Conservation Equations with Mesh Motion

Fluid Mechanics

Mesh motion modifies the conservation equations by introducing an additional flux in the convective terms. Also, if motion is defined with respect to a moving reference frame, the momentum equation accounts for the fictitious forces arising from the non-inertial reference frame.

Flow Equations for Moving Meshes

Consider the most general case where the mesh motion is defined with respect to a reference frame that is moving relative to the laboratory reference frame. The conservation equations (Eqn. (654), Eqn. (655), and Eqn. (658)) can be written in terms of the fluid velocity in the laboratory frame (also called absolute velocity) as:

$$\frac{\partial }{\partial t}\underset{V}{\int }\rho dV+{\oint }_A\rho (v_r-v_g)\cdot da=\underset{V}{\int }{S}_{u}dV$$

(4862)

$$\begin{array}{c}\frac{\partial }{\partial t}{\int }_V\rho \mathbf{\text{v}}dV+\underset{A}{\overset{}{\oint }}\rho v\otimes (v_r-v_g)\cdot d\mathbf{\text{a }}=\end{array}{\oint }_A\sigma \cdot da+{\int }_V{f}_{b}dV-{\int }_V\rho \omega \times vdV$$

(4863)

$$\frac{\partial }{\partial t}{\int }_V^{}\rho EdV+{\oint }_A^{}\rho E(v_r-v_g)\cdot d\mathbf{\text{a}}=-{\oint }_A^{}q\cdot d\mathbf{\text{a}}+{\oint }_A^{}(\mathbf{\text{v}}\cdot \sigma )\cdot d\mathbf{\text{a}}+{\int }_V^{}{\mathbf{\text{f}}}_b\mathbf{\cdot }\mathbf{\text{v}}dV+{\int }_V^{}{S}_{E}dV$$

(4864)

where $v_g$ is the grid velocity in the laboratory reference frame, and $v_r$ is the relative velocity defined in Eqn. (4860). The last term on the right-hand side of Eqn. (4863) is the fictitious force introduced by the non-inertial moving reference frame. The fictitious force consists of coriolis force and centrifugal force. If the reference frame is not moving, this fictitious force vanishes and $v_r=v$ .

For stationary meshes in the Laboratory frame, Eqn. (4862), Eqn. (4863), and Eqn. (4864) reduce to Eqn. (664), Eqn. (665), and Eqn. (666).

Conservation of Space

When the mesh is moving, the shape and position of its cells can vary with time. For motions that move the mesh vertices directly, such as morphing or user-defined motions, Simcenter STAR-CCM+ solves an additional equation to enforce space conservation:

$$\frac{d}{dt}\underset{V}{\int }dV=\underset{A}{\int }{v}_g\cdot da$$

(4865)

In this way, Simcenter STAR-CCM+ ensures that the rate of change of a cell volume balances the motion of its bounding surface.

Grid Flux Calculation

In Eqn. (4862), Eqn. (4863), and Eqn. (4864), the convective terms contain additional contributions due to the mesh velocity and moving reference frame.

Considering the discretized form of the convective terms (see Eqn. (882)), these contributions translate into an additional flux $G$, referred to as the grid flux. The modified convective flux at a face $f$ can then be written as:

$${\left[\varphi \rho (\mathbf{\text{v}}\cdot \mathbf{\text{a}}-G)\right]}_f={\dot{m}}_f{\varphi }_f$$

(4866)

where:

$$G=G_{MRF}+G_g$$

(4867)

The contribution due to a moving reference frame, $G_{MRF}$, is calculated as:

$$G_{MRF}=(v_{MRF,t}+{\omega }_{MRF}\times r)\cdot {\mathbf{\text{a}}}_f$$

(4868)

When the reference frame is not moving, $G_{MRF}=0$.

For first-order time approximation, the contribution due to mesh motion is calculated as:

$$G_g=\frac{\delta V_f^n}{\Delta t}$$

(4869)

For second-order time approximation:

$$G_g=\frac{\left({\alpha }^2-1\right){\delta V}_f^{n+1}-{\delta V}_f^n}{\alpha \left(\alpha -1\right)\Delta t^{n+1}}$$

(4870)

with:

$$\begin{array}{ccc}\alpha & =& 1+\frac{\Delta t^{n+1}}{\Delta t^n}\\ \mathrm{\Delta }{t}^{n+1}& =& t^{n+1}-t^n\\ \mathrm{\Delta }{t}^n& =& t^n-t^{n-1}\end{array}$$

(4871)

where $\delta V_f^{n+1}$ is the volume swept by the face $f$ over the time-step $\Delta t^{n+1}$ and $\delta V_f^n$ is the volume swept by the face $f$ over the time-step $\Delta t^n$.

The swept volumes are calculated from the grid position at two time levels. $n$ denotes the current time level, $n-1$ the previous time level, and $n+1$ the next time level. $\Delta t$ is the time-step.

When the mesh is stationary, $G_g=0$.

Solid Mechanics

In solid mechanics applications, you model the deformation of solid structures in response to applied loads. When the solid has prescribed motion, Simcenter STAR-CCM+ also accounts for the deformation caused by inertia forces due to acceleration of the solid structure.

Solid mechanics applications are compatible with the Rotation, Translation, and Rotation and Translation rigid motion models. These motions can only be defined with respect to the Laboratory reference frame.

Position, Velocity, and Acceleration

Simcenter STAR-CCM+ supports multi-level superposed rotations and translations, where a lower level motion is defined with respect to the parent motion. After applying $n$ arbitrary superposing rotations and translations, the position of a mesh vertex can be written as:

$$X_n=X^{c_1}+R_n(X-X^{c_n})+{\displaystyle \sum _{i=1}^n{d}_i}+{\displaystyle \sum _{i=1}^{n-1}{R}_i(X^{c_{(i+1)}}-X^{c_i})}$$

(4872)

where $X^{c_i}$ is the position of the origin for the $i$-th motion, $X$ is the initial position of the vertex, $d_i=v_{i}t$ is the displacement of the vertex for the $i$-th motion due to translation (with $v_i$ being the translational velocity), and $R_i$ is the compound rotation matrix, defined as:

$$R_i=R\left[\left({\displaystyle \sum _{k=1}^i}{\omega }_k\right)t\right]$$

(4873)

Differentiating Eqn. (4872) provides the rigid mesh velocity $v_g$:

$$v_g={\displaystyle \sum _{i=1}^n[v_i+{\displaystyle {\omega }_i\times (X-X^{c_i})}]}$$

(4874)

where ${\omega }_i$ is the compound angular velocity for the $i$-th motion:

$${\omega }_i={\displaystyle \sum _{k=1}^i{\omega }_k}$$

(4875)

Differentiating Eqn. (4874) provides the rigid mesh acceleration ${\dot{v}}_g$:

$${\dot{v}}_g={\displaystyle \sum _{i=1}^n{\dot{v}}_i+\sum _{i=1}^{n}2{\omega }_i\times v_i+\sum _{i=1}^n{\dot{\omega }}_i\times (X-X^{c_i})+\sum _{i=1}^n\left[{\omega }_i\times (\sum _{i=1}^n[{\omega }_i\times (X-X^{c_i}\left)\right]-\sum _{j=1}^i[{\omega }_j\times (X^{c_i}-X^{c_j})])\right]}$$

(4876)

where the first term is the translational acceleration, the second is the Coriolis acceleration, the third term is the centrifugal acceleration, and the last term is the tangential acceleration.

Momentum Equation for Rigidly Moving Meshes

With prescribed mesh motion, the momentum equation (Eqn. (4460)) can be written as:

$$\rho \ddot{u}-\nabla \cdot \sigma -f_b+\rho {\dot{v}}_g=0$$

(4877)

where ${\dot{v}}_g$ is the mesh acceleration due to the prescribed motion (see Eqn. (4876)). A similar change applies to the virtual work equation (Eqn. (4462)).

The total displacement can then be written as:

$$u_T=u_R+u$$

(4878)

where $u_R$ is the rigid displacement resulting from the prescribed rotations and translations and $u$ is the displacement resulting from non-rigid deformations, as calculated by the solid stress solver. In the absence of rotations and translations, $u_T=u$.

For FSI applications, you can also move the solid mesh according to the displacements calculated from Eqn. (4877) (see Solid Displacement Motion).

For stationary meshes in the Laboratory frame, Eqn. (4877) reduces to Eqn. (4460). For compatibility with fluids, solid mechanics applications also support stationary meshes in moving references frames. In the case of mesh motion, all results are displayed on the moving mesh with respect to the Laboratory frame. For stationary meshes in moving reference frame, all results are displayed on the stationary mesh with respect to the moving reference frame.