Potential Body Forces Research Articles

On a New Spatial Discretization for a Regularized 3D Compressible Isothermal Navier–Stokes–Cahn–Hilliard System of Equations with Boundary Conditions

We construct a new spatial finite-difference discretization for a regularized 3D Navier–Stokes–Cahn–Hilliard system of equations. The system can be attributed to phase field type models and describes flows of a viscous compressible isothermal two-component two-phase fluid with surface effects; the potential body force is also taken into account. In the discretization, the main sought functions are defined on one and the same mesh, and an original approximation of the solid wall boundary conditions (homogeneous with the discretization of equations) is suggested. The discretization has an important property of the total energy dissipativity allowing one to eliminate completely the so-called spurious currents. The discrete total mass and component mass conservation laws hold as well, and the discretization is also well-balanced for the equilibrium solutions. To ensure that the concentration C remains within the physically meaningful interval (0, 1), the non-convex part of the Helmholtz free energy is taken in a special logarithmic form (the Flory–Huggins potential). The speed of sound can depend on C that leads to different equilibrium mass densities of the “pure” phases. The results of numerical 3D simulations are also presented including those with a gravitational-type force. The positive role of the relaxation parameter is discussed too.

AN ENERGY DISSIPATIVE SPATIAL DISCRETIZATION FOR THE REGULARIZED COMPRESSIBLE NAVIER-STOKES-CAHN-HILLIARD SYSTEM OF EQUATIONS

We consider the regularized 3D Navier-Stokes-Cahn-Hilliard equations describing isothermal flows of viscous compressible two-component fluids with interphase effects. We construct for them a new energy dissipative finite-difference discretization in space, i.e., with the non-increasing total energy in time. This property is preserved in the absence of a regularization. In addition, the discretization is well-balanced for equilibrium flows and the potential body force. The sought total density, mixture velocity and concentration of one of the components are defined at nodes of one and the same grid. The results of computer simulation of several 2D test problems are presented. They demonstrate advantages of the constructed discretization including the absence of the so-called parasitic currents.

On the energy dissipative spatial discretization of the barotropic quasi-gasdynamic and compressible Navier–Stokes equations in polar coordinates

Abstract The barotropic quasi-gasdynamic system of equations in polar coordinates is treated. It can be considered a kinetically motivated parabolic regularization of the compressible Navier–Stokes system involving additional 2nd order terms with a regularizing parameter τ > 0. A potential body force is taken into account. The energy equality is proved ensuring that the total energy is non-increasing in time. This is the crucial physical property. The main result is the construction of symmetric spatial discretization on a non-uniform mesh in a ring such that the property is preserved. The unknown density and velocity are defined on the same mesh whereas the mass flux and the viscous stress tensor are defined on the staggered meshes. Additional difficulties in comparison with the Cartesian coordinates are overcome, and a number of novel elements are implemented to this end, in particular, a self-adjoint and positive definite discretization for the Navier–Stokes viscous stress, special discretizations of the pressure gradient and regularizing terms using enthalpy, non-standard mesh averages for various products of functions, etc. The discretization is also well-balanced. The main results are valid for τ = 0 as well, i.e., for the barotropic compressible Navier–Stokes system.

Asymptotic expansion in Gevrey spaces for solutions of Navier–Stokes equations

In this paper, we study the asymptotic behavior of solutions to the threedimensional incompressible Navier-Stokes equations (NSE) with periodic boundary conditions and potential body forces. In particular, we prove that the Foias-Saut asymptotic expansion for the regular solutions of the NSE in fact holds in all Gevrey classes . This strengthens the previous result obtained in Sobolev spaces by Foias-Saut. By using the Gevrey-norm technique of Foias-Temam, the proof of our improved result simplifies the original argument of Foias-Saut, thereby, increasing its adaptability to other dissipative systems. Moreover, the expansion is extended to Leray-Hopf weak solutions and the exponential decay rate for each of its remainders is calculated explicitly.

On some properties of multidimensional hyperbolic quasi-gasdynamic systems of equations

We study a multidimensional hyperbolic quasi-gasdynamic (HQGD) system of equations containing terms with a regularizing parameter τ > 0 and 2nd order space and time derivatives; the body force is taken into account. We transform it to a form close to the compressible Navier–Stokes system of equations. Then we derive the entropy balance equation and show that the entropy production is similar to the latter system plus a term of the order of O(τ2). We analyze an equation for the total entropy as well. We also show that the corresponding residuals in the HQGD equations with respect to the compressible Navier–Stokes ones are of the order of O(τ2) too. Finally we treat the simplified barotropic HQGD system of equations with the general state equation and the stationary potential body force and obtain the corresponding results for it.

On conservative spatial discretizations of the barotropic quasi-gasdynamic system of equations with a potential body force

A multidimensional barotropic quasi-gasdynamic system of equations in the form of mass and momentum conservation laws with a general gas equation of state p = p(ρ) with p′(ρ) > 0 and a potential body force is considered. For this system, two new symmetric spatial discretizations on nonuniform rectangular grids are constructed (in which the density and velocity are defined on the basic grid, while the components of the regularized mass flux and the viscous stress tensor are defined on staggered grids). These discretizations involve nonstandard approximations for ∇p(ρ), div(ρu), and ρ. As a result, a discrete total mass conservation law and a discrete energy inequality guaranteeing that the total energy does not grow with time can be derived. Importantly, these discretizations have the additional property of being well-balanced for equilibrium solutions. Another conservative discretization is discussed in which all mass flux components and viscous stresses are defined on the same grid. For the simpler barotropic quasi-hydrodynamic system of equations, the corresponding simplifications of the constructed discretizations have similar properties.

Asymptotic integration of Navier–Stokes equations with potential forces. II. An explicit Poincaré–Dulac normal form

We study the incompressible Navier–Stokes equations with potential body forces on the three-dimensional torus. We show that the normalization introduced in the paper [C. Foias, J.-C. Saut, Linearization and normal form of the Navier–Stokes equations with potential forces, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1) (1987) 1–47], produces a Poincaré–Dulac normal form which is obtained by an explicit change of variable. This change is the formal power series expansion of the inverse of the normalization map. Each homogeneous term of a finite degree in the series is proved to be well-defined in appropriate Sobolev spaces and is estimated recursively by using a family of homogeneous gauges which is suitable for estimating homogeneous polynomials in infinite dimensional spaces.

On the Helicity in 3D-Periodic Navier–Stokes Equations II: The Statistical Case

We study the asymptotic behavior of the statistical solutions to the Navier–Stokes equations using the normalization map [9]. It is then applied to the study of mean energy, mean dissipation rate of energy, and mean helicity of the spatial periodic flows driven by potential body forces. The statistical distribution of the asymptotic Beltrami flows are also investigated. We connect our mathematical analysis with the empirical theory of decaying turbulence. With appropriate mathematically defined ensemble averages, the Kolmogorov universal features are shown to be transient in time. We provide an estimate for the time interval in which those features may still be present. Our collaborator and friend Basil Nicolaenko passed away in September of 2007, after this work was completed. Honoring his contribution and friendship, we dedicate this article to him.

The normal form of the Navier–Stokes equations in suitable normed spaces

We consider solutions to the incompressible Navier–Stokes equations on the periodic domain Ω=[0,2π]3 with potential body forces. Let R⊆H1(Ω)3 denote the set of all initial data that lead to regular solutions. Our main result is to construct a suitable Banach space SA⋆ such that the normalization map W:R→SA⋆ is continuous, and such that the normal form of the Navier–Stokes equations is a well-posed system in all of SA⋆. We also show that SA⋆ may be seen as a subset of a larger Banach space V⋆ and that the extended Navier–Stokes equations, which are known to have global solutions, are well-posed in V⋆.

On the helicity in 3D-periodic Navier-Stokes equations I: the non-statistical case

We study the asymptotic behavior of the statistical solutions to the Navier–Stokes equations using the normalization map [9]. It is then applied to the study of mean energy, mean dissipation rate of energy, and mean helicity of the spatial periodic flows driven by potential body forces. The statistical distribution of the asymptotic Beltrami flows are also investigated. We connect our mathematical analysis with the empirical theory of decaying turbulence. With appropriate mathematically defined ensemble averages, the Kolmogorov universal features are shown to be transient in time. We provide an estimate for the time interval in which those features may still be present. Our collaborator and friend Basil Nicolaenko passed away in September of 2007, after this work was completed. Honoring his contribution and friendship, we dedicate this article to him.

Three-dimensional structures of magnetostatic atmospheres. III - A general formulation

This paper presents a mathematical description of the three-dimensional structures of static, magnetized atmospheres in terms of the electric-current flux surfaces. In the first paper of this series, a class of three-dimensional magnetostatic states was presented. These states are characterized by electric currents flowing perpendicular to gravity which was taken to be either uniform in space or spherically symmetric about a point mass. A limitation of these solutions is that the magnetic field in general cannot have a net twist. The new formulation generalizes these solutions in two significant ways, namely, to allow for other general potential body forces than the special gravitational forces assumed in the first paper and to include the presence of a net magnetic twist. This theoretical development opens up opportunities in modeling realistic solar magnetic fields as well as other astrophysical magnetic fields. The obliquely rotating magnetosphere and self-gravitating magnetized plasma clouds are two astrophysical examples for which the governing equations are derived. The complete analytic solution to a specific problem is presented, which illustrates the anchoring of a fully three-dimensional force-free magnetic field in the solar atmosphere to the base of the atmosphere through a non-force-free boundary layer where the magnetic field interacts strongly with the plasma. This and a rich variety of other three-dimensional magnetic fields will be studied in the subsequent papers of this series.

Establishment of Concentration Profile for the Flow of a Binary Fluid Mixture

A study is made of the establishment of concentration profile of the heavy species for a binary fluid mixture under a body force from a uniformly mixed upstream condition. An inviscid hydrodynamical model is adopted. Two cases are considered. In the first case the flow is a swirling motion and the body force is provided by the centrifugal action of the swirl. In the second case the flow is a duct flow confined between two parallel walls with potential body force. Use is made of a perturbation procedure in terms of an expansion of the reciprocal of a diffusive Péclet number to obtain a closed form zeroth-order solution. The orders of the higher-order correction terms are explicitly given. The interplay between Fickian and baro-diffusion is brought out, and asymptotic solutions far downstream of the inlet are explicitly calculated for a class of swirling flows. The problem bears on various phenomena where a knowledge of the concentration of the heavier component is important, such as in gaseous nuclear cavity reactor propulsion devices.