Multi-material DG hydrodynamics
The stiff velocity equilibrium multi-material hydrodynamic equations are considered in this work. In this system, although all the materials are advected with the same velocity , they possess different pressures and internal energies. Finite pressure relaxation source terms are added to the material volume fraction and material total energy equations based on the material bulk-modulii. The Eulerian form of this multi-material system of equations is,
where and is the number of materials. is the volume-fraction of material- . Bulk properties such as density , pressure , and internal energy are defined as,
where is the material density , material pressure , or material internal energy as required. The specific total energy of material- is, , and its specific total enthalpy is, , where is the specific internal energy of the material. is the mass fraction of material- . The source term consists of the finite pressure relaxation. Note that although the material total energy equations are written in a non-conservative form, the mixture total energy equation ( ) is a conservative equation (non-conservative terms cancel when summed over all materials). Thus, total energy of the system is conserved. The system is closed by specifying equations of state for each material, usually in the form of .
A mixed-cell pressure closure has to be specified for the above system of equations to be complete. This term accounts for volume fraction redistribution due to differential compaction. This term attempts to model the different amount of compression in materials caused by unequal material compressibilities, by redistributing pressure-induced volume changes based on material bulk modulii. This redistribution of volume fractions results in a finite amount of relaxation between material pressures. The functional form is similar to Tipton's closure. The volume redistribution is given by,
where is the equilibrium pressure that the multi-material cell is expected to reach after sufficient time:
is the material's bulk modulus, and is the pressure-equilibration time-scale,
The constant adjusts the pressure equilibration rate of the materials in mixed cells. It controls how fast the pressure relaxation takes place in a mixed cell, as compared to the sound-crossing speed in that cell . The value of can be specified in the input file.
Currently, the stiffened-gas equation of state (SG-EoS) is used to close the PDE system. This EoS can be used to describe materials ranging from ideal gases to stiff liquid-like fluids. The internal energy, temperature and speed of sound for material- using the SG-EoS are:
where , , , and are the stiffness parameter, heat capacity ratio, specific heat at constant pressure, and temperature for material-$k$ respectively.
More complex equations of state are a topic of future work.
The temporal and spatial derivatives are discretized separately resulting in a method of lines. A 3rd order total variation diminishing (TVD) Runge-Kutta scheme is used to discretize the time derivatives.
A modal Reconstructed Discontinuous Galerkin (rDG) method is used to discretize the multi-material system in space. The specific rDG method implemented in Quinoa uses Dubiner's orthogonal basis functions (see the DG page for more details). Reconstruction to higher order is via a least-squares procedure based on node-neighbors. Fluxes through cell-faces are calculated using multi-material Riemann solvers. Non-linear instabilities are suppresed using a vertex-based limiter. Further details can be found in the Inciter papers.
Tangent of Hyperbola for INterface Capturing (THINC) is used to algebraically reconstruct material interfaces in the domain. THINC assumes a smooth distribution of volume fractions across the interface, using the function. This results in sharp interface capturing, without expensive geometric procedures. The THINC scheme can be tuned to capture material interfaces in 2-4 cells. Due to the smooth nature of the function, stability concerns are mitigated.
For further details about the numerical method, see Inciter papers.
For an example test set up see the example gas impact problem.