compressible fluids, multiphase flows and fluids with density constraints

All-speed schemes for compressible fluids

Compressible fluids under small Mach-number (i.e. when the fluid motion is slow compared to the acoustic speed) are subject to very fast acoustic waves which contribute to instantaneously equilibrate the pressure over the whole domain. AP-methods (aka 'all-speed' schemes) are necessary when the Mach-number is small in some regions and order unity in other regions. Such methods have been developed for the isentropic and full Euler equations with perfect gas and real gas equations-of-state.

Below are comparisons between the AP and classical Roe schemes (top and bottom respectively) for the backwards facing step test case. The picture depicts the flow lines. While the recirculation zone is clearly visible with the AP-scheme, it is totally lost with the classical Roe scheme.

AP-schemes for multi-phase flows

AP-methods have also been applied to multi-phase flows in regimes where one of the phases appears or dissappears. Such regimes are usually very challenging for classical schemes. Below is the result of an AP-method for a Tee-junction case. The two-phase (water and vapour) fluid circulates in the horizontal branch from left to right. Due to differences of inertia, segregation of the water and vapour occurs: water flows preferentially in the horizontal branch while vapour concentrates in the vertical branch. The color code indicates the vapor fraction (or void fraction). The yellow and red colors show an increase of the void fraction in the vertical branch. The blue color along the top boundary of the horizontal branch highlights the local decrease of the void fraction in this area. The AP-method reproduces this phenomenon quite well. It has been impossible for us to produce any meaningful result with a classical method in this case.

In future work, the combination of these two methods will be implemented in view of the simulation of low-Mach multiphase flows.

Fluids with density constraints and the modeling of herding phenomena

The numerical simulation of fluids with density constraints can be treated by similar methods. Such fluids arise in various types of complex systems modeling herding phenomena such as herds, crowds, etc. Their numerical simulation is delicate. They show a transition between a compressible (or uncongested) region, where the density constraint is not reached and an incompressible (or congested) one, where the density constraint is saturated. The geometry and topology of the congestion boundary evolves in time in a non-explicit way.

To avoid the numerical tracking of the congestion boundary, which can be computationally intensive, we use the approximate model with regularized pressure as introduced here. However, the acoustic speed is very large close to the congestion density. The CFL stability condition imposes very small time steps and leads to unpractical numerical simulations unless an Asymptotic-Preserving (AP) method is used. To this aim, the all-speed schemes presented above has been used.

The two videos below show the collision of two counter-propagating blobs in the isentropic Euler model with maximal density constraint (the first video shows the density as a function of the 2-dimensional spatial coordinates, colour coded ; the second video shows the momentum vector field). The congestion constraint deflects the blobs in an almost perpendicular direction).

Current work concerns the extension of this method to the density-constrained Self-Organized Hydrodynamic model for the modeling of herds.

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