Semi-implicit Source

The fvOptions library in OpenFOAM provides a flexible way to add source terms to governing equations. These sources can represent, for example, heat addition to internal energy (like ignition, or electronic component heating), a species injection such as a pollutant source or fuel inejction, or momentum input such as a propeller.

However, source terms can also make a computation unstable or even cause divergence. For that reason, implicit or semi-implicit scheme is often used.

Simple System

Consider a simple system without any source term:

\[\frac{dx}{dt}=0\]

The solution is trivial.

\[x=constant\]

Now suppose a source term is addded in the form

\[S(x)=S_u+S_p\cdot x\]

where $S_u$ is the independent or unaffected part, $S_px$ the proportional part with a scalar factor $S_p$. The system becomes

\[\frac{dx}{dt}=S_u+S_p\cdot x\]

$S_u$ has the same unit as $dx/dt$, and $S_p$ is scalar.

If $S_u\gt 0$, then $x$ wil increase with time. So $S_u$ acts as a source. If $S_u \lt 0$, it is a sink. In some situations, this growth may become too rapid and lead to numerical instability. A negative $S_p$ can act like a damping. In this way, the proportional term helps stabilize the solution and prevents uncontrollable increase of x.

Momentum Source Term

Froude’s actuation disk theory provides the thrust formula

\[T=2\rho A |U_\infty \cdot n|^2 a (1-a)\]

When using fvOptions with type actuationDiskSource, $S_u$ and $S_p$ are calculated internally via function $addSup$.

\[S_u=\frac{T}{V_{disk}}\]

This is added to the RHS of the velocity equation(UEqn). The proportional part doesn’t have an explicit value unless the model includes velocity-dependent linearization. The solve may internally add a small value to stabilize the computation.

Issues with rhoCentralFoam

Unlike other solvers, rhocentralFoam solve $\rho U$ instead of $U$.

For density-weighted momentum sourcs, a modified type, such as rhoActuationDiskSource is required. This variant calculates $\rho T$ instead of $T$, ensuring consistency with the solver’s equation.