Preconditioned finite element algorithms for 3D Stokes flows

Preconditioned conjugate gradient algorithms for solving 3D Stokes problems by stable piecewise discontinuous pressure finite elements are presented. The emphasis is on the preconditioning schemes and their numerical implementation for use with Hermitian based discontinuous pressure elements. For the piecewise constant discontinuous pressure elements, a variant implementation of the preconditioner proposed by Cahouet and Chabard for the continuous pressure elements is employed. For the piecewise linear discontinuous pressure elements, a new preconditioner is presented. Numerical examples are presented for the cubic lid-driven cavity problem with two representative elements, i.e. the Q2-PO and the Q2-P1 brick elements. Numerical results show that the preconditioning schemes are very effective in reducing the number of pressure iterations at very reasonable costs. It is also shown that they are insensitive to the mesh Reynolds number except for nearly steady flows (Re_m → 0) and are almost independent of mesh sizes. It is demonstrated that the schemes perform reasonably well on non-uniform meshes.     

On the solution of Stokes equations for plane boundaries

In a recent paper(Li et al., Acta Mech. Sin. 31,32–44, 2015), the authors claimed that the general solution of steady Stokes flows can be compactly expressed using only two harmonic functions. They present two cases of a flat plate translating through a viscous fluid. The present paper shows that such a two-harmonic solution does not describe the rotation of a circular plate in an unbounded fluid and thus confirms that at least three independent harmonics are required to express the general solution of Stokes equations.     

Investigation of solution of Navier–Stokes equations using a variational formulation

A variational formulation for the solution of two dimensional, incompressible viscous flows has been developed by one of the authors.¹ The main objective of the present paper is to demonstrate the applicability of this approach for the solution of practical problems and in particular to investigate the introduction of boundary conditions to the Navier-Stokes equations through a variational formulation. The application of boundary conditions for typical internal and external flow problems is presented. Sample cases include flow around a cylinder and flow through a stepped channel. Quadrilateral, bilinear isoparametric elements are utilized in the formulation. A single-step, implicit, and fully coupled numerical integration scheme based on the variational principle is employed. Presented results include sample cases with different Reynolds numbers for laminar and turbulent flows. Turbulence is modelled using a simple mixing length model. Numerical results show good agreement with existing solutions.     

On the Controllability of the Hydrostatic Stokes Equations

This paper is devoted to present some results on the controllability of the hydrostatic Stokes equations. The first main result of this paper states that the approximate null controllability of this system holds. This is proved whatever the boundary conditions are. Then, we extend this result to an exact null controllability result when the boundary conditions are ∂ _z u = 0 (the vertical derivative of the horizontal velocity) at the bottom.      

A comparison of visualization of Stokes flow in a V-shaped and rectangular channel

A comparative study of visualization of Stokes flow induced by rotation of a circular cylinder in a V-shaped and rectangular channel was performed. The visualization was carried out using solid tracers of magnesium. Some quantitative data were deduced from visualization photographs and a comparison was made.     

On the motion of thin plates and shells subject to Stokes damping

The motion of a homogeneous, thin rigid plate or shell subject to Stokes damping over its entire surface and to any additional system of workless forces, including those that are equipollent to zero, is studied. It is shown that the equation for the single-degree of freedom motion of every thin body having the same mass density per unit area is universal; the result, except for sufficient mathematical smoothness essential to integration over the body, is independent of any other physical features or dimensions of the body. A few examples illustrate the general problem and the universal nature of the equation of motion for thin plates and shells.     

Analysis of incompressible massively separated viscous flows using unsteady Navier–Stokes equations

The unsteady incompressible Navier–Stokes equations are formulated in terms of vorticity and stream-function in generalized curvilinear orthogonal co-ordinates to facilitate analysis of flow configurations with general geometries. The numerical method developed solves the conservative form of the vorticity transport equation using the alternating direction implicit method, whereas the streamfunction equation is solved by direct block Gaussian elimination. The method is applied to a model problem of flow over a backstep in a doubly infinite channel, using clustered conformal co-ordinates. One-dimensional stretching functions, dependent on the Reynolds number and the asymptotic behaviour of the flow, are used to provide suitable grid distribution in the separation and reattachment regions, as well as in the inflow and outflow regions. The optimum grid distribution selected attempts to honour the multiple length scales of the separated flow model problem. The asymptotic behaviour of the finite differenced transport equation near infinity is examined and the numerical method is carefully developed so as to lead to spatially second-order-accurate wiggle-free solutions, i.e. with minimum dispersive error. Results have been obtained in the entire laminar range for the backstep channel and are in good agreement with the available experimental data for this flow problem, prior to the onset of three-dimensionality in the experiment.     

Stokes paradox for power-law flow around a cylinder

A simple analysis for power-law fluids shows that the Stokes paradox for creeping flow around a cylinder is removed for shear-thinning (n < 1) but not for shear-thickening (n 1) fluids. An approximate drag value is found for n < 1 and is compared with computed results.     

A finite element procedure for viscoelastic flows

A finite element method for the simulation of viscoelastic flows has been developed. It uses a weak formulation of the method of characteristics to treat the viscoelastic constitutive law. Numerical results in a 4:1 contraction are presented and are discussed with respect to previous computations. New phenomena are put in evidence and new questions are opened in this already controversial problem.     

The addition of quasi-three-dimensional terms into a finite element method for transonic turbomachinery blade-to-blade flows

This paper describes the extension of a purely two-dimensional finite element method for the calculation of transonic turbomachinery blade-to-blade flows to include the quasi-three-dimensional terms. These terms account for the effect of variations in streamline radius, stream-tube height and blade rotation. By approximating the stream surface as a piecewise linear function, then using a local developed cone transformation on an element basis, the finite element equations are shown to remain of the same form as the two-dimensional equations. The numerical results presented demonstrate that the stream-tube height, streamline radius and blade rotation terms must be included if the prediction of the Mach number distribution around a gas turbine blade is to be calculated correctly.     

A staggered grid‐based explicit jump immersed interface method for two‐dimensional Stokes flows

In this paper the explicit jump immersed interface method (EJIIM) is applied to stationary Stokes flows. The boundary value problem in a general, non‐grid aligned domain is reduced by the EJIIM to a sequence of problems in a rectangular domain, where staggered grid‐based finite differences for velocity and pressure variables are used. Each of these subproblems is solved by the fast Stokes solver, consisting of the pressure equation (known also as conjugate gradient Uzawa) method and a fast Fourier transform‐based Poisson solver. This results in an effective algorithm with second‐order convergence for the velocity and first order for the pressure. In contrast to the earlier versions of the EJIIM, the Dirichlét boundary value problem is solved very efficiently also in the case when the computational domain is not simply connected. Copyright © 2007 John Wiley & Sons, Ltd.     

Boundary conditions for stokes flows near a porous membrane

A theoretical development is carried out to model the boundary conditions for Stokes flows near a porous membrane, which, in general, allows non-zero slip as well as normal flow at the surface. Two types of models are treated: an infinitesimally thin plate with a periodic array of circular apertures and a series of parallel slits. For Stokes flows, the mean normal flux and slip velocity are proportional to the pressure difference across the membrane and the average shear stress at the membrane, respectively. The appropriate proportionality constants which depend on the membrane geometry are calculated as functions of the porosity. An interesting feature of the results is that the slip at the membrane has, in general, a direction different from that of the applied shear for these models.     

Simulation of polymeric flows in the injection moulding process

Recent progress in the simulation of polymeric flows of two key problems in the injection moulding process, carried out by a team at Cornell University, is briefly described. For the filling of cooled thin cavities, the fluid is characterized by a power-law viscosity with exponential temperature dependence, and interaction between the transient thermal boundary-layer and the core flow in a domain with moving boundary is essential. The earlier procedure of Hieber and Shen is modified in two aspects: a boundary-integral formulation replaces the finite-element treatment of the pressure, and an ‘energy integral’ approach is used for the transient temperature. The second problem is the steady visco-elastic flow in the juncture region where sudden changes of the geometry and large strain rates occur. The constitutive equation is postulated according to the Leonov model. The main features in the numerical implementation are: integration along a streamline to determine the elastic deformation tensors for a given velocity field, and finite-element treatment (in time-dependent form) of the pressure and fields for given stresses. In an example where the contraction ratio is 7:1, results for nominal Deborah number exceeding 100 show no numerical instability. (However, for this problem, the true Weissenberg number, i.e. the ratio of local first-normal-stress difference to shear stress turns out to be generally O(10).) The predictions also correlate very well with experimental birefringence measurements.     

Stokes Flow through an Array of Staggered Strips

The Stokes flow through a periodic array of thin staggered strips is studied. The method of eigenfunction expansion and collocation is used to obtain detailed flow and pressure fields. The permeabilities in the three principle directions are found to be different and their characters depend heavily on the geometry. Approximate formulas are also obtained.     

Numerical computation of transient shocked chemically reactive flows

Using the ability of the method of characteristics to evaluate shock fronts in an accurate manner, a formulation is presented which incorporates the effect of rapid exothermic chemical reactions in the flow. The formulation is applied to the computation of the unsteady reactive flow field behind a cylindrial expanding blast wave propagating in a mixture of hydrogen and oxygen. Details of the computational procedure are described. Results are presented for a sample problem and compared with those for the non-reactive case to illustrate the influence of chemical reactions.     

The effect of the slip condition on Stokes and Couette flows due to an oscillating wall: exact solutions

Stokes and Couette flows produced by an oscillatory motion of a wall are analyzed under conditions where the no-slip assumption between the wall and the fluid is no longer valid. The motion of the wall is assumed to have a generic sinusoidal behavior. The exact solutions include both steady periodic and transient velocity profiles. It is found that slip conditions between the wall and the fluid produces lower amplitudes of oscillations in the flow near the oscillating wall than when no-slip assumption is utilized. Further, the relative velocity between the fluid layer at the wall and the speed of the wall is found to overshoot at a specific oscillating slip parameter or vibrational Reynolds number at certain times. In addition, it is found that wall slip reduces the transient velocity for Stokes flow while minimum transient effects for Couette flow is achieved only for large and small values of the wall slip coefficient and the gap thickness, respectively. The time needed to reach to steady periodic Stokes flow due to sine oscillations is greater than that for cosine oscillations with both wall slip and no-slip conditions.     

A theorem for a shear stress-free plane boundary in Stokes flow

A theorem for non-axisymmetrical Stokes flow about a shear stress-free plane boundary is established by using a representation for the velocity and pressure fields for the same flow in terms of biharmonic and harmonic functions. A corollary of the theorem is derived which gives the axisymmetrical Stokes flow in terms of the Stokes function about the same boundary. The formulae for drag and torque on the boundary are also given. A few illustrative examples are presented.     

A note on general solution of Stokes equations

Li et al. (2015) claim that it is sufficient to use two harmonic functions to express the general solution of Stokes equations. In this paper, we demonstrate that this is not true in a general case and that we in fact need three scalar harmonic functions to represent the general solution of Stokes equations (Venkatalaxmi et al., 2004).