Stokes Equations In Terms Research Articles

Navier–Stokes equations and forward–backward SDEs on the group of diffeomorphisms of a torus

We establish a connection between the strong solution to the spatially periodic Navier–Stokes equations and a solution to a system of forward–backward stochastic differential equations (FBSDEs) on the group of volume-preserving diffeomorphisms of a flat torus. We construct representations of the strong solution to the Navier–Stokes equations in terms of diffusion processes.

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.

Theoretical investigation of the flow in a stenosis

AbstractThe theoretical investigation of the flow in stenosis is presented. We use a new formulation of the incompressible Navier–Stokes equation in terms of an auxiliary field that differs from the velocity by a gauge transformation [1]. The gauge freedom allows us to formulate simple boundary conditions for the auxiliary field and the gauge field as well. The gauge field eliminates the pressure distribution in the Navier–Stokes equation. The numerical investigation of the creeping flow, depending on the geometrical parameters of the system, is performed. The influence of the pressure drop has been taken into account. An excellent agreement with the analytical results in frames of the film theory was observed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Solution to transient Navier–Stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method

AbstractThe two‐dimensional time‐dependent Navier–Stokes equations in terms of the vorticity and the stream function are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in space with the differential quadrature method (DQM) in time. In DRBEM application, the convective and the time derivative terms in the vorticity transport equation are considered as the nonhomogeneity in the equation and are approximated by radial basis functions. The solution to the Poisson equation, which links stream function and vorticity with an initial vorticity guess, produces velocity components in turn for the solution to vorticity transport equation. The DRBEM formulation of the vorticity transport equation results in an initial value problem represented by a system of first‐order ordinary differential equations in time. When the DQM discretizes this system in time direction, we obtain a system of linear algebraic equations, which gives the solution vector for vorticity at any required time level. The procedure outlined here is also applied to solve the problem of two‐dimensional natural convection in a cavity by utilizing an iteration among the stream function, the vorticity transport and the energy equations as well. The test problems include two‐dimensional flow in a cavity when a force is present, the lid‐driven cavity and the natural convection in a square cavity. The numerical results are visualized in terms of stream function, vorticity and temperature contours for several values of Reynolds (Re) and Rayleigh (Ra) numbers. Copyright © 2008 John Wiley & Sons, Ltd.

On regularity criteria for the n-dimensional Navier–Stokes equations in terms of the pressure

We study the Cauchy problem for the n-dimensional Navier–Stokes equations ( n ⩾ 3 ), and prove some regularity criteria involving the integrability of the pressure or the pressure gradient for weak solutions in the Morrey, Besov and multiplier spaces.

Regularity Criterion for Solutions of Three-Dimensional Turbulent Channel Flows

In this paper we consider the three-dimensional Navier–Stokes equations in infinite channel. We provide a regularity criterion for solutions of the three-dimensional Navier–Stokes equations in terms of the vertical component of the velocity field.

Estimates for the two-dimensional Navier–Stokes equations in terms of the Reynolds number

The tradition in Navier–Stokes analysis of finding estimates in terms of the Grashof number Gr, whose character depends on the ratio of the forcing to the viscosity ν, means that it is difficult to make comparisons with other results expressed in terms of Reynolds number Re, whose character depends on the fluid response to the forcing. The first task of this paper is to apply the approach of Doering and Foias [C. R. Doering and C. Foias, J. Fluid Mech. 467, 289 (2002)] to the two-dimensional Navier–Stokes equations on a periodic domain [0,L]2 by estimating quantities of physical relevance, particularly long-time averages ⟨∙⟩, in terms of the Reynolds number Re=Uℓ∕ν, where U2=L−2⟨∥u∥22⟩ and ℓ is the forcing scale. In particular, the Constantin–Foias–Temam upper bound [P. Constantin, C. Foias, and R. Temam, Physica D 30, 284 (1988)] on the attractor dimension converts to aℓ2Re(1+lnRe)1∕3, while the estimate for the inverse Kraichnan length is (aℓ2Re)1∕2, where aℓ is the aspect ratio of the forcing. Other inverse length scales, based on time averages, and associated with higher derivatives, are estimated in a similar manner. The second task is to address the issue of intermittency: it is shown how the time axis is broken up into very short intervals on which various quantities have lower bounds, larger than long time averages, which are themselves interspersed by longer, more quiescent, intervals of time.

On a Serrin-Type Regularity Criterion for the Navier–Stokes Equations in Terms of the Pressure

We prove a Serrin-type regularity result for Leray–Hopf solutions to the Navier–Stokes equations, extending a recent result of Zhou [28].

Effect of boundary vorticity discretization on explicit stream-function vorticity calculations

The numerical solution of the time-dependent Navier–Stokes equations in terms of the vorticity and a stream function is a well tested process to describe two-dimensional incompressible flows, both for fluid mixing applications and for studies in theoretical fluid mechanics. In this paper, we consider the interaction between the unsteady advection–diffusion equation for the vorticity, the Poisson equation linking vorticity and stream function and the approximation of the boundary vorticity, examining from a practical viewpoint, global iteration stability and error. Our results show that most schemes have very similar global stability constraints although there may be small stability gains from the choice of method to determine boundary vorticity. Concerning accuracy, for one model problem we observe that there were cases where the boundary vorticity discretization did not propagate to the interior, but for the usual cavity flow all the schemes tested had error close to second order. Copyright © 2005 John Wiley & Sons, Ltd.

Bivariate splines for fluid flows

We discuss numerical approximations of the 2D steady-state Navier–Stokes equations in stream function formulation using bivariate splines of arbitrary degree d and arbitrary smoothness r with r< d. We derive the discrete Navier–Stokes equations in terms of B-coefficients of bivariate splines over a triangulation, with curved boundary edges, of any given domain. Smoothness conditions and boundary conditions are enforced through Lagrange multipliers. The pressure is computed by solving a Poisson equation with Neumann boundary conditions. We have implemented this approach in MATLAB and our numerical experiments show that our method is effective. Numerical simulations of several fluid flows will be included to demonstrate the effectiveness of the bivariate spline method.

An Analysis of Electrophoresis of Concentrated Suspensions of Colloidal Particles

An analysis of the electrophoretic motion of charged colloidal particles in a concentrated suspension is developed to predict the electrophoretic mobility of the particles and the electrical conductivity of the suspension. The analysis is based on a unit cell model that takes into account particle–particle hydrodynamic interactions and includes relatively thick electric double layers. The fluid motion in the unit cell is treated by writing the relevant Navier–Stokes equation in terms of the stream function and vorticity. The governing equations were then solved by a finite-difference method. The calculated electrophoretic mobilities are in agreement with prior analytical solutions for moderately concentrated suspensions, and the theory reduces to the result of O'Brien and White for low to moderate zeta potentials and dilute suspensions and to the classical result of Smoluchowski for thin double layers and dilute suspensions. A parametric study shows that the electrical conductivity of the suspension relative to a free electrolyte solution is affected by the counterion to co-ion diffusivity ratio, the double-layer thickness, and the volume fraction of particles. For a dispersion of moderately charged particles (moderate zeta potentials) with thick double layers, the numerical model predicts the electrical conductivity in agreement with experimental values reported in the literature.

A mixed formulation of the Stokes equation in terms of (ω,p,u)

The aim of this paper is to give a mixed formulation of the Stokes problem of fluid mechanics in terms of the vorticity, the pressure and the velocity that does not need an artificial boundary condition for vorticity but only the usual compatibility conditions between the pressure space and the velocity space.

On the Velocity-Vorticity Formulation of the Two-Dimensional Stokes System and its Finite Element Approximation

This paper deals with the numerical study of the two-dimensional Stokes equations in terms of the velocity and the vorticity. We take one of the possible sets of boundary conditions, for which the equivalence between the standard velocity-pressure Stokes system and such velocity-vorticity formulation holds. We consider the corresponding rigorous variational framework and an uncoupled solution scheme. In this context, the authors proposed several families of Galerkin finite element approximations, for which they derived optimal or suboptimal convergence results. Here their numerical approach is illustrated by several numerical examples. As a byproduct, hints are given on the best way to interpolate the different fields involved in the analogous uncoupling scheme, for solving the incompressible Navier-Stokes equations.

Space Analyticity for the Navier–Stokes and Related Equations with Initial Data inLp

We introduce a method of estimating the space analyticity radius of solutions for the Navier–Stokes and related equations in terms ofLpandL∞norms of the initial data. The method enables us to express the space analyticity radius for 3D Navier–Stokes equations in terms of the Reynolds number of the flow. Also, for the Kuramoto–Sivashinsky equation, we give a partial answer to a conjecture that the radius of space analyticity on the attractor is independent of the spatial period.

A BEM-BDF Scheme for Curvature Driven Moving Stokes Flows

A backward differences formulae (BDF) scheme, is proposed to simulate the deformation of a viscous incompressible Newtonian fluid domain in time, which is driven solely by the boundary curvature. The boundary velocity field of the fluid domain is obtained by writing the governing Stokes equations in terms of an integral formulation that is solved by a boundary element method. The motion of the boundary is modelled by considering the boundary curve as material points. The trajectories of those points are followed by applying the Lagrangian representation for the velocity. Substituting this representation into the discretized version of the integral equation yields a system of non-linear ODEs. Here the numerical integration of this system of ODEs is outlined. It is shown that, depending on the geometrical shape, the system can be stiff. Hence, a BDF-scheme is applied to solve those equations. Some important features with respect to the numerical implementation of this method are high-lighted, like the approximation of the Jacobian matrix and the continuation of integration after a mesh redistribution. The usefulness of the method for both two-dimensional and axisymmetric problems is demonstrated.

Transport in ordered and disordered porous media II: Generalized volume averaging

In this paper we develop the averaged form of the Stokes equations in terms of weighting functions. The analysis clearly indicates at what point one must choose a media-specific weighting function in order to achieve spatially smoothed transport equations. The form of the weighting function that produces the cellular average is derived, and some important geometrical theorems are presented.

A multigrid method for solution of vorticity—velocity form of 3‐D navier—stokes equations

AbstractA multigrid method has been developed and applied to the simulation of unsteady three‐dimensional incompressible flow, using the Navier—Stokes equations in terms of vorticity and velocity variables. A system of equations for the velocity problem, consisting of the continuity equation and three curl equations, is first formulated as a uniquely determined, non‐singular system. An efficient and robust technique, namely the multigrid distributive Gauss—Seidel scheme, is developed for solution of this system of algebraic equations. The method is applied to study 3‐D flow in a shear‐driven cavity.

Turbulence: the filtering approach

Explicit or implicit filtered representations of chaotic fields like spectral cut-offs or numerical discretizations are commonly used in the study of turbulence and particularly in the so-called large-eddy simulations. Peculiar to these representations is that they are produced by different filtering operators at different levels of resolution, and they can be hierarchically organized in terms of a characteristic parameter like a grid length or a spectral truncation mode. Unfortunately, in the case of a general implicit or explicit filtering operator the Reynolds rules of the mean are no longer valid, and the classical analysis of the turbulence in terms of mean values and fluctuations is not so simple.In this paper a new operatorial approach to the study of turbulence based on the general algebraic properties of the filtered representations of a turbulence field at different levels is presented. The main results of this analysis are the averaging invariance of the filtered Navier—Stokes equations in terms of the generalized central moments, and an algebraic identity that relates the turbulent stresses at different levels. The statistical approach uses the idea of a decomposition in mean values and fluctuations, and the original turbulent field is seen as the sum of different contributions. On the other hand this operatorial approach is based on the comparison of different representations of the turbulent field at different levels, and, in the opinion of the author, it is particularly fitted to study the similarity between the turbulence at different filtering levels. The best field of application of this approach is the numerical large-eddy simulation of turbulent flows where the large scale of the turbulent field is captured and the residual small scale is modelled. It is natural to define and to extract from the resolved field the resolved turbulence and to use the information that it contains to adapt the subgrid model to the real turbulent field. Following these ideas the application of this approach to the large-eddy simulation of the turbulent flow has been produced (Germano et al. 1991). It consists in a dynamic subgrid-scale eddy viscosity model that samples the resolved scale and uses this information to adjust locally the Smagorinsky constant to the local turbulence.