Compressible Hydrodynamic Equations Research Articles

Piston driven shock waves in non-homogeneous planar media

In this work, we analyze in detail the problem of piston driven shock waves in planar media. Similarity solutions to the compressible hydrodynamics equations are developed, for a strong shock wave, generated by a time dependent pressure piston, propagating in a non-homogeneous planar medium consisting of an ideal gas. Power law temporal and spatial dependency is assumed for the piston pressure and initial medium density, respectively. The similarity solutions are written in both Lagrangian and Eulerian coordinates. It is shown that the solutions take various qualitatively different forms according to the value of the pressure and density exponents. We show that there exist different families of solutions, for which the shock propagates at a constant speed, accelerates, or slows down. Similarly, we show that there exist different types of solutions, for which the density near the piston is either finite, vanishes, or diverges. Finally, we perform a comprehensive comparison between the planar shock solutions and Lagrangian hydrodynamic simulations, by setting proper initial and boundary conditions. A very good agreement is reached, which demonstrates the usefulness of the analytic solutions as a code verification test problem.

Modelling multiphase gases in cosmological simulations using compressible multifluid hydrodynamics

ABSTRACT The diffuse medium in and around galaxies can exist in a multiphase state: small, cold gas clouds contributing significantly to the total mass embedded in pressure equilibrium with a hotter, more diffuse volume-filling component. Modelling this multiphase state in cosmological simulations poses a significant challenge due to the requirements to spatially resolve the clouds and consequently the interactions between the phases. In this paper, we present a novel method to model this gas state in cosmological hydrodynamical simulations. We solve the compressible two-fluid hydrodynamic equations using a moving-mesh finite-volume method and define mass, momentum, and energy exchange terms between the phases as operator-split source terms. Using a stratified flow model, our implementation is able to maintain volume fraction discontinuities in pressure equilibrium to machine precision, allowing for the treatment of both resolved and unresolved multiphase fluids. The solver remains second order accurate on smooth hydrodynamics problems. We use the source and sink terms of an existing two-phase model for the interstellar medium to demonstrate the value of this type of approach in simulations of galaxy formation, compare it to its effective equation of state implementation, and discuss its advantages in future large-scale simulations of galaxy formation.

On regular solutions to compressible radiation hydrodynamic equations with far field vacuum

Abstract The Cauchy problem for three-dimensional (3D) isentropic compressible radiation hydrodynamic equations is considered. When both shear and bulk viscosity coefficients depend on the mass density ρ \rho in a power law ρ δ {\rho }^{\delta } (with 0 < δ < 1 0\lt \delta \lt 1 ), based on some elaborate analysis of this system’s intrinsic singular structures, we establish the local-in-time well-posedness of regular solution with arbitrarily large initial data and far field vacuum in some inhomogeneous Sobolev spaces by introducing some new variables and initial compatibility conditions. Note that due to the appearance of the vacuum, the momentum equations are degenerate both in the time evolution and viscous stress tensor, which, along with the strong coupling between the fluid and the radiation field, make the study on corresponding well-posedness challenging. For proving the existence, we first introduce an enlarged reformulated structure by considering some new variables, which can transfer the degeneracies of the radiation hydrodynamic equations to the possible singularities of some special source terms, and then carry out some singularly weighted energy estimates carefully designed for this reformulated system.

Application of Two-dimensional Finite Volume Method to Protoplanetary Disks

Many fascinating astrophysical phenomena can be simulated insufficiently by standard numerical schemes for the compressible hydrodynamics equations. In the present work, a high performant 2D hydrodynamical code has been developed. The model is designed for the planetary formation that consists of momentum, continuity and energy equations. Since the two-phase model seems to be hardly executed, we will show in a simplified form, the implementation of this model in one-phase. It is applied to the Solar System that such stars can form planets. The finite volume method (FVM) is used in this model. We aim to develop a first-order well-balanced scheme for the Euler equations in the the radial direction, combined with second-order centered ux following the radial direction. This conception is devoted to balance the uxes, and guarantee hydrostatic equilibrium preserving. Then the model is used on simplified examples in order to show its ca- pability to maintain steady-state solutions with a good precision. Additionally, we demonstrate the performance of the numerical code through simulations. In particularly, the time evolution of gas orbited around the star, and some proper- ties of the Rossby wave instability are analyzed. The resulting scheme shows consequently that this model is robust and simple enough to be easily implemented.

Energy-consistent finite difference schemes for compressible hydrodynamics and magnetohydrodynamics using nonlinear filtering

In this paper, an energy-consistent finite difference formulation for the compressible hydrodynamic and magnetohydrodynamic (MHD) equations is introduced. For the compressible magnetohydrodynamics, an energy-consistent finite difference formulation is derived using the product rule for the spatial difference. The conservation properties of the internal, kinetic, and magnetic energy equations can be satisfied in the discrete level without explicitly solving the total energy equation. The shock waves and discontinuities in the numerical solution are stabilized by nonlinear filtering schemes. An energy-consistent discretization of the filtering schemes is also derived by introducing the viscous and resistive heating rates. The resulting energy-consistent formulation can be implemented with the various kinds of central difference, nonlinear filtering, and time integration schemes. The second- and fifth-order schemes are implemented based on the proposed formulation. The conservation properties and the robustness of the present schemes are demonstrated via one- and two-dimensional numerical tests. The proposed schemes successfully handle the most stringent problems in extremely high Mach number and low beta conditions.

Local well‐posedness of compressible radiation hydrodynamic equations with density‐dependent viscosities and vacuum

In this paper, we consider the Cauchy problem for three‐dimensional isentropic compressible radiation hydrodynamic equations with density‐dependent viscosity coefficients. When the viscosity coefficients are given as power of density (ρδ with δ > 1), we establish the local‐in‐time existence of classical solutions containing a vacuum for large initial data. Here, we point out that the initial layer compatibility conditions are not necessary.

Time periodic solutions for the non-isentropic compressible quantum hydrodynamic equations with viscosity in R^3

This article concerns the existence and uniqueness of a time periodic solution for the non-isentropic quantum hydrodynamic equations with viscosity. By applying the Leray-Schauder theory, subtle energy estimates and a limiting method, we obtain the existence of time periodic solutions under some smallness assumptions on the time periodic external force in \(\mathbb{R}^3\). The uniqueness can be proved by similar energy estimates. In particular, the quantum effects and the energy equation are taken into account in this paper which play a significant role in the uniform (in the domain R and the positive constant \(\epsilon\)) estimates, especially in the selection of the norm.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/103/abstr.html

Local existence and Serrin-type blow-up criterion for strong solutions to the radiation hydrodynamic equations

We consider the Cauchy problem for the three-dimensional, compressible radiation hydrodynamic equations. We establish the existence and uniqueness of local strong solutions for large initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Moreover, we establish a Serrin-type blow-up criterion, which is stated in terms of the velocity and density variables [Formula: see text] and is independent of the temperature and the radiation intensity.

Group-invariant solutions for one dimensional, inviscid hydrodynamics

In this paper, the results of the Lie group method carried out by Ovsiannikov are utilized to study the one-dimensional hydrodynamic equations governing compressible, inviscid fluid flow in the absence of heat conduction. One-parameter subgroups of the admissible R-parameter Lie group of point transformations of the system are applied to reduce the first-order, non-linear system of partial differential equations (PDE)s to a first-order system of ordinary differential equations (ODE)s. Closed-form solutions to the reduced ODE systems are subsequently determined using a linear velocity profile ansatz. These solutions are valid in one-dimensional (1D) planar, cylindrical and spherical geometries and are connected to solutions of the governing system of PDEs through an inverse map. The linear velocity type solutions were first considered by Sedov in 1953 and constitute a subclass of all possible similarity solutions of the compressible hydrodynamics equations. They further serve as illustrative examples of using solutions obtained for the reduced ODEs to find solutions of the associated system of PDEs. Consequently, the reduced systems of ODEs are provided in their entirety inviting additional solutions to be determined via alternative explicit or numerical solution techniques.

Blow-up criteria for three-dimensional compressible radiation hydrodynamics equations with vacuum

We consider the blow-up criteria for the Cauchy problems of three-dimensional compressible radiation fluids with vacuum. It is shown to own the same BKM-type criterion as the compressible Navier–Stokes equations [14], while the Lp˜(p˜∈[2,3]) norm of the density gradient should be involved for the Serrin-type criterion.

Global weak solutions to one-dimensional compressible viscous hydrodynamic equations

In this article, we are concerned with the global weak solutions to the 1D compressible viscous hydrodynamic equations with dispersion correction δ2 ρ((φ(ρ))xxφ′(ρ)x with φ(ρ)=ρα. The model consists of viscous stabilizations because of quantum Fokker-Planck operator in the Wigner equation and is supplemented with periodic boundary and initial conditions. The diffusion term ɛuxx in the momentum equation may be interpreted as a classical conservative friction term because of particle interactions. We extend the existence result in [1](α=12)to 0<α≤1. In addition, we perform the limit ɛ→ 0 with respect to 0 < α ≤ ½.

Critical role of blockage ratio for flame acceleration in channels with tightly spaced obstacles

A conceptually laminar mechanism of extremely fast flame acceleration in obstructed channels, identified by Bychkov et al. [“Physical mechanism of ultrafast flame acceleration,” Phys. Rev. Lett. 101, 164501 (2008)], is further studied by means of analytical endeavors and computational simulations of compressible hydrodynamic and combustion equations. Specifically, it is shown how the obstacles length, distance between the obstacles, channel width, and thermal boundary conditions at the walls modify flame propagation through a comb-shaped array of parallel thin obstacles. Adiabatic and isothermal (cold and preheated) side walls are considered, obtaining minor difference between these cases, which opposes the unobstructed channel case, where adiabatic and isothermal walls provide qualitatively different regimes of flame propagation. Variations of the obstructed channel width also provide a minor influence on flame propagation, justifying a scale-invariant nature of this acceleration mechanism. In contrast, the spacing between obstacles has a significant role, although it is weaker than that of the blockage ratio (defined as the fraction of the channel blocked by obstacles), which is the key parameter of the problem. Evolution of the burning velocity and the dependence of the flame acceleration rate on the blockage ratio are quantified. The critical blockage ratio, providing the limitations for the acceleration mechanism in channels with comb-shaped obstacles array, is found analytically and numerically, with good agreement between both approaches. Additionally, this comb-shaped obstacles-driven acceleration is compared to finger flame acceleration and to that produced by wall friction.

A well-balanced finite volume scheme for the Euler equations with gravitation

Context. Many problems in astrophysics feature flows which are close to hydrostatic equilibrium. However, standard numerical schemes for compressible hydrodynamics may be deficient in approximating this stationary state, where the pressure gradient is nearly balanced by gravitational forces.Aims. We aim to develop a second-order well-balanced scheme for the Euler equations. The scheme is designed to mimic a discrete version of the hydrostatic balance. It therefore can resolve a discrete hydrostatic equilibrium exactly (up to machine precision) and propagate perturbations, on top of this equilibrium, very accurately.Methods. A local second-order hydrostatic equilibrium preserving pressure reconstruction is developed. Combined with a standard central gravitational source term discretization and numerical fluxes that resolve stationary contact discontinuities exactly, the well-balanced property is achieved.Results. The resulting well-balanced scheme is robust and simple enough to be very easily implemented within any existing computer code that solves time explicitly or implicitly the compressible hydrodynamics equations. We demonstrate the performance of the well-balanced scheme for several astrophysically relevant applications: wave propagation in stellar atmospheres, a toy model for core-collapse supernovae, convection in carbon shell burning, and a realistic proto-neutron star.

A semi-implicit spectral method for compressible convection of rotating and density-stratified flows in Cartesian geometry

In this paper, we have described a 'stratified' semi-implicit spectral method to study compressible convection in Cartesian geometry. The full set of compressible hydrodynamic equations are solved in conservative forms. The numerical scheme is accurate and efficient, based on fast Fourier/sin/cos spectral transforms in the horizontal directions, Chebyshev spectral transform or second-order finite difference scheme in the vertical direction, and second order semi-implicit scheme in time marching of linear terms. We have checked the validity of both the fully pseudo-spectral scheme and the mixed finite-difference pseudo-spectral scheme by studying the onset of compressible convection. The difference of the critical Rayleigh number between our numerical result and the linear stability analysis is within two percent. Besides, we have computed the Mach numbers with different Rayleigh numbers in compressible convection. It shows good agreement with the numerical results of finite difference methods and finite volume method. This model has wide application in studying laminar and turbulent flow. Illustrative examples of application on horizontal convection, gravity waves, and long-lived vortex are given in this paper.

Existence Results and Blow-Up Criterion of Compressible Radiation Hydrodynamic Equations

In this paper, we consider the 3D compressible radiation hydrodynamic equations with thermal conductivity in a bounded domain. The existence of unique local strong solutions with vacuum is firstly established when the initial data are arbitrarily large and satisfy some initial layer compatibility condition. Moreover, we show that if the initial vacuum domain is not so irregular, then the compatibility condition is necessary and sufficient to guarantee the existence of the unique strong solution. Finally, a Beale–Kato–Majda type blow-up criterion is shown in terms of $$(\nabla I,\rho ,\theta )$$ .

Existence results for compressible radiation hydrodynamic equations with vacuum

In this paper, we consider the three-dimensional compressible isentropic radiation hydrodynamic (RHD) equations. The existence of unique local strong solutions is firstly proved when the initial data are arbitrarily large, contain vacuum and satisfy some initial layer compatibility condition. The initial mass density does not need to be bounded away from zero and may vanish in some open set. We also prove that if the initial vacuum is not so irregular, then the initial layer compatibility condition is necessary and sufficient to guarantee the existence of a unique strong solution. Finally, we establish a blow-up criterion for the strong solution that we obtained. The similar results also hold for the barotropic flow with general pressure law $p_m=p_m(\rho)\in C^1(\mathbb{\overline{R}}^+)$.

Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum

We study the Cauchy problem for multi-dimensional compressible radiation hydrodynamics equations with vacuum. First, we present some sufficient conditions on the blow-up of smooth solutions in multi-dimensional space. Then, we obtain the invariance of the support of density for the smooth solutions with compactly supported initial mass density by the property of the system under the vacuum state. Based on the above-mentioned results, we prove that we cannot get a global classical solution, no matter how small the initial data are, as long as the initial mass density is of compact support. Finally, we will see that some of the results that we obtained are still valid for the isentropic flows with degenerate viscosity coefficients as well as for one-dimensional case.

Three-dimensional numerical simulation of convection in giant planets: Effects of solid core size

By solving the compressible hydrodynamic equations, the effects of solid core size on planetary winds are studied with a three-dimensional spherical spectral method. Four cases with solid core sizes χ (core radius ratio)=0, 0.1, 0.35 and 0.75 are discussed. A prograde equatorial jet and two adjacent retrograde jets are prominently produced in each model. The surface speed of the prograde equatorial jet increases as the solid core size increases from χ=0 to 0.35, then drops as χ further increases to 0.75. Alternating zonal wind bands in the mid latitudes are produced only in the case where the convection zone is most shallow. The velocity fields are significantly controlled by the Taylor–Proudman theorem (in the sense that the mean zonal velocities do not change much along the angular velocity direction). Three-dimensional effects are essential for the dynamics, as both the meridional-zonal and radial-zonal components of the Reynolds stress play key roles in driving the equatorial jet. The solid core affects the flows through changing the spatial geometry inside the tangent cylinder. As a result, the tangent cylinder controls the width of the prograde equatorial jet when the core becomes sufficiently large.

Numerical calculation of convection with reduced speed of sound technique

Context. The anelastic approximation is often adopted in numerical calculation with low Mach number, such as stellar internal convection. This approximation requires frequent global communication, because of an elliptic partial differential equation. Frequent global communication is negative factor for the parallel computing with a large number of CPUs. Aims. The main purpose of this paper is to test the validity of a method that artificially reduces the speed of sound for the compressible fluid equations in the context of stellar internal convection. The reduction of speed of sound allows for larger time steps in spite of low Mach number, while the numerical scheme remains fully explicit and the mathematical system is hyperbolic and thus does not require frequent global communication. Methods. Two and three dimensional compressible hydrodynamic equations are solved numerically. Some statistical quantities of solutions computed with different effective Mach numbers (due to reduction of speed of sound) are compared to test the validity of our approach. Results. Numerical simulations with artificially reduced speed of sound are a valid approach as long as the effective Mach number (based on the reduced speed of sound) remains less than 0.7.

High-Order Curvilinear Finite Element Methods for Lagrangian Hydrodynamics

The numerical approximation of the Euler equations of gas dynamics in a movingLagrangian frame is at the heart of many multiphysics simulation algorithms. In this paper, we present a general framework for high-order Lagrangian discretization of these compressible shock hydrodynamics equations using curvilinear finite elements. This method is an extension of the approach outlined in [Dobrev et al., Internat. J. Numer. Methods Fluids, 65 (2010), pp. 1295--1310] and can be formulated for any finite dimensional approximation of the kinematic and thermodynamic fields, including generic finite elements on two- and three-dimensional meshes with triangular, quadrilateral, tetrahedral, or hexahedral zones. We discretize the kinematic variables of position and velocity using a continuous high-order basis function expansion of arbitrary polynomial degree which is obtained via a corresponding high-order parametric mapping from a standard reference element. This enables the use of curvilinear zone geometry, higher-order approximations for fields within a zone, and a pointwise definition of mass conservation which we refer to as strong mass conservation. We discretize the internal energy using a piecewise discontinuous high-order basis function expansion which is also of arbitrary polynomial degree. This facilitates multimaterial hydrodynamics by treating material properties, such as equations of state and constitutive models, as piecewise discontinuous functions which vary within a zone. To satisfy the Rankine--Hugoniot jump conditions at a shock boundary and generate the appropriate entropy, we introduce a general tensor artificial viscosity which takes advantage of the high-order kinematic and thermodynamic information available in each zone. Finally, we apply a generic high-order time discretization process to the semidiscrete equations to develop the fully discrete numerical algorithm. Our method can be viewed as the high-order generalization of the so-called staggered-grid hydrodynamics (SGH) approach and we show that under specific low-order assumptions, we exactly recover the classical SGH method. We present numerical results from an extensive series of verification tests that demonstrate several important practical advantages of using high-order finite elements in this context.