Showing papers on "Navier–Stokes equations published in 1996"

Artificial boundaries and flux and pressure conditions for the incompressible navier–stokes equations

Abstract: Fluid dynamical problems are often conceptualized in unbounded domains. However, most methods of numerical simulation then require a truncation of the conceptual domain to a bounded one, thereby introducing artificial boundaries. Here we analyse out experience in choosing artificial boundary conditions implicitly through the choice of variational formulations. We deal particularly with a class of problems that involve the prescription of pressure drops and/or net flux conditions.

Two-layer approximate boundary conditions for large-eddy simulations

Abstract: A new method to model the effect of the solid boundaries on the rest of the flowfield in large-eddy simulations is proposed The filtered Navier-Stokes equations are solved up to the first computational point From there to the wall, a simplified set of equations is solved, and an estimate of the instantaneous wall shear stress required to impose boundary conditions is obtained Computations performed for the plane channel, square duct, and the rotating channel flow cases gave improved results compared with existing models The additional computing time required by the model is on the order of 10-15% of the overall computing time The mean flow quantities and low-order statistics, which are of primary interest in engineering calculations, are in very good agreement with the reference data available in the literature

Boundary conditions for the lattice Boltzmann method

Abstract: When the Lattice Boltzmann Method (LBM) is used for simulating continuum fluid flow, the discrete mass distribution must satisfy imposed constraints for density and momentum along the boundaries of the lattice. These constraints uniquely determine the three‐dimensional (3‐D) mass distribution for boundary nodes only when the number of external (inward‐pointing) lattice links does not exceed four. We propose supplementary rules for computing the boundary distribution where the number of external links does exceed four, which is the case for all except simple rectangular lattices. Results obtained with 3‐D body‐centered‐cubic lattices are presented for Poiseuille flow, porous‐plate Couette flow, pipe flow, and rectangular duct flow. The accuracy of the two‐dimensional (2‐D) Poiseuille and Couette flows persists even when the mean free path between collisions is large, but that of the 3‐D duct flow deteriorates markedly when the mean free path exceeds the lattice spacing. Accuracy in general decreases with Knudsen number and Mach number, and the product of these two quantities is a useful index for the applicability of LBM to 3‐D low‐Reynolds‐number flow.

Variational bounds on energy dissipation in incompressible flows. III. Convection

Abstract: Building on a method of analysis for the Navier-Stokes equations introduced by Hopf [Math. Ann. 117, 764 (1941)], a variational principle for upper bounds on the largest possible time averaged convective heat flux is derived from the Boussinesq equations of motion. When supplied with appropriate test background fields satisfying a spectral constraint, reminiscent of an energy stability condition, the variational formulation produces rigorous upper bounds on the Nusselt number (Nu) as a function of the Rayleigh number (Ra). For the case of vertical heat convection between parallel plates in the absence of sidewalls, a simplified (but rigorous) formulation of the optimization problem yields the large Rayleigh number bound Nu\ensuremath{\le}0.167 ${\mathrm{Ra}}^{1/2}$-1. Nonlinear Euler-Lagrange equations for the optimal background fields are also derived, which allow us to make contact with the upper bound theory of Howard [J. Fluid Mech. 17, 405 (1963)] for statistically stationary flows. The structure of solutions of the Euler-Lagrange equations are elucidated from the geometry of the variational constraints, which sheds light on Busse's [J. Fluid Mech. 37, 457 (1969)] asymptotic analysis of general solutions to Howard's Euler-Lagrange equations. The results of our analysis are discussed in the context of theory, recent experiments, and direct numerical simulations. \textcopyright{} 1996 The American Physical Society.

Numerical simulation of particle interactions with wall turbulence

Abstract: This paper presents the results of a numerical investigation of the effects of near‐neutral density solid particles on turbulent liquid flow in a channel. Interactions of particles, in a size range about the dissipative length scale, with wall turbulence have been simulated at low volume fractions (average volume fraction less than 4×10−4). Fluid motion is calculated by directly solving the Navier‐Stokes equations by a pseudo‐spectral method and resolving all scales of motion. Particles are moved in a Lagrangian frame through the action of forces imposed by the fluid and gravity. Particle effects on fluid motion are fed back at each time step by calculating the velocity disturbance caused by the particles assuming the flow around them is locally Stokesian. Particle‐particle interactions are not considered. The slightly heavier‐than‐fluid particles of the size range considered are found to preferentially accumulate in the low‐speed streaks, as reported in several other investigations. It is also found that particles smaller than the dissipative length scale reduce turbulence intensities and Reynolds stress, whereas particles that are somewhat larger increase intensities and stress. By examining higher order turbulence statistics and doing a quadrant analysis of the Reynolds stress, it is found that the ejection‐sweep cycle is affected—primarily through suppression of sweeps by the smaller particles and enhancement of sweep activity by the larger particles. A preliminary assessment of the impact of these findings on scalar transfer is made, as enhancement of transfer rate is a motivation of the overall work on this subject. For the case investigated, comparison of the calculations with an existing experiment was possible, and shows good agreement.

Global attractors for the three-dimensional Navier-Stokes equations

Abstract: In this paper we show that the weak solutions of the Navier-Stokes equations on any bounded, smooth three-dimensional domain have a global attractor for any positive value of the viscosity. The proof of this result, which bypasses the two issues of the possible nonuniqueness of the weak solutions and the possible lack of global regularity of the strong solutions, is based on a new point of view for the construction of the semiflow generated by these equations. We also show that, under added assumptions, this global attractor consists entirely of strong solutions.

A Numerical Method for the Incompressible Navier--Stokes Equations Based on an Approximate Projection

Abstract: In this method we present a fractional step discretization of the time-dependent incompressible Navier--Stokes equations. The method is based on a projection formulation in which we first solve diffusion--convection equations to predict intermediate velocities, which are then projected onto the space of divergence-free vector fields. Our treatment of the diffusion--convection step uses a specialized second-order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high Reynolds number. In contrast to conventional projection-type discretizations that impose a discrete form of the divergence-free constraint, we only approximately impose the constraint; i.e., the velocity field we compute is not exactly divergence-free. The approximate projection is computed using a conventional discretization of the Laplacian and the resulting linear system is solved using conventional multigrid methods. Numerical examples are presented to validate the second-order convergence of the method for Euler, finite Reynolds number, and Stokes flow. A second example illustrating the behavior of the algorithm on an unstable shear layer is also presented.

An approximate projection scheme for incompressible flow using spectral elements

Abstract: SUMMARY An approximate projection scheme based on the pressure correction method is proposed to solve the NavierStokes equations for incompressible flow. The algorithm is applied to the continuous equations; however, there are no problems concerning the choice of boundary conditions of the pressure step. The resulting velocity and pressure are consistent with the original system. For the spatial discretization a high-order spectral element method is chosen. The high-order accuracy allows the use of a diagonal mass matrix, resulting in a very efficient algorithm. The properties of the scheme are extensively tested by means of an analytical test example. The scheme is Mer validated by simulating the laminar flow over a backward-facing step. The solution of the Navier-Stokes equations for unsteady incompressible fluid flow is still a major challenge in the field of computational fluid dynamics. An overview of the most important aspects with respect to the solution of the incompressible Navier-Stokes equations can be found in References 1-5. The Navier-Stokes equations form a set of coupled equations for both velocity and pressure (or, better, the gradient of the pressure). One of the main problems related to the numerical solution of these equations is the imposition of the incompressibility constraint and consequently the calculation of the pressure. The pressure is not a thermodynamic variable, as there is no equation of state for an incompressible fluid. It is an implicit variable which instantaneously 'adjusts itself' in such a way that the velocity remains divergence-free. The gradient of the pressure, on the other hand, is a relevant physical quantity: a force per unit volume. The mathematical importance of the pressure in an incompressible flow lies in the theory of saddle-point problems (of which the steady Stokes equations are an example), where it acts as a Lagrangian multiplier that constrains the velocity to remain divergence-free. There are numerous approaches to solve the Navier-Stokes equations. For the solution of unsteady Navier-Stokes flow perhaps one of the most successful approaches to date is provided by the class of projection methods.&' Projection methods have been developed as a useful way of obtaining an efficient solution algorithm for unsteady incompressible flow. In this paper, projection methods are considered that are applied to the set of continuous equations, yielding very efficient and simple-toimplement algorithms. By decoupling the treatment of velocity and pressure terms, a set of easier-tosolve equations arises: a convectiondiffusion problem for the velocity, yielding an intermediate velocity which is not divergence-free; and a Poisson equation for the pressure (or a related quantity).

On error estimates of the projection methods for the Navier-Stokes equations: second-order schemes

Abstract: We present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible Navier-Stokes equations. The error analysis is accomplished by interpreting the respective projection schemes as second-order time discretizations of a perturbed system which approximates the Navier-Stokes equations. Numerical results in agreement with the error analysis are also presented.

Robustness of coupled Algebraic Multigrid for the Navier-Stokes equations

Abstract: A new implementation of the Algebraic Multigrid method is presented. It is applied to the coupled linearized discrete equations arising from an implicit pressure-based finite volume discretization of the 3D Navier-Stokes equations on structured or unstructured meshes. It employs a grid-coarsening algorithm based on an evaluation of relative coefficient strengths. The restriction and prolongation operators are those implied by the Additive Correction Multigrid method. The relaxation scheme is a coupled Incomplete Lower Upper factorization, and the W-Cycle is used. The results of the coarsening algorithm applied to two 2D model problems is visualized and discussed. Verification of performance is made on several very different CFD applications relevant to the aerospace field. These include incompressible to supersonic, external and internal, stationary and rotating, reacting and nonreacting flows. In all cases the exact same multigrid method is used. Thus the feasibility of using Algebraic Multigrid as the linear equation solver for an implicit CFD method that must handle diverse applications is shown. The limitations and weaknesses of the method are also described. (Author)

Coarse‐grained description of thermo‐capillary flow

Abstract: A mesoscopic or coarse‐grained approach is presented to study thermo‐capillary induced flows. An order parameter representation of a two‐phase binary fluid is used in which the interfacial region separating the phases naturally occupies a transition zone of small width. The order parameter satisfies the Cahn–Hilliard equation with advective transport. A modified Navier–Stokes equation that incorporates an explicit coupling to the order parameter field governs fluid flow. It reduces, in the limit of an infinitely thin interface, to the Navier–Stokes equation within the bulk phases and to two interfacial forces: a normal capillary force proportional to the surface tension and the mean curvature of the surface, and a tangential force proportional to the tangential derivative of the surface tension. The method is illustrated in two cases: thermo‐capillary migration of drops and phase separation via spinodal decomposition, both in an externally imposed temperature gradient.

On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions

Abstract: For boundary or distributed controls, we get an approximate controllability result for the Navier-Stokes equations in dimension 2 in the case where the fluid is incompressible and slips on the boundary in agreement with the Navier slip boundary conditions.

Data-parallel lower-upper relaxation method for the navier-stokes equations

Abstract: The lower-upper symmetric Gauss-Seidel method is modified for the simulation of viscous flows on massively parallel computers. The resulting diagonal data-parallel lower-upper relaxation (DP-LUR) method is shown to have good convergence properties on many problems. However, the convergence rate decreases on the high cell aspect ratio grids required to simulate high Reynolds number flows. Therefore, the diagonal approximation is relaxed, and a full matrix version of the DP-LUR method is derived. The full matrix method retains the data-parallel properties of the original and reduces the sensitivity of the convergence rate to the aspect ratio of the computational grid. Both methods are implemented on the Thinking Machines CM-5, and a large fraction of the peak theoretical performance of the machine is obtained. The low memory use and high parallel efficiency of the methods make them attractive for large-scale simulation of viscous flows.

Self-similar solutions for navier-stokes equations in

Abstract: We construct self-similar solutions for three-dimensional incompressible Navier-Stokes equations, providing some examples of functional spaces where this can be done. We apply our results to a Part...

On the decay of higher-order norms of the solutions of Navier–Stokes equations

Abstract: We show that an energy decay ∥u(t)∥2 = O(t−µ) for solutions of the Navier–Stokes equations on ℝn, n ≦ 5, implies a decay of the higher order norms, e.g. ∥Dα u(t)∥2 = O(t−µ −|α|/2) and ∥u(t)|∞ = O(t−µ −n/4).

Modelling of bypass transition with conditioned Navier-Stokes equations coupled to an intermittency transport equation

Abstract: A differential method is proposed to simulate bypass transition. The intermittency in the transition zone is taken into account by conditioned averages. These are averages taken during the fraction of time the flow is turbulent or laminar respectively. Starting from the Navier-Stokes equations, conditioned continuity, momentum and energy equations are derived for the larninar and turbulent parts of the intermittent flow. The turbulence is described by a classical k-e model. The supplementary parameter, the intermittency factor, is determined by a transport equation applicable for zero, favourable and adverse pressure gradients. Results for these pressure gradients are given.

Solution-Adaptive Cartesian Cell Approach for Viscous and Inviscid Flows

Abstract: A Cartesian cell-based approach for adaptively refined solutions of the Euler and Navier-Stokes equations in two dimensions is presented. Grids about geometrically complicated bodies are generated automatically, by the recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, polygonal cut cells are created using modified polygon-clipping algorithms. The grid is stored in a binary tree data structure that provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite volume formulation. The convective terms are upwinded: A linear reconstruction of the primitive variables is performed, providing input states to an approximate Riemann solver for computing the fluxes between neighboring cells. The results of a study comparing the accuracy and positivity of two classes of cell-centered, viscous gradient reconstruction procedures is briefly summarized. Adaptively refined solutions of the Navier-Stokes equations are shown using the more robust of these gradient reconstruction procedures, where the results computed by the Cartesian approach are compared to theory, experiment, and other accepted computational results for a series of low and moderate Reynolds number flows.

A stable penalty method for the compressible navier-stokes equations. i. open boundary conditions

Abstract: The purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible Navier--Stokes equations in three spatial dimensions The treatment uses the conservation form of the Navier--Stokes equations and utilizes linearization and localization at the boundaries based on these variables The proposed boundary conditions are applied through a penalty procedure, thus ensuring correct behavior of the scheme as the Reynolds number tends to infinity The versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder

New efficient boundary conditions for incompressible Navier-Stokes equations : a well-posedness result

Abstract: Efficient natural condition s on open boundaries for incompressible flows are derived from a weak formulation of Navier-Stokes equations. Energy estimates in velocity-pressure are established from a mixed formulation and a rigourous proof of existence of solutions is given. As an illustration the conditions are written down for the flow behind an obstacle in a channel. Moreover numerical tests have shown the accuracy and robustness of such conditions.

Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in $\mathbb {R}^3$

Abstract: We construct global strong solutions of the Navier-Stokes equations with sufficiently oscillating initial data. We will show that the condition is for the norm in some Besov space to be small enough.

a Comparative Study of Time-Stepping Techniques for the Incompressible Navier-Stokes Equations: from Fully Implicit Non-Linear Schemes to Semi-Implicit Projection Methods

Abstract: We present a numerical comparison of some time-stepping schemes for the discretization and solution of the non-stationary incompressible Navier-Stokes equations. The spatial discretization is by non-conforming quadrilateral finite elements which satisfy the LBB condition. The major focus is on the differences in accuracy and efficiency between the backward Euler, Crank-Nicolson and fractional-step θ schemes used in discretizing the momentum equations. Further, the differences between fully coupled solvers and operator-splitting techniques (projection methods) and the influence of the treatment of the nonlinear advection term are considered. The combination of both discrete projection schemes and non-conforming finite elements allows the comparison of schemes which are representative for many methods used in practice. On Cartesian grids this approach encompasses some well-known staggered grid finite difference discretizations too. The results which are obtained for several typical flow problems are thought to be representative and should be helpful for a fair rating of solution schemes, particularly in long-time simulations.

Pinching threads, singularities and the number 0.0304...

Abstract: The dynamics of capillary pinching of a fluid thread are described by similarity solutions of the Navier–Stokes equations. Eggers [Phys. Rev. Lett. 71, 3458 (1993)] recently proposed a single universal similarity solution for a viscous thread pinching with an inertial–viscous–capillary balance in an inviscid environment. In this paper it is shown that there is actually a countably infinite family of such similarity solutions which are each an asymptotic solution to the Navier–Stokes equations. The solutions all have axial scale t′1/2 and radial scale t′, where t′ is the time to pinching. The solution obtained by Eggers appears to be special in that it is selected by the dynamics for most initial conditions by virtue of being less susceptible to finite‐amplitude instabilities. The analogous problem of a thread pinching in the absence of inertia is also investigated and it is shown that there is a countably infinite family of similarity solutions with axial scale t′β and radial scale t′, where each solution...