Entropy Consistent Methods for the Navier–Stokes Equations

Abstract

The concept of entropy conservation, stability, and consistency is applied to systems of hyperbolic equations to create new flux functions for the scalar and systems of conservation laws. Firstly, Burgers’ equation is modelled, followed by the Navier–Stokes equations. The new models are compared with the pre-existing entropy consistent fluxes at selected viscosity levels; it is found that the system flux requires additional entropy production at low viscosities, but not at higher viscosity values. Initial results herein demonstrate that the accuracy of the first order systems approach are comparable to the results produced by the original entropy-consistent Navier–Stokes flux.

Appendices

Appendix 1: Entropy Conserving Flux

The entropy conserving flux $\mathbf f _i$ from Eq. (50) satisfies

$$\begin{aligned} \mathbf v ^\mathbf{T } \mathbf f _i = [\rho u] \end{aligned}$$

(67)

and is calculated based on averaged quantities of

$$\begin{aligned} \mathbf f _i (\mathbf u _L, \mathbf u _R) = \begin{bmatrix} \hat{\rho } \hat{u} \\ \hat{\rho } \hat{u}^2 +\hat{p_1} \\ \hat{\rho } \hat{u} \hat{H} \end{bmatrix} \end{aligned}$$

(68)

To determine the averaged quantities, we firstly define $z_1 = \sqrt{\frac{\rho }{p}}, \, z_2 = \sqrt{\frac{\rho }{p}u}, \, z_3 = \sqrt{\rho p}$. The averaged quantities are composed from functions of arithmetic mean $\bar{a} = \frac{a_L + a_R}{2}$ and logarithmic mean as defined in “Appendix 2”. Based on Eq. (67), the quantities used in the flux are as follows

$$\begin{aligned} \hat{u}&= \frac{\bar{z_2}}{\bar{z_1}}, \quad \hat{\rho } = \bar{z_1} z_3^{ln}, \quad \hat{p_1} = \frac{\bar{z_3}}{\bar{z_1}}, \quad \hat{p_2} = \frac{\gamma + 1}{2 \gamma } \frac{z_3^{ln}}{z_1^{ln}} + \frac{\gamma - 1}{2 \gamma } \frac{\bar{z_3}}{\bar{z_1}}\end{aligned}$$

(69)

$$\begin{aligned} \hat{a}&= \left( \frac{\gamma \hat{p_2}}{\hat{\rho }}\right) ^{\frac{1}{2}}, \quad \hat{H} = \frac{\hat{a}^2}{\gamma - 1} + \frac{\hat{u}^2}{2} \end{aligned}$$

(70)

Appendix 2: Logarithmic Mean

Let $\zeta = \frac{a_L}{a_R}$. Define $a^{ln}(L, R) = \frac{a_L + a_R}{ln (\zeta )} \frac{\zeta - 1}{\zeta + 1}$ where $ln(\zeta ) = 2 (\frac{1 - \zeta }{1 + \zeta } + \frac{1}{3} \frac{(1 - \zeta )^3}{(1 + \zeta )^3} + \frac{1}{5} \frac{(1 - \zeta )^5}{(1 + \zeta )^5} + \frac{1}{7} \frac{(1 - \zeta )^7}{(1 + \zeta )^7} + O(\zeta ^9))$. To calculate the logarithmic mean we use the following subroutine:

• 1. Set the following: $\zeta = \frac{a_L}{a_R}, \quad f = \frac{\zeta - 1}{\zeta + 1}, \quad u = f * f$

• 2. If $(u < \epsilon )$

  $$\begin{aligned} F = 1.0 + u/3.0 + u * u/5.0 + u * u * u/7.0 \end{aligned}$$

• 3. Else

  $$\begin{aligned} F = ln(\zeta )/2.0/(f) \end{aligned}$$

  thus

  $$\begin{aligned} a^{ln}(L, R) = \frac{a_L + a_R}{2F}, \quad \epsilon = 10^{-2} \end{aligned}$$