Moving Frame of Reference

Why do we need MRF?

A Moving Reference Frame (MRF) is a very straightforward, reliable, and effective steady-state Computational Fluid Dynamics (CFD) modeling tool to simulate rotating machinery. A quadcopter's rotors, for instance, can be modeled using MRFs. An MRF presupposes that the non-wall boundaries are surfaces of revolution and that the allotted volume rotates at a constant speed (e.g., cylindrical, spherical, conical). The volumes between the rotor blades in the quadcopter example are identified as MRFs, given rotational speeds, and embedded within a multi-volume fluid domain.

MRF is comparable to running a rotational simulation and observing the outcomes at the instant corresponding to the rotor's position inside the MRF. In MRF, the surrounding stationary volumes and the MRF volume are assumed to have only a minimal interaction. The moving/sliding mesh approach, which can handle significant interactions between the moving volume and the surrounding stationary volumes, is an alternative to MRF. However, because the moving mesh technique depends on calculations at intersections between the rotating and stationary volumes, it frequently encounters robustness issues in practise. Moreover, the moving mesh method is a transient (unsteady) simulation, leading to typically very large runtimes. Given the moving mesh technique's severe disadvantages, the MRF is the preferred method within those constraints.

The majority of CFD applications involve fluid that is inside or around still objects. Hence, meshes are corrected. Incompressible If the flow is incompressible, steady Navier-Stokes equations can be solved. Two relatively common uses include flow inside pipes and flow around airfoils. The incompressible stable Navier-Stokes equations are utilized to solve the meshes since both the airfoil and pipe are stationary. The Moving Reference Frame (MRF) methodology is a steady-state method used in industrial computational fluid dynamics to model problems involving rotating components. It is believed to use less computer power while still being effective in most industrial situations. The fundamental ideas behind this tactic are simple but beautiful. During the meshing phase, a small volumetric zone of mesh cells is produced around the spinning body, and the MRF zone encompasses this area.

During the simulation phase, the body is kept stationary while the MRF zone is rotated around the body's axis. The governing equations are solved in a reference frame that rotates or moves at the same rate as the rotating or moving geometry. On a physical level, it means that while sitting on the moving body, we can perceive the flow field surrounding it. As a result, the flow field becomes stable with respect to the geometry.

The flow environment surrounding a rotating turbine appears ephemeral to us when we are on the ground. The flow field surrounding us or the turbine would appear steady if we were sitting on a turbine blade and rotating with it as opposed to viewing it from a distance.

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.

How is it done?

The MRF zone is rotated around the body's axis while the body is kept still during the simulation phase. The body's rotational velocity is reversed in the MRF zone to maintain the direction of the Coriolis forces acting on it.

Consider a coordinate system which is rotating steadily with angular velocity relative to a stationary (inertial) reference frame. The origin of the rotating system is located by a position vector .

The axis of rotation is defined by a unit direction vector such that.

The computational domain for the CFD problem is defined with respect to the rotating frame such that an arbitrary point in the CFD domain is located by a position vector from the origin of the rotating frame.

 Where 

In the above, is the relative velocity (the velocity viewed from the rotating frame), is the absolute velocity (the velocity viewed from the stationary frame), and is the "whirl" velocity (the velocity due to the moving frame).

When the equations of motion are solved in the rotating reference frame, the acceleration of the fluid is augmented by additional terms that appear in the momentum equations. Moreover, the equations can be formulated in two different ways:

• Expressing the momentum equations using the relative velocities as dependent variables (known as the relative velocity formulation).
• Expressing the momentum equations using the absolute velocities as dependent variables in the momentum equations (known as the absolute velocity formulation).

Relative velocity formulation

For the relative velocity formulation, the governing equations of fluid flow for a steadily rotating frame can be written as follows:

Conservation of mass:

Conservation of momentum:

Conservation of energy:

The momentum equation contains two additional acceleration terms: the Coriolis acceleration and the centripetal acceleration. In addition, the viscous stress is identical, except that relative velocity derivatives are used. The energy equation is written in terms of the relative internal energy and the relative total enthalpy, also known as the rothalpy. These variables are defined as:

Absolute velocity formulation

For the absolute velocity formulation, the governing equations of fluid flow for a steadily rotating frame can be written as follows:

Conservation of mass:

Conservation of momentum:

Conservation of energy:

In this formulation, the Coriolis and centripetal accelerations can be collapsed into a single term.