New Algorithm for the Lid-driven Cavity Flow Problem with Boussinesq-Stokes Suspension

2018 ◽  
pp. 462-472 ◽  
Keyword(s):  
Numerical Solutions ◽  
Suspended Particle ◽  
Finite Difference Methods ◽  
Reynolds Numbers ◽  
Driven Cavity ◽  
Time Stepping ◽  
Differential Transform ◽  
High Reynolds ◽  
Newtonian Liquids ◽  
Limiting Case

In the present investigation, a streamfunction-vorticity form for Boussinesq-Stokes liquids (with suspended particles) is suitably used to examine the problem of 2-D unsteady incompressible flow in a square cavity with moving top and bottom wall. A new algorithm is used for this form in order to compute the numerical solutions for high Reynolds numbers up to Re=2500. This algorithm is conducted as a combination of the multi-time-stepping temporal differential transform and the spatial finite difference methods. Convergence of the time-series solution is ensured by multi-time-stepping method. The classical benchmark results of the Newtonian liquid are recovered as a limiting case and the decelerating influence of the suspended particle on the Newtonian liquids’ flow field is clearly elaborated.

On the spatial instability of piecewise linear free shear layers

1987 ◽  
Vol 174 ◽  
pp. 553-563 ◽  
Author(s):  
T. F. Balsa
Keyword(s):  
Velocity Profile ◽  
Numerical Solutions ◽  
Piecewise Linear ◽  
Reynolds Numbers ◽  
Shear Layers ◽  
Spatial Instability ◽  
Free Shear Layers ◽  
Linear Shear ◽  
High Reynolds ◽  
Sommerfeld Equation

The main goal of this paper is to clarify the spatial instability of a piecewise linear free shear flow. We do this by obtaining numerical solutions to the Orr–Sommerfeld equation at high Reynolds numbers. The velocity profile chosen is very much like a piecewise linear one, with the exception that the corners have been rounded so that the entire profile is infinitely differentiable. We find that the (viscous) spatial instability of this modified profile is virtually identical to the inviscid spatial instability of the piecewise linear profile and agrees qualitatively with the inviscid results for the tanh profile when the shear layers are convectively unstable. The unphysical features, previously identified for the piecewise linear velocity profile, arise only when the flow is absolutely unstable. In a nutshell, we see nothing wrong with the inviscid spatial instability of piecewise linear shear flows.

A two-dimensional vortex condensate at high Reynolds number

2013 ◽  
Vol 715 ◽  
pp. 359-388 ◽  
Author(s):  
Basile Gallet ◽  
William R. Young
Keyword(s):  
Stokes Equation ◽  
Numerical Solutions ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Finite Limit ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Quasilinear Theory ◽  
Zero Integral ◽  
High Reynolds

AbstractWe investigate solutions of the two-dimensional Navier–Stokes equation in a $\lrm{\pi} \ensuremath{\times} \lrm{\pi} $ square box with stress-free boundary conditions. The flow is steadily forced by the addition of a source $\sin nx\sin ny$ to the vorticity equation; attention is restricted to even $n$ so that the forcing has zero integral. Numerical solutions with $n= 2$ and $6$ show that at high Reynolds numbers the solution is a domain-scale vortex condensate with a strong projection on the gravest mode, $\sin x\sin y$. The sign of the vortex condensate is selected by a symmetry-breaking instability. We show that the amplitude of the vortex condensate has a finite limit as $\nu \ensuremath{\rightarrow} 0$. Using a quasilinear approximation we make an analytic prediction of the amplitude of the condensate and show that the amplitude is determined by viscous selection of a particular solution from a family of solutions to the forced two-dimensional Euler equation. This theory indicates that the condensate amplitude will depend sensitively on the form of the dissipation, even in the undamped limit. This prediction is verified by considering the addition of a drag term to the Navier–Stokes equation and comparing the quasilinear theory with numerical solution.

On the motion of laminar wing wakes in a stratified fluid

1996 ◽  
Vol 327 ◽  
pp. 139-160 ◽  
Author(s):  
Philippe R. Spalart
Keyword(s):  
Numerical Solutions ◽  
Practical Importance ◽  
Reynolds Numbers ◽  
Boussinesq Approximation ◽  
Stratified Fluid ◽  
Small Scale ◽  
Aspect Ratios ◽  
Sufficient Energy ◽  
High Reynolds ◽  
Vortex Systems

We present numerical solutions for two-dimensional laminar symmetric vortex systems descending in a stably stratified fluid, within the Boussinesq approximation. Three types of flows are considered: (I) tight vortices; (II) those deriving from an elliptical wing lift distribution; and (III) those deriving from a ‘high-lift’ distribution, with a part-span flap on the wing. The non-dimensional stratification ranges from zero to moderate, as it does for airliners. For Types I and II, with high Reynolds numbers and weak stratification, the solutions confirm the theory of Scorer & Davenport (1970) (their article lacks a crucial link which we provide, equivalent to one of Crow (1974)). Contrary to common conceptions and observations in small-scale experiments, the descent velocity increases exponentially with time, as the distance between vortices decreases and the circulation of the vortices proper is conserved. With moderate stratification, wakes with sufficient energy also attain the accelerating régime, until the vortex cores make contact. However, they first experience a rebound, which is both of practical importance and out of reach of simple formulas. Type III wakes produce two durable vortex pairs which tumble, and mitigate the buoyancy effect by exchanging fluid with the surroundings. These phenomena are obscured by low wing aspect ratios, Reynolds numbers below about 105, or appreciable surrounding turbulence; this may explain why neither a clear rebound nor an acceleration can be reconciled with experiments to date. We argue that airliner wakes have very little inherent diffusion, and that a rapid end to the wake's descent must reveal effects other than simple buoyancy. In particular, stratification promotes the Crow instability.

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