Dirichlet Boundary Condition For Velocity Research Articles

High-order velocity and pressure wall boundary conditions in Eulerian incompressible SPH

High-order velocity and pressure boundary conditions are presented in Eulerian incompressible smoothed particle hydrodynamics (ISPH). While the high-order convergence of Eulerian ISPH has been demonstrated by the authors for periodic internal flows using Gaussian kernels this was limited by first to second-order accuracy for cases with solid boundaries. Since the SPH interpolation method is numerically robust there is potential for obtaining high-order accuracy in topologically complex domains with robust high-order accurate boundary conditions. In this paper high-order finite-difference extrapolation methods at solid boundaries are developed in Eulerian ISPH to allow for enforcement of the Dirichlet boundary condition for velocity and the Neumann boundary condition for pressure with high-order accuracy. Convergence up to fourth-order is demonstrated for 2-D Taylor-Couette flow and 3-D simulations of Taylor-Couette cellular flow structures are used to demonstrate accuracy and robustness. The order of accuracy may be extended to even higher-order using the analysis presented. Compact fourth-order Wendland-type kernels have also been derived to reduce the particle support region thereby lowering computational effort without loss of high-order convergence. The proposed formulation is therefore entirely high order.

The Boundary Layer Problem of MHD System with the Non-characteristic Dirichlet Boundary Condition for Velocity

In this paper, we study the boundary layer problem for the two-dimensional incompressible MHD system with the non-characteristic Dirichlet boundary condition for the velocity and the perfect conducting wall boundary condition for the magnetic field. Using multiscale analysis and the elaborate energy methods, we rigorously prove that the solutions of the viscous and diffusive MHD system are approximated by the inner solutions with the boundary layers in the sense of $L^{2}$ norm when the viscosity and diffusion coefficient tend to zero.

Flux-preserving enforcement of inhomogeneous Dirichlet boundary conditions for strongly divergence-free mixed finite element methods for flow problems

We investigate a flux-preserving enforcement of inhomogeneous Dirichlet boundary conditions for velocity, u|∂Ω=g, for use with finite element methods for incompressible flow problems that strongly enforce mass conservation. Typical enforcement via nodal interpolation is not flux-preserving in general, and it can create divergence error even when divergence-free elements are used. We show with analysis and numerical tests that by slightly (and locally) changing nodal interpolation, the enforcement recovers flux-preservation, is optimally accurate, and delivers divergence-free solutions when used with divergence-free finite elements.

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.

Stability of Optimal Controls for the Stationary Boussinesq Equations

The stationary Boussinesq equations describing the heat transfer in the viscous heat‐conducting fluid under inhomogeneous Dirichlet boundary conditions for velocity and mixed boundary conditions for temperature are considered. The optimal control problems for these equations with tracking‐type functionals are formulated. A local stability of the concrete control problem solutions with respect to some disturbances of both cost functionals and state equation is proved.

Numerical Accuracy of Fluid Solver Using Signed Distance Function for Shape Representation (Difference and Interpolation Methods near Interfaces Based on Distance and Normal)

A fluid solver using signed distance function (SDF) for shape representation was developed based on the immersed boundary method to simulate incompressible viscous flows. The forcing velocities near boundary are extrapolated by trilinear interpolation with taking into account a boundary condition using SDF. SMAC method is employed for solving basic equations for unsteady incompressible flows. The equations are discretized in space by 2nd-order central difference method, where the discretization near boundary is improved by SDF to satisfy a Dirichlet boundary condition for velocity. The fluid solver was verified in both steady and oscillating three-dimensional Poiseuille flows. As the grid spacing decreases, L2 and L∞ norm of the error of the axial velocity profile respectively decrease by the order of 1.96 and 1.89 for the oscillating flow. Therefore, the fluid solver enables to analyze the Poiseuille flow using Cartesian mesh by 2nd-order of accuracy in space.

Extremum problems of boundary control for a stationary thermal convection model

151 Recently, the control theory of thermal and hydro� dynamic fields in continuous media has been inten� sively developed. The mathematical description of such problems includes three components: a goal, the control mechanisms used to achieve this goal, and constraints imposed on the state and controls of the system under study. The constraints are specified by the equations of the used continuous medium model together with boundary conditions set on the bound� ary of the domain, while the desired goal is achieved by minimizing a certain cost functional. Control problems and inverse extremum problems for stationary models of heat and mass transfer have been theoretically studied, for example, in [1–8], where the solvability of optimization problems was analyzed and optimality systems describing necessary extremum conditions were derived and examined. In [6–8], based on an analysis of an optimality system, sufficient conditions for the uniqueness and stability of solutions to control problems were established in spe� cial cases corresponding to purely hydrodynamic or temperature cost functionals and controls. The con� trol of viscous flows by applying heat sources and the control of thermal processes by applying hydrody� namic sources have been investigated to a lesser degree. At the same time, computations show that effective mechanisms for the control of viscous flows can be found in this direction. The goal of this paper is to study such control prob� lems for the Oberbeck–Boussinesq thermal convec� tion model. First, we formulate the general control problem for this model and present a theorem provid� ing sufficient conditions on the initial data under which the solution of the control problem is unique and stable with respect to small perturbations in the cost functional and the boundary function involved in the Dirichlet boundary condition for velocity. Next, we describe a numerical solution algorithm for the control problem based on Newton’s method and dis� cuss numerical results. They confirm that the algo� rithm is highly effective in a wide range of the basic parameters of the boundary value problem.