A variational framework for flow optimization using semi-norm constraints

When considering a general system of equations describing the space-time evolution (flow) of one or several variables, the problem of the optimization over a finite period of time of a measure of the state variable at the final time is a problem of great interest in many fields. Methods already exist in order to solve this kind of optimization problem, but sometimes fail when the constraint bounding the state vector at the initial time is not a norm, meaning that some part of the state vector remains unbounded and might cause the optimization procedure to diverge. In order to regularize this problem, we propose a general method which extends the existing optimization framework in a self-consistent manner. We first derive this framework extension, and then apply it to a problem of interest. Our demonstration problem considers the transient stability properties of a one-dimensional (in space) averaged turbulent model with a space- and time-dependent model "turbulent viscosity". We believe this work has a lot of potential applications in the fluid dynamics domain for problems in which we want to control the influence of separate components of the state vector in the optimization process.Comment: 30 page

The effects of Prandtl number on the nonlinear dynamics of Kelvin-Helmholtz instability in two dimensions

It is known that the pitchfork bifurcation of Kelvin-Helmholtz instability occurring at minimum gradient Richardson number in viscous stratified shear flows can be subcritical or supercritical depending on the value of the Prandtl number,. Here, we study stratified shear flow restricted to two dimensions at finite Reynolds number, continuously forced to have a constant background density gradient and a hyperbolic tangent shear profile, corresponding to the 'Drazin model' base flow. Bifurcation diagrams are produced for fluids with (typical for air), 3 and (typical for water). For and, steady billow-like solutions are found to exist for strongly stable stratification of beyond. Interestingly, these solutions are not a direct product of a Kelvin-Helmholtz instability, having half the wavelength of the linear instability, and arising through a superharmonic bifurcation. These short-wavelength states can be tracked down to at least and act as instigators of complex dynamics, even in strongly stratified flows. Direct numerical simulations of forced and unforced two-dimensional flows are performed, which support the results of the bifurcation analyses. Perturbations are observed to grow approximately exponentially from random initial conditions where no modal instability is predicted by a linear stability analysis.</p

Optimal mixing in two-dimensional plane Poiseuille flow at finite Peclet number

International audienceWe consider the nonlinear optimisation of the mixing of a passive scalar, initially arranged in two layers, in a two-dimensional plane Poiseuille flow at finite Reynolds and Péclet numbers, below the linear instability threshold. We use a nonlinear-adjoint-looping approach to identify optimal perturbations leading to maximum time-averaged energy as well as maximum mixing in a freely evolving flow, measured through the minimisation of either the passive scalar variance or the so-called mix-norm, as defined by Mathew, Mezić & Petzold (Physica D, vol. 211, 2005, pp. 23-46). We show that energy optimisation appears to lead to very weak mixing of the scalar field whereas the optimal mixing initial perturbations, despite being less energetic, are able to homogenise the scalar field very effectively. For sufficiently long time horizons, minimising the mix-norm identifies optimal initial perturbations which are very similar to those which minimise scalar variance, demonstrating that minimisation of the mix-norm is an excellent proxy for effective mixing in this finite-Péclet-number bounded flow. By analysing the time evolution from initial perturbations of several optimal mixing solutions, we demonstrate that our optimisation method can identify the dominant underlying mixing mechanism, which appears to be classical Taylor dispersion, i.e. shear-augmented diffusion. The optimal mixing proceeds in three stages. First, the optimal mixing perturbation, energised through transient amplitude growth, transports the scalar field across the channel width. In a second stage, the mean flow shear acts to disperse the scalar distribution leading to enhanced diffusion. In a final third stage, linear relaxation diffusion is observed. We also demonstrate the usefulness of the developed variational framework in a more realistic control case: mixing optimisation by prescribed streamwise velocity boundary conditions

Optimal perturbation growth on a breaking internal gravity wave

The breaking of internal gravity waves in the abyssal ocean is thought to be responsible for much of the mixing necessary to close oceanic buoyancy budgets. The exact mechanism by which these waves break down into turbulence remains an active area of research and can have significant implications on the mixing efficiency. Recent evidence has suggested that both shear instabilities and convective instabilities play a significant role in the breaking of an internal gravity wave in a high Richardson number mean shear flow. We perform a systematic analysis of the stability of a configuration of an internal gravity wave superimposed on a background shear flow first considered by Howland et al. (J. Fluid Mech., vol. 921, 2021, A24), using direct–adjoint looping to find the perturbation giving maximal energy growth on this evolving flow. We find that three-dimensional, convective mechanisms produce greater energy growth than their two-dimensional counterparts. In particular, we find close agreement with the direct numerical simulations of Howland et al. (J. Fluid Mech., 2021, in press), which demonstrated a clear three-dimensional mechanism causing breakdown to turbulence. The results are shown to hold at realistic Prandtl numbers. At low mean Richardson numbers, two-dimensional, shear-driven mechanisms produce greater energy growth

Optimal perturbation growth on a breaking internal gravity wave

The breaking of internal gravity waves in the abyssal ocean is thought to be responsible for much of the mixing necessary to close oceanic buoyancy budgets. The exact mechanism by which these waves break down into turbulence remains an active area of research and can have significant implications on the mixing efficiency. Recent evidence has suggested that both shear instabilities and convective instabilities play a significant role in the breaking of an internal gravity wave in a high Richardson number mean shear flow. We perform a systematic analysis of the stability of a configuration of an internal gravity wave superimposed on a background shear flow first considered by Howland et al. (J. Fluid Mech., vol. 921, 2021, A24), using direct–adjoint looping to find the perturbation giving maximal energy growth on this evolving flow. We find that three-dimensional, convective mechanisms produce greater energy growth than their two-dimensional counterparts. In particular, we find close agreement with the direct numerical simulations of Howland et al. (J. Fluid Mech., 2021, in press), which demonstrated a clear three-dimensional mechanism causing breakdown to turbulence. The results are shown to hold at realistic Prandtl numbers. At low mean Richardson numbers, two-dimensional, shear-driven mechanisms produce greater energy growth

Shear-induced breaking of internal gravity waves

Motivated by observations of turbulence in the strongly stratified ocean thermocline, we use direct numerical simulations to investigate the interaction of a sinusoidal shear flow and a large-amplitude internal gravity wave. Despite strong nonlinearities in the flow and a lack of scale separation, we find that linear ray tracing theory is qualitatively useful in describing the early development of the flow as the wave is refracted by the shear. Consistent with the linear theory, the energy of the wave accumulates in regions of negative mean shear where we observe evidence of convective and shear instabilities. Streamwise-aligned convective rolls emerge the fastest, but their contribution to irreversible mixing is dwarfed by shear-driven billow structures that develop later. Although the wave strongly distorts the buoyancy field on which these billows develop, the mixing efficiency of the subsequent turbulence is similar to that arising from Kelvin-Helmholtz instability in a stratified shear layer. We run simulations at Reynolds numbers of 5000 and 8000, and vary the initial amplitude of the internal gravity wave. For high values of initial wave amplitude, the results are qualitatively independent of $Re$. Smaller initial wave amplitudes delay the onset of the instabilities, and allow for significant laminar diffusion of the internal wave, leading to reduced turbulent activity. We discuss the complex interaction between the mean flow, internal gravity wave and turbulence, and its implications for internal wave-driven mixing in the ocean.Comment: 27 pages, 12 figures, accepted to J. Fluid. Mec

Quantifying mixing and available potential energy in vertically periodic simulations of stratified flows

Turbulent mixing exerts a significant influence on many physical processes in the ocean. In a stably stratified Boussinesq fluid, this irreversible mixing describes the conversion of available potential energy (APE) to background potential energy (BPE). In some settings the APE framework is difficult to apply and approximate measures are used to estimate irreversible mixing. For example, numerical simulations of stratified turbulence often use triply periodic domains to increase computational efficiency. In this setup however, BPE is not uniquely defined and the method of Winters et al. (1995, J. Fluid Mech., 289) cannot be directly applied to calculate the APE. We propose a new technique to calculate APE in periodic domains with a mean stratification. By defining a control volume bounded by surfaces of constant buoyancy, we can construct an appropriate background buoyancy profile $b_\ast(z,t)$ and accurately quantify diapycnal mixing in such systems. This technique also permits the accurate calculation of a finite amplitude local APE density in periodic domains. The evolution of APE is analysed in various turbulent stratified flow simulations. We show that the mean dissipation rate of buoyancy variance $\chi$ provides a good approximation to the mean diapycnal mixing rate, even in flows with significant variations in local stratification. When quantifying measures of mixing efficiency in transient flows, we find significant variation depending on whether laminar diffusion of a mean flow is included in the kinetic energy dissipation rate. We discuss how best to interpret these results in the context of quantifying diapycnal diffusivity in real oceanographic flows.Comment: 28 pages, 10 figures, accepted to J. Fluid Mec

Two-color holography concept (T-CHI)

The Material Processing in the Space Program of NASA-MSFC was active in developing numerous optical techniques for the characterization of fluids in the vicinity of various materials during crystallization and/or solidification. Two-color holographic interferometry demonstrates that temperature and concentration separation in transparent (T-CHI) model systems is possible. The experiments were performed for particular (succinonitrile) systems. Several solutions are possible in Microgravity Sciences and Applications (MSA) experiments on future Shuttle missions. The theory of the T-CHI concept is evaluated. Although particular cases are used for explanations, the concepts developed will be universal. A breadboard system design is also presented for ultimate fabrication and testing of theoretical findings. New developments in holography involving optical fibers and diode lasers are also incorporated

Recent developments in the determination of the amplitude and phase of quantum oscillations for the linear chain of coupled orbits

De Haas-van Alphen oscillations are studied for Fermi surfaces (FS) illustrating the model proposed by Pippard in the early sixties, namely the linear chain of orbits coupled by magnetic breakdown. This FS topology is relevant for many multiband quasi-two dimensional (q-2D) organic metals such as $\kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$ and $\theta$-(BEDT-TTF)$_4$CoBr$_4$(C$_6$H$_4$Cl$_2$) which are considered in detail. Whereas the Lifshits-Kosevich model only involves a first order development of field- and temperature-dependent damping factors, second order terms may have significant contribution on the Fourier components amplitude for such q-2D systems at high magnetic field and low temperature. The strength of these second order terms depends on the relative value of the involved damping factors, which are in turns strongly dependent on parameters such as the magnetic breakdown field, effective masses and, most of all, effective Land\'{e} factors. In addition, the influence of field-dependent Onsager phase factors on the oscillation spectra is considered.Comment: arXiv admin note: text overlap with arXiv:1304.665