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Concept

Reynolds-averaged Navier–Stokes equations

The Reynolds-averaged Navier–Stokes equations (RANS equations) are time-averaged 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. The RANS equations are primarily used to describe turbulent flows. These equations can be used with approximations based on knowledge of the properties of flow turbulence to give approximate time-averaged solutions to the Navier–Stokes equations. For a stationary flow of an incompressible Newtonian fluid, these equations can be written in Einstein notation in Cartesian coordinates as: \rho\bar{u}_j \frac{\partial \bar{u}_i }{\partial x_j} = \rho \bar{f}_i

\frac{\partial}{\partial x_j} \left[ - \bar{p}\delta_{ij}
\mu \left( \frac{\partial \bar{u}_i}{\partial x_j} + \frac{\partial \bar{u}_j}{\partial x_i} \right)

\rho \overline{u_i^\prime u_j^\prim

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Summary

\frac{\partial}{\partial x_j} \left[ - \bar{p}\delta_{ij}
\mu \left( \frac{\partial \bar{u}_i}{\partial x_j} + \frac{\partial \bar{u}_j}{\partial x_i} \right)

\rho \overline{u_i^\prime u_j^\prim

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About this result

Related publications (8)

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Related concepts (8)

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Numerical simulation and flow analysis of an elbow diffuser

Sebastiano Mauri

Numerical simulation of the unsteady turbulent flow in a three-dimensional elbow diffuser is performed. The investigation is carried out with a commercial finite volume solver implementing the Reynolds averaged Navier-Stokes equations. Against the background of current research in DNS and LES, the modeling of most practically relevant turbulent flows continues to be based on this system of equations. For this reason it is important to evaluate the limitations of the Reynolds averaging approach with the associated turbulence modeling, in particular for the prediction of time-dependent flows. Verification and validation are presented; detailed measurements are compared with computations. While a great deal of research has focused on draft tube design, relatively little is known about the complex flow features present. The flow is analyzed over a wide range of operating conditions including part load. Topological changes in the flow patterns with the global characteristics of the diffuser are presented. Visualization provides extra insight into the complex flow. Forced and self-sustained time-dependent flow phenomena are captured. Falling into these categories are flow field fluctuations introduced by the runner, self-sustained vortex shedding phenomena, and the typical rotating helical vortex observed at part load. Additionally, the linear stability of measured inlet profiles is investigated, providing a fuller understanding of the basic instability mechanism.

EPFL2002

Numerical approximation of the unsteady Navier-Stokes equations with an application to the flow past a square cylinder

Francesco Casalegno

The goal of this project is to numerically solve the Navier-Stokes equations by using different numerical methods with particular emphasis on solving the problem of the flow past a square cylinder. In particular, we use the finite element method based on P2/P1 elements for the velocity and pressure fields for the spatial approximation, while the backward Euler method (with semi-implicit treatment of the nonlinear term) for the time discretization. We firstly test the numerical schemes on a benchmark problem with known exact solution. Then, we discuss in detail the advantages, in terms of computational costs, in using the algebraic Chorin-Temam method with additional implementation improvements. We finally investigate the problem of the two-dimensional flow past a square cylinder, focusing our attention on the range 0.1-300 for the Reynolds number (Re). We describe the two different regimes associated to the steady and the unsteady flows and we remark as the latter is in fact due to a Hopf bifurcation of the system. We also discuss the relation between the Strouhal and Reynolds numbers concluding that the Strouhal number attains its maximum value in the range 169-170 for the Reynolds number. In particular, a cubic model is proposed, showing very good matching with observed data and a better fitting than other models available in literature.

2013

Parallel preconditioners for the unsteady Navier-Stokes equations and applications to hemodynamics simulations

Simone Deparis, Gwenol Grandperrin, Alfio Quarteroni

We are interested in the numerical solution of the unsteady Navier-Stokes equations on large scale parallel architectures. We consider efficient preconditioners, such as the Pressure Convection-Diffusion (PCD), the Yosida preconditioner, the SIMPLE preconditioner, and the algebraic additive Schwarz preconditioner, for the linear systems arising from finite element discretizations using tetrahedral unstructured meshes and time advancing finite difference schemes. To achieve parallel efficiency, we introduce approximate versions of these preconditioners, based on their factorizations where each factor can be either inverted exactly or using an add-hoc preconditioner. We investigate their strong scalability for both classical benchmark problems and simulations relevant to hemodynamics, using up to 8192 cores. (C) 2013 Elsevier Ltd. All rights reserved.

Elsevier2014

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