Navier–Stokes equations

About: Navier–Stokes equations is a research topic. Over the lifetime, 18180 publications have been published within this topic receiving 552555 citations. The topic is also known as: Navier-Stokes equations.

Chapter 3 - Harmonic Analysis Tools for Solving the Incompressible Navier–Stokes Equations

Abstract: Introduction Section 1: Preliminaries 1.1 The Navier-Stokes equations 1.2 Classical, mild and weak solutions 1.3 Navier meets Fourier Section 2: Functional setting of the equations 2.1 The Littlewood-Paley decomposition 2.2 The Besov spaces 2.3 The paraproduct rule 2.4 The wavelet decomposition 2.5 Other useful function spaces Section 3: Existence theorems 3.1 The fixed point theorem 3.2 Scaling invariance 3.3 Super-critical case

Transition in shear flows. Nonlinear normality versus non‐normal linearity

Abstract: A critique is presented of recent works promoting the concept of non‐normal operators and transient growth as the key to understanding transition to turbulence in shear flows. The focus is in particular on a simple model [Baggett et al., Phys. Fluids 7, 883 (1995)] illustrating that view. It is argued that the question of transition is really a question of existence and basin of attraction of nonlinear self‐sustaining solutions that have little contact with the non‐normal linear problem. An alternative nonlinear point of view [Hamilton et al., J. Fluid Mech. 287, 317 (1995)] that seeks to isolate a self‐sustaining nonlinear process, and the critical Reynolds number below which it ceases to exist, is discussed and illustrated by a simple model. Connections with the Navier–Stokes equations and observations are highlighted throughout.

Transition stages of Rayleigh{Taylor instability between miscible fluids

Abstract: Direct numerical simulations (DNS) are presented of three-dimensional, Rayleigh–Taylor instability (RTI) between two incompressible, miscible fluids, with a 3:1 density ratio. Periodic boundary conditions are imposed in the horizontal directions of a rectangular domain, with no-slip top and bottom walls. Solutions are obtained for the Navier–Stokes equations, augmented by a species transport-diffusion equation, with various initial perturbations. The DNS achieved outer-scale Reynolds numbers, based on mixing-zone height and its rate of growth, in excess of 3000. Initial growth is diffusive and independent of the initial perturbations. The onset of nonlinear growth is not predicted by available linear-stability theory. Following the diffusive-growth stage, growth rates are found to depend on the initial perturbations, up to the end of the simulations. Mixing is found to be even more sensitive to initial conditions than growth rates. Taylor microscales and Reynolds numbers are anisotropic throughout the simulations. Improved collapse of many statistics is achieved if the height of the mixing zone, rather than time, is used as the scaling or progress variable. Mixing has dynamical consequences for this flow, since it is driven by the action of the imposed acceleration field on local density differences.

Finite element analysis of the compressible Euler and Navier-Stokes equations

Abstract: A space-time finite element method is presented for solving the compressible Euler and Navier-Stokes equations. The proposed formulation includes the variational equation, predictor multi-corrector algorithms, boundary conditions, and solution strategies. The variational equation is based on the time-discontinuous Galerkin method, in which the physical entropy variables are employed. A least-squares operator and a discontinuity-capturing operator are added, resulting in a high-order accurate and unconditionally stable method. Implicit/explicit predictor multi-corrector algorithms, applicable to steady as well as unsteady problems, are presented; techniques are developed to enhance their efficiency. Implementation of boundary conditions is addressed; in particular, a technique is introduced to satisfy nonlinear essential boundary conditions, and a consistent method is presented to calculate boundary fluxes. A multi-element group, domain decomposition algorithm is presented for solving the nonsymmetric linear systems. This algorithm employs an iterative strategy based on the generalized minimal residual (GMRES) procedure. A two-level preconditioning technique is presented, which significantly accelerates the convergence of the GMRES procedure. Numerical results are presented to demonstrate the performance of the method.