The phenomenon of chattering may occur when discontinuities in the model variables are present. This phenomenon can lead to extremely slow simulation, or to simulation failures because the computed variables exceed acceptable boundaries.

In a discretized two-phase flow models, the main discontinuity is often the density derivative on the liquid saturation curve. This is shown in the animation below: when travelling from the critical point down to lower pressures on the liquid saturation line, the partial derivative of the density w.r.t. enthalpy presents a growing discontinuity.

Simulation failure or stiff systems can occur if the cell-generated (and purely numerical) flow rate due to this discontinuity causes a flow reversal in one of the nodes (i.e. the computation of h su and h ex switches from one value to the other in the discretization scheme.

Therefore, to ensure the robustness of the simulation and to avoid chattering, numeric flow reversals should be avoided. This can be expressed by an inequality stating that numerically cell-generated flow rates must be lower than the flow rate circulating through the cycle, which can be written (for a single cell):

According to the previous equation, flow reversals and thus chattering or simulation failures are likely to occur if:

- The number of cells (N ) is low
- The working fluid flow rate (Mext ) is low
- The internal volume (V ) is high
- The working conditions are highly transient (i.e. dp/dt and dh/dt are high)
- The discontinuity in the partial derivatives of the density is significant

# Proposed Robustness Methods

The solutions implemented in ThermoCycle to avoid numerical flow reversals are the following: