scholarly journals Solutions of the Navier–Stokes Equation at Large Reynolds Number

1975 ◽  
Vol 28 (1) ◽  
pp. 202-214 ◽  
Author(s):  
Paco A. Lagerstrom
Keyword(s):  
Reynolds Number ◽  
Stokes Equation ◽  
Navier Stokes ◽  
Large Reynolds Number ◽  
Navier Stokes Equation

A note on the solution of the Navier-Stokes equations for a spherically symmetric expansion into a very low pressure

1973 ◽  
Vol 59 (2) ◽  
pp. 391-396 ◽  
Author(s):  
N. C. Freeman ◽  
S. Kumar
Keyword(s):  
Shock Wave ◽  
Reynolds Number ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Low Pressure ◽  
Navier Stokes ◽  
Spherically Symmetric ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Type Flow

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

scholarly journals Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation

2013 ◽  
Vol 729 ◽  
pp. 285-308 ◽  
Author(s):  
Maciej J. Balajewicz ◽  
Earl H. Dowell ◽  
Bernd R. Noack
Keyword(s):  
Reynolds Number ◽  
Galerkin Method ◽  
Stokes Equation ◽  
High Reynolds Number ◽  
Transient Flow ◽  
Navier Stokes ◽  
Power Balance ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
High Reynolds

AbstractWe generalize the POD-based Galerkin method for post-transient flow data by incorporating Navier–Stokes equation constraints. In this method, the derived Galerkin expansion minimizes the residual like POD, but with the power balance equation for the resolved turbulent kinetic energy as an additional optimization constraint. Thus, the projection of the Navier–Stokes equation on to the expansion modes yields a Galerkin system that respects the power balance on the attractor. The resulting dynamical system requires no stabilizing eddy-viscosity term – contrary to other POD models of high-Reynolds-number flows. The proposed Galerkin method is illustrated with two test cases: two-dimensional flow inside a square lid-driven cavity and a two-dimensional mixing layer. Generalizations for more Navier–Stokes constraints, e.g. Reynolds equations, can be achieved in straightforward variation of the presented results.

scholarly journals Exact two-dimensionalization of rapidly rotating large-Reynolds-number flows

2015 ◽  
Vol 783 ◽  
pp. 412-447 ◽  
Author(s):  
Basile Gallet
Keyword(s):  
Reynolds Number ◽  
Stokes Equation ◽  
Three Dimensional ◽  
Threshold Value ◽  
Navier Stokes ◽  
Rossby Number ◽  
Navier Stokes Equation ◽  
Rotating Turbulence ◽  
2D Flow ◽  
Long Time

We consider the flow of a Newtonian fluid in a three-dimensional domain, rotating about a vertical axis and driven by a vertically invariant horizontal body force. This system admits vertically invariant solutions that satisfy the 2D Navier–Stokes equation. At high Reynolds number and without global rotation, such solutions are usually unstable to three-dimensional perturbations. By contrast, for strong enough global rotation, we prove rigorously that the 2D (and possibly turbulent) solutions are stable to vertically dependent perturbations. We first consider the 3D rotating Navier–Stokes equation linearized around a statistically steady 2D flow solution. We show that this base flow is linearly stable to vertically dependent perturbations when the global rotation is fast enough: under a Reynolds-number-dependent threshold value$Ro_{c}(Re)$of the Rossby number, the flow becomes exactly 2D in the long-time limit, provided that the initial 3D perturbations are small. We call this property linear two-dimensionalization. We compute explicit lower bounds on$Ro_{c}(Re)$and therefore determine regions of the parameter space$(Re,Ro)$where such exact two-dimensionalization takes place. We present similar results in terms of the forcing strength instead of the root-mean-square velocity: the global attractor of the 2D Navier–Stokes equation is linearly stable to vertically dependent perturbations when the forcing-based Rossby number$Ro^{(f)}$is lower than a Grashof-number-dependent threshold value$Ro_{c}^{(f)}(Gr)$. We then consider the fully nonlinear 3D rotating Navier–Stokes equation and prove absolute two-dimensionalization: we show that, below some threshold value$Ro_{\mathit{abs}}^{(f)}(Gr)$of the forcing-based Rossby number, the flow becomes two-dimensional in the long-time limit, regardless of the initial condition (including initial 3D perturbations of arbitrarily large amplitude). These results shed some light on several fundamental questions of rotating turbulence: for arbitrary Reynolds number$Re$and small enough Rossby number, the system is attracted towards purely 2D flow solutions, which display no energy dissipation anomaly and no cyclone–anticyclone asymmetry. Finally, these results challenge the applicability of wave turbulence theory to describe stationary rotating turbulence in bounded domains.

History forces and the unsteady wake of a cylinder

1999 ◽  
Vol 393 ◽  
pp. 99-121 ◽  
Author(s):  
J. R. CHAPLIN
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Stokes Equation ◽  
Oscillatory Flow ◽  
Numerical Solutions ◽  
Asymptotic Properties ◽  
Quantitative Agreement ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Unsteady Wake

History forces on a stationary cylinder in arbitrary unsteady rectilinear flow are calculated by means of a model based on the asymptotic properties of the steady-state wake. The results capture many features found in numerical solutions of the Navier–Stokes equation for the same flows, though quantitative agreement deteriorates as the Reynolds number increases over the range 2 to 40. The cases studied are the impulsive start, stop, and reverse, and oscillatory flow.

scholarly journals A self-sustaining process model of inertial layer dynamics in high Reynolds number turbulent wall flows

2017 ◽  
Vol 375 (2089) ◽  
pp. 20160090 ◽  
Author(s):  
G. P. Chini ◽  
B. Montemuro ◽  
C. M. White ◽  
J. Klewicki
Keyword(s):  
Reynolds Number ◽  
Process Model ◽  
Outer Region ◽  
Wall Turbulence ◽  
Navier Stokes ◽  
Large Reynolds Number ◽  
Navier Stokes Equation ◽  
Wall Flows ◽  
High Reynolds ◽  
Turbulent Wall

Field observations and laboratory experiments suggest that at high Reynolds numbers Re the outer region of turbulent boundary layers self-organizes into quasi-uniform momentum zones (UMZs) separated by internal shear layers termed ‘vortical fissures’ (VFs). Motivated by this emergent structure, a conceptual model is proposed with dynamical components that collectively have the potential to generate a self-sustaining interaction between a single VF and adjacent UMZs. A large- Re asymptotic analysis of the governing incompressible Navier–Stokes equation is performed to derive reduced equation sets for the streamwise-averaged and streamwise-fluctuating flow within the VF and UMZs. The simplified equations reveal the dominant physics within—and isolate possible coupling mechanisms among—these different regions of the flow. This article is part of the themed issue ‘Toward the development of high-fidelity models of wall turbulence at large Reynolds number’.

A proposal concerning laminar wakes behind bluff bodies at large Reynolds number

1956 ◽  
Vol 1 (4) ◽  
pp. 388-398 ◽  
Author(s):  
G. K. Batchelor
Keyword(s):  
Reynolds Number ◽  
Steady Flow ◽  
Bluff Body ◽  
The Body ◽  
Navier Stokes ◽  
Large Reynolds Number ◽  
Bluff Bodies ◽  
Proper Solution ◽  
Navier Stokes Equation ◽  
Limit Solution

This note advocates a model of the steady flow about a bluff body at large Reynolds number which is different from the classical free-streamline model of Helmholtz and Kirchhoff. It is suggested that, although the free-streamline model may be a proper solution of the Navier-Stokes equation with μ = 0, it is unlikely to be the limit, as μ → 0, of the solution describing the steady flow due to the presence of a bluff body in an otherwise uniform stream. The limit solution proposed here is one which gives a closed wake.A closed wake contains a standing eddy, or eddies, whose general features can be inferred from the results of an earlier investigation of steady flow in a closed region at large Reynolds number. In all cases, the drag (coefficient) on the body tends to zero as the Reynolds number tends to infinity. The proccedure for finding the details of the closed wake behind two-dimensional and axisymmetrical bodies is described, although no particular case has yet been worked out.

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