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The incompressible Navier–Stokes equation is a differential algebraic equation, having the inconvenient feature that there is no explicit mechanism for advancing the pressure in time. Consequently, much effort has been expended to eliminate the pressure from all or part of the computational process.
The Reynolds-averaged Navier–Stokes equations (RANS equations) are time-averaged [a] equations of motion for fluid flow. The idea behind the equations is Reynolds decomposition , whereby an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities, an idea first proposed by Osborne Reynolds . [ 1 ]
In mathematics, the Navier–Stokes equations are a system of nonlinear partial differential equations for abstract vector fields of any size. In physics and engineering, they are a system of equations that model the motion of liquids or non-rarefied gases (in which the mean free path is short enough so that it can be thought of as a continuum mean instead of a collection of particles) using ...
The streamline upwind Petrov–Galerkin pressure-stabilizing Petrov–Galerkin formulation for incompressible Navier–Stokes equations can be used for finite element computations of high Reynolds number incompressible flow using equal order of finite element space (i.e. ) by introducing additional stabilization terms in the Navier–Stokes Galerkin formulation.
The small time behavior of the flow is then found through simplification of the incompressible Navier–Stokes equations using the initial flow to give a step-by-step solution as time progresses. An exact solution in two spatial dimensions is known, and is presented below. Animation of a Taylor-Green Vortex using colour coded Lagrangian tracers
This is an exact solution to the incompressible form of the Navier-Stokes equations, which was first discovered by Lev Landau in 1944 [1] [2] and later by Herbert Squire in 1951. [3] The self-similar equation was in fact first derived by N. A. Slezkin in 1934, [ 4 ] but never applied to the jet.
In fluid dynamics, Rayleigh problem also known as Stokes first problem is a problem of determining the flow created by a sudden movement of an infinitely long plate from rest, named after Lord Rayleigh and Sir George Stokes. This is considered as one of the simplest unsteady problems that have an exact solution for the Navier-Stokes equations.
The equation of motion for Stokes flow can be obtained by linearizing the steady state Navier–Stokes equations.The inertial forces are assumed to be negligible in comparison to the viscous forces, and eliminating the inertial terms of the momentum balance in the Navier–Stokes equations reduces it to the momentum balance in the Stokes equations: [1]