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10.3.1 The Multiple Reference Frame Model



Overview


The MRF model [ 223] is, perhaps, the simplest of the two approaches for multiple zones. It is a steady-state approximation in which individual cell zones move at different rotational and/or translational speeds. The flow in each moving cell zone is solved using the moving reference frame equations (see Section  10.2). If the zone is stationary ( $\omega = 0$), the stationary equations are used. At the interfaces between cell zones, a local reference frame transformation is performed to enable flow variables in one zone to be used to calculate fluxes at the boundary of the adjacent zone. The MRF interface formulation will be discussed in more detail in Section  10.3.1.

It should be noted that the MRF approach does not account for the relative motion of a moving zone with respect to adjacent zones (which may be moving or stationary); the grid remains fixed for the computation. This is analogous to freezing the motion of the moving part in a specific position and observing the instantaneous flowfield with the rotor in that position. Hence, the MRF is often referred to as the "frozen rotor approach."

While the MRF approach is clearly an approximation, it can provide a reasonable model of the flow for many applications. For example, the MRF model can be used for turbomachinery applications in which rotor-stator interaction is relatively weak, and the flow is relatively uncomplicated at the interface between the moving and stationary zones. In mixing tanks, for example, since the impeller-baffle interactions are relatively weak, large-scale transient effects are not present and the MRF model can be used.

Another potential use of the MRF model is to compute a flow field that can be used as an initial condition for a transient sliding mesh calculation. This eliminates the need for a startup calculation. The multiple reference frame model should not be used, however, if it is necessary to actually simulate the transients that may occur in strong rotor-stator interactions, the sliding mesh model alone should be used (see Section  11.2).



Examples


For a mixing tank with a single impeller, you can define a rotating reference frame that encompasses the impeller and the flow surrounding it, and use a stationary frame for the flow outside the impeller region. An example of this configuration is illustrated in Figure  10.3.1. (The dashes denote the interface between the two reference frames.) Steady-state flow conditions are assumed at the interface between the two reference frames. That is, the velocity at the interface must be the same (in absolute terms) for each reference frame. The grid does not move.

Figure 10.3.1: Geometry with One Rotating Impeller
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You can also model a problem that includes more than one rotating reference frame. Figure  10.3.2 shows a geometry that contains two rotating impellers side by side. This problem would be modeled using three reference frames: the stationary frame outside both impeller regions and two separate rotating reference frames for the two impellers. (As noted above, the dashes denote the interfaces between reference frames.)

Figure 10.3.2: Geometry with Two Rotating Impellers
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Limitations


The following limitations exist when using the MRF approach:

figure   

You can switch from the MRF model to the sliding mesh model for a more robust solution. See Section  11.4 for details on how to make this change in the fluid's boundary conditions. Currently, this switch is not possible when running in parallel.



The MRF Interface Formulation


The MRF formulation that is applied to the interfaces will depend on the velocity formulation being used. The specific approaches will be discussed below for each case. It should be noted that the interface treatment applies to the velocity and velocity gradients, since these vector quantities change with a change in reference frame. Scalar quantities, such as temperature, pressure, density, turbulent kinetic energy, etc., do no require any special treatment, and thus are passed locally without any change.

Interface Treatment: Relative Velocity Formulation

In FLUENT's implementation of the MRF model, the calculation domain is divided into subdomains, each of which may be rotating and/or translating with respect to the laboratory (inertial) frame. The governing equations in each subdomain are written with respect to that subdomain's reference frame. Thus, the flow in stationary and translating subdomains is governed by the equations in Section  9.2, while the flow in rotating subdomains is governed by the equations presented in Section  10.2.2.

At the boundary between two subdomains, the diffusion and other terms in the governing equations in one subdomain require values for the velocities in the adjacent subdomain (see Figure  10.3.3). FLUENT enforces the continuity of the absolute velocity, ${\vec v}$, to provide the correct neighbor values of velocity for the subdomain under consideration. (This approach differs from the mixing plane approach described in Section  10.3.2, where a circumferential averaging technique is used.)

When the relative velocity formulation is used, velocities in each subdomain are computed relative to the motion of the subdomain. Velocities and velocity gradients are converted from a moving reference frame to the absolute inertial frame using Equation  10.3-1.

Figure 10.3.3: Interface Treatment for the MRF Model
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For a translational velocity $\vec{v_t}$, we have


 {\vec v} = {\vec v}_r + ({\vec \omega} \times {\vec r}) + {\vec v}_t (10.3-1)

From Equation  10.3-1, the gradient of the absolute velocity vector can be shown to be


 \nabla {\vec v} = \nabla {\vec v}_r + \nabla \left( {\vec \omega} \times {\vec r} \right) (10.3-2)

Note that scalar quantities such as density, static pressure, static temperature, species mass fractions, etc., are simply obtained locally from adjacent cells.

Interface Treatment: Absolute Velocity Formulation

When the absolute velocity formulation is used, the governing equations in each subdomain are written with respect to that subdomain's reference frame, but the velocities are stored in the absolute frame. Therefore, no special transformation is required at the interface between two subdomains. Again, scalar quantities are determined locally from adjacent cells.


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