An Arbitrarily High Order and Asymptotic Preserving Kinetic Scheme in Compressible Fluid Dynamic

We present a class of arbitrarily high order fully explicit kinetic numerical methods in compressible fluid dynamics,both in time and space,which include the relaxation schemes by Jin and Xin.These methods can use the CFL number larger or equal to unity on regular Cartesian meshes for the multi-dimensional case.These kinetic models depend on a small parameter that can be seen as a"Knudsen"number.The method is asymptotic preserving in this Knudsen number.Also,the computational costs of the method are of the same order of a fully explicit scheme.This work is the extension of Abgrall et al.(2022)[3]to multidimensional systems.We have assessed our method on several problems for two-dimensional scalar problems and Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.

UNIFIED COMPUTATION OF FLOW WITH COMPRESSIBLE AND INCOMPRESSIBLE FLUID BASED ON ROE'S SCHEME

A unified numerical scheme for the solutions of the compressible and incompressible Navier-Stokes equations is investigated based on a time-derivative preconditioning algorithm. The primitive variables are pressure, velocities and temperature. The time integration scheme is used in conjunction with a finite volume discretization. The preconditioning is coupled with a high order implicit upwind scheme based on the definition of a Roe＇s type matrix. Computational capabilities are demonstrated through computations of high Mach number, middle Mach number, very low Mach number, and incompressible flow. It has also been demonstrated that the discontinuous surface in flow field can be captured for the implementation Roe＇s scheme.

GLOBAL EXISTENCE OF STRONG SOLUTIONS OF NAVIER-STOKES EQUATIONS WITH NON-NEWTONIAN POTENTIAL FOR ONE-DIMENSIONAL ISENTROPIC COMPRESSIBLE FLUIDS

The aims of this paper are to discuss global existence and uniqueness of strong solution for a class of isentropic compressible navier-Stokes equations with non-Newtonian in one-dimensional bounded intervals. We prove two global existence results on strong solutions of isentropic compressible Navier-Stokes equations. The first result shows only the existence. And the second one shows the existence and uniqueness result based on the first result, but the uniqueness requires some compatibility condition.

Preliminary results of the spilling effects on the oscillation of a pipe with compressible fluid inside

It is known to all, the spilling of pipeline may cause serious problems, especially when the pipe conveying petroleum, natural gas or other toxic substance. There are countless accidents during past century. Once the spilling occurs, the vibration of the pipe would aggravate spill situation and even result in crack of the pipe. The consequence will be more severe when the fluid inside is compressible. To prevent the detriment of the spilling model is developed by assuming the leakages as orifices or nozzles and a 2-D vertical simply supported pipe is selected to analyze the phenomena of the oscillation. Combining these two models, the oscillation model for the pipe with leakage is set up and the spilling effect is analyzed by numerical method. The amplitude of the pipe oscillation and the normal stress enlarge as the internal velocity increased, while the shear stress changes very little.

A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations

Lagrangianmethods arewidely used inmany fields formulti-material compressible flow simulations such as in astrophysics and inertial confinement fusion(ICF),due to their distinguished advantage in capturing material interfaces automatically.In some of these applications,multiple internal energy equations such as those for electron,ion and radiation are involved.In the past decades,several staggeredgrid based Lagrangian schemes have been developed which are designed to solve the internal energy equation directly.These schemes can be easily extended to solve problems with multiple internal energy equations.However such schemes are typically not conservative for the total energy.Recently,significant progress has been made in developing cell-centered Lagrangian schemes which have several good properties such as conservation for all the conserved variables and easiness for remapping.However,these schemes are commonly designed to solve the Euler equations in the form of the total energy,therefore they cannot be directly applied to the solution of either the single internal energy equation or the multiple internal energy equations without significant modifications.Such modifications,if not designed carefully,may lead to the loss of some of the nice properties of the original schemes such as conservation of the total energy.In this paper,we establish an equivalency relationship between the cell-centered discretizations of the Euler equations in the forms of the total energy and of the internal energy.By a carefully designed modification in the implementation,the cell-centered Lagrangian scheme can be used to solve the compressible fluid flow with one or multiple internal energy equations and meanwhile it does not lose its total energy conservation property.An advantage of this approach is that it can be easily applied to many existing large application codes which are based on the framework of solving multiple internal energy equations.Several two dimensional numerical examples for both Euler equations and three-temperature hydrodynamic equations in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry preserving,accuracy and non-oscillatory performance.

Domain Decomposition Preconditioners for Discontinuous Galerkin Discretizations of Compressible Fluid Flows

In this article we consider the application of Schwarz-type domain decomposition preconditioners to the discontinuous Galerkin finite element approximation of the compressible Navier-Stokes equations.To discretize this system of conservation laws,we exploit the(adjoint consistent)symmetric version of the interior penalty discontinuous Galerkin finite element method.To define the necessary coarse-level solver required for the definition of the proposed preconditioner,we exploit ideas from composite finite element methods,which allow for the definition of finite element schemes on general meshes consisting of polygonal(agglomerated)elements.The practical performance of the proposed preconditioner is demonstrated for a series of viscous test cases in both two-and three-dimensions.

Blow-up of Smooth Solutions to the Compressible Fluid Models of Korteweg Type

We show the blow-up of smooth solutions to a non-isothermal model of capillary compressible fluids in arbitrary space dimensions with initial density of compact support. This is an extension of Xin＇s result [Xin, Z.： Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density. Comm. Pure Appl. Math., 51, 229-240 （1998）] to the capillary case but we do not need the condition that the entropy is bounded below. Moreover, from the proof of Theorem 1.2, we also obtain the exact relationship between the size of support of the initial density and the life span of the solutions. We also present a sufficient condition on the blow-up of smooth solutions to the compressible fluid models of Korteweg type when the initial density is positive but has a decay at infinity.

SPECTRAL-DIFFERENCE METHOD FOR TWO-DIMENSIONAL COMPRESSIBLE FLUID FLOW

We develop a combined Fourier spectral-finite difference method for solving 2-dimensional, semi-periodic compressible fluid how problem. The error estimation, as well as the convergence rate, is presented.

A Functional Equation Governing the Motion of a Compressible Fluid Flow

The fundamental problem of the statistical dynamics of a turbulent flow, formulated in terms of characteristic functionals, has already been pointed out in the work of E. Hopf. In his work he deduced a functional equation governing the evolution of the characteristic functional of a turbulent velocity field in an incompressible field. In this paper we present a derivation of a dynamical equation governing the evolution of the characteristic functional of a turbulent velocity field in a compressible field. However, the characteristic functional equations we derived are governing the motions of an ideal gas and van der Waals gas.

A Geometry-Preserving Finite Volume Method for Compressible Fluids on Schwarzschild Spacetime

We consider the relativistic Euler equations governing spherically symmetric,perfect fluid flows on the outer domain of communication of Schwarzschild spacetime,and we introduce a version of the finite volume method which is formulated from the geometric formulation(and thus takes the geometry into account at the discretization level)and is well-balanced,in the sense that it preserves steady solutions to the Euler equations on the curved geometry under consideration.In order to formulate our method,we first derive a closed formula describing all steady and spherically symmetric solutions to the Euler equations posed on Schwarzschild spacetime.Second,we describe a geometry-preserving,finite volume method which is based from the family of steady solutions to the Euler system.Our scheme is second-order accurate and,as required,preserves the family of steady solutions at the discrete level.Numerical experiments are presented which demonstrate the efficiency and robustness of the proposed method even for solutions containing shock waves and nonlinear interacting wave patterns.As an application,we investigate the late-time asymptotics of perturbed steady solutions and demonstrate its convergence for late time toward another steady solution,taking the overall effect of the perturbation into account.

ASecond-Order Finite-Difference Method for Compressible Fluids in Domains with Moving Boundaries

In this paper,we describe how to construct a finite-difference shockcapturing method for the numerical solution of the Euler equation of gas dynamics on arbitrary two-dimensional domainΩ,possibly with moving boundary.The boundaries of the domain are assumed to be changing due to the movement of solid objects/obstacles/walls.Although the motion of the boundary could be coupled with the fluid,all of the numerical tests are performed assuming that such a motion is prescribed and independent of the fluid flow.The method is based on discretizing the equation on a regular Cartesian grid in a rectangular domainΩ_(R)⊃Ω.Ωe identify inner and ghost points.The inner points are the grid points located insideΩ,while the ghost points are the grid points that are outsideΩbut have at least one neighbor insideΩ.The evolution equations for inner points data are obtained from the discretization of the governing equation,while the data at the ghost points are obtained by a suitable extrapolation of the primitive variables(density,velocities and pressure).Particular care is devoted to a proper description of the boundary conditions for both fixed and time dependent domains.Several numerical experiments are conducted to illustrate the validity of themethod.Ωe demonstrate that the second order of accuracy is numerically assessed on genuinely two-dimensional problems.

ON THE EXISTENCE OF LOCAL CLASSICAL SOLUTION FOR A CLASS OF ONE-DIMENSIONAL COMPRESSIBLE NON-NEWTONIAN FLUIDS

In this paper, the aim is to establish the local existence of classical solutions for a class of compressible non-Newtonian fluids with vacuum in one-dimensional bounded intervals, under the assumption that the data satisfies a natural compatibility condition. For the results, the initial density does not need to be bounded below away from zero.

Vibration of a Two-Layer“Metal+PZT”Plate Contacting with Viscous Fluid

The present work investigates the mechanically forced vibration of the hydro-elasto-piezoelectric system consisting of a two-layer plate“elastic+PZT”,a compressible viscous fluid,and a rigid wall.It is assumed that the PZT(piezoelectric)layer of the plate is in contact with the fluid and time-harmonic linear forces act on the free surface of the elastic-metallic layer.This study is valuable because it considers for the first time the mechanical vibration of the metal+piezoelectric bilayer plate in contact with a fluid.It is also the first time that the influence of the volumetric concentration of the constituents on the vibration of the hydro-elasto-piezoelectric system is studied.Another value of the present work is the use of the exact equations and relations of elasto-electrodynamics for elastic and piezoelectric materials to describe the motion of the plate layers within the framework of the piecewise homogeneous body model and the use of the linearized Navier-Stokes equations to describe the flow of the compressible viscous fluid.The plane-strain state in the plate and the plane flow in the fluid take place.For the solution of the corresponding boundary-value problem,the Fourier transform is used with respect to the spatial coordinate on the axis along the laying direction of the plate.The analytical expressions of the Fourier transform of all the sought values of each component of the system are determined.The origins of the searched values are determined numerically,after which numerical results on the stress on the fluid and plate interface planes are presented and discussed.These results are obtained for the case where PZT-2 is chosen as the piezoelectric material,steel and aluminum as the elastic metal materials,and Glycerin as the fluid.Analysis of these results allows conclusions to be drawn about the character of the problem parameters on the frequency response of the interfacial stress.In particular,it was found that after a certain value of the vibration frequency,the presence of the metal layer in the two-layer plate led to an increase in the absolute values of the above interfacial stress.

ANALYSIS AND APPLICATION OF ELLIPTICITY OF STABILITY EQUATIONS ON FLUID MECHANICS

By using characteristic analysis of the linear and nonlinear parabolic stability equations ( PSE), PSE of primitive disturbance variables are proved to be parabolic intotal. By using sub- characteristic analysis of PSE, the linear PSE are proved to be elliptical and hyperbolic-parabolic for velocity U, in subsonic and supersonic, respectively; the nonlinear PSE are proved to be elliptical and hyperbolic-parabolic for relocity U + u in subsonic and supersonic., respectively . The methods are gained that the remained ellipticity is removed from the PSE by characteristic and sub-characteristic theories , the results for the linear PSE are consistent with the known results, and the influence of the Mach number is also given out. At the same time , the methods of removing the remained ellipticity are further obtained from the nonlinear PSE .

WEAK TIME-PERIODIC SOLUTIONS TO THE COMPRESSIBLE NAVIER-STOKES EQUATIONS

The compressible Navier-Stokes equations driven by a time-periodic external force are considered in this article. We establish the existence of weak time-periodic solutions and improve the result from [3] in the following sense： we extend the class of pressure functions, that is, we consider lower exponent γ.

The Asymmetric Collapse of Bubbles in Compressible Liquid

The analytical solution of a bubble collapse close to a solid boundary in a compressible water is investigated by means of a perturbation method to first order in the bubble wall Mach number. It is shown, in this paper, that it is the Rayleigh?Plesset equation for incompressible liquid to zero order solution or similar to the Gilmore equation for compressible water to first order solution when the effect of solid boundary is negligibly small enough, i.e., sufficiently far away from the bubble center.

FORMATION OF SINGULARITY FOR COMPRESSIBLE VISCOELASTICITY

The formation of singularity and breakdown of classical solutions to the three- dimensional compressible viscoelasticity and inviscid elasticity are considered. For the compressible inviscid elastic fluids, the finite-time formation of singularity in classical solu- tions is proved for certain initial data. For the compressible viscoelastic fluids, a criterion in term of the temporal integral of the velocity gradient is obtained for the breakdown of smooth solutions.

Numerical investigation of non-uniform temperature fields for proppant and fluid phases in supercritical CO_(2)fracturing

The non-uniform temperature distribution in supercritical CO_(2)(Sc-CO_(2))fracturing influences the density,viscosity,and volume expansion or shrinkage rate of Sc-CO_(2),impacting proppant migration.This study presents a coupled computational fluid dynamics-discrete element method and heat transfer model to examine the effects of proppant bed shape and the heat transfers of proppant-wall,proppant-fluid,and fluid-wall on the fluid and proppant temperature fields.The Sc-CO_(2)volume expansion is assessed under various temperature conditions by evaluating the volume-averaged Sc-CO_(2)density.Several factors,including proppant size,shape,thermal conductivity,concentration,temperature difference,and injection velocity,are carefully analyzed to elucidate their impacts.The findings elucidate the existence of four distinct zones in the fluid temperature field.Each zone exhibits different magnitudes of temperature change under diverse conditions and undergoes dynamic transformations with the development of the proppant bed.The fluid-wall heat transfer and the fluid temperatures in Zones C and D are significantly subject to the fluid injection velocity(governing the heating duration),the temperature difference between fluid and formation(impacting the magnitude of heat flux),and the proppant bed shape(controlling the effective heating area).Additionally,the proppant-wall and proppant-fluid heat transfers determine the temperatures of both the proppant bed and the fluid within Zone B,showing a strong correlation with proppant thermal conductivity,proppant size,injection velocity,and temperature difference.The proposed coupled model provides valuable insights into the temperature distributions and flow behaviors of temperature-dependent fracturing fluids and proppants.

Subsea Compensation of Pressure Based on Reducer Bellows

In this study,the pressure compensation mechanism of a reducer bellows is analyzed.This device is typically used to reduce the size of undersea instruments and improve related pressure resistance and sealing capabilities.Here,its axial stiffness is studied through a multi-fold approach based on theory,simulations and experiments.The results indicate that the mechanical strength of the reducer bellows,together with the oil volume and temperature are the main factors influencing its performances.In particular,the wall thickness,wave number,middle distance,and wave height are the most influential parameters.For a certain type of reducer bellows,the compensation capacity attains a maximum when the wave number ratio is between 6:6 and 8:4,the wall thickness is 0.3 mm,and the wave height is between 4–5 mm and 5–6 mm.Moreover,the maximum allowable ambient pres-sure of the optimized reducer bellows can reach 62.6 MPa without failure,and the maximum working water depth is 6284 m.

Analysis of natural vibration of truncated conical shells partially filled with fluid

In this paper,we study the vibrational behavior of shells in the form of truncated cones containing an ideal compressible fluid.The sloshing effect on the free surface of the fluid is neglected.The dynamic behavior of the elastic structure is investigated based on the classical shell theory,the constitutive relations of which represent a system of ordinary differential equations written for new unknowns.Small fluid vibrations are described in terms of acoustic approximation using the wave equation for hydrodynamic pressure written in spherical coordinates.Its transformation into the system of ordinary differential equations is carried out by applying the generalized differential quadrature method.The formulated boundary value problem is solved by Godunov's orthogonal sweep method.Natural frequencies of shell vibrations are calculated using the stepwise procedure and the Muller method.The accuracy and reliability of the obtained results are estimated by making a comparison with the known numerical and analytical solutions.The dependencies of the lowest frequency on the fluid level and cone angle of shells under different combinations of boundary conditions(simply supported,rigidly clamped,and cantilevered shells)have been studied comprehensively.For conical straight and inverted shells,a numerical analysis has been performed to estimate the possibility of finding configurations at which the lowest natural frequencies exceed the corresponding values of the equivalent cylindrical shell.