Theory for the Mixing Plane Feature
The Mixing Plane boundary condition is typically applied to interior boundaries at the intersection of rotating and nonrotating domains. The boundary condition intends to obtain a steady-state solution by taking into consideration all possible spatial configurations corresponding to different relative positionings of the rotating and nonrotating domains. The word ‘mixing’ refers to the averaging operation performed on flow field variables in the direction of rotation, while the word ‘plane’ refers to the interior boundary on which the condition is applied.
The solution thus obtained from the Mixing Plane condition is independent of the relative positioning of the rotating and nonrotating domains. It alleviates the need to perform time-averaging operation on solution steps from computationally expensive time dependent studies. Moreover, it allows users to take advantage of symmetry planes in the geometry of the problem.
The Mixing Plane duplicates the degrees of freedom at the boundary, thus allowing for discontinuous solution values on either sides of the boundary. Consequently, flow and turbulence quantities may be defined on the up and down sides, which are denoted by subscripts ‘u’ and ‘d’, respectively. Also, the normals are defined as and .
Using the framework described in the previous paragraph, mixing is introduced as follows. In the case of subsonic flows, modeled using one of the Single Phase Flow interfaces, Lagrange multipliers are utilized to apply the following Dirichlet condition at the side with incoming flow:
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Here, the ordered subscript, , should be read as ‘down side or up side’ and the operator ‘mix()’ performs averaging along the direction of rotation. When expanded, the above equation is equivalent to:
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Meanwhile, at the side with outgoing flow, an appropriate Neumann condition is applied, as follows:
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Also, the conservation of normal mass flux is ensured. That is,
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When the Nonisothermal Flow multiphysics coupling feature is active, similar conditions on the temperature apply. They are given by:
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In addition, the flow direction is considered for upwinding of convective fluxes.
For high-speed flows in the transonic and supersonic flow regimes, using one of the High-Mach Number Flow interfaces, one must take circumferentially-averaged values of eigenvalues of the inviscid flux and flow variables into consideration when computing the characteristic variables. In other words, we replace , , , , , and with , , , , , and , respectively, in Equation 5-11 and Equation 5-12. The primitive variables (denoted by subscript ‘face’), obtained from transformation of the these characteristic variables using Equation 5-13, are used to apply appropriate conditions at the ‘up side’ and ‘down side’ boundaries of the mixing plane.
At the side with incoming flow, the following conditions are applied using the Nitsche’s method:
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At the side with outgoing flow, appropriate conditions based on the local Mach number are applied:
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The conditions on the turbulence variables, for the k-ε model, low Reynolds number k-ε model, and Realizable k-ε model are:
The conditions on the turbulence variables, for the k-ω model are:
The conditions on the turbulence variables, for the SST model are:
If transition modeling is included, then,
The conditions on the turbulence variable, for the Spalart–Allmaras model are:
The conditions on the turbulence variables, for the v2-f model are:
See Mixing Plane for the feature node details.