Navier Stokes Fourier Equations Research Articles

Improved compressible hybrid lattice Boltzmann method on standard lattice for subsonic and supersonic flows

A D2Q9 Hybrid Lattice Boltzmann Method (HLBM) is proposed for the simulation of both compressible subsonic and supersonic flows. This HLBM is an extension of the model of Feng et al. [1], which has been found, via different test cases, to be unstable for supersonic regimes. To circumvent this limitation, we propose:: (1) a new discretization of the lattice closure correction term that makes possible the simulation of supersonic flows, (2) a corrected viscous stress tensor that takes into account polyatomic gases, and (3) a novel discretization of the viscous heat production term fitting with the regularized formalism. The result is a hybrid method that resolves the mass and momentum equations with an LBM algorithm, and resolves the entropy-based energy equation with a finite volume method. This approach fully recovers the physics of the Navier–Stokes–Fourier equations with the ideal gas equation of state, and is valid from subsonic to supersonic regimes. It is then successfully assessed with both smooth flows and flows involving shocks. The proposed model is shown to be an efficient, accurate, and robust alternative to classic Navier–Stokes methods for the simulation of compressible flows.

Zero viscosity and thermal diffusivity limit of the linearized compressible Navier–Stokes–Fourier equations in the half plane

We study the zero viscosity and thermal diffusivity limit of an initial boundary problem for the linearized Navier–Stokes–Fourier equations of a compressible viscous and heat conducting fluid in the half plane. We consider the case that the viscosity and thermal diffusivity converge to zero at the same order. The approximate solution of the linearized Navier–Stokes–Fourier equations with inner and boundary expansion terms is analyzed formally first by multiscale analysis. Then the pointwise estimates of the error terms of the approximate solution are obtained by energy methods. Thus establish the uniform stability for the linearized Navier–Stokes–Fourier equations in the zero viscosity and heat conductivity limit. This work is based on (Comm. Pure Appl. Math. 52 (1999), 479–541) and generalize their results from isentropic case to the general compressible fluid with thermal diffusive effect. Besides the viscous layer as in (Comm. Pure Appl. Math. 52 (1999), 479–541), the thermal layer appears and couples with the viscous layer linearly.

Nonlinear axisymmetric Taylor–Couette flow in a dilute gas: multiroll transition and the role of compressibility

Abstract

Role of Atwood number on flow morphology of a planar shock-accelerated square bubble: A numerical study

The Atwood number plays a critical role in describing the physics of fluids behind the hydrodynamic instabilities in gas dynamics. In order to investigate the impacts of the Atwood number (At), the evolution of a shock-accelerated square bubble containing either SF6, Kr, Ar, Ne, or He and surrounded by N2 is investigated numerically. For this purpose, the unsteady compressible Navier–Stokes–Fourier equations are solved using an explicit modal discontinuous Galerkin method. For validation, the numerical results are compared with available experimental results and are found to be in good agreement. The results demonstrate that the Atwood number has a significant influence on flow morphology with wave patterns, vortex creation, vorticity generation, and bubble deformation. For At > 0, the speed of the shock wave traveling along with the bubble inner surface is often less than that of the incident shock wave and greater than that of the transmitted shock wave. Moreover, vortex pairs from the upstream and downstream corners are generated, and the former vortex pair ultimately dominates the flow morphology. For At ≈ 0, the incident and transmitted shock waves move at the same speeds, whereas for At < 0, the transmitted shock wave travels faster than the incident shock wave. Moreover, only one vortex pair at the upstream corners is generated, which dominates the flow morphology. Furthermore, a detailed study of Atwood number impacts is investigated through the vorticity generation at interfaces. A quantitative analysis based on the shock trajectories, the interface features, and the integral diagnostics is also studied in detail to investigate the impacts of the Atwood number on the flow structure. Finally, a comparative study of the flow physics between the shock-accelerated square and cylindrical bubbles is conducted to examine their natural differences.

Compressibility in lattice Boltzmann on standard stencils: effects of deviation from reference temperature.

With growing interest in the simulation of compressible flows using the lattice Boltzmann (LB) method, a number of different approaches have been developed. These methods can be classified as pertaining to one of two major categories: (i) solvers relying on high-order stencils recovering the Navier-Stokes-Fourier equations, and (ii) approaches relying on classical first-neighbour stencils for the compressible Navier-Stokes equations coupled to an additional (LB-based or classical) solver for the energy balance equation. In most cases, the latter relies on a thermal Hermite expansion of the continuous equilibrium distribution function (EDF) to allow for compressibility. Even though recovering the correct equation of state at the Euler level, it has been observed that deviations of local flow temperature from the reference can result in instabilities and/or over-dissipation. The aim of the present study is to evaluate the stability domain of different EDFs, different collision models, with and without the correction terms for the third-order moments. The study is first based on a linear von Neumann analysis. The correction term for the space- and time-discretized equations is derived via a Chapman-Enskog analysis and further corroborated through spectral dispersion-dissipation curves. Finally, a number of numerical simulations are performed to illustrate the proposed theoretical study. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Hypersonic Aerodynamics of Blunt Plates in Near-Continuum Regime by Improved Navier–Stokes Model

Hypersonic argon flows over circularly blunt flat plates in near-continuum regime are numerically investigated in this study. The flows are, respectively, simulated using the direct simulation Monte Carlo (DSMC) method, the Navier–Stokes–Fourier (NSF) equations, and the DSMC data-improved Navier–Stokes (DiNS) model that is an improved continuum model with consideration of local shear nonequilibrium effects. The influences of nose radius, angle of attack, and freestream density on the local nonequilibrium characteristics and body surface aerodynamics are investigated in detail. The numerical results show that both the nose radius and the angle of attack present significant influences on the local nonequilibrium effects. Furthermore, it is found that the inaccuracy of the NSF equations for predicting aerodynamics mainly results in the error of the axial force. Compared with the NSF equations, the DiNS model exhibits remarkable improvements for predicting surface aerodynamic properties, especially the axial force, in all the simulated cases.

On the total enthalpy behavior inside a shock wave

The total enthalpy behavior inside a shock wave in a dilute monatomic gas has been numerically studied for various values of Mach and Prandtl numbers with the continuum (the Navier–Stokes–Fourier equations) and kinetic (the Shakhov model and the direct simulation Monte Carlo method) approaches. A significant difference between the results by the continuum and kinetic approaches has been observed for the internal shock wave structure. In a wide range of the free-stream Mach numbers, the continuum approach predicts qualitatively similar behavior of total enthalpy distributions that can be of a concave, constant, or convex shape depending on the Prandtl number. The more sophisticated kinetic approach predicts a more complicated form of total enthalpy profiles: e.g., an inflection point for Mach numbers around two and Prandtl numbers close to unity. The evolution of the total enthalpy in the shock is determined by the balance of heat conduction and mechanical work of normal viscous stress—processes that are predicted inaccurately by using the Navier–Stokes–Fourier equations at high Mach numbers.

A Blow-up Criterion for the Modified Navier–Stokes–Fourier Equations

We study the initial boundary value problem for the three-dimensional modified compressible Navier–Stokes–Fourier equations proposed by Brenner. We establish a blow-up criterion for local strong solutions in terms of the density $$\rho $$ and the temperature $$\theta $$ only.

Topology of the second-order constitutive model based on the Boltzmann–Curtiss kinetic equation for diatomic and polyatomic gases

The topological aspects of fluid flows have long been fascinating subjects in the study of the physics of fluids. In this study, the topology of the second-order Boltzmann–Curtiss constitutive model beyond the conventional Navier–Stokes–Fourier equations and Stokes’s hypothesis was investigated. In the case of velocity shear, the topology of the second-order constitutive model was shown to be governed by a simple algebraic form. The bulk viscosity ratio in diatomic and polyatomic gases was found to play an essential role in determining the type of topology: from an ellipse to a circle, to a parabola, and then finally to a hyperbola. The topology identified in the model has also been echoed in other branches of science, notably in the orbits of planets and comets and Dirac cones found in electronic band structures of two-dimensional materials. The ultimate origin of the existence of the topology was traced to the coupling of viscous stress and velocity gradient and its subtle interplay with the bulk viscosity ratio. In the case of compression and expansion, the topology of the second-order constitutive model was also found to be governed by a hyperbola. The trajectories of solutions of two representative flow problems—a force-driven Poiseuille gas flow and the inner structure of shock waves—were then plotted on the topology of the constitutive model, demonstrating the indispensable role of the topology of the constitutive model in fluid dynamics.

Combined Effects of Thermal Non-equilibrium and Chemical Reactions on Hypersonic Air Flows Around An Orbital Reentry Vehicle

The present work computationally investigated the combined effects of thermal non-equilibrium and chemical reactions on hypersonic air flows around an orbital reentry vehicle during its reentry. The chemically reacting gas flow around the orbital vehicle was simulated for actual reentry trajectories, with a computational solver based on the Navier–Stokes–Fourier equations. Hypersonic flows for a wide range of flying altitudes from 40 to 80 km were examined. We first analyzed important properties such as Mach number, trans-rotational temperature, vibrational temperature, chemical species composition, and surface properties such as the maximum heat flux for various altitudes. We then investigated the distribution of heat flux on the surface of the orbital reentry vehicle and the coupled effects of thermal non-equilibrium and chemical reactions on the hypersonic air flows. The computed heat flux results were compared with actual flight test data obtained during reentry of the orbital vehicle. The computed results were found to be in excellent agreement with the flight test data until an altitude of 60 km. Finally, we propose an explanation why heat flux was over-predicted around 70 km altitude but returned to better agreement at the higher altitude of 80 km. The phenomenon is based on two competing effects: the shortcoming of the first-order law of heat flux in the NSF equations and the inaccurate description of the effects of thermal non-equilibrium on chemical reactions in Park’s two temperature model.

Comprehensive assessment of newly-developed slip-jump boundary conditions in high-speed rarefied gas flow simulations

In this paper we numerically evaluate the recently developed Aoki et al. slip and jump conditions in high-speed rarefied gas flows for the first time. These slip and jump conditions are developed to be employed with the Navier–Stokes–Fourier equations. They were derived based on the Boltzmann equation with the first order Chapman–Enskog solution, and the analysis of the Knudsen layer. Four aerodynamic configurations are selected for a comprehensive evaluation of these conditions such as sharp-leading-edge flat plate, vertical plate, wedge and circular cylinder in cross-flow with the Knudsen number varying from 0.004 to 0.07, and argon as the working gas. The simulation results using the Aoki et al. boundary conditions show suitable agreement with the DSMC data for slip velocity and surface gas temperature. The accuracy of these boundary conditions is superior to the conventional Maxwell, Smoluchowski and Le boundary conditions.

Visualization of supersonic flows with bow shock using transversal discharges

Transonic and supersonic flows near the cones were visualized with two types of discharges. Pulse volume discharge with pre-ionization by plasma sheets was used to visualize the flow with discontinuities with time resolution in nanosecond range. Bow shock angle was measured by the glow images, and flow Mach number calculated. Stationary discharge was used to record a non-changing weakly luminous micro-nozzle jet flow spreading into a high vacuum region and interacting with a conical metal skimmer, which is essential for gas cluster equipment. A fusiform Mach barrel at the nozzle exit and the bow shock at the obstacle are visualized. The computational flow dynamics model based on Navier–Stokes–Fourier equations was verified by comparison with the experimental images and showed a good agreement with the observed flow structure.

Mathematical analysis of the motion of a rigid body in a compressible Navier–Stokes–Fourier fluid

AbstractWe study an initial and boundary value problem modelling the motion of a rigid body in a heat conducting gas. The solid is supposed to be a perfect thermal insulator. The gas is described by the compressible Navier–Stokes–Fourier equations, whereas the motion of the solid is governed by Newton's laws. The main results assert the existence of strong solutions, in an ‐ setting, both locally in time and globally in time for small data. The proof is essentially using the maximal regularity property of associated linear systems. This property is checked by proving the ‐sectoriality of the corresponding operators, which in turn is obtained by a perturbation method.

Coupled constitutive relations: a second law based higher-order closure for hydrodynamics.

In the classical framework, the Navier–Stokes–Fourier equations are obtained through the linear uncoupled thermodynamic force-flux relations which guarantee the non-negativity of the entropy production. However, the conventional thermodynamic descrip- tion is only valid when the Knudsen number is sufficiently small. Here, it is shown that the range of validity of the Navier–Stokes–Fourier equations can be extended by incorporating the nonlinear coupling among the thermodynamic forces and fluxes. The resulting system of conservation laws closed with the coupled constitutive relations is able to describe many interesting rarefaction effects, such as Knudsen paradox, transpiration flows, thermal stress, heat flux without temperature gradients, etc., which cannot be predicted by the classical Navier–Stokes–Fourier equations. For this system of equations, a set of phenomenological boundary conditions, which respect the second law of thermodynamics, is also derived. Some of the benchmark problems in fluid mechanics are studied to show the applicability of the derived equations and boundary conditions.

혼합 Modal 불연속 Galerkin 기법을 이용한 Lid-Driven Cavity 내의 희박유동 전산 연구

With the advancement of fabrication technology and miniaturization, fluid flows at micro- and nanoscales has received considerable attention. Flow characteristics in these systems significantly vary from those of macro- scale devices, due to geometric restrictions. In such cases, Navier-Stokes-Fourier(NSF) equations with no-slip condition may not be valid for studying gas flows inside a micro- cavity. This article therefore investigates the cavity flows using modified NSF equations with velocity slip and temperature jump conditions. In the present case, a monatomic gas is considered for modelling gas flows. To accurately predict the flow physics, mixed modal discontinuous Galerkin(DG) method is being developed. The flow characteristics of monatomic gas is studied by varying the Reynolds number(Re) and Knudsen number(Kn), respectively. Results obtained are compared with the previous results and are found to be in good agreement with them.

Asymptotic Stability of Homogeneous States in the Relativistic Dynamics of Viscous, Heat-Conductive Fluids

This paper shows global-in-time existence and asymptotic decay of small solutions to the Navier–Stokes–Fourier equations for a class of viscous, heat-conductive relativistic fluids. As this second-order system is symmetric hyperbolic, existence and uniqueness on a short time interval follow from the work of Hughes, Kato and Marsden. In this paper it is proven that solutions which are close to a homogeneous reference state can be extended globally and decay to the reference state. The proof combines decay results for the linearization with refined Kawashima-type estimates of the nonlinear terms.

Numerical modeling of nozzle gas flow using continuum approach in transition regime

This study is devoted to the numerical modelling of two-dimensional micro nozzle gas flow in continuum and transition regime with continuum mathematical models. Regularized 13-moment (R13) set of equations and Navier-Stokes-Fourier (NSF) equations are used as continuum approaches here. The numerical results for two-dimensional nozzle flow of the R13 equations and NSF equations have been compared for two cases of input and output pressures.

Fundamental solutions to the regularised 13-moment equations: efficient computation of three-dimensional kinetic effects

Fundamental solutions (Green’s functions) are derived for the regularised 13-moment system (R13) of rarefied gas dynamics, for small departures from equilibrium; these solutions show the presence of Knudsen layers, associated with exponential decay terms, that do not feature in the solution of lower-order systems (e.g. the Navier–Stokes–Fourier equations). Incorporation of these new fundamental solutions into a numerical framework based on the method of fundamental solutions (MFS) allows for efficient computation of three-dimensional gas microflows at remarkably low computational cost. The R13-MFS approach accurately recovers analytic solutions for low-speed flow around a stationary sphere and heat transfer from a hot sphere (for which a new analytic solution has been derived), capturing non-equilibrium flow phenomena missing from lower-order solutions. To demonstrate the potential of the new approach, the influence of kinetic effects on the hydrodynamic interaction between approaching solid microparticles is calculated. Finally, a programme of future work based on the initial steps taken in this article is outlined.

A posteriori analysis for dynamic model adaptation in convection-dominated problems

In this work, we present an a posteriori error indicator for approximation schemes of Runge–Kutta–discontinuous–Galerkin type arising in applications of compressible fluid flows. The purpose of this indicator is not only for mesh adaptivity, we also make use of this to drive model adaptivity. This is perhaps where a costly complex model and a cheaper simple model are solved over different parts of the domain. The a posteriori bound we derive indicates the regions where the complex model can be relatively well approximated with the cheaper one. One such example which we choose to highlight is that of the Navier–Stokes–Fourier equations approximated by Euler’s equations.

On the Vanishing Dissipation Limit for the Full Navier–Stokes–Fourier System with Non-slip Condition

In this paper, we study the vanishing dissipation limit problem for the full Navier–Stokes–Fourier equations with non-slip boundary condition in a smooth bounded domain $$\Omega \subseteq \mathbb {R}^{3}$$ . By using Kato’s idea (Math Sci Res Inst Publ 2:85–98, 1984) of constructing an artificial boundary layer, we obtain a sufficient condition for the convergence of the solution of the full Navier–Stokes–Fourier equations to the solution of the compressible Euler equations in the energy space $$L^{2}(\Omega )$$ uniformly in time.