Solution of three-dimensional viscous flows using integral velocity–vorticity formulation

In this paper, a numerical approach is presented to solve the velocity–vorticity integro-differential formulations for three-dimensional incompressible viscous flow. Both the velocity and pressure are solved in integral formulations and the general numerical method is based on standard finite volume scheme. The velocities needed at the vertexes of each control volume are calculated by a so-called generalized Biot–Savart formula combined with a fast multipole algorithm, which makes the velocity boundary conditions implicitly satisfied by maintaining the kinematic compatibility of the velocity and vorticity fields. The well-known fractional step approaches are used to solve the vorticity transport equation. No-flux boundary conditions on solid objects are satisfied as vorticity Helmholtz equation is solved. The diffusion term in the transport equation is treated implicitly using a conservative finite update. The diffusive fluxes of vorticity into flow domain from solid boundaries are determined by an iterative process in order to satisfy the no tangential-flow boundary condition. As an application example, the impulsively started flow through a sphere with different Reynolds numbers is computed using the method. The calculated results are compared with the experimental data and other numerical results and show good agreement.     

Combined single domain and subdomain BEM for 3D laminar viscous flow

A subdomain boundary element method (BEM) using a continuous quadratic interpolation of function and discontinuous linear interpolation of flux is presented for the solution of the vorticity transport equation and the kinematics equation in 3D. By employing compatibility conditions between subdomains an over-determined system of linear equations is obtained, which is solved in a least squares manner. The method, combined with the single domain BEM, is used to solve laminar viscous flows using the velocity vorticity formulation of Navier–Stokes equations. The versatility and accuracy of the method are proven using the 3D lid driven cavity test case.     

A velocity-decomposition formulation for the incompressible Navier–Stokes equations

The principle of velocity decomposition is used to combine field discretization and boundary-element techniques to solve for steady, viscous, external flows around bodies. The decomposition modifies the Navier–Stokes boundary-value problem and produces a Laplace problem for a viscous potential, and a new Navier–Stokes sub-problem that can be solved on the portion of the domain where the total velocity has rotation. The key development in the decomposition is the formulation for the boundary condition on the viscous potential that couples the two components of velocity. An iterative numerical scheme is described to solve the decomposed problem. Results are shown for the steady laminar flow over a sectional airfoil, a circular cylinder with separation, and the turbulent flow around a slender body-of-revolution. The results show the viscous potential is obtainable even for massively separated flows, and the field discretization must only encompass the vortical region of the total velocity.     

High Reynolds number flows with closed streamlines

Summary In this paper we consider high Reynolds number flows with closed streamliness within which an inviscid region of uniform vorticity is separated from the containing boundary by viscous boundary layers. From numerical solutions of the boundary-layer equations we are able to determine that value of the core vorticity for which inviscid core and boundary layer are compatible.     

Existence of periodic solutions of the equations of incompressible micropolar fluid flow

The equations of incompressible micropolar fluid flow are a coupled system of vector differential equations involving the two basic vectors, viz. the velocity and the microrotation of the fluid elements. Let D = D (t) be a bounded region in space, and let a flow velocity and a microrotation be prescribed at each point of the boundary of D(t). Assume that D(t) as well as the assigned velocity and microrotation vectors depend periodically on the time t and that the condition (2μ+k)j−4a 0 is satisfied (equation (25) in the text). Further assumptions are that (i) to every continuous initial distribution of the flow fields over D, there corresponds a solution of the field equations for all time t 0 satisfying the prescribed boundary conditions; (ii) there is one solution for which the Reynolds numbers Re, Rm satisfy the condition Re² + Rm² < 80 and this solution is equicontinuous in for all t. Then there exists a unique, stable, periodic solution of the micropolar flow equations in D(t) taking the prescribed values on the boundary. The proof of the theorem rests on a formula describing the rate of decay of the kinetic energy of the difference of two micropolar flows in the domain subject to the same boundary conditions.     

A vorticity-only formulation and a low-order asymptotic expansion solution near Hopf bifurcation

A vorticity-only formulation is used in order to study the behavior of the solutions of the Navier-Stokes equations for two-dimensional incompressible flows as the Reynolds number is increased. This approach allows one to limit the numerical-solution domain to the vortical region of the flow, thereby reducing the number of the state variables of the system. The vorticity-only formulation is obtained from a vorticity-stream function formulation by inverting the Poisson equation relating the vorticity to the stream function and substituting the expression for the velocity in the vorticity-transport equation. The vorticity at the solid boundary is determined from the boundary conditions. The resulting dynamical system consists of a set of first-order ordinary differential equations having only quadratic nonlinearities. This system is then used to address the behavior of the solution beyond the stability boundary, within the context of the theory of dynamical systems. This part of the paper is general and is based on the use of a singular-perturbation technique, known as the method of multiple time scales; formulae for the nonlinear analysis near a Hopf bifurcation are given explicitly in terms of the coefficients of the dynamical system. Preliminary numerical results for the validation of the formulation are presented for the limited case of two-dimensional flows around a circular cylinder. These results include the steady-state solution with varying Reynolds number, the eigenvalues and the eigenvectors of the related stability matrix, and the characteristics of the corresponding limit cycle.     

A combined BEM–FEM model for the velocity–vorticity formulation of the Navier–Stokes equations in three dimensions

This paper describes a combined boundary element and finite element model for the solution of velocity–vorticity formulation of the Navier–Stokes equations in three dimensions. In the velocity–vorticity formulation of the Navier–Stokes equations, the Poisson type velocity equations are solved using the boundary element method (BEM) and the vorticity transport equations are solved using the finite element method (FEM) and both are combined to form an iterative scheme. The vorticity boundary conditions for the solution of vorticity transport equations are exactly obtained directly from the BEM solution of the velocity Poisson equations. Here the results of medium Reynolds number of up to 1000, in a typical cubic cavity flow are presented and compared with other numerical models. The combined BEM–FEM model are generally in fairly close agreement with the results of other numerical models, even for a coarse mesh.     

Electroosmotic flows in microchannels with finite inertial and pressure forces

Emerging microfluidic systems have spurred an interest in the study of electrokinetic flow phenomena in complex geometries and a variety of flow conditions. This paper presents an analysis of the effects of fluid inertia and pressure on the velocity and vorticity field of electroosmotic flows. In typical on-chip electrokinetics applications, the flow field can be separated into an inner flow region dominated by viscous and electrostatic forces and an outer flow region dominated by inertial and pressure forces. These two regions are separated by a slip velocity condition determined by the Helmholtz-Smoulochowski equation. The validity of this assumption is investigated by analyzing the velocity field in a pressure-driven, two-dimensional flow channel with an impulsively started electric field. The regime for which the inner/outer flow model is valid is described in terms of nondimensional parameters derived from this example problem. Next, the inertial forces, surface conditions, and pressure-gradient conditions for a full-field similarity between the electric and velocity fields in electroosmotic flows are discussed. A sufficient set of conditions for this similarity to hold in arbitrarily shaped, insulating wall microchannels is the following: uniform surface charge, low Reynolds number, low Reynolds and Strouhal number product, uniform fluid properties, and zero pressure differences between inlets and outlets. Last, simple relations describing the generation of vorticity in electroosmotic flow are derived using a wall-local, streamline coordinate system.     

A numerical investigation of the effect of boundary motion on viscous channel flow

The steady, laminar, incompressible flow past a moving boundary in the entry region of a two-dimensional channel is considered in this study. The formation of a separated region during the upstream motion of a section of the lower boundary is of particular interest. The size of the separated region depends on the Reynolds number, and on the velocity and length of the moving boundary as well. A numerical solution is obtained from the continuity and the complete Navier-Stokes equations, subject to the appropriate boundary conditions. The describing equations are expressed in terms of the vorticity and the stream function. The alternating-direction implicit method is used to solve the vorticity equation while the successive over-relaxation method is used to solve, the stream function equation. The present numerical scheme is second-order-accurate since both the finite-difference equations and boundary conditions have second-order accuracy.     

Stagnant vortex flow

With the aid of the vorticity transport equation it is shown that in inviscid, incompressible, axially symmetric vortex flow the axial vorticity component near the axis of the vortex approaches zero if the axial velocity component approaches a stagnation point, and vice versa, the axial vorticity component is increased, if the axial flow is accelerated. This result, obtained in earlier investigations by simplifying the momentum equations for the neighborhood of the axis of the vortex, is already contained in the vorticity transport equation as formulated by von Helmholtz in 1858. In laminar flow, with viscous forces acting near the stagnation point, the angular velocity does not necessarily vanish with the axial velocity component. These questions are discussed in the following.     

Micropolar fluid jet impingement on a curved surface

This paper deals with an analysis of a normally impinging micropolar fluid jet on a curved surface. The flow near the stagnation point in the impingement region is divided into inviscid and viscous flow regions. The inviscid flow solution is governed by Euler's equations of motion expressed in curvilinear coordinate system. The viscous flow solution is governed by the zeroth and the first order boundary layer equations. These boundary layer equations are solved by assuming power series expansions for both velocity and microrotation fields which give rise to two systems of ordinary coupled differential equations. The effects of surface curvature and material parameters on boundary layer characteristics have been studied and presented graphically. The gradients of zeroth order velocity and microrotation at the wall decrease and the zeroth order displacement and momentum thicknesses increase with decrease in the value of surface curvature. The reduction in curvature results in the reduction in the gradients of first order velocity and microrotation at the wall as well as first order displacement and momentum thicknesses.     

Stokes flow past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition

The solution of the problem of symmetrical creeping flow of an incompressible viscous fluid past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition is investigated. The Brinkman equation for the flow inside the porous region and the Stokes equation for the outside region in their stream function formulations are used. As boundary conditions, continuity of velocity and surface stresses across the porous surface and Kuwabara boundary condition on the cell surface are employed. Explicit expressions are investigated for both inside and outside flow fields to the first order in a small parameter characterizing the deformation. As a particular case, the flow past a swarm of porous oblate spheroidal particles is considered and the drag force experienced by each porous oblate spheroid in a cell is evaluated. The dependence of the drag coefficient on permeability for a porous oblate spheroid in an unbounded medium and for a solid oblate spheroid in a cell on the solid volume fraction is discussed numerically an and graphically for various values of the deformation parameter. The earlier known results are then also deduced from the present analysis.     

Pressure correction DRBEM solution for heat transfer and fluid flow in incompressible viscous fluids

The Dual Reciprocity Boundary Element Method (DRBEM) is used to solve incompressible laminar viscous fluid flows and heat transfer. The DRBEM is extended to develop a pressure correction scheme to solve the incompressible Navier-Stokes equations. The velocity field is then used as input to the DRBEM solution of the energy transport equation, thereby retaining the boundary only discretization feature of the BEM for the solution of this problem. Numerical results for the proposed DRBEM solution for laminar flow and heat transfer in a channel are obtained for several Reynolds numbers and compare well with previously published data.     

A parallel domain decomposition BEM for diffusion problems with surface reactions

A parallel domain decomposition boundary element method (BEM) is developed for the solution of three-dimensional multispecies diffusion problems. The chemical species are uncoupled in the interior of the domain but couple at the boundary through a nonlinear surface reaction equation. The method of lines is used whereby time is discretized using the finite difference method and space is discretized using the boundary element method. The original problem is transformed into a sequence of nonhomogeneous modified Helmholtz equations. A Schwarz Neumann–Neumann iteration scheme is used to satisfy interfacial boundary conditions between subdomains. A segregated solver based on a quasi-predictor–corrector time integrator is used to satisfy the nonlinear boundary conditions on the reactive surfaces. The accuracy and parallel efficiency of the method is demonstrated through a benchmark problem.     

Boundary element analysis of wave-current interaction around a large structure

A numerical approach for wave-current interaction around a large structure is investigated, based on potential flow theory, linear waves and small current velocity approximation. The velocity potential in a wave-current coexisting field is separated into a steady current potential and an unsteady wave potential. The boundary element method was then employed to compute the unsteady wave potential with effects of both a uniform current and a large body taken into consideration. It is demonstrated that the steady current potential can be expressed as the sum of a uniform current and a steady disturbance due to the presence of the object. The variation of current velocity in the vicinity of the object is then calculated by using a surface vorticity boundary integral meethod. Boundary element analysis is also used for the numerical solutions of the surface vorticity method. Substituting both unsteady wave potential and current velocity into the first-order dynamic surface boundary condition, the water surface elevation around a large structure in a wave-current coexisting field can then be obtained. Comparisons of numerical predictions with experimental results ar also made; qualitative good agreements are obtained.     

NUMERICAL SIMULATION OF GRANULAR COUETTE FLOWS BETWEEN TWO ROUGH PARALLEL PLATES

This paper presents a numerical method for calculating the granular Couette flows between two parallel plates. A kinetic model which includes the frictional energy loss effects is employed, and the equations of motion are solved using a numerical iterative method. The boundary conditions are satisfied by ensuring the balance of momentum and energy at such boundaries. The mean velocity, the fluctuation kinetic energy and the solid volume fraction profiles are evaluated under a variety of conditions. The mean velocity profiles are compared with the molecular dynamic simulation results, and good agreement is observed. The study shows that the slip velocity may vary considerably depending on the surface roughness, coefficient of restitution and friction coefficient.     

DRBEM solution of the thermo-solutal buoyancy induced mixed convection flow problems

Mixed convective heat and solutal transport is important in engineering applications such as nuclear waste disposal, crystal growth and oceanography. Mixed convective in a lid-driven cavity and through channels with backward-facing step is studied by solving the equations of conservation of mass, momentum, energy and solutal concentration numerically. The governing equations are solved by using dual reciprocity boundary element method (DRBEM) with constant elements in terms of stream function, vorticity, temperature and concentration. Vorticity, energy and concentration equations are transformed to the form of modified Helmholtz equations by utilizing forward difference with relaxation parameters for the time derivatives, and also approximating Laplacian terms at two consecutive time levels. The DRBEM application is carried out with the fundamental solution of modified Helmholtz equation, resulting in linear algebraic systems for the time dependent unknowns. Inhomogeneities in modified Helmholtz equations are approximated with the thin plated radial basis functions. Computations are carried out for several values of Richardson number, buoyancy ratio and Reynolds number. When the temperature and solutal concentration boundary conditions are changed, the thermal and solutal buoyancy forces can oppose or aid each other. The effects of these parameters on the flow behavior and heat transfer are shown in terms of graphics.     

Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation

In this paper, the boundary knot method is extended to the solution of inhomogeneous equations, and it is applied to the Cauchy problem associated with the inhomogeneous Helmholtz equation. Here, we assume that the boundary condition is specified only on a part of the boundary, and the boundary conditions on the remaining part of the boundary are to be determined with the assistance of additional data. Since the resulting matrix equation is highly ill-conditioned, a regularized solution is obtained by employing the truncated singular value decomposition to solve the matrix equation arising from the boundary knot method, with the regularization parameter determined by the L-curve method. Numerical results are presented for several examples with smooth and piecewise smooth boundaries. The numerical verification shows that the proposed numerical scheme is accurate, stable with respect to data noise, and convergent with respect to decreasing the amount of noise in the data.     

An implicit potential method for incompressible flows

In this paper, we consider a novel numerical scheme for solving incompressible flows on collocated grids. The implicit potential method utilizes an implicit potential velocity obtained from a Helmholtz decomposition for the mass conservation and employs a modified form of Bernoulli's law for the coupling of the velocity–pressure corrections. It requires the solution only of the momentum equations, does not involve the solution of additional partial differential equations for the pressure, and is applied on a collocated grid. The accuracy of the method is tested through comparison with analytical, experimental, and numerical data from the literature, and its efficiency and robustness are evaluated by solving several benchmark problems such as flow around a circular cylinder and in curved square and circular ducts.Copyright © 2013 John Wiley & Sons, Ltd.     

Boundary integral equation solution of viscous flows with free surfaces

Summary solutions of the biharmonic equation governing steady two-dimensional viscous flow of an incompressible Newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green's theorem is used to reformulate the differential equation as a pair of coupled integral equations which are applied only on the boundary of the solution domain.An iterative modification of the classical BBIE is presented which is able to solve a large class of (nonlinear) viscous free surface flows for a wide range of surface tensions. The method requires a knowledge of the asymptotic behaviour of the free surface profile in the limiting case of infinite surface tension but this can usually be obtained from a perturbation analysis. Unlike space discretisation techniques such as finite difference or finite element, the BBIE evaluates only boundary information on each iteration. Once the solution is evaluated on the boundary the solution at interior points can easily be obtained.