scholarly journals Optimal perturbation growth in axisymmetric intrusions

2016 ◽  
Vol 811 ◽  
Author(s):  
Bruce R. Sutherland ◽  
C. P. Caulfield
Keyword(s):  
Gravity Current ◽  
Optimal Growth ◽  
Base Flow ◽  
Optimal Perturbation ◽  
Target Time ◽  
Experimental Parameters ◽  
Radial Spokes ◽  
Perturbation Energy ◽  
The One ◽  
Perturbation Growth

The cylindrical lock-release laboratory experiments of Sutherland & Nault (J. Fluid Mech., vol. 586, 2007, pp. 109–118) showed that a radially advancing symmetric intrusive gravity current spreads not as an expanding annulus (as is the case for bottom-propagating gravity currents), but rather predominantly along azimuthally periodic radial ‘spokes’. Here, we investigate whether the spokes are associated with azimuthal perturbations that undergo ‘optimal’ growth. We use a nonlinear axisymmetric numerical simulation initialised with the experimental parameters to compute the time-evolving axisymmetric base state of the collapsing lock fluid. Using fields from this rapidly evolving base state together with the linearised perturbation equations and their adjoint, the ‘direct–adjoint looping’ method is employed to identify, as a function of the azimuthal wavenumber $m$, the vertical–radial structure of the set of initial perturbations that exhibit the largest total perturbation energy gain over a target time $T$. Of this set of perturbations, the one that extracts energy fastest, and so is expected to be observed to emerge first from the base flow, has azimuthal wavenumber comparable to the number of spokes observed in the experiment.

scholarly journals Hairpin-like optimal perturbations in plane Poiseuille flow

2015 ◽  
Vol 775 ◽  
Author(s):  
Mirko Farano ◽  
Stefania Cherubini ◽  
Jean-Christophe Robinet ◽  
Pietro De Palma
Keyword(s):  
Poiseuille Flow ◽  
Nonlinear Effects ◽  
Optimal Growth ◽  
Turbulent Shear ◽  
Hairpin Vortex ◽  
Plane Poiseuille Flow ◽  
Hairpin Vortices ◽  
Optimal Perturbation ◽  
Target Time ◽  
Lagrange Multiplier Technique

In this work it is shown that hairpin vortex structures can be the outcome of a nonlinear optimal growth process, in a similar way as streaky structures can be the result of a linear optimal growth mechanism. With this purpose, nonlinear optimizations based on a Lagrange multiplier technique coupled with a direct-adjoint iterative procedure are performed in a plane Poiseuille flow at subcritical values of the Reynolds number, aiming at quickly triggering nonlinear effects. Choosing a suitable time scale for such an optimization process, it is found that the initial optimal perturbation is composed of sweeps and ejections resulting in a hairpin vortex structure at the target time. These alternating sweeps and ejections create an inflectional instability occurring in a localized region away from the wall, generating the head of the primary and secondary hairpin structures, quickly inducing transition to turbulent flow. This result could explain why transitional and turbulent shear flows are characterized by a high density of hairpin vortices.

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.

Stability of Viscoelastic Channel Flow

2007 ◽  
Author(s):  
Nariman Ashrafi
Keyword(s):  
Reynolds Number ◽  
Viscosity Ratio ◽  
Weissenberg Number ◽  
Base Flow ◽  
Galerkin Projection ◽  
Large Reynolds Number ◽  
One Dimensional ◽  
Stress Effects ◽  
The Stability ◽  
The One

The nonlinear stability and bifurcation of the one-dimensional channel (Poiseuille) flow is examined for a Johnson-Segalman fluid. The velocity and stress are represented by orthonormal functions in the transverse direction to the flow. The flow field is obtained from the conservation and constitutive equations using the Galerkin projection method. Both inertia and normal stress effects are included. The stability picture is dramatically influenced by the viscosity ratio. The range of shear rate or Weissenberg number for which the base flow is unstable increases from zero as the fluid deviates from the Newtonian limit as decreases. Typically, two turning points are observed near the critical Weissenberg numbers. The transient response is heavily influenced by the level of inertia. It is found that the flow responds oscillatorily. When the Reynolds number is small, and monotonically at large Reynolds number when elastic effects are dominated by inertia.

Characteristics of the leading Lyapunov vector in a turbulent channel flow

2018 ◽  
Vol 849 ◽  
pp. 942-967 ◽  
Author(s):  
Nikolay Nikitin
Keyword(s):  
Turbulent Flow ◽  
Buffer Layer ◽  
Stokes Equations ◽  
Initial Conditions ◽  
Spatial Scales ◽  
Plane Channel ◽  
Base Flow ◽  
Lyapunov Vector ◽  
Perturbation Growth ◽  
Near Wall

The values of the highest Lyapunov exponent (HLE)$\unicode[STIX]{x1D706}_{1}$for turbulent flow in a plane channel at Reynolds numbers up to$Re_{\unicode[STIX]{x1D70F}}=586$are determined. The instantaneous and statistical properties of the corresponding leading Lyapunov vector (LLV) are investigated. The LLV is calculated by numerical solution of the Navier–Stokes equations linearized about the non-stationary base solution corresponding to the developed turbulent flow. The base turbulent flow is calculated in parallel with the calculation of the evolution of the perturbations. For arbitrary initial conditions, the regime of exponential growth${\sim}\exp (\unicode[STIX]{x1D706}_{1}t)$which corresponds to the approaching of the perturbation to the LLV is achieved already at$t^{+}<50$. It is found that the HLE increases with increasing Reynolds number from$\unicode[STIX]{x1D706}_{1}^{+}\approx 0.021$at$Re_{\unicode[STIX]{x1D70F}}=180$to$\unicode[STIX]{x1D706}_{1}^{+}\approx 0.026$at$Re_{\unicode[STIX]{x1D70F}}=586$. The LLV structures are concentrated mainly in a region of the buffer layer and are manifested in the form of spots of increased fluctuation intensity localized both in time and space. The root-mean-square (r.m.s.) profiles of the velocity and vorticity intensities in the LLV are qualitatively close to the corresponding profiles in the base flow with artificially removed near-wall streaks. The difference is the larger concentration of LLV perturbations in the vicinity of the buffer layer and a relatively larger (by approximately 80 %) amplitude of the vorticity pulsations. Based on the energy spectra of velocity and vorticity pulsations, the integral spatial scales of the LLV structures are determined. It is found that LLV structures are on average twice narrower and twice shorter than the corresponding structures of the base flow. The contribution of each of the terms entering into the expression for the production of the perturbation kinetic energy is determined. It is shown that the process of perturbation development is essentially dictated by the inhomogeneity of the base flow, as well as by the presence of transversal motion in it. Neglecting of these factors leads to a significant underestimation of the perturbation growth rate. The presence of near-wall streaks in the base flow, on the contrary, does not play a significant role in the development of the LLV perturbations. Artificial removal of streaks from the base flow does not change the character of the perturbation growth.

Modal and non-modal stability of boundary layers forced by spanwise wall oscillations

2015 ◽  
Vol 778 ◽  
pp. 389-427 ◽  
Author(s):  
M. J. Philipp Hack ◽  
Tamer A. Zaki
Keyword(s):  
Boundary Layers ◽  
Base Flow ◽  
Normal Velocity ◽  
Lower Growth ◽  
Stokes Layer ◽  
Time Harmonic ◽  
Inflection Points ◽  
Lower Growth Rate ◽  
Inviscid Instability ◽  
Perturbation Growth

Modal and non-modal perturbation growth in boundary layers subjected to time-harmonic spanwise wall motion are examined. The superposition of the streamwise Blasius flow and the spanwise Stokes layer can lead to strong modal amplification during intervals of the base-flow period. Linear stability analysis of frozen phases of the base state demonstrates that this growth is due to an inviscid instability, which is related to the inflection points of the spanwise Stokes layer. The generation of new inflection points at the wall and their propagation towards the free stream leads to mode crossing when tracing the most unstable mode as a function of phase. The fundamental mode computed in Floquet analysis has a considerably lower growth rate than the instantaneous eigenfunctions. Furthermore, the algebraic lift-up mechanism that causes the formation of Klebanoff streaks is examined in transient growth analyses. The wall forcing significantly weakens the wall-normal velocity perturbations associated with lift-up. This effect is attributed to the formation of a pressure field which redistributes energy from the wall-normal to the spanwise velocity perturbations. The results from linear theory explain observations from direct numerical simulations of breakdown to turbulence in the same flow configuration by Hack & Zaki (J. Fluid Mech., vol. 760, 2014a, pp. 63–94). When bypass mechanisms are dominant, the flow is stabilized due to the weaker non-modal growth. However, at high amplitudes of wall oscillation, transition is promoted due to fast growth of the modal instability.

OPTIMAL PERTURBATION OF UNCERTAIN SYSTEMS

2002 ◽  
Vol 02 (03) ◽  
pp. 395-402 ◽  
Author(s):  
BRIAN F. FARRELL ◽  
PETROS J. IOANNOU
Keyword(s):  
Dynamical System ◽  
Uncertain Systems ◽  
Initial Condition ◽  
Statistical Characterization ◽  
Optimal Perturbation ◽  
Physical Systems ◽  
Perturbation Problem ◽  
Ensemble Mean ◽  
Perturbation Growth

In studies of perturbation dynamics in physical systems, certain specification of the governing perturbation dynamical system is generally lacking, either because the perturbation system is imperfectly known or because its specification is intrinsically uncertain, while a statistical characterization of the perturbation dynamical system is often available. In this report exact and asymptotically valid equations are derived for the ensemble mean and moment dynamics of uncertain systems. These results are used to extend the concept of optimal deterministic perturbation of certain systems to uncertain systems. Remarkably, the optimal perturbation problem has a simple solution: In uncertain systems there is a sure initial condition producing the greatest expected second moment perturbation growth.

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.

Summary-The Weaning Process

PEDIATRICS ◽  
1985 ◽  
Vol 75 (1) ◽  
pp. 214-215
Author(s):  
Laurence Finberg
Keyword(s):  
Human Milk ◽  
Breast Feeding ◽  
Growth Curves ◽  
Optimal Growth ◽  
Health Statistics ◽  
Modern Times ◽  
The Family ◽  
The One ◽  
Breast Fed

Early in this symposium, Räihä (p 136) called attention to the biologic truism that breast-feeding evolved so that the composition of human milk would be optimal for the survival of the species, which must include the mother's survival as well as the infant's. However, optimal for survival in infancy may not be the same as optimal for life in the fifth, sixth, and subsequent decades of modern life. Therefore, as we consider approaches to infant feeding, these considerations, unanswerable at present, should be continuously in our thinking, at no time more than when we consider the weaning period, which, though variously defined, presents the most important nutritional challenge of modern times. Whitehead has provided us with the news that our understanding of energy requirements for infants is surprisingly imperfect. His data suggest that sole breast-feeding by well-nourished mothers has sufficient calories for optimal growth on the average for 3 months. After this point, the average infant requires another source of calories. This conclusion comes from calculations derived from the National Center for Health Statistics growth curves on the one hand and actual milk production on the other. Although there is room for improvement of the data base, there is certainly a suggestion that weaning should begin at about 3 months of age in the breast-fed infant. This point is underscored by Underwood, who views weaning as a long-term process in which calories from other foods complement human milk until the infant moves to the family diet. In some societies, such a process may extend over several years, and the presence in human milk of the digestive enzymes amylase and lipase adds an aura of naturalness to such a process.

scholarly journals Adjoint Sensitivity of North Pacific Atmospheric River Forecasts

2019 ◽  
Vol 147 (6) ◽  
pp. 1871-1897 ◽  
Author(s):  
Carolyn A. Reynolds ◽  
James D. Doyle ◽  
F. Martin Ralph ◽  
Reuben Demirdjian
Keyword(s):  
Kinetic Energy ◽  
Forecast Error ◽  
Meridional Wind ◽  
Precipitation Forecast ◽  
Initial State ◽  
Adjoint Sensitivity ◽  
Optimal Perturbation ◽  
Low Level ◽  
Atmospheric River ◽  
Perturbation Growth

Abstract The initial-state sensitivity and optimal perturbation growth for 24- and 36-h forecasts of low-level kinetic energy and precipitation over California during a series of atmospheric river (AR) events that took place in early 2017 are explored using adjoint-based tools from the Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS). This time period was part of the record-breaking winter of 2016–17 in which several high-impact ARs made landfall in California. The adjoint sensitivity indicates that both low-level winds and precipitation are most sensitive to mid- to lower-tropospheric perturbations in the initial state in and near the ARs. A case study indicates that the optimal moist perturbations occur most typically along the subsaturated edges of the ARs, in a warm conveyor belt region. The sensitivity to moisture is largest, followed by temperature and winds. A 1 g kg−1 perturbation to moisture may elicit twice as large a response in kinetic energy and precipitation as a 1 m s−1 perturbation to the zonal or meridional wind. In an average sense, the sensitivity and related optimal perturbations are very similar for the kinetic energy and precipitation response functions. However, on a case-by-case basis, differences in the sensitivity magnitude and optimal perturbation structures result in substantially different forecast perturbations, suggesting that optimal adaptive observing strategies should be metric dependent. While the nonlinear evolved perturbations are usually smaller (by about 20%, on average) than the expected linear perturbations, the optimal perturbations are still capable of producing rapid nonlinear perturbation growth. The positive correlation between sensitivity magnitude and wind speed forecast error or precipitation forecast differences supports the relevance of adjoint-based calculations for predictability studies.

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