scholarly journals Onset of secondary instabilities on the zigzag instability in stratified fluids

2011 ◽  
Vol 682 ◽  
pp. 120-131 ◽  
Author(s):  
PIERRE AUGIER ◽  
PAUL BILLANT
Keyword(s):  
Numerical Simulations ◽  
Gravitational Instability ◽  
Dominant Role ◽  
Vortex Pair ◽  
Secondary Instability ◽  
Shear Instability ◽  
Small Scale ◽  
Scaling Analysis ◽  
Wide Range ◽  
Secondary Instabilities

Recently, Deloncle, Billant & Chomaz (J. Fluid Mech., vol. 599, 2008, p. 229) and Waite & Smolarkiewicz (J. Fluid Mech., vol. 606, 2008, p. 239) have performed numerical simulations of the nonlinear evolution of the zigzag instability of a pair of counter-rotating vertical vortices in a stratified fluid. Both studies report the development of a small-scale secondary instability when the vortices are strongly bent if the Reynolds number Re is sufficiently high. However, the two papers are at variance about the nature of this secondary instability: it is a shear instability according to Deloncle et al. (J. Fluid Mech., vol. 599, 2008, p. 229) and a gravitational instability according to Waite & Smolarkiewicz (J. Fluid Mech., vol. 606, 2008, p. 239). They also profoundly disagree about the condition for the onset of the secondary instability: ReF2h > O(1) according to the former or ReFh > 80 according to the latter, where Fh is the horizontal Froude number. In order to understand the origin of these discrepancies, we have carried out direct numerical simulations of the zigzag instability of a Lamb–Chaplygin vortex pair for a wide range of Reynolds and Froude numbers. The threshold for the onset of a secondary instability is found to be ReF2h ≃ 4 for Re ≳ 3000 and ReFh = 80 for Re ≲ 1000 in agreement with both previous studies. We show that the scaling analysis of Deloncle et al. (J. Fluid Mech., vol. 599, 2008, p. 229) can be refined to obtain a universal threshold: (Re − Re0)F2h ≃ 4, with Re0 ≃ 400, which works for all Re. Two different regimes for the secondary instabilities are observed: when (Re − Re0)F2h ≃ 4, only the shear instability develops while when (Re − Re0)F2h ≫ 4, both shear and gravitational instabilities appear almost simultaneously in distinct regions of the vortices. However, the shear instability seems to play a dominant role in the breakdown into small scales in the range of parameters investigated.

scholarly journals Spectral analysis of the transition to turbulence from a dipole in stratified fluid

2012 ◽  
Vol 713 ◽  
pp. 86-108 ◽  
Author(s):  
Pierre Augier ◽  
Jean-Marc Chomaz ◽  
Paul Billant
Keyword(s):  
Reynolds Number ◽  
Spectral Analysis ◽  
Kinetic Energy ◽  
Numerical Simulations ◽  
Scaling Laws ◽  
Stratified Fluid ◽  
Shear Instability ◽  
Transition To Turbulence ◽  
Small Scale ◽  
Wide Range

AbstractWe investigate the spectral properties of the turbulence generated during the nonlinear evolution of a Lamb–Chaplygin dipole in a stratified fluid for a high Reynolds number $Re= 28\hspace{0.167em} 000$ and a wide range of horizontal Froude number ${F}_{h} \in [0. 0225~0. 135] $ and buoyancy Reynolds number $\mathscr{R}= Re{{F}_{h} }^{2} \in [14~510] $. The numerical simulations use a weak hyperviscosity and are therefore almost direct numerical simulations (DNS). After the nonlinear development of the zigzag instability, both shear and gravitational instabilities develop and lead to a transition to small scales. A spectral analysis shows that this transition is dominated by two kinds of transfer: first, the shear instability induces a direct non-local transfer toward horizontal wavelengths of the order of the buoyancy scale ${L}_{b} = U/ N$, where $U$ is the characteristic horizontal velocity of the dipole and $N$ the Brunt–Väisälä frequency; second, the destabilization of the Kelvin–Helmholtz billows and the gravitational instability lead to small-scale weakly stratified turbulence. The horizontal spectrum of kinetic energy exhibits a ${{\varepsilon }_{K} }^{2/ 3} { k}_{h}^{\ensuremath{-} 5/ 3} $ power law (where ${k}_{h} $ is the horizontal wavenumber and ${\varepsilon }_{K} $ is the dissipation rate of kinetic energy) from ${k}_{b} = 2\lrm{\pi} / {L}_{b} $ to the dissipative scales, with an energy deficit between the integral scale and ${k}_{b} $ and an excess around ${k}_{b} $. The vertical spectrum of kinetic energy can be expressed as $E({k}_{z} )= {C}_{N} {N}^{2} { k}_{z}^{\ensuremath{-} 3} + C{{\varepsilon }_{K} }^{2/ 3} { k}_{z}^{\ensuremath{-} 5/ 3} $ where ${C}_{N} $ and $C$ are two constants of order unity and ${k}_{z} $ is the vertical wavenumber. It is therefore very steep near the buoyancy scale with an ${N}^{2} { k}_{z}^{\ensuremath{-} 3} $ shape and approaches the ${{\varepsilon }_{K} }^{2/ 3} { k}_{z}^{\ensuremath{-} 5/ 3} $ spectrum for ${k}_{z} \gt {k}_{o} $, ${k}_{o} $ being the Ozmidov wavenumber, which is the cross-over between the two scaling laws. A decomposition of the vertical spectra depending on the horizontal wavenumber value shows that the ${N}^{2} { k}_{z}^{\ensuremath{-} 3} $ spectrum is associated with large horizontal scales $\vert {\mathbi{k}}_{h} \vert \lt {k}_{b} $ and the ${{\varepsilon }_{K} }^{2/ 3} { k}_{z}^{\ensuremath{-} 5/ 3} $ spectrum with the scales $\vert {\mathbi{k}}_{h} \vert \gt {k}_{b} $.

scholarly journals Role of overturns in optimal mixing in stratified mixing layers

2017 ◽  
Vol 826 ◽  
pp. 522-552 ◽  
Author(s):  
A. Mashayek ◽  
C. P. Caulfield ◽  
W. R. Peltier
Keyword(s):  
Numerical Simulations ◽  
Ocean Circulation ◽  
Turbulent Mixing ◽  
Isotropic Turbulence ◽  
Life Cycles ◽  
Shear Instability ◽  
Mixing Efficiency ◽  
Small Scale ◽  
Mean Flow

Turbulent mixing plays a major role in enabling the large-scale ocean circulation. The accuracy of mixing rates estimated from observations depends on our understanding of basic fluid mechanical processes underlying the nature of turbulence in a stratified fluid. Several of the key assumptions made in conventional mixing parameterizations have been increasingly scrutinized in recent years, primarily on the basis of adequately high resolution numerical simulations. We add to this evidence by compiling results from a suite of numerical simulations of the turbulence generated through stratified shear instability processes. We study the inherently intermittent and time-dependent nature of wave-induced turbulent life cycles and more specifically the tight coupling between inherently anisotropic scales upon which small-scale isotropic turbulence grows. The anisotropic scales stir and stretch fluid filaments enhancing irreversible diffusive mixing at smaller scales. We show that the characteristics of turbulent mixing depend on the relative time evolution of the Ozmidov length scale $L_{O}$ compared to the so-called Thorpe overturning scale $L_{T}$ which represents the scale containing available potential energy upon which turbulence feeds and grows. We find that when $L_{T}\sim L_{O}$, the mixing is most active and efficient since stirring by the largest overturns becomes ‘optimal’ in the sense that it is not suppressed by ambient stratification. We argue that the high mixing efficiency associated with this phase, along with observations of $L_{O}/L_{T}\sim 1$ in oceanic turbulent patches, together point to the potential for systematically underestimating mixing in the ocean if the role of overturns is neglected. This neglect, arising through the assumption of a clear separation of scales between the background mean flow and small-scale quasi-isotropic turbulence, leads to the exclusion of an highly efficient mixing phase from conventional parameterizations of the vertical transport of density. Such an exclusion may well be significant if the mechanism of shear-induced turbulence is assumed to be representative of at least some turbulent events in the ocean. While our results are based upon simulations of shear instability, we show that they are potentially more generic by making direct comparisons with $L_{T}-L_{O}$ data from ocean and lake observations which represent a much wider range of turbulence-inducing physical processes.

The ‘zoo’ of secondary instabilities precursory to stratified shear flow transition. Part 2 The influence of stratification

2012 ◽  
Vol 708 ◽  
pp. 45-70 ◽  
Author(s):  
A. Mashayek ◽  
W. R. Peltier
Keyword(s):  
Three Dimensional ◽  
Reynolds Numbers ◽  
Shear Instability ◽  
Developed Turbulence ◽  
Wide Range ◽  
Stratified Shear Flow ◽  
High Reynolds ◽  
The Individual ◽  
Secondary Instabilities ◽  
Alternate Routes

AbstractThe linear stability analyses described in Mashayek & Peltier (J. Fluid Mech., vol. 708, 2012, 5–44, hereafter MP1) are extended herein in an investigation of the influence of stratification on the evolution of secondary instabilities to which an evolving Kelvin–Helmholtz (KH) wave is susceptible in an initially unstable parallel stratified shear layer. We show that over a wide range of background stratification levels, the braid shear instability has a higher probability of emerging at early stages of the flow evolution while the secondary convective instability (SCI), which occurs in the eyelids of the individual Kelvin ‘cats eyes’, will remain a relevant and dominant instability at high Reynolds numbers. The evolution of both modes is greatly influenced by the background stratification. Various other three-dimensional secondary instabilities are found to exist over a wide range of stratification levels. In particular, the stagnation point instability (SPI), which was discussed in detail in MP1, may be of great potential importance providing alternate routes for transition of an initially two-dimensional KH wave into fully developed turbulence. The energetics of the secondary instabilities revealed by our simulations are analysed in detail and the preturbulent mixing properties are studied.

The influence of stratification on secondary instability in free shear layers

1991 ◽  
Vol 227 ◽  
pp. 71-106 ◽  
Author(s):  
G. P. Klaassen ◽  
W. R. Peltier
Keyword(s):  
Kinetic Energy ◽  
Initial Density ◽  
Three Dimensional ◽  
Primary Source ◽  
Underlying Mechanism ◽  
Finite Amplitude ◽  
Secondary Instability ◽  
Shear Instability ◽  
Complete Analysis ◽  
Secondary Instabilities

We analyse the stability of horizontally periodic, two-dimensional, finite-amplitude Kelvin-Helmholtz billows with respect to infinitesimal three-dimensional perturbations having the same streamwise wavelength for several different levels of the initial density stratification. A complete analysis of the energy budget for this class of secondary instabilities establishes that the contribution to their growth from shear conversion of the basic-state kinetic energy is relatively insensitive to the strength of the stratification over the range of values considered, suggesting that dynamical shear instability constitutes the basic underlying mechanism. Indeed, during the initial stages of their growth, secondary instabilities derive their energy predominantly from shear conversion. However, for initial Richardson numbers between 0.065 and 0.13, the primary source of kinetic energy for secondary instabilities at the time the parent wave climaxes is in fact the conversion of potential energy by convective overturning in the cores of the individual billows. A comparison between the secondary instability properties of unstratified Kelvin-Helmholtz billows and Stuart vortices is made, as the latter have often been assumed to provide an adequate approximation to the former. Our analyses suggest that the Stuart vortex model has several shortcomings in this regard.

Low-Emission, Liquid Fuel Combustion System for Conventional and Alternative Fuels Developed by the Scaling Analysis

2015 ◽  
Vol 138 (4) ◽  
Author(s):  
Yonas Niguse ◽  
Ajay Agrawal
Keyword(s):  
Alternative Fuels ◽  
Large Scale ◽  
Small Scale ◽  
Scaling Analysis ◽  
Combustion System ◽  
Low Emission ◽  
Wide Range ◽  
Air Mixing ◽  
Scaling Parameters ◽  
Fine Spray

The objective of this study is to develop a theoretical basis for scalability considerations and design of a large-scale combustor utilizing flow blurring (FB) atomization. FB atomization is a recently discovered twin-fluid atomization concept, reported to produce fine spray of liquids with wide range of viscosities. Previously, we have developed and investigated a small-scale swirl-stabilized combustor of 7-kWth capacity. Spray measurements have shown that the FB injector's atomization capability is superior when compared to other techniques, such as air blast atomization. However, despite these favorable results, scalability of the FB injector and associated combustor design has never been explored for large capacity; for example, for gas turbine applications. In this study, a number of dimensionless scaling parameters that affect the processes of atomization, fuel–air mixing, and combustion are analyzed, and scaling criteria for the different components of the combustion system are selected. Constant velocity criterion is used to scale key geometric components of the system. Scaling of the nonlinear dimensions and complex geometries, such as swirler vanes and internal parts of the injector is undertaken through phenomenological analysis of the flow processes associated with the scaled component. A scaled-up 60-kWth capacity combustor with FB injector is developed and investigated for combustion performance using diesel and vegetable oil (VO) (soybean oil) as fuels. Results show that the scaled-up injector's performance is comparable to the smaller scale system in terms of flame quality, emission levels, and static flame stability. Visual flame images at different atomizing air-to-liquid ratio by mass (ALR) show mainly blue flames, especially for ALR > 2.8. Emission measurements show a general trend of lower CO and NOx levels at higher ALRs, replicating the performance of the small-scale combustion system. Flame liftoff height at different ALRs is similar for both scales. The scaled-up combustor with FB injector preformed robustly with uncompromised stability for the range of firing rates (FRs) above 50% of the design capacity. Experimental results corroborate with the scaling methodology developed in this research.

scholarly journals Scaling Analysis of the Unsteady Natural Convection Boundary Layer Adjacent to an Inclined Plate for Pr > 1 Following Instantaneous Heating

2011 ◽  
Vol 133 (11) ◽  
Author(s):  
Suvash C. Saha ◽  
Feng Xu ◽  
Md Mamun Molla
Keyword(s):  
Boundary Layer ◽  
Natural Convection ◽  
Numerical Simulations ◽  
Direct Numerical Simulations ◽  
Convection Boundary Layer ◽  
Scaling Analysis ◽  
Inclined Plate ◽  
Wide Range ◽  
Convection Boundary ◽  
Unsteady Natural Convection

The unsteady natural convection boundary layer adjacent to an instantaneously heated inclined plate is investigated using an improved scaling analysis and direct numerical simulations. The development of the unsteady natural convection boundary layer following instantaneous heating may be classified into three distinct stages including a start-up stage, a transitional stage, and a steady state stage, which can be clearly identified in the analytical and numerical results. Major scaling relations of the velocity and thicknesses and the flow development time of the natural convection boundary layer are obtained using triple-layer integral solutions and verified by direct numerical simulations over a wide range of flow parameters.

scholarly journals Transition to turbulence through steep global-modes cascade in an open rotating cavity

2011 ◽  
Vol 688 ◽  
pp. 493-506 ◽  
Author(s):  
Bertrand Viaud ◽  
Eric Serre ◽  
Jean-Marc Chomaz
Keyword(s):  
Threshold Value ◽  
Secondary Instability ◽  
Base Flow ◽  
Transition To Turbulence ◽  
Small Scale ◽  
Global Mode ◽  
Rotating Boundary ◽  
The Stability ◽  
Real Flow ◽  
Secondary Instabilities

AbstractThe transition to turbulence in a rotating boundary layer is analysed via direct numerical simulation (DNS) in an annular cavity made of two parallel corotating discs of finite radial extent, with a forced inflow at the hub and free outflow at the rim. In a former numerical investigation (Viaud, Serre & Chomaz J. Fluid Mech., vol. 598, 2008, pp. 451–464) realized in a sectorial cavity of azimuthal extent $2\lrm{\pi} / 68$, we have established the existence of a primary bifurcation to nonlinear global mode with angular phase velocity and radial envelope coherent with the so-called elephant mode theory. The former study has demonstrated the subcritical nature of this primary bifurcation with a base flow that keeps being linearly stable for all Reynolds numbers studied. The present work investigates the stability of this elephant mode by extending the cavity both in the radial and azimuthal direction. When the Reynolds number based on the forced throughflow is increased above a threshold value for the existence of the nonlinear global mode, a large-amplitude impulsive perturbation gives rise to a self-sustained saturated wave with characteristics identical to the 68-fold global elephant mode obtained in the smaller cavity. This saturated wave is itself globally unstable and a second front appears in the lee of the primary where small-scale instability develops. These secondary instabilities are identical for the $2\lrm{\pi} / 68$ and the $2\lrm{\pi} / 4$ long sectorial cavities, indicating that transition involves a Floquet mode of zero azimuthal wavenumber. This secondary instability leads to a very disorganized state, defining the transition to turbulence. The observed transition to turbulence linked to the secondary instability of a global mode confirms, for the first time on a real flow, the possibility of a direct transition to turbulence through an elephant mode cascade, a scenario that was up to now only observed on the Ginzburg–Landau model.

Stability transitions and turbulence in horizontal convection

2014 ◽  
Vol 751 ◽  
pp. 698-724 ◽  
Author(s):  
Bishakhdatta Gayen ◽  
Ross W. Griffiths ◽  
Graham O. Hughes
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Large Scale ◽  
Mechanical Energy ◽  
Shear Instability ◽  
Momentum Balance ◽  
Small Scale ◽  
Wide Range ◽  
Small Scales ◽  
Rayleigh Numbers

AbstractRecent results have shown that convection forced by a temperature gradient along one horizontal boundary of a rectangular domain at a large Rayleigh number can be turbulent in parts of the flow field. However, the conditions for onset of turbulence, the dependence of flow and heat transport on Rayleigh number, and the roles of large and small scales in the flow, have not been established. We use three-dimensional direct numerical simulation (DNS) and large-eddy simulation (LES) over a wide range of Rayleigh numbers,$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}Ra\sim 10^8\mbox{--}10^{15}$, for Prandtl number$Pr=5$and a small aspect ratio, and show that a sequence of several stability transitions at$Ra \sim 10^{10}\mbox{--} 10^{11}$defines a change from laminar to turbulent flow. The Prandtl number dependence too is examined at$Ra = 5.86 \times 10^{11}$. At the smallest$Ra$considered the thermal boundary layer is characterized by a balance of viscous stress and buoyancy, whereas inertia and buoyancy dominate in the large-$Ra$regime. The change in the momentum balance is accompanied by turbulent enhancement of the overall heat transfer, although both laminar and turbulent regimes give$Nu\sim Ra^{1/5}$. The results support both viscous and inviscid theoretical scaling models from previous work. The mechanical energy budget for an intermediate range of Rayleigh numbers above onset of instability ($10^{10}<Ra<10^{13}$) reveals that the small scales of motion are produced predominantly by thermal convection, whereas at$Ra \ge 10^{14}$shear instability of the large-scale flow begins to play a dominant role in sustaining the small-scale turbulence. Extrapolation to ocean conditions requires knowledge of the inertial regime identified here, but the simulations show that the corresponding asymptotic balance has not been fully realized by$Ra \sim 10^{15}$.

scholarly journals Validity domains and parametrizations for white-noise and multiscale models in turbulence and wave-turbulence interactions

2021 ◽  
Author(s):  
valentin resseguier ◽  
Erwan Hascoet ◽  
Bertrand Chapron ◽  
Baylor Fox-Kemper
Keyword(s):  
Numerical Simulations ◽  
Fluid Transport ◽  
Parametric Models ◽  
Small Scale ◽  
Numerical Representation ◽  
Spatial Correlations ◽  
Self Similarity ◽  
Fluid Velocities ◽  
Wide Range ◽  
Spatio Temporal

&lt;p&gt;Geophysical fluid dynamics systems generally involve a wide range of spatio-temporal scales. Numerical representation can only simulate some of the scales. The others, at the unresolved scales of motion, must be parameterized for each type of phenomenon (wave, eddy, current), in terms of expected effects on the resolved scales. Most developments then assume that the fluid transport velocity has a time-uncorrelated noisy component with zero mean and stationary statistics. These approximations generally simplify theoretical descriptions, numerical simulations, data comparisons or more recently model error quantifications for data assimilation.&lt;/p&gt;&lt;p&gt;In the present work, we will discuss the applicability of such approximations through two examples: a surface oceanic current dynamics and swell refractions by surface currents.&lt;/p&gt;&lt;p&gt;When the time-decorrelation assumption is valid, we propose simple and tuning-free parametric models to represent the spatial correlations of the white-in-time small-scale velocity to help simulate the geophysical system of interest. These parametric models relies on turbulence space self-similarity and their statistical properties (e.g. spectral slope) can be easily estimated from observations of larger scale fluid velocities.&lt;/p&gt;&lt;p&gt;When the white-in-time approximation is not valid, we extend the previous parametric models to follow self-similarity properties in both time and space.&lt;/p&gt;&lt;p&gt;Numerical simulations will illustrate these theoretical developments along the presentation.&lt;/p&gt;

scholarly journals Scaling analysis and simulation of strongly stratified turbulent flows

2007 ◽  
Vol 585 ◽  
pp. 343-368 ◽  
Author(s):  
G. BRETHOUWER ◽  
P. BILLANT ◽  
E. LINDBORG ◽  
J.-M. CHOMAZ
Keyword(s):  
Reynolds Number ◽  
Numerical Simulations ◽  
Turbulent Flows ◽  
Scaling Laws ◽  
Three Dimensional ◽  
Direct Numerical Simulations ◽  
Small Scale ◽  
Scaling Analysis ◽  
Length Scale ◽  
Stratified Turbulence

Direct numerical simulations of stably and strongly stratified turbulent flows with Reynolds number Re ≫ 1 and horizontal Froude number Fh ≪ 1 are presented. The results are interpreted on the basis of a scaling analysis of the governing equations. The analysis suggests that there are two different strongly stratified regimes according to the parameter $\mathcal{R} \,{=}\, \hbox{\it Re} F^2_h$. When $\mathcal{R} \,{\gg}\, 1$, viscous forces are unimportant and lv scales as lv ∼ U/N (U is a characteristic horizontal velocity and N is the Brunt–Väisälä frequency) so that the dynamics of the flow is inherently three-dimensional but strongly anisotropic. When $\mathcal{R} \,{\ll}\, 1$, vertical viscous shearing is important so that $l_v \,{\sim}\, l_h/\hbox{\it Re}^{1/2}$ (lh is a characteristic horizontal length scale). The parameter $\cal R$ is further shown to be related to the buoyancy Reynolds number and proportional to (lO/η)4/3, where lO is the Ozmidov length scale and η the Kolmogorov length scale. This implies that there are simultaneously two distinct ranges in strongly stratified turbulence when $\mathcal{R} \,{\gg}\, 1$: the scales larger than lO are strongly influenced by the stratification while those between lO and η are weakly affected by stratification. The direct numerical simulations with forced large-scale horizontal two-dimensional motions and uniform stratification cover a wide Re and Fh range and support the main parameter controlling strongly stratified turbulence being $\cal R$. The numerical results are in good agreement with the scaling laws for the vertical length scale. Thin horizontal layers are observed independently of the value of $\cal R$ but they tend to be smooth for $\cal R$< 1, while for $\cal R$ > 1 small-scale three-dimensional turbulent disturbances are increasingly superimposed. The dissipation of kinetic energy is mostly due to vertical shearing for $\cal R$ < 1 but tends to isotropy as $\cal R$ increases above unity. When $\mathcal{R}$ < 1, the horizontal and vertical energy spectra are very steep while, when $\cal R$ > 1, the horizontal spectra of kinetic and potential energy exhibit an approximate k−5/3h-power-law range and a clear forward energy cascade is observed.

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