Governing Equations
In the Flow module, Simerics-MP solves conservation equations for mass and momentum. For laminar flows, the instantaneous continuity and Navier-Stokes equations are directly solved. For turbulent flows, the instantaneous equations are time-averaged (incompressible flows) or Favre-averaged (compressible flows) resulting in additional turbulence terms. The RANS turbulence models, discussed in the Turbulence module, are used to close the system of the averaged governing equations. In addition, for flows involving heat transfer or compressibility, an additional equation for energy conservation is also solved, as described in the Heat module.
Basic Governing Equations
Description
In a Cartesian coordinate system 
, the vector, 
, is introduced to represent the velocity field (instantaneous in laminar flows, while averaged in turbulent flows), and the variable 
is commonly used to stand for the Instantaneous/Reynolds-averaged static pressure. The conservation equations of mass and momentum have the following general forms:
Mass Conservation (Continuity Equation)
|
where 
is the Instantaneous/Reynolds-averaged flow density; 
represents the external/user mass source term, and 
represents time.
Momentum Conservation (Navier-Stokes Equations)
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where the stress tensor 
is related to the strain rate by
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and 
is the effective viscosity. For laminar flows, 
, the fluid dynamic viscosity; For turbulent flows, 
,where 
is the turbulent viscosity , obtained from one of the turbulence models. The negative term in equation 5.14 on the right-hand side is the effect of volume dilation, and the term 
is the unit tensor of second order.
In the last two terms in equation 5.13, 
is the body force including the gravity, buoyancy, or the extra terms in a non-inertia reference frame (centrifugal and Coriolis effects); and 
contains other model-dependent source terms such as porous-media momentum resistances and user-defined sources.
In a finite volume CFD solver, the conservation equations are numerically solved in their integral forms. With respect to a computational mesh, the integral form of the continuity and momentum equations, on an arbitrary control volume, 
, whose boundary surface is 
, can be expressed as:
Continuity:
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Momentum:
|
where
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Flow velocity  in m/s. The three components are solved in Flow. |
|
Control volume, which is a function of time in dynamic meshes. |
|
Surface of the control volume  . |
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Normal vector of the surface  . |
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Volume integral. |
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Surface integral. In equation 5.15 and equation 5.16, the Green-Gauss Theorem is applied to convert the volume integral to surface internal. |
In Simerics-MP, equation 5.15 and equation 5.16 have been discretized into a set of algebraic equations and solved iteratively. The numerical algorithms have been discussed in Numerics and Convergence.
Flows with Moving Domains
Simerics-MP solves the discretized flow governing equations, by default, in a stationary (and inertial) reference frame. In many fluid machineries such as pumps and impellers, however, there are moving (translational and/or rotating) parts or walls involved, and it is the flow relative to these moving objects that is of primary interest.
Flow in Moving Reference Frames
Many unsteady flows in the inertial reference frame can be considered as steady-state with respective to the moving parts. Therefore, it is of practical importance to solve such flows in moving or non-inertial reference frames.
Furthermore, for many complex geometries consisting of both stationary and moving parts, or the moving parts with different speeds, it may not be possible or desirable to refer the entire computational domain to a single reference frame (SRF). Instead, the more appropriate approach is to break the flow domain into multiple volumetric zones with well-defined zone-to-zone interfaces, and to solve each zone using different reference frames. The methods to treat the interfaces result in different models. The simplest formulation is the Multiple Reference Frame (MRF). This approach, along with the moving reference frames, will be discussed in this subsection.
In a moving (non-inertial or relative) reference frame, the motion of the reference frame can be translation and/or rotation. The main advantage for using a moving reference frame is to solve the intrinsically unsteady flows under the stationary (inertial or absolute) frame in a steady-state fashion with respect to the moving frame (non-inertial). In general, however, an unsteady simulation can also be carried out in a moving reference frame.
Considering a coordinate system with a translational linear velocity
, and rotational angular velocity
relative to the absolute stationary reference frame, we have the non-inertial reference frame moving with the velocity:
where
is the vector connecting the point of the interest (cell center, face or vertices) to the center of the rotating axis. It should be noted that
and
can be the functions of time.
The fluid velocities can then be transformed from the stationary frame to the moving frame using the following relation:
where
is the relative velocity viewed from the moving frame.
In Simerics-MP, when the governing equations are solved in a moving reference frame, the equations are formulated based on the absolute velocity formulation, which have the following forms:
Mass Conservation
Momentum Conservation
In this formulation, the Coriolis and centripetal accelerations can be simplified into a single term (the last term on the right-hand side),
. It is treated as an additional source term.
The Multiple Reference Frame (MRF) Model
The Multiple Reference Frame (MRF) model is a steady-state approximation in which the individual volumetric zones can be assigned different translational and/or rotational speeds. In this model, the mesh on each zone remains fixed for the simulation, and the neighboring zones can be connected using the Mismatched Grid Interfaces (MGI) algorithm. The flow governing equations in each zone are solved under different reference frames: if the zone is stationary or translating, the governing equations are solved in the absolute/inertial reference frame, while the flow in a moving zone is governed by continuity and momentum equations in the relative/non-inertial reference frame. At the interfaces between cell zones, it is assumed to have no relative motion of a moving zone with respect to adjacent zones (which may be moving or stationary). With the use of the absolute velocity formulation, equation 5.19 and equation 5.20, the velocities are stored in the absolute frame so that no special transformation is required for velocity and velocity gradients. For the scalar quantities such as temperature, pressure, density and turbulent kinetic energy, they are always determined locally from adjacent cells.
It may be noted that while the MRF approach is clearly an approximation, it can provide a reasonable model of the flow for applications such as centrifugal pumps and mixing tanks, where the interactions between the moving and stationary parts are relatively weak, and the flow at the interfaces are relatively uncomplicated. Another potential use of the MRF model is to compute a flow field that can be used as an initial condition for a transient simulation with mismatched meshes at the interfaces.
Flows Using Dynamic Meshes
In both single and multiple moving reference frames, the computational domains and meshes remain unchanged. In many practical situations, however, the flow domains not only move relative to each other, but the boundaries may also move and/or deform with time to cause the volumetric zones changing accordingly. As a result, the volume mesh of the computational domain should be adjusted to accommodate the geometric change. To model this type of flows, it is necessary to introduce a dynamic mesh algorithm, allowing the volumetric mesh to vary with the motion of its boundaries.
Obviously, since the geometries and meshes change with time, the dynamic mesh solutions are inherently unsteady. The governing equations, unlike those used for describing the fluid motion over a fixed domain and mesh in both stationary and moving reference frames, are based on moving/deforming control volumes. With respect to a dynamic mesh, the integral form of the continuity and momentum equations, on an arbitrary control volume,
, whose boundary
is moving with the grid velocity
, can be expressed as:
Continuity:
Momentum:
where
Flow velocity in m/s. The three components are solved in Flow.
Mesh velocity. It is zero if there is no geometry change or mesh movement. 
Control volume, which is a function of time in dynamic meshes . 
Surface of the control volume .
Normal vector of the surface .
Volume integral. 
Surface integral. In equation 5.21 and equation 5.22, the Green-Gauss Theorem is applied to convert the volume integral to surface internal.
Comparing with equation 5.15 and equation 5.16, one can see that equation 5.21 and equation 5.22 are the universal expressions of the integrated governing equations. For example, when cell volume
remains a constant, and the mesh velocity
is set to zero (no geometry changes or mesh movement), then equation 5.21 and equation 5.22 return to equation 5.15 and equation 5.16.
In Simerics-MP, therefore, the generalized governing equations equation 5.21 and equation 5.22 have been discretized into a set of algebraic equations and solved iteratively. The numerical algorithms have been discussed in Numerics and Convergence.
Mesh Conservation
When the cell volume
changes with time, the net rate of change
must be equal to the total volume swept out by the control volume faces to satisfy the mesh conservation law:
where
Area of surface “ ”:
.
Control volume face. 
Number of surfaces enclosing the cell. Introducing
to represent the volume swept out by the control volume face “
" over the time-step
, then we have the term
on the face “
” calculated from:
where
depends on the temporal schemes employed. As discussed in Numerics and Convergence, all the temporal discretization schemes are applicable for the computation of the cell volume change due to the grid motion. For example, with the implicit time schemes, this term can be written in the following general form:
where
is a blending parameter and has the value between 0 and 1.