PCNSFVHLLC
The derivation of the porous HLLC discretization that follows is based extensively on the material in (Toro, 2009), drawing mostly from chapters 2, 3, and 10. Details pertinent to the MOOSE implementation of the free-flow HLLC discretization can be found in CNSFVHLLCBase.
Solution Properties Across the Contact Wave
HLLC restores the middle contact wave to the HLL formulation. Generalized Riemann Invariants reveal what quantities change or are constant across the wave. We perform the Generalized Riemann Invariants analysis on the porous Euler equations in the following way: we convert the term into and ignore the latter term when composing the flux vector (the term is instead treated as part of a source vector ). Then we define our conserved variable set (for one-dimension for simplicity here)
(1)
and our flux vector
and our source vector
Armed with these definitions we can write the Euler equations succinctly as
(2)
We can also write Eq. (2) in a quasi-linear form
where is the Jacobian matrix of partial derivatives of with respect to . It can be shown for an ideal gas that the eigenvalues of are
where is the speed of sound in the medium. The corresponding eigenvectors are
(3)
where is the total specific enthalpy. The second eigenvector corresponds to the middle contact wave. For a general system, with variable set:
and eigenvectors
the Generalized Riemann Invariants are given by the ODEs:
(4)
Taking our conserved variable set (Eq. (1)) and the contact wave eigenvector from Eq. (3) and substituting into Eq. (4) yields the relations
These equalities can be algebraically manipulated to yield the following relationships across the contact wave:
(5)
We will use Eq. (5) when constructing the porous HLLC fluxes below.
Porous HLLC Fluxes
For discontinuous wave solutions over a wave-speed associated with the characteristic, the Rankine-Hugoniot conditions state that the flux changes according to
We can apply the Rankine-Hugoniot conditions to help us establish a discretization for the porous HLLC fluxes. Applying Rankine-Hugoniot conditions over the left wave results in
(6)
Analogously over the center contact wave:
(7)
and the right wave
(8)
The star fluxes can be written = , where denotes or and preliminarily we will write
(9)
Our goal is to eventually express and consequently in terms of known left and right quantities, We can leverage the information from Eq. (5) to help us construct star region solutions
(10)
Per (Toro, 2009) it is justifiable to select that the middle wave speed be equal to the star region velocity
(11)
Manipulating the mass, momentum, and energy components of Eq. (6) and Eq. (8), we can construct equations for the star region density, pressure, and total specific energy as functions of the known left and right states and the middle wave speed (where we have substituted anyplace we encountered or ). The star region density relationships are given by
(12)
the star region pressure relationships are given by
(13)
and the total specific energy relationships are given by
(14)
Substituting Eq. (12), Eq. (13), and Eq. (14) into Eq. (9) and using from Eq. (11), we arrive at the vector expression for :
We must now establish an equation for . Combining Eq. (13) with the pressure information from Eq. (10), we arrive at:
(15)
Left and right wave speeds and are computed in the same way as for free flow as outlined in CNSFVHLLCBase. With and the final HLLC flux can be constructed:
(16)
Note that although in the derivation above we assumed a one-dimensional setup, the intermediate solution states can be generalized to three-dimensions in a way analogously to the multidimensional free-flow intermediate solutions expressed in CNSFVHLLCBase. Indeed, the porous intermediate states can be simply expressed as
and with expressed according to Eq. (15).
The mass component of Eq. (16) is implemented in PCNSFVMassHLLC, momentum in PCNSFVMomentumHLLC, and fluid energy in PCNSFVFluidEnergyHLLC.
References
- Eleuterio F Toro.
Riemann solvers and numerical methods for fluid dynamics: a practical introduction.
Springer Science & Business Media, 3rd edition, 2009.[BibTeX]