24. Arbitrary Lagrangian-Eulerian description#

Reynold’s transport theorem allows for the formulation of intergal equations, and grid-based methods like FVM, on moving grids and changing domains. Rules for derivatives of composite functions provide the relations between time derivatives in a Lagrangian, Eulerian, or arbitrary description,

\[\begin{split}\begin{aligned} & \left.\dfrac{\partial f}{\partial t}\right|_{\vec{r}_0} = \left.\dfrac{\partial f}{\partial t}\right|_{\vec{r}} + \vec{u} \cdot \nabla f \\ & \left.\dfrac{\partial f}{\partial t}\right|_{\vec{r}_b} = \left.\dfrac{\partial f}{\partial t}\right|_{\vec{r}} + \vec{u}_b \cdot \nabla f \\ \end{aligned}\end{split}\]

Equations governing the motion of the grid are usually required as well. E.g.:

  • known and prescribed motion of the grid;

  • boundary conditions only without changing grids (for small displacements)

  • pseudo-elastic deformation (usually good for small strain and displacement;

  • for large displacements of/or models with complex geometry, sliding and/or overlapping grids could an option for grid-based methods.

24.1. Integral problem#

Application of Reynolds theorem to the balance equation of the quantity \(\mathbf{u}\) for a material volume \(V_t\)

\[\dfrac{d}{dt} \int_{V_t} \rho \mathbf{u} = \int_{V_t} \rho \mathbf{f} + \oint_{\partial V_t} \hat{n} \cdot \mathbf{T} \ .\]

provides the expression of the balance equation for a geometrical volume \(v_t\) in arbitrary motion,

\[\dfrac{d}{dt} \int_{v_t} \rho \mathbf{u} + \oint_{\partial v_t} \rho \mathbf{u} \left( \vec{u} - \vec{u}_b \right) \cdot \hat{n} = \int_{v_t} \rho \mathbf{f} + \oint_{\partial v_t} \hat{n} \cdot \mathbf{T} \ .\]

Here, the integral forulation of the problem will be applied to each element of the grid in arbitrary motion, for domains with variable geometry.

24.2. Differential problem#

Rules for derivatives of composite functions allows to write the differential w.r.t. the variables associated with the points of a moving grid. A balance equation in convective form can be written as

\[\begin{split}\begin{aligned} \rho \dfrac{D \mathbf{u}}{D t} & = \rho \mathbf{f} + \nabla \cdot \mathbf{T} \\ \rho \left[ \dfrac{\partial \mathbf{u}}{\partial t} + \vec{u} \cdot \nabla \mathbf{u} \right] & = \\ \rho \left[ \left.\dfrac{\partial \mathbf{u}}{\partial t}\right|_{\vec{r}_b} + \left( \vec{u} - \vec{u}_b \right) \cdot \nabla \mathbf{u} \right] & = \\ \end{aligned}\end{split}\]