Discretization of incompressible vorticity–velocity equations on triangular meshes

This paper describes a new approach to discretizing first- and second-order partial differential equations. It combines the advantages of finite elements and finite differences in having both unstructured (triangular/tetrahedral) meshes and low-order physically intuitive schemes. In this ‘co-volume’ framework, the discretized gradient, divergence, curl, (scalar) Laplacian, and vector Laplacian operators satisfy relationships found in standard vector field theory, such as a Helmholtz decomposition. This article focuses on the vorticity–velocity formulation for planar incompressible flows. The algorithm is described and some supporting numerical evidence is provided.     

An improved Vorticity–Potential method for three-dimensional duct flow simulations

An improved Vorticity–Potential method is presented for the numerical solutions of three-dimensional duct flow problems. The solution procedure requires first a potential solution. Then the viscous effects are added through the vorticity transport equation. By using body-fitted coordinates, the method is applied to simulate the incompressible laminar flows in a square elbow and in a twisted square elbow.     

Vorticity–velocity formulation of the 3D Navier–Stokes equations in cylindrical co‐ordinates

A finite difference method is presented for solving the 3D Navier–Stokes equations in vorticity–velocity form. The method involves solving the vorticity transport equations in ‘curl‐form’ along with a set of Cauchy–Riemann type equations for the velocity. The equations are formulated in cylindrical co‐ordinates and discretized using a staggered grid arrangement. The discretized Cauchy–Riemann type equations are overdetermined and their solution is accomplished by employing a conjugate gradient method on the normal equations. The vorticity transport equations are solved in time using a semi‐implicit Crank–Nicolson/Adams–Bashforth scheme combined with a second‐order accurate spatial discretization scheme. Special emphasis is put on the treatment of the polar singularity. Numerical results of axisymmetric as well as non‐axisymmetric flows in a pipe and in a closed cylinder are presented. Comparison with measurements are carried out for the axisymmetric flow cases. Copyright © 2003 John Wiley & Sons, Ltd.     

Pseudostress–velocity formulation for incompressible Navier–Stokes equations

This paper presents a numerical algorithm using the pseudostress–velocity formulation to solve incompressible Newtonian flows. The pseudostress–velocity formulation is a variation of the stress–velocity formulation, which does not require symmetric tensor spaces in the finite element discretization. Hence its discretization is greatly simplified. The discrete system is further decoupled into an H ( div ) problem for the pseudostress and a post‐process resolving the velocity. This can be done conveniently by using the penalty method for steady‐state flows or by using the time discretization for nonsteady‐state flows. We apply this formulation to the 2D lid‐driven cavity problem and study its grid convergence rate. Also, computational results of the time‐dependent‐driven cavity problem and the flow past rectangular problem are reported. Copyright © 2009 John Wiley & Sons, Ltd.     

Fundamental solutions of the streamfunction–vorticity formulation of the Navier–Stokes equations

A new boundary element procedure is developed for the solution of the streamfunction–vorticity formulation of the Navier–Stokes equations in two dimensions. The differential equations are stated in their transient version and then discretized via finite differences with respect to time. In this discretization, the non-linear inertial terms are evaluated in a previous time step, thus making the scheme explicit with respect to them. In the resulting discretized equations, fundamental solutions that take into account the coupling between the equations are developed by treating the non-linear terms as in homogeneities. The resulting boundary integral equations are solved by the regular boundary element method, in which the singular points are placed outside the solution domain.     

On an attempt to simplify the Quartapelle–Napolitano approach to computation of hydrodynamic forces in open flows

In this paper we are interested in the Quartapelle–Napolitano approach to calculation of forces in viscous incompressible flows in exterior domains. We study the possibility of deriving a simpler formulation of this approach which might lead to a more convenient expression for the hydrodynamic force, but conclude that such a simplification is, within the family of approaches considered, impossible. This shows that the original Quartapelle–Napolitano formula is in fact “optimal” within this class of approaches.     

The steady Navier–Stokes/energy system with temperature‐dependent viscosity—Part 1: Analysis of the continuous problem

In this first part we propose and analyse a model for the study of two‐dimensional incompressible Navier–Stokes equations with a temperature‐dependent viscosity. The flow is supposed in a mixed convection regime and considers an outflow region, leading to a strongly coupled problem between the Navier–Stokes and energy equations, which will be justified theoretically. The coupling in the continuous problem is treated by an outer temperature fixed point strategy. Existence results for a particular variational formulation follows from this study. Further, a particular uniqueness result for small data is also obtained. Copyright © 2007 John Wiley & Sons, Ltd.     

Parallel finite element computation of unsteady incompressible flows

A parallel semi-explicit iterative finite element computational procedure for modelling unsteady incompressible fluid flows is presented. During the procedure, element flux vectors are calculated in parallel and then assembled into global flux vectors. Equilibrium iterations which introduce some ‘local implicitness’ are performed at each time step. The number of equilibrium iterations is governed by an implicitness parameter. The present technique retains the advantages of purely explicit schemes, namely (i) the parallel speed-up is equal to the number of parallel processors if the small communication overhead associated with purely explicit schemes is ignored and (ii) the computation time as well as the core memory required is linearly proportional to the number of elements. The incompressibility condition is imposed by using the artificial compressibility technique. A pressure-averaging technique which allows the use of equal-order interpolations for both velocity and pressure, this simplifying the formulation, is employed. Using a standard Galerkin approximation, three benchmark steady and unsteady problems are solved to demonstrate the accuracy of the procedure. In all calculations the Reynolds number is less than 500. At these Reynolds numbers it was found that the physical dissipation is sufficient to stabilize the convective term with no need for additional upwind-type dissipation. © 1998 John Wiley & Sons, Ltd.     

An operator splitting scheme for the stream‐function formulation of unsteady Navier–Stokes equations

A fictitious time is introduced into the unsteady equation of the stream function rendering it into a higher‐order ultra‐parabolic equation. The convergence with respect to the fictitious time (we call the latter ‘internal iterations’) allows one to obtain fully implicit nonlinear scheme in full time steps for the physical‐time variable. For particular choice of the artificial time increment, the scheme in full time steps is of second‐order of approximation. For the solution of the internal iteration, a fractional‐step scheme is proposed based on the splitting of the combination of the Laplace, bi‐harmonic and advection operators. A judicious choice for the time staggering of the different parts of the nonlinear advective terms allows us to prove that the internal iterations are unconditionally stable and convergent. We assess the number of operations needed per time step and show computational effectiveness of the proposed scheme. We prove that when the internal iterations converge, the scheme is second‐order in physical time and space, nonlinear, implicit and absolutely stable. The performance of the scheme is demonstrated for the flow created by oscillatory motion of the lid of a square cavity. All theoretical findings are demonstrated practically. Copyright © 2006 John Wiley & Sons, Ltd.     

A streamfunction–velocity approach for 2D transient incompressible viscous flows

Electrodeposition is a widely used technique for the fabrication of high aspect ratio microstructures. In recent years, much research has been focused within this area aiming to understand the physics behind the filling of high aspect ratio vias and trenches on substrates and in particular how they can be made without the formation of voids in the deposited material. This paper reports on the fundamental work towards the advancement of numerical algorithms that can predict the electrodeposition process in micron scaled features. Two different numerical approaches have been developed, which capture the motion of the deposition interface and 2‐D simulations are presented for both methods under two deposition regimes: those where surface kinetics is governed by Ohm's law and the Butler–Volmer equation, respectively. In the last part of this paper the modelling of acoustic forces and their subsequent impact on the deposition profile through convection is examined. Copyright © 2009 John Wiley & Sons, Ltd.     

Projection‐based variational multiscale method for incompressible Navier–Stokes equations in time‐dependent domains

A variational multiscale method for computations of incompressible Navier–Stokes equations in time‐dependent domains is presented. The proposed scheme is a three‐scale variational multiscale method with a projection‐based scale separation that uses an additional tensor valued space for the large scales. The resolved large and small scales are computed in a coupled way with the effects of unresolved scales confined to the resolved small scales. In particular, the Smagorinsky eddy viscosity model is used to model the effects of unresolved scales. The deforming domain is handled by the arbitrary Lagrangian–Eulerian approach and by using an elastic mesh update technique with a mesh‐dependent stiffness. Further, the choice of orthogonal finite element basis function for the resolved large scale leads to a computationally efficient scheme. Simulations of flow around a static beam attached to a square base, around an oscillating beam and around a plunging aerofoil are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

A new stable space–time formulation for two‐dimensional and three‐dimensional incompressible viscous flow

A space–time finite element method for the incompressible Navier–Stokes equations in a bounded domain in ?^d (with d=2 or 3) is presented. The method is based on the time‐discontinuous Galerkin method with the use of simplex‐type meshes together with the requirement that the space–time finite element discretization for the velocity and the pressure satisfy the inf–sup stability condition of Brezzi and Babu?ka. The finite element discretization for the pressure consists of piecewise linear functions, while piecewise linear functions enriched with a bubble function are used for the velocity. The stability proof and numerical results for some two‐dimensional problems are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Upwind discretization of the steady Navier–Stokes equations

A discretization method is presented for the full, steady, compressible Navier–Stokes equations. The method makes use of quadrilateral finite volumes and consists of an upwind discretization of the convective part and a central discretization of the diffusive part. In the present paper the emphasis lies on the discretization of the convective part. The solution method applied solves the steady equations directly by means of a non-linear relaxation method accelerated by multigrid. The solution method requires the discretization to be continuously differentiable. For two upwind schemes which satisfy this requirement (Osher's and van Leer's scheme), results of a quantitative error analysis are presented. Osher's scheme appears to be increasingly more accurate than van Leer's scheme with increasing Reynolds number. A suitable higher-order accurate discretization of the convection terms is derived. On the basis of this higher-order scheme, to preserve monotonicity, a new limiter is constructed. Numerical results are presented for a subsonic flat plate flow and a supersonic flat plate flow with oblique shock wave–boundary layer interaction. The results obtained agree with the predictions made. Useful properties of the discretization method are that it allows an easy check of false diffusion and that it needs no tuning of parameters.     

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.     

Optimal control of unsteady compressible viscous flows

The control of complex, unsteady flows is a pacing technology for advances in fluid mechanics. Recently, optimal control theory has become popular as a means of predicting best case controls that can guide the design of practical flow control systems. However, most of the prior work in this area has focused on incompressible flow which precludes many of the important physical flow phenomena that must be controlled in practice including the coupling of fluid dynamics, acoustics, and heat transfer. This paper presents the formulation and numerical solution of a class of optimal boundary control problems governed by the unsteady two‐dimensional compressible Navier–Stokes equations. Fundamental issues including the choice of the control space and the associated regularization term in the objective function, as well as issues in the gradient computation via the adjoint equation method are discussed. Numerical results are presented for a model problem consisting of two counter‐rotating viscous vortices above an infinite wall which, due to the self‐induced velocity field, propagate downward and interact with the wall. The wall boundary control is the temporal and spatial distribution of wall‐normal velocity. Optimal controls for objective functions that target kinetic energy, heat transfer, and wall shear stress are presented along with the influence of control regularization for each case. Copyright © 2002 John Wiley & Sons, Ltd.     

Solution of the incompressible Navier–Stokes equations on domains with one or several open boundaries1

The aim of this paper is to develop a methodology for solving the incompressible Navier–Stokes equations in the presence of one or several open boundaries. A new set of open boundary conditions is first proposed. This has been developed in the context of the velocity–vorticity formulation, but it is also emphasized how it can be formally extended to the equations in primitive variables. The case of a domain involving several independent open boundaries is considered next. An influence matrix technique is applied such that the inlet mass flux is split onto the several outlets in order to enforce the prescribed mean pressure at each outlet. Both approaches are validated by numerical test cases. Copyright © 1999 John Wiley & Sons, Ltd.     

A methodology for high performance computation of fully inhomogeneous turbulent flows

A large‐eddy simulation methodology for high performance parallel computation of statistically fully inhomogeneous turbulent flows on structured grids is presented. Strategies and algorithms to improve the memory efficiency as well as the parallel performance of the subgrid‐scale model, the factored scheme, and the Poisson solver on shared‐memory parallel platforms are proposed and evaluated. A novel combination of one‐dimensional red–black/line Gauss–Seidel and two‐dimensional red–black/line Gauss–Seidel methods is shown to provide high efficiency and performance for multigrid relaxation of the Poisson equation. Parallel speedups are measured on various shared‐distributed memory systems. Validations of the code are performed in large‐eddy simulations of turbulent flows through a straight channel and a square duct. Results obtained from the present solver employing a Lagrangian dynamic subgrid‐scale model show good agreements with other available data. The capability of the code for more complex flows is assessed by performing a large‐eddy simulation of the tip‐leakage flow in a linear cascade. Copyright © 2006 John Wiley & Sons, Ltd.     

Integral transform solution for the lid-driven cavity flow problem in streamfunction-only formulation

The basic ideas in the generalized integral transform technique are further advanced to allow for the hybrid numerical-analytical solution of the two-dimensional steady Navier-Stokes equations in streamfunction-only formulation. The classical lid-driven square cavity problem is selected for illustration of the approach. The corresponding biharmonic-type non-linear partial differential equation for the streamfunction is integral transformed in one of the co-ordinates and an infinite system of coupled non-linear ODEs for the transformed potential results in the other independent variable. Upon truncation to an appropriate finite order, the ODE system is numerically solved by well-established algorithms with automatic error control devices. The convergence behaviour of the eigenfunction expansion is demonstrated and reference results are provided for typical values of Reynolds number.     

A locally DIV‐free projection scheme for incompressible flows based on non‐conforming finite elements

We present a projection scheme whose end‐of‐step velocity is locally pointwise divergence free, using a continuous ?₁ approximation for the velocity in the momentum equation, a first‐order Crouzeix–Raviart approximation at the projection step, and a ?₀ approximation for the pressure in both steps. The analysis of the scheme is done only for grids that guarantee the existence of a divergence free conforming ?₁ interpolant for the velocity. Optimal estimates for the velocity error in L²‐ and H¹‐norms are deduced. The numerical results demonstrate that these estimates should also hold on grids on which the continuous ?₁ approximation for the velocity locks. Since the end‐of‐step velocity is locally solenoidal, the scheme is recommendable for problems requiring good mass conservation. Copyright © 2005 John Wiley & Sons, Ltd.