Instability and breakdown of a vertical vortex pair in a strongly stratified fluid

2008 ◽  
Vol 606 ◽  
pp. 239-273 ◽  
Author(s):  
MICHAEL L. WAITE ◽  
PIOTR K. SMOLARKIEWICZ
Keyword(s):  
Reynolds Number ◽  
Laboratory Experiments ◽  
Gravitational Instability ◽  
Initial Perturbation ◽  
Vortex Pair ◽  
Horizontal Displacement ◽  
Transition To Turbulence ◽  
Large Reynolds Number ◽  
Length Scales ◽  
Moderate Reynolds Number

The dynamics of a counter-rotating pair of columnar vortices aligned parallel to a stable density gradient are investigated. By means of numerical simulation, we extend the linear analyses and laboratory experiments of Billant & Chomaz (J. Fluid Mech. vol. 418, p. 167; vol. 419, pp. 29, 65 (2000)) to the fully nonlinear, large-Reynolds-number regime. A range of stratifications and vertical length scales is considered, with Frh < 0.2 and 0.1 < Frz < 10. Here Frh ≡ U/(NR) and Frz ≡ Ukz/N are the horizontal and vertical Froude numbers, U and R are the horizontal velocity and length scales of the vortices, N is the Brunt–Väisälä frequency, and 2π/kz is the vertical wavelength of a small initial perturbation. At early times with Frz < 1, linear predictions for the zigzag instability are reproduced. Short-wavelength perturbations with Frz > 1 are found to be unstable as well, with growth rates only slightly less than those of the zigzag instability but with very different structure. At later times, the large-Reynolds-number evolution diverges profoundly from the moderate-Reynolds-number laboratory experiments as the instabilities transition to turbulence. For the zigzag instability, this transition occurs when density perturbations generated by the vortex bending become gravitationally unstable. The resulting turbulence rapidly destroys the vortex pair. We derive the criterion η/R ≈ 0.2/Frz for the onset of gravitational instability, where η is the maximum horizontal displacement of the bent vortices, and refine it to account for a finite twisting disturbance. Our simulations agree for the fastest growing wavelengths 0.3 < Frz < 0.8. Short perturbations with Frz > 1 saturate at low amplitude, preserving the columnar structure of the vortices well after the generation of turbulence. Viscosity is shown to suppress the transition to turbulence for Reynolds number Re ≲ 80/Frh, yielding laminar dynamics and, under certain conditions, pancake vortices like those observed in the laboratory.

scholarly journals When is high Reynolds number shear flow not turbulent?

2017 ◽  
Vol 824 ◽  
pp. 1-4 ◽  
Author(s):  
Steven A. Balbus
Keyword(s):  
Reynolds Number ◽  
Shear Flow ◽  
High Reynolds Number ◽  
Laboratory Experiments ◽  
Numerical Study ◽  
Accretion Discs ◽  
Large Reynolds Number ◽  
Rotating Flow ◽  
General Tendency ◽  
High Reynolds

Rotating flow in which the angular velocity decreases outward while the angular momentum increases is known as ‘quasi-Keplerian’. Despite the general tendency of shear flow to break down into turbulence, this type of flow seems to maintain stability at very large Reynolds number, even when nonlinearly perturbed, a behaviour that strongly influences our understanding of astrophysical accretion discs. Investigating these flows in the laboratory is difficult because secondary Ekman flows, caused by the retaining Couette cylinders, can become turbulent on their own. A recent high Reynolds number numerical study by Lopez & Avila (J. Fluid Mech., vol. 817, 2017, pp. 21–34) reconciles apparently discrepant laboratory experiments by confirming that this secondary flow recedes toward the axial boundaries of the container as the Reynolds number is increased, a result that enhances our understanding of nonlinear quasi-Keplerian flow stability.

scholarly journals Turbulent mixing due to the Holmboe wave instability at high Reynolds number

2016 ◽  
Vol 803 ◽  
pp. 591-621 ◽  
Author(s):  
Hesam Salehipour ◽  
C. P. Caulfield ◽  
W. R. Peltier
Keyword(s):  
Reynolds Number ◽  
Richardson Number ◽  
Initial Density ◽  
Finite Amplitude ◽  
Transition To Turbulence ◽  
Wave Instability ◽  
Length Scales ◽  
Density Interface ◽  
Gradient Richardson Number ◽  
High Reynolds

We consider numerically the transition to turbulence and associated mixing in stratified shear flows with initial velocity distribution $\overline{U}(z,0)\,\boldsymbol{e}_{x}=U_{0}\,\boldsymbol{e}_{x}\tanh (z/d)$ and initial density distribution $\overline{\unicode[STIX]{x1D70C}}(z,0)=\unicode[STIX]{x1D70C}_{0}[1-\tanh (z/\unicode[STIX]{x1D6FF})]$ away from a hydrostatic reference state $\unicode[STIX]{x1D70C}_{r}\gg \unicode[STIX]{x1D70C}_{0}$. When the ratio $R=d/\unicode[STIX]{x1D6FF}$ of the characteristic length scales over which the velocity and density vary is equal to one, this flow is primarily susceptible to the classic well-known Kelvin–Helmholtz instability (KHI). This instability, which typically manifests at finite amplitude as an array of elliptical vortices, strongly ‘overturns’ the density interface of strong initial gradient, which nevertheless is the location of minimum initial gradient Richardson number $Ri_{g}(0)=Ri_{b}=g\unicode[STIX]{x1D70C}_{0}d/\unicode[STIX]{x1D70C}_{r}U_{0}^{2}$, where $Ri_{g}(z)=-([g/\unicode[STIX]{x1D70C}_{r}]\,\text{d}\overline{\unicode[STIX]{x1D70C}}/\text{d}z)/(\text{d}\overline{U}/\text{d}z)^{2}$ and $Ri_{b}$ is a bulk Richardson number. As is well known, at sufficiently high Reynolds numbers ($Re$), the primary KHI induces a vigorous but inherently transient burst of turbulence and associated irreversible mixing localised in the vicinity of the density interface, leading to a relatively well-mixed region bounded by stronger density gradients above and below. We explore the qualitatively different behaviour that arises when $R\gg 1$, and so the density interface is sharp, with $Ri_{g}(z)$ now being maximum at the density interface $Ri_{g}(0)=RRi_{b}$. This flow is primarily susceptible to Holmboe wave instability (HWI) (Holmboe, Geophys. Publ., vol. 24, 1962, pp. 67–113), which manifests at finite amplitude in this symmetric flow as counter-propagating trains of elliptical vortices above and below the density interface, thus perturbing the interface so as to exhibit characteristically cusped interfacial waves which thereby ‘scour’ the density interface. Unlike previous lower-$Re$ experimental and numerical studies, when $Re$ is sufficiently high the primary HWI becomes increasingly more three-dimensional due to the emergence of shear-aligned secondary convective instabilities. As $Re$ increases, (i) the growth rate of secondary instabilities appears to saturate and (ii) the perturbation kinetic energy exhibits a $k^{-5/3}$ spectrum for sufficiently large length scales that are influenced by anisotropic buoyancy effects. Therefore, at sufficiently high $Re$, vigorous turbulence is triggered that also significantly ‘scours’ the primary density interface, leading to substantial irreversible mixing and vertical transport of mass above and below the (robust) primary density interface. Furthermore, HWI produces a markedly more long-lived turbulence event compared to KHI at a similarly high $Re$. Despite their vastly different mechanics (i.e. ‘overturning’ versus ‘scouring’) and localisation, the mixing induced by KHI and HWI is comparable in both absolute terms and relative efficiency. Our results establish that, provided the flow Reynolds number is sufficiently high, shear layers with sharp density interfaces and associated locally high values of the gradient Richardson number may still be sites of substantial and efficient irreversible mixing.

On the linear stability of Stokes layers

2008 ◽  
Vol 366 (1876) ◽  
pp. 2685-2697 ◽  
Author(s):  
P.J Blennerhassett ◽  
Andrew P Bassom
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Practical Interest ◽  
Transition To Turbulence ◽  
Large Reynolds Number ◽  
Poor Agreement ◽  
Oscillatory Flows ◽  
Stokes Layer ◽  
The Stability ◽  
Time Periodic

Oscillatory flows occur naturally, with applications ranging across many disciplines from engineering to physiology. Transition to turbulence in such flows is a topic of practical interest and this article discusses some recent work that has furthered our understanding of the stability of a class of time-periodic fluid motions. Our study starts with an examination of the linear stability of a classical flat Stokes layer. Although experiments conducted over many years have demonstrated conclusively that this layer is unstable at a sufficiently large Reynolds number, it has only been relatively recently that rigorous theoretical confirmation of this behaviour has been obtained. The analysis and numerical calculations for the planar Stokes layer were subsequently extended to flows in channels and pipes and for the flow within a torsionally oscillating circular cylinder. We discuss why our predictions for the onset of instability in these geometries are in disappointingly poor agreement with experimental results. Finally, some suggestions for future experimental work are given and some areas for future theoretical analysis outlined.

scholarly journals Turbulent transition in a truncated one-dimensional model for shear flow

2011 ◽  
Vol 467 (2135) ◽  
pp. 3066-3087 ◽  
Author(s):  
J. H. P. Dawes ◽  
W. J. Giles
Keyword(s):  
Reynolds Number ◽  
Shear Flow ◽  
Turbulent Flows ◽  
Temporal Dynamics ◽  
Initial Conditions ◽  
Initial Perturbation ◽  
Initial Energy ◽  
Chaotic Regime ◽  
Transition To Turbulence ◽  
Spanwise Direction

We present a reduced model for the transition to turbulence in shear flow that is simple enough to admit a thorough numerical investigation, while allowing spatio-temporal dynamics that are substantially more complex than those allowed in previous modal truncations. Our model allows a comparison of the dynamics resulting from initial perturbations that are localized in the spanwise direction with those resulting from sinusoidal perturbations. For spanwise-localized initial conditions, the subcritical transition to a ‘turbulent’ state (i) takes place more abruptly, with a boundary between laminar and turbulent flows that appears to be much less ‘structured’ and (ii) results in a spatio-temporally chaotic regime within which the lifetimes of spatio-temporally complicated transients are longer, and are even more sensitive to initial conditions. The minimum initial energy E 0 required for a spanwise-localized initial perturbation to excite a chaotic transient has a power-law scaling with the Reynolds number E 0 ∼ Re p with p ≈−4.3. The exponent p depends only weakly on the width of the localized perturbation and is lower than that commonly observed in previous low-dimensional models where typically p ≈−2. The distributions of lifetimes of chaotic transients at the fixed Reynolds number are found to be consistent with exponential distributions.

Turbulence in supersonic boundary layers at moderate Reynolds number

2011 ◽  
Vol 688 ◽  
pp. 120-168 ◽  
Author(s):  
Sergio Pirozzoli ◽  
Matteo Bernardini
Keyword(s):  
Reynolds Number ◽  
Boundary Layers ◽  
Outer Layer ◽  
Large Reynolds Number ◽  
Wall Region ◽  
Normal Velocity ◽  
Momentum Thickness ◽  
Mean Velocity ◽  
Velocity Statistics ◽  
Moderate Reynolds Number

AbstractWe study the organization of turbulence in supersonic boundary layers through large-scale direct numerical simulations (DNS) at ${M}_{\infty } = 2$, and momentum-thickness Reynolds number up to ${\mathit{Re}}_{{\delta }_{2} } \approx 3900$ (corresponding to ${\mathit{Re}}_{\tau } \approx 1120$) which significantly extend the current envelope of DNS in the supersonic regime. The numerical strategy relies on high-order, non-dissipative discretization of the convective terms in the Navier–Stokes equations, and it implements a recycling/rescaling strategy to stimulate the inflow turbulence. Comparison of the velocity statistics up to fourth order shows nearly exact agreement with reference incompressible data, provided the momentum-thickness Reynolds number is matched, and provided the mean velocity and the velocity fluctuations are scaled to incorporate the effects of mean density variation, as postulated by Morkovin’s hypothesis. As also found in the incompressible regime, we observe quite a different behaviour of the second-order flow statistics at sufficiently large Reynolds number, most of which show the onset of a range with logarithmic variation, typical of ‘attached’ variables, whereas the wall-normal velocity exhibits a plateau away from the wall, which is typical of ‘detached’ variables. The modifications of the structure of the flow field that underlie this change of behaviour are highlighted through visualizations of the velocity and temperature fields, which substantiate the formation of large jet-like and wake-like motions in the outer part of the boundary layer. It is found that the typical size of the attached eddies roughly scales with the local mean velocity gradient, rather than being proportional to the wall distance, as happens for the wall-detached variables. The interactions of the large eddies in the outer layer with the near-wall region are quantified through a two-point amplitude modulation covariance, which characterizes the modulating action of energetic outer-layer eddies.

An example of steady laminar flow at large Reynolds number

1960 ◽  
Vol 9 (4) ◽  
pp. 593-602 ◽  
Author(s):  
Iam Proudman
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Boundary Layers ◽  
Tangential Velocity ◽  
Fluid Flows ◽  
The Other ◽  
Large Reynolds Number ◽  
Rotational Flows ◽  
Flow Domain ◽  
Source Of Information

The purpose of this note is to describe a particular class of steady fluid flows, for which the techniques of classical hydrodynamics and boundary-layer theory determine uniquely the asymptotic flow for large Reynolds number for each of a continuously varied set of boundary conditions. The flows involve viscous layers in the interior of the flow domain, as well as boundary layers, and the investigation is unusual in that the position and structure of all the viscous layers are determined uniquely. The note is intended to be an illustration of the principles that lead to this determination, not a source of information of practical value.The flows take place in a two-dimensional channel with porous walls through which fluid is uniformly injected or extracted. When fluid is extracted through both walls there are boundary layers on both walls and the flow outside these layers is irrotational. When fluid is extracted through one wall and injected through the other, there is a boundary layer only on the former wall and the inviscid rotational flow outside this layer satisfies the no-slip condition on the other wall. When fluid is injected through both walls there are no boundary layers, but there is a viscous layer in the interior of the channel, across which the second derivative of the tangential velocity is discontinous, and the position of this layer is determined by the requirement that the inviscid rotational flows on either side of it must satisfy the no-slip conditions on the walls.

Vortex shedding from a circular cylinder in moderate-Reynolds-number shear flow

1980 ◽  
Vol 101 (4) ◽  
pp. 721-735 ◽  
Author(s):  
Masaru Kiya ◽  
Hisataka Tamura ◽  
Mikio Arie
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Shear Flow ◽  
Vortex Shedding ◽  
Transverse Velocity ◽  
Number Range ◽  
Uniform Shear ◽  
Reynolds Number Range ◽  
Moderate Reynolds Number ◽  
Shear Parameter

The frequency of vortex shedding from a circular cylinder in a uniform shear flow and the flow patterns around it were experimentally investigated. The Reynolds number Re, which was defined in terms of the cylinder diameter and the approaching velocity at its centre, ranged from 35 to 1500. The shear parameter, which is the transverse velocity gradient of the shear flow non-dimensionalized by the above two quantities, was varied from 0 to 0·25. The critical Reynolds number beyond which vortex shedding from the cylinder occurred was found to be higher than that for a uniform stream and increased approximately linearly with increasing shear parameter when it was larger than about 0·06. In the Reynolds-number range 43 < Re < 220, the vortex shedding disappeared for sufficiently large shear parameters. Moreover, in the Reynolds-number range 100 < Re < 1000, the Strouhal number increased as the shear parameter increased beyond about 0·1.

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