Polynomial Filtration Laws for Low Reynolds Number Flows Through Porous Media

In this study, we use the method of homogenization to develop a filtration law in porous media that includes the effects of inertia at finite Reynolds numbers. The result is much different than the empirically observed quadratic Forchheimer equation. First, the correction to Darcy’s law is initially cubic (not quadratic) for isotropic media. This is consistent with several other authors (Mei and Auriault, J Fluid Mech 222:647–663, 1991; Wodié and Levy, CR Acad Sci Paris t.312:157–161, 1991; Couland et al. J Fluid Mech 190:393–407, 1988; Rojas and Koplik, Phys Rev 58:4776–4782, 1988) who have solved the Navier–Stokes equations analytically and numerically. Second, the resulting filtration model is an infinite series polynomial in velocity, instead of a single corrective term to Darcy’s law. Although the model is only valid up to the local Reynolds number, at the most, of order 1, the findings are important from a fundamental perspective because it shows that the often-used quadratic Forchheimer equation is not a universal law for laminar flow, but rather an empirical one that is useful in a limited range of velocities. Moreover, as stated by Mei and Auriault (J Fluid Mech 222:647–663, 1991) and Barree and Conway (SPE Annual technical conference and exhibition, 2004), even if the quadratic model were valid at moderate Reynolds numbers in the laminar flow regime, then the permeability extrapolated on a Forchheimer plot would not be the intrinsic Darcy permeability. A major contribution of this study is that the coefficients of the polynomial law can be derived a priori, by solving sequential Stokes problems. In each case, the solution to the Stokes problem is used to calculate a coefficient in the polynomial, and the velocity field is an input of the forcing function, F, to subsequent problems. While numerical solutions must be utilized to compute each coefficient in the polynomial, these problems are much simpler and robust than solving the full Navier–Stokes equations.     

Numerical investigation of the convective flow in the toroidal gap

A numerical investigation of the convective flow in the toroidal gap is presented. A new formulation of the incompressible Navier–Stokes equation in terms of an auxiliary field that differs from the velocity by a gauge transformation [Weinen and Liu in Commun Math Sci 1(2):317–332, 2003] has been used. The gauge freedom allows simple boundary conditions to be formulated for the auxiliary field, as well as the gauge field. The gauge field eliminates the pressure distribution in the Navier–Stokes equation. The influence of the geometric parameters and the Prandtl number is discussed.     

Derivation of a Darcy’s Law for a Porous Medium Composed of Two Solid Phases Saturated by a Single- Phase Fluid: A Homogenization Approach

The objective of this article is to derive a macroscopic Darcy’s law for a fluid-saturated moving porous medium whose matrix is composed of two solid phases which are not in direct contact with each other (weakly coupled solid phases). An example of this composite medium is the case of a solid matrix, unfrozen water, and an ice matrix within the pore space. The macroscopic equations for this type of saturated porous material are obtained using two-space homogenization techniques from microscopic periodic structures. The pore size is assumed to be small compared to the macroscopic scale under consideration. At the microscopic scale the two weakly coupled solids are described by the linear elastic equations, and the fluid by the linearized Navier–Stokes equations with appropriate boundary conditions at the solid–fluid interfaces. The derived Darcy’s law contains three permeability tensors whose properties are analyzed. Also, a formal relation with a previous macroscopic fluid flow equation obtained using a phenomenological approach is given. Moreover, a constructive proof of the existence of the three permeability tensors allows for their explicit computation employing finite elements or analogous numerical procedures.     

Computational Modeling of Fluid Flow through a Fracture in Permeable Rock

Laminar, single-phase, finite-volume solutions to the Navier–Stokes equations of fluid flow through a fracture within permeable media have been obtained. The fracture geometry was acquired from computed tomography scans of a fracture in Berea sandstone, capturing the small-scale roughness of these natural fluid conduits. First, the roughness of the two-dimensional fracture profiles was analyzed and shown to be similar to Brownian fractal structures. The permeability and tortuosity of each fracture profile was determined from simulations of fluid flow through these geometries with impermeable fracture walls. A surrounding permeable medium, assumed to obey Darcy’s Law with permeabilities from 0.2 to 2,000 millidarcies, was then included in the analysis. A series of simulations for flows in fractured permeable rocks was performed, and the results were used to develop a relationship between the flow rate and pressure loss for fractures in porous rocks. The resulting friction-factor, which accounts for the fracture geometric properties, is similar to the cubic law; it has the potential to be of use in discrete fracture reservoir-scale simulations of fluid flow through highly fractured geologic formations with appreciable matrix permeability. The observed fluid flow from the surrounding permeable medium to the fracture was significant when the resistance within the fracture and the medium were of the same order. An increase in the volumetric flow rate within the fracture profile increased by more than 5% was observed for flows within high permeability-fractured porous media.     

Energy-Conserving Simulation of Incompressible Electro-Osmotic and Pressure-Driven Flow

A numerical model for electro-osmotic flow is described. The advecting velocity field is computed by solving the incompressible Navier–Stokes equation. The method uses a semi-implicit multigrid algorithm to compute the divergence-free velocity at each grid point. The finite differences are second-order accurate and centered in space; however, the traditional second-order compact finite differencing of the Poisson equation for the pressure field is shown not to conserve energy in the inviscid limit. We have designed a non-compact finite differencing for the Laplacian in the pressure equation that allows exact energy conservation and affords second-order accuracy. The model also incorporates a new numerical method for passive scalar advection, called parcel advection, which accurately predicts the evolution of a passively traveling scalar pulse without requiring the addition of any artificial diffusion. The algorithm is used to confirm the experimentally observed asymmetric concentration profile that arises when an external pressure drop is imposed on electro-osmotic flow. Received 25 January 2001 and accepted 10 May 2002 Published online 30 October 2002 Communicated by H.J.S. Fernando     

On the MHD flow of fractional generalized Burgers＇ fluid with modified Darcy＇s law

This work is concerned with applying the fractional calculus approach to the magnetohydrodynamic (MHD) pipe flow of a fractional generalized Burgers’ fluid in a porous space by using modified Darcy’s relationship. The fluid is electrically conducting in the presence of a constant applied magnetic field in the transverse direction. Exact solution for the velocity distribution is developed with the help of Fourier transform for fractional calculus. The solutions for a Navier–Stokes, second grade, Maxwell, Oldroyd-B and Burgers’ fluids appear as the limiting cases of the present analysis.     

A New Derivation of Jeffery’s Equation

In this article, we present a modern derivation of Jeffery’s equation for the motion of a small rigid body immersed in a Navier–Stokes flow, using methods of asymptotic analysis. While Jeffery’s result represents the leading order equations of a singularly perturbed flow problem involving ellipsoidal bodies, our formulation is for bodies of general shape and we also derive the equations of the next relevant order.      

Numerical simulation of shock wave propagation in microchannels using continuum and kinetic approaches

Numerical simulations of shock wave propagation in microchannels and microtubes (viscous shock tube problem) have been performed using three different approaches: the Navier–Stokes equations with the velocity slip and temperature jump boundary conditions, the statistical Direct Simulation Monte Carlo method for the Boltzmann equation, and the model kinetic Bhatnagar–Gross–Krook equation with the Shakhov equilibrium distribution function. Effects of flow rarefaction and dissipation are investigated and the results obtained with different approaches are compared. A parametric study of the problem for different Knudsen numbers and initial shock strengths is carried out using the Navier–Stokes computations.      

Kramers’ problem and the Knudsen minimum: a theoretical analysis using a linearized 26-moment approach

A set of linearized 26 moment equations, along with their wall boundary conditions, are derived and used to study low-speed gas flows dominated by Knudsen layers. Analytical solutions are obtained for Kramers’ defect velocity and the velocity-slip coefficient. These results are compared to the numerical solution of the BGK kinetic equation. From the analysis, a new effective viscosity model for the Navier–Stokes equations is proposed. In addition, an analytical expression for the velocity field in planar pressure-driven Poiseuille flow is derived. The mass flow rate obtained from integrating the velocity profile shows good agreement with the results from the numerical solution of the linearized Boltzmann equation. These results are good for Knudsen numbers up to 3 and for a wide range of accommodation coefficients. The Knudsen minimum phenomenon is also well captured by the present linearized 26-moment equations.     

Stability for the 3D Navier–Stokes Equations with Nonzero far Field Velocity on Exterior Domains

Concerning to the non-stationary Navier–Stokes flow with a nonzero constant velocity at infinity, just a few results have been obtained, while most of the results are for the flow with the zero velocity at infinity. The temporal stability of stationary solutions for the Navier–Stokes flow with a nonzero constant velocity at infinity has been studied by Enomoto and Shibata (J Math Fluid Mech 7:339–367, 2005), in L ^p spaces for p ≥ 3. In this article, we first extend their result to the case \frac32 < p{\frac{3}{2} < p} by modifying the method in Bae and Jin (J Math Fluid Mech 10:423–433, 2008) that was used to obtain weighted estimates for the Navier–Stokes flow with the zero velocity at infinity. Then, by using our generalized temporal estimates we obtain the weighted stability of stationary solutions for the Navier–Stokes flow with a nonzero velocity at infinity.     

Comparison of low Mach number models for natural convection problems

 We investigate in this paper two numerical methods for solving low Mach number compressible flows and their application to single-phase natural convection flow problems. The first method is based on an asymptotic model of the Navier–Stokes equations valid for small Mach numbers, whereas the second is based on the full compressible Navier–Stokes equations with particular care given to the discretization at low Mach numbers. These models are more general than the Boussinesq incompressible flow model, in the sense that they are valid even for cases in which the fluid is subjected to large temperature differences, that is when the compressibility of the fluid manifests itself through low Mach number effects. Numerical solutions are computed for a series of test problems with fixed Rayleigh number and increasing temperature differences, as well as for varying Rayleigh number for a given temperature difference. Numerical difficulties associated with low Mach number effects are discussed, as well as the accuracy of the approximations. Received on 17 January 2000     

On the stability and extension of reduced-order Galerkin models in incompressible flows

Proper orthogonal decomposition (POD) has been used to develop a reduced-order model of the hydrodynamic forces acting on a circular cylinder. Direct numerical simulations of the incompressible Navier–Stokes equations have been performed using a parallel computational fluid dynamics (CFD) code to simulate the flow past a circular cylinder. Snapshots of the velocity and pressure fields are used to calculate the divergence-free velocity and pressure modes, respectively. We use the dominant of these velocity POD modes (a small number of eigenfunctions or modes) in a Galerkin procedure to project the Navier–Stokes equations onto a low-dimensional space, thereby reducing the distributed-parameter problem into a finite-dimensional nonlinear dynamical system in time. The solution of the reduced dynamical system is a limit cycle corresponding to vortex shedding. We investigate the stability of the limit cycle by using long-time integration and propose to use a shooting technique to home on the system limit cycle. We obtain the pressure-Poisson equation by taking the divergence of the Navier–Stokes equation and then projecting it onto the pressure POD modes. The pressure is then decomposed into lift and drag components and compared with the CFD results.     

Asymptotic Properties of Axis-Symmetric D-Solutions of the Navier–Stokes Equations

We consider asymptotic behavior of Leray’s solution which expresses axis-symmetric incompressible Navier–Stokes flow past an axis-symmetric body. When the velocity at infinity is prescribed to be nonzero constant, Leray’s solution is known to have optimum decay rate, which is in the class of physically reasonable solution. When the velocity at infinity is prescribed to be zero, the decay rate at infinity has been shown under certain restrictions such as smallness on the data. Here we find an explicit decay rate when the flow is axis-symmetric by decoupling the axial velocity and the horizontal velocities. The first author was supported by KRF-2006-312-C00466. The second author was supported by KRF-2006-531-C00009.     

Homogenization of Wall-Slip Gas Flow Through Porous Media

The permeability of reservoir rocks is most commonly measured with an atmospheric gas. Permeability is greater for a gas than for a liquid. The Klinkenberg equation gives a semi-empirical relation between the liquid and gas permeabilities. In this paper, the wall-slip gas flow problem is homogenized. This problem is described by the steady state, low velocity Navier–Stokes equations for a compressible gas with a small Knudsen number. Darcy's law with a permeability tensor equal to that of liquid flow is shown to be valid to the lowest order. The lowest order wall-slip correction is a local tensorial form of the Klinkenberg equation. The Klinkenberg permeability is a positive tensor. It is in general not symmetric, but may under some conditions, which we specify, be symmetric. Our result reduces to the Klinkenberg equation for constant viscosity gas flow in isotropic media.     

Unsteady PTV velocity field past an airfoil solved with DNS: Part 1. Algorithm of hybrid simulation and hybrid velocity field at <Emphasis Type="Italic">Re</Emphasis> ≈ 10<Superscript>3</Superscript>

We develop a hybrid unsteady-flow simulation technique combining direct numerical simulation (DNS) and particle tracking velocimetry (PTV) and demonstrate its capabilities by investigating flows past an airfoil. We rectify instantaneous PTV velocity fields in a least-squares sense so that they satisfy the equation of continuity, and feed them to the DNS by equating the computational time step with the frame rate of the time-resolved PTV system. As a result, we can reconstruct unsteady velocity fields that satisfy the governing equations based on experimental data, with the resolution comparable to numerical simulation. In addition, unsteady pressure distribution can be solved simultaneously. In this study, particle velocities are acquired on a laser-light sheet in a water tunnel, and unsteady flow fields are reconstructed with the hybrid algorithm solving the incompressible Navier–Stokes equations in two dimensions. By performing the hybrid simulation, we investigate nominally two-dimensional flows past the NACA0012 airfoil at low Reynolds numbers. In part 1, we introduce the algorithm of the proposed technique and discuss the characteristics of hybrid velocity fields. In particular, we focus on a vortex shedding phenomenon under a deep stall condition (α = 15°) at Reynolds numbers of Re = 1000 and 1300, and compare the hybrid velocity fields with those computed with two-dimensional DNS. In part 2, the extension to higher Reynolds numbers is considered. The accuracy of the hybrid simulation is evaluated by comparing with independent experimental results at various angles of attack and Reynolds numbers up to Re = 10⁴. The capabilities of the hybrid simulation are also compared with two-dimensional unsteady Reynolds-Averaged Navier–Stokes (URANS) solutions in part 2. In the first part of these twin papers, we demonstrate that the hybrid velocity field approaches the PTV velocity field over time. We find that intensive alternate vortex shedding past the airfoil, which is predicted by the two-dimensional DNS, is substantially suppressed in the hybrid simulation and the resultant flow field is similar to the PTV velocity field, which is projection of the three-dimensional velocity field on the streamwise plane. We attempt to identify the motion that originates three-dimensional flow patterns by highlighting the deviation of the PTV velocity field from the two-dimensional governing equations at each snapshot. The results indicate that the intensive spots of the deviation appear in the regions in which three-dimensional instabilities are induced in the shear layer separated from the pressure side.     

Quasi-Neutral Limit for a Model of Viscous Plasma

We perform a rigorous analysis of the quasi-neutral limit for a model of viscous plasma represented by the Navier–Stokes–Poisson system of equations. It is shown that the limit problem is the Navier–Stokes system describing a barotropic fluid flow, with the pressure augmented by a component related to the nonlinearity in the original Poisson equation.     

2D Navier–Stokes Equations in a Time Dependent Domain with Neumann Type Boundary Conditions

In this paper the two-dimensional Navier–Stokes system for incompressible fluid coupled with a parabolic equation through the Neumann type boundary condition for the second component of the velocity is considered. Navier–Stokes equations are defined on a given time dependent domain. We prove the existence of a weak solution for this system. In addition, we prove the continuous dependence of solutions on the data for a regularized version of this system. For a special case of this regularized system also a problem with an unknown interface is solved.     

Large Eddy Simulations Using the Subgrid-Scale Estimation Model and Truncated Navier–Stokes Dynamics

We describe a procedure for large eddy simulations of turbulence which uses the subgrid-scale estimation model and truncated Navier–Stokes dynamics. In the procedure the large eddy simulation equations are advanced in time with the subgrid-scale stress tensor calculated from the parallel solution of the truncated Navier–Stokes equations on a mesh two times smaller in each Cartesian direction than the mesh employed for a discretization of the resolved quantities. The truncated Navier–Stokes equations are solved through a sequence of runs, each initialized using the subgrid-scale estimation model. The modeling procedure is evaluated by comparing results of large eddy simulations for isotropic turbulence and turbulent channel flow with the corresponding results of experiments, theory, direct numerical simulations, and other large eddy simulations. Subsequently, simplifications of the general procedure are discussed and evaluated. In particular, it is possible to formulate the procedure entirely in terms of the truncated Navier–Stokes equation and a periodic processing of the small-scale component of its solution. Received 27 April 2001 and accepted 16 December 2001     

Finite-Amplitude Equilibrium Solutions for Plane Poiseuille–Couette Flow

Two-dimensional nonlinear equilibrium solutions for the plane Poiseuille–Couette flow are computed by directly solving the full Navier–Stokes equations as a nonlinear eigenvalue problem. The equations are solved using the two-point fourth-order compact scheme and the Newton–Raphson iteration technique. The linear eigenvalue computations show that the combined Poiseuille–Couette flow is stable at all Reynolds numbers when the Couette velocity component σ₂ exceeds 0.34552. Starting with the neutral solution for the plane Poiseuille flow, the nonlinear neutral surfaces for the combined Poiseuille–Couette flow were mapped out by gradually increasing the velocity component σ₂. It is found that, for small σ₂, the neutral surfaces stay in the same family as that for the plane Poiseuille flow, and the nonlinear critical Reynolds number gradually increases with increasing σ₂. When the Couette velocity component is increased further, the neutral curve deviates from that for the Poiseuille flow with an appearance of a new loop at low wave numbers and at very low energy. By gradually increasing the σ₂ values at a constant Reynolds number, the nonlinear critical Reynolds numbers were determined as a function of σ₂. The results show that the nonlinear neutral curve is similar in shape to a linear case. The critical Reynolds number increases slowly up to σ₂∼ 0.2 and remains constant until σ₂∼ 0.58. Beyond σ₂ > 0.59, the critical Reynolds number increases sharply. From the computed results it is concluded that two-dimensional nonlinear equilibrium solutions do not exist beyond a critical σ₂ value of about 0.59. Received: 26 November 1996 and accepted 12 May 1997     

Quadrant analysis of persistent spatial velocity perturbations over square-bar roughness

We explore a new application of the quadrant method in the context of the double-averaged Navier–Stokes equations for studying open channel flow near rough beds. Quadrant analysis is applied to spatial disturbances of time-averaged velocity components, using the experimental data from flow over two-dimensional regular transverse square-bar roughness. The spatial velocity disturbances change periodically performing a full cycle over a single roughness element, so that the quadrant diagrams are regular closed orbits. A colour code is used to produce a quadrant map of the flow cross-section, which reveals contributions from each quadrant to the time-averaged momentum transfer.