Transient growth in strongly stratified shear layers

2014 ◽  
Vol 758 ◽  
Author(s):  
A. K. Kaminski ◽  
C. P. Caulfield ◽  
J. R. Taylor
Keyword(s):  
Shear Layer ◽  
Normal Mode ◽  
Richardson Number ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Buoyancy Frequency ◽  
Transient Growth ◽  
Time Intervals ◽  
Optimal Perturbation ◽  
Target Time

AbstractWe investigate numerically transient linear growth of three-dimensional perturbations in a stratified shear layer to determine which perturbations optimize the growth of the total kinetic and potential energy over a range of finite target time intervals. The stratified shear layer has an initial parallel hyperbolic tangent velocity distribution with Reynolds number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}=U_0 h/\nu =1000$ and Prandtl number $\nu /\kappa =1$, where $\nu $ is the kinematic viscosity of the fluid and $\kappa $ is the diffusivity of the density. The initial stable buoyancy distribution has constant buoyancy frequency $N_0$, and we consider a range of flows with different bulk Richardson number ${\mathit{Ri}}_b=N_0^2h^2/U_0^2$, which also corresponds to the minimum gradient Richardson number ${\mathit{Ri}}_g(z)=N_0^2/(\mathrm{d}U/\mathrm{d} z)^2$ at the midpoint of the shear layer. For short target times, the optimal perturbations are inherently three-dimensional, while for sufficiently long target times and small ${\mathit{Ri}}_b$ the optimal perturbations are closely related to the normal-mode ‘Kelvin–Helmholtz’ (KH) instability, consistent with analogous calculations in an unstratified mixing layer recently reported by Arratia et al. (J. Fluid Mech., vol. 717, 2013, pp. 90–133). However, we demonstrate that non-trivial transient growth occurs even when the Richardson number is sufficiently high to stabilize all normal-mode instabilities, with the optimal perturbation exciting internal waves at some distance from the midpoint of the shear layer.

scholarly journals Transient perturbation growth in time-dependent mixing layers

2013 ◽  
Vol 717 ◽  
pp. 90-133 ◽  
Author(s):  
C. Arratia ◽  
C. P. Caulfield ◽  
J.-M. Chomaz
Keyword(s):  
Velocity Distribution ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Base Flow ◽  
Time Dependent ◽  
Time Interval ◽  
Start Time ◽  
Time Intervals ◽  
Optimal Perturbation ◽  
2D Flow

AbstractWe investigate numerically the transient linear growth of three-dimensional (3D) perturbations in a homogeneous time-evolving mixing layer in order to identify which perturbations are optimal in terms of their kinetic energy gain over a finite, predetermined time interval. We model the mixing layer with an initial parallel velocity distribution $\mathbi{U}(y)= {U}_{0} \tanh (y/ d)\mathbi{e}_{x}$ with Reynolds number $Re= {U}_{0} d/ \nu = 1000$, where $\nu $ is the kinematic viscosity of the fluid. We consider a range of time intervals on both a constant ‘frozen’ base flow and a time-dependent two-dimensional (2D) flow associated with the growth and nonlinear saturation of two wavelengths of the most-unstable eigenmode of linear theory of the initial parallel velocity distribution, which rolls up into two classical Rayleigh instabilities commonly referred to as Kelvin–Helmholtz (KH) billows, which eventually pair to form a larger vortex. For short times, the most-amplified perturbations on the frozen $\tanh $ profile are inherently 3D, and are most appropriately described as oblique wave ‘OL’ perturbations which grow through a combination of the Orr and lift-up mechanisms, while for longer times, the optimal perturbations are 2D and similar to the KH normal mode, with a slight enhancement of gain. For the time-evolving KH base flow, OL perturbations continue to dominate over sufficiently short time intervals. However, for longer time intervals which involve substantial evolution of the primary KH billows, two broad classes of inherently 3D linear optimal perturbation arise, associated at low wavenumbers with the well-known core-centred elliptical translative instability, and at higher wavenumbers with the braid-centred hyperbolic instability. The hyperbolic perturbation is relatively inefficient in exploiting the gain of the OL perturbations, and so only dominates the smaller wavenumber (ultimately) core-centred perturbations when the time evolution of the base flow or the start time of the optimization interval does not allow the OL perturbations much opportunity to grow. When the OL perturbations can grow, they initially grow in the braid, and then trigger an elliptical core-centred perturbation by a strong coupling with the primary KH billow. If the optimization time interval includes pairing of the primary billows, the secondary elliptical perturbations are strongly suppressed during the pairing event, due to the significant disruption of the primary billow cores during pairing.

Evolution of a stratified rotating shear layer with horizontal shear. Part 2. Nonlinear evolution

2013 ◽  
Vol 732 ◽  
pp. 373-400 ◽  
Author(s):  
Eric Arobone ◽  
Sutanu Sarkar
Keyword(s):  
Shear Layer ◽  
Rotation Rate ◽  
Nonlinear Evolution ◽  
Mixing Layer ◽  
Vertical Mixing ◽  
Three Dimensional ◽  
Vertical Axis ◽  
Buoyancy Frequency ◽  
Omega 3 ◽  
Horizontal Shear

AbstractDirect numerical simulation is used to investigate the nonlinear evolution of a horizontally oriented mixing layer with uniform stable stratification and coordinate system rotation about the vertical axis. The important dimensional parameters governing inviscid dynamics are maximum shear $S(t)$, buoyancy frequency $N$, angular velocity of rotation $\Omega $ and characteristic shear thickness $L(t)$. The effect of rotation rate, $\Omega $, on the development of fluctuations in the shear layer is systematically studied in a regime of strong stratification. An instability mechanism, qualitatively distinct from the inertial instability, is found to deform columnar vortex cores in vertical planes for a strongly stratified rotating mixing layer. This mechanism emerges when centreline absolute vertical vorticity, $\langle {\omega }_{3} \rangle (t)+ 2\Omega $, is nearly zero as predicted by the linear stability analysis in Part 1 (J. Fluid. Mech., vol. 703, 2012, pp. 29–48). When the initial rotation rate is moderately anticyclonic, strong destabilization and a cascade to small scales is observed, consistent with prior studies involving horizontally sheared flow in the presence of rotation. Examination of enstrophy budgets in cases which are initially inertially unstable reveal the importance of baroclinic torque in maintaining lateral enstrophy fluctuations substantially beyond the time when the flow becomes inertially stable. The cyclonic stratified cases show weak nonlinearity in vortex dynamics. At high Reynolds number, despite the strong stratification, the flow exhibits three-dimensional, nonlinear dynamics and significant vertical mixing except for cases where the rotation is stabilizing.

Effective transition of steady flow over a square leading-edge plate

2012 ◽  
Vol 698 ◽  
pp. 335-357 ◽  
Author(s):  
Mark C. Thompson
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Three Dimensional ◽  
Leading Edge ◽  
Maximum Energy ◽  
Transient Growth ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Perturbation Mode ◽  
Optimal Mode

AbstractPrevious experimental studies have shown that the steady recirculation bubble that forms as the flow separates at the leading-edge corner of a long plate, becomes unsteady at relatively low Reynolds numbers of only a few hundreds. The reattaching shear layer irregularly releases two-dimensional vortices, which quickly undergo three-dimensional transition. Similar to the flow over a backward-facing step, this flow is globally stable at such Reynolds numbers, with transition to a steady three-dimensional flow as the first global instability to occur as the Reynolds number is increased to 393. Hence, it appears that the observed flow behaviour is governed by transient growth of optimal two-dimensional transiently growing perturbations (constructed from damped global modes) rather than a single three-dimensional unstable global mode. This paper quantifies the details of the transient growth of two- and three-dimensional optimal perturbations, and compares the predictions to other related cases examined recently. The optimal perturbation modes are shown to be highly concentrated in amplitude in the vicinity of the leading-edge corners and evolve to take the local shape of a Kelvin–Helmholtz shear-layer instability further downstream. However, the dominant mode reaches a maximum amplitude downstream of the position of the reattachment point of the shear layer. The maximum energy growth increases at 2.5 decades for each increment in Reynolds number of 100. Maximum energy growth of the optimal perturbation mode at a Reynolds number of 350 is greater than $1{0}^{4} $, which is typically an upper limit of the Reynolds number range over which it is possible to observe steady flow experimentally. While transient growth analysis concentrates on the evolution of wavepackets rather than continuous forcing, this appears consistent with longitudinal turbulence levels of up to 1 % for some water tunnels, and the fact that the optimal mode is highly concentrated close to the leading-edge corner so that an instantaneous projection of a perturbation field from a noisy inflow onto the optimal mode can be significant. Indeed, direct simulations with inflow noise reveal that a root-mean-square noise level of just 0.1 % is sufficient to trigger some unsteadiness at $\mathit{Re}= 350$, while a 0.5 % level results in sustained shedding. Three-dimensional optimal perturbation mode analysis was also performed showing that at $\mathit{Re}= 350$, the optimal mode has a spanwise wavelength of 11.7 plate thicknesses and is amplified 20 % more than the two-dimensional optimal disturbance. The evolved three-dimensional mode shows strong streamwise vortical structures aligned at a shallow angle to the plate top surface.

scholarly journals Non-Modal Three-Dimensional Optimal Perturbation Growth in Thermally Stratified Mixing Layers

Fluids ◽  
2021 ◽  
Vol 6 (1) ◽  
pp. 37
Author(s):  
Helena Vitoshkin ◽  
Alexander Gelfgat
Keyword(s):  
Mixing Layer ◽  
Three Dimensional ◽  
Stable Stratification ◽  
Base Flow ◽  
Transient Growth ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Dimensional Growth ◽  
Layer Flow ◽  
Perturbation Growth

A non-modal transient disturbances growth in a stably stratified mixing layer flow is studied numerically. The model accounts for a density gradient within a shear region, implying a heavier layer at the bottom. Numerical analysis of non-modal stability is followed by a full three-dimensional direct numerical simulation (DNS) with the optimally perturbed base flow. It is found that the transient growth of two-dimensional disturbances diminishes with the strengthening of stratification, while three-dimensional disturbances cause significant non-modal growth, even for a strong, stable stratification. This non-modal growth is governed mainly by the Holmboe modes and does not necessarily weaken with the increase of the Richardson number. The optimal perturbation consists of two waves traveling in opposite directions. Compared to the two-dimensional transient growth, the three-dimensional growth is found to be larger, taking place at shorter times. The non-modal growth is observed in linearly stable regimes and, in slightly linearly supercritical regimes, is steeper than that defined by the most unstable eigenmode. The DNS analysis confirms the presence of the structures determined by the transient growth analysis.

Three-dimensional instabilities in the boundary-layer flow over a long rectangular plate

2011 ◽  
Vol 681 ◽  
pp. 411-433 ◽  
Author(s):  
HEMANT K. CHAURASIA ◽  
MARK C. THOMPSON
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Vortical Flow ◽  
Forced Flow ◽  
Recirculation Region ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Dimensional Instability

A detailed numerical study of the separating and reattaching flow over a square leading-edge plate is presented, examining the instability modes governing transition from two- to three-dimensional flow. Under the influence of background noise, experiments show that the transition scenario typically is incompletely described by either global stability analysis or the transient growth of dominant optimal perturbation modes. Instead two-dimensional transition effectively can be triggered by the convective Kelvin–Helmholtz (KH) shear-layer instability; although it may be possible that this could be described alternatively in terms of higher-order optimal perturbation modes. At least in some experiments, observed transition occurs by either: (i) KH vortices shedding downstream directly and then almost immediately undergoing three-dimensional transition or (ii) at higher Reynolds numbers, larger vortical structures are shed that are also three-dimensionally unstable. These two paths lead to distinctly different three-dimensional arrangements of vortical flow structures. This paper focuses on the mechanisms underlying these three-dimensional transitions. Floquet analysis of weakly periodically forced flow, mimicking the observed two-dimensional quasi-periodic base flow, indicates that the two-dimensional vortex rollers shed from the recirculation region become globally three-dimensionally unstable at a Reynolds number of approximately 380. This transition Reynolds number and the predicted wavelength and flow symmetries match well with those of the experiments. The instability appears to be elliptical in nature with the perturbation field mainly restricted to the cores of the shed rollers and showing the spatial vorticity distribution expected for that instability type. Indeed an estimate of the theoretical predicted wavelength is also a good match to the prediction from Floquet analysis and theoretical estimates indicate the growth rate is positive. Fully three-dimensional simulations are also undertaken to explore the nonlinear development of the three-dimensional instability. These show the development of the characteristic upright hairpins observed in the experimental dye visualisations. The three-dimensional instability that manifests at lower Reynolds numbers is shown to be consistent with an elliptic instability of the KH shear-layer vortices in both symmetry and spanwise wavelength.

A numerical study of shear layer characteristics of low-speed transverse jets

2016 ◽  
Vol 790 ◽  
pp. 275-307 ◽  
Author(s):  
Prahladh S. Iyer ◽  
Krishnan Mahesh
Keyword(s):  
Velocity Profile ◽  
Shear Layer ◽  
Convective Instability ◽  
Numerical Study ◽  
Velocity Ratio ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
Dominant Frequencies

Direct numerical simulation (DNS) and dynamic mode decomposition (DMD) are used to study the shear layer characteristics of a jet in a crossflow. Experimental observations by Megerian et al. (J. Fluid Mech., vol. 593, 2007, pp. 93–129) at velocity ratios ($R=\overline{v}_{j}/u_{\infty }$) of 2 and 4 and Reynolds number ($Re=\overline{v}_{j}D/{\it\nu}$) of 2000 on the transition from absolute to convective instability of the upstream shear layer are reproduced. Point velocity spectra at different points along the shear layer show excellent agreement with experiments. The same frequency ($St=0.65$) is dominant along the length of the shear layer for $R=2$, whereas the dominant frequencies change along the shear layer for $R=4$. DMD of the full three-dimensional flow field is able to reproduce the dominant frequencies observed from DNS and shows that the shear layer modes are dominant for both the conditions simulated. The spatial modes obtained from DMD are used to study the nature of the shear layer instability. It is found that a counter-current mixing layer is obtained in the upstream shear layer. The corresponding mixing velocity ratio is obtained, and seen to delineate the two regimes of absolute or convective instability. The effect of the nozzle is evaluated by performing simulations without the nozzle while requiring the jet to have the same inlet velocity profile as that obtained at the nozzle exit in the simulations including the nozzle. The shear layer spectra show good agreement with the simulations including the nozzle. The effect of shear layer thickness is studied at a velocity ratio of 2 based on peak and mean jet velocity. The dominant frequencies and spatial shear layer modes from DNS/DMD are significantly altered by the jet exit velocity profile.

Early-period dynamics of an incompressible mixing layer

1994 ◽  
Vol 273 ◽  
pp. 31-82 ◽  
Author(s):  
Y. Bun ◽  
W. O. Criminale
Keyword(s):  
Kinetic Energy ◽  
Normal Mode ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Early Period ◽  
Early Time ◽  
Mode Analysis ◽  
Inviscid Fluid ◽  
Initial Growth ◽  
Initial Value

The evolution of three-dimensional disturbances in an incompressible mixing layer in an inviscid fluid is investigated as an initial-value problem. A Green's function approach is used to obtain a general space–time solution to the problem using a piecewise linear model for the basic flow, thereby making it possible to determine complete and closed-form analytical expressions for the variables with arbitrary input. Structure, kinetic energy, vorticity, and the evolution of material particles can be ascertained in detail. Moreover, these solutions represent the full three-dimensional disturbances that can grow exponentially or algebraically in time. For large time, the behaviour of these disturbances is dominated by the exponentially increasing discrete modes. For the early time, the behaviour is controlled by the algebraic variation due to the continuous spectrum. Contrary to Squire's theorem for normal mode analysis, the early-time behaviour indicates growth at comparable rates for all values of the wavenumbers and the initial growth of these disturbances is shown to rapidly increase. In particular, the disturbance kinetic energy can rise to a level approximately ten times its initial value before the exponentially growing normal mode prevails. As a result, the transient behaviour can trigger the roll-up of the mixing layer and its development into the well-known pattern that has been observed experimentally.

Turbulent wake behind side-by-side flat plates: computational study of interference effects

2018 ◽  
Vol 855 ◽  
pp. 1040-1073 ◽  
Author(s):  
Fatemeh H. Dadmarzi ◽  
Vagesh D. Narasimhamurthy ◽  
Helge I. Andersson ◽  
Bjørnar Pettersen
Keyword(s):  
Shear Layer ◽  
Computational Study ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Flat Plates ◽  
Layer Transition ◽  
Initial Flow ◽  
Plane Mixing Layer ◽  
Gap Flow ◽  
Dimensional Instability

The complex wake behind two side-by-side flat plates placed normal to the inflow direction has been explored in a direct numerical simulation study. Two gaps, $g=0.5d$ and $1.0d$ , were considered, both at a Reynolds number of 1000 based on the plate width $d$ and the inflow velocity. For gap ratio $g/d=0.5$ , the biased gap flow resulted in an asymmetric flow configuration consisting of a narrow wake with strong vortex shedding and a wide wake with no periodic near-wake shedding. Shear-layer transition vortices were observed in the wide wake, with characteristic frequency 0.6. For $g/d=1.0$ , two simulations were performed, started from a symmetric and an asymmetric initial flow field. A symmetric configuration of Kármán vortices resulted from the first simulation. Surprisingly, however, two different three-dimensional instability features were observed simultaneously along the span of the upper and lower plates. The spanwise wavelengths of these secondary streamwise vortices, formed in the braid regions of the primary Kármán vortices, were approximately $1d$ and $2d$ , respectively. The wake bursts into turbulence some $5d$ – $10d$ downstream. The second simulation resulted in an asymmetric wake configuration similar to the asymmetric wake found for the narrow gap $0.5d$ , with the appearance of shear-layer instabilities in the wide wake. The analogy between a plane mixing layer and the separated shear layer in the wide wake was examined. The shear-layer frequencies obtained were in close agreement with the frequency of the most amplified wave based on linear stability analysis of a plane mixing layer.

scholarly journals Optimal Perturbations of an Oceanic Vortex Lens

Fluids ◽  
2018 ◽  
Vol 3 (3) ◽  
pp. 63 ◽  
Author(s):  
Thomas Meunier ◽  
Claire Ménesguen ◽  
Xavier Carton ◽  
Sylvie Le Gentil ◽  
Richard Schopp
Keyword(s):  
Normal Mode ◽  
Growth Rates ◽  
Time Interval ◽  
Time Intervals ◽  
Horizontal Structure ◽  
Azimuthal Wave ◽  
Wave Numbers ◽  
Non Linear ◽  
Long Time ◽  
Short Time

The stability properties of a vortex lens are studied in the quasi geostrophic (QG) framework using the generalized stability theory. Optimal perturbations are obtained using a tangent linear QG model and its adjoint. Their fine-scale spatial structures are studied in details. Growth rates of optimal perturbations are shown to be extremely sensitive to the time interval of optimization: The most unstable perturbations are found for time intervals of about 3 days, while the growth rates continuously decrease towards the most unstable normal mode, which is reached after about 170 days. The horizontal structure of the optimal perturbations consists of an intense counter-shear spiralling. It is also extremely sensitive to time interval: for short time intervals, the optimal perturbations are made of a broad spectrum of high azimuthal wave numbers. As the time interval increases, only low azimuthal wave numbers are found. The vertical structures of optimal perturbations exhibit strong layering associated with high vertical wave numbers whatever the time interval. However, the latter parameter plays an important role in the width of the vertical spectrum of the perturbation: short time interval perturbations have a narrow vertical spectrum while long time interval perturbations show a broad range of vertical scales. Optimal perturbations were set as initial perturbations of the vortex lens in a fully non linear QG model. It appears that for short time intervals, the perturbations decay after an initial transient growth, while for longer time intervals, the optimal perturbation keeps on growing, quickly leading to a non-linear regime or exciting lower azimuthal modes, consistent with normal mode instability. Very long time intervals simply behave like the most unstable normal mode. The possible impact of optimal perturbations on layering is also discussed.

Aspects of Shear Layer Unsteadiness in a Three-Dimensional Supersonic Wake

2005 ◽  
Vol 127 (6) ◽  
pp. 1085-1094 ◽  
Author(s):  
Alan L. Kastengren ◽  
J. Craig Dutton
Keyword(s):  
Shear Layer ◽  
Stream Flow ◽  
Free Stream ◽  
Angle Of Attack ◽  
Three Dimensional ◽  
Base Flow ◽  
Recirculation Region ◽  
Side View ◽  
Near Wake ◽  
Supersonic Wake

The near wake of a blunt-base cylinder at 10° angle-of-attack to a Mach 2.46 free-stream flow is visualized at several locations to study unsteady aspects of its structure. In both side-view and end-view images, the shear layer flapping grows monotonically as the shear layer develops, similar to the trends seen in a corresponding axisymmetric supersonic base flow. The interface convolution, a measure of the tortuousness of the shear layer, peaks for side-view and end-view images during recompression. The high convolution for a septum of fluid seen in the middle of the wake indicates that the septum actively entrains fluid from the recirculation region, which helps to explain the low base pressure for this wake compared to that for a corresponding axisymmetric wake.

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