A fractional-step method for the incompressible Navier–Stokes equations related to a predictor–multicorrector algorithm

An implicit fractional-step method for the numerical solution of the time-dependent incompressible Navier–Stokes equations in primitive variables is studied in this paper. The method, which is first-order-accurate in the time step, is shown to converge to an exact solution of the equations. By adequately splitting the viscous term, it allows the enforcement of full Dirichlet boundary conditions on the velocity in all substeps of the scheme, unlike standard projection methods. The consideration of this method was actually motivated by the study of a well-known predictor–multicorrector algorithm, when this is applied to the incompressible Navier–Stokes equations. A new derivation of the algorithm in a general setting is provided, showing in what sense it can also be understood as a fractional-step method; this justifies, in particular, why the original boundary conditions of the problem can be enforced in this algorithm. Two different finite element interpolations are considered for the space discretization, and numerical results obtained with them for standard benchmark cases are presented. © 1998 John Wiley & Sons, Ltd.     

AN ARTIFICIAL BOUNDARY CONDITION FOR TWO-DIMENSIONAL INCOMPRESSIBLE VISCOUS FLOWS USING THE METHOD OF LINES

We design an artificial boundary condition for the steady incompressible Navier–Stokes equations in streamfunction–vorticity formulation in a flat channel with slip boundary conditions on the wall. The new boundary condition is derived from the Oseen equations and the method of lines. A numerical experiment for the non-linear Navier–Stokes equations is presented. The artificial boundary condition is compared with Dirichlet and Neumann boundary conditions for the flow past a rectangular cylinder in a flat channel. The numerical results show that our boundary condition is more accurate.     

The Navier–Stokes Equations Under a Unilateral Boundary Condition of Signorini’s Type

We propose a new outflow boundary condition, a unilateral condition of Signorini’s type, for the incompressible Navier–Stokes equations. The condition is a generalization of the standard free-traction condition. Its variational formulation is given by a variational inequality. We also consider a penalty approximation, a kind of the Robin condition, to deduce a suitable formulation for numerical computations. Under those conditions, we can obtain energy inequalities that are key features for numerical computations. The main contribution of this paper is to establish the well-posedness of the Navier–Stokes equations under those boundary conditions. Particularly, we prove the unique existence of strong solutions of Ladyzhenskaya’s class using the standard Galerkin’s method. For the proof of the existence of pressures, however, we offer a new method of analysis.     

Incompressible Fluids in Thin Domains with Navier Friction Boundary Conditions (II)

We study the Navier–Stokes equations for incompressible fluids in a three-dimensional thin two-layer domain whose top, bottom and interface boundaries are not flat. In addition to the Navier friction boundary conditions on the top and bottom boundaries of the domain, and the periodicity condition on the sides, the fluid velocities are subject to an interface boundary condition which relates the normal stress of each fluid to the relative velocity between them on the common boundary. We prove that the strong solutions exist for all time if the initial data and body force, measured in relevant norms, are appropriately large as the domain becomes very thin. In our analysis, the interface boundary condition is interpreted as a variation of the Navier boundary conditions containing an interaction part. The effect of that interaction on the Stokes operator and the nonlinear term of the Navier–Stokes equations is expressed and carefully estimated in different ways in order to obtain suitable inequalities.     

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.     

Least‐squares meshfree method for incompressible Navier–Stokes problems

A least‐squares meshfree method based on the first‐order velocity–pressure–vorticity formulation for two‐dimensional incompressible Navier–Stokes problem is presented. The convective term is linearized by successive substitution or Newton's method. The discretization of all governing equations is implemented by the least‐squares method. Equal‐order moving least‐squares approximation is employed with Gauss quadrature in the background cells. The boundary conditions are enforced by the penalty method. The matrix‐free element‐by‐element Jacobi preconditioned conjugate method is applied to solve the discretized linear systems. Cavity flow for steady Navier–Stokes problem and the flow over a square obstacle for time‐dependent Navier–Stokes problem are investigated for the presented least‐squares meshfree method. The effects of inaccurate integration on the accuracy of the solution are investigated. Copyright © 2004 John Wiley & Sons, Ltd.     

A numerical study of projection algorithms in the finite element simulation of three‐dimensional viscous incompressible flow

In this work a comparative study of two versions of the projection algorithm used either for time integration or as an iterative method to solve the three‐dimensional incompressible Navier–Stokes equations is presented. It is also shown that these projection algorithms combined with the finite element method are particularly suited for the treatment of outflow boundary conditions in the context of external flows. This assertion is illustrated by means of some numerical examples in which five types of boundary conditions are compared. The scheme is applied to simulate the flow past a cylinder clamped on two fixed parallel solid walls. Comparison with experimental data available for this problem shows good agreement of the velocity and pressure fields, both computed with continuous piecewise linear elements. Copyright © 1999 John Wiley & Sons, Ltd.     

Effective downstream boundary conditions for incompressible Navier–Stokes equations

The aim of this paper is to give open boundary conditions for the incompressible Navier–Stokes equations. From a weak formulation in velocity–pressure variables, some natural boundary conditions involving the traction or pseudotraction and inertial terms are established. Numerical experiments on the flow behind a cylinder show the efficiency of these conditions, which convey properly the vortices downstream. Comparisons with other boundary conditions for the velocity and pressure are also performed.     

On the Inviscid Limit for the Compressible Navier–Stokes System in an Impermeable Bounded Domain

In this paper we investigate the issue of the inviscid limit for the compressible Navier–Stokes system in an impermeable fixed bounded domain. We consider two kinds of boundary conditions. The first one is the no-slip condition. In this case we extend the famous conditional result (Kato, T.: Remarks on zero viscosity limit for nonstationary Navier–Stokes flows with boundary. In: Seminar on nonlinear partial differential equations, vol. 2, pp. 85–98. Math. Sci. Res. Inst. Publ., Berkeley 1984) obtained by Kato in the homogeneous incompressible case. Kato proved that if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes then the solutions of the incompressible Navier–Stokes equations converge to some solutions of the incompressible Euler equations in the energy space. We provide here a natural extension of this result to the compressible case. The other case is the Navier condition which encodes that the fluid slips with some friction on the boundary. In this case we show that the convergence to the Euler equations holds true in the energy space, as least when the friction is not too large. In both cases we use in a crucial way some relative energy estimates proved recently by Feireisl et al. in J. Math. Fluid Mech. 14:717–730 (2012).     

Solution of the incompressible Navier–Stokes equations by the method of lines

The numerical method of lines (NUMOL) is a numerical technique used to solve efficiently partial differential equations. In this paper, the NUMOL is applied to the solution of the two‐dimensional unsteady Navier–Stokes equations for incompressible laminar flows in Cartesian coordinates. The Navier–Stokes equations are first discretized (in space) on a staggered grid as in the Marker and Cell scheme. The discretized Navier–Stokes equations form an index 2 system of differential algebraic equations, which are afterwards reduced to a system of ordinary differential equations (ODEs), using the discretized form of the continuity equation. The pressure field is computed solving a discrete pressure Poisson equation. Finally, the resulting ODEs are solved using the backward differentiation formulas. The proposed method is illustrated with Dirichlet boundary conditions through applications to the driven cavity flow and to the backward facing step flow. Copyright © 2015 John Wiley & Sons, Ltd.     

A Newton's method scheme for solving free-surface flow problems

A Newton's method scheme is described for solving the system of non-linear algebraic equations arising when finite difference approximations are applied to the Navier–Stokes equations and their associated boundary conditions. The problem studied here is the steady, buoyancy-driven motion of a deformable bubble, assumed to consist of an inviscid, incompressible gas. The linear Newton system is solved using both direct and iterative equation solvers. The numerical results are in excellent agreement with previous work, and the method achieves quadratic convergence.     

Fourth‐order accurate compact‐difference discretization method for Helmholtz and incompressible Navier–Stokes equations

A fourth‐order accurate solution method for the three‐dimensional Helmholtz equations is described that is based on a compact finite‐difference stencil for the Laplace operator. Similar discretization methods for the Poisson equation have been presented by various researchers for Dirichlet boundary conditions. Here, the complicated issue of imposing Neumann boundary conditions is described in detail. The method is then applied to model Helmholtz problems to verify the accuracy of the discretization method. The implementation of the solution method is also described. The Helmholtz solver is used as the basis for a fourth‐order accurate solver for the incompressible Navier–Stokes equations. Numerical results obtained with this Navier–Stokes solver for the temporal evolution of a three‐dimensional instability in a counter‐rotating vortex pair are discussed. The time‐accurate Navier–Stokes simulations show the resolving properties of the developed discretization method and the correct prediction of the initial growth rate of the three‐dimensional instability in the vortex pair. Copyright © 2004 John Wiley & Sons, Ltd.     

Incompressible Roe‐like scheme for turbulent flows

In the current study, numerical investigation of incompressible turbulent flow is presented. By the artificial compressibility method, momentum and continuity equations are coupled. Considering Reynolds averaged Navier–Stokes equations, the Spalart–Allmaras turbulence model, which has accurate results in two‐dimensional problems, is used to calculate Reynolds stresses. For convective fluxes a Roe‐like scheme is proposed for the steady Reynolds averaged Navier–Stokes equations. Also, Jameson averaging method was implemented. In comparison, the proposed characteristics‐based upwind incompressible turbulent Roe‐like scheme, demonstrated very accurate results, high stability, and fast convergence. The fifth‐order Runge–Kutta scheme is used for time discretization. The local time stepping and implicit residual smoothing were applied as the convergence acceleration techniques. Suitable boundary conditions have been implemented considering flow behavior. The problem has been studied at high Reynolds numbers for cross flow around the horizontal circular cylinder and NACA0012 hydrofoil. Results were compared with those of others and a good agreement has been observed. Copyright © 2012 John Wiley & Sons, Ltd.     

A Fourier–Chebyshev spectral collocation method for simulating flow past spheres and spheroids

An accurate Fourier–Chebyshev spectral collocation method has been developed for simulating flow past prolate spheroids. The incompressible Navier–Stokes equations are transformed to the prolate spheroidal co‐ordinate system and discretized on an orthogonal body fitted mesh. The infinite flow domain is truncated to a finite extent and a Chebyshev discretization is used in the wall‐normal direction. The azimuthal direction is periodic and a conventional Fourier expansion is used in this direction. The other wall‐tangential direction requires special treatment and a restricted Fourier expansion that satisfies the parity conditions across the poles is used. Issues including spatial and temporal discretization, efficient inversion of the pressure Poisson equation, outflow boundary condition and stability restriction at the pole are discussed. The solver has been validated primarily by simulating steady and unsteady flow past a sphere at various Reynolds numbers and comparing key quantities with corresponding data from experiments and other numerical simulations. Copyright © 1999 John Wiley & Sons, Ltd.     

Optimal flow control for Navier–Stokes equations: drag minimization

Optimal control and shape optimization techniques have an increasing role in Fluid Dynamics problems governed by partial differential equations (PDEs). In this paper, we consider the problem of drag minimization for a body in relative motion in a fluid by controlling the velocity through the body boundary. With this aim, we handle with an optimal control approach applied to the steady incompressible Navier–Stokes equations. We use the Lagrangian functional approach and we consider the Lagrangian multiplier method for the treatment of the Dirichlet boundary conditions, which include the control function itself. Moreover, we express the drag coefficient, which is the functional to be minimized, through the variational form of the Navier–Stokes equations. In this way, we can derive, in a straightforward manner, the adjoint and sensitivity equations associated with the optimal control problem, even in the presence of Dirichlet control functions. The problem is solved numerically by an iterative optimization procedure applied to state and adjoint PDEs which we approximate by the finite element method. Copyright © 2007 John Wiley & Sons, Ltd.     

Solution of the Incompressible Unsteady Navier–Stokes Equations Only in Terms of the Velocity Components

A method which uses only the velocity components as primitive variables is described for solution of the incompressible unsteady Navier–Stokes equations. The method involves the multiplication of the primitive variable-based Navier–Stokes equations with the unit normal vector of finite volume elements and the integration of the resulting equations along the boundaries of four-node quadrilateral finite volume elements. Therefore, the pressure term is eliminated from the governing equations and any difficulty associated with pressure or vorticity boundary conditions is avoided. The equations are discretized on four-node quadrilateral finite volume elements by using the second-order-accurate central finite differences with the mid-point integral rule in space and the first-order-accurate backward finite differences in time. The resulting system of algebraic equations is solved in coupled form using a direct solver. As a test case, an impulsively accelerated lid-driven cavity flow in a square enclosure is solved in order to verify the accuracy of the present method.     

Some uses of Green's theorem in solving the Navier–Stokes equations

This paper gives a review of methods where Green's theorem may be employed in solving numerically the Navier–Stokes equations for incompressible fluid motion. They are based on the concept of using the theorem to transform local boundary conditions given on the boundary of a closed region in the solution domain into global, or integral, conditions taken over it. Two formulations of the Navier–Stokes equations are considered: that in terms of the streamfunction and vorticity for two-dimensional motion and that in terms of the primitive variables of the velocity components and the pressure. In the first formulation overspecification of conditions for the streamfunction is utilized to obtain conditions of integral type for the vorticity and in the second formulation integral conditions for the pressure are found. Some illustrations of the principle of the method are given in one space dimension, including some derived from two-dimensional flows using the series truncation method. In particular, an illustration is given of the calculation of surface vorticity for two-dimensional flow normal to a flat plate. An account is also given of the implementation of these methods for general two-dimensional flows in both of the mentioned formulations and a numerical illustration is given.     

Numerical simulations of falling leaves using a pseudo-spectral method with volume penalization

The dynamics of falling leaves is studied by means of numerical simulations. The two-dimensional incompressible Navier–Stokes equations, coupled with the equations governing solid body dynamics, are solved using a Fourier pseudo-spectral method with volume penalization to impose no-slip boundary conditions. Comparison with other numerical methods is made. Simulations performed for different values of the Reynolds number show that its decrease stabilizes the free fall motion.     

An implicit edge‐based ALE method for the incompressible Navier–Stokes equations

A new finite volume method for the incompressible Navier–Stokes equations, expressed in arbitrary Lagrangian–Eulerian (ALE) form, is presented. The method uses a staggered storage arrangement for the pressure and velocity variables and adopts an edge‐based data structure and assembly procedure which is valid for arbitrary n‐sided polygonal meshes. Edge formulas are presented for assembling the ALE form of the momentum and pressure equations. An implicit multi‐stage time integrator is constructed that is geometrically conservative to the precision of the arithmetic used in the computation. The method is shown to be second‐order‐accurate in time and space for general time‐dependent polygonal meshes. The method is first evaluated using several well‐known unsteady incompressible Navier–Stokes problems before being applied to a periodically forced aeroelastic problem and a transient free surface problem. Published in 2003 by John Wiley & Sons, Ltd.     

Higher order spectral/hp finite element models of the Navier–Stokes equations

In this paper, we present higher order least-squares finite element formulations for viscous, incompressible, isothermal Navier–Stokes equations using spectral/hp basis functions. The second-order Navier–Stokes equations are recast as first-order system of equations using stresses as auxiliary variables. Both steady-state and transient problems are considered. For a better coupling of pressure and velocity, especially in transient flows, an iterative penalisation strategy is employed. The outflow-type boundary conditions are applied in a weak sense through the least-squares functional. The formulation is verified by solving various benchmark problems like the lid-driven cavity, backward-facing step and flow over cylinder problems using direct serial solver UMFPACK.