Showing papers on "Incompressible flow published in 2009"

Modeling and simulation of pore-scale multiphase fluid flow and reactive transport in fractured and porous media

Abstract: In the subsurface fluids play a critical role by transporting dissolved minerals, colloids and contaminants (sometimes over long distances), by mediating dissolution and precipitation processes and enabling chemical transformations in solution and at mineral surfaces. Although the complex geometries of fracture apertures, fracture networks and pore spaces may make it difficult to accurately predict fluid flow in saturated (single-phase) subsurface systems, well developed methods are available. The simulation of multiphase fluid flow in the subsurface is much more challenging because of the large density and/or viscosity ratios found in important applications (water/air in the vadose zone, water/oil, water/gas, gas/oil and water/oil/gas in oil reservoirs, water/air/non-aqueous phase liquids (NAPL) in contaminated vadose zone systems and gas/molten rock in volcanic systems, for example). In addition, the complex behavior of fluid-fluid-solid contact lines, and its impact on dynamic contact angles, must also be taken into account, and coupled with the fluid flow. Pore network models and simple statistical physics based models such as the invasion percolation and diffusion-limited aggregation models have been used quite extensively. However, these models for multiphase fluid flow are based on simplified models for pore space geometries and simplified physics. Other methods such a lattice Boltzmann and lattice gasmore » models, molecular dynamics, Monte Carlo methods, and particle methods such as dissipative particle dynamics and smoothed particle hydrodynamics are based more firmly on first principles, and they do not require simplified pore and/or fracture geometries. However, they are less (in some cases very much less) computationally efficient that pore network and statistical physics models. Recently a combination of continuum computation fluid dynamics, fluid-fluid interface tracking or capturing and simple models for the dependence of contact angles on fluid velocity at the contact line has been used to simulate multiphase fluid flow in fracture apertures, fracture networks and pore spaces. Fundamental conservation principles - conservation of momentum, and conservation of mass (or conservation of volume for incompressible fluids) and conservation of energy, as well as symmetries (Galilean invariance and isotropy) are central to the physics of fluids and the models used to simulate them. In molecular and mesoscale models observance of these conservation principles and symmetries at the microscopic level leads to macroscopic fluid dynamics that can be represented by the Navier Stokes equation. The remarkable fact that the flow of all simpe fluids, irrespective of their chemical nature, can be described by the Navier-Stokes equation is a result of these conservation principles and symmetries acting on the molecular level.« less

On a Diffuse Interface Model for Two-Phase Flows of Viscous, Incompressible Fluids with Matched Densities

Abstract: We study a diffuse interface model for the flow of two viscous incompressible Newtonian fluids of the same density in a bounded domain. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account. This leads to a coupled Navier–Stokes/Cahn–Hilliard system, which is capable of describing the evolution of droplet formation and collision during the flow. We prove the existence of weak solutions of the non-stationary system in two and three space dimensions for a class of physical relevant and singular free energy densities, which ensures—in contrast to the usual case of a smooth free energy density—that the concentration stays in the physical reasonable interval. Furthermore, we find that unique “strong” solutions exist in two dimensions globally in time and in three dimensions locally in time. Moreover, we show that for any weak solution the concentration is uniformly continuous in space and time. Because of this regularity, we are able to show that any weak solution becomes regular for large times and converges as t → ∞ to a solution of the stationary system. These results are based on a regularity theory for the Cahn–Hilliard equation with convection and singular potentials in spaces of fractional time regularity as well as on maximal regularity of a Stokes system with variable viscosity and forces in L2(0, ∞; Hs(Ω)), \({s \in [0, \frac12)}\) , which are new themselves.

Added Mass Effects of Compressible and Incompressible Flows in Fluid-Structure Interaction

Abstract: The subiteration method which forms the basic iterative procedure for solving fluid structure-interaction problems is based on a partitioning of the fluid-structure system into a fluidic part and a structural part. In fluid-structure interaction, on short time scales the fluid appears as an added mass to the structural operator, and the stability and convergence properties of the subiteration process depend significantly on the ratio of this apparent added mass to the actual structural mass. In the present paper, we establish that the added-mass effects corresponding to compressible and incompressible flows are fundamentally different. For a model problem, we show that on increasingly small time intervals, the added mass of a compressible flow is proportional to the length of the time interval, whereas the added mass of an incompressible flow approaches a constant. We then consider the implications of this difference in proportionality for the stability and convergence properties of the subiteration process, and for the stability and accuracy of loosely-coupled staggered time-integration methods.

BKM's Criterion and Global Weak Solutions for Magnetohydrodynamics with Zero Viscosity

Abstract: In this paper we derive a criterion for the breakdown of classical solutions to the incompressible magnetohydrodynamic equations with zero viscosity and positive resistivity in $\mathbb{R}^3$. This result is analogous to the celebrated Beale-Kato-Majda's breakdown criterion for the inviscid Eluer equations of incompressible fluids. In $\mathbb{R}^2$ we establish global weak solutions to the magnetohydrodynamic equations with zero viscosity and positive resistivity for initial data in Sobolev space $H^1(\mathbb{R}^2)$.

An Introduction to the Mechanics of Fluids

Abstract: Preface- General References- Bodies, Configurations, and Motions- Kinematics and Basic Laws- Constitutive Equations, Reduced Constitutive Equations, and Internal Constraints- Simple Fluids- Flows of Incompressible Fluids in General- Some Flows of Particular Nonlinear Fluids- Some Flows of Fluids of Grade 2- Navier-Stokes Fluids- Incompressible Euler Fluids- Compressible Euler Fluids- Singular Surfaces and Waves- Some Elementary Results from Real Analysis- Index

BKM's Criterion and Global Weak Solutions for Magnetohydrodynamics with Zero Viscosity

Abstract: In this paper we derive a criterion for the breakdown of classical solutions to the incompressible magnetohydrodynamic equations with zero viscosity and positive resistivity in $\mathbb{R}^3$. This result is analogous to the celebrated Beale-Kato-Majda's breakdown criterion for the inviscid Eluer equations of incompressible fluids. In $\mathbb{R}^2$ we establish global weak solutions to the magnetohydrodynamic equations with zero viscosity and positive resistivity for initial data in Sobolev space $H^1(\mathbb{R}^2)$.

Structure-Preserving Discretization of Incompressible Fluids

Abstract: The geometric nature of Euler fluids has been clearly identified and extensively studied over the years, culminating with Lagrangian and Hamiltonian descriptions of fluid dynamics where the configuration space is defined as the volume-preserving diffeomorphisms, and Kelvin's circulation theorem is viewed as a consequence of Noether's theorem associated with the particle relabeling symmetry of fluid mechanics. However computational approaches to fluid mechanics have been largely derived from a numerical-analytic point of view, and are rarely designed with structure preservation in mind, and often suffer from spurious numerical artifacts such as energy and circulation drift. In contrast, this paper geometrically derives discrete equations of motion for fluid dynamics from first principles in a purely Eulerian form. Our approach approximates the group of volume-preserving diffeomorphisms using a finite dimensional Lie group, and associated discrete Euler equations are derived from a variational principle with non-holonomic constraints. The resulting discrete equations of motion yield a structure-preserving time integrator with good long-term energy behavior and for which an exact discrete Kelvin's circulation theorem holds.

Existence of Weak Solutions for a Diffuse Interface Model for Viscous, Incompressible Fluids with General Densities

Abstract: We study a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account. In contrast to previous works, we study the general case that the fluids have different densities. This leads to an inhomogeneous Navier-Stokes system coupled to a Cahn-Hilliard system, where the density of the mixture depends on the concentration, the velocity field is no longer divergence free, and the pressure enters the equation for the chemical potential. We prove existence of weak solutions for the non-stationary system in two and three space dimensions.

A Stable and Efficient Method for Treating Surface Tension in Incompressible Two-Phase Flow

Abstract: A new approach based on volume preserving motion by mean curvature for treating surface tension in two-phase flows is introduced. Many numerical tests and a theoretical justification are included which provide evidence regarding the efficacy of the new approach. For many flows, which exhibit stiff surface tension effects, the new approach gives a factor of at least three and sometimes five or more speed-up for a given accuracy. The new method is easy to implement in the context of (1) level set methods, or coupled level set and volume-of-fluid methods, (2) complicated interfaces separating gas from liquid, and (3) three-dimensional axisymmetric, or fully three-dimensional adaptive mesh refinement.

A Navier–Stokes–Fourier system for incompressible fluids with temperature dependent material coefficients

Abstract: We consider a complete thermodynamic model for unsteady flows of incompressible homogeneous Newtonian fluids in a fixed bounded three-dimensional domain. The model comprises evolutionary equations for the velocity, pressure and temperature fields that satisfy the balance of linear momentum and the balance of energy on any (measurable) subset of the domain, and is completed by the incompressibility constraint. Finding a solution in such a framework is tantamount to looking for a weak solution to the relevant equations of continuum physics. If in addition the entropy inequality is required to hold on any subset of the domain, the solution that fulfills all these requirements is called the suitable weak solution . In our setting, both the viscosity and the coefficient of the thermal conductivity are functions of the temperature. We deal with Navier’s slip boundary conditions for the velocity that yield a globally integrable pressure, and we consider zero heat flux across the boundary. For such a problem, we establish the large-data and long-time existence of weak as well as suitable weak solutions, extending thus Leray [J. Leray, Sur le mouvement d’un liquide visquex emplissant l’espace, Acta Math. 63 (1934) 193–248] and Caffarelli, Kohn and Nirenberg [L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. 35 (6) (1982) 771–831] results, that deal with the problem in a purely mechanical context, to the problem formulated in a fully thermodynamic setting.

Fluid–structure interaction problems with strong added‐mass effect

Abstract: In this paper, the so-called added-mass effect is investigated from a different point of view of previous publications. The monolithic fluid–structure problem is partitioned using a static condensation of the velocity terms. Following this procedure the classical stabilized projection method for incompressible fluid flows is introduced. The procedure allows obtaining a new pressure segregated scheme for fluid–structure interaction problems, which has good convergent characteristics even for biomechanical application, where the added-mass effect is strong. The procedure reveals its power when it is shown that the same projection technique must be implemented in staggered FSI methods. Copyright © 2009 John Wiley & Sons, Ltd.

Mathematical analysis of unsteady flows of fluids with pressure, shear-rate, and temperature dependent material moduli that slip at solid boundaries

Abstract: In Bridgman's treatise [The Physics of High Pressures, MacMillan, New York, 1931], he carefully documented that the viscosity and the thermal conductivity of most liquids depend on the pressure and the temperature. The relevant experimental studies show that even at high pressures the variations of the values in the density are insignificant in comparison to that of the viscosity, and it is thus reasonable to assume that the liquids in question are incompressible fluids with pressure dependent viscosities. We rigorously investigate the mathematical properties of unsteady three-dimensional internal flows of such incompressible fluids. The model is expressed through a system of partial differential equations representing the balance of mass, the balance of linear momentum, the balance of energy, and the equation for the entropy production. Assuming that we have Navier's slip at the impermeable boundary we establish the long-time existence of a (suitable) weak solution when the data are large.

A point-based method for animating incompressible flow

Abstract: In this paper, we present a point-based method for animating incompressible flow. The advection term is handled by moving the sample points through the flow in a Lagrangian fashion. However, unlike most previous approaches, the pressure term is handled by performing a projection onto a divergence-free field. To perform the pressure projection, we compute a Voronoi diagram with the sample points as input. Borrowing from Finite Volume Methods, we then invoke the divergence theorem and ensure that each Voronoi cell is divergence free. To handle complex boundary conditions, Voronoi cells are clipped against obstacle boundaries and free surfaces. The method is stable, flexible and combines many of the desirable features of point-based and grid-based methods. We demonstrate our approach on several examples of splashing and streaming liquid and swirling smoke.

Convergence to equilibrium for a phase-field model for the mixture of two viscous incompressible fluids

Abstract: In this paper, we study the existence and long-time behavior of global strong solutions to a system describing the mixture of two viscous incompressible Newtonian fluids of the same density The system consists of a coupling of Navier-Stokes and Cahn-Hilliard equations We first show the global existence of strong solutions in several cases Then we prove that the global strong solution of our system will converge to a steady state as time goes to infinity We also provide an estimate on the convergence rate

Incompressible Computational Fluid Dynamics: Trends and Advances

Abstract: 1. A few tools for turbulence models in Navier-Stokes equations B. Cardot, B. Mohammadi and O. Pironneau 2. On some finite element methods for the numerical simulation of incompressible viscous flow Edward J. Dean and Roland Glowinski 3. CFD - an industrial perspective Michael S. Engelman 4. Stabilized finite element methods Leopoldo P. Franca, Thomas J. R. Hughes and Rolf Stenberg 5. Optimal control and optimization of viscous, incompressible flows Max D. Gunzburger, L. Steven Hou and Thomas P. Svobodny 6. A fully-coupled finite element algorithm, using direct and iterative solvers, for the incompressible Navier-Stokes equations W. G. Habashi, M. F. Peeters, M. P. Robichaud and V-N. Nguyen 7. Numerical solution of the incompressible Navier-Stokes equations in primitive variables on unstaggered grids M. Hafez and M. Soliman 8. Spectral element and lattice gas methods for incompressible fluid dynamics George Em Karniadakis, Steven A. Orszag, Einar M Ronquist and Anthony T. Patera 9. Design of incompressible flow solvers: practical aspects Rainald Loehner 10. The covolume approach to computing incompressible flows R. A. Nicolaides 11. Vortex methods: an introduction and survey of selected research topics Elbridge Gerry Puckett 12. New emerging methods in numerical analysis: applications to fluid mechanics Roger Temam 13. The finite element method for three-dimensional incompressible flow R. W. Thatcher 14. A posteriori error estimators and adaptive mesh-refinement techniques for the Navier-Stokes equation R. Verfurth.