Mean-Square Approximation of Navier-Stokes Equations with Additive Noise in Vorticity-Velocity Formulation

We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions and additive noise in the vorticity-velocity formulation. The approximation is based on freezing the velocity on time subintervals resulting in a linear stochastic parabolic equation for vorticity. At each time step, the velocity is expressed via vorticity using a formula corresponding to the Biot-Savart-type law. We prove the first mean-square convergence order of the vorticity approximation.     

A High-Order Kernel-Free Boundary Integral Method for Incompressible Flow Equations in Two Space Dimensions

This paper presents a fourth-order kernel-free boundary integral method for the time-dependent, incompressible Stokes and Navier-Stokes equations defined on irregular bounded domains. By the stream function-vorticity formulation, the incompressible flow equations are interpreted as vorticity evolution equations. Time discretization methods for the evolution equations lead to a modified Helmholtz equation for the vorticity, or alternatively, a modified biharmonic equation for the stream function with two clamped boundary conditions. The resulting fourth-order elliptic boundary value problem is solved by a fourth-order kernel-free boundary integral method, with which integrals in the reformulated boundary integral equation are evaluated by solving corresponding equivalent interface problems, regardless of the exact expression of the involved Green's function. To solve the unsteady Stokes equations, a four-stage composite backward differential formula of the same order accuracy is employed for time integration. For the Navier-Stokes equations, a three-stage third-order semi-implicit Runge-Kutta method is utilized to guarantee the global numerical solution has at least third-order convergence rate. Numerical results for the unsteady Stokes equations and the Navier-Stokes equations are presented to validate efficiency and accuracy of the proposed method.     

Viscosity-splitting scheme for the Navier-Stokes equations

A viscosity-splitting scheme for the initial boundary value problems of the Navier-Stokes equations is considered. In the scheme, the Stokes equation is solved in conjunction with a nonhomogeneous boundary condition which connects the tangent flow with a no-slip flow. Convergence is proved.     

Convergence of euler-stokes splitting of the navier-stokes equations

We consider approximation by partial time steps of a smooth solution of the Navier-Stokes equations in a smooth domain in two or three space dimensions with no-slip boundary condition. For small k > 0, we alternate the solution for time k of the inviscid Euler equations, with tangential boundary condition, and the solution of the linear Stokes equations for time k, with the no-slip condition imposed. We show that this approximation remains bounded in H^2,p and is accurate to order k in L^p for p > ∞. The principal difficulty is that the initial state for each Stokes step has tangential velocity at the boundary generated during the Euler step, and thus does not satisfy the boundary condition for the Stokes step. The validity of such a fractional step method or splitting is an underlying principle for some computational methods. © 1994 John Wiley & Sons, Inc.     

Finite element approximation on incompressible Navier-Stokes equations with slip boundary condition

Summary We consider a mixed finite element approximation of the stationary, incompressible Navier-Stokes equations with slip boundary condition, which plays an important rôle in the simulation of flows with free surfaces and incompressible viscous flows at high angles of attack and high Reynold's numbers. The central point is a saddle-point formulation of the boundary conditions which avoids the well-known Babuka paradox when approximating smooth domains by polyhedrons. We prove that for the new formulation one can use any stable mixed finite element for the Navier-Stokes equations with no-slip boundary condition provided suitable bubble functions on the boundary are added to the velocity space. We obtain optimal error estimates under minimal regularity assumptions for the solution of the continous problem. The techniques apply as well to the more general Navier boundary condition.     

The inviscid limit for an inflow problem of compressible viscous gas in presence of both shocks and boundary layers

In this paper, we study the inviscid limit problem for the Navier-Stokes equations of one-dimensional compressible viscous gas on half plane. We prove that if the solution of the inviscid Euler system on half plane is piecewise smooth with a single shock satisfying the entropy condition, then there exist solutions to Navier-Stokes equations which converge to the inviscid solution away from the shock discontinuity and the boundary at an optimal rate of ε¹ as the viscosity ε tends to zero.     

Navier-Stokes equations: Green's matrices, vorticity direction, and regularity up to the boundary

We consider the initial-boundary value problem for the 3D Navier-Stokes equations. The physical domain is a bounded open set with a smooth boundary on which we assume a condition of free-boundary type. We show that if a suitable hypothesis on the vorticity direction is assumed, then weak solutions are regular. The main tool we use in the proof is an explicit representation of the velocity in terms of the vorticity, by means of Green's matrices.     

Homogenisation for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes

We prove the homogenisation to the Brinkman equations for the incompressible Stokes equations in a bounded domain which is perforated by a random collection of small spherical holes. The fluid satisfies a no-slip boundary condition at the holes. The balls generating the holes have centres distributed according to a Poisson point process and i.i.d. unbounded radii satisfying a suitable moment condition. We stress that our assumption on the distribution of the radii does not exclude that, with overwhelming probability, the holes contain clusters made by many overlapping balls. We show that the formation of these clusters has no effect on the limit Brinkman equations. Due to the incompressibility condition and the lack of a maximum principle for the Stokes equations, our proof requires a very careful study of the geometry of the random holes generated by the class of probability measures considered.     

AN ARTIFICIAL BOUNDARY CONDITION FOR THE INCOMPRESSIBLE VISCOUS FLOWS IN A NO-SLIP CHANNEL

1.IntroductionWhencomputingthenumericals0luti0nsofviscousfluidfl0wproblemsinallun-boundedd0main,0neoftenintroducesartificialboundaries,andsetsupanartificialbopundarycondition0nthem;thenthe0riginalproblemisreducedtoaproblemonab0undedc0mputationald0main.InordertoIimitthecomputatio11alcost,theseboundariesmustnotbet00farfromthedomainofinterest.Theref0re,theartificialboundaryc0nditi0nsmustbegoodapprotimationt0the"exact"boundaryconditions(sothatthes0lutionoftheproblemintheboundeddonlainisequaltothes…     

Backward stochastic differential equations associated with the vorticity equations

In this paper we derive a non-linear version of the Feynman–Kac formula for the solutions of the vorticity equation in dimension 2 with space periodic boundary conditions. We prove the existence (global in time) and uniqueness for a stochastic terminal value problem associated with the vorticity equation in dimension 2. A particular class of terminal values provide, via these probabilistic methods, solutions for the vorticity equation.     

COMBINED FINITE ELEMENT AND PSEUDOSPECTRAL METHOD FOR THE THREE-DIMENSIONAL NAVIER-STOKES EQUATIONS

This paper analyzes a combined method with artificial compression for solving thethree-dimensional evolutionary Navier-Stokes equations with periodic and no-slipboundary condition.A Fourier pseudospectral method with a control operator is used inthe periodic direction and a standard finite element method in the two others.The gener-alized stability of the scheme and optimal rate of convergence of the velocity in L~2-normare proved on the assumption that the BB condition of the fiinite element approximation forthe two-dimensional Stokes equations is satisfied.     

Spectral approximation of the periodic-nonperiodic Navier-Stokes equations

Summary In order to approximate the Navier-Stokes equations with periodic boundary conditions in two directions and a no-slip boundary condition in the third direction by spectral methods, we justify by theoretical arguments an appropriate choice of discrete spaces for the velocity and the pressure. The compatibility between these two spaces is checked via an infsup condition. We analyze a spectral and a collocation pseudo-spectral method for the Stokes problem and a collocation pseudo-spectral method for the Navier-Stokes equations. We derive error bounds of spectral type, i.e. which behave likeM ^– whereM depends on the number of degrees of freedom of the method and represents the regularity of the data.     

Effective boundary condition at a rough surface starting from a slip condition

We consider the homogenization of the Navier-Stokes equation, set in a channel with a rough boundary, of small amplitude and wavelength ?. It was shown recently that, for any non-degenerate roughness pattern, and for any reasonable condition imposed at the rough boundary, the homogenized boundary condition in the limit ε=0 is always no-slip. We give in this paper error estimates for this homogenized no-slip condition, and provide a more accurate effective boundary condition, of Navier type. Our result extends those obtained in Basson and Gérard-Varet (2008) [6] and Gerard-Varet and Masmoudi (2010) [13], in which the special case of a Dirichlet condition at the rough boundary was examined.     

On the Inviscid Limit Problem of the Vorticity Equations for Viscous Incompressible Flows in the Half‐Plane

We consider the Navier‐Stokes equations for viscous incompressible flows in the half‐plane under the no‐slip boundary condition. By using the vorticity formulation we prove the local‐in‐time convergence of the Navier‐Stokes flows to the Euler flows outside a boundary layer and to the Prandtl flows in the boundary layer in the inviscid limit when the initial vorticity is located away from the boundary. © 2014 Wiley Periodicals, Inc.     

Formulation of boundary conditions for vorticity in viscous incompressible flow problems

For the stream function-vorticity formulation of the Navier-Stokes equations, vorticity boundary conditions are required on the body surface and the far-field boundary. A two-parameter approximating formula is derived that relates the velocity and vorticity on the outer boundary of the computational domain. The formula is used to construct an algorithm for correcting the conventional far-field boundary conditions. Specifically, a soft boundary condition is set for the vorticity and a uniform flux is specified for the transversal velocity. A third-order accurate three-parameter formula for the vorticity on the wall is derived. The use of the formula does not degrade the convergence of the iterative process of finding the vorticity as compared with a previously derived and tested two-parameter formula.     

Solving the Dirichlet problem for Navier–Stokes equations by probabilistic approach

We construct a number of layer methods for Navier-Stokes equations (NSEs) with no-slip boundary conditions. The methods are obtained using probabilistic representations of solutions to NSEs and exploiting ideas of the weak sense numerical integration of stochastic differential equations. Despite their probabilistic nature, the proposed methods are nevertheless deterministic.     

A FOURIER-CHEBYSHEV SPECTRAL METHOD FOR THREE-DIMENSIONAL VORTICITY EQUATIONS WITH UNILATERAL PERIODIC BOUNDARY CONDITIONS

A mixed Fourier-Chebyshev spectral method is constructed for three-dimensional vorticity equations with unilateral periodic boundary condition. The generalized stability and the convergence are analyzed. The optimal error estimation is given. The technique in this paper is also suitable for other nonlinear problems.     

Geometric depletion of vortex stretch in 3D viscous incompressible flow

New geometric constraints on vorticity are obtained which suppress possible development of finite-time singularity from the nonlinear vortex stretching mechanism. We find a new condition on the smoothness of the direction of vorticity in the vortical region which yields regularity. We also detect a regularity condition of isotropy type on vorticity in the intensive vorticity region via a new cancellation principle. This is in contrast with the one of isotropy type on the curl of vorticity obtained recently by A. Ruzmaikina and Z. Gruji? [A. Ruzmaikina, Z. Gruji?, On depletion of the vortex-stretching term in the 3D Navier-Stokes equations, Comm. Math. Phys. 247 (2004) 601-611]. We improve as well all of their results by eliminating their assumption that the initial vorticity ω₀ is required to be in L¹.     

Boundary Shape Control of the Navier-Stokes Equations and Applications

In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the Gateaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.