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11.1.1 Conservation Equations

With respect to dynamic meshes, the integral form of the conservation equation for a general scalar, $\phi$, on an arbitrary control volume, $V$, whose boundary is moving can be written as

 {d\over dt}\int_V\rho\phi dV + \int_{\partial V}\rho\phi\lef... ...{\partial V}\Gamma\nabla\phi \cdot d\vec{A} + \int_V S_\phi dV (11.1-1)

where $\rho$ is the fluid density
  $\vec{u}$ is the flow velocity vector
  $\vec{u}_g$ is the grid velocity of the moving mesh
  $\Gamma$ is the diffusion coefficient
  $S_\phi$ is the source term of $\phi$

Here $\partial V$ is used to represent the boundary of the control volume $V$.

The time derivative term in Equation  11.1-1 can be written, using a first-order backward difference formula, as

 {d\over{\sl dt}}\int_V\rho\phi dV = {\left(\rho\phi V\right)^{n+1} - \left(\rho\phi V\right)^{n} \over \Delta t} (11.1-2)

where $n$ and $n + 1$ denote the respective quantity at the current and next time level. The $(n + 1)$th time level volume $V^{n+1}$ is computed from

 V^{n+1} = V^{n} + {dV\over dt}\Delta t (11.1-3)

where $dV/dt$ is the volume time derivative of the control volume. In order to satisfy the grid conservation law, the volume time derivative of the control volume is computed from

 {dV\over dt} = \int_{\partial V}\vec{u}_g\cdot d\vec{A} = \sum_j^{n_f} \vec{u}_{g,j}\cdot \vec{A}_j (11.1-4)

where $n_f$ is the number of faces on the control volume and $\vec{A}_j$ is the $j$ face area vector. The dot product $\vec{u}_{g,j}\cdot\vec{A}_j$ on each control volume face is calculated from

 \vec{u}_{g,j}\cdot\vec{A}_j = {\delta V_j\over\Delta t} (11.1-5)

where $\delta V_j$ is the volume swept out by the control volume face $j$ over the time step $\Delta t$.

In the case of the sliding mesh, the motion of moving zones is tracked relative to the stationary frame. Therefore, no moving reference frames are attached to the computational domain, simplifying the flux transfers across the interfaces. In the sliding mesh formulation, the control volume remains constant, therefore from Equation  11.1-3, ${dV\over dt} = 0$ and $V^{n+1} = V^n$. Equation  11.1-2 can now be expressed as follows:

 {d\over{\sl dt}}\int_V\rho\phi dV = {[\left(\rho\phi \right)^{n+1} - \left(\rho\phi \right)^{n}]V \over \Delta t} (11.1-6)

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