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.     

Vorticity‐velocity formulations of the Stokes problem in 3D

This work studies the three‐dimensional Stokes problem expressed in terms of vorticity and velocity variables. We make general assumptions on the regularity and the topological structure of the flow domain: the boundary is Lipschitz and possibly non‐connected and the flow domain may be multiply connected. Upon introducing a new variational space for the vorticity, five weak formulations of the Stokes problem are obtained. All the formulations are shown to lead to well‐posed problems and to be equivalent to the primitive variable formulation. The various formulations are discussed by interpreting the test functions for the vorticity (resp. velocity) equation as vector potentials for the velocity (resp. vorticity). Of the five sets of boundary conditions derived in the paper, three are already known, but only for domains with a trivial topological structure, while the remaining two are new. Copyright © 1999 John Wiley & Sons, Ltd.     

Two-parameter extremum problems of boundary control for stationary thermal convection equations

Two-parameter extremum problems of boundary control are formulated for the stationary thermal convection equations with Dirichlet boundary conditions for velocity and with mixed boundary conditions for temperature. The cost functional is defined as the root mean square integral deviation of the desired velocity (vorticity, or pressure) field from one given in some part of the flow region. Controls are the boundary functions involved in the Dirichlet condition for velocity on the boundary of the flow region and in the Neumann condition for temperature on part of the boundary. The uniqueness of the extremum problems is analyzed, and the stability of solutions with respect to certain perturbations in the cost functional and one of the functional parameters of the original model is estimated. Numerical results for a control problem associated with the minimization of the vorticity norm aimed at drag reduction are discussed.     

Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation

Summary. In this paper, we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressible Navier-Stokes equations in vorticity-stream function formulation. The no-slip boundary condition for the velocity is converted into local vorticity boundary conditions. Thom's formula, Wilkes' formula, or other local formulas in the earlier literature can be used in the second order method; while high order formulas, such as Briley's formula, can be used in the fourth order compact difference scheme proposed by E and Liu. The stability analysis of these long-stencil formulas cannot be directly derived from straightforward manipulations since more than one interior point is involved in the formula. The main idea of the stability analysis is to control local terms by global quantities via discrete elliptic regularity for stream function. We choose to analyze the second order scheme with Wilkes' formula in detail. In this case, we can avoid the complicated technique necessitated by the Strang-type high order expansions. As a consequence, our analysis results in almost optimal regularity assumption for the exact solution. The above methodology is very general. We also give a detailed analysis for the fourth order scheme using a 1-D Stokes model. Received December 10, 1999 / Revised version received November 5, 2000 / Published online August 17, 2001     

The Velocity of Viscous Vortex Rings

The motion of vortex rings of small cross section is considered. A formula is derived for the velocity of a ring in an ideal fluid with an arbitrary distribution of vorticity in the core. The effects of viscosity are then examined. A definition of the velocity of an unsteady diffusing ring is given and it is shown that the method used to calculate the ring speed in an ideal fluid can be extended to give the velocity of a vortex ring subject to viscous diffusion.     

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.     

Subsonic solutions of hydrodynamic model for semiconductors

This paper is concerned with the existence and uniqueness of the steady-state solution of hydrodynamic model for semiconductor devices. Boundary conditions are prescribed by vorticity on inflow boundary as well as by electron density, temperature, and normal component of electron velocity on whole boundary. If the ambient temperature is large, and if both vorticity on inflow boundary and the variation of density on boundary are small, a unique subsonic solution exists. © 1997 by B.G. Teubner Stuttgart–John Wiley & Sons Ltd.     

A new formulation of the water wave problem for Stokes waves of constant vorticity

We consider steady symmetric gravity water waves on finite depth with constant vorticity and a monotone surface profile between crests and troughs. The problem is transformed into one concerning the vertical velocity. A representation formula for the stream function in terms of the surface and the vorticity is presented, and we show that the surface can be determined from the vertical velocity.     

Group theoretic method for unsteady free convection flow of a micropolar fluid along a vertical plate in a thermally stratified medium

The group theoretic approach is applied for solving the problem of unsteady natural convection flow of micropolar fluid along a vertical flat plate in a thermally stratified medium. The application of two-parameter transformation group reduces the number of independent variables in the governing system consisting of partial differential equations and a set of auxiliary conditions from three to only one independent variable, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. Numerical solution of the velocity, microrotation and heat transfer have been obtained. The possible forms of the ambient temperature variation with position and time are derived.     

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. This revised version was published online in June 2006 with corrections to the Cover Date.     

Vorticity–velocity-pressure and stream function-vorticity formulations for the Stokes problem

We study the Stokes problem of incompressible fluid dynamics in two and three-dimension spaces, for general bounded domains with smooth boundary. We use the vorticity–velocity-pressure formulation and introduce a new Hilbert space for the vorticity. We develop an abstract mixed formulation that gives a precise variational frame and conducts to a well-posed Stokes problem involving a new velocity–vorticity boundary condition. In the particular case of simply connected bidimensional domains with homogeneous boundary conditions, the link with the classical stream function-vorticity formulation is completely described, and we show that the vorticity–velocity-pressure formulation is a natural mathematical extension of the previous one.     

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

The velocity–vorticity formulation of the 3D Navier–Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier–Stokes equations, which we call the 3D velocity–vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity–vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier–Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier–Stokes equations based on this inviscid regularization.     

Vorticity-velocity-pressure formulation for Stokes problem

We propose a three-field formulation for efficiently solving a two-dimensional Stokes problem in the case of nonstandard boundary conditions. More specifically, we consider the case where the pressure and either normal or tangential components of the velocity are prescribed at some given parts of the boundary. The proposed computational methodology consists in reformulating the considered boundary value problem via a mixed-type formulation where the pressure and the vorticity are the principal unknowns while the velocity is the Lagrange multiplier. The obtained formulation is then discretized and a convergence analysis is performed. A priori error estimates are established, and some numerical results are presented to highlight the perfomance of the proposed computational methodology.

     

Meshless solution of two-dimensional incompressible flow problems using the radial basis integral equation method

The two-dimensional incompressible fluid flow problems governed by the velocity–vorticity formulation of the Navier–Stokes equations were solved using the radial basis integral (RBIE) equation method. The RBIE is a meshless method based on the multi-domain boundary element method with overlapping subdomains. It solves at each node for the potential and its spatial derivatives. This feature of the RBIE is advantageous in solving the velocity–vorticity formulation of the Navier–Stokes equations since the calculated velocity gradients can be used to compute the vorticity that is prescribed as a boundary condition to the vorticity transport equation. The accuracy of the numerical solution was examined by solving the test problem with known analytical solution. Two benchmark problems, i.e. the lid driven cavity flow and the thermally driven cavity flow were also solved. The numerical results obtained using the RBIE showed very good agreement with the benchmark solutions.     

Robin-type boundary control problems for the nonlinear Boussinesq type equations

In this paper, we study a linear and a nonlinear boundary control problems arising from viscous flows. The equations are of nonlinear Navier-Stokes type for the velocity and pressure, of transport-diffusion type for the temperature and the salinity. The essential difficulties are due to the nonlinear nature of a part of the boundary conditions and to the nature of the equations: time-dependent, coupled and nonlinear. The existence and the conditions of the uniqueness of the solution, for the variational problem, are studied. The control is of linear or nonlinear Robin-type and acts on a part of the boundary during a time T. The cost function measures the distance between the observed and the computed vorticity. The existence of an optimal control in the admissible set of states and controls is proved. A first order necessary conditions of optimality are obtained.     

Long-time behaviour of absorbing boundary conditions

A new class of computational far-field boundary conditions for hyperbolic partial differential equations was recently introduced by the authors. These boundary conditions combine properties of absorbing conditions for transient solutions and properties of far-field conditions for steady states. This paper analyses the properties of the wave equation coupled with these new boundary conditions: well-posedness, dissipativity and convergence in time.     

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.     

双参数地基上板弯曲问题的边界积分方程

本文应用广义函数的Fourrier积分变换导出了双参数地基上板弯曲问题的基本解,并将基本解展成一致收敛的级数形式.在此基础上,应用广义Rayleigh-Green公式建立了适用于任意形状、任意边界条件情形的两个边界积分方程,为边界元法在这一问题中的应用提供了理论基础.     

The Far-Field Equations in Linear Elasticity for Disconnected Rigid Bodies and Cavities

In this paper the far-field equations in linear elasticity for scattering from disjoint rigid bodies and cavities are considered. The direct scattering problem is formulated in differential and integral form. The boundary integral equations are constructed using a combination of single- and double-layer potentials. Using a Fredholm type theory it is proved that these boundary integral equations are uniquely solvable. Assuming that the incident field is produced by a superposition of plane incident waves in all directions of propagation and polarization it is established that the scattered field is also expressed as the superposition of the corresponding scattered fields. A pair of integral equations of the first kind which hold independently of the boundary conditions are constructed for the far-field region. The properties of the Herglotz functions are used to derive solvability conditions for the far-field equations. It is also proved that the far-field operators, in terms of which we can express the far-field equations, are injective and have dense range. An analytical example for spheres illuminates the theoretical results.