Fluid Of Constant Density Research Articles

Buchdahl limit for d-dimensional spherical solutions with a cosmological constant

We investigate the conditions for which a d-dimensional perfect fluid solution is in hydrostatic equilibrium with a cosmological constant. We find a generalization of Buchdahl inequality and obtain an upper bound for the degree of compactification. Using the Tolman–Oppenheimer–Volkoff equation to get a lower bound for the degree of compactification we analyse the regions where the solution is in hydrostatic equilibrium. We obtain the inner metric solution and the pressure for the constant fluid density model.

The rotation of Io with a liquid core

In a previous paper (Henrard, Celest. Mech. Dyn. Astron. 178, 144–153, 2005c) we have developed an analytical theory of the rotation of the Galilean satellite Io, considered as a rigid body and based on a synthetic theory of its orbital motion due to Lainey (Theorie Dynamique des Satellites Galileens. PhD dissertation, Observatoire de Paris, 2002) (see also Lainey et al., AA A&A, 427, 371–376, 2004b). One of the most important causes of departure of the actual rotation from the rigid theory is thought to be the existence of a liquid core, the size of which is unknown but would be an important piece of information concerning the structure of the interior of the satellite. In this contribution we develop the analytical theory of a liquid core contained in a cavity filled by an inviscid fluid of constant uniform density and vorticity. The theory is based on Poincare (Bull. Astron. 27, 321–356, 1910) model and is developed by a Lie transform perturbation method, very much like in our previous contribution. Our main conclusion is that the addition of a degree of freedom (the spin of the core) with a frequency close to the orbital frequency multiplies the possibility of resonances and that for some particular size of the core one may expect a (possibly small) region of chaotic behaviour in the vicinity of the Cassini state.

Simulation of natural convection with the collocated clustered finite volume scheme

This paper presents numerical results obtained in the case of natural convection within non constant fluid density, using the collocated clustered finite volume (CCFV) scheme. The continuous equations are first given in a dimensionless form. Then we present the finite volume scheme with the principles and the spatial discretization used. Analytical tests illustrate the numerical behavior of this scheme according to the type of grid, of the pressure stabilization method and check the robustness of this scheme. Next, the results obtained on the square thermally driven cavity under large temperature differences show that the CCFV scheme accurately fits the reference results.

CFD Modelling of Liquid Metal Flow and Heat Transfer in Blast Furnace Hearth

An in-depth understanding of the liquid metal flow and heat transfer is essential in order to identify the key mechanisms for the hearth erosion of a blast furnace. In this study, a comprehensive computational fluid dynamics model is described which predicts the flow and temperature distributions of liquid iron in blast furnace hearth, and the temperature distribution in the refractories. The new model addresses conjugate heat transfer, natural convection and turbulent flow through porous media, with its main features including improved transport equations (a modified k–ε turbulence model and thermal dispersion term) and a three-dimensional, high-resolution grid. The new turbulence model and terms take account of the effect of microscopic flows around coke particles and allow unified treatment of coke bed and coke-free layer. The predicted results show a well-organized flow pattern: two large-scale recirculation zones are separated vertically at the taphole level. This flow pattern controls the temperature distribution in the liquid phase, so that the temperature remains nearly uniform in the upper zone, but changes mainly across the lower zone. The effects of several factors were examined, such as cases comparing fluid buoyancy with constant fluid density as well as the shape and position of the coke free zone (i.e. based on reported dissection studies). Natural convection is found to be most important for the liquid metal flow patterns observed. Comparison with the plant data shows that the refractory pad temperature is under-predicted when assuming intact hearth lining. The pad temperature is very sensitive to the erosion of protection layer in the hearth lining.

Local geometry of isoscalar surfaces

An inert dynamically passive scalar in a constant density fluid forced by a statistically homogeneous field of turbulence has been investigated using the results of a 256(3) grid direct numerical simulation. Mixing characteristics are characterized in terms of either principal curvatures or mean and Gauss curvatures. The most probable small-scale scalar geometries are flat and tilelike isosurfaces. Preliminary correlations between flow and scalar small-scale structures associate highly curved saddle points with large-strain regions and elliptic points with vorticity-dominated zones. The concavity of the scalar profiles along the isosurface normal coordinate xn correlates well with negative mean curvatures, Gauss curvatures displaying any sign, which correspond to scalar minima, tiles, or saddle points; on the other hand, convexity along xn is associated with positive mean curvatures, Gauss curvatures ranging from negative to positive signs, featuring maxima, tiles, or saddle points; inflection points along xn correlate well with small values of the mean curvature and zero or negative values of kg, corresponding to plane isosurfaces or saddle points with curvatures of equal and opposite signs. Small values of the scalar gradient are associated with elliptic points, either concave or convex (kg>0) , for both concave and convex scalar profiles along xn. Large values of the scalar gradient (or, equivalently, scalar fluctuation dissipation rates) are generally connected with small values of the Gauss curvature (either flat or moderate-curvature tilelike local geometries), with both concave and convex scalar profiles along xn equally probable. Vortical local flow structures correlate well with small and moderate values of the scalar gradient, while strain-dominated regions are associated with large values.

A new method for Stokes problems with circular boundaries using degenerate kernel and Fourier series

AbstractThis study is concerned with the Stokes flow of an incompressible fluid of constant density and viscosity with circular boundaries. To fully capture the circular boundary, the boundary densities in the direct and indirect boundary integral equations (BIEs) are expanded in terms of Fourier series. The kernel functions in either the direct BIE or the indirect BIE are expanded to degenerate kernels by using the separation of field and source points. Consequently, the improper integrals are transformed to series sum and are easily calculated. The linear algebraic system can be established by matching the boundary conditions at the collocation points. Then, the unknown Fourier coefficients can be easily determined. Finally, several examples including circular and eccentric domains are presented to demonstrate the validity of the present method. Five gains were obtained: (1) meshless approach; (2) free of boundary‐layer effect; (3) singularity free; (4) exponential convergence; and (5) well‐posed model. Copyright © 2007 John Wiley & Sons, Ltd.

A note on the flow of a homogeneous intrusion into a two-layer fluid

The intrusion of a constant density fluid at the interface of a two-layer fluid is considered. Numerical solutions are computed for a model of a steady intrusion resulting from flow down a bank and across a broad lake or reservoir. The incoming fluid is homogeneous and spreads across the lake at its level of neutral buoyancy. Solutions are obtained for a range of different inflow angles, flow rate and density differences. Except in extreme cases, the nature of the solution is predicted quite well by linear theory, with the wavelength at any Froude number given by a dispersion relation and wave steepness determined largely by entry angle. However, some extreme solutions with rounded meandering flows and non-unique solutions in the parameter space are also obtained.

On dimensionality of attainable region construction for isothermal reactor networks

In this paper, a number of novel properties of reactor network synthesis (RNS) problems are presented and proved within the Infinite DimEnsionAl State-space (IDEAS) framework for isothermal reactor networks with and without the constant density assumption. These properties can then be used to reduce the dimension of the composition space (i.e., concentration space for constant density fluid (CDF) reactor networks, mass fraction space for variable density fluid (VDF) reactor networks) within which RNS is conducted and subsequently the attainable region (AR) construction. This procedure may be used to overcome the difficulty preventing the AR methodology and other RNS methods from being used for real industrial size problems.

Improved FVM for two-layer shallow-water models: Application to the Strait of Gibraltar

This paper deals with the numerical simulation of flows of stratified fluids through channels with irregular geometry. Channel cross-sections are supposed to be symmetric but not necessarily rectangular. The fluid is supposed to be composed of two shallow layers of immiscible fluids of constant densities, and the flow is assumed to be one-dimensional. Therefore, the equations to be solved are a coupled system composed of two Shallow Water models with source terms involving depth and breadth functions. Extensions of Roe’s Q-scheme are proposed where a suitable treatment of the coupling and source terms is performed by adapting the techniques developed in [Vázquez-Cendón ME. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J Comp Phys 1999;148:497–526; García-Navarro P, Vázquez-Cendón ME. On numerical treatment of the source terms in the shallow water equations. Comput Fluids 2000;29(8):17–45; Castro MJ, Macías J, Parés C. A Q-Scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system. Math Model Numer An 2001;35(1):107–27]. Finally we apply the numerical scheme to the simulation of the flow through the Strait of Gibraltar. Real bathymetric and coast-line data are considered to include in the model the main features of the abrupt geometry of this natural strait connecting the Atlantic Ocean and the Mediterranean Sea. A steady state solution is obtained from lock-exchange initial conditions. This solution is then used as initial condition to simulate the main semidiurnal and diurnal tidal waves in the Strait of Gibraltar through the imposition of suitable boundary conditions obtained from observed tidal data. Comparisons between numerical results and observed data and some tests on friction sensitivity are also presented.

Internal symmetric waves of two-layer fluids over an obstruction

We study two-dimensional capillary-gravity waves on the interface between two immiscible, inviscid and incompressible fluids of different constant densities bounded by two horizontal rigid boundaries with small obstructions with compact support. A forced modified Korteweg--de Vries equation is derived as a model equation without assuming that the fluid is of constant depth at far upstream. Various new types of steady solutions have been obtained numerically.

A Series-Solution Method for Free-Boundary Problems Arising from Flow Over Topography

An analytical series method is presented for steady, two-dimensional, irrotational flow of a single layer of constant-density fluid over topography. This problem is formulated as a Laplacian free-boundary problem with fully nonlinear boundary conditions. The method is an iterative scheme that allows the calculation of analytical series solutions for supercritical, transcritical and subcritical flow regimes over arbitrary topography. By an appropriate choice of the free-boundary representation, exponential convergence of the series solution is achieved. With this accuracy, the issue of apparent dual transcritical/subcritical solutions previously obtained by boundary-integral-equation methods (BIEM) is resolved. Results are compared with solutions previously obtained using BIEM, and solutions are presented for flow over asymmetric and arbitrarily shaped obstacles.

Trapped modes in a channel containing three layers of fluids and a submerged cylinder

The problem of existence of trapped waves in fluids due to a cylinder is investigated for the hydrodynamic set-up which involves a horizontal channel of infinite length and depth and of finite width containing three layers of incompressible fluids of different constant densities. The set-up also contains a cylinder which is impermeable, fully immersed in the bottom (lower-most) fluid layer of infinite depth, and extends across the channel with its generators perpendicular to the side walls of the channel. When the ratios of the densities of the adjacent fluids differ from unity by sufficiently small quantities, the underlying mathematical problem reduces to a generalized nonlinear eigenvalue problem involving a cubic polynomial-cum-operator equation. The perturbation analysis of this eigenvalue problem suggests existence of three distinct modes with different frequencies: one of the order of one persisting at the free surface, and the other two of the order of the density ratio (except for modulo one) persisting at the two internal interfaces. The correlation between these results for the three-layer case and very recent numerical results of other authors in the two-layer case has also been addressed.

Two-layer hydraulic falls over an obstacle

Motions in a forced channel flow of two contiguous homogeneous fluids of different constant densities and different thicknesses are considered. The total depth is finite and the upper surface is constrained to be planar (rigid lid approximation). The forcing is due to a bottom obstruction. The existence of a critical thickness ratio, obtained when the square of the thickness ratio is equal to the density ratio, leads to major differences with the one-layer case. The present study concentrates on this critical case. Moreover it is restricted to hydraulic falls, which are steady flows over an obstacle providing a transition between a subcritical and a supercritical flow. A weakly nonlinear analysis is performed. At leading order the problem reduces to a forced modified Korteweg–de Vries equation which can be integrated exactly. The weakly nonlinear results are validated by comparison with a numerical integration of the full governing equations. The numerical method is based on boundary integral equation techniques. The differences with the one-layer case are the existence of a second family of subcritical hydraulic falls when the thickness ratio is below critical, and the existence of supercritical hydraulic falls described by four parameters instead of three for all thickness ratios.

Numerical simulation of two-layer shallow water flows through channels with irregular geometry

This paper deals with the numerical simulation of flows of stratified fluids through channels with irregular geometry. Channel cross-sections are supposed to be symmetric but not necessarily rectangular. The fluid is supposed to be composed of two shallow layers of immiscible fluids of constant densities, and the flow is assumed to be one-dimensional. Therefore, the equations to be solved are a coupled system composed of two Shallow Water models with source terms involving depth and breadth functions. Extensions of the Q-schemes of van Leer and Roe are proposed where a suitable treatment of the coupling and source terms is performed by adapting the techniques developed in [J. Comput. Phys. 148 (1999) 497; Comput. Fluids 29(8) (2000) 17; Math. Model. Numer. Anal. 35(1) (2001) 107]. An enhanced consistency condition, the so-called C-property, introduced in [Comput. Fluids 23(8) (1994) 1049] is extended to this case and a general result providing sufficient conditions to ensure this property is shown. Then, some numerical tests to validate the resulting schemes are presented. First we verify that, in practice, the numerical schemes satisfy the C-property, even for extremely irregular channels. Then, in order to validate the schemes, we compare some approximate steady solutions obtained with the generalized Q-scheme of Van Leer with those obtained by using the asymptotic techniques developed by Armi and Farmer for channels with simplified geometries. Finally we apply the numerical scheme to the simulation of the flow through the Strait of Gibraltar. Real bathymetric and coast-line data are considered to include in the model the main features of the abrupt geometry of this natural strait connecting the Atlantic Ocean and the Mediterranean Sea. A steady-state solution is obtained from lock-exchange initial conditions. This solution is then used as initial condition to simulate the main semidiurnal and diurnal tidal waves in the Strait of Gibraltar through the imposition of suitable boundary conditions obtained from observed tidal data. Comparisons between numerical results and observed data are also presented.

Numerical simulation of mold filling in foam reaction injection molding

AbstractA numerical simulation of reaction injection molding (RIM) of polymeric foam is developed, using a finite volume method (FVM). In this study we predict mold filling with a variable‐density fluid that fills a mold by self‐expansion. We deal with two‐dimensional, isothermal cases. With the assumptions of ideal mixing and rapid bubble nucleation, the foam is modelled as a continuum with a time‐dependent density. The continuum is assumed to be a Newtonian fluid. We develop a pressure‐based FVM for unstructured meshes that includes the SIMPLE algorithm with treatment of fluid compressibility. Cell‐based, co‐located storage is used for all physical variables. To treat the moving interface, an explicit high‐resolution interface capturing method is used. Foam flow in a slit is investigated, and the numerical calculations are in good agreement with an approximate analytic solution. For fountain flow in a rectangular cavity, the shape of the flow front is flatter and the traces of the particles are more complicated for an expanding foam than for a constant‐density fluid. An example of mold filling by an expanding foam demonstrates the geometric flexibility of the method. Copyright © 2003 John Wiley & Sons, Ltd.

In-situ measurement of chemical shrinkage of MY750 epoxy resin by a novel gravimetric method

A novel approach has been developed to measure in-situ chemical shrinkage of epoxy resins at the temperature of cure, during which the epoxy resins pass through liquid, rubbery and glassy states. A small sample of MY750/HY917/DY073 epoxy resin system, sealed in a thin-walled silicone bag, was suspended in a pot of silicone fluid and weighed independently of the silicone bath. The buoyancy of the sample was monitored as its density increases with respect to the constant density fluid during isothermal cures at three different temperatures. The relationship between the chemical shrinkage and degree of cure was deduced from a cure kinetics model for the resin. The good match of the results for the three different cure cycles suggests that chemical shrinkage is only a function of degree of cure regardless of time and temperature. A bi-linear relationship was fitted to the experimental data. The break point is at a degree of cure of about 28%, with corresponding chemical shrinkage of 3%. This point is linked to the gel point of the resin, which was measured as approximately 25% degree of cure in previous work. Total cure shrinkage of 6.9% for the MY750 resin system was obtained by extrapolating the results to a degree of cure of 100%. The method is sensitive, reliable and keeps the resin stress-free; therefore it should be applicable to a wide range of materials.

Thermodynamics with long-range interactions: from Ising models to black holes.

Methods are presented which enables one to analyze the thermodynamics of systems with long-range interactions. Generically, such systems have entropies which are nonextensive (do not scale with the size of the system). We show how to calculate the degree of nonextensivity for such a system. We find that a system interacting with a heat reservoir is in a probability distribution of canonical ensembles. The system still possesses a parameter akin to a global temperature, which is constant throughout the substance. There is also a useful quantity which acts like a local temperatures and it varies throughout the substance. These quantities are closely related to counterparts found in general relativity. A lattice model with long-range spin-spin coupling is studied. This is compared with systems such as those encountered in general relativity and gravitating systems with Newtonian-type interactions. A long-range lattice model is presented which can be seen as a black hole analog. One finds that the analog's temperature and entropy have many properties which are found in black holes. Finally, the entropy scaling behavior of a gravitating perfect fluid of constant density is calculated. For weak interactions, the entropy scales like the volume of the system. As the interactions become stronger, the entropy becomes higher near the surface of the system, and becomes more area scaling.

Lattice Boltzmann model with nearly constant density.

An improved lattice Boltzmann model is developed to simulate fluid flow with nearly constant fluid density. The ingredient is to incorporate an extra relaxation for fluid density, which is realized by introducing a feedback equation in the equilibrium distribution functions. The pressure is dominated by the moving particles at a node, while the fluid density is kept nearly constant and explicit mass conservation is retained as well. Numerical simulation based on the present model for the (steady) plane Poiseuille flow and the (unsteady) two-dimensional Womersley flow shows a great improvement in simulation results over the previous models. In particular, the density fluctuation has been reduced effectively while achieving a relatively large pressure gradient.

A numerical study of the interactions between viscous flow, transport and kinetics in fixed bed reactors

A numerical method is tailored for the solution of a reduced set of model equations developed for the description of reactive flows in chemical reactors. The numerical method synthesis intends to utilize experience both in chemical engineering solving dispersion models containing complex reaction kinetics, phase equilibria and non-ideal thermodynamics, and in fluid dynamics applying classical numerical algorithms constructed for pure flow calculations. The two types of model equations involved in reactive flow simulations reflect very different physics. A modular method is therefore suggested enabling a split between the flow- and chemistry model parts. In this way more optimal solution methods can be adopted for each operator, in contrast to more traditional computational fluid dynamics (CFD) methods where all equations (and operators) are normally solved by the numerical solvers originally intended for pure flow calculations. In addition, emphasis has been put on modularity enabling the same model framework to be used both for the more traditional 1D and 2D dispersion models with simple 1D flow calculations, as well as for the more elaborate 2D and 3D reactive flow calculations. In the flow part of the algorithm a fractional-step algorithm is constructed for the simulation of dynamic reactive flows in chemical reactors. The numerical scheme is based on the compressible transport equations in the low-Mach-number limit. The method can handle real gas (and liquid) mixtures with variable density as well as constant density fluids. The velocity field is advanced using a projection scheme, which consists of a partial convection–diffusion update followed by a pressure correction step with an intermediate an-elastic filter. A variable density corrector step is implemented for variable density systems in order to couple the evolution of the density and the velocity fields. In the chemistry part of the model all scalar fields are updated using Strang-type operator-split integration steps that combine several explicit convection and semi-implicit diffusion transport operators with a suitable solver tailored for non-linear sink/source terms. Diffusion terms are discretized with a second order central difference scheme in space. Convection terms are discretized with a second order TVD scheme in space. The temperature advection is discretized using a second order upwind scheme. The chemical reaction part of the model is discretized by an implicit Euler approximation, and the resulting set of algebraic equations is solved using a Broyden subroutine. The other source terms are solved using an explicit Euler approximation. The performance and behavior of the operator-split scheme are assessed based on simulations of two industrial chemical processes (i.e. the synthesis gas and methanol production processes) performed in multi-tube fixed bed reactors. These processes are important parts of the Statoil methanol plant at Tjeldbergodden in mid Norway. Both 1D and 2D pseudo-homogeneous dispersion models (i.e. heat and mass balances) with prescribed velocity, mixture density and total pressure profiles are used describing the reactor performance. The predicted profiles for both the methanol and synthesis gas processes were in accordance with results reported in the literature. An oscillatory radial void fraction distribution was then implemented in the 2D model. It was found that the non-uniform void fraction distribution had no significant effect on the temperature and mole fraction profiles. A 1D CFD simulation was performed to evaluate the effect of the variations in velocity, pressure and mixture density on the reactor performance. The changes in these variables significantly effect the composition and conversion in the reactor. A 2D CFD model was then developed for further studies of the multi-tube fixed bed reactor for the synthesis gas process, analyzing the influence of non-uniform void fraction distributions on the flow and chemical conversion. The non-uniform void fraction distribution induces a significant reduction in axial pressure drop and a much higher fluid velocity close to the wall. However, the influence on the temperature and mole fraction profiles was hardly noticeable. Mass- and enthalpy budgets were implemented for the heat and mass quantities in the program. These budgets show that the enthalpy and both the mixture and component mass balances are fulfilled in the simulations. The numerical CFD algorithm developed in this paper has been found to be both computationally stable and accurate. Computational efficiency analysis shows that the Poisson solver requires about 75% of the total computational time. The computational time spend on the chemistry part of the model is about 15% of the total time. About 10% of the cost used on the chemistry part is spend on the chemistry solver.

A model for layer formation in stably stratified turbulence

Stably stratified turbulent flows are common in geophysics and astrophysics, and frequently exhibit layered structures in which large regions of nearly constant fluid density are separated by sharp density gradients. Experiments have demonstrated that, under suitable conditions, the stirring of a stably stratified fluid generates these layer structures. In this paper, a stochastic one-dimensional model is used to study layer formation in stably stratified turbulence. The results support mixing length arguments previously proposed to describe layers in steady state.