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10.3.2 The Mixing Plane Model

The mixing plane model in FLUENT provides an alternative to the multiple reference frame and sliding mesh models for simulating flow through domains with one or more regions in relative motion. This section provides a brief overview of the model and a list of its limitations.



Overview


As discussed in Section  10.3.1, the MRF model is applicable when the flow at the boundary between adjacent zones that move at different speeds is nearly uniform ("mixed out''). If the flow at this boundary is not uniform, the MRF model may not provide a physically meaningful solution. The sliding mesh model (see Section  11.2) may be appropriate for such cases, but in many situations it is not practical to employ a sliding mesh. For example, in a multistage turbomachine, if the number of blades is different for each blade row, a large number of blade passages is required in order to maintain circumferential periodicity. Moreover, sliding mesh calculations are necessarily unsteady, and thus require significantly more computation to achieve a final, time-periodic solution. For situations where using the sliding mesh model is not feasible, the mixing plane model can be a cost-effective alternative.

In the mixing plane approach, each fluid zone is treated as a steady-state problem. Flow-field data from adjacent zones are passed as boundary conditions that are spatially averaged or "mixed'' at the mixing plane interface. This mixing removes any unsteadiness that would arise due to circumferential variations in the passage-to-passage flow field (e.g., wakes, shock waves, separated flow), thus yielding a steady-state result. Despite the simplifications inherent in the mixing plane model, the resulting solutions can provide reasonable approximations of the time-averaged flow field.



Limitations


Note the following limitations of the mixing plane model:



Rotor and Stator Domains


Consider the turbomachine stages shown schematically in Figures  10.3.4 and 10.3.5. Figure  10.3.4 shows a constant radial plane within a single stage of an axial machine, while Figure  10.3.5 shows a constant $\theta$ plane within a mixed-flow device. In each case, the stage consists of two flow domains: the rotor domain, which is rotating at a prescribed angular velocity, followed by the stator domain, which is stationary. The order of the rotor and stator is arbitrary (that is, a situation where the rotor is downstream of the stator is equally valid).

Figure 10.3.4: Axial Rotor-Stator Interaction (Schematic Illustrating the Mixing Plane Concept)
figure

Figure 10.3.5: Radial Rotor-Stator Interaction (Schematic Illustrating the Mixing Plane Concept)
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In a numerical simulation, each domain will be represented by a separate mesh. The flow information between these domains will be coupled at the mixing plane interface (as shown in Figures  10.3.4 and 10.3.5) using the mixing plane model. Note that you may couple any number of fluid zones in this manner; for example, four blade passages can be coupled using three mixing planes.

figure   

Note that the stator and rotor meshes do not have to be conformal; that is, the nodes on the stator exit boundary do not have to match the nodes on the rotor inlet boundary. In addition, the meshes can be of different types (e.g., the stator can have a hexahedral mesh while the rotor has a tetrahedral mesh).



The Mixing Plane Concept


The essential idea behind the mixing plane concept is that each fluid zone is solved as a steady-state problem. At some prescribed iteration interval, the flow data at the mixing plane interface are averaged in the circumferential direction on both the stator outlet and the rotor inlet boundaries. The FLUENT implementation uses area-weighted averages. By performing circumferential averages at specified radial or axial stations, "profiles'' of flow properties can be defined. These profiles--which will be functions of either the axial or the radial coordinate, depending on the orientation of the mixing plane--are then used to update boundary conditions along the two zones of the mixing plane interface. In the examples shown in Figures  10.3.4 and 10.3.5, profiles of averaged total pressure ( $p_0$), direction cosines of the local flow angles in the radial, tangential, and axial directions ( $\alpha_r, \alpha_t, \alpha_z$), total temperature ( $T_0$), turbulence kinetic energy ( $k$), and turbulence dissipation rate ( $\epsilon$) are computed at the rotor exit and used to update boundary conditions at the stator inlet. Likewise, a profile of static pressure ( $p_s$), direction cosines of the local flow angles in the radial, tangential, and axial directions ( $\alpha_r, \alpha_t, \alpha_z$), are computed at the stator inlet and used as a boundary condition on the rotor exit.

Passing profiles in the manner described above assumes specific boundary condition types have been defined at the mixing plane interface. The coupling of an upstream outlet boundary zone with a downstream inlet boundary zone is called a "mixing plane pair''. In order to create mixing plane pairs in FLUENT, the boundary zones must be of the following types:


Upstream Downstream
pressure outlet pressure inlet
pressure outlet velocity inlet
pressure outlet mass flow inlet

Specific instructions for setting up mixing planes are provided in Section  10.10.2.



FLUENT's Mixing Plane Algorithm


FLUENT's basic mixing plane algorithm can be described as follows:

1.   Update the flow field solutions in the stator and rotor domains.

2.   Average the flow properties at the stator exit and rotor inlet boundaries, obtaining profiles for use in updating boundary conditions.

3.   Pass the profiles to the boundary condition inputs required for the stator exit and rotor inlet.

4.   Repeat steps 1-3 until convergence.

figure   

Note that it may be desirable to under-relax the changes in boundary condition values in order to prevent divergence of the solution (especially early in the computation). FLUENT allows you to control the under-relaxation of the mixing plane variables.



Mass Conservation


Note that the algorithm described above will not rigorously conserve mass flow across the mixing plane if it is represented by a pressure inlet and pressure outlet mixing plane pair. If you use a mass flow inlet and pressure outlet pair instead, FLUENT will force mass conservation across the mixing plane. The basic technique consists of computing the mass flow rate across the upstream zone (pressure outlet) and adjusting the mass flux profile applied at the mass flow inlet such that the downstream mass flow matches the upstream mass flow. This adjustment occurs at every iteration, thus ensuring rigorous conservation of mass flow throughout the course of the calculation.

figure   

Note that, since mass flow is being fixed in this case, there will be a jump in total pressure across the mixing plane. The magnitude of this jump is usually small compared with total pressure variations elsewhere in the flow field.



Swirl Conservation


By default, FLUENT does not conserve swirl across the mixing plane. For applications such as torque converters, where the sum of the torques acting on the components should be zero, enforcing swirl conservation across the mixing plane is essential, and is available in FLUENT as a modeling option. Ensuring conservation of swirl is important because, otherwise, sources or sinks of tangential momentum will be present at the mixing plane interface.

Consider a control volume containing a stationary or rotating component (e.g., a pump impeller or turbine vane). Using the moment of momentum equation from fluid mechanics, it can be shown that for steady flow,


 T=\int\!\!\!\int_S r v_\theta \rho \vec{v} \cdot \hat{n} dS (10.3-3)

where $T$ is the torque of the fluid acting on the component, $r$ is the radial distance from the axis of rotation, $v_\theta$ is the absolute tangential velocity, $\vec{v}$ is the total absolute velocity, and $S$ is the boundary surface. (The product $r v_\theta$ is referred to as swirl.)

For a circumferentially periodic domain, with well-defined inlet and outlet boundaries, Equation  10.3-3 becomes


 T=\int\!\!\!\int_{\rm outlet} r v_\theta \rho \vec{v} \cdot ... ...!\!\!\int_{\rm inlet} r v_\theta \rho \vec{v} \cdot \hat{n} dS (10.3-4)

where inlet and outlet denote the inlet and outlet boundary surfaces.

Now consider the mixing plane interface to have a finite streamwise thickness. Applying Equation  10.3-4 to this zone and noting that, in the limit as the thickness shrinks to zero, the torque should vanish, the equation becomes


 \int\!\!\!\int_{\rm downstream} r v_\theta \rho \vec{v} \cdo... ...\!\int_{\rm upstream} r v_\theta \rho \vec{v} \cdot \hat{n} dS (10.3-5)

where upstream and downstream denote the upstream and downstream sides of the mixing plane interface. Note that Equation  10.3-5 applies to the full area (360 degrees) at the mixing plane interface.

Equation  10.3-5 provides a rational means of determining the tangential velocity component. That is, FLUENT computes a profile of tangential velocity and then uniformly adjusts the profile such that the swirl integral is satisfied. Note that interpolating the tangential (and radial) velocity component profiles at the mixing plane does not affect mass conservation because these velocity components are orthogonal to the face-normal velocity used in computing the mass flux.



Total Enthalpy Conservation


By default, FLUENT does not conserve total enthalpy across the mixing plane. For some applications, total enthalpy conservation across the mixing plane is very desirable, because global parameters such as efficiency are directly related to the change in total enthalpy across a blade row or stage. This is available in FLUENT as a modeling option.

The procedure for ensuring conservation of total enthalpy simply involves adjusting the downstream total temperature profile such that the integrated total enthalpy matches the upstream integrated total enthalpy. For multiphase flows, conservation of mass, swirl, and enthalpy are calculated for each phase. However, for the Eulerian multiphase model, since mass flow inlets are not permissible, conservation of the above quantities does not occur.


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