Equations Of Compressible Fluid Flow Research Articles

The limiting behavior of Riemann solutions to the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force

In this paper, we investigate the limiting behavior of Riemann solutions to the Euler equations of compressible fluid flow for modified Chaplygin gas with the body force as the two parameters tend to zero. The formation of delta shock waves and the vacuum states is identified and analyzed during the process of vanishing pressure in the Riemann solutions. The concentration and cavitation are fundamental and physical phenomena in fluid dynamics, which can be mathematically described by delta shock waves and vacuums, respectively. In this paper, our main objective is to rigorously investigate the formation of delta shock waves and vacuums and observe the concentration and cavitation phenomena. First, the Riemann problem of the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force is solved. Second, we rigorously confirm that, as the pressure vanishes, any two shock Riemann solution to the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force tends to a δ-shock solution to the pressureless gas dynamics model with a body force, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the δ-shock; any two-rarefaction-wave Riemann solution to the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force tends to a solution consisting of four contact discontinuities together with vacuum states with three different virtual velocities in the limiting situation.

Flux approximation to the Euler equations of compressible fluid flow for extended Chaplygin gas

The concentration and cavitation are fundamental and physical phenomena in fluid dynamics, which can be mathematically described by delta shock waves and vacuums, respectively. In this paper, we are concerned with the Euler equations of compressible fluid flow when the state equation is governed by the extended Chaplygin gas, an important candidate for describing dark matter and dark energy. Our main objective is to apply the flux-approximation method to rigorously investigate the formation of delta shock waves and vacuums and observe the concentration and cavitation phenomena. First, the Riemann problem of the compressible fluid flow with n + 2 parameters including flux and pressure is solved. Then, two kinds of flux approximation are discussed, that is, both the flux perturbation and pressure tend to zero, or only the flux perturbation vanishes while the extended Chaplygin gas pressure partly tends to the Chaplygin gas pressure. The results indicate that different manners of flux approximation have their respective effects on the formation of delta shock waves. Finally, several numerical results are presented to confirm the theoretical analysis.

Piston problem for the generalized Chaplygin Euler equations of compressible fluid flow

We study the piston problem for the compressible fluid flow of generalized Chaplygin gas, for both cases that the piston rushes into or recedes from the uniform still gas at a constant speed. The global existence of a solution to the piston problem is established. For the proceeding piston problem, we find that the front shock can be detached or adhered to the surface of the piston depending on the range of Mach numbers. Then, a Radon measure solution, with density containing a Dirac measure supported on the piston is rigorously defined and proved to be a reasonable solution to deal with the concentration of mass on the piston. When the piston recedes from the gas, we show strictly that only a first-family rarefaction wave exists in front of the piston and that concentration will never occur. Moreover, we also analyzed the occurrence of the vacuum state and the degeneration of the compressible fluid flow in the receding case as the Mach number tends to infinity.

The limit behavior of Riemann solutions to the Euler equations of compressible fluid flow for the modified Chaplygin gas

In this paper, we study the formation of delta shock waves and vacuum states in a pressureless limit of solutions to the Euler equations of compressible fluid flow for modified Chaplygin gas as two parameters tend to zero. At first, the Riemann problem of Euler equations of compressible fluid flow for modified Chaplygin gas is solved. Then, as the pressure drops to zero, we strictly confirm that the two-shock Riemann solution converges to a delta shock wave solution and the two-rarefaction wave Riemann solution converges to a two-contact discontinuity solution with the intermediate state approaching a vacuum state. At last, some numerical simulations are given to verify the above analysis.

The limits of Riemann solutions to Euler equations of compressible fluid flow with a source term

In this paper, we investigate the limits of Riemann solutions to the Euler equations of compressible fluid flow with a source term as the adiabatic exponent tends to one. The source term can represent friction or gravity or both in Engineering. For instance, a concrete physical model is a model of gas dynamics in a gravitational field with entropy assumed to be a constant. The body force source term is presented if there is some external force acting on the fluid. The force assumed here is the gravity. Different from the homogeneous equations, the Riemann solutions of the inhomogeneous system are non self-similar. We rigorously proved that, as the adiabatic exponent tends to one, any two-shock Riemann solution tends to a delta shock solution of the pressureless Euler system with a Coulomb-like friction term, and the intermediate density between the two shocks tends to a weighted $$\delta $$ -mesaure which forms the delta shock; while any two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution of the pressureless Euler system with a Coulomb-like friction term, whose intermediate state between the two contact discontinuities is a vacuum state. Moreover, we also give some numerical simulations to confirm the theoretical analysis.

Mitigation of aerodynamic sound for a laminar flow past a square cylinder using a pair of cowl plates

A new arrangement of splitter plates has been proposed for the mitigation of aeroacoustic noise generated by the two-dimensional laminar flow over a square cylinder at the Reynolds number Re = 100 and the Mach number M = 0.2. The proposed arrangement involves a pair of cowl plates (arc-shaped splitter plates) symmetrically positioned on either side of the wake center-line near the rear corners of the square cylinder. Direct numerical simulations have been carried out to analyze the nature of flow and flow induced sound fields. Unsteady, two-dimensional, compressible fluid flow equations are solved using high-resolution, space–time accurate, dispersion relation preserving schemes. Simulations have been performed for various radial locations of the cowl plates. It is observed that the maximum reduction in sound pressure level of around 24 dB is possible using the proposed cylinder and cowl plate arrangement. Based on the observed directivity patterns, we have classified the sound fields into three different regions.

Limiting behavior of scaled general Euler equations of compressible fluid flow

The aim of this article is to study the limiting behavior of the solutions for the scaled generalized Euler equations of compressible fluid flow. When the initial data is of Riemann type, we showed the existence of solution which consists of shock waves and rarefaction waves and that the distributional limit of the solutions for this system converges to the solution of a non-strictly hyperbolic system, called one dimensional model for large scale structure formation of universe as the scaling parameter vanishes. An explicit entropy and entropy flux pair are also constructed for the particular flux function (Brio system) and it is shown that the solution constructed is entropy admissible. This is a continuation of our work[23].

Comment on the paper “Interaction of delta shock waves for the Chaplygin Euler equations of compressible fluid flow with split delta functions, Yu Zhang, Yanyan Zhang, Jinhuan Wang Mathematical Methods in the Applied Sciences , 2018; 41 :7678–7697”

Comment on the paper “Interaction of delta shock waves for the Chaplygin Euler equations of compressible fluid flow with split delta functions, Yu Zhang, Yanyan Zhang, Jinhuan Wang <i>Mathematical Methods in the Applied Sciences</i> , 2018; <b>41</b> :7678–7697”

Concentration and cavitation in the vanishing pressure limit of solutions to the generalized Chaplygin Euler equations of compressible fluid flow

Abstract The phenomena of concentration and cavitation and the formation of delta shock waves and vacuum states in vanishing pressure limits of solutions to the generalized Chaplygin Euler equations of compressible fluid flow are analyzed. It is proved that, as the pressure vanishes, any two-shock-wave Riemann solution of the generalized Chaplygin Euler equations of compressible fluid flow tends to a delta-shock solution to the transport equations, and the intermediate density between them tends to a weighted δ -measure that forms a delta shock wave; any two-rarefaction-wave solution is shown to tend to two contact discontinuities connecting the constant states and vacuum states, which form a vacuum solution of the transport equations. Moreover, some numerical simulations completely coinciding with the theoretical analysis are presented.

Interaction of delta shock waves for the Chaplygin Euler equations of compressible fluid flow with split delta functions

We investigate the interaction of delta shock waves and contact discontinuities for the Chaplygin Euler equations of compressible fluid flow with split delta functions. The perturbed Riemann problem when initial data are three piecewise constant states is constructively solved, and the global structure and large time‐asymptotic behaviors of solutions are discussed case by case via deriving how the solution continues beyond points of interaction. It is shown that the Riemann solutions are stable for such small perturbations with initial data by letting perturbed parameter ε tend to zero. Moreover, some numerical simulations completely coinciding with the theoretical analysis are also exhibited.

Stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow

We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than $1$, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.

Experimental Studies on the Effect of Suction Chamber Angle on the Entrainment of Passive Fluid in a Steam Ejector

In this experimental study, the impact of suction chamber angle (SCA) on the entrainment ratio of a steam ejector refrigeration system (ERS) of 700 W was investigated. The basic dimensions of the ejector were derived from the compressible fluid flow equations using matlab. The system was tested with six different SCAs with various operating conditions, and its performance was analyzed. It is inferred that the entrainment of passive fluid from the evaporator is the strong function of the SCA. For all the active steam pressures, the entrainment of the passive fluid increases up to 12 deg of SCA, and above that the performance decreases significantly. Optimum angle of suction chamber increases the entrainment ratio for at least 49.96%. It is also found that the SCA has a minor influence on the back pressure.

Viscous flow simulation of geological bodies

The results of the numerical simulation of geological processes that differ by ten orders of magnitude in flow rate are presented. In one case, it is the gold placer dynamics in a river valley. In another case, it is the rapid movement of the ice-debris mudflow. In both cases, the moving material was considered as an independent flow of solid particles free from host rocks. In order to mathematically describe such a solid flow, a system of viscous compressible fluid flow equations is used. The comparison of the results of the numerical experiments with field data for the real placer and mudflow has shown their sufficient similarity.

Stability of Finite Difference Discretizations of Multi-Physics Interface Conditions

AbstractWe consider multi-physics computations where the Navier-Stokes equations of compressible fluid flow on some parts of the computational domain are coupled to the equations of elasticity on other parts of the computational domain. The different subdomains are separated by well-defined interfaces. We consider time accurate computations resolving all time scales. For such computations, explicit time stepping is very efficient. We address the issue of discrete interface conditions between the two domains of different physics that do not lead to instability, or to a significant reduction of the stable time step size. Finding such interface conditions is non-trivial.We discretize the problem with high order centered difference approximations with summation by parts boundary closure. We derive L2 stable interface conditions for the linearized one dimensional discretized problem. Furthermore, we generalize the interface conditions to the full non-linear equations and numerically demonstrate their stable and accurate performance on a simple model problem. The energy stable interface conditions derived here through symmetrization of the equations contain the interface conditions derived through normal mode analysis by Banks and Sjögreen in [8] as a special case.

A Normal Mode Stability Analysis of Numerical Interface Conditions for Fluid/Structure Interaction

AbstractIn multi physics computations where a compressible fluid is coupled with a linearly elastic solid, it is standard to enforce continuity of the normal velocities and of the normal stresses at the interface between the fluid and the solid. In a numerical scheme, there are many ways that velocity- and stress-continuity can be enforced in the discrete approximation. This paper performs a normal mode stability analysis of the linearized problem to investigate the stability of different numerical interface conditions for a model problem approximated by upwind type finite difference schemes. The analysis shows that depending on the ratio of densities between the solid and the fluid, some numerical interface conditions are stable up to the maximal CFL-limit, while other numerical interface conditions suffer from a severe reduction of the stable CFL-limit. The paper also presents a new interface condition, obtained as a simplified characteristic boundary condition, that is proved to not suffer from any reduction of the stable CFL-limit. Numerical experiments in one space dimension show that the new interface condition is stable also for computations with the non-linear Euler equations of compressible fluid flow coupled with a linearly elastic solid.

A Cartesian Embedded Boundary Method for the Compressible Navier-Stokes Equations

We here generalize the embedded boundary method that was developed for boundary discretizations of the wave equation in second order formulation in Kreiss et al. (SIAM J. Numer. Anal. 40(5):1940---1967, 2002) and for the Euler equations of compressible fluid flow in Sjogreen and Peterson (Commun. Comput. Phys. 2:1199---1219, 2007), to the compressible Navier-Stokes equations. We describe the method and we implement it on a parallel computer. The implementation is tested for accuracy and correctness. The ability of the embedded boundary technique to resolve boundary layers is investigated by computing skin-friction profiles along the surfaces of the embedded objects. The accuracy is assessed by comparing the computed skin-friction profiles with those obtained by a body fitted discretization.

The Perturbation and ADAE Index of a Degenerated Hyperbolic System

We consider a linear version of the zero Mach-number limit of the Euler equations of compressible fluid flow. This system turns out to be a coupled hyperbolic/parabolic equation with coupled, time-dependent boundary conditions. Using the theory of abstract differential-algebraic equations it is shown that the frozen coefficient system has ADAE index 1. Moreover, the much stronger result is proven that the system has time-perturbation index one and space-perturbation index two even in the case of time-dependent boundary conditions. The results are stated in terms of the original physical variables. The estimates agree well with numerical experiments.

Simulation of transition and turbulence decay in the Taylor–Green vortex

Conventional large-eddy simulation (LES) and monotone integrated LES (MILES) are tested in emulating the dynamics of transition to turbulence in the Taylor–Green vortex (TGV). A variety of subgrid scale (SGS) models and high-resolution numerical methods are implemented in the framework of both incompressible and compressible fluid flow equations. Comparisons of the evolution of characteristic TGV integral measures are made with previously reported and new direct numerical simulation (DNS) data. The computations demonstrate that the convective numerical diffusion effects in the MILES methods can consistently capture the physics of flow transition and turbulence decay without resorting to an explicit SGS model, while providing accurate prediction of established theoretical findings for the kinetic energy dissipation, energy spectra, enstrophy and kinetic energy decay. All approaches tested provided fairly robust computational frameworks.

Aerodynamic shape design using evolutionary algorithms and new gradient-assisted metamodels

Aerodynamic shape design and optimization problems based on evolutionary algorithms and surrogate evaluation tools, i.e., the so-called metamodels, have recently found widespread use. Using metamodels, trained either separately from or during the optimization loop, a considerable reduction in the overall computing cost can be achieved. To support metamodel-based evolutionary algorithms, a class of new metamodels which utilize both known responses and response gradients for their training is proposed. The new gradient-assisted metamodels are extensions of standard multi-layer perceptrons and radial basis function networks. To demonstrate the prediction capabilities of the proposed metamodels and investigate different implementation modes within search algorithms along with the relevant CPU cost, a number of 2D and 3D aerodynamic shape (namely airfoils and turbomachinery blades) design problems are analyzed. Single- and two-objective problems, aiming at designing shapes that reproduce known pressure distributions at specific operating points, are considered. The exact evaluation tool is a numerical solver of the compressible fluid flow equations. The necessary gradient of the objective function is obtained by formulating and numerically solving adjoint equations.

Stability Analysis of a Reduced Heat Exchanger Model

AbstractWe consider the mathematical model of a heat exchanger in a heat pumping system using carbon dioxide as refridgerant. From the Euler equations of compressible fluid flow we derive a reduced model, called the zero Mach‐number limit, by eliminating time scales associated with soundwaves. The reduced model is a highly nonlinear partial differential‐algebraic equation (PDAE). We analyse the stability of this model and show that it is possible to use simple spatial discretisations within the method of lines (MOL), a standard practice in many system simulation software, to simulate the system. (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)