Showing papers on "Navier–Stokes equations published in 2006"

Kinetic theory representation of hydrodynamics: a way beyond the Navier–Stokes equation

Abstract: We present in detail a theoretical framework for representing hydrodynamic systems through a systematic discretization of the Boltzmann kinetic equation. The work is an extension of a previously proposed formulation. Conventional lattice Boltzmann models can be shown to be directly derivable from this systematic approach. Furthermore, we provide here a clear and rigorous procedure for obtaining higher-order approximations to the continuum Boltzmann equation. The resulting macroscopic moment equations at each level of the systematic discretization give rise to the Navier–Stokes hydrodynamics and those beyond. In addition, theoretical indications to the order of accuracy requirements are given for each discrete approximation, for thermohydrodynamic systems, and for fluid systems involving long-range interactions. All these are important for complex and micro-scale flows and are missing in the conventional Navier–Stokes order descriptions. The resulting discrete Boltzmann models are based on a kinetic representation of the fluid dynamics, hence the drawbacks in conventional higher-order hydrodynamic formulations can be avoided.

Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing

Abstract: The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in L 2 0 (T 2 ). Unlike previous works, this class is independent of the viscosity and the strength of the noise. The two main tools of our analysis are the asymptotic strong Feller property, introduced in this work, and an approximate integration by parts formula. The first, when combined with a weak type of irreducibility, is shown to ensure that the dynamics is ergodic. The second is used to show that the first holds un

Partially-Averaged Navier-Stokes Model for Turbulence: A Reynolds-Averaged Navier-Stokes to Direct Numerical Simulation Bridging Method

Abstract: A turbulence bridging method purported for any filter-width or scale resolution-fully averaged to completely resolved-is developed. The method is given the name partially averaged Navier-Stokes (PANS) method. In PANS, the model filter width (extent of partial averaging) is controlled through two parameters: the unresolved-to-total ratios of kinetic energy (f k ) and dissipation (f e ). The PANS closure model is derived formally from the Reynolds-averaged Navier-Stokes (RANS) model equations by addressing the following question: if RANS represents the closure for fully averaged statistics, what is the corresponding closure for partially averaged statistics? The PANS equations vary smoothly from RANS equations to Navier-Stokes (direct numerical simulation) equations, depending on the values of the filter-width control parameters. Preliminary results are very encouraging.

Steady solutions of the Navier-Stokes equations by selective frequency damping

Abstract: A new method, enabling the computation of steady solutions of the Navier-Stokes equations in globally unstable configurations, is presented. We show that it is possible to reach a steady state by damping the unstable (temporal) frequencies. This is achieved by adding a dissipative relaxation term proportional to the high-frequency content of the velocity fluctuations. Results are presented for cavity-driven boundary-layer separation and a separation bubble induced by an external pressure gradient.

The Navier-Stokes equations : a classification of flows and exact solutions

Abstract: The Navier-Stokes equations were firmly established in the 19th Century as the system of nonlinear partial differential equations which describe the motion of most commonly occurring fluids in air and water, and since that time exact solutions have been sought by scientists. Collectively these solutions allow a clear insight into the behavior of fluids, providing a vehicle for novel mathematical methods and a useful check for computations in fluid dynamics, a field in which theoretical research is now dominated by computational methods. This 2006 book draws together exact solutions from widely differing sources and presents them in a coherent manner, in part by classifying solutions via their temporal and geometric constraints. It will prove to be a valuable resource to all who have an interest in the subject of fluid mechanics, and in particular to those who are learning or teaching the subject at the senior undergraduate and graduate levels.

The Equations of Magnetohydrodynamics: On the Interaction Between Matter and Radiation in the Evolution of Gaseous Stars

Abstract: We prove existence of global-in-time weak solutions to the equations of magnetohydrodynamics, specifically, the Navier-Stokes-Fourier system describing the evolution of a compressible, viscous, and heat conducting fluid coupled with the Maxwell equations governing the behaviour of the magnetic field. The result applies to any finite energy data posed on a bounded spatial domain in R3, supplemented with conservative boundary conditions.

Low Mach Number Limit of the Full Navier-Stokes Equations

Abstract: The low Mach number limit for classical solutions of the full Navier-Stokes equations is here studied. The combined effects of large temperature variations and thermal conduction are taken into account. In particular, we consider general initial data. The equations lead to a singular problem, depending on a small scaling parameter, whose linearized system is not uniformly well-posed. Yet, it is proved that solutions exist and they are uniformly bounded for a time interval which is independent of the Mach number Ma ∈ (0,1], the Reynolds number Re ∈ [1,+∞] and the Peclet number Pe ∈ [1,+∞]. Based on uniform estimates in Sobolev spaces, and using a theorem of G. Metivier & S. Schochet [30], we next prove that the penalized terms converge strongly to zero. This allows us to rigorously justify, at least in the whole space case, the well-known computations given in the introduction of P.-L. Lions' book [26].

Fully Discrete Finite Element Approximations of the Navier--Stokes--Cahn-Hilliard Diffuse Interface Model for Two-Phase Fluid Flows

Abstract: This paper develops and analyzes some fully discrete finite element methods for a parabolic system consisting of the Navier--Stokes equations and the Cahn--Hilliard equation, which arises as a diffuse interface model for the flow of two immiscible and incompressible fluids. In the model the two sets of equations are coupled through an extra phase induced stress term in the Navier--Stokes equations and a fluid induced transport term in the Cahn--Hilliard equation. Fully discrete mixed finite element methods are proposed for approximating the coupled system, it is shown that the proposed numerical methods satisfy a mass conservation law, and a discrete energy law which is analogous to the basic energy law for the phase field model. The convergence of the numerical solutions to the solutions of the phase field model and its sharp interface limit is established by utilizing the discrete energy law. As a by-product, the convergence result also provides a constructive proof of the existence of weak solutions to the Navier--Stokes-Cahn--Hilliard phase field model. Numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach.

Lower Bounds for an Integral Involving Fractional Laplacians and the Generalized Navier-Stokes Equations in Besov Spaces

Abstract: When estimating solutions of dissipative partial differential equations in L p -related spaces, we often need lower bounds for an integral involving the dissipative term. If the dissipative term is given by the usual Laplacian −Δ, lower bounds can be derived through integration by parts and embedding inequalities. However, when the Laplacian is replaced by the fractional Laplacian (−Δ) α , the approach of integration by parts no longer applies. In this paper, we obtain lower bounds for the integral involving (−Δ) α by combining pointwise inequalities for (−Δ) α with Bernstein's inequalities for fractional derivatives. As an application of these lower bounds, we establish the existence and uniqueness of solutions to the generalized Navier-Stokes equations in Besov spaces. The generalized Navier-Stokes equations are the equations resulting from replacing −Δ in the Navier-Stokes equations by (−Δ) α .

The Interaction between Quasilinear Elastodynamics and the Navier-Stokes Equations

Abstract: The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations. Unlike our approach in [5] for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial viscosity term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in strong norms of our sequence of regularized problems.

Euler Equations, Navier-Stokes Equations and Turbulence

Abstract: In 2004 the mathematical world will mark 120 years since the advent of turbulence theory ([80]). In his 1884 paper Reynolds introduced the decomposition of turbulent flow into mean and fluctuation and derived the equations that describe the interaction between them. The Reynolds equations are still a riddle. They are based on the Navier-Stokes equations, which are a still a mystery. The Navier-Stokes equations are a viscous regularization of the Euler equations, which are still an enigma. Turbulence is a riddle wrapped in a mystery inside an enigma ([11]).

High‐order stable interpolations for immersed boundary methods

Abstract: The analysis and improvement of an immersed boundary method (IBM) for simulating turbulent flows over complex geometries are presented. Direct forcing is employed. It consists in interpolating boundary conditions from the solid body to the Cartesian mesh on which the computation is performed. Lagrange and least squares high-order interpolations are considered. The direct forcing IBM is implemented in an incompressible finite volume Navier–Stokes solver for direct numerical simulations (DNS) and large eddy simulations (LES) on staggered grids. An algorithm to identify the body and construct the interpolation schemes for arbitrarily complex geometries consisting of triangular elements is presented. A matrix stability analysis of both interpolation schemes demonstrates the superiority of least squares interpolation over Lagrange interpolation in terms of stability. Preservation of time and space accuracy of the original solver is proven with the laminar two-dimensional Taylor–Couette flow. Finally, practicability of the method for simulating complex flows is demonstrated with the computation of the fully turbulent three-dimensional flow in an air-conditioning exhaust pipe. Copyright © 2006 John Wiley & Sons, Ltd.

One component regularity for the Navier–Stokes equations

Abstract: We establish sufficient conditions for the regularity of solutions of the Navier–Stokes system based on conditions on one component of the velocity. The first result states that if , where and 54/23 ≤ r ≤ 18/5, then the solution is regular. The second result is that if , where and 24/5 ≤ r ≤ ∞, then the solution is regular. These statements improve earlier results on one component regularity.

Capturing Knudsen layer phenomena using a lattice Boltzmann model.

Abstract: In recent years, lattice Boltzmann methods have been increasingly used to simulate rarefied gas flows in microscale and nanoscale devices This is partly due to the fact that the method is computationally efficient, particularly when compared to solution techniques such as the direct simulation Monte Carlo approach However, lattice Boltzmann models developed for rarefied gas flows have difficulty in capturing the nonlinear relationship between the shear stress and strain rate within the Knudsen layer As a consequence, these models are equivalent to slip-flow solutions of the Navier-Stokes equations In this paper, we propose an effective mean-free path to address the Knudsen layer effect, so that the capabilities of lattice Boltzmann methods can be extended beyond the slip-flow regime The model has been applied to rarefied shear-driven and pressure-driven flows between parallel plates at Knudsen numbers between 001 and 1 Our results show that the proposed approach significantly improves the near-wall accuracy of the lattice Boltzmann method and provides a computationally economic solution technique over a wide range of Knudsen numbers