Manuel Torrilhon

Regularization of Grad’s 13 moment equations: Derivation and linear analysis

A new closure for Grad’s 13 moment equations is presented that adds terms of Super-Burnett order to the balances of pressure deviator and heat flux vector. The additional terms are derived from equations for higher moments by means of the distribution function for 13 moments. The resulting system of equations contains the Burnett and Super-Burnett equations when expanded in a series in the Knudsen number. However, other than the Burnett and Super-Burnett equations, the new set of equations is linearly stable for all wavelengths and frequencies. Dispersion relation and damping for the new equations agree better with experimental data than those for the Navier–Stokes–Fourier equations, or the original 13 moments system. The new equations also allow the description of Knudsen boundary layers.

Regularized 13-moment equations: shock structure calculations and comparison to Burnett models

Recently a new system of field equations for the accurate description of flows in rarefied gases, called regularized 13-moment equations, was obtained by means of a hybrid gas kinetic approach. The first part of this paper discusses the relationship of the new system to classical high-order theories like the Burnett and super-Burnett equations as well as to modified models like the augmented and regularized Burnett equations. In the second part, shock structure calculations with the new theory are presented and compared to direct-simulation Monte Carlo (DSMC) solutions and to solutions of the Burnett models. Owing to additional higher-order dissipation in the system, the profiles are smooth for any Mach number, in contrast to the results of Grad’s 13-moment case. The results show reliable quantitative agreement with DSMC simulations for Mach numbers up to M0 ≈ 3.0. The agreement is better for Maxwell molecules than for hard spheres. The results of the augmented Burnett equations are comparable, but these equations are shown to be spatially unstable. Additionally, a validiation procedure for the new equations is presented by investigating the positivity of Grad’s distribution function.

Boundary conditions for regularized 13-moment-equations for micro-channel-flows

Boundary conditions are the major obstacle in simulations based on advanced continuum models of rarefied and micro-flows of gases. In this paper, we present a theory how to combine the regularized 13-moment-equations derived from Boltzmanns equation with boundary conditions obtained from Maxwells kinetic accommodation model. While for the linear case these kinetic boundary conditions suffice, we need additional conditions in the non-linear case. These are provided by the bulk solutions obtained after properly transforming the equations while keeping their asymptotic accuracy with respect to Boltzmanns equation.After finding a suitable set of boundary conditions and equations, a numerical method for generic shear flow problems is formulated. Several test simulations demonstrate the stable and oscillation-free performance of the new approach.

Couette and Poiseuille microflows : Analytical solutions for regularized 13-moment equations

The regularized 13-moment equations for rarefied gas flows are considered for planar microchannel flows. The governing equations and corresponding kinetic boundary conditions are partly linearized, such that analytical solutions become feasible. The nonlinear terms include contributions of the shear stress and shear rate, which describe the coupling between velocity and temperature fields. Solutions for Couette and force-driven Poiseuille flows show good agreement with direct simulation Monte Carlo data. Typical rarefaction effects, e.g., heat flux parallel to the wall and the characteristic dip in the temperature profile in Poiseuille flow, are reproduced accurately. Furthermore, boundary effects such as velocity slip, temperature jump, and Knudsen boundary layers are predicted correctly.

A robust numerical method for the R13 equations of rarefied gas dynamics: Application to lid driven cavity

In this work we present a finite difference scheme to compute steady state solutions of the regularized 13 moment (R13) equations of rarefied gas dynamics. The scheme allows fast solutions for 2D and 3D boundary value problems (BVPs) with velocity slip and temperature jump boundary conditions. The scheme is applied to the lid driven cavity problem for Knudsen numbers up to 0.7. The results compare well with those obtained from more costly solvers for rarefied gas dynamics, such as the Integro Moment Method (IMM) and the Direct Simulation Monte Carlo (DSMC) method. The R13 equations yield better results than the classical Navier-Stokes-Fourier equations for this boundary value problem, since they give an approximate description of Knudsen boundary layers at moderate Knudsen numbers. The R13 based numerical solutions are computationally economical and may be considered as a reliable alternative mathematical model for complex industrial problems at moderate Knudsen numbers.

Uniqueness conditions for Riemann problems of ideal magnetohydrodynamics

The equations of ideal magnetohydrodynamics (MHD) form a non-strict hyperbolic system with a non-convex flux function and admit non-regular, so-called intermediate shocks. The presence of non-regular waves in the MHD system causes the Riemann problem to be not unique in some cases. This paper investigates the uniqueness of Riemann solutions of ideal MHD. To determine uniqueness conditions we discuss the correspondence of non-regular solutions and non-uniqueness. Additionally the structure of the Hugoniot curves and its non-regular behaviour are demonstrated. It follows that the degree of freedom for solving a Riemann problem is reduced in the case of a non-regular solution. From this, we can deduce uniqueness conditions depending on the initial conditions of an MHD Riemann problem. The results also allow one to construct non-unique solutions. We give an example for the case of non-planar initial conditions.

A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion

In this paper, a stochastic model is presented to simulate the flow of gases, which are not in thermodynamic equilibrium, like in rarefied or micro situations. For the interaction of a particle with others, statistical moments of the local ensemble have to be evaluated, but unlike in molecular dynamics simulations or DSMC, no collisions between computational particles are considered. In addition, a novel integration technique allows for time steps independent of the stochastic time scale. The stochastic model represents a Fokker-Planck equation in the kinetic description, which can be viewed as an approximation to the Boltzmann equation. This allows for a rigorous investigation of the relation between the new model and classical fluid and kinetic equations. The fluid dynamic equations of Navier-Stokes and Fourier are fully recovered for small relaxation times, while for larger values the new model extents into the kinetic regime. Numerical studies demonstrate that the stochastic model is consistent with Navier-Stokes in that limit, but also that the results become significantly different, if the conditions for equilibrium are invalid. The application to the Knudsen paradox demonstrates the correctness and relevance of this development, and comparisons with existing kinetic equations and standard solution algorithms reveal its advantages. Moreover, results of a test case with geometrically complex boundaries are presented.

The shock tube study in extended thermodynamics

In this paper we investigate the shock tube experiment with extended thermodynamics. Extended thermodynamics (ET) provides dissipative field equations for monatomic gases which are symmetrically hyperbolic. The theory relies on the extension of the set of variables in order to describe extreme nonequilibrium processes. As an example for such a process we focus on the start-up phase of the shock tube experiment. We show numerically that ET succeeds to describe this short time behavior. For small times more and more variables are needed for a physically valid description. In the limit of very small times the solution of ET for the start-up phase converges to the solution of the free-flight-equation. Additionally it turns out that the system of Navier–Stokes and Fourier fails to describe the start-up phase of a shock tube even qualitatively.

Constraint-Preserving Upwind Methods for Multidimensional Advection Equations

A general framework for constructing constraint-preserving numerical methods is presented and applied to a multidimensional divergence-constrained advection equation. This equation is part of a set of hyperbolic equations that evolve a vector field while locally preserving either its divergence or its curl. We discuss the properties of these equations and their relation to ordinary advection. Due to the constraint, such equations form model equations for general evolution equations with intrinsic constraints which appear frequently in physics. The general framework allows the construction of numerical methods that preserve \emph{exactly} the discretized constraint by special flux distribution. Assuming a rectangular, two-dimensional grid as a first approach, application of this framework leads to a locally constraint-preserving multidimensional upwind method. We prove consistency and stability of the new method and present several numerical experiments. Finally, extensions of the method to the three-dimensional case are described.

Essentially optimal explicit Runge–Kutta methods with application to hyperbolic–parabolic equations

Optimal explicit Runge–Kutta methods consider more stages in order to include a particular spectrum in their stability domain and thus reduce time-step restrictions. This idea, so far used mostly for real-line spectra, is generalized to more general spectra in the form of a thin region. In thin regions the eigenvalues may extend away from the real axis into the imaginary plane. We give a direct characterization of optimal stability polynomials containing a maximal thin region and calculate these polynomials for various cases. Semi-discretizations of hyperbolic–parabolic equations are a relevant application which exhibit a thin region spectrum. As a model, linear, scalar advection–diffusion is investigated. The second-order-stabilized explicit Runge–Kutta methods derived from the stability polynomials are applied to advection–diffusion and compressible, viscous fluid dynamics in numerical experiments. Due to the stabilization the time step can be controlled solely from the hyperbolic CFL condition even in the presence of viscous fluxes.