## 11.1.1 Conservation Equations

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

 (11.1-1)

 where is the fluid density is the flow velocity vector is the grid velocity of the moving mesh is the diffusion coefficient is the source term of

Here is used to represent the boundary of the control volume .

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

 (11.1-2)

where and denote the respective quantity at the current and next time level. The th time level volume is computed from

 (11.1-3)

where 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

 (11.1-4)

where is the number of faces on the control volume and is the face area vector. The dot product on each control volume face is calculated from

 (11.1-5)

where is the volume swept out by the control volume face over the time step .

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, and . Equation  11.1-2 can now be expressed as follows:

 (11.1-6)

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