CNSFVHLLCBase is the base class from which all Harten-Lax-Van Leer-Contact (HLLC) residual computing objects inherit from. It computes the wave speeds for the HLLC formulation. Because it is the base class for all HLLC residual objects, we will use its documentation page to outline the HLLC equations. The HLLC flux is defined as:
where FL,R are the left and right evaluations respectively of the convective flux:
F=⎣⎡ρanρanu+pnxρanv+pnyρanw+pnzρet+p⎦⎤
where an=a⋅n^ where a={u,v,w} and n^ is the normal vector at the face. nx, ny, and nz represent the x, y, and z components of the normal vector respectively, while u, v, and w represent the x, y, and z components of the velocity a respectively. p is the static pressure and et=e+(v⋅v)/2 where e is the specific internal energy. ρ is the density. SL, S∗, and SR are the left, middle, and right wave speeds respectively. S∗ is given by (Toro, 2009):
The final piece to finish the definition of the HLLC flux is to specify the evaluation of the left and right wave speeds, SL and SR. These can be evaluated in a variety of ways, but the current CNSFVHLLCBase implementation evaluates them as follows, following (Batten et al., 1997):
where c denotes the local speed of sound in the medium and Roe denotes Roe-averaged quantities which are computed via
bRoe=ρL+ρRρLbL+ρRbR
where b can be any quantity such as an or c.
References
P Batten, MA Leschziner, and UC Goldberg.
Average-state Jacobians and implicit methods for compressible viscous and turbulent flows.
Journal of Computational Physics, 137(1):38–78, 1997.[BibTeX]
@article{batten1997average,
author = "Batten, P and Leschziner, MA and Goldberg, UC",
title = "{Average-state Jacobians and implicit methods for compressible viscous and turbulent flows}",
journal = "Journal of Computational Physics",
volume = "137",
number = "1",
pages = "38--78",
year = "1997",
publisher = "Elsevier"
}
Eleuterio F Toro.
Riemann solvers and numerical methods for fluid dynamics: a practical introduction.
Springer Science & Business Media, 3rd edition, 2009.[BibTeX]
@book{toro2009riemann,
author = "Toro, Eleuterio F",
title = "Riemann solvers and numerical methods for fluid dynamics: a practical introduction",
year = "2009",
publisher = "Springer Science \\& Business Media",
edition = "3rd"
}