Application of lattice Boltzmann method for incompressible viscous flows

Because of the presence of corner eddies that change in number and pattern the lid-driven cavity problem has been found suitable to study various aspects of the performance of solution algorithms for incompressible viscous flows. It retains all the difficult flow physics and is characterized by a large primary eddy at the centre and secondary eddies located near the cavity corners. In this work, lid-driven cavity flow is simulated by lattice Boltzmann method with single-relaxation-time and it is compared with those by lattice Boltzmann method with multi-relaxation-time and finite difference method. The effects of the Reynolds number on the size, centre position and number of vortices are studied in detail together with the flow pattern in the cavity. The close agreement of the results bears testimony to the validity of this relatively new approach. However lattice Boltzmann method with multi-relaxation-time model is seen to remove the difficulties faces by the lattice Boltzmann method with single-relaxation-time at higher Reynolds numbers.     

Spectral element solution of steady incompressible viscous free-surface flows

In this paper we present a new approach for the solution of the steady incompressible Navier-Stokes equations in a domain bounded in part by a free surface. In the spatial discretization procedure, a Legendre spectral element method is used to generate the discrete equations. For effective solution of the set of algebraic equations, the geometry is decoupled from the fluid velocity and pressure. In addition, two different algorithms are proposed depending on the importance of surface tension effects. Numerical results are presented to demonstrate the effectiveness of the proposed algorithms.     

The method of lines for the computation of incompressible viscous flows

A common approach to finding numerical solutions of the time-dependent incompressible Navier-Stokes equations is considered within the method of lines framework [1]. For the determination of viscous incompressible flows the stream-function formulation for the fourth-order equation [2, 3], an artificial compressibility method [4], and a modified velocity correction method [5] are designed. Some improved and extended results of numerical simulations obtained by the author in the previous works are presented. Test calculations have been done for various flows inside square, triangular and semicircular cavities with one moving wall, the backward-facing step, double bent channels and for the flow around an aerofoil at large angle of attack. An alternative and practical methodology for resolving the Navier-Stokes equations in arbitrarily complex geometries using Cartesian meshes is proposed. Some of complex geometrical configurations can be decomposed into a set of subdomains. The simplest approach for specifying boundary conditions near curved or irregular boundaries is to transfer all the variables from the boundaries to the nearest grid knots. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A variant of the finite superelement method for computing viscous incompressible flows

A variant of the finite superelement method (FSEM) is proposed for computing viscous incompressible convection-dominated flows on a triangular unstructured mesh. To construct the high-order FSEM scheme, vector polynomial test functions (SE basis) are calculated in each grid cell by solving the linearized Navier-Stokes equations with special boundary conditions in the form of basis functions in the trace space.     

On the stationary flows of viscous,incompressible and heat-conducting fluids

We consider a boundary-value problem describing the motion of viscous, incompressible and heat-conducting fluids in a bounded domain in ?³. We admit non-homogeneous boundary conditions, the appearance of exterior forces and heat sources. Our aim is to prove the existence of a solution of the problem in Sobolev spaces.     

A penalization method to take into account obstacles in incompressible viscous flows

From the Navier-Stokes/Brinkman model, a penalization method has been derived by several authors to compute incompressible Navier-Stokes equations around obstacles. In this paper, convergence theorems and error estimates are derived for two kinds of penalization. The first one corresponds to penalization inducing a Darcy equation in the solid body, the second one corresponds to a penalization and induces a Brinkman equation in the body. Numerical tests are performed to confirm the efficiency and accuracy of the method. Received August 10, 1997     

Friction boundary conditions on thermal incompressible viscous flows

This work adresses an unsteady heat flow problem involving friction and convective heat transfer behaviors on a part of the boundary. The problem is constituted by a variational motion inequality with energy dependent coefficients, and the energy equation in the framework of L ¹-theory for the dissipative term. Using the duality theory of convex analysis, it also envolves the existence of Lagrange multipliers. Weak solutions of an approximate coupled system are proven by a fixed point argument for multivalued mappings and compactness methods. Then the existence result for the initial coupled system is proven by the passage to the limit. This work was partially supported by FCT research program POCTI (Portugal/FEDER-EU).     

Influence matrix technique for the numerical spectral simulation of viscous incompressible flows

A spectral algorithm based on the influence matrix technique is desclibed for solving numerically the flow of incompressible viscous fluids. The algorithmic development is for both Newtonian and non-Newtonian flows. To investigate the performance of the method several test problems are solved. Accurate results are obtained with relatively few degrees of freedom.     

Numerical solution of stationary viscous incompressible fluid problems by the method of fictitious domains

We consider some issues of numerical implementation of the fictitious domain method for viscous incompressible fluid problems. Plane stationary problems are solved by successive approximations in the nonlinearity. Plane heat convection problems in the Boussinesq approximation are also considered. Solution examples of some specimen problems are presented.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 255–262, 1985.     

Generalized stochastic flows and applications to incompressible viscous fluids

We introduce a notion of generalized stochastic flows on manifolds, that extends to the viscous case the one defined by Brenier for perfect fluids. Their kinetic energy extends the classical kinetic energy to Brownian flows, defined as the L²$L^2$ norm of their drift. We prove that there exists a generalized flow which realizes the infimum of the kinetic energy among all generalized flows with prescribed initial and final configuration. We also construct generalized flows with prescribed drift and kinetic energy smaller than the L²$L^2$ norm of the drift.     

Solutions to the mixed problem of viscous incompressible flows in a channel

We study regularity of viscous incompressible fluid flows in a 2D channel with “do nothing” outflow boundary condition on the output for the steady Stokes and Navier–Stokes equations.     

On viscous incompressible flows of nonsymmetric fluids with mixed boundary conditions

In this paper we are concerned with the initial boundary value problem for the micropolar fluid system in nonsmooth domains with mixed boundary conditions. The considered boundary conditions are of two types: Navier’s slip conditions on solid surfaces and Neumann-type boundary conditions on free surfaces. The Dirichlet boundary condition for the microrotation of the fluid is commonly used in practice. However, the well-posedness of problems with different types of boundary conditions for microrotation are completely unexplored. The present paper is devoted to the proof of the existence, regularity and uniqueness of the solution in distribution spaces.     

On the interface boundary conditions between two interacting incompressible viscous fluid flows

We consider two incompressible viscous fluid flows interacting through thin non-Newtonian boundary layers of higher Reynolds? number. We study the asymptotic behaviour of the problem, with respect to the vanishing thickness of the layers, using Γ-convergence methods. We derive general interfacial boundary conditions between the two fluid flows. These boundary conditions are specified for some particular cases including periodic or fractal structures of layers.     

On the decay of higher order derivatives of solutions to Ladyzhenskaya model for incompressible viscous flows

This article concerns large time behavior of Ladyzhenskaya model for incompressible viscous flows in R~3.Based on linear L~P-L~q estimates,the auxiliary decay properties of the solutions and generalized Gronwall type arguments,some optimal upper and lower bounds for the decay of higher order derivatives of solutions are derived without assuming any decay properties of solutions and using Fourier splitting technology.     

A vortex method for computing two-dimensional inviscid incompressible flows

A vortex method is suggested for computing two-dimensional inviscid incompressible flows in a closed domain with a possible flow through it. An algorithm for searching for stable steady vortex configurations is described. The method developed is used to study the dynamics of the Chaplygin-Lamb dipole in a rectangular channel in various flow regimes.