Clarify sign of radial-velocity-squared term in Euler–Euler HLLC modified mixture pressure

[rj-scharff]

Summary

I found an apparent source/documentation divergence in the Euler–Euler modified
mixture pressure and would appreciate clarification of the intended sign and
derivation.

I am reporting this as a formulation question, not as a confirmed software bug.

Versions checked

Audited commit:

ca26c4cd48c3d98927a2879a3fc16e53e546a555

Current upstream commit checked:

b281bfeab5ddc1f66d374d130e05d16132884d30

The relevant source and documentation files are byte-identical between these
commits:

• src/simulation/m_riemann_solver_hllc.fpp
• docs/documentation/equations.md

Source expression

In m_riemann_solver_hllc:s_hllc_riemann_solver, the arithmetic-average
Euler–Euler branch contains:

pres_L = pres_L - alpha_L(num_fluids) * &
    (pres_L - PbwR3Lbar/R3Lbar - rho_L*R3V2Lbar/R3Lbar)

The right-state expression has the same form.

Define

$$ A = \frac{\left\langle p_{bw}R^3\right\rangle} {\left\langle R^3\right\rangle}, \qquad K = \rho_\ell \frac{\left\langle R^3\dot R^2\right\rangle} {\left\langle R^3\right\rangle}. $$

The source expression expands to

$$ \begin{aligned} p_{\mathrm{new}} &=p_\ell-\alpha(p_\ell-A-K) \\ &=(1-\alpha)p_\ell+\alpha A+\alpha K. \end{aligned} $$

The radial-velocity-squared contribution is therefore positive:

$$ +\alpha\rho_\ell \frac{\left\langle R^3\dot R^2\right\rangle} {\left\langle R^3\right\rangle}. $$

Documented expression

The “Modified mixture pressure” equation in
docs/documentation/equations.md gives

$$ p=(1-\alpha)p_\ell+ \alpha\left[ \frac{\left\langle R^3p_{bw}\right\rangle} {\left\langle R^3\right\rangle} -\rho_\ell \frac{\left\langle R^3\dot R^2\right\rangle} {\left\langle R^3\right\rangle} \right]. $$

which has a negative radial-velocity-squared contribution.

Downstream trace

I did not identify an explicit downstream term that reverses or compensates for
the sign:

• the modified pressure enters the HLLC momentum and energy fluxes directly;
• face energy is constructed earlier from the unmodified primitive pressure;
• the bubble source updates do not apply a carrier momentum or energy
  counterterm involving this quantity; and
• R3V2 appears only in this HLLC routine.

Question

Is the positive source expression intentional—for example, because pres_L and
pres_R represent an intermediate or transformed pressure?

If so, could you point to the derivation or compensating identity that produces
the documented final mixture pressure?

Otherwise, which sign is intended?

[sbryngelson]

@rj-scharff Thanks for this. It resolves the other way: the source is right and the documentation is wrong.

Ando (Caltech thesis, 2010) writes the mixture momentum flux as pl - ptilde in Eq. (2.2), with

ptilde = alpha (pl - <R^3 pbw>/<R^3> - rho <R^3 Rdot^2>/<R^3>)   Eq. (2.5)

Both terms are negative inside the parenthesis, so

pl - ptilde = (1 - alpha) pl + alpha (<R^3 pbw>/<R^3> + rho <R^3 Rdot^2>/<R^3>)

and the radial-velocity contribution is positive. Eq. (A.71) of the same thesis carries the two contributions in a single bracket with the same sign, which is an independent check.

That is what the HLLC branch computes: it overwrites pres_L in place with pl - ptilde. So your guess about an intermediate or transformed pressure was the right reading.

#1740 fixes the sign in equations.md. The solver does not change and no golden values move.