Incompressible Viscous Flow Problems Research Articles

Numerical solution of a viscous incompressible flow problem through an orifice by Adomian decomposition method

A steady flow problem of a viscous incompressible fluid through an orifice is widely applicable to many physical phenomena and has been studied previously by many researchers. A problem of such type has been solved by applying LAD method given by Roache [Comp. Fluids 3 (1975) 179]. The resulting system of linear equations is solved by Adomian decomposition method [Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, MA, 1994]. We study an analytic solution of a steady flow problem of a viscous incompressible fluid through an orifice by using the Adomian's decomposition method. The numerical results show the effectiveness of the method for this type of equations, they show that the present approach is relatively easy and highly accurate.

On streamline diffusion arising in Galerkin FEM with predictor/multi‐corrector time integration

AbstractIn the present paper, the author shows that the predictor/multi‐corrector (PMC) time integration for the advection–diffusion equations induces numerical diffusivity acting only in the streamline direction, even though the equations are spatially discretized by the conventional Galerkin finite element method (GFEM). The transient 2‐D and 3‐D advection problems are solved with the PMC scheme using both the GFEM and the streamline upwind/Petrov Galerkin (SUPG) as the spatial discretization methods for comparison. The solutions of the SUPG‐PMC turned out to be overly diffusive due to the additional PMC streamline diffusion, while the solutions of the GFEM‐PMC were comparatively accurate without significant damping and phase error. A similar tendency was seen also in the quasi‐steady solutions to the incompressible viscous flow problems: 2‐D driven cavity flow and natural convection in a square cavity. Copyright © 2002 John Wiley & Sons, Ltd.

Parallelization of a vorticity formulation for the analysis of incompressible viscous fluid flows

AbstractA parallel computer implementation of a vorticity formulation for the analysis of incompressible viscous fluid flow problems is presented. The vorticity formulation involves a three‐step process, two kinematic steps followed by a kinetic step. The first kinematic step determines vortex sheet strengths along the boundary of the domain from a Galerkin implementation of the generalized Helmholtz decomposition. The vortex sheet strengths are related to the vorticity flux boundary conditions. The second kinematic step determines the interior velocity field from the regular form of the generalized Helmholtz decomposition. The third kinetic step solves the vorticity equation using a Galerkin finite element method with boundary conditions determined in the first step and velocities determined in the second step. The accuracy of the numerical algorithm is demonstrated through the driven‐cavity problem and the 2‐D cylinder in a free‐stream problem, which represent both internal and external flows. Each of the three steps requires a unique parallelization effort, which are evaluated in terms of parallel efficiency. Copyright © 2002 John Wiley & Sons, Ltd.

Analysis of à‐posteriori error indicator in viscous flows

An à‐posteriori error indicator for solving viscous incompressible flow problems is analyzed in this paper. The indicator named “velocity angle error estimator” is based on the spatial derivative of velocity direction fields and it can detect local flow features, such as vortices and separation, and resolve flow details precisely. The refinement indicator corresponds to the antisymmetric part of the deformation‐rate‐tensor, and it is sensitive to the second derivative of the velocity angle field. Rationality discussions reveal that the à‐posteriori error indicator is a curvature error indicator, and its value reflects the accuracy of streamline curves. It is also found that the velocity angle error indicator contains the nonlinear convective term of the Navier–Stokes equations, and it identifies and computes the direction difference when the convective acceleration direction and the flow velocity direction have a disparity. Numerical simulation is presented to illustrate the use of the velocity angle error indicator.

Stochastic Problem of Viscous Incompressible Fluid Flow with Heat Transfer

The paper presented is devoted to the problem of stochastic viscous incompressible fluid flow with heat transfer. The material parameters of the medium are treated as random variables and are defined by use of the first two probabilistic moments. The second order perturbation stochastic second central moment approach is used to improve zeroth, first, and second order differential equations; it makes it possible to compute temperature, pressure, and flow velocity probabilistic moments, i.e. their expected values and cross-covariances. Next the problem is discretized using stochastic finite element method technique, well-known from solid mechanics, and it is illustrated with an example of plane Couette flow.

Numerical solution of Navier–Stokes equations using multiquadric radial basis function networks

AbstractA numerical method based on radial basis function networks (RBFNs) for solving steady incompressible viscous flow problems (including Boussinesq materials) is presented in this paper. The method uses a ‘universal approximator’ based on neural network methodology to represent the solutions. The method is easy to implement and does not require any kind of ‘finite element‐type’ discretization of the domain and its boundary. Instead, two sets of random points distributed throughout the domain and on the boundary are required. The first set defines the centres of the RBFNs and the second defines the collocation points. The two sets of points can be different; however, experience shows that if the two sets are the same better results are obtained. In this work the two sets are identical and hence commonly referred to as the set of centres. Planar Poiseuille, driven cavity and natural convection flows are simulated to verify the method. The numerical solutions obtained using only relatively low densities of centres are in good agreement with analytical and benchmark solutions available in the literature. With uniformly distributed centres, the method achieves Reynolds number Re = 100 000 for the Poiseuille flow (assuming that laminar flow can be maintained) using the density of $11\times 11$, Re = 400 for the driven cavity flow with a density of $33\times 33$ and Rayleigh number Ra = 1 000 000 for the natural convection flow with a density of $27\times 27$. Copyright © 2001 John Wiley & Sons, Ltd.

A coupled BEM and arbitrary Lagrangian–Eulerian FEM model for the solution of two‐dimensional laminar flows in external flow fields

AbstractThis paper describes a new computational model developed to solve two‐dimensional incompressible viscous flow problems in external flow fields. The model based on the Navier–Stokes equations in primitive variables is able to solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the pressure projection method. The external flow field is simulated using the boundary element method by solving a pressure Poisson equation that assumes the pressure as zero at the infinite boundary. The momentum equation of the flow motion is solved using the three‐step finite element method. The arbitrary Lagrangian–Eulerian method is incorporated into the model, to solve the moving boundary problems. The present model is applied to simulate various external flow problems like flow across circular cylinder, acceleration and deceleration of the circular cylinder moving in a still fluid and vibration of the circular cylinder induced by the vortex shedding. The simulation results are found to be very reasonable and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd.

A Fourth-Order-Accurate Finite Volume Compact Method for the Incompressible Navier–Stokes Solutions

This paper presents a finite volume fourth-order-accurate compact scheme for discretization of the incompressible Navier–Stokes equations in primitive variable formulation. The numerical method of integrating the Navier–Stokes equations comprises a compact finite volume formulation of the average convective and diffusive fluxes. The pressure–velocity coupling is achieved via the coupled solution of the resulting system of equations. The solution of the coupled set of equations is performed with an implicit Newton–Krylov matrix-free method for stationary problems. For simulation of unsteady flows, a standard fourth-order Runge–Kutta method was used for temporal discretization and the velocity–pressure coupling was ensured at each stage also using the matrix-free method. Several incompressible viscous steady and unsteady flow problems have been computed to assess the robustness and accuracy of the proposed method.

Solution of the Navier–Stokes equations in velocity–vorticity form using a Eulerian–Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection-dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two-dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.

Upwind local differential quadrature method for solving incompressible viscous flow

The differential quadrature method (DQM) is able to obtain quite accurate numerical solutions of differential equations with few grid points and less computational effort. The successful applications of DQM have been reported to solve various problems of fluid mechanics. But, owing to the limitation of its quadrature rules, DQM is convenient only for regular regions and lacks upwind mechanism to characterize the convection of the fluid flow. In the present paper, a local differential quadrature method (LDQM) with upwind mechanism is proposed to solve incompressible viscous fluid flow problems in irregular regions. As its application, planar contraction flows in irregular regions is numerically solved and excellent numerical results are obtained for coarse meshes.

AN ADAPTIVE MESH REFINEMENT USING À-POSTERIORI FINITE ELEMENT ERROR ESTIMATION

An a-posteriori adaptive estimator is presented and employed for solving viscous incompressible flow problems. In an effort to detect local flow features and resolve flow details, an error estimation that is based on velocity angle is investigated, analyzed and benchmarked by an exact solution which is known as Kovasznay flow. It is found that the estimator is sensitive to the variations of the derivative of the velocity direction field, and it can capture the region and refine grids where the velocity direction has abrupt changes. Unstructured grids are adapted by employing local cell division as well as unrefinement of transition cells. The adaptive scheme is applied to flow over a cavity, flow past a backward-facing step, and flow past an obstacle at different Reynolds numbers. The pressure oscillation which usually occurs in advection-dominated flow cases is suppressed by adding more nodes at the most appropriate regions by using the velocity angle estimator. The results exhibit good accuracy and just...

Solution of the Navier-Stokes equations in velocity-vorticity form using a Eulerian-Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection-dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two-dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.

Numerical Study of Flows of Two Immiscible Liquids at Low Reynolds Number

We review our recent results on fingering, bamboo waves, and drop breakup in low Reynolds number flows composed of two viscous liquids under shear.

Effect of coriolis forces on nonstationary incompressible viscous liquid flow in a channel

We present a solution of the problem of nonstationary viscous incompressible liquid flow in a channel, taking account of the action of Coriolis forces. In this case the solution is shown to be determined by the sole dimensionless Taylor parameter, with an increase of which the velocity regime of nonstationary flow changes from a monotonic to a pulsating one.

VELOCITY-VORTICITY FORMULATION WITH VORTEX PARTICLE-IN-CELL METHOD FOR INCOMPRESSIBLE VISCOUS FLOW SIMULATION, PART I: FORMULATION AND VALIDATION

A new velocity-vorticity formulation along with a vortex particle-in-cell method (PIC) is developed to solve the incompressible viscous steady and unsteady flow problems for a closed domain. The article demonstrates that the vortex PIC method has some of the advantages of Lagrangian particle methods for computing convection, while the use of a Eulerian grid to compute the diffusion enables viscous flows to be computed. The solution procedure that is described takes advantage of fast Poisson solvers for computational efficiency. Two-dimensional driven cavity flows with impulsively started and oscillating lids are chosen to test the method. The computation results are in good agreement with those of the numerical literature.

Domain Decomposition Methods for Sensitivity Analysis of a Nonlinear Aeroelasticity Problem

We consider the nonlinear aeroelasticity problem of the interaction between a viscous, incompressible fluid and Lin elastic solid undergoing large displacement. The non-linearities of the problem formulation include the solid and fluid governing equations. as well as thc dependence of the How geometry on the solid deformation. The resulting coupling is thus two-way. We develop domain-decomposition methods for solution and sensitivity analysis of the coupled problem. The domain decomposition is in the form of a block-Gauss-Seidel-like prcconditioncr that decomposes ihc coupled-domain problem into distinct nonovcrlapping fluid and solid subdotnain problems. The preconditioner thus enables exploitation or single-domain algorithms for solid and fluid mechanics discretization and solution. On the other hand, two-way fluid-solid coupling is retained within the residuals, which is essential for correct sensitivities. Sensitivities of field quantities can be found with little additional work beyond that required for solving the coupled fluid-solid system. The methodology developed here is illustrated by the solution of a problem of viscous incompressible flow about an infinite clastic cylinder. Sensitivities of the resulting velocity and displacement fields with respect to elastic modulus and fluid viscosity are computed.

Parallel adaptive finite element analysis of viscous flows based on a combined compressible‐incompressible formulation

Describes a new finite element scheme for the large‐scale analysis of compressible and incompressible viscous flows. The scheme is based on a combined compressible‐ incompressible Galerkin least‐squares (GLS) space‐time variational formulation. Three‐ dimensional unstructured meshes are employed, with piecewise‐constant temporal interpolation, local time‐stepping for steady flows, and linear continuous spatial interpolation in all the variables. The scheme incorporates automatic adaptive mesh refinement, with a choice of various error indicators. It is implemented on a distributed‐memory parallel computer, and includes an automatic load‐balancing procedure. Demonstrates the ability to solve both compressible and incompressible viscous flow problems using the parallel adaptive framework via numerical examples. These include Mach 3 flow over a flat plate, and a divergence‐free buoyancy‐driven flow in a cavity. The latter is a model for the steady melt flow in a Czochralski crystal growth process.

On a compact mixed-order finite element for solving the three-dimensional incompressible Navier-Stokes equations

Our work is an extension of the previously proposed multivariant element. We assign this refined element as a compact mixed-order element in the sense that use of this element offers a much smaller bandwidth. The analysis is implemented on quadratic hexahedral elements with a view to analysing a three-dimensional incompressible viscous flow problem using a method formulated within the mixed finite element context. The idea of constructing such a stable element is to bring the marker-and-cell (MAC) grid lay-out to the finite element context. This multivariant element can thus be classified as a discontinuous pressure element. We have several reasons for advocating the proposed multivariant element. The primary advantage gained is its ability to reduce the bandwidth of the matrix equation, as compared with its univariant counterparts, so that it can be effectively stored in a compressed row storage (CRS) format. The resulting matrix equation can be solved efficiently by a multifrontal solver owing to its reduced bandwidth. The coding is, however, complicated by the appearance of restricted degrees of freedom at mid-face nodes. Through analytic study this compact multivariant element has a marked advantage over the multivariant element of Gupta et al. in that both bandwidth and computation time have been drastically reduced. © 1997 John Wiley & Sons, Ltd.

A Numerical Method for Solving Incompressible Viscous Flow Problems

A numerical method for solving incompressible viscous flow problems is introduced. This method uses the velocities and the pressure as variables and is equally applicable to problems in two and three space dimensions. The principle of the method lies in the introduction of an artificial compressibility δ into the equations of motion, in such a way that the final results do not depend on δ. An application to thermal convection problems is presented.

Parallel overlapping scheme for viscous incompressible flows

SUMMARY A 3D parallel overlapping scheme for viscous incompressible flow problems is presented that combines the finite element method, which is best suited for analysing flow in any arbitrarily shaped flow geometry, with the finite difference method, which is advantageous in terms of both computing time and computer storage. A modified ABMAC method is used as the solution algorithm, to which a sophisticated time integration scheme proposed by the present authors has been applied. Parallelization is based on the domain decomposition method. The RGB (recursive graph bisection) algorithm is used for the decomposition of the FEM mesh and simple slice decomposition is used for the FDM mesh. Some estimates of the parallel performance of FEM, FDM and overlapping computations are presented. 1997 by John Wiley & Sons, Ltd.