Structures and Instabilities in Reaction Fronts Separating Fluids of Different Densities

Density gradients across reaction fronts propagating vertically can lead to Rayleigh–Taylor instabilities. Reaction fronts can also become unstable due to diffusive instabilities, regardless the presence of a mass density gradient. In this paper, we study the interaction between density driven convection and fronts with diffusive instabilities. We focus in fluids confined in Hele–Shaw cells or porous media, with the hydrodynamics modeled by Brinkman’s equation. The time evolution of the front is described with a Kuramoto–Sivashinsky (KS) equation coupled to the fluid velocity. A linear stability analysis shows a transition to convection that depends on the density differences between reacted and unreacted fluids. A stabilizing density gradient can surpress the effects of diffusive instabilities. The two-dimensional numerical solutions of the nonlinear equations show an increase of speed due to convection. Brinkman’s equation lead to the same results as Darcy’s laws for narrow gap Hele–Shaw cells. For large gaps, modeling the hydrodynamics using Stokes’ flow lead to the same results.

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Modeling Transient Flows in Heterogeneous Layered Porous Media Using the Space–Time Trefftz Method

Applied Sciences ◽

10.3390/app11083421 ◽

2021 ◽

Vol 11 (8) ◽

pp. 3421

Author(s):

Cheng-Yu Ku ◽

Li-Dan Hong ◽

Chih-Yu Liu ◽

Jing-En Xiao ◽

Wei-Po Huang

Keyword(s):

Porous Media ◽

Time Domain ◽

Numerical Solutions ◽

Space Time ◽

Two Dimensional ◽

Transient Flows ◽

Layered Porous Media ◽

Time Marching ◽

Time Basis ◽

Marching Scheme

In this study, we developed a novel boundary-type meshless approach for dealing with two-dimensional transient flows in heterogeneous layered porous media. The novelty of the proposed method is that we derived the Trefftz space–time basis function for the two-dimensional diffusion equation in layered porous media in the space–time domain. The continuity conditions at the interface of the subdomains were satisfied in terms of the domain decomposition method. Numerical solutions were approximated based on the superposition principle utilizing the space–time basis functions of the governing equation. Using the space–time collocation scheme, the numerical solutions of the problem were solved with boundary and initial data assigned on the space–time boundaries, which combined spatial and temporal discretizations in the space–time manifold. Accordingly, the transient flows through the heterogeneous layered porous media in the space–time domain could be solved without using a time-marching scheme. Numerical examples and a convergence analysis were carried out to validate the accuracy and the stability of the method. The results illustrate that an excellent agreement with the analytical solution was obtained. Additionally, the proposed method was relatively simple because we only needed to deal with the boundary data, even for the problems in the heterogeneous layered porous media. Finally, when compared with the conventional time-marching scheme, highly accurate solutions were obtained and the error accumulation from the time-marching scheme was avoided.

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Fluid Flow and Lateral Fluid Dispersion in Bounded Granular Beds

Heat Transfer, Volume 1 ◽

10.1115/imece2002-39395 ◽

2002 ◽

Author(s):

Azita Soleymani ◽

Eveliina Takasuo ◽

Piroz Zamankhan ◽

William Polashenski

Keyword(s):

Reynolds Number ◽

Porous Media ◽

Numerical Solutions ◽

Stokes Equations ◽

Numerical Study ◽

Fluid Velocity ◽

Three Dimensional ◽

Numerical Results ◽

Transition To Turbulence ◽

Wide Range

Results are presented from a numerical study examining the flow of a viscous, incompressible fluid through random packing of nonoverlapping spheres at moderate Reynolds numbers (based on pore permeability and interstitial fluid velocity), spanning a wide range of flow conditions for porous media. By using a laminar model including inertial terms and assuming rough walls, numerical solutions of the Navier-Stokes equations in three-dimensional porous packed beds resulted in dimensionless pressure drops in excellent agreement with those reported in a previous study (Fand et al., 1987). This observation suggests that no transition to turbulence could occur in the range of Reynolds number studied. For flows in the Forchheimer regime, numerical results are presented of the lateral dispersivity of solute continuously injected into a three-dimensional bounded granular bed at moderate Peclet numbers. Lateral fluid dispersion coefficients are calculated by comparing the concentration profiles obtained from numerical and analytical methods. Comparing the present numerical results with data available in the literature, no evidence has been found to support the speculations by others for a transition from laminar to turbulent regimes in porous media at a critical Reynolds number.

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Fluid Flows Through Two-Dimensional Irregular Composite Channels

10.1115/imece2000-2105 ◽

2000 ◽

Author(s):

A. K. Al-Hadhrami ◽

L. Elliott ◽

D. B. Ingham ◽

X. Wen

Keyword(s):

Porous Media ◽

Fluid Flow ◽

Numerical Solutions ◽

Control Volume ◽

Brinkman Equation ◽

Channel Height ◽

Two Dimensional ◽

Composite Channels ◽

Constant Height ◽

Flow Through

Abstract The present analysis is concerned with the study of two-dimensional fluid flow problems through channels of irregular composite materials. The fluid is assumed to be steady, incompressible, with a negligible gravitational force, and is constrained to flow in an infinite long channel in which the height assumes a series of piecewise constant values. An analytical study in the fully developed section of the composite channel is presented when the channel is of constant height and composed of several layers of porous media, each of uniform porosity. Numerical solutions are utilised using CFD based on the control volume method to solve the Brinkman equation, which governs fluid flow through porous media. In the fully developed flow regime the analytical and numerical solutions are graphically indistinguishable. A geometrical configuration involving several discontinuities of channel height, and where the entry and exit sections are layered, is considered and the effect of different permeabilities is demonstrated. Several numerical investigations which form a first attempt to mathematically model some geological structures, e.g. a fault or a fracture, are performed. Further, flow through fractures composed of randomly generated permeability values are also discussed and the effect on the overall pressure gradient is considered.

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Solution of two dimensional Stokes flow with elliptical coordinates and its application to permeability of porous media

Journal of Applied Mechanics ◽

10.1115/1.4050687 ◽

2021 ◽

pp. 1-18

Author(s):

Mustapha Hellou ◽

Franck Lominé ◽

Mohamed Khaled Bourbatache ◽

Mohamed Hajjam

Keyword(s):

Porous Media ◽

Stokes Flow ◽

Biharmonic Equation ◽

Fluid Flows ◽

Gegenbauer Polynomials ◽

Series Expansions ◽

Two Dimensional ◽

Elliptic Coordinates ◽

Elliptical Coordinates

Abstract Analytical developments of the biharmonic equation representing bi-dimensional Stokes flow are realized with elliptic coordinates. It's found that the streamfunction is expressed with series expansions based on Gegenbauer polynomials of first and second kinds with order one Cn1and Dn1. A term corresponding to order n=-1 is added in view to create drag on a body around which the fluid flows. Application to an array of elliptic cylinders is made and the permeability of this medium is determined as a function of porosity.

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Microscopic flow near the surface of two-dimensional porous media. Part 2. Transverse flow

Journal of Fluid Mechanics ◽

10.1017/s0022112087001149 ◽

1987 ◽

Vol 178 ◽

pp. 119-136 ◽

Cited By ~ 138

Author(s):

R. E. Larson ◽

J. J. L. Higdon

Keyword(s):

Porous Media ◽

Model Problem ◽

Axial Flow ◽

Transverse Flow ◽

Two Dimensional ◽

Brinkman’S Equation ◽

Cylindrical Inclusions ◽

Microscopic Flow ◽

Slip Coefficients

A model problem is analysed to study the microscopic flow near the surface of porous media. In the idealized system, we consider two-dimensional media consisting of infinite and semi-infinite periodic lattices of cylindrical inclusions. In Part 1, results for axial flow were presented. In this work results for transverse flow are presented and discussed in the context of macroscopic approaches such as slip coefficients and Brinkman's equation.

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A New Two Phase Extension of Modified Brinkman Formulation for Fluid Flow through Porous Media

Fluids Engineering ◽

10.1115/imece2005-81777 ◽

2005 ◽

Author(s):

Hadi Belhaj ◽

Shabbir Mustafiz ◽

Fuxi Ma ◽

M. R. Islam

Keyword(s):

Porous Media ◽

Fluid Flow ◽

Petroleum Reservoirs ◽

Two Dimensional ◽

Transient Pressure ◽

Two Phase ◽

Phase Saturation ◽

Brinkman’S Equation ◽

The Difference ◽

Relative Permeability Curves

In porous media research, Modified Brinkman’s equation is a very recent development. It is important as it incorporates the concept of viscous effect to inertial effect in a fluid flow system when Darcy’s, Forchheimer’s and Brinkman’s terms are brought all together. So far, researchers have developed the modified equation in its two-dimensional forms; however, limited to only one phase. In reality, petroleum reservoirs experience the multiphase conditions. Therefore, the simulation of a multidimensional, multiphase scenario is mostly desired, the highlight of this paper. The paper presents the formulation of two-dimensional, transient pressure and saturation equations for oil and water phases, one equation for each phase. The difference between phases is noticeable explicitly in their respective saturation, permeability, viscosity and velocity terms. The equations are then solved numerically to generate relative permeability curves. The simultaneous solution of pressure and saturation terms in the governing equations required additional relationships: the phase saturation constraint and capillary pressure as function of saturation. Finally, the numerical results are compared and validated with the experimental results. The implication of this study is manifold. The formulated equations including the solution part for the multiphase conditions are new. The new comprehensive model will describe fluid flow in reservoirs prone to high velocity or fractures more accurately than ever described by Darcy’s or other aforementioned equations.

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Two-dimensional Brinkman flows and their relation to analogous Stokes flows

IMA Journal of Applied Mathematics ◽

10.1093/imamat/hxz020 ◽

2019 ◽

Vol 84 (5) ◽

pp. 912-929 ◽

Cited By ~ 1

Author(s):

P A Martin

Keyword(s):

Circular Cylinder ◽

Stokes Flow ◽

Fluid Velocity ◽

Additional Term ◽

Far Field ◽

Two Dimensional ◽

Logarithmic Divergence ◽

Stokes Flows ◽

Flow Past ◽

Logarithmic Growth

Abstract 2D Stokes flows often exhibit the Stokes paradox: logarithmic growth of the fluid velocity in the far field. Analogous Brinkman flows are governed by the same equations apart from an additional term involving a parameter $\alpha$. Although these equations reduce to those for Stokes flow when $\alpha =0$, we show that the Brinkman solutions do not approach the corresponding Stokes solutions as $\alpha \to 0$; instead, logarithmic divergence with $\alpha$ is found. We also show that Brinkman flows do not exhibit a Stokes-like paradox. These results are given in detail for two specific problems, namely flow past a rigid circular cylinder and flow past a thin rigid strip.

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Two-Dimensional Incompressible Flows by Velocity-Vorticity Formulation and Finite Element Method

Journal of Mechanics ◽

10.1017/s1727719100002379 ◽

2001 ◽

Vol 17 (1) ◽

pp. 13-20 ◽

Cited By ~ 1

Author(s):

Der-Chang Lo ◽

Der-Liang Young

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Stokes Flow ◽

Incompressible Flows ◽

Numerical Solutions ◽

Computational Fluid Mechanics ◽

Two Dimensional ◽

Square Cavity ◽

Element Analysis ◽

Element Method

ABSTRACTIn this study, the motion of incompressible viscous fluid in a two-dimensional domain is solved by the finite element method using the velocity-vorticity formulation. To demonstrate the model feasibility, first of all the steady Stokes flow in a square cavity are computed. The results of square cavity flow are comparable with the numerical solutions of Burggraff (1966, FDM) [1]. Then the unsteady Navier-Stokes flow are computed and compared with other models [1∼4]. The results reveal that finite element analysis is a very powerful approach in the realm of computational fluid mechanics.

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