Compressible Fluid Flow Research Articles

Concentration of mass in the pressureless limit of the Euler equations of one-dimensional compressible fluid flow

In this paper, we study the limiting behavior of Riemann solutions to the Euler equations of one-dimensional compressible fluid flow as γ tends to one. We show that the limit solution forms the delta wave to the pressureless Euler system of one-dimensional compressible fluid flow in the distribution sense. Some numerical results exhibiting the phenomena of concentration are also presented.

Accuracy assessment of the Non-Ideal Computational Fluid Dynamics model for siloxane MDM from the open-source SU2 suite

Abstract The first-ever accuracy assessment of a computational model for Non-Ideal Compressible-Fluid Dynamics (NICFD) flows is presented. The assessment relies on a comparison between numerical predictions, from the open-source suite SU2, and pressure and Mach number measurements of compressible fluid flows in the non-ideal regime. Namely, measurements regard supersonic flows of siloxane MDM (Octamethyltrisiloxane, C 8 H24O 2 Si 3 ) vapor expanding along isentropes in the close proximity of the liquid–vapor saturation curve. The model accuracy assessment takes advantage of an Uncertainty Quantification (UQ) analysis, to compute the variability of the numerical solution with respect the uncertainties affecting the test-rig operating conditions. This allows for an uncertainty-based assessment of the accuracy of numerical predictions. The test set is representative of typical operating conditions of Organic Rankine Cycle systems and it includes compressible flows expanding through a converging–diverging nozzle in mildly-to-highly non-ideal conditions. All the considered flows are well represented by the computational model. Therefore, the reliability of the numerical implementation and the predictiveness of the NICFD model are confirmed.

Homogenized model with memory for two-phase compressible flow in double-porosity media

A completely averaged model of two-phase flow of compressible fluids in a medium with double porosity is developed. The variational asymptotic two-scale averaging method with splitting the nonlocality and nonlinearity is presented. Several mechanisms of delay are detected, as the nonequilibrium capillary redistribution of phases, pressure field relaxation caused by the compressibility, and the cross effects of fluid extrusion from pores due to rock compaction and fluid expansion. A generalized nonequilibrium capillary equation is obtained. All characteristic times of delay are explicitly defined as functions of saturation.

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.

Completed repeated Richardson extrapolation for compressible fluid flows

Abstract Richardson extrapolation is a powerful approach for reducing the spatial discretization errors and increasing, in this way, the accuracy of the computed solution obtained by using many numerical methods for solving different scientific and engineering problems. This approach has been used in a variety of computational fluid dynamics problems to reduce the numerical error, but it has been mainly restricted to the computation of incompressible fluid flows and on grids with coincident nodes. The purpose of this work is to present a completed repeated Richardson extrapolation (CRRE) procedure for a more generic type of grids not necessarily with coincident nodes, and test it on compressible fluid flows. Three tests are performed for one- and quasi-one-dimensional Euler equations, i.e. (i) the Rayleigh flow, (ii) the isentropic flow, and (iii) the adiabatic flow through a nozzle. The last test presents a normal shock wave. In order to build a simple CRRE solver, these problems are solved using a first-order upwind-type finite difference method as the base scheme. The normal shock wave problem is also solved using a high-order weighted essentially non-oscillatory (WENO) scheme to compare it with the CRRE procedure. The procedure we have proposed can increase the achieved accuracy and significantly decrease the magnitude of the spatial error in all three tests. Its performance can be best demonstrated in the Rayleigh flow test, where the spatial discretization error is reduced by seven orders of magnitude and the achieved accuracy is increased from 0.998 to 6.62 on the grid with 10240 nodes. Similar performance can be observed for the isentropic flow, for which the spatial discretization error is reduced by nine orders of magnitude and the achieved accuracy is again increased from 0.996 to 6.73 on the grid with 10240 nodes. Finally, in the adiabatic flow with a normal shock wave, the procedure can reduce the spatial discretization error both upstream and downstream of the shock. We do remark, however, that the more expensive high-order WENO scheme has errors of lower magnitude upstream of the shock and has a sharper shock transition for this shocked test case.

On the 1D modeling of fluid flowing through a Junction

<p style='text-indent:20px;'>A compressible fluid flows through a junction between two different pipes. Its evolution is described by the 2D or 3D Euler equations, whose analytical theory is far from complete and whose numerical treatment may be rather costly. This note compares different 1D approaches to this phenomenon.

Limiting behavior of some strictly hyperbolic systems of conservation laws

The objective of this paper is two fold. Firstly, a strictly hyperbolic system of conservation laws known as Euler equation of compressible fluid flow is considered and limiting behavior of its solutions as the pressure like term vanishes is studied.

A Compressible Fluid Flow with Double-Deck Structure Inside an Axially Symmetric Wavy-Wall Pipe

The problem of viscous compressible fluid flow in an axially symmetric pipe with small periodic irregularities on the wall is considered for large Reynolds numbers. An asymptotic solution with double-deck structure of the boundary layer and unperturbed core flow is obtained. Numerical investigations of the influence of the density of the core flow on the flow behavior in the near-wall region are presented.

A second order fully-discrete linear energy stable scheme for a binary compressible viscous fluid model

We present a linear, second order fully discrete numerical scheme on a staggered grid for a thermodynamically consistent hydrodynamic phase field model of binary compressible fluid flows, derived from the generalized Onsager Principle. The hydrodynamic model possesses not only the variational structure in its constitutive equation, but also warrants the mass, linear momentum conservation as well as energy dissipation. We first reformulate the model using the energy quadratization method into an equivalent form and then discretize the reformulated model to obtain a semidiscrete partial differential equation system using the Crank-Nicolson method in time. The semi-discrete numerical scheme preserves the mass conservation and energy dissipation law in time. Then, we discretize the semi-discrete PDE system on a staggered grid in space to arrive at a fully discrete scheme using 2nd order finite difference methods, which respects a discrete energy dissipation law. We prove the unique solvability of the linear system resulting from the fully discrete scheme. Mesh refinements and numerical examples on phase separation due to spinodal decomposition in binary polymeric fluids and interface evolution in the gas-liquid mixture are presented to show the convergence property and the usefulness of the new scheme in applications, respectively.

Vanishing Viscosity Limit of the Navier–Stokes Equations to the Euler Equations for Compressible Fluid Flow with Vacuum

We establish the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for three-dimensional compressible isentropic flow in the whole space. It is shown that there exists a unique regular solution of compressible Navier-Stokes equations with density-dependent viscosities, arbitrarily large initial data and vacuum, whose life span is uniformly positive in the vanishing viscosity limit. It is worth paying special attention that, via introducing a "quasi-symmetric hyperbolic"--"degenerate elliptic" coupled structure, we can also give some uniformly bounded estimates of $\displaystyle\Big(\rho^{\frac{\gamma-1}{2}}, u\Big)$ in $H^3$ space and $\rho^{\frac{\delta-1}{2}}$ in $H^2$ space (adiabatic exponent $\gamma>1$ and $1<\delta \leq \min\{3, \gamma\}$), which lead the strong convergence of the regular solution of the viscous flow to that of the inviscid flow in $L^{\infty}([0, T]; H^{s'})$ (for any $s'\in [2, 3)$) space with the rate of $\epsilon^{2(1-s'/3)}$. Further more, we point out that our framework in this paper is applicable to other physical dimensions, say 1 and 2, with some minor modifications. This paper is based on our early preprint in 2015.

Strong solutions to 1D compressible Navier‐Stokes/Allen‐Cahn system with free boundary

This paper is concerned with a diffuse interface model for two‐phase flow of compressible fluids with a type of free boundary. We establish the existence and uniqueness of global strong solutions of a coupled Navier‐Stokes/Allen‐Cahn system in 1D.

An implicit integral formulation to model inviscid fluid flows in obstructed media

We focus here on a technique to compute compressible fluid flows in physical domains cluttered up with many small obstacles. This technique, referred to here as the integral formulation, consists in integrating the flow governing equations over the fluid part of control volumes including both fluid and solid zones; doing so, the integral of fluxes over solid boundaries may appear, for which expressions as a function of discrete variables must be provided. The integral formulation presents two essential advantages: first, we naturally recover the standard fluid approach when the mesh is refined; second, fluid/solid interactions may be, to some extent, modelled to recover the singular head losses at the interface between a free and a congested part of the computational domain. We apply here this approach to the Euler equations, using a collocated space discretization and a pressure correction algorithm, preserving the positivity of both the density and the internal energy. Verification test cases are performed, including a Riemann problem in a free domain and a shock wave reflection on a wall, using an equation of state which is suitable for weakly compressible fluid flows. Finally, we address a two-dimensional situation, where a shock wave impacts a set of obstacles; we observe a very encouraging agreement between the integral approach results and a CFD reference solution obtained with a pure fluid approach on a fine mesh.

Scrutinizing the sources of inefficiencies in the piston-cylinder clearance of an oil-free linear compressor

The piston of linear compressors is subjected to low side loads and metal contact can be avoided by feeding high-pressure refrigerant gas from the discharge system into the piston-cylinder clearance. This paper reports a thorough study on the inefficiencies associated with leakage and viscous friction brought about by the transient compressible fluid flow in the piston-cylinder clearance of an oil-free linear compressor. The main objective is to understand the effect of design parameters, such as compressor speed, piston oscillation amplitude, pressure ratio and radial clearance, on the volumetric and isentropic efficiencies. The compressor efficiency was found to be mainly affected by the clearance and pressure ratio. A reduction of 2.5% in the volumetric efficiency and 4.2% in the isentropic efficiency, for instance, were predicted when the radial clearance was varied from 2.5 to 10.0 µm.

Compressible lattice Boltzmann simulations on high‐performance and low‐cost GeForce GPU

SummaryThe feasibility of compressible lattice Boltzmann simulations on high‐performance and low‐cost NVIDIA GeForce GPU is studied to enhance computational efficiency of compressible lattice Boltzmann by using much cheaper computing hardware. The NVIDIA GeForce GTX 1080 GPU, which has similar single precision operation computing performance with NVIDIA scientific computational Tesla K80 GPU and much lower price (seven to ten times) than the latter, is used in this work. The problem of GeForce GPUs is with low double precision operation performance and lack of error‐correcting code memory, which will influence the accuracy of compressible lattice Boltzmann simulations. To evaluate its feasibility, the double‐distribution‐function lattice Boltzmann method is adopted for compressible fluid flows. Three typical compressible problems are tested, ie, the Riemann problem, the regular shock reflection problem, and the supersonic boundary layer problem. The results by GeForce GPU with single precision operation have numerical fluctuations in the simulations with shock waves. Moreover, GeForce GPUs have similar numerical accuracy with CPUs in the regular shock reflection problem and supersonic boundary layer problem.

Pressure-driven gas flow in viscously deformable porous media: application to lava domes

The behaviour of low-viscosity, pressure-driven compressible pore fluid flows in viscously deformable porous media is studied here with specific application to gas flow in lava domes. The combined flow of gas and lava is shown to be governed by a two-equation set of nonlinear mixed hyperbolic–parabolic type partial differential equations describing the evolution of gas pore pressure and lava porosity. Steady state solution of this system is achieved when the gas pore pressure is magmastatic and the porosity profile accommodates the magmastatic pressure condition by increased compaction of the medium with depth. A one-dimensional (vertical) numerical linear stability analysis (LSA) is presented here. As a consequence of the pore-fluid compressibility and the presence of gravitation compaction, the gradients present in the steady-state solution cause variable coefficients in the linearized equations which generate instability in the LSA despite the diffusion-like and dissipative terms in the original system. The onset of this instability is shown to be strongly controlled by the thickness of the flow and the maximum porosity, itself a function of the mass flow rate of gas. Numerical solutions of the fully nonlinear system are also presented and exhibit nonlinear wave propagation features such as shock formation. As applied to gas flow within lava domes, the details of this dynamics help explain observations of cyclic lava dome extrusion and explosion episodes. Because the instability is stronger in thicker flows, the continued extrusion and thickening of a lava dome constitutes an increasing likelihood of instability onset, pressure wave growth and ultimately explosion.

Transient boundary layer flow of viscous compressible fluid past a heated sphere with the effect of Mach number

Abstract A two-dimensional transient boundary layer flow of an unsteady, viscous, compressible fluid past a heated sphere with the effect of Mach number is considered. The governing boundary layer equations are reduced to nonsimilar form, and these are solved numerically by three methods, namely, the finite difference method for all time regime, perturbation solutions for small time regime, and the asymptotic solutions for large time regime. Comparisons of the results show a good agreement between the above mentioned methods. Effects of various practical values of the surface temperature parameter, α and the Mach number, M 0 on the local shear stress, the rate of heat transfer, streamlines and isoenthalpies are shown graphically. Further, it is observed that increase in the value of Mach number, M 0 , leads to an increase in the local shear stress and the rate of heat transfer; whereas opposite effect is found in case of local shear stress once the point of separation occured. On the other hand size of the vortex found to be decreased with an increase in Mach number.

A predictive formula for the Nusselt number of compressible laminar fluid flow passing the thermal developing zone of a hot tube

AbstractMany of the industrial and engineering problems need to know the value of the convective heat transfer coefficient of fluid flows within various tools, systems, or tubes. In the present study, a predictive formula has been obtained for the Nusselt number for compressible laminar flows passing the velocity developed and thermal developing zone of a hot tube with different angles with the horizon. A fractional factorial experimental design method and an optimization method have been utilized for obtaining a width range and accurate formula. Finally, the ability of this formula has been examined using a set of testing data.

Long-Term Regularity of the Periodic Euler–Poisson System for Electrons in 2D

We study a basic plasma physics model--the one-fluid Euler--Poisson system on the square torus, in which a compressible electron fluid flows under its own electrostatic field. In this paper we prove long-term regularity of periodic solutions of this system in 2 spatial dimensions. Our main conclusion is that on a square torus of side length $R$, if the initial data is sufficiently close to a constant solution, then the solution is wellposed for a time at least $R/\epsilon^2(\log R)^{O(1)}$, where $\epsilon$ is the size of the initial data.

Numerical analysis and modeling of multiscale Forchheimer–Forchheimer coupled model for compressible fluid flow in fractured media aquifer system

A multiscale coupled model is developed to simulate the compressible fluid flow in fractured media aquifer system, where the flow is governed by Forchheimer’s law in the fracture and continuum porous medium. Due to the fact that the thickness of fracture is much smaller than characteristic diameters of surrounding porous medium, the fracture is reduced to a lower dimensional interface and a more complicated transmission condition is derived on the fracture-interface. The coupled model is numerically solved by the finite difference method with an implicit iteration procedure. The fewest nodal points are used to construct the optimal scheme for approximating the multiscale Forchheimer–Forchheimer coupled model. Different degrees of freedom are located on both sides of fracture-interface in order to capture the jump of velocity. Second-order error estimates in discrete norms are derived on nonuniform staggered grids for both pressure and velocity. The proposed scheme can also be extended to high dimensional model without loss of accuracy. Numerical experiments are performed to demonstrate the efficiency and accuracy of the numerical method.

A combined momentum-interpolation and advection upstream splitting pressure-correction algorithm for simulation of convective and acoustic transport at all levels of Mach number

A pressure-correction algorithm is presented for compressible fluid flow regimes. It is well-suited to simulate flows at all levels of Mach number with smooth and discontinuous flow field changes, by providing a precise representation of convective transport and acoustic propagation. The co-located finite volume space discretization is used with the AUSM flux splitting. It is demonstrated that two ingredients are essential for obtaining good quality solutions: the presence of an inertia term in the face velocity expression; a velocity difference diffusive term in the face pressure expression, with a correct Mach number scaling to recover the hydrodynamic and acoustic low Mach number limits. To meet these two requirements, a new flux scheme, named MIAU, for Momentum Interpolation with Advection Upstream splitting is proposed.