Meshless Galerkin analysis of Stokes slip flow with boundary integral equations

This paper presents a novel meshless Galerkin scheme for modeling incompressible slip Stokes flows in 2D. The boundary value problem is reformulated as boundary integral equations of the first kind which is then converted into an equivalent variational problem with constraint. We introduce a Lagrangian multiplier to incorporate the constraint and apply the moving least‐squares approximations to generate trial and test functions. In this boundary‐type meshless method, boundary conditions can be implemented exactly and system matrices are symmetric. Unlike the domain‐type method, this Galerkin scheme requires only a nodal structure on the bounding surface of a body for approximation of boundary unknowns. The convergence and abstract error estimates of this new approach are given. Numerical examples are also presented to show the efficiency of the method. Copyright © 2009 John Wiley & Sons, Ltd.     

平面Laplace外边值问题

证明平面调和函数的Dirichlet外问题解存在唯一的充要条件,在此基础上,确立外问题的等价边界积分方程,首次给出外域上的极值原理,对第一类Fredholm边界的积分方程的可解性进行了讨论。     

Efficient quadrature for a boundary element method to compute three-dimensional Stokes flow

A collocation-type boundary element method based on bilinear B-splines is used for the numerical solution of the Stokes Dirichlet problem in bounded domains D ? R³. The computation of the influence matrix requires the numerical evaluation of weakly singular integrals on the domain boundary if the usual double-layer potential ansatz is chosen. Here mostly standard methods with disjoint grids for collocation and integration are used. We develop a special integration scheme based on triangular co-ordinates near the singularity and show its efficiency compared with the method mentioned above.     

A high‐order scheme for the incompressible Navier–Stokes equations with open boundary condition

We present a numerical scheme to solve the incompressible Navier–Stokes equations with open boundary condition. After replacing the incompressibility constraint by the pressure Poisson equation, the key is how to give an appropriate boundary condition for the pressure Poisson equation. We propose a new boundary condition for the pressure on the open boundary. Some numerical experiments are presented to verify the accuracy and stability of scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

Towards a transparent boundary condition for compressible Navier–Stokes equations

A new artificial boundary condition for two‐dimensional subsonic flows governed by the compressible Navier–Stokes equations is derived. It is based on the hyperbolic part of the equations, according to the way of propagation of the characteristic waves. A reference flow, as well as a convection velocity, is used to properly discretize the terms corresponding to the entering waves. Numerical tests on various classical model problems, whose solutions are known, and comparisons with other boundary conditions (BCs), show the efficiency of the BC. Direct numerical simulations of more complex flows over a dihedral plate are simulated, without creation of acoustic waves going back in the flow. Copyright © 2001 John Wiley & Sons, Ltd.     

On the uniqueness of boundary integral equation for the exterior Helmholtz problem

From the point of view of energy analysis,the cause that the uniqueness of theboundary integral equation induced from the exterior Helmholtz problem does not hold isinvestigated in this paper.It is proved that the Sommerfeld’s condition at the infinity ischanged so that it is suitable not only for the radiative wave but also for the absorptive wavewhen we use the boundary integral equation to describe the exterior Helmholtz problem.Therefore,the total energy of the system is conservative.The mathematical dealings toguarantee the uniqueness are discussed based upon this explanation     

Stokes flow before plane boundary with mixed stick-slip boundary conditions

A general theorem for the Stokes flow over a plane boundary with mixed stick-slip boundary conditions is established. This is done by using a representation for the velocity and pressure fields in the three-dimensional Stokes flow in terms of a biharmonic function and a harmonic function. The earlier theorem for the Stokes flow due to fundamental singularities before a no-slip plane boundary is shown to be a special case of the present theorem. Furthermore, in terms of the Stokes stream function, a corollary of the theorem is also derived, providing a solution to the problem of the axisymmetric Stokes flow along a rigid plane with stick-slip boundary conditions. The formulae for the drag and torque exerted by the fluid on the boundary are established. An illustrative example is given.     

A Sinc‐collocation method with boundary treatment for the Stokes equations

In Stokes equations the velocity u and the pressure p are coupled together by the imcompressibility condition div u =0 which makes the equations difficult to solve numerically. In this paper, a method named Sinc‐collocation method with boundary treatment (SCMBT) is applied to the Stokes equations. The numerical results show that our method is of high accuracy, of good convergence with little computational effort. Copyright © 2006 John Wiley & Sons, Ltd.     

Penalty finite element method for Navier–Stokes equations with nonlinear slip boundary conditions

The penalty finite element method for Navier–Stokes equations with nonlinear slip boundary conditions is investigated in this paper. Since this class of nonlinear slip boundary conditions include the subdifferential property, the weak variational formulation is a variational inequality problem of the second kind. Using the penalty finite element approximation, we obtain optimal error estimates between the exact solution and the finite element approximation solution. Finally, we show the numerical results which are in full agreement with the theoretical results. Copyright © 2011 John Wiley & Sons, Ltd.     

A complete boundary integral formulation for compressible Navier–Stokes equations

A complete boundary integral formulation for compressible Navier–Stokes equations with time discretization by operator splitting is developed using the fundamental solutions of the Helmholtz operator equation with different order. The numerical results for wall pressure and wall skin friction of two‐dimensional compressible laminar viscous flow around airfoils are in good agreement with field numerical methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Nonlinear Galerkin method for the exterior nonstationary Navier-Stokes equations

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔΩｃｏｎｔａｉｎｉｎｇｚｅｒｏｐｏｉｎｔｂｅａｓｉｍｐｌｙ＿ｃｏｎｎｅｃｔｅｄｂｏｕｎｄｅｄｏｐｅｎｓｅｔｏｆＲ2 ｗｉｔｈｓｍｏｏｔｈｂｏｕｎｄａｒｙΓａｎｄｌｅｔΩ′ｄｅｎｏｔｅｔｈｅｃｏｍｐｌｅｍｅｎｔｏｆΩ ∪Γ .ＴｈｅｅｘｔｅｒｉｏｒｎｏｎｓｔａｔｉｏｎａｒｙＮａｖｉｅｒ＿ＳｔｏｋｅｓｐｒｏｂｌｅｍｆｏｒａｆｌｕｉｄｏｃｃｕｐｙｉｎｇΩ′ｃｏｎｓｉｓｔｓｉｎｆｉｎｄｉｎｇｔｈｅｖｅｌｏｃｉｔｙ ｕ(ｘ,ｔ)ｏｆｔｈｅｆｌｕｉｄａｎｄｉｔｓｐｒｅｓｓｕｒｅ ｐ(ｘ ,…     

Stokes flow of micro-polar fluids by peristaltic pumping through tube with slip boundary condition

This paper studies the Stokes flow of micro-polar fluids by peristaltic pumping through the cylindrical tube under the effect of the slip boundary condition. The motion of the wall is governed by the sinusoidal wave equation. The analytical and numerical solutions for the axial velocity, the micro-polar vector, the stream function, the pressure gradient, the friction force, and the mechanical efficiency are obtained by using the lubrication theory under the low Reynolds number and long wavelength approximations. The impacts of the emerging parameters, such as the coupling number, the micro-polar parameter, the slip parameter on pumping characteristics, the friction force, the velocity profile, the mechanical efficiency, and the trapping phenomenon are depicted graphically. The numerical results infer that large pressure is required for peristaltic pumping when the coupling number is large, while opposite behaviors are found for the micro-polar parameter and the slip parameter. The size of the trapped bolus reduces with the increase in the coupling number and the micro-polar parameter, whereas it blows up with the increase in the slip parameter.     

Stokes equations with penalised slip boundary conditions

We consider the finite-element approximation of Stokes equations with slip boundary conditions imposed with the penalty method. In the case of a smooth curved boundary, our numerical results suggest that curved finite elements, regularised normal vectors or reduced integration techniques can be used to avoid a Babuska’s-type paradox and ensure the convergence of finite-element approximations to the exact solution. Convergence orders with these remedies are also compared.     

Simulation of the Navier–Stokes equations in three dimensions with a spectral collocation method

This paper describes a nonlinear, three‐dimensional spectral collocation method for the simulation of the incompressible Navier–Stokes equations under the Boussinesq approximation, motivated by geophysical and environmental flows. These flows are driven by the interaction of stratified fluid with topography, which this model accurately accounts for by using a mapped coordinate system. The spectral collocation method is implemented with both a Fourier trigonometric expansion and the Chebyshev polynomials, as appropriate for the domain boundary conditions. The coordinate mapping prohibits the use of existing, fast solution methods that rely on the separation of variables, so a preconditioner based on the approximate solution of a corresponding finite‐difference problem with geometric multigrid is used. The model is parallelized with the Message Passing Interface library, and it runs effectively on shared and distributed‐memory systems. Copyright © 2013 John Wiley & Sons, Ltd.     

Weak imposition of boundary conditions for the Navier–Stokes equations by a penalty method

We prove convergence of the finite element method for the Navier–Stokes equations in which the no‐slip condition and no‐penetration condition on the flow boundary are imposed via a penalty method. This approach has been previously studied for the Stokes problem by Liakos (Weak imposition of boundary conditions in the Stokes problem. Ph.D. Thesis, University of Pittsburgh, 1999). Since, in most realistic applications, inertial effects dominate, it is crucial to extend the validity of the method to the nonlinear Navier–Stokes case. This report includes the analysis of this extension, as well as numerical results validating their analytical counterparts. Specifically, we show that optimal order of convergence can be achieved if the computational boundary follows the real flow boundary exactly. Copyright © 2008 John Wiley & Sons, Ltd.     

A domain decomposition method for modelling Stokes flow in porous materials

An algorithm is presented for solving the Stokes equation in large disordered two‐dimensional porous domains. In this work, it is applied to random packings of discs, but the geometry can be essentially arbitrary. The approach includes the subdivision of the domain and a subsequent application of boundary integral equations to the subdomains. This gives a block diagonal matrix with sparse off‐block components that arise from shared variables on internal subdomain boundaries. The global problem is solved using a biconjugate gradient routine with preconditioning. Results show that the effectiveness of the preconditioner is strongly affected by the subdomain structure, from which a methodology is proposed for the domain decomposition step. A minimum is observed in the solution time versus subdomain size, which is governed by the time required for preconditioning, the time for vector multiplications in the biconjugate gradient routine, the iterative convergence rate and issues related to memory allocation. The method is demonstrated on various domains including a random 1000‐particle domain. The solution can be used for efficient recovery of point velocities, which is discussed in the context of stochastic modelling of solute transport. Copyright © 2002 John Wiley & Sons, Ltd.     

Fitted finite element discretization of two‐phase Stokes flow

We propose a novel fitted finite element method for two‐phase Stokes flow problems that uses piecewise linear finite elements to approximate the moving interface. The method can be shown to be unconditionally stable. Moreover, spherical stationary solutions are captured exactly by the numerical approximation. In addition, the meshes describing the discrete interface in general do not deteriorate in time, which means that in numerical simulations, a smoothing or a remeshing of the interface mesh is not necessary. We present several numerical experiments for our numerical method, which demonstrate the accuracy and robustness of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

A well-posed problem for the exterior Stokes equations in two and three dimensions

This paper treats the Stokes problem in exterior Lipschitz-continuous domains of ² and ³. Using the weighted Sobolev spaces of Hanouzet (in ³) and Giroire (in ²), we establish the inf-sup condition between the velocity and pressure spaces. This fundamental result shows that the variational Stokes problem is well-posed in those spaces. In the last paragraph, we obtain additional regularity of the solution when the data are smoother.     

A boundary element method for steady viscous fluid flow using penalty function formulation

A new boundary element method is presented for steady incompressible flow at moderate and high Reynolds numbers. The whole domain is discretized into a number of eight-noded cells, for each of which the governing boundary integral equation is formulated exclusively in terms of velocities and tractions. The kernels used in this paper are the fundamental solutions of the linearized Navier–Stokes equations with artificial compressibility. Significant attention is given to the numerical evaluation of the integrals over quadratic boundary elements as well as over quadratic quadrilateral volume cells in order to ensure a high accuracy level at high Reynolds numbers. As an illustration, square driven cavity flows are considered for Reynolds numbers up to 1000. Numerical results demonstrate both the high convergence rate, even when using simple (direct) iterations, and the appropriate level of accuracy of the proposed method. Although the method yields a high level of accuracy in the primary vortex region, the secondary vortices are not properly resolved. © 1997 John Wiley & Sons, Ltd.     

Adaptive variational multiscale method for the Stokes equations

An adaptive variational multiscale method for the Stokes equations is presented in this paper. We solve the coarse scale problem on the coarse mesh and approximate the fine scale solution by solving a series of local residual equations defined on some local fine grids, which can be implemented in parallel. In addition, we also propose a reliable local a posteriori error estimator and construct an adaptive algorithm based on the corresponding a posterior error estimate. Finally, numerical examples are presented to verify the algorithm.Copyright © 2012 John Wiley & Sons, Ltd.