Showing papers on "Navier–Stokes equations published in 1997"

A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers

Abstract: The numerical implementation of an ocean model based on the incompressible Navier Stokes equations which is designed for studies of the ocean circulation on horizontal scales less than the depth of the ocean right up to global scale is described. A "pressure correction" method is used which is solved as a Poisson equation for the pressure field with Neumann boundary conditions in a geometry as complicated as that of the ocean basins. A major objective of the study is to make this inversion, and hence nonhydrostatic ocean modeling, efficient on parallel computers. The pressure field is separated into surface, hydrostatic, and nonhydrostatic components. First, as in hydrostatic models, a two-dimensional problem is inverted for the surface pressure which is then made use of in the three-dimensional inversion for the nonhydrostatic pressure. Preconditioned conjugate-gradient iteration is used to invert symmetric elliptic operators in both two and three dimensions. Physically motivated preconditioners are designed which are efficient at reducing computation and minimizing communication between processors. Our method exploits the fact that as the horizontal scale of the motion becomes very much larger than the vertical scale, the motion becomes more and more hydrostatic and the three- dimensional Poisson operator becomes increasingly anisotropic and dominated by the vertical axis. Accordingly, a preconditioner is used which, in the hydrostatic limit, is an exact integral of the Poisson operator and so leads to a single algorithm that seamlessly moves from nonhydrostatic to hydrostatic limits. Thus in the hydrostatic limit the model is "fast," competitive with the fastest ocean climate models in use today based on the hydrostatic primitive equations. But as the resolution is increased, the model dynamics asymptote smoothly to the Navier Stokes equations and so can be used to address small- scale processes. A "finite-volume" approach is employed to discretize the model in space in which property fluxes are defined normal to faces that delineate the volumes. The method makes possible a novel treatment of the boundary in which cells abutting the bottom or coast may take on irregular shapes and be "shaved" to fit the boundary. The algorithm can conveniently exploit massively parallel computers and suggests a domain decomposition which allocates vertical columns of ocean to each processing unit. The resulting model, which can handle arbitrarily complex geometry, is efficient and scalable and has been mapped on to massively parallel multiprocessors such as the Connection Machine (CM5) using data-parallel FORTRAN and the Massachusetts Institute of Technology data-flow machine MONSOON using the implicitly parallel language Id. Details of the numerical implementation of a model which has been designed for the study of dynamical processes in the ocean from the convective, through the geostrophic eddy, up to global scale are set out. The "kernel" algorithm solves the incompressible Navier Stokes equations on the sphere, in a geometry as complicated as that of the ocean basins with ir- regular coastlines and islands. (Here we use the term "Navier Stokes" to signify that the full nonhydrostatic equations are being employed; it does not imply a particular constitutive relation. The relevant equations for modeling the full complex- ity of the ocean include, as here, active tracers such as tem- perature and salt.) It builds on ideas developed in the compu- tational fluid community. The numerical challenge is to ensure that the evolving velocity field remains nondivergent. Most

Hydrostatic, quasi‐hydrostatic, and nonhydrostatic ocean modeling

Abstract: Ocean models based on consistent hydrostatic, quasi-hydrostatic, and nonhydrostatic equation sets are formulated and discussed. The quasi-hydrostatic and nonhydrostatic sets are more accurate than the widely used hydrostatic primitive equations. Quasi-hydrostatic models relax the precise balance between gravity and pressure gradient forces by including in a consistent manner cosine-of-latitude Coriolis terms which are neglected in primitive equation models. Nonhydrostatic models employ the full incompressible Navier Stokes equations; they are required in the study of small-scale phenomena in the ocean which are not in hydrostatic balance. We outline a solution strategy for the Navier Stokes model on the sphere that performs efficiently across the whole range of scales in the ocean, from the convective scale to the global scale, and so leads to a model of great versatility. In the hydrostatic limit the Navier Stokes model involves no more computational effort than those models which assume strict hydrostatic balance on all scales. The strategy is illustrated in simulations of laboratory experiments in rotating convection on scales of a few centimeters, simulations of convective and baroclinic instability of the mixed layer on the 1- to 10-km scale, and simulations of the global circulation of the ocean.

Gaseous slip flow in long microchannels

Abstract: An analytic and experimental investigation into gaseous flow with slight rarefaction through long microchannels is undertaken. A two-dimensional (2-D) analysis of the Navier-Stokes equations with a first-order slip-velocity boundary condition demonstrates that both compressibility and rarefied effects are present in long microchannels. By undertaking a perturbation expansion in /spl epsiv/, the height-to-length ratio of the channel, and using the ideal gas equation of state, it is shown that the zeroth-order analytic solution for the streamwise mass flow corresponds well with the experimental results. Also, the effect of slip upon the pressure distribution is derived, and it is obtained that this slip velocity leads directly to a wall-normal migration of mass. The fabrication of wafer-bonded microchannels that possess well-controlled surface structure is described, and a means for accurately measuring the mass how through the channels is presented. Experimental results obtained with this mass-flow measurement technique for streamwise helium mass flow through microchannels 52.25-/spl mu/m wide, 1.33-/spl mu/m deep, and 7500-/spl mu/m long for a pressure range of 1.6-4.2 atmospheres (outlet pressures at atmospheric) are presented and shown to compare favorably with the analysis.

Zero-Pressure-Gradient Turbulent Boundary Layer

Abstract: Of the many aspects of the long-studied field of turbulence, the zero-pressure-gradient boundary layer is probably the most investigated, and perhaps also the most reviewed. Turbulence is a fluid-dynamical phenomenon for which the dynamical equations are generally believed to be the Navier-Stokes equations, at least for a single-phase, Newtonian fluid. Despite this fact, these governing equations have been used in only the most cursory manner in the development of theories for the boundary layer, or in the validation of experimental data-bases. This article uses the Reynolds-averaged Navier-Stokes equations as the primary tool for evaluating theories and experiments for the zero-pressure-gradient turbulent boundary layer. Both classical and new theoretical ideas are reviewed, and most are found wanting. The experimental data as well is shown to have been contaminated by too much effort to confirm the classical theory and too little regard for the governing equations. Theoretical concepts and experiments are identified, however, which are consistent-both with each other and with the governing equations. This article has 77 references.

Proposed Inflow/Outflow Boundary Condition for Direct Computation of Aerodynamic Sound

Abstract: A simple new zonal boundary condition has been proposed. It is based upon the addition of dissipative and convective terms to the compressible Navier Stokes equations

A generalization of a theorem by Kato on Navier-Stokes equations

Abstract: We generalize a classical result of T. Kato on the existence of global solutions to the Navier-Stokes system in C([0,8);L3(R3)). More precisely, we show that if the initial data are sufficiently oscillating, in a suitable Besov space, then Kato's solution exists globally. As a corollary to this result, we obtain a theory of existence of self-similar solutions for the Navier-Stokes equations.

A vortex-based subgrid stress model for large-eddy simulation

Abstract: A class of subgrid stress (SGS) models for large-eddy simulation (LES) is presented based on the idea of structure-based Reynolds-stress closure. The subgrid structure of the turbulence is assumed to consist of stretched vortices whose orientations are determined by the resolved velocity field. An equation which relates the subgrid stress to the structure orientation and the subgrid kinetic energy, together with an assumed Kolmogorov energy spectrum for the subgrid vortices, gives a closed coupling of the SGS model dynamics to the filtered Navier-Stokes equations for the resolved flow quantities. The subgrid energy is calculated directly by use of a local balance between the total dissipation and the sum of the resolved-scale dissipation and production by the resolved scales. Simple one- and two-vortex models are proposed and tested in which the subgrid vortex orientations are either fixed by the local resolved velocity gradients, or rotate in response to the evolution of the gradient field. These models are not of the eddy viscosity type. LES calculations with the present models are described for 32^(3) decaying turbulence and also for forced 32^(3) box turbulence at Taylor Reynolds numbers R-lambda in the range R(lambda)similar or equal to 30 (fully resolved) to R-lambda=infinity. The models give good agreement with experiment for decaying turbulence and produce negligible SGS dissipation for forced turbulence in the limit of fully resolved flow.

A Detailed Analysis of Film–Cooling Physics: Part I — Streamwise Injection With Cylindrical Holes

Abstract: A previously documented systematic computational methodology is implemented and applied to a jet-in-crossflow problem in order to document all of the pertinent flow physics associated with a film-cooling flowfield Numerical results are compared to experimental data for the case of a row of three-dimensional, inclined jets with length-to-diameter ratios similar to a realistic film-cooling application A novel vorticity-based approach is included in the analysis of the flow physics Particular attention has been paid to the downstream coolant structures and to the source and influence of counterrotating vortices in the crossflow region It is shown that the vorticity in the boundary layers within the film hole is primarily responsible for this secondary motion Important aspects of the study include: (1) a systematic treatment of the key numerical issues, including accurate computational modeling of the physical problem, exact geometry and high-quality grid generation techniques, higher-order numerical discretization, and accurate evaluation of turbulence model performance; (2) vorticity-based analysis and documentation of the physical mechanisms of jet-crossflow interaction and their influence on film-cooling performance; (3) a comparison of computational results to experimental data; and (4) comparison of results using a two-layer model near-wall treatment versus generalized wall functions Solution of the steady, time-averaged Navier-Stokes equations were obtained for all cases using an unstructured/adaptive grid, fully explicit, time-marching code with multigrid, local time stepping, and residual smoothing acceleration techniques For the case using the two-layer model, the solution was obtained with an implicit, pressure-correction solver with multigrid The three-dimensional test case was examined for two different film-hole length-to-diameter ratios of 175 and 35, and three different blowing ratios, from 05 to 20 All of the simulations had a density ratio of 20, and an injection angle of 35 deg An improved understanding of the flow physics has provided insight into future advances to film-cooling configuration design In addition, the advantages and disadvantages of the two-layer turbulence model are highlighted for this class of problems

Vacuum states for compressible flow

Abstract: In this paper we study the evolutions of the interfaces between gases and the vacuum for both inviscid and viscous one dimensional isentropic gas motions. The local (in time) existence of solutions for both inviscid and viscous models with initial data containing vacuum states is proved and some singular properties on the free surfaces separating the gas and the vacuum are obtained. It is found that the Euler equations are better behaved near the vacuum than the compressible Navier-Stokes equations. The Navier-Stokes equations with viscosity depending on density are introduced, which is shown to be well-posed (at least locally) and yield the desired solutions near vacuum.

Axisymmetric free boundary problems

Abstract: We present a number of three-dimensional axisymmetric free boundary problems for two immiscible fluids, such as air and water. A level set method is used where the interface is the zero level set of a continuous function while the two fluids are solutions of the incompressible Navier–Stokes equation. We examine the rise and distortion of an initially spherical bubble into cap bubbles and toroidal bubbles. Steady solutions for gas bubbles rising in a liquid are computed, with favourable comparisons to experimental data. We also study the inviscid limit and compare our results with a boundary integral method. The problems of an air bubble bursting at a free surface and a liquid drop hitting a free surface are also computed.

Discontinuous Solutions of the Navier-Stokes Equations for Multidimensional Flows of Heat-Conducting Fluids

Abstract: We prove the global existence of weak solutions of the Navier-Stokes equations for compressible, heat-conducting fluids in two and three space dimensions when the initial density is close to a constant in L 2∩L ∞, the initial temperature is close to a constant in L 2, and the initial velocity is small in H s ∩L 4, where s=0 when n=2 and when n=3. (The L p norms must be weighted slightly when n=2.) In particular, the initial data may be discontinuous across a hypersurface of n . A great deal of qualitative information about the solution is obtained. For example, we show that the velocity, vorticity, and temperature are relatively smooth in positive time, as is the “effective viscous flux”F, which is the divergence of the velocity minus a certain multiple of the pressure. We find that F plays a central role in the entire analysis, particularly in closing the required energy estimates and in understanding rates of regularization near the initial layer. Moreover, F is precisely the quantity through which the hyperbolicity of the corresponding equations for inviscid fluids shows itself, an effect which is crucial for obtaining time-independent pointwise bounds for the density.

Numerical Simulation of Submarine Landslides and Their Hydraulic Effects

Abstract: The submarine flow slides and their hydraulic effects are studied by numerical means. These types of landslides are assumed to separate into a dense flow close to the bed and a turbulent dispersion above it. A two-dimensional fluid mechanics mixture model based on Navier-Stokes' equations has been developed to study water waves generated by these landslides. The dense part is considered as a viscoplastic fluid, whereas the dispersed part is modeled by an ideal fluid. The rheological parameters of the model comprise a diffusion coefficient, a Bingham yield stress, a viscosity coefficient, and friction on the slope. First, the numerical model is validated with an analytical solution for a viscous and a Bingham flow. Then, it has been tested for a rigid box sliding into water along an inclined plane. The results of this simulation have been compared with experiments conducted in a channel. Finally, laboratory experiments consisting in the slide of a gravel mass have been carried out. The results of these experiments have shown the importance of the sediment rheology and the diffusion. The model parameters are adjusted by trial and error to match the observed landslide flow.

Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves

Abstract: In [2], we determined a unique "effective artificial viscosity" system approximating the behavior of the compressible Navier-Stokes equations. Here, we derive a detailed, pointwise description of the Green's function for this system. This Green's function generalizes the notion of "diffusion wave" introduced by Liu in the one-dimensional case, being expressible as a nonstandard heat kernel convected by the hyperbolic solution operator of the linearized compressible Euler equations. It dominates the asymptotic behavior of solutions of the (nonlinear) compressible Navier-Stokes equations with localized initial data. The problem reduces to determining estimates on the wave equation, with initial data consisting of various combinations of heat and Riesz kernels; however, the calculations turn out to be surprisingly subtle, involving cancellation not captured by standard \(L^p\) estimates for the wave equation.

Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations

Abstract: Introduction - Preliminareis - Stationary Quasi-Compressibility Methods: The Penalty Method and the Pressure Stabilization Method - Nonstationary Quasi-Compressibility Methods - Mixed Quasi-Compressibility Methods - The Projection Scheme of Chorin - The Projection Scheme of Van Kan - Two Modified Chorin Schemes - Multi-Component Schemes - Time Discretization on Time-Grids with Structure from Euler to Revised Projection Schemes

Regularity and integrability of 3D Euler and Navier–Stokes equations for rotating fluids

Abstract: We consider 3D Euler and Navier–Stokes equations describing dynamics of uniformly rotating fluids. Periodic (as well as zero vertical flux) boundary conditions are imposed, the ratios of domain periods are assumed to be generic (nonresonant). We show that solutions of 3D Euler/Navier–Stokes equations can be decomposed as U(t, x1, x2, x3) = Ũ(t, x1, x2) +V(t, x1, x2, x3) + r, where Ũ is a solution of the 2D Euler/Navier–Stokes system with vertically averaged initial data (axis of rotation is taken along the vertical e3). The vector field V(t, x1, x2, x3) is exactly solved in terms of the phases Ωt, τ1(t) and τ2(t). The phases τ1(t) and τ2(t) explicitly expressed in terms of vertically averaged vertical vorticity curl U(t) ·e3 and velocity U 3 (t). The remainder r is uniformly estimated from above by a majorant of order a3/Ω, a3 is the vertical aspect ratio (shallowness) and Ω is non-dimensional rotation parameter based on horizontal scales. The resolution of resonances and a non-standard small divisor problem for 3D rotating Euler are the basis for error estimates. Contribution of 3-wave resonances is estimated in terms of the measure of almost resonant aspect ratios. Global solvability of the limit equations and estimates of the error r are used to prove existence on a long time interval T ∗ of regular solutions to 3D Euler equations (T ∗ → +∞, as 1/Ω → 0); and existence on infinite time interval of regular solutions to 3D Navier–Stokes equations with smooth arbitrary initial data in the case of small 1/Ω.

Direct numerical simulation of turbulent flow over a wavy wall

Abstract: The Navier–Stokes equations have been solved, by a pseudospectral method, for pressure-driven flows between a no-slip wavy wall and a slip flat wall. Periodic boundary conditions were used in the streamwise and spanwise directions. The physical domain is mapped into a computational domain that is a rectangular parallelepiped using a nonorthogonal transformation. The pseudospectral solution procedure employed in previous studies, for example, Lam and Banerjee [Phys. Fluids A 4, 306 (1992)], eliminated the pressure and solved for the wall–normal velocity and vorticity. The other velocity components were calculated using the definition of vorticity, and the continuity equation. This procedure leads to oscillations in the pressure field when solutions were attempted in the mapped computational domain. To overcome the problem, the procedure had to be modified and the pressure solved for directly using a fractional time step technique. For the cases examined here, these modifications resulted in spectral accuracy being maintained. Flow over sinusoidal wave trains has been simulated and the results compare well with available experiments. The simulations show significant effects of the wavy boundary on the mean flow and the turbulence statistics. The mean velocity profile differs substantially from the profile for the flat-wall case, particularly in the buffer region where the fluid is under the influence of both the wavy wall and the slip boundary. The velocity fluctuations in the streamwise direction decrease in the buffer region. This effect becomes more pronounced when the wave amplitude increases. Most of the redistribution of energy, from the streamwise direction to the spanwise and wall–normal directions, occurs in a thin layer close to the boundary, downstream of the wave troughs. The energy primarily redistributes into spanwise fluctuations. High shear stress regions form downstream of the wave troughs, and streaky structures and quasi-streamwise vortices are also seen to initiate in these regions. The length of the streaks, and the extent of the quasi-streamwise vortices, scale with wave length for the two cases investigated.

Numerical simulations of laminar flow over a 3D backward‐facing step

Abstract: A numerical investigation of laminar flow over a three-dimensional backward-facing step is presented with comparisons with detailed experimental data, available in the literature, serving to validate the numerical results. The continuity constraint method, implemented via a finite element weak statement, was employed to solve the unsteady three-dimensional Navier–Stokes equations for incompressible laminar isothermal flow. Two-dimensional numerical simulations of this step geometry underestimate the experimentally determined extent of the primary separation region for Reynolds numbers Re greater than 400. It has been postulated that this disagreement between physical and computational experiments is due to the onset of three-dimensional flow near Re ≈ 400. This paper presents a full three-dimensional simulation of the step geometry for 100⩽ Re⩽ 800 and correctly predicts the primary reattachment lengths, thus confirming the influence of three-dimensionality. Previous numerical studies have discussed possible instability modes which could induce a sudden onset of three-dimensional flow at certain critical Reynolds numbers. The current study explores the influence of the sidewall on the development of three-dimensional flow for Re greater than 400. Of particular interest is the characterization of three-dimensional vortices in the primary separation region immediately downstream of the step. The complex interaction of a wall jet, located at the step plane near the sidewall, with the mainstream flow reveals a mechanism for the increasing penetration (with increasing Reynolds number) of three-dimensional flow structures into a region of essentially two-dimensional flow near the midplane of the channel. The character and extent of the sidewall-induced flow are investigated for 100⩽Re⩽ 800. © 1997 John Wiley & Sons, Ltd.

Regularity of weak solutions of the compressible isentropic Navier-Stokes equations

Abstract: Regularity of weak solutions of the compressible isentropic Navier-Stokes equations is proven for small time in dimension N = 2 or 3 under periodic boundary conditions. In this paper, the initial density is not required to have a positive lower bound and the pressure law is assumed to satisfy a condition that reduces to τ > 1 when N = 2 and p(φ) = aφτ. Moreover,weak solutions in T2turn out to be smooth as long as the density remains bounded in L∞( T2).

Stochastic cascades and 3-dimensional Navier-Stokes equations

Abstract: In this article, we study the incompressible Navier–Stokes equations in ℝ3. The non linear integral equation satisfied by the Fourier transform of the Laplacian of the velocity field can be interpreted in terms of a branching process and a composition rule along the associated tree. We derive from this representation new classes where global existence and uniqueness can be proven.

A Self-Contained, Automated Methodology for Optimal Flow Control

Abstract: This paper describes a self-contained automated methodology for active flow control which couples the time-dependent Navier-Stokes system with an adjoint Navier-Stokes system and optimality conditions from which optimal states, i.e., unsteady flow fields and controls (e.g., actuators), may be determined. The problem of boundary-layer instability suppression through wave cancellation is used as the intital validation case to test the methodology. Here, the objective of control is to match the stress vector along a portion of the boundary to a steady base flow. Control is effected through the injection or suction of fluid through a single orifice on the boundary. The results demonstrate that instability suppression can be achieved without any a priori knowledge of the flow unsteadiness such as frequencies, instability type, etc. The present methodology has been extended to three dimensions and may potentially be applied to separation control, relaminarization, and turbulence control applications using one to many sensors and actuators.