Navier Stokes Fourier Equations Research Articles

A theoretical derivation of slip boundary conditions based on the Cercignani–Lampis–Lord scattering model

To characterize fluid flow in the slip regime, the use of Navier–Stokes–Fourier (NSF) equations with slip boundary conditions is prevalent. This trend underscores the necessity of developing reliable and accurate slip boundary conditions. According to kinetic theory, slip behaviours are intrinsically linked to the gas scattering processes at the surface. The widely used Maxwell scattering model, which employs a single accommodation coefficient to describe gas scattering processes, reveals its limitations when the difference between accommodation coefficients in the tangential and normal directions becomes significant. In this work, we provide a derivation of velocity slip and temperature jump boundary conditions based on the Cercignani–Lampis–Lord scattering model, which applies two independent accommodation coefficients to describe the gas scattering process. A Knudsen layer correction term is introduced to account for the impact of the surface on the velocity distribution function, which is associated with the scattering model. The governing equation of the correction term is established based on the linearized Boltzmann equation. Additionally, two moments are derived to capture the collision effect in the Knudsen layer: a conserving moment of collision invariants, and an approximate higher-order conserving moment. These moments are then employed to determine the coefficients in the correction term. We demonstrate that the derived slip coefficients align closely with numerical results obtained by solving the Boltzmann equation in the Knudsen layer. Besides, we apply the derived slip boundary conditions within the framework of the NSF equations, yielding numerical results that exhibit excellent consistency with those obtained through molecular-level simulations.

Explicit exact solutions for plane shock waves in dilute polyatomic gases

AbstractThe exact solutions for the Navier–Stokes–Fourier equations in the case of plane shock waves for dilute monatomic gases with constant transport coefficients were found by Becker in 1922. The solutions obtained are limited for a Prandtl's number given by . Besides the solutions for the speed and temperature, profiles were given in an implicit way. In this paper, we consider Becker's model to find some exact explicit solutions for dilute polyatomic gases.

Efficient parallel solver for rarefied gas flow using GSIS

Recently, the general synthetic iterative scheme (GSIS) has been proposed to find the steady-state solution of the Boltzmann equation in the whole range of gas rarefaction, where its fast-converging and asymptotic-preserving properties lead to the significant reduction of iteration numbers and spatial cells in the near-continuum flow regime. However, the efficiency and accuracy of GSIS have only been demonstrated in two-dimensional problems with small numbers of spatial cells and discrete velocities. Here, a large-scale parallel computing strategy is designed to extend the GSIS to three-dimensional flow problems, including the supersonic flows which are usually difficult to solve by the discrete velocity method. Since the GSIS involves the calculation of the mesoscopic kinetic equation which is defined in six-dimensional phase-space, and the macroscopic high-temperature Navier–Stokes–Fourier equations in three-dimensional physical space, the proper partition of the spatial and velocity spaces, and the allocation of CPU cores to the mesoscopic and macroscopic solvers, are the keys to improving the overall computational efficiency. These factors are systematically tested to achieve optimal performance, up to 100 billion spatial and velocity grids. For hypersonic flows around the Apollo reentry capsule, the X38-like vehicle, and the space station, our parallel solver can obtain the converged solution within one hour.

Regular reflection of shock waves in steady flows: viscous and non-equilibrium effects

Numerical analysis of a steady monatomic gas flow about the point of the regular reflection of a strong oblique shock wave from the symmetry plane is conducted with the Navier–Stokes–Fourier (NSF) equations, the regularized Grad 13-moment (R13) equations and the direct simulation Monte Carlo (DSMC) method. In contrast to the inviscid solution to this problem completely defined by the Rankine–Hugoniot (RH) relations, all three models predict a complicated flow structure with strong thermal non-equilibrium and a long wake with flow parameters not predicted by the RH relations. The temperature $T_y$ related to thermal motion of molecules in the direction normal to the symmetry plane has a maximum inside the reflection zone while in a planar shock wave the maximum is observed for the $T_x$ temperature. The R13 equations predict these features much better than the NSF equations and are in good agreement with the benchmark DSMC results. An analysis of the flow with the conservation equations was conducted in order to evaluate the effects of various processes on a fluid element moving along the symmetry plane. In contrast to the shock wave where effects of viscosity and heat conduction are one-dimensional with zeroth net contribution to the fluid-element energy across the shock, the flow across the zone of the shock reflection is dominated by two-dimensional effects with positive net contribution of viscosity and negative contribution of heat conduction to the fluid-element energy. These effects are believed to be the main source of the wake with parameters deviating from the RH values.

A Viewpoint on Thermally-Induced Transport in Rarefied Gases through the Method of Fundamental Solutions

Some phenomena pertaining to rarefied gases are beyond the reach of traditional fluid dynamics described, e.g., by the Euler or Navier–Stokes–Fourier equations. Therefore we adopt a recently developed model—referred to as the CCR model—to investigate thermally-induced transport in rarefied gases. To this end, the method of fundamental solutions is employed on the CCR model to investigate two problems: (i) a rarefied gas flow confined between two coaxial cylinders having different temperatures with the inner cylinder being circular while the outer being elliptical, and (ii) evaporation/condensation process in a rarefied vapor confined between two coaxial cylinders, again with the inner cylinder being circular and the outer being elliptical. Through a comprehensive analysis, the efficiency of the method of fundamental solutions is assessed. The work contributes toward a better understanding of thermally-induced confined rarefied gas flows.

On the shock wave boundary layer interaction in slightly rarefied gas

The shock wave and boundary layer interaction (SWBLI) plays an important role in the design of hypersonic vehicles. However, discrepancies between the numerical results of high-temperature gas dynamics and experiment data have not been fully addressed. It is believed that the rarefaction effects are important in SWBLI, but the systematic analysis of the temperature-jump boundary conditions and the role of translational/rotational/vibrational heat conductivities are lacking. In this paper, we derive the three-temperature Navier–Stokes–Fourier (NSF) equations from the gas kinetic theory, with special attention paid to the components of heat conductivity. With proper temperature-jump boundary conditions, we simulate the SWBLI in the double cone experiment. Our numerical results show that, when the three heat conductivities are properly recovered, the NSF equations can capture the position and peak value of the surface heat flux, in both low- and high-enthalpy inflow conditions. Moreover, the separation bubble induced by the separated shock and the reattachment point induced by impact between transmitted shock and boundary layer are found to agree with the experimental measurement.

Energy equality of weak solutions of the Navier–Stokes–Fourier equations allowing vacuum

In this paper, we are concerned with the conservation of energy criterion of the weak solutions to the Navier–Stokes–Fourier equations on the torus allowing vacuum. We establish the energy conservation criteria via the gradient of velocity for the weak solutions of this system, which generalizes the corresponding recent energy conservation criteria in terms of the velocity obtained by Aoki and Iwabuchi in Aoki and Iwabuchi (2022). Moreover, sufficient conditions for weak solutions keeping the energy balance allowing vacuum are shown for the full Navier–Stokes equations.

Numerical Study of Hypersonic Near-Continuum Flow by Improved Slip Boundary Condition

Appropriate numerical models and boundary conditions are required to accurately model hypersonic flows involving local nonequilibrium effects, which can be induced by the wall effect and large gradients of flow properties in the flowfield. Considering the strong density gradient near the wall for hypersonic flows, an improved Gökçen slip model is proposed in this paper by introducing a scaled mean free path to the conventional Gökçen’s model. Hypersonic argon flows over a circular cylinder and a blunt plate in the near-continuum regime are simulated by the proposed slip model with the conventional Navier–Stokes–Fourier equations and an extended continuum model, which was recently developed to model the shear nonequilibrium effect in high-speed flows. The accuracy of the new boundary model is evaluated by comparing the results with the direct simulation Monte Carlo method. The effects of Knudsen number, Mach number, and wall temperature on the nonequilibrium degree and aerodynamic predictions are also investigated. The numerical results show that the improved model can well predict the surface friction and heat transfer coefficients, exhibiting much better performance than the existing continuum models.

General synthetic iterative scheme for polyatomic rarefied gas flows

Recently, the general synthetic iterative scheme (GSIS) has been proposed to solve the Boltzmann equation efficiently and accurately in the whole range of gas rarefaction. Here, the GSIS is extended and further improved to find the steady-state solutions in nonlinear polyatomic gas flows. The numerical method is developed within the unstructured finite volume framework, where the kinetic equation and synthetic equations are solved alternately. The synthetic equations, which are derived from the modified Rykov kinetic model, reduce to the two-temperature (translational and rotational) Navier–Stokes–Fourier equations in the continuum flow regime, and include high-order constitutive relations for rarefaction effects in other flow regimes. These high-order terms are obtained by solving the kinetic model with the discrete velocity method. When the synthetic equations are solved, their macroscopic quantities are used to guide the evolution of velocity distribution functions in the kinetic model. As a consequence of this two-way coupling, GSIS not only finds steady-state solutions quickly but also relieves the constraints on spatial cell size. Furthermore, the robustness of GSIS is improved by modifying the boundary macroscopic fluxes in the macroscopic implicit iteration. The efficiency and multiscale property of GSIS is demonstrated through several challenging test cases, including the one-dimensional normal shock wave with large internal-translational relaxation times, the two-dimensional hypersonic flow passing a cylinder, and the pump-driven flow in Europe’s demonstration fusion power plant.

A second-order slip/jump boundary condition modified by nonlinear Rayleigh–Onsager dissipation factor

A newly heuristic form of second-order slip/jump boundary conditions (BCs) for the Navier–Stokes–Fourier (NSF) equations is proposed from the viewpoint of generalized hydrodynamic equations (GHE) to extend the capability of the NSF equations for moderately rarefied gas flows. The nonlinear Rayleigh–Onsager dissipation function appearing in the GHE, which contains useful information about the nonequilibrium flow fields of interest, is introduced into the proposed BCs named the simplified generalized hydrodynamic (SGH) BCs as a correction parameter. Compared with the classical Maxwell/Smoluchowski (MS) BCs, the SGH BCs may be more sensitive to capture the nonequilibrium information of flows adaptively and produce physically consistent solutions near the wall. Subsequently, the SGH BCs are implemented in the NSF equations for planar micro-Couette gas flows over a wide range of Knudsen numbers. The results indicate that the SGH BCs make impressive improvements against the MS BCs for diatomic and monatomic gases at the slip region and early transition regime, particularly in terms of capturing precisely the temperature and normal heat flux profiles in the flow and the temperature jump on the wall. More importantly, the SGH BCs conducted in NSF equations with less computational cost still can obtain well-pleased results comparable to the non-Newton–Fourier equations, such as several Burnett-type equations and regularized 13-moment equations, and even perform better than these models near the wall compared with direct simulation Monte Carlo data for the Couette flows to some extent.

Navier–Stokes–Fourier equations as a parabolic limit of a general hyperbolic system of rational extended thermodynamics

This paper aims to prove that a general system of 14 balance laws for a compressible, possibly dense, gas that satisfies the universal principles of Rational Extended Thermodynamics (RET) converges to the Navier–Stokes–Fourier equations in the first step of the Maxwellian iteration. Moreover, in a theory not far from equilibrium, we show that the production terms of the hyperbolic system are uniquely determined as soon as the heat conductivity, the shear viscosity, and the bulk viscosity are assigned. The obtained results are tested on the RET theories for rarefied monatomic and polyatomic gases.

Continuum Models for Bulk Viscosity and Relaxation in Polyatomic Gases

Bulk viscosity and acoustic wave propagation in polyatomic gases and their mixtures are studied in the frame of one-temperature and multi-temperature continuum models developed using the generalized Chapman–Enskog method. Governing equations and constitutive relations for both models are written, and the dispersion equations are derived. In the vibrationally nonequilibrium multi-component gas mixture, wave attenuation mechanisms include viscosity, thermal conductivity, bulk viscosity, diffusion, thermal diffusion, and vibrational relaxation; in the proposed approach these mechanisms are fully coupled contrarily to commonly used models based on the separation of classical Stokes–Kirchhoff attenuation and relaxation. Contributions of rotational and vibrational modes to the bulk viscosity coefficient are evaluated. In the one-temperature approach, artificial separation of rotational and vibrational modes causes great overestimation of bulk viscosity whereas using the effective internal energy relaxation time yields good agreement with experimental data and molecular-dynamic simulations. In the multi-temperature approach, the bulk viscosity is specified only by rotational modes. The developed two-temperature model provides excellent agreement of theoretical and experimental attenuation coefficients in polyatomic gases; both the location and the value of its maximum are predicted correctly. One-temperature dispersion relations do not reproduce the non-monotonic behavior of the attenuation coefficient; large bulk viscosity improves its accuracy only in the very limited frequency range. It is emphasized that implementing large bulk viscosity in the one-temperature Navier–Stokes–Fourier equations may lead to unphysical results.

Nonlinear coupled constitutive relation model for monatomic gas in the OpenFOAM framework

In high-speed rarefied gas flow simulations, the linear constitutive relations of the classical Navier-Stokes-Fourier (NSF) equations are no longer successfully applied. In the present work, the two-dimensional nonlinear coupled constitutive relations (NCCR) model of the monatomic gas in an algebraic-type form is successfully implemented in the rhoCentralFoam in OpenFOAM to simulate the high-speed rarefied gas flows. Methods: This solver initially resolves the NSF equations using high-resolution central schemes in the finite volume method. The implicit NCCR model is solved by the iterative process with the initial values from the linear constitutive relations of the NSF model. After the convergence, these nonlinear constitutive relations are embedded in the NSF equations to continue the solution. Results: This new modified solver would be validated for the argon gas flows past a circular cylinder in cross-flow at Mach of 5 and Kn = 0.05 and the sharp leading edge flat plate at Mach of 4 and Kn = 0.0042. Conclusion: The simulation results show that the NCCR model agrees well with the DSMC data for the temperature contours for both cases and the temperature and velocity along the shock stand-off distance and is better than the NSF model in the high-speed rarefied gas simulations for the circular cylinder case.

Global Existence of Weak Solutions for Compresssible Navier—Stokes—Fourier Equations with the Truncated Virial Pressure Law

Abstract This paper concerns the existence of global weak solutions á la Leray for compressible Navier–Stokes–Fourier systems with periodic boundary conditions and the truncated virial pressure law which is assumed to be thermodynamically unstable. More precisely, the main novelty is that the pressure law is not assumed to be monotone with respect to the density. This provides the first global weak solutions result for the compressible Navier-Stokes-Fourier system with such kind of pressure law which is strongly used as a generalization of the perfect gas law. The paper is based on a new construction of approximate solutions through an iterative scheme and fixed point procedure which could be very helpful to design efficient numerical schemes. Note that our method involves the recent paper by the authors published in Nonlinearity (2021) for the compactness of the density when the temperature is given.

Using gene expression programming to discover macroscopic governing equations hidden in the data of molecular simulations

The unprecedented amount of data and the advancement of machine learning methods are driving the rapid development of data-driven modeling in the community of fluid mechanics. In this work, a data-driven strategy is developed by the combination of the direct simulation Monte Carlo (DSMC) method and the gene expression programming (GEP) method. DSMC is a molecular simulation method without any assumed macroscopic governing equations a priori and is employed to generate data of flow fields, while the enhanced GEP method is leveraged to discover governing equations. We first validate our idea using two benchmarks, such as the Burgers equation and Sine–Gordon equation. Then, we apply the strategy to discover governing equations hidden in the complex fluid dynamics. Our results demonstrate that in the continuum regime, the discovered equations are consistent with the traditional ones with linear constitutive relations, while in the non-continuum regime such as shock wave, the discovered equation comprises of high-order constitutive relations, which are similar to those in the Burnett equation but with modified coefficients. Compared to the Navier–Stokes–Fourier equations and the Burnett equation, the prediction of the viscous stress and heat flux in the shock wave via the presented data-driven model has the best match to the DSMC data. It is promising to extend the proposed data-driven strategy to more complex problems and discover hidden governing equations which may be unknown so far.

Steady Compressible Navier–Stokes–Fourier Equations with Dirichlet Boundary Condition for the Temperature

Steady Compressible Navier–Stokes–Fourier Equations with Dirichlet Boundary Condition for the Temperature

A multiple-relaxation-time collision model by Hermite expansion

The Bhatnagar-Gross-Krook (BGK) single-relaxation-time collision model for the Boltzmann equation serves as the foundation of the lattice BGK (LBGK) method developed in recent years. The description of the collision as a uniform relaxation process of the distribution function towards its equilibrium is, in many scenarios, simplistic. Based on a previous series of papers, we present a collision model formulated as independent relaxations of the irreducible components of the Hermite coefficients in the reference frame moving with the fluid. These components, corresponding to the irreducible representation of the rotation group, are the minimum tensor components that can be separately relaxed without violating rotation symmetry. For the 2nd, 3rd and 4th moments, respectively, two, two and three independent relaxation rates can exist, giving rise to the shear and bulk viscosity, thermal diffusivity and some high-order relaxation process not explicitly manifested in the Navier-Stokes-Fourier equations. Using the binomial transform, the Hermite coefficients are evaluated in the absolute frame to avoid the numerical dissipation introduced by interpolation. Extensive numerical verification is also provided. This article is part of the theme issue 'Progress in mesoscale methods for fluid dynamics simulation'.

Development of unsteady natural convection in a square cavity under large temperature difference

To investigate how the nonuniform fluid density distribution caused by large temperature variations affects the development of unsteady natural convection, we perform a series of direct numerical simulations of two-dimensional compressible natural convection in an air-filled square cavity. The cavity has a hot wall on the left and a cold wall on the right, and two horizontal walls are adiabatic. The simulations are done using a kinetic approach based on a modeled Boltzmann equation, from which the fully compressible Navier–Stokes–Fourier equations are recovered. No Boussinesq approximation or low Mach number approximation is made. An extra source term is introduced to adjust the fluid Prandtl number. Simulations are performed for a range of Rayleigh numbers (107−109) with a fixed dimensionless temperature difference of ε=0.6 to determine the critical Rayleigh number and study the development of unsteady flow. To illustrate the instability mechanism, instantaneous fluctuation field, time trace of temperature, and velocity at selected monitoring points, the spectrum and other statistics are presented and discussed. As expected, significant differences are observed between the instability of compressible natural convection and the Boussinesq-type natural convection. With a large temperature difference, the transition to unsteady flow is asymmetric for the flows near the hot wall and cold wall. For the Rayleigh number range we studied, the cold wall region is dominated by low-frequency impact instability of the boundary thermal jet at the bottom corner. For the hot wall region, besides the upper corner impact instability, a boundary layer instability featuring high-frequency oscillations is observed.

Extended lattice Boltzmann model for gas dynamics

We propose a two-population lattice Boltzmann model on standard lattices for the simulation of compressible flows. The model is fully on-lattice and uses the single relaxation time Bhatnagar–Gross–Krook kinetic equations along with appropriate correction terms to recover the Navier–Stokes–Fourier equations. The accuracy and performance of the model are analyzed through simulations of compressible benchmark cases including Sod shock tube, sound generation in shock–vortex interaction, and compressible decaying turbulence in a box with eddy shocklets. It is demonstrated that the present model provides an accurate representation of compressible flows, even in the presence of turbulence and shock waves.

Revisiting the Lattice Boltzmann Method Through a Nonequilibrium Thermodynamics Perspective

Abstract In this paper, we incorporate a nonequilibrium thermodynamics perspective that is consistent with the Onsager reciprocity principle into the lattice Boltzmann framework to propose a novel regularized lattice Boltzmann formulation for modeling the Navier–Stokes–Fourier equations. The new method is applied to one-dimensional (1D) isothermal situations wherein the advantages of incorporating such a nonequilibrium perspective can be explicitly appreciated. In such situations, the nonequilibrium contribution of the lattice populations obtained by the new method completely vanishes, and the lattice update is entirely reduced to evaluating the equilibrium distribution function. Such a counterintuitive 1D mesoscopic description is not obtained in any other existing lattice Boltzmann scheme. We therefore numerically test the proposed formulation on two complex problems, namely, shockwave and nonlinear wave propagation, and compare results with analytical results along with six existing lattice Boltzmann schemes; it is found that the new method indeed yields results that are more stable and accurate. These results highlight the potency of the nonequilibrium thermodynamics-based approach for obtaining accurate and stable lattice Boltzmann computations, and provide new insights into established lattice Boltzmann simulation methods.