Showing papers on "Compressibility published in 1995"

Preconditioning Applied to Variable and Constant Density Flows

Abstract: A time-derivative preconditioning of the Navier-Stokes equations, suitable for both variable and constant density fluids, is developed. The ideas of low-Mach-number preconditioning and artificial compressibility are combined into a unified approach designed to enhance convergence rates of density-based, time-marching schemes for solving flows of incompressible and variable density fluids at all speeds. The preconditioning is coupled with a dual time-stepping scheme implemented within an explicit, multistage algorithm for solving time-accurate flows. The resultant time integration scheme is used in conjunction with a finite volume discretization designed for unstructured, solution-adaptive mesh topologies. This method is shown to provide accurate steady-state solutions for transonic and low-speed flow of variable density fluids. The time-accurate solution of unsteady, incompressible flow is also demonstrated.

The effects of an elastic solid surface layer on the radial pulsations of gas bubbles

Abstract: Most previous theoretical investigations of gas bubble dynamics have assumed an uncontaminated gas–liquid interface. Recently, however, the potential importance of layers of surface active agents on bubble dynamics has been increasingly recognized. In this work it is assumed that a continuous layer of incompressible, solid elastic material separates the gas from the bulk Newtonian liquid. Elasticity is modeled to include viscous damping. A Rayleigh–Plesset‐like equation describing the dynamics of such surface‐contaminated gas bubbles is derived. The equation predicts that the surface layer supports a strain that counters the Laplace pressure and thereby stabilizes the bubble against dissolution. An analytical solution to this equation which includes both the fundamental and second‐harmonic response is presented. The dispersion relation describing the propagation of linear pressure waves in liquids containing suspensions of these bubbles also is presented. It is found that (1) the resonance frequencies of ...

Compressible turbulent channel flows: DNS results and modelling

Abstract: The present paper addresses some topical issues in modelling compressible turbulent shear flows. The work is based on direct numerical simulation (DNS) of two supersonic fully developed channel flows between very cold isothermal walls. Detailed decomposition and analysis of terms appearing in the mean momentum and energy equations are presented. The simulation results are used to provide insights into differences between conventional Reynolds and Favre averaging of the mean-flow and turbulent quantities. Study of the turbulence energy budget for the two cases shows that compressibility effects due to turbulent density and pressure fluctuations are insignificant. In particular, the dilatational dissipation and the mean product of the pressure and dilatation fluctuations are very small, contrary to the results of simulations for sheared homogeneous compressible turbulence and to recent proposals for models for general compressible turbulent flows. This provides a possible explanation of why the Van Driest density-weighted transformation (which ignores any true turbulent compressibility effects) is so successful in correlating compressible boundary-layer data. Finally, it is found that the DNS data do not support the strong Reynolds analogy. A more general representation of the analogy is analysed and shown to match the DNS data very well.

Large-scale structure and entrainment in the supersonic mixing layer

Abstract: Experiments were conducted in a two-stream planar mixing layer at convective Mach numbers, Mc, of 0.28, 0.42, 0.50, 0.62 and 0.79. Planar laser Mie scattering (PLMS) from a condensed alcohol fog and planar laser-induced fluorescence (PLIF) of nitric oxide were used for flow visualization in the side, plan and end views. The PLIF signals were also used to characterize the turbulent mixture fraction fluctuations.Visualizations using PLMS indicate a transition in the turbulent structure from quasi-two-dimensionality at low convective Mach number, to more random three-dimensionality for . A transition is also observed in the core and braid regions of the spanwise rollers as the convective Mach number increases from 0.28 to 0.62. A change in the entrainment mechanism with increasing compressibility is also indicated by signal intensity profiles and perspective views of the PLMS and PLIF images. These show that at Mc = 0.28 the instantaneous mixture fraction field typically exhibits a gradient in the streamwise direction, but is more uniform in the cross-stream direction. At Mc = 0.62 and 0.79, however, the mixture fraction field is more streamwise uniform and with a gradient in the cross-stream direction. This change in the composition of the structures is indicative of different entrainment motions at the different compressibility conditions. The statistical results are consistent with the qualitative observations and suggest that compressibility acts to reduce the magnitude of the mixture fraction fluctuations, particularly on the high-speed edge of the layer.

Thermodynamic and transport properties of sodium liquid and vapor

Abstract: Data have been reviewed to obtain thermodynamically consistent equations for thermodynamic and transport properties of saturated sodium liquid and vapor Recently published Russian recommendations and results of equation of state calculations on thermophysical properties of sodium have been included in this critical assessment Thermodynamic properties of sodium liquid and vapor that have been assessed include: enthalpy, heat capacity at constant pressure, heat capacity at constant volume, vapor pressure, boiling point, enthalpy of vaporization, density, thermal expansion, adiabatic and isothermal compressibility, speed of sound, critical parameters, and surface tension Transport properties of liquid sodium that have been assessed include: viscosity and thermal conductivity For each property, recommended values and their uncertainties are graphed and tabulated as functions of temperature Detailed discussions of the analyses and determinations of the recommended equations include comparisons with recommendations given in other assessments and explanations of consistency requirements The rationale and methods used in determining the uncertainties in the recommended values are also discussed

The elastic coefficients of double‐porosity models for fluid transport in jointed rock

Abstract: Phenomenological equations (with coefficients to be determined by specified experiments) for the poroelastic behavior of a dual porosity medium are formulated, and the coefficients in these linear equations are identified. The generalization from the single-porosity case increases the number of independent coefficients for volume deformation from three to six for an isotropic applied stress. The physical interpretations are based upon considerations of different temporal and spatial scales. For very short times, both matrix and fractures behave in an undrained fashion. For very long times, the double-porosity medium behaves like an equivalent single-porosity medium. At the macroscopic spatial level, the pertinent parameters (such as the total compressibility) may be determined by appropriate field tests. At an intermediate or mesoscopic scale, pertinent parameters of the rock matrix can be determined directly through laboratory measurements on core, and the compressibility can be measured for a single fracture. All six coefficients are determined from the three poroelastic matrix coefficients and the fracture compressibility from the single assumption that the solid grain modulus of the composite is approximately the same as that of the matrix for a small fracture porosity. Under this assumption, the total compressibility and three-dimensional storage coefficient of the composite are the volume averages of the matrix and fracture contributions.

Pressure-impulse theory for liquid impact problems

Abstract: A mathematical model is presented for the high pressures and sudden velocity changes which may occur in the impact between a region of incompressible liquid and either a solid surface or a second liquid region. The theory rests upon the well-known idea of pressure impulse, for the sudden initiation of fluid motion in incompressible fluids. We consider the impulsive pressure field which occurs when a moving fluid region collides with a fixed target, such as when an ocean wave strikes a sea wall. The boundary conditions are given for modelling liquid-solid and liquid-liquid impact problems. For a given fluid domain, and a given velocity field just before impact, the theory gives information on the peak pressure distribution, and the velocity after impact. Solutions for problems in simple domains are presented, which give insight into the peak pressures exerted by a wave breaking against a sea wall, and a wave impacting in a confined space. An example of liquid-liquid impact is also examined. Results of particular interest include a relative insensitivity to the shape of the incident wave, and an increased pressure impulse when impact occurs in a confined space. The theory predicts that energy is lost from the bulk fluid motion and we suggest that this energy can be transferred to a thin jet of liquid which is projected away from the impact region.

Direct and Large-Eddy Simulation of the compressible turbulent mixing layer

Abstract: The Large-Eddy Simulation technique of compressible flows and the effect of compressibility on mixing layers are the main subjects of this thesis. Direct Numerical Simulations (DNS) and Large-Eddy Simulations (LES) of the temporal compressible mixing layer at various Mach and Reynolds numbers have been conducted to investigate these subjects. With respect to the LES technique, Large-Eddy Simulations have been performed at convective Mach numbers 0.2, 0.6 and 1.2 and the results have been compared with filtered DNS-data. It appeared that the dynamic subgrid-models lead to relatively accurate results compared to the other models tested. The dynamic approach turned out to yield acceptable results too in LES of a mixing layer that currently cannot be simulated using DNS. Care has to be taken to ensure that the numerical errors in LES are sufficiently small. It was found that these errors are usually sufficiently small if the filter width equals twice the grid-spacing. In addition to modelling the turbulent stress tensor, compressible LES formally requires the modelling of the subgrid-terms in the energy equation, which do not occur in incompressible LES. However, the compressible Large- Eddy Simulations demonstrated that the turbulent stress tensor is the dominant subgrid-term, even at convective Mach number 1.2. This important subgrid-term was also investigated from a theoretical point of view and realizability conditions for this tensor were derived. Regarding compressibility effects in the mixing layer, shock-waves were found in the three-dimensional DNS at convective Mach number 1.2. Furthermore, we have investigated the cause of the mixing layer growth rate reduction with increasing compressibility, using four DNS-databases covering the range of convective Mach numbers from 0.2 to 1.2. It was found that the growth rate reduction cannot be explained by the dilatational terms, but rather by the reduced pressure fluctuations, leading to reduced pressure strain and turbulent production terms.

Direct computation of the sound from a compressible co-rotating vortex pair

Abstract: The far-field sound from corotating vortices is computed by direct computation of the unsteady, compressible Navier-Stokes equations on a computational mesh that extends to two acoustic wavelengths in all directions. The vortices undergo a period of corotation followed by a sudden merger. A 2D version of Moehring's equation is developed and used in conjunction with source terms computed in the simulation to predict the far-field sound. The prediction agrees with the simulation to within 3 percent. Results of far-field pressure fluctuations for an acoustically noncompact case are also presented for which the prediction is 66 percent too high. Results also indicate that the monopole contribution of 'viscous sound' is negligible for this flow.

A improved incompressible lattice Boltzmann model for time-independent flows

Abstract: It is well known that the lattice Boltzmann equation method (LBE) can model the incompressible Navier-Stokes (NS) equations in the limit where density goes to a constant. In a LBE simulation, however, the density cannot be constant because pressure is equal to density times the square of sound speed, hence a compressibility error seems inevitable for the LBE to model incompressible flows. This work uses a modified equilibrium distribution and a modified velocity to construct an LBE which models time-independent (steady) incompressible flows with significantly reduced compressibility error. Computational results in 2D cavity flow and in a 2D flow with an exact solution are reported.

Seismic pore space compressibility and Gassmann’s relation

Abstract: The pore space compressibility of a rock provides a robust, model-independent descriptor of porosity and pore fluid effects on effective moduli. The pore space compressibility is also the direct physical link between the dry and fluid-saturated moduli, and is therefore the basis of Gassmann's equation for fluid substitution. For a fixed porosity, an increase in pore space compressibility increases the sensitivity of the modulus to fluid substitution. Two simple techniques, based on pore compressibility, are presented for graphically applying Gassmann's relation for fluid substitution. In the first method, the pore compressibility is simply reweighted with a factor that depends only on the ratio of fluid to mineral bulk modulus. In the second technique, the rock moduli are rescaled using the Reuss average, which again depends only on the fluid and mineral moduli.

Melting transition in two dimensions: A finite-size scaling analysis of bond-orientational order in hard disks.

Abstract: We describe a general and efficient method, based on computer simulations and applicable to a general class of fluids, that allows us to determine (i) bounds on the transition densities of the melting transition that are valid in the thermodynamic limit and (ii) the order of the phase transition. The bond-orientational order parameter, its susceptibility, and the compressibility are measured simulataneously on many length scales, and the latter two quantities are extrapolated to the thermodynamic limit by application of the subblock analysis method of finite-size scaling. We include a detailed analysis, related to the subblock method, of the cross correlations of the fluctuations of the density and the order parameter. The behavior of the extrapolated order parameter susceptibility yields precise upper and lower bounds for the melting and freezing densities, respectively. We apply these techniques to the two-dimensional melting transition in large systems of 16 384 hard disks using canonical Monte Carlo computer simulations. The measured bond-orientational susceptibilities are found to be incompatible with predictions of the Halperin-Nelson-Young theory of two-dimensional melting. This and the behavior of the bond-orientational cumulant are two strong pieces of evidence of a first-order phase transition.

Compressible cavity flow oscillation due to shear layer instabilities and pressure feedback

Abstract: A computational analysis was performed on a compressible flow oscillation due to shear layer instabilities over a cavity and pressure feedback in a cavity of length-to-depth ratio 3 at Mach 1.5 and 2.5. The mass-averaged NavierStokes equations were solved. Turbulence closure was achieved using a k-u> model with compressibility corrections. Self-sustained oscillations were produced. Negative form drag coefficient was observed within an oscillatory cycle due to mass ejection from the cavity near the trailing edge and vortex production near the leading edge. The shock wave-expansion wave interaction patterns, modes of the oscillation, sound pressure level, and time-averaged surface pressure were compared with experimental results of previous investigations and good agreement was achieved, particularly the time-averaged pressure. The prediction showed a marked improvement over earlier analysis.

A fast, accurate real gas equation of state for fluid dynamic analysis applications

Abstract: A modified form of the Redlich-Kwong two-parameter equation of state is presented. The modified equation employs the acentric factor and the critical point compressibility factor as additional parameters to improve its accuracy and to extend its application range to include the critical point. This modified equation is as simple as the original form, yet achieves substantially better prediction accuracy, including thermodynamic parameters such as enthalpy and entropy. Results from this equation, the original equation, and three other popular modified forms are compared with gas property data for several compounds to demonstrate its improved accuracy and increased application range. Practical application limits to the other modified forms are identified to guide current users of those methods.

Physical, Mechanical, and Thermal Properties of Starch-Based Plastic Foams

Abstract: Normal corn starch was extruded with polystyrene or polymethyl methacrylate and blowing and cross-linking agents in a single screw C. W. Brabender laboratory scale extruder at 140° C barrel temperature, 140 rpm screw speed and 16% moisture content. Expansion ratio, unit density, water solubility index, spring index, compressibility, specific heat and thermal conductivity of the resultant products were studied. Physical, mechanical, and thermal properties depended upon the type of additives used. The results indicated suitability of different foams for packaging and insulation applications. Keywords. Extrusion, Starch-based, Plastic, Foam, Properties.

A simple expression for radial distribution functions of pure fluids and mixtures

Abstract: A simple expression for the radial distribution function (RDF) of pure fluids and mixtures is presented. It satisfies the limiting conditions of zero density and infinite distance imposed by statistical thermodynamics. The equation contains seven adjustable parameters; they have been fitted to extensive literature data of RDF’s for a Lennard‐Jones fluid at different values of temperature and density. These in turn have been expressed as functions of reduced temperature and density, thus allowing a complete parametrization with respect to these variables using 21 parameters altogether with fairly good accuracy. The values of the reduced pressure and internal energy calculated by numerical integration of the completely parametrized equation compare fairly with literature molecular dynamics simulation results. The capability of the expression to fit to RDF’s of mixtures has been checked against some of the extensive RDF simulation data of binary mixtures of Lennard‐Jones fluids with different diameters available in the literature. Data pertaining to different molar fractions as well as to different eAA/eBB ratios have been considered, and the agreement between calculated and simulation curves has resulted satisfactory. The proposed expression can be used to calculate by integration related quantities such as compressibility, internal energy, pressure and, using the Kirkwood–Buff theory, the chemical potentials and partial molar volumes of the components of mixtures for which RDF data are available.

An accurate and simple equation of state for hard disks

Abstract: An equation of state for a fluid of hard disks is proposed: Z=[1−2η+(2η0−1)(η/η0)2]−1. The exact fit of the second virial coefficient and the existence of a single pole singularity at the close‐packing fraction η0 are the only requirements imposed on its construction. A comparison of the prediction of virial coefficients and of the values of the compressibility factor Z with those stemming out of other known equations of state is made. The overall performance of this very simple equation of state is quite satisfactory.

Pressure structure functions and spectra for locally isotropic turbulence

Abstract: Beginning with the known relationship between the pressure structure function and the fourth-order two-point correlation of velocity derivatives, we obtain a new theory relating the pressure structure function and spectrum to fourth-order velocity structure functions. This new theory is valid for all Reynolds numbers and for all spatial separations and wavenumbers. We do not use the joint Gaussian assumption that was used in previous theory. The only assumptions are local homogeneity, local isotropy, incompressibility, and use of the Navier-Stokes equation. Specific formulae are given for the mean-squared pressure gradient, the correlation of pressure gradients, the viscous range of the pressure structure function, and the pressure variance. Of course, pressure variance is a descriptor of the energy-containing range. Therefore, for any Reynolds number, the formula for pressure variance requires the more restrictive assumption of isotropy. For the case of large Reynolds numbers, formulae are given for the inertial range of the pressure structure function and spectrum and of the pressure-gradient correlation; these are valid on the basis of local isotropy, as are the formulae for mean-squared pressure gradient and the viscous range of the pressure structure function. Using the experimentally verified extension to fourth-order velocity structure functions of Kolmogorov’s theory, we obtain r4I3 and kp7/3 laws for the inertial range of the pressure structure function and spectrum. The modifications of these power laws to account for the effects of turbulence intermittency are also given. New universal constants are defined; these require experimental evaluation. The pressure structure function is sensitive to slight departures from local isotropy, implying stringent conditions on experimental data, but applicability of the previous theory is likewise constrained. The results are also sensitive to compressibility.

Coupled fluid flow and geomechanics in reservoir study. I. Theory and governing equations

Abstract: The purpose of this study is to examine Biot`s two-phase (fluid and rock), isothermal, linear poroelastic theory from the conventional porous fluid-flow modeling point of view Not`s theory and the published applications are oriented more toward rock mechanics than fluid flow Our goal is to preserve the commonly used systematic porous fluid-flow modeling and include geomechanics as an additional module By developing such an approach, complex reservoir situations involving geomechanical issues (eg, naturally fractured reservoirs, stress-sensitive reservoirs) can be pursued more systematically and easily We show how the conventional fluid-flow formulations is extended to a coupled fluid-flow-geomechanics model Consistent interpretation of various rock compressibilities and the effective stress law are shown to be critical in achieving the coupling The {open_quotes}total (or system) compressibility{close_quotes} commonly used in reservoir engineering is shown to be a function of boundary conditions Under the simplest case (isotropic homogeneous material properties), the fluid pressure satisfies a fourth-order equation instead of the conventional second-order diffusion equation Limiting cases include nondeformable, incompressible fluid and solid, and constant mean normal stress are analyzed

High-pressure structural study of muscovite

Abstract: The compressibility and structural variations of two 2M1 muscovites having compositions (Na0.07K0.90 Ba0.01□0.02)(Al1.84Ti0.04Fe0.07Mg0.04)(Si3.02Al0.98) O10 (OH)2 (7 mole % paragonite) and (Na0.37K0.60□0.03)(Al1.84Ti0.02 Fe0.10Mg0.06)(Si3.03Al0.97) O10(OH)2 (37 mole % paragonite) were determined at pressures between 1 bar and 35 kbar, by single-crystal X-ray diffraction using a Merrill-Bassett diamond anvil cell. Isothermal bulk moduli, setting K′ = 4, were 490 and 540 (± 30) kbar for the Na-poor and Na-rich samples respectively. Both samples show highly anisotropic compressibility patterns, with β a ∶β b ∶β c = 1∶1.15∶3.95 for the Na-poor sample and β a ∶β b ∶β c = 1∶1.19∶3.46 for the Na-rich one. HP structural refinements showed that the different compressibility was largely due to the partial substitution of Na for K in the interlayer region. Moreover, the different compressibility of the tetrahedral and octahedral layers, observed in both micas, increased the a rotation of the tetrahedral layer by about 2° in 28 kbar, as also indicated by the evolution of interlayer cation bond lengths. This increases the repulsion of oxygens of the basal layers and between the high-charged cations of the tetrahedral layer. As a consequence, phengitic substitution, reducing α rotation, would increase the baric stability of mica. Comparison between the HP structures of muscovite and phlogopite indicated the lower compressibility of the latter, mainly due to the greater compressibility of the dioctahedral layer with respect to that of the trioctahedral layer. The HT and HP behaviour of di- and trioctahedral micas showed an anisotropy in the compressional pattern which was markedly greater than that observed in the dilatation pattern. This unexpected result was explained by the different evolution with P and T of alkaliO bond lengths. By combining HP and HT data, a tentative equation of state of muscovite is proposed.

Accurate solution algorithms for incompressible multiphase flows

Abstract: A number of advances in modeling multiphase incompressible flow are described. These advances include high-order Godunov projection methods, piecewise linear interface reconstruction and tracking and the continuum surface force model. Examples are given.

Thermal expansivities and compressibilities of hydrous phases in the system MgO–SiO2–H2O: talc, phase A and 10-Å phase

Abstract: We have measured the thermal expansivity of talc, Mg3Si4O10(OH)2, and phase A, Mg7Si2O8(OH)6, and the compressibility of talc, phase A and 10-A phase, Mg3Si4O10(OH)2 ⋅ xH2O, using powder X-ray diffraction. The thermal expansivity of talc and phase A were measured at temperatures up to 810° C and 600° C, respectively. Volumes of both phases increase linearly with temperature, and can be described as follows: Talc: V/V0=1+2.15 (±0.05)×10-5 (T−298), V0=136.52 (±0.03) cm3 mol-1; Phase A: V/V0=1+4.86 (±0.18)×10-5 (T– 298), V0=154.42 (±0.09) cm3 mol-1. Compressibility measurements of talc, 10-A phase and phase A were made at pressures up to 6.05, 8.52 and 9.85 GPa, respectively. Values of the isothermal bulk modulus K298 and its pressure derivative K′, obtained by fitting the compressibility data to the Murnaghan equation, are as follows: Talc: K298=41.6±0.9 GPa, K′=6.5±0.4; 10-A phase: K298=32.2±5.5 GPa, K′=9.2±2.8; Phase A: K298=145±5 GPa (assuming that K′=4). Combining the new talc data with existing thermodynamic data provides a more accurate thermodynamic description of talc than previously available, enabling its high-pressure, high-temperature phase relations to be calculated. The data for 10-A phase are consistent with a positive slope for its dehydration reaction, making 10-A phase a good candidate for H2O storage in subducting slabs. The measurements of the thermal expansivity and compressibility of phase A allow its enthalpy of formation and entropy to be derived from the results of phase equilibrium experiments on phase A.

Universal compressibility behavior of dense phases.

Abstract: We recently proposed an isothermal equation of state that was successfully applied to study the high-pressure behavior of molecular liquids. In this work we extend its applicability to liquid metals, polymers, molten salts, and solids. The possibility of considering this equation as an alternative to long-standing equations such as those of Tait and Birch-Murnaghan is emphasized. This suggestion is firmly supported by comparisons with experimental data up to pressures of several GPa. A new physical interpretation of the pressure coefficient of the Tait equation ${\mathit{B}}_{\mathit{T}}$ is given. It can be identified to the divergence pressure along a pseudospinodal curve. We also compare the performance of our equation with the most successful equations of state used to represent isothermal data of solids, with excellent results. Our equation can be applied to systems with phase transitions. An interesting observation is that it seems that the different pseudospinodal curves obtained for the different phases can be put together into the same curve except for characteristic jumps occurring at the phase transitions.