Showing papers on "Navier–Stokes equations published in 1999"

Direct numerical simulation of free-surface and interfacial flow

Abstract: The numerical simulation of flows with interfaces and free-surface flows is a vast topic, with applications to domains as varied as environment, geophysics, engineering, and fundamental physics. In engineering, as well as in other disciplines, the study of liquid-gas interfaces is important in combustion problems with liquid and gas reagents. The formation of droplet clouds or sprays that subsequently burn in combustion chambers originates in interfacial instabilities, such as the Kelvin-Helmholtz instability. What can numerical simulations do to improve our understanding of these phenomena? The limitations of numerical techniques make it impossible to consider more than a few droplets or bubbles. They also force us to stay at low Reynolds or Weber numbers, which prevent us from finding a direct solution to the breakup problem. However, these methods are potentially important. First, the continuous improvement of computational power (or, what amounts to the same, the drop in megaflop price) continuously extends the range of affordable problems. Second, and more importantly, the phenomena we consider often happen on scales of space and time where experimental visualization is difficult or impossible. In such cases, numerical simulation may be a useful prod to the intuition of the physicist, the engineer, or the mathematician. A typical example of interfacial flow is the collision between two liquid droplets. Finding the flow involves the study not only of hydrodynamic fields in the air and water phases but also of the air-water interface. This latter part

A front-tracking algorithm for accurate representation of surface tension

Abstract: We present a front tracking algorithm for the solution of the 2D incompressible Navier-Stokes equations with interfaces and surface forces. More particularly, we focus our attention on the accurate description of the surface tension terms and the associated pressure jump. We consider the stationary Laplace solution for a bubble with surface tension. A careful treatment of the pressure gradient terms at the interface allows us to reduce the spurious currents to the machine precision. Good results are obtained for the oscillation of a capil- lary wave compared with the linear viscous theory. A classical test of Rayleigh-Taylor instability is presented.

Preconditioning techniques in computational fluid dynamics

Abstract: ▪ Abstract An overview of preconditioning for the steady-state compressible inviscid fluid dynamic equations is presented. Extensions to the Navier-Stokes equations are also considered. These preconditioners are necessary for many algorithms in order to have the correct behavior at low speeds and to converge to the solution of the incompressible equations as the Mach number goes to zero. In addition, the preconditioning accelerates the convergence to a steady state for problems in which a significant portion of the flow is low speed. This low speed preconditioner can be combined with Jacobi and line preconditioners to damp high frequencies at all speeds. This is necessary for use with multigrid methods. Such combined methods are also better at accelerating problems with high aspect ratios. Details of the implementation are presented including several different variants for the preconditioning matrix.

Monotonically integrated large eddy simulation of free shear flows

Abstract: With a view to ensure that proper interaction between resolvable or grid scale and subgrid scale (SGS) motions are mimicked, it is vital to determine the necessary physics that must be built into the SGS models. In ordinary large eddy simulation (LES) approaches, models are introduced for closure in the low-pass filtered Navier-Stokes equations (NSEs), which are the ones solved numerically. A promising LES approach is monotonically integrated LES (MILES), which involves solving the unfiltered NSE using high-resolution monotone algorithms; in this approach, implicit SGS models, provided by intrinsic nonlinear high-frequency filters built into the convection discretization, are coupled naturally to the resolvable scales of the flow. Formal properties of the effectual SGS modeling using MILES are documented using databases of simulated homogeneous turbulence and transitional freejets; mathematical and physical aspects of (implicit) SGS modeling through the use of nonlinear flux limiters are addressed in this context

A discontinuous hp finite element method for the Euler and Navier–Stokes equations

Abstract: We introduce a new method for the solution of the Euler and Navier-Stokes equations, which is based on the application of a recently developed discontinuous Galerkin technique to obtain a compact, higher-order accurate and stable solver. The method involves a weak imposition of continuity conditions on the state variables and on inviscid and diffusive fluxes across interelement and domain boundaries. Within each element the field variables are approximated using polynomial expansions with local support; therefore, this method is particularly amenable to adaptive refinements and polynomial enrichment. Moreover, the order of spectral approximation on each element can be adaptively controlled according to the regularity of the solution. The particular formulation on which the method is based makes possible a consistent implementation of boundary conditions, and the approximate solutions are locally (elementwise) conservative. The results of numerical experiments for representative benchmarks suggest that the method is robust, capable of delivering high rates of convergence, and well suited to be implemented in parallel computers

On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional Navier-Stokes equations

Abstract: We prove a criterion of local Holder continuity for suitable weak solutions to the Navier—Stokes equations. One of the main part of the proof, based on a blow-up procedure, has quite general nature and can be applied to other problems in spaces of solenoidal vector fields.

Computational and experimental analysis of dynamics of drop formation

Abstract: Dynamics of formation of a drop of a Newtonian liquid from a capillary tube into an ambient gas phase is studied computationally and experimentally. While this problem has previously been studied computationally either (a) using a set of one-dimensional equations or (b) treating the dynamics as that of irrotational flow of an inviscid fluid or creeping flow, here the full nonlinear, transient Navier–Stokes system subject to appropriate initial and boundary conditions is solved in two dimensions to analyze the dynamics at finite Reynolds numbers. The success of the computations rests on a finite element algorithm incorporating a multiregion mesh which conforms to and evolves with the changing shape of the drop. The new algorithm is able to capture both the gross features of the phenomenon, such as the limiting length of a drop at breakup and the volume of the primary drop, and its fine features, such as a microthread that develops from a main thread or a neck in a viscous drop approaching breakup. The accuracy of the new calculations is verified by comparison of computed predictions to old and new experiments. With the new algorithm, it is shown for the first time that the interface of a viscous drop can overturn before the drop breaks. Calculations have also been carried out to determine the range of parameters over which algorithms that treat the drop liquid as inviscid and the flow inside it as irrotational can accurately predict the dynamics of formation of drops of low viscosity liquids. Limiting lengths of drops and primary drop volumes are computed over a wide range of the parameter space spanned by the relevant dimensionless groups.

Implicit Solution of Preconditioned Navier- Stokes Equations Using Algebraic Multigrid

Abstract: We describe an algorithm for the solution of the Navier-Stokes equations on unstructured meshes that employs a coupled algebraic multigrid method to accelerate a point-implicit symmetric Gauss-Seidel relaxation scheme. The equations are preconditioned to permit solution of both compressible and incompressible flows. A cell-based, finite volume discretization is used in conjunction with flux-difference splitting and a linear reconstruction of variables. We present results for flowfields representing a range of Mach numbers and Reynolds numbers. The scheme remains stable up to infinite Courant number and exhibits CPU usage that scales linearly with cell count

Steady and Unsteady Flow Simulations Using the Hybrid Flow Solver AVBP

Abstract: The unstructured flow solver AVBP of CERFACS is presented. The basic concepts of the program are described, and various computational results are presented for a large spectrum of applications ranging from steady-state external aerodynamics to unsteady turbulent flows with and without combustion. The code solves the compressible Navier-Stokes equations on hybrid grids of arbitrary cell type. The code is built on a modular software library and has been ported to a wide range of parallel computers.

Preconditioning for the Steady-State Navier--Stokes Equations with Low Viscosity

Abstract: We introduce a preconditioner for the linearized Navier--Stokes equations that is effective when either the discretization mesh size or the viscosity approaches zero. For constant coefficient problems with periodic boundary conditions, we show that the preconditioning yields a system with a single eigenvalue equal to 1, so that performance is independent of both viscosity and mesh size. For other boundary conditions, we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon. We also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration, with virtually no degradation of performance.

An unsteady incompressible navier-stokes solver for large eddy simulation of turbulent flows

Abstract: SUMMARY An unsteady incompressible Navier‐Stokes solver that uses a dual time stepping method combined with spatially high-order-accurate finite differences, is developed for large eddy simulation (LES) of turbulent flows. The present solver uses a primitive variable formulation that is based on the artificial compressibility method and various convergence‐acceleration techniques are incorporated to efficiently simulate unsteady flows. A localized dynamic subgrid model, which is formulated using the subgrid kinetic energy, is employed for subgrid turbulence modeling. To evaluate the accuracy and the efficiency of the new solver, a posteriori tests for various turbulent flows are carried out and the resulting turbulence statistics are compared with existing experimental and direct numerical simulation (DNS) data. Copyright © 1999 John Wiley & Sons, Ltd.

Two-dimensional navier-stokes simulation of breaking waves

Abstract: Numerical simulations describing plunging breakers including the splash-up phenomenon are presented. The motion is governed by the classical, incompressible, two-dimensional Navier–Stokes equation. The numerical modeling of this two-phase flow is based on a piecewise linear version of the volume of fluid method. Capillary effects are taken into account such as a nonisotropic stress tensor concentrated near the interface. Results concerning the time evolution of liquid–gas interface and velocity field are given for short waves, showing how an initial steep wave undergoes breaking and successive splash-up cycles. Breaking processes including overturning, splash-up and gas entrainment, and breaking induced vortex-like motion beneath the surface and energy dissipation, are presented and discussed. It is found that strong vorticities are generated during the breaking process, and that more than 80% of the total pre-breaking wave energy is dissipated within three wave periods. The numerical results are compared with some laboratory measurements, and a favorable agreement is found.

Airfoil Design on Unstructured Grids for Turbulent Flows

Abstract: An aerodynamic design algorithm for turbulent e ows using unstructured grids is described. The current approachusesadjoint (costate)variablestoobtainderivativesofthecostfunction.Thesolutionoftheadjointequations is obtained by using an implicit formulation in which the turbulence model is fully coupled with the e ow equations when solving for the costate variables. The accuracy of the derivatives is demonstrated by comparison with e nite difference gradients, and a few sample computations are shown. Recommendations on directions of further research into the Navier ‐Stokes design process are made. Nomenclature A = area of control volume a = speed of sound C ¤ = constant used in Sutherland’ s law for viscosity cb1;cb2;cv1; = constants used in Spalart ‐Allmaras cw1;cw2;cw3 turbulence model cd = drag cl = lift c1

The characteristic-based-split procedure: an efficient and accurate algorithm for fluid problems

Abstract: In 1995 the two senior authors of the present paper introduced a new algorithm designed to replace the Taylor–Galerkin (or Lax–Wendroff) methods, used by them so far in the solution of compressible flow problems. The new algorithm was applicable to a wide variety of situations, including fully incompressible flows and shallow water equations, as well as supersonic and hypersonic situations, and has proved to be always at least as accurate as other algorithms currently used. The algorithm is based on the solution of conservation equations of fluid mechanics to avoid any possibility of spurious solutions that may otherwise result. The main aspect of the procedure is to split the equations into two parts, (1) a part that is a set of simple scalar equations of convective–diffusion type for which it is well known that the characteristic Galerkin procedure yields an optimal solution; and (2) the part where the equations are self-adjoint and therefore discretized optimally by the Galerkin procedure. It is possible to solve both the first and second parts of the system explicitly, retaining there the time step limitations of the Taylor–Galerkin procedure. But it is also possible to use semi-implicit processes where in the first part we use a much bigger time step generally governed by the Peclet number of the system while the second part is solved implicitly and is unconditionally stable. It turns out that the characteristic-based-split (CBS) process allows equal interpolation to be used for all system variables without difficulties when the incompressible or nearly incompressible stage is reached. It is hoped that the paper will help to make the algorithm more widely available and understood by the profession and that its advantages can be widely realised. Copyright © 1999 John Wiley & Sons, Ltd.

Global Regularity of 3D Rotating Navier-Stokes Equations for Resonant Domains

Abstract: We prove existence on infinite time intervals of regular solutions to the 3D rotating Navier-Stokes equations in the limit of strong rotation (large Coriolis parameter Ω). This uniform existence is proven for periodic or stress-free boundary conditions for all domain aspect ratios, including the case of three wave resonances which yield nonlinear “ 21 2 - dimensional ” limit equations; smoothness assumptions are the same as for local existence theorems. The global existence is proven using techniques of the Littlewood-Paley dyadic decomposition. Infinite time regularity for solutions of the 3D rotating Navier-Stokes equations is obtained by bootstrapping from global regularity of the limit equations and convergence theorems.

Effect of Free Stream Turbulence and Other Vortical Disturbances on a Laminar Boundary Layer

Abstract: This paper is concerned with the effect of free-stream turbulence on the pretransitional flat-plate boundary layer. It is assumed that either the turbulence Reynolds number or the downstream distance (or both) is small enough so that the flow can be linearized. The dominant disturbances in the boundary layer, which are of the Klebanoff type, are governed by the linearized unsteady boundary-region equations, i.e., the Navier Stokes equations with the streamwise derivatives neglected in the viscous and pressure-gradient terms. The turbulence is represented as a superposition of vortical free-stream Fourier modes, and the corresponding individual Fourier component solutions to the boundary-region equations are obtained numerically. The results are then superposed to compute the root mean square of the fluctuating streamwise velocity in the boundary layer produced by the actual free-stream turbulence. The calculated boundary-layer disturbances are in good quantitative agreement with the experimentally observed Klebanoff modes when strong low-frequency anisotropic effects are included in the free-stream turbulence spectrum. We discuss some additional effects that may need to be accounted for in order to obtain a complete description of the Klebanoff modes.

A Psuedo-Temporal Multi-Grid Relaxation Scheme for Solving the Parabolized Navier-Stokes Equations

Abstract: A multi-grid, flux-difference-split, finite-volume code, VULCAN, is presented for solving the eliptic and parabolized form of the equations governing three-dimensional, turbulent, calorically perfect and non-equilibrium chemically reacting flows. The space marching algorithms developed to improve convergence rate and or reduce computational cost are emphasized. The algorithms presented are extensions to the class of implicit pseudo-time iterative, upwind space-marching schemes. A full approximate storage, full multi-grid scheme is also described which is used to accelerate the convergence of a Gauss-Seidel relaxation method. The multi-grid algorithm is shown to significantly improve convergence on high ratio grids.

A Pseudo-Temporal Multi-Grid Relaxation Scheme for Solving the Parabolized Navier-Stokes Equations

Abstract: A multi-grid, flux-difference-split, finite-volume code, VULCAN, is presented for solving the elliptic and parabolized form of the equations governing three-dimensional, turbulent, calorically perfect and non-equilibrium chemically reacting flows. The space marching algorithms developed to improve convergence rate and or reduce computational cost are emphasized. The algorithms presented are extensions to the class of implicit pseudo-time iterative, upwind space-marching schemes. A full approximate storage, full multi-grid scheme is also described which is used to accelerate the convergence of a Gauss-Seidel relaxation method. The multi-grid algorithm is shown to significantly improve convergence on high aspect ratio grids.

The effect of the slip boundary condition on the flow of fluids in a channel

Abstract: The assumption that a liquid adheres to a solid boundary (“no-slip” boundary condition) is one of the central tenets of the Navier-Stokes theory. However, there are situations wherein this assumption does not hold. In this paper we investigate the consequences of slip at the wall on the flow of a linearly viscous fluid in a channel. Usually, the slip is assumed to depend on the shear stress at the wall. However, a number of experiments suggests that the slip velocity also depends on the normal stress. Thus, we investigate the flow of a linearly viscous fluid when the slip depends on both the shear stress and the normal stress. In regions where the slip velocity depends strongly on the normal stress, the flow field in a channel is not fully developed and rectilinear flow is not possible. Also, it is shown that, in general, traditional methods such as the Mooney method cannot be used for calculating the slip velocity.

Secondary breakup of axisymmetric liquid drops. I. Acceleration by a constant body force

Abstract: The secondary breakup of liquid drops, accelerated by a constant body force, is examined for small density differences between the drops and the surrounding fluid. Two cases are examined in detail: a density ratio close to unity (ρd/ρo=1.15, where the Boussinesq approximation is valid) and a density ratio of ten. A finite difference/front tracking numerical technique is used to solve the unsteady Navier–Stokes equations for both the drops and the surrounding fluid. The breakup is controlled by the Eotvos number (Eo), the Ohnesorge number (Oh), and the viscosity and density ratios. If viscous effects are small (small Oh), the Eotvos number is the main controlling parameter. In the Boussinesq limit, as Eo increases the drops break up in a backward facing bag, transient breakup, and a forward facing bag mode. At a density ratio of ten, similar breakup modes are observed, with the exception that the forward facing bag mode is replaced by a shear breakup mode. Similar breakup modes have been seen experimentally for much larger density ratios. Although a backward facing bag is seen at low Oh, where viscous effects are small, comparisons with simulations of inviscid flows show that the bag breakup is a viscous phenomenon, due to boundary layer separation and the formation of a wake. At higher Oh, where viscous effects modify the evolution, the simulations show that the main effect of increasing Oh is to move the boundary between the different breakup modes to higher Eo. The results are summarized by “breakup maps” where the different breakup modes are shown in the Eo–Oh plane for different values of the viscosity and the density ratios.

A distributed Lagrange multiplier/fictitious domain method for flows around moving rigid bodies : Application to particulate flow

Abstract: This article discusses the application of a Lagrange multiplier-based fictitious domain method to the numerical simulation of incompressible viscous flow modeled by the Navier–Stokes equations around moving rigid bodies; the rigid body motions are due to hydrodynamical forces and gravity. The solution method combines finite element approximations, time discretization by operator splitting and conjugate gradient algorithms for the solution of the linearly constrained quadratic minimization problems coming from the splitting method. The study concludes with the presentation of numerical results concerning four test problems, namely the simulation of an incompressible viscous flow around a NACA0012 airfoil with a fixed center but free to rotate, then the sedimentation of 200 and 1008 cylinders in a two-dimensional channel, and finally the sedimentation of two spherical balls in a rectangular cylinder. Copyright © 1999 John Wiley & Sons, Ltd.

Ergodicity of 2D Navier–Stokes Equations with¶Random Forcing and Large Viscosity

Abstract: The stochastically forced, two-dimensional, incompressable Navier–Stokes equations are shown to possess an unique invariant measure if the viscosity is taken large enough. This result follows from a stronger result showing that at high viscosity there is a unique stationary solution which attracts solutions started from arbitrary initial conditions. That is to say, the system has a trivial random attractor. Along the way, results controling the expectation and averaging time of the energy and enstrophy are given.

A generalization of the Navier-Stokes equations to two-phase flows

Abstract: A modified Allen-Cahn equation is combined with the compressible Navier-Stokes system. We show that, after a modification of the stress tensor, the second law of thermodynamics is valid for the resulting equations. We give a physical motivation of this alteration of the stress tensor and compare the new equations with the well known phase-field approach. The model can be used to describe cavitation in a flowing liquid.

Gradient estimation on Navier–Stokes equations

Abstract: In this note, we present a prior uniform gradient estimates on solutions to the 3-dimensional Navier-Stokes equations. It is shown that the gradient of the velocity field is locally uniformly bounded in L^-norm provided that either the scaled local L-norm of the vorticity or the scaled local total energy is small. In particular, our results imply that the smooth solutions to 3-dimensional NavierStokes equations cannot develop finite time singularity and suitable weak solutions are in fact regular if either the scaled local L-norm of the vorticity or the scaled local energy is small.

Finite Volume Discretization Aspects for Viscous Flows on Mixed Unstructured Grids

Abstract: A solution method for compressible turbulent flows on unstructured grids in two dimensions is described. The method can be used on grids consisting of triangular and/or quadrilateral cells. Control volumes are constructed from dual cells, and the solution variables are stored at the vertices of the grid. Grid-transparent algorithms are developed that do not require knowledge of cell types, leading to simple discretization schemes on mixed grids. The inviscid fluxes are computed from limited high-resolution schemes originally developed for unstructured triangular grids. They are easily applied to quadrilateral or mixed grids and are grid transparent. The discretization of the viscous fluxes is studied in detail. A positive, grid-transparent discretization of Laplace's equation is developed. The existence of tangential derivatives in the viscous terms prevents grid transparency. By neglecting tangential derivatives, an approximate form of the viscous fluxes is developed, which recovers grid transparency. The approximate form is shown to be similar to the thin-shear-layer approximation. Results are obtained for a transonic inviscid flow, a laminar separated flow, and a transonic turbulent flow