High-order fully implicit solver for all-speed fluid dynamics

Abstract

We introduce a novel Newton–Krylov (NK)-based fully implicit algorithm for solving fluid flows in a wide range of flow conditions—from variable density nearly incompressible to supersonic shock dynamics. The key enabling feature of our all-speed solver is the ability to efficiently solve conservation laws by choosing a set of independent variables that produce a well-conditioned Jacobian matrix for the linear iterations of the global nonlinear iterative solver. In particular, instead of choosing to discretize the conservative variables (density, momentum, total energy), which is traditionally used in Eulerian high-speed compressible fluid dynamics, we demonstrate superior performance by discretizing the primitive variables—pressure–velocity–temperature in the very low-Mach flow limits or density–velocity–temperature/entropy in the shock dynamics range. Moreover, our method allows us to avoid direct inversion of the mass matrix in discrete time derivatives, which is usually an additional source for stiffness, especially pronounced when going to very high-order schemes with non-orthogonal basis functions. Here, we show robust solutions obtained for discontinuous finite element discretization up to seventh-order accuracy. Another important aspect of the solution algorithm is the Advection Upstream Splitting Method (AUSM), adopted to compute numerical fluxes within our reconstructed discontinuous Galerkin (rDG) spatial discretization scheme. The use of the low-Mach modification of the hyperbolic flux operator is found to be necessary for enabling robust simulations of very stiff liquids and metals for Mach numbers below $M=10^{-5}$, which is well known to be very computationally challenging for compressible solvers. We demonstrate that our fully implicit rDG-NK solver with the ${\mathrm{AUSM}}^{+}$-up flux treatment produces efficient and high-resolution numerical solutions at all speeds, ranging from vanishing Mach numbers to transonic and supersonic, without substantial modifications of the solution procedures. (At high speed, we add limiting and use a simpler preconditioning of the Krylov solver.) Numerical examples include nearly incompressible constant-property flow past a backward-facing step with heat transfer, low-Mach variable-property channel flow of water at supercritical state, phase change and melt pool dynamics for laser spot welding and selective laser melting in additive manufacturing, and Mach 3 flow in a wind tunnel with a step.

Appendix: IAPWS-IF97

To represent water in a supercritical thermodynamic state, we use the “International Association for the Properties of Water and Steam, Industrial Formulation 1997” (IAPWS-IF97) [31]. Above the fluid’s critical point, i.e.,

$$\begin{aligned} \left[ \begin{array}{cll} P &{}\ge &{} P_{_\mathrm{c}}=22.064~\mathrm{MPa} \\ T &{}\ge &{} T_{_\mathrm{c}}=647.096~\mathrm{K} \\ &{}&{}\rho _{_{_\mathrm{c}}}=322~\mathrm{\frac{kg}{m^3}} \end{array} \right] , \end{aligned}$$

(60)

the pressure can be represented as a function of density and temperature, as

$$\begin{aligned} \tilde{P} \left( \hat{\rho }, \tilde{T} \right)= & {} \frac{\rho _{_{_{\mathrm{c}}}} T_{_{\mathrm{c}}} R}{\bar{P}} \delta ^{^2} \phi _{_{\delta }}-\bar{P}_{_{\mathrm{R}}}, \end{aligned}$$

(61)

where $R=461.526$$\mathrm{\frac{J}{kg~K}}$ is the specific gas constant, while dimensionless density ($\delta$) and inverse of temperature ($\theta$) are defined as:

$$\begin{aligned} \delta \left( \hat{\rho }\right) = \frac{\hat{\rho }}{\hat{ \rho }_{_{_{\mathrm{c}}}}} \text{ and } \theta \left( \tilde{T} \right) =\frac{\hat{T}_{_{\mathrm{c}}}}{\tilde{T}+\hat{T}_{_R}}. \end{aligned}$$

(62)

The dimensionless specific internal energy ($\tilde{{\mathfrak {u}}}$), specific isobaric heat capacity ($\hat{C}_p$), speed of sound ($\hat{c}_s$), and isobaric compressibility ($\hat{\beta }$) can be written as:

$$\begin{aligned} \begin{aligned}&\tilde{{\mathfrak {u}}} = \frac{T_{_c} R}{\bar{{\mathfrak {u}}}} \phi _{_{\theta }}- \hat{{\mathfrak {u}}}_{_R},\\&\hat{C}_p = \left( \frac{\delta \phi _{_{\delta }} \left( 1-\theta \frac{\phi _{_{\delta \theta }}}{\phi _{_{\delta }}}\right) ^2}{2+ \delta \frac{\phi _{_{\delta \delta }}}{\phi _{_{\delta }}}}-\theta ^2 \phi _{_{\theta \theta }} \right) \frac{R}{\bar{C}_{_p}},\\&\hat{c}_s = \left( 2 \phi _{_{\delta }}+ \delta \left[ \phi _{_{\delta \delta }}- \frac{\left( \phi _{_{\delta }} - \theta \phi _{_{\delta \theta }} \right) ^2 }{\theta ^2 \phi _{_{\theta \theta }}} \right] \right) \frac{\delta }{\theta } \frac{T_{_{\mathrm{c}}} R}{ \bar{v}^2},\\&\hat{\beta } =-\frac{\theta \phi _{_{\delta }}}{\left( 2 \phi _{_{\delta }}+\delta \phi _{_{\delta \delta }}\right) \hat{T}_{_{\mathrm{c}}}}, \end{aligned} \end{aligned}$$

(63)

where $\phi$ is the dimensionless Helmholtz free energy:

$$\begin{aligned} \phi \left( \delta , \theta \right) = a_{_{(1)}} \ln \delta + \sum \limits _{i=2}^{40} a_{_{(i)}} \delta ^{^{b_{_{(i)}}}} \theta ^{^{c_{_{(i)}}}}, \end{aligned}$$

(64)

while $\phi _{_{\delta }}$, $\phi _{_{\theta }}$, $\phi _{_{\delta \delta }}$, $\phi _{_{\delta \theta }}$, and $\phi _{_{\theta \theta }}$ are its first and second derivatives relative to $\delta$ and $\theta$. The coefficients $a_{_{(n)}}$ and exponents $b_{_{(n)}}$ and $c_{_{(n)}}$ are given in [31].

Dynamic viscosity and thermal conductivity are nonlinear functions of temperature and density and defined as

$$\begin{aligned} \begin{aligned}&\tilde{\mu } \left( \hat{\rho }, \tilde{T}\right) = \frac{\mu _{_\mathrm{dg}} \left( \tilde{T}\right) \times \mu _{_\mathrm{fd}}\left( \hat{\rho }, \tilde{T}\right) \times \mu _{_\mathrm{ce}} }{\bar{\mu } \bar{\mathrm{Re}}},\\&\tilde{\kappa } \left( \hat{\rho }, \tilde{T}\right) = \frac{\kappa _{_\mathrm{dg}} \left( \tilde{T}\right) \times \kappa _{_\mathrm{fd}}\left( \hat{\rho }, \tilde{T}\right) + \kappa _{_\mathrm{ce}} }{\bar{\kappa } \bar{\mathrm{Re}} \bar{\mathrm{Pr}} }. \end{aligned} \end{aligned}$$

(65)

Viscosities and thermal conductivities in the dilute-gas limit ($\mu _{_\mathrm{dg}}$ and $\kappa _{_\mathrm{dg}}$), and contributions due to finite density ($\mu _{_\mathrm{fd}}$ and $\kappa _{_\mathrm{fd}}$) are described in [75, 76]. In the present study, we will ignore critical enhancement factors, $\mu _{_\mathrm{ce}}=1$ and $\kappa _{_\mathrm{ce}}=0$, as these are relatively unimportant in the range of thermodynamic states of interest.

Special care must be taken to properly code (63)–(65), as these might become prohibitively expensive to be used in implicit solvers. We have optimized all these computations to be compatible with discontinuous Galerkin-based residual evaluations (Sect. 3.1). As described in Sect. 3.1, our nonlinear solver is formulated in terms of DG degrees of freedom for pressure ($\tilde{P}$), velocity ($\hat{{\mathbf{v }}}$), and temperature ($\tilde{T}$). This would require the application of a Newton solver to first evaluate $\hat{\rho }\left( \tilde{P}, \tilde{T}\right)$, for (61), before evaluating the rest of the thermodynamic variables using analytical (63). We used quadruple precision (16-bit) for (61)–(65), as this improves the conditioning and overall convergence of the Newton–Krylov algorithm. The performance of our nonlinear solver using this optimized equation of state is comparable (only two or three times more expensive) to the performance of the solver employed with simpler analytical forms of equation of state (e.g., Sect. 2.3.1).

Finally, we would like to note that the scaling properties $\bar{\rho }$, $\bar{\mu }$, $\bar{\kappa }$, and $\bar{C}_{_p}$ are chosen as evaluated at the reference state $\left( P_{_{\mathrm{R}}}, T_{_{\mathrm{R}}}\right)$.