CMPSTheory & Implementation Manual
Coordinate Systems, Body Forces and Rotating Reference Frames
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Coordinate Systems, Body Forces and Rotating Reference Frames

Implementation convention. Rotating zones use an absolute-velocity formulation. The transported velocity is the absolute velocity; grid-relative velocity is introduced only where transport crosses a moving/rotating face.

Planar and axisymmetric coordinates

CMPS supports planar and two-dimensional axisymmetric problem definitions. In an axisymmetric problem, the second coordinate represents radius and finite-volume geometric measures are radius weighted. Axis faces are treated specially, and axisymmetric turbulence/strain expressions add the required radial contributions such as \(u_r/r\) in the model-specific source evaluation rather than hiding them in the planar gradient tensor.

Rotating-zone grid velocity

For a rotating cell zone with angular-velocity vector \(\boldsymbol\Omega\) and center \(\mathbf x_c\), the face grid velocity is

\[\mathbf v_g(\mathbf x_f)=\boldsymbol\Omega\times(\mathbf x_f-\mathbf x_c).\]

The relative transport velocity used in moving-face convection is

\[\mathbf u_{rel}=\mathbf u_{abs}-\mathbf v_g.\]

Thus mass convection is grid relative while momentum and energy remain expressed with the absolute state.

The face-normal relative speed is

\[u_{n,rel}=(\mathbf u_{abs}-\mathbf v_g)\cdot\mathbf n_f,\]

so a generic convective mass flux has the grid-relative form

\[\dot m_f=\rho_f u_{n,rel}A_f.\]

Absolute-energy moving-face flux

For the compressible AUSM carrier path, the absolute total-energy flux has the implementation form

\[F_E=\dot m_f H_{abs,upw}+p_f\,(\mathbf v_{g,f}\cdot\mathbf n_f).\]

The second term is the moving-face pressure work required by the absolute-energy formulation. The artificial-compressibility face operator follows the same grid-relative transport principle.

Rotating-zone momentum source

The current absolute-velocity rotating-zone assembly adds the volumetric momentum term

\[\mathbf S_{rot}=\rho\,\boldsymbol\Omega\times\mathbf u_{abs}.\]

Its cell contribution is assembled with the solver residual sign convention and differentiated consistently into the local block. No separate rotational total-energy volume source is added because \(\mathbf u_{abs}\cdot(\boldsymbol\Omega\times\mathbf u_{abs})=0\); moving-face pressure work is carried by the face energy flux.

The skew-symmetric cross-product operator \([\boldsymbol\Omega]_\times\) satisfies

\[\boldsymbol\Omega\times\mathbf u=[\boldsymbol\Omega]_\times\mathbf u,\qquad [\boldsymbol\Omega]_\times=\begin{bmatrix}0&-\Omega_z&\Omega_y\\\Omega_z&0&-\Omega_x\\-\Omega_y&\Omega_x&0\end{bmatrix}.\]

Its kinetic-energy contribution vanishes identically,

\[\mathbf u\cdot(\boldsymbol\Omega\times\mathbf u)=0.\]

Gravity

When gravity is enabled, carrier momentum receives

\[\mathbf S_g=\rho\mathbf g.\]

The dilute dispersed phase receives its corresponding \(\rho_d\mathbf g\) momentum source. In the current artificial-compressibility carrier branch, body-force work \(\rho\mathbf g\cdot\mathbf u\) is also assembled into the transported total-energy row. The current compressible gravity assembly contributes the momentum source without an additional gravity-power term in the energy row; this distinction is an implementation property of the present source.

For a compressible total-energy equation, the corresponding gravitational work is

\[S_{E,g}=\rho\,\mathbf g\cdot\mathbf u.\]

Rotating walls

A no-slip wall attached to a rotating zone uses the absolute wall velocity

\[\mathbf u_{wall,abs}=\boldsymbol\Omega\times(\mathbf x_f-\mathbf x_c),\]

so the wall-relative velocity vanishes. A wall in a stationary zone uses its prescribed absolute wall velocity.

Current compatibility limits