1.1.9.3. 2D Finite Volume Method: Spatial and Numerical Discretisation

1.1.9.3.1. Spatial Discretisation

1.1.9.3.1.1. Types of Spatial Discretisation

The FVM can handle any type of spatial discretisation

For the same mesh we can have different definitions of the control volumes:

Cell-centred scheme

  • Control volumes are identical with the grid cells

  • Flow variables are located at the centres of grid cells - \(q_{i,j}\) and \(u_{i,j}\) can be an average

  • Fluxes are located at the volume surfaces (red)

../_images/centred_scheme.png

Cell-vertex scheme

  • Control volume is 4 cells that share the grid point \(i,j\)

  • Control volumes may overlap

../_images/vertex_scheme.png

Hexagonal control volume scheme

../_images/hexagonal_scheme.png

Trapezoidal control volume scheme

../_images/trapezoidal_scheme.png

1.1.9.3.1.2. For a Consistent Scheme

  1. The sum of all CVs should cover the entire domain

  2. The sub-domains \(\Omega_J\) are allowed to overlap, as long as each portion of the surface \(S_J\) have to appear as a part of an even number of different surfaces - so that the fluxes cancel out, e.g. 2 surfaces, 4 surfaces etc.

  3. Computing fluxes along a cell surface has to be independent of the cell in which they are considered - ensures that conservation is satisfied

1.1.9.3.2. Numerical Discretisation

Apply integral conservation law to subvolumes (\(\Omega_J\) = sub-volume, \(S_J\) = sub-surface)

(1)\[{\partial \over \partial t} \int_{\Omega_J} U d \Omega + \oint_{S_J} \mathbf{F} \cdot d \mathbf{S} = \int_{\Omega_J} Q d \Omega\]

Discrete Form, for small volumes (J)

(2)\[{\partial \over \partial t} (\overline{U_J} \Omega_J) + \sum_{faces} \mathbf{F} \cdot \Delta \mathbf{S_J} = \overline{Q_J} \Omega_J\]

Where:

\(\overline{U_J}\) and \(\overline{Q_J}\) are averaged values over the cell

If integrating between time level \(n\) and \(n+1\), then using 1st order Euler (we could use Runge-Kutta etc)

(3)\[(\overline{U_J} \Omega_J)^{n+1} = (\overline{U_J} \Omega_J)^{n} - \Delta t \sum_{faces} \mathbf{F^*} \cdot \Delta \mathbf{S_J} + \Delta t \overline{Q_J^*} \Omega_J\]

What are the Cell Averaged Values and the Star Values?

Cell-Averaged Values

(4)\[\overline{U_J^n} = \left. {1 \over \Omega_J} \int_{\Omega_J} U d \Omega_J \right|^n\]
(5)\[\overline{Q_J} = \left. {1 \over \Omega_J} \int_{\Omega_J} Q d \Omega_J \right|^n\]

Star Values

(6)\[\mathbf{F^*} \cdot \Delta \mathbf{S_J} = {1 \over \Delta t} \int_n^{n+1} \mathbf{F} \cdot \mathbf{\Delta S_J} dt\]
(7)\[\overline{Q_J^*} = {1 \over \Delta t} \int_n^{n+1} \overline{Q_J} dt\]

The reason we omitted the time index in (3) for the balance of fluxes and the sources was to indicate the choice:

  • If \(n\) was chosen, it would have been an explicit scheme

  • If \(n+1\) was chosen it would have been an implicit scheme

A scheme is identified by the way the numerical flux \(\mathbf{F^*}\) approximates the time averaged physical flux across each cell face.

Leaving open the choice of time integrator:

\[{d \over dt} (\overline{U_J} \Omega_J) = - \sum_{faces} \mathbf{F^*} \cdot \Delta \mathbf{S_J} + \overline{Q_J^*} \Omega_J = -R_J\]

RHS defines the “residual” \(R_J\)