Equations Of Compressible Fluid Dynamics Research Articles

On the Equations of Compressible Fluid Dynamics With Cattaneo‐Type Extensions for the Heat Flux: Symmetrizability and Relaxation Structure

ABSTRACTThe aim of this work is twofold. From a mathematical point of view, we show the existence of a hyperbolic system of equations that is not symmetrizable in the sense of Friedrichs. Such system appears in the theory of compressible fluid dynamics with Cattaneo‐type extensions for the heat flux. In contrast, the linearizations of such system around constant equilibrium solutions have Friedrichs symmetrizers. Then, from a physical perspective, we aim to understand the relaxation term appearing in this system. By noticing the violation of the Kawashima–Shizuta condition, locally and smoothly, with respect to the Fourier frequencies, we construct persistent waves, that is, solutions preserving the norm for all times that are not dissipated by the relaxation terms.

Classical solutions for the Euler equations of compressible fluid dynamics: A new topological approach

In this article we study a class of Euler equations of compressible fluid dynamics. We give conditions under which the considered equations have at least one and at least two classical solutions. To prove our main results we propose a new approach based upon recent theoretical results.

Evaluation of Various Approximations in Atmosphere and Ocean Modeling Based on an Exact Treatment of Gravity Wave Dispersion

Abstract Various approximations of the governing equations of compressible fluid dynamics are commonly used in both atmospheric and ocean modeling. Their main purpose is to eliminate the acoustic waves that are potentially responsible for inefficiency in the numerical solution, leaving behind gravity waves. The author carries out a detailed study of gravity wave dispersion for seven such approximations, individually and in combination, to exactly evaluate some of the often subtle errors. The atmospheric and oceanic cases are qualitatively and quantitatively different because, although they solve the same equations, their boundary conditions are entirely different and they operate in distinctly different parameter regimes. The atmospheric case is much more sensitive to approximation. The recent “unified” approximation of Arakawa and Konor is one of the most accurate. Remarkably, a simpler approximation, the combined Boussinesq–dynamically rigid approximation turns out to be exactly equivalent to the unified approximation with respect to gravity waves. The oceanic case is insensitive to the effects of any of the approximations, except for the hydrostatic approximation. The hydrostatic approximation is inaccurate at large wavenumbers in both the atmospheric and oceanic cases because it eliminates the entire buoyancy oscillation flow regime and is therefore to be restricted to low aspect ratio flows. For oceanic applications, certain approximations, such as the unified, dynamically rigid, and dynamically stiff approximations, are particularly interesting because they are accurate and approximately conserve mass, which is important for the treatment of sea level rise.

Special issues on moment methods in kinetic gas theory

This two-volume special issue of ‘Continuum Mechanics and Thermodynamics’ is the result of an international meeting on moment methods in kinetic gas theory that took place in November 2008 at ETH Zurich [1]. There have been over 30 participants from 10 different countries attending 13 invited lectures on various developments in moment methods. The workshop was hosted jointly by the ‘Research Institute for Mathematics (FIM)’ and the ‘Seminar for Applied Mathematics (SAM)’ at ETH Zurich, as well as supported by the ‘Swiss National Science Foundation (SNF)’. In recent years, moment methods in kinetic gas theory proved to be a successful tool to extend classical fluid dynamics for applications in microor rarefied gas flows. The classical equations of compressible fluid dynamics, known as the system ofNavier–Stokes and Fourier, loose their validity in extreme physical processes involving rarefaction or micro-scales. The lack of sufficient particle collisions is expressed in larger values of the Knudsen number K n—the ratio between the mean free path and an observation scale—and leads to thermal non-equilibrium which can only be described using refined mathematical models. Kinetic gas theory based on a statistical description of the gas provides a valid framework to model processes in a rarefied regime or at small scales [2,3]. The main variable used to describe the gas is the distribution function or probability density of the particle velocities. However, in situations of intermediate non-equilibrium this detailed statistical approach still yields a far too complex description of the gas. In these situations it is desirable to have a continuum model based on partial differential equations for the fluid mechanical field variables. The classical fluid dynamics equations can be derived fromBoltzmann’s equation as the first order contribution in a Chapman–Enskog expansion [4], an asymptotic expansion in the Knudsen number. It seemed natural to look for higher order contributions as an extension for the laws of Navier–Stokes and Fourier. However, the hope for useful equations was dashed by the observation of Bobylev [5] stating that these higher order models are inherently unstable in the general case. In the 1940s Harold Grad [6,7] developed an alternative to the Chapman–Enskog expansion by producing moment equations from Boltzmann’s equation. The fundamental idea of moment approximations is to replace the variable of a high dimensional distribution function f (x, t, c) for the velocity c ∈ R3 by a finite setmoment variables {Fn (x, t)}n=0,1,2,... at each (x, t). The working hypothesis is that at (x, t) a finite number of moments are sufficient to encode the details of f as a function of c. The first moments are the conservative variables of gas dynamics, the density, the momentum density and total energy density. Obviously, the motivation is that

Continuous compression waves in the two-dimensional Riemann problem

The interaction between a plane shock wave in a plate and a wedge is considered within the framework of the nondissipative compressible fluid dynamic equations. The wedge is filled with a material that may differ from that of the plate. Based on the numerical solution of the original equations, self-similar solutions are obtained for several versions of the problem with an iron plate and a wedge filled with aluminum and for the interaction of a shock wave in air with a rigid wedge. The behavior of the solids at high pressures is approximately described by a two-term equation of state. In all the problems, a two-dimensional continuous compression wave develops as a wave reflected from the wedge or as a wave adjacent to the reflected shock. In contrast to a gradient catastrophe typical of one-dimensional continuous compression waves, the spatial gradient of a two-dimensional compression wave decreases over time due to the self-similarity of the solution. It is conjectured that a phenomenon opposite to the gradient catastrophe can occur in an actual flow with dissipative processes like viscosity and heat conduction. Specifically, an initial shock wave is transformed over time into a continuous compression wave of the same amplitude.

Generalized time-periodic solutions to the Euler equations of compressible fluids

S. Nea) Abstract. We introduce a notion of generalized time-periodic solutions to first-order hyperbolic systems in one space dimension and we establish the existence of such solutions to the Euler equations of compressible fluid dynamics for a large class of equations of state.

Finite energy solutions to the isentropic Euler equations with geometric effects

Considering the isentropic Euler equations of compressible fluid dynamics with geometric effects included, we establish the existence of entropy solutions for a large class of initial data. We cover fluid flows in a nozzle or in spherical symmetry when the origin r = 0 is included. These partial differential equations are hyperbolic, but fail to be strictly hyperbolic when the fluid mass density vanishes and vacuum is reached. Furthermore, when geometric effects are taken into account, the sup-norm of solutions can not be controlled since there exist no invariant regions. To overcome these difficulties and to establish an existence theory for solutions with arbitrarily large amplitude, we search for solutions with finite mass and total energy. Our strategy of proof takes advantage of the particular structure of the Euler equations, and leads to a versatile framework covering general compressible fluid problems. We establish first higher-integrability estimates for the mass density and the total energy. Next, we use arguments from the theory of compensated compactness and Young measures, extended here to sequences of solutions with finite mass and total energy. The third ingredient of the proof is a characterization of the unbounded support of entropy admissible Young measures. This requires the study of singular products involving measures and principal values.

Pulsing Inertial Oscillation, Supercell Storms, and Surface Mesonetwork Data

The pulsing inertial oscillation (PIO) model is a nonlinear, time-dependent, translating vortex solution of the inviscid, compressible fluid dynamic equations in the middle troposphere. The translation of this vortex during a pulse is strikingly similar to that of a supercell storm – a rotating thunderstorm that can generate tornadoes and hail. Two studies were performed to test the hypothesis that some supercell storms are manifestations of a PIO pulse. The first study applied the model to an intense interior draft whose buoyancy was bounded by a temperature excess of ± 12 K. The peak updraft speed achieved was 41·5 m s−1 and the peak Rossby number was 92·9. The study also pointed to an advanced concept for attaining higher values. The second study applied the PIO model to a supercell storm as a whole and succeeded in replicating its bulk properties, such as mesocylonic circulation, net mass and moisture influxes, and time track. This study also identified a critical feature of the PIO model that could be tested against storm data: The average vertical draft is downward before the turn in the storm track and upward afterwards. In the conventional theory, the average vertical draft is upward from storm inception until dissipation. These differing draft predictions were compared with the best available data, which are surface mesonetwork data. These data were found to support the PIO model. However, surface data alone are not conclusive, and further measurements are warranted.

Analytic theory of Richtmyer–Meshkov instability for the case of reflected rarefaction wave

An analytic theory of the Richtmyer–Meshkov (RM) instability for the case of reflected rarefaction wave is presented. The exact solutions of the linearized equations of compressible fluid dynamics are obtained by the method used previously for the reflected shock wave case of the RM instability and for stability analysis of a ‘‘stand-alone’’ rarefaction wave. The time histories of perturbations and asymptotic growth rates given by the analytic theory are shown to be in good agreement with earlier linear and nonlinear numerical results. Applicability of the prescriptions based on the impulsive model is discussed. The theory is applied to analyze stability of solutions of the Riemann problem, for the case of two rarefaction waves emerging after interaction. The RM instability is demonstrated to develop with fully symmetrical initial conditions of the unperturbed Riemann problem, identically zero density difference across the contact interface both before and after interaction, and zero normal acceleration of the interface. This confirms that the RM instability is not caused by the instant normal acceleration of the interface, and hence, is not a type of Rayleigh–Taylor instability. The RM instability is related to the growth of initial transverse velocity perturbations at the interface, which may be either present initially as in symmetrical Riemann problem, or be induced by a shock passing a corrugated interface.

Review of preconditioning methods for fluid dynamics

We consider the use of preconditioning methods to accelerate the convergence to a steady state for both the incompressible and compressible fluid dynamic equations. Most of the analysis relies on the inviscid equations though some applications for viscous flow are considered. The preconditioning can consist of either a matrix or a differential operator acting on the time derivatives. Hence, in the steady state the original steady solution is obtained. For finite difference methods the preconditioning can change and improve the steady-state solutions. Several preconditioners previously discussed are reviewed and some new approaches are presented.

Compressible flow in a hovercraft air cushion

A solution of the compressible fluid-dynamic equations describing the two-dimensional, inviscid, subsonic fluid-dynamic field in the peripheral jets of a hovercraft at zero forward speed is presented. The analysis is made in the hodograph plane in which the motion equation is linear and enables one to write the solution of the problem as a sum of elementary separable solutions of the equation. The stream function is expressed as a Fourier series whose coefficients can be calculated numerically with ease. The comparison of the results with those of a different representation, valid only in the incompressible case, shows that the two solutions coincide

A finite difference scheme for inviscid flows with non-equilibrium chemistry and internal energy

A finite difference scheme is presented for the inviscid terms of the equations of compressible fluid dynamics with general non-equilibrium chemistry and internal energy.

Stable Viscosities and Shock Profiles for Systems of Conservation Laws

Abstract : Many equations of mathematical physics take the form of nonlinear hyperbolic systems of conservation laws. With small dissipative effects neglected, typically smooth solutions must develop discontinuities (shocks) in finite time. Re-incorporating dissipation helps select those discontinuities which are physically relevant. For this purpose, many different sorts of dissipation will do; in particular, the physical viscosity is typically degenerate and not convenient. In this paper the author provide an understanding of what high order viscosity terms smooth the physical discontinuities. A natural class of admissible viscosity terms is determined based on a simple linearized stability criterion. In addition, they determine a class of degenerate second order viscosity terms of physical type which are admissible. These results are applied to the equations of compressible fluid dynamics, to determine what conditions ensure the existence of the shock layer with viscosity and heat conduction. This should be of interest to others interested in general equations of state for compressible fluids, such as those investigating phase transitions.

An extension ofA-stability to alternating direction implicit methods

Completely implicit, noniterative, finite-difference schemes have recently been developed by several authors for nonlinear, multidimensional systems of hyperbolic and mixed hyperbolic-parabolic partial differential equations. The method of Douglas and Gunn or the method of approximate factorization can be used to reduce the computational problem to a sequence of one-dimensional or alternating direction implicit (ADI) steps. Since the eigenvalues of partial differential equations (for example, the equations of compressible fluid dynamics) are often widely distributed with large imaginary parts,A-stable integration formulas provide ideal time-differencing approximations. In this paper it is shown that if anA-stable linear multistep method is used to integrate a model two-dimensional hyperbolic-parabolic partial differential equation, then one can always construct an ADI scheme by the method of approximate factorization which is alsoA-stable, i.e., unconditionally stable. A more restrictive result is given for three spatial dimensions. Since necessary and sufficient conditions forA-stability can easily be determined by using the theory of positive real functions, the stability analysis of the factored partial difference equations is reduced to a simple algebraic test.

Stellar convection theory. I - The anelastic modal equations

Methods are developed for dealing with the various dynamical problems that arise because of convective zones in stars. A system of equations for stellar convection is derived from the full equations of compressible fluid dynamics with the aid of two major approximations. The first of these is the anelastic approximation, which involves both the filtering out of acoustic waves and a suitable linearization of the fluctuating thermodynamic variables. The second one approximates the horizontal structure of convection by expanding the motion in a set of horizontal cellular platforms and severely truncating the expansion. The resulting system of partial differential equations, referred to as the anelastic modal equations, is outlined along with suggested boundary conditions and techniques for solving the equations. Ways of assessing the overall validity of the present treatment are discussed.