Asymptotic Dynamics of High Dynamic Range Stratified Turbulence.

Direct numerical simulations of homogeneous sheared and stably stratified turbulence are considered to probe the asymptotic high dynamic range regime suggested by Gargett et al. J. Fluid Mech. 144, 231 (1984)10.1017/S0022112084001592 and Shih et al. J. Fluid Mech. 525, 193 (1999)10.1017/S0022112004002587. We consider statistically stationary configurations of the flow that span three decades in dynamic range defined by the separation between the Ozmidov length scale L_{O}=sqrt[ε/N^{3}] and the Kolmogorov length scale L_{K}=(ν^{3}/ε)^{1/4}, up to Re_{b}≡(L_{O}/L_{K})^{4/3}=ε/(νN^{2})∼O(1000), where ε is the mean turbulent kinetic energy dissipation rate, ν is the kinematic viscosity, and N is the buoyancy frequency. We isolate the effects of Re_{b}, particularly on irreversible mixing, from the effects of other flow parameters of stratified and sheared turbulence. Specifically, we evaluate the influence of dynamic range independent of initial conditions. We present evidence that the flow approaches an asymptotic state for Re_{b}⪆300, characterized both by an asymptotic partitioning between the potential and kinetic energies and by the approach of components of the dissipation rate to their expected values under the assumption of isotropy. As Re_{b} increases above 100, there is a slight decrease in the turbulent flux coefficient Γ=χ/ε, where χ is the dissipation rate of buoyancy variance, but, for this flow, there is no evidence of the commonly suggested Γ∝Re_{b}^{-1/2} dependence when 100≤Re_{b}≤1000.

Detrainment of plumes from vertically distributed sources.

We present experimental results demonstrating that, for the turbulent plume from a buoyancy source that is vertically distributed over the full area of a wall, detrainment qualitatively changes the shape of the ambient buoyancy profile that develops in a sealed space. Theoretical models with one-way-entrainment predict stratifications that are qualitatively different from the stratifications measured in experiments. A peeling plume model, where density and vertical velocity vary linearly across the width of the plume, so that plume fluid "peels" off into the ambient at intermediate heights, more accurately captures the shape of the ambient buoyancy profiles measured in experiments than a conventional one-way-entrainment model does.

Instabilities of interacting vortex rings generated by an oscillating disk.

We propose a natural model to probe in a controlled fashion the instability of interacting vortex rings shed from the edge of an oblate spheroid disk of major diameter c, undergoing oscillations of frequency f_{0} and amplitude A. We perform a Floquet stability analysis to determine the characteristics of the instability modes, which depend strongly on the azimuthal (integer) wave number m. We vary two key control parameters, the Keulegan-Carpenter number K_{C}=2πA/c and the Stokes number ß=f_{0}c^{2}/ν, where ν is the kinematic viscosity of the fluid. We observe two distinct flow regimes. First, for sufficiently small ß, and hence low frequency of oscillation corresponding to relatively weak interaction between sequentially shedding vortex rings, symmetry breaking occurs directly to a single unstable mode with m=1. Second, for sufficiently large yet fixed values of ß, corresponding to a higher oscillation frequency and hence stronger ring-ring interaction, the onset of asymmetry is predicted to occur due to two branches of high m instabilities as the amplitude is increased, with m=1 structures being dominant only for sufficiently large values of K_{C}. These two branches can be distinguished by the phase properties of the vortical structures above and below the disk. The region in (K_{C},ß) parameter space where these two high m instability branches arise can be described accurately in terms of naturally defined Reynolds numbers, using appropriately chosen characteristic length scales. We subsequently carry out direct numerical simulations of the fully three-dimensional flow to verify the principal characteristics of the Floquet analysis, in particular demonstrating that high wave-number symmetry-breaking generically occurs when vortex rings sequentially interact sufficiently strongly.

Three-dimensional transition after wake deflection behind a flapping foil.

We report the inherently three-dimensional linear instabilities of a propulsive wake, produced by a flapping foil, mimicking the caudal fin of a fish or the wing of a flying animal. For the base flow, three sequential wake patterns appear as we increase the flapping amplitude: Bénard-von Kármán (BvK) vortex streets; reverse BvK vortex streets; and deflected wakes. Imposing a three-dimensional spanwise periodic perturbation, we find that the resulting Floquet multiplier |µ| indicates an unstable "short wavelength" mode at wave number ß=30, or wavelength λ=0.21 (nondimensionalized by the chord length) at sufficiently high flow Reynolds number Re=Uc/ν≃600, where U is the upstream flow velocity, c is the chord length, and ν is the kinematic viscosity of the fluid. Another, "long wavelength" mode at ß=6 (λ=1.05) becomes critical at somewhat higher Reynolds number, although we do not expect that this mode would be observed physically because its growth rate is always less than the short wavelength mode, at least for the parameters we have considered. The long wavelength mode has certain similarities with the so-called mode A in the drag wake of a fixed bluff body, while the short wavelength mode appears to have a period of the order of twice that of the base flow, in that its structure seems to repeat approximately only every second cycle of the base flow. Whether it is appropriate to classify this mode as a truly subharmonic mode or as a quasiperiodic mode is still an open question however, worthy of a detailed parametric study with various flapping amplitudes and frequencies.

Variational framework for flow optimization using seminorm constraints.

We present a general variational framework designed to consider constrained optimization and sensitivity analysis of spatially and temporally evolving flows defined as solutions of partial differential equations. We particularly focus on seminorm constraints which naturally arise for instance when the quantity which we wish to optimize can have contributions from several terms in the PDE through different physical mechanisms in a specific physical system. We show that this case implicitly requires that constraints be placed on the magnitude of complementary (with respect to the first constraining seminorm) seminorms of initial perturbations such that the sum of these complementary seminorms defines a total "true" norm of the state vector. A simple (true) norm constraint naturally satisfies this property. Therefore, the use of this framework requires the introduction of new parameters which describe the relative magnitude of the initial perturbation state vector calculated using the various constrained complementary seminorms to the magnitude calculated using the true norm, even for linear problems. We demonstrate that any required optimization has to be carried out by prescribing these new parameters as initial conditions on the admissible perturbations; the influence and significance of each seminorm component, partitioning the initial total norm of the perturbation, can then be considered quantitatively. To demonstrate the utility of this framework, we consider an idealized problem, the (linear) nonmodal stability analysis of a mean flow given by a "Reynolds averaging" of the one-dimensional stochastically forced Burgers equation. We close the mean flow equation by introducing a turbulent viscosity to model the turbulent mixing, which we allow to evolve subject to a new transport equation. Since we are interested in optimizing the relative amplification of the perturbation kinetic energy (i.e., the perturbation's "gain") this problem naturally requires the use of our new framework, as the kinetic energy is a seminorm of the full state velocity-viscosity vector, with a new adjustable parameter, describing the ratio of an appropriate viscosity seminorm to the sum of this viscosity seminorm and the kinetic energy seminorm. Using this framework, we demonstrate that the dynamics of the full system, allowing the turbulent viscosity to evolve subject to its transport equation, is qualitatively different from the behavior when the turbulent viscosity is "frozen" at a fixed, mean value, since a new mechanism of perturbation energy production appears, through the coupling of the evolving turbulent viscosity perturbation and the mean velocity field.