Divergence schemes

Table of Contents

• Usage
• Options
  • NVD/TVD convection schemes
  • Non-NVD/TVD convection schemes
• Special cases
  • Steady state
  • Bounded scalars
  • V schemes
• Example
• Further information

The divergence of a property \(Q\) describes the net rate at which it changes as a function of space, represented using the notation:

\[ \div Q \]

If \( Q \) is a vector quantity, this equates to

\[ \div \vec{Q} = \frac{\partial Q_x}{\partial x} + \frac{\partial Q_y}{\partial y} + \frac{\partial Q_z}{\partial z} \]

Usage

Divergence schemes are specified in the fvSchemes file under the divSchemes sub-dictionary using the general syntax:

divSchemes
{
    default         none;
    div(Q)          Gauss <interpolation scheme>;
}

A typical use is for convection schemes, which transport a property, \( Q \) under the influence of a velocity field \(\phi\), specified using:

divSchemes
{
    default         none;
    div(phi,Q)      Gauss <interpolation scheme>;
}

The phi keyword is typically used to represent the flux (flow) across cell faces, i.e.

• volumetric flux: \( \phi = \u_f \dprod \vec{S}_f \)
• mass flux: \( \phi = \rho_f \left( \u_f \dprod \vec{S}_f \right) \)

Options

NVD/TVD convection schemes

Many of the convection schemes available in OpenFOAM are based on the TVD and NVD [PROVIDE REF] For further information, see the page NVD and TVD schemes

• Limited linear divergence scheme
• Linear divergence scheme
• Linear-upwind divergence scheme
• Mid-point divergence scheme
• Minmod divergence scheme
• MUSCL divergence scheme
• QUICK divergence scheme
• UMIST divergence scheme
• Upwind divergence scheme
• Van Leer divergence scheme

Non-NVD/TVD convection schemes

• Courant number blended divergence scheme
• DES hybrid divergence scheme
• Filtered Linear (2) divergence scheme
• LUST divergence scheme

Special cases

Several options are available for convection schemes for certain flow cases, including for steady state, bounded scalar transport and limited schemes.

Steady state

For steady state cases the bounded form can be applied:

divSchemes
{
    default         none;
    div(phi,Q)      bounded Gauss <interpolation scheme>;
}

This adds a linearised, implicit source contribution to the transport equation of the form

\[ \ddt{\u} + \div (\u \otimes \u ) - \color{red}{( \div \u ) \u} = \div ( \Gamma \grad \u ) + S_u \]

i.e. it removes a component proportional to the continuity error. This acts as a convergence aid to tend towards a bounded solution as the calculation proceeds. At convergence, this term becomes zero and does not contribute to the final solution.

Bounded scalars

The limited<interpolation scheme> options bound scalar field values to user-specified limits, e.g. for the vanLeer scheme

limitedVanLeer -1 3

Many fields require bounding between 0 and 1, for which the following syntax is used:

<interpolation scheme>01

V schemes

A set of schemes for vector fields where the limiter is applied in the direction of greatest change.

Example

See this example to see the relative performance of the schemes.

Further information

See the implementation details to see how the schemes are coded.

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