Finite element analysis of viscous,incompressible fluid flow: Part 2: Applications

The first part of this paper described a general numerical procedure for the analysis of two-dimensional flows of viscous, incompressible fluids, using the finite element method. A number of special computational procedures were also discussed that allowed significant reductions to be made in the computational effort required in the solution of problems. The present paper is devoted to demonstrating the utility of the methods described by the solution of several example problems. The illustrative examples consist of flow in a plane 90° T, flow in a cavity and flow around a circular cylinder.     

Finite element analysis of viscous fluid flow

A finite element model for the analysis of two dimensional viscous flows is formulated using the virtual work method. The model is in part based on a finite element shell model, using the same reduced integration of quadratic interpolations for all variables[1]. Differences from preceding formulations are that integration by parts is applied to the continuity equation, yielding different loading terms which are more easily defined in some problems, and a new approach is used for the convective inertia terms, giving a clearer interpretation of their effects which are distributed to both sides of the nonlinear recurrence relation. In the case of compressible flow, for which comparatively few formulations have been proposed to date, the thermal energy equation is used to form a two stage solution and here this seems the most natural and economical approach.     

Finite element solutions of nonsteady incompressible viscous flow between two rotating concentric spheres

The problem of unsteady, incompressible viscous flow between two rotating concentric spheres has been investigated here. The full Navier-Stokes equations in terms of the velocity components u, v w and the pressure p, using spherical coordinates for axially symmetric flow, were solved by means of the finite element method in the spatial dimension and the alternating-direction method in the time dimension using Glowinski's algorithm. The element used is an annular-sector-type element with a bilinear approximation for the velocity components and with constant pressure within the element. Reynolds numbers in the range from 1 to 1000, gap size 0.5 and different combinations of the angular velocity of the inner and outer spheres were studied. In some of these cases a steady-state solution was possible, while in others only a transient solution was possible. This method proved to be successful and powerful in predicting the behavior of the flow for these nonlinear-type problems.     

Finite difference methods for viscous incompressible global stability analysis

In the present article some high-order finite-difference schemes and in particularly dispersion-relation-preserving (DRP) family schemes, initially developed by Tam and Webb [Dispersion-relation-preserving finite difference schemes for computational acoustics, J. Comput. Phys. 107 (1993) 262-281.] for computational aeroacoustic problems, are used for global stability issue. (The term global is not used in weakly-non-parallel framework but rather for fully non-parallel flows. Some authors like Theofilis [Advances in global linear instability analysis of non-parallel and three-dimensional flows, Progress in Aerospace Sciences 39 (2003) 249-315] refer to this approach as “BiGlobal”.) These DRP schemes are compared with different classical schemes as second and fourth-order finite-difference schemes, seven-order compact schemes and spectral collocation scheme which is usually employed in such stability problems. A detailed comparative study of these schemes for incompressible flows over two academic configurations (square lid-driven cavity and separated boundary layer at different Reynolds numbers) is presented, and we intend to show that these schemes are sufficiently accurate to perform global stability analyses.     

Theory and implementation of a semi-implicit finite element method for viscous incompressible flow

A numerical procedure for solving the time-dependent, incompressible Navier-Stokes equations is derived based on the operator-splitting technique. This operator split allows separate operations on each of the variable fields to enable pressure-velocity coupling. Discretizations of the equations are formed on a nonstaggered finite element mesh and the solutions are obtained in a time-marching fashion. Several benchmark problems, including a standing vortex problem, a lid-driven cavity and a flow around a rectangular cylinder, are studied to demonstrate the robustness and accuracy of the present algorithm.     

Error control and mesh optimization for high-order finite element approximation of incompressible viscous flow

We discuss the use of a posteriori error estimates for high-order finite element methods during simulation of the flow of incompressible viscous fluids. The correlation between the error estimator and actual error is used as a criterion for the error analysis efficiency. We show how to use the error estimator for mesh optimization which improves computational efficiency for both steady-state and unsteady flows. The method is applied to two-dimensional problems with known analytical solutions (Jeffrey-Hamel flow) and more complex flows around a body, both in a channel and in an open domain.     

Numerical solution of an electrically conducting viscous incompressible fluid

This paper investigates the flow pattern of an axisymmetric, electrically conducting viscous and incompressible fluid in a pipe, having a throat at its centre. An iterative finite difference method, based on Picards approximations for the spatial derivatives of the governing equations, is used to obtain the streamline patterns of the fluid flow field in presence of a magnetic field. An initial parabolic velocity profile is assumed at the entrance of the pipe. Numerical solutions which are computed for the values of Reynolds number lying between 50 and 2000 and the magnetic pressure number varying from 0 to 80 show the occurrence of a secondary flow in the downstream region and a recirculating flow in the upstream region of the pipe.     

Improved solution of steady,viscous, incompressible flow between two rotating spheres

A refined solution is presented for the analysis of viscous, incompressible, steady flow between two rotating spheres. A new method, used previously for simpler problems only, is adapted to this problem. The method allows the use of small grid spacing and thus yields improved accuracy.     

Numerical studies of steady,viscous, incompressible flow between two rotating spheres

A new numerical method, developed for the study of secondary flow in a curved tube, is adapted and extended to the study of viscous, incompressible, steady flow between two rotating spheres. The Navier-Stokes equations are approximated by a triple sequence of linear problems, each of which has a diagonally dominant coefficient matrix. Computer examples are described and discussed.     

Finite element methods for the solution of some incompressible non-newtonian fluid mechanics problems with free surfaces

A finite element scheme suitable for incompressible fluid flow in plane and axisymmetric geometries is constructed by using a Galerkin method. Principal features of the associated computer program are that it can be used for problems with or without free surfaces, with or without inertia, and for many types of non-Newtonian flow. Here we describe the use of the program in solving Poiseuille flow, a 2:1 contraction tube flow, free non-Newtonian jets without inertia, and a Newtonian free jet problem at a Reynolds number of 10. Comparison with exact solutions, other computer solutions and experiment is given where such information is available.     

Finite element modelling of flow,heat and mass transfer in fluid saturated porous media

Summary In this article, a short review of numerical modelling of porous medium flow, heat and mass transfer is presented. The focus of the article is mainly on the use of finite element method and velocity correction algorithm. In addition to a detailed discussion on the velocity correction scheme, some essential fundamental and application problems are solved and results are presented. Many of these results are compared against available experimental and numerical data.     

Assessment of grid interface treatments for multi-block incompressible viscous flow computation

In the multi-block computation of the Navier-Stokes equations, the interface treatment is a key issue. In the present work, we investigate this issue in the context of a pressure-based method using a non-orthogonal grid. For the momentum equations, a straightforward bilinear interpolation seems satisfactory as the interface treatment; on the other hand, because the pressure field depends on the satisfaction of the mass continuity equation, a conservative interface treatment has been found necessary for the pressure-correction equation. Two alternative interface treatments for the pressure-correction equation, one employing the Neumann boundary condition in both grid blocks, based on explicit local, cell-by-cell mass flux conservation, and the other utilizing Neumann-Dirichlet boundary conditions, allowing the interface condition in one block to be derived by interpolating the pressure field from the adjacent block, are assessed in the present work. To evaluate these interface schemes, the laminar flow inside a lid-driven cavity flow, and the turbulent flow around cascades of multiple airfoils have been investigated. For the case tested, both interface treatments give comparable accuracy. The finding that more than one type of interface treatment can work well allows one to devise a flexible multi-block strategy for complex flow computations.     

A Legendre spectral element method for simulation of unsteady incompressible viscous free-surface flows

A new Legendre spectral element method is presented for the solution of viscous incompressible free-surface flows. It is based on the following extensions of the fixed-domain spectral element method: use of the full viscous stress tensor for natural imposition of traction (surface tension) boundary conditions; use of arbitrary-Lagrangian-Eulerian methods for accurate representation of moving boundaries; and use of semi-implicit time-stepping procedures to partially decouple the free-surface evolution and interior Navier-Stokes equations. For purposes of analysis and clarity of presentation, attention is focused on the stability of falling films. Analysis of the spectrum of the linear stability problem (Orr-Sommerfeld equation) associated with film flow reveals physical effects that limit the stability of semi-implicit schemes and suggests optimal formulas for temporal discretization of the spectral element equations. Detailed results are presented for the spectral element simulation of the film flow problem.     

Mixed finite element method for analysis of viscoelastic fluid flow

In the present paper, numerical analysis of incompressible viscoelastic fluid flow is discussed using mixed finite element Galerkin method. Because Maxwellian viscoelasticity is assumed as the constitutive equation, stress components could not be eliminated from the governing equation system. Because of this, mixed finite element method is utilized to discretize the basic equations. For the solution procedures to solve discretized equation system, Newton-Raphson method for steady flow and perturbation method for unsteady flow is employed. As the numerical examples, comparison was made on the finite element computational results between by direct method and by mixed method. Effects of the viscoelasticity is analyzed for the flows at Reynold's numbers 30, 50 and 70.     

Finite element solution of viscous jet flows with surface tension

A finite element method is used to study the effect of Reynolds number and surface tension on the expansion and contraction of jets of Newtonian liquids. For values of Reynolds numbers (based on tube diameter), below 14 the jets expand, and when Re > 14 the jets contract. For higher Reynolds numbers the jet diameter approaches a limiting value. It is also found that the surface tension has a considerable effect on low Reynolds number jet flows, becoming negligible at higher Reynolds numbers. As an example, if the surface tension parameter $\text{σ}\text{η}\text{u}$ is equal to unity, the creeping flow jet expansion is reduced by 4% relative to the case with no surface tension but when Re is equal to 20 and 50 the final jet diameters increase by only 0.2%. The calculated jet shapes are compared with available experimental results.     

Boundary control of a laminar flow of a viscous incompressible fluid in the generalized Couette cell. The asymptotic approach

In this paper, a limiting problem for an optimal boundary control problem of a laminar flow of a viscous incompressible fluid in the generalized Couette cell, when the number of inner cylinders unrestrictedly increases, is obtained.     

Critical rotational speed,critical velocity of fluid flow and free vibration analysis of a spinning SWCNT conveying viscous fluid

In this article, the influences of rotational speed and velocity of viscous fluid flow on free vibration behavior of spinning single-walled carbon nanotubes (SWCNTs) are investigated using the modified couple stress theory (MCST). Taking attention to the first-order shear deformation theory, the modeled rotating SWCNT and its equations of motion are derived using Hamilton’s principle. The formulations include Coriolis, centrifugal and initial hoop tension effects due to rotation of the SWCNT. This system is conveying viscous fluid, and the related force is calculated by modified Navier–Stokes relation considering slip boundary condition and Knudsen number. The accuracy of the presented model is validated with some cases in the literatures. Novelty of this study is considering the effects of spinning, conveying viscous flow and MCST in addition to considering the various boundary conditions of the SWCNT. Generalized differential quadrature method is used to approximately discretize the model and to approximate the equations of motion. Then, influence of material length scale parameter, velocity of viscous fluid flow, angular velocity, length, length-to-radius ratio, radius-to-thickness ratio and boundary conditions on critical speed, critical velocity and natural frequency of the rotating SWCNT conveying viscous fluid flow are investigated.     

Finite element solution of incompressible flows using an explicit segregated approach

Summary An explicit finite element method for solving the incompressible Navier-Stokes equations for laminar and turbulent, newtonian, nonisothermal flow is presented. This method is based on the segregated velocity pressure formulation which has seen considerable development in the last decade. An endeavour has been made to include beneficial features from much of the relevant published work in the developed code. Some of the main features include, the use of the velocity correction method (segregation at the differential equation level), equal order interpolation of velocity and pressure, splitting of advection and diffusion terms, Taylor-Galerkin method for discretizing the advection terms, lumped-explicit solution of diffusion, and iterative-explicit solution of advection. In addition to these a consistent treatment of the natural boundary conditions for the pressure Poisson equation has been presented. Full details of the formulation are given with examples demonstrating the method.