We have to solve totally for 0.0167 seconds. This is our total solution time, 0.0167 seconds.

Here we have to focus on this – 1 revolution is happening in 0.0016 seconds. In order to capture full transient flow physics, we have to keep the time step extremely small. For example, let’s say it is only allowed to mesh move 10 degrees 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 the initial condition? If the given initial conditions are poor. We have to further increase the total solution time so that errors in the initial solution can be eliminated. Sometimes the poor initial conditions can lead to divergence also.

Another issue is when there is mesh in motion. The interface between the moving and stationary regions will generally going to be non-conformal. In order to transfer the solution from the moving region to the stationary region, the CFD solver has 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 the flow field around the turbine would be transient from our perspective.

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

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

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

Reference: ANSYS Officials

Steady-state problems in the moving frame are easier to solve than the transient problems with the moving mesh.

This approach significantly reduces the computational time & cost.

Now, the Equation of fluid flow motion is defined with respect to Moving Reference Frame (MRF). We must account for additional accelerations terms that model the effect of fluid motion in the moving frame.

Fixed Vector in a Rotating Frame

What if vector A 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 times vector(A).

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

Acceleration in a Rotating Frame

Momentum Conservation

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

Centrifugal Acceleration – Experience when sitting on a rotating frame. 

Translation Acceleration –  If we are sitting on the translating reference frame.

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

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

Navier Stokes in Moving Reference Frame

Recall: above Eqn [1] is already derived in “Acceleration in a Rotating Frame” section

Let’s look at the left-hand side of the momentum equation of Eqn [2], by taking into account Eqn [1] 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 a single set of Navier stokes equations in the entire domain but with an extra source term in the rotating region only. As we can see the image above, we define the rotating region separately and assign it as a MRF region. By applying vector identities and doing rearrangements we can arrive at the Navier-Stokes equations in the relative frame with absolute velocity.  

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

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

In the CFD codes of the Finite volume method, We discretize the convection term in the usual way which is by integrating the terms and applying Gauss’s divergence theorem. Finally, We arrive at “face volume fluxes” which is shown below:-

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

Finally we arrive at :-

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

The second term is “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

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