Hayder Salman

A Method for Assimilating Lagrangian Data into a Shallow-Water-Equation Ocean Model

Lagrangian measurements provide a significant portion of the data collected in the ocean. Difficulties arise in their assimilation, however, since Lagrangian data are described in a moving frame of reference that does not correspond to the fixed grid locations used to forecast the prognostic flow variables. A new method is presented for assimilating Lagrangian data into models of the ocean that removes the need for any commonly used approximations. This is accomplished by augmenting the state vector of the prognostic variables with the Lagrangian drifter coordinates at assimilation. It is shown that this method is best formulated using the ensemble Kalman filter, resulting in an algorithm that is essentially transparent for assimilating Lagrangian data. The method is tested using a set of twin experiments on the shallow-water system of equations for an unsteady double-gyre flow configuration. Numerical simulations show that this method is capable of correcting the flow even if the assimilation time interval is of the order of the Lagrangian autocorrelation time scale (TL) of the flow. These results clearly demonstrate the benefits of this method over other techniques that require assimilation times of 20%–50% of TL, a direct consequence of the approximations introduced in assimilating their Lagrangian data. Detailed parametric studies show that this method is particularly effective if the classical ideas of localization developed for the ensemble Kalman filter are extended to the Lagrangian formulation used here. The method that has been developed, therefore, provides an approach that allows one to fully realize the potential of Lagrangian data for assimilation in more realistic ocean models.

Using flow geometry for drifter deployment in Lagrangian data assimilation

Methods of Lagrangian data assimilation (LaDA) require carefully chosen sites for optimal drifter deployments. In this work, we investigate a directed drifter deployment strategy with a recently developed LaDA method employing an augmented state vector formulation for an Ensemble Kalman filter. We test our directed drifter deployment strategy by targeting Lagrangian coherent flow structures of an unsteady double gyre flow to analyse how different release sites influence the performance of the method.We consider four different launch methods; a uniform launch, a saddle launch in which hyperbolic trajectories are targeted, a vortex centre launch, and a mixed launch targeting both saddles and centres. We show that global errors in the flow field require good dispersion of the drifters which can be realized with the saddle launch. Local errors on the other hand are effectively reduced by targeting specific flow features. In general, we conclude that it is best to target the strongest hyperbolic trajectories for shorter forecasts although vortex centres can produce good drifter dispersion upon bifurcating on longer time-scales.

Lagrangian simulation of evaporating droplet sprays

In this paper we present the purely Lagrangian approach for the high fidelity simulation of evaporating sprays as a potentially advantageous alternative to the widely used Lagrangian–Eulerian approach. We motivate our arguments using analytical solutions which we derive from a set of simplified spray (droplet and vapor) equations, albeit ones that retain key physical complexities of the two-phase flow. These solutions are obtained for two model flows: the stagnation point flow and the point vortex flow. By comparing numerical results with these analytical solutions we demonstrate limitations of the Lagrangian–Eulerian approach in providing solutions that are consistent with the governing equations. The problem alluded to here is specific to evaporating sprays and is over and above the well known issue of the point source approximation. Moreover, while it is related to the presence of interpolations between the disperse and continuous phases, it is not inherent in them—an interpolation scheme that effectiv...

Breathers on quantized superfluid vortices

We consider the propagation of breathers along a quantized superfluid vortex. Using the correspondence between the local induction approximation (LIA) and the nonlinear Schrödinger equation, we identify a set of initial conditions corresponding to breather solutions of vortex motion governed by the LIA. These initial conditions, which give rise to a long-wavelength modulational instability, result in the emergence of large amplitude perturbations that are localized in both space and time. The emergent structures on the vortex filament are analogous to loop solitons but arise from the dual action of bending and twisting of the vortex. Although the breather solutions we study are exact solutions of the LIA equations, we demonstrate through full numerical simulations that their key emergent attributes carry over to vortex dynamics governed by the Biot-Savart law and to quantized vortices described by the Gross-Pitaevskii equation. The breather excitations can lead to self-reconnections, a mechanism that can play an important role within the crossover range of scales in superfluid turbulence. Moreover, the observation of breather solutions on vortices in a field model suggests that these solutions are expected to arise in a wide range of other physical contexts from classical vortices to cosmological strings.

A numerical study of passive scalar evolution in peripheral regions

We study the effect of slip and no-slip wall boundaries on the decay rate of a passive scalar in a spatially smooth and random in time velocity field. Numerical simulations are carried out to verify the effect of the peripheral (near-wall) regions on the decay of the scalar variance. Using two kinematic flow models with simple velocity fields, we show that, in the case of slip boundaries, the passive scalar is characterized by an initial rapid stirring followed by an exponential decay of the scalar variance. In stark contrast, results for the case with no-slip boundaries show that, following an initial rapid stirring of the scalar within the bulk, there is an intermediate-time regime where the variance follows a power-law decay. This intermediate regime is established as a result of the trapping of the scalar in the peripheral regions near the no-slip walls. Finally, the behavior of the scalar variance switches to a final regime that is characterized by an exponential decay rate. The results presented here indicate that the recent ensemble-based theories regarding the evolution of a passive scalar in the peripheral regions correctly predict the main stages of the scalar evolution that arise in a single flow realization.

Wind generated rogue waves in an annular wave flume

We investigate experimentally the statistical properties of a wind-generated wave field and the spontaneous formation of rogue waves in an annular flume. Unlike many experiments on rogue waves where waves are mechanically generated, here the wave field is forced naturally by wind as it is in the ocean. What is unique about the present experiment is that the annular geometry of the tank makes waves propagating circularly in an unlimited-fetch condition. Within this peculiar framework, we discuss the temporal evolution of the statistical properties of the surface elevation. We show that rogue waves and heavy-tail statistics may develop naturally during the growth of the waves just before the wave height reaches a stationary condition. Our results shed new light on the formation of rogue waves in a natural environment.

Condensation of classical nonlinear waves in a two-component system

We study the formation of large-scale coherent structures (condensates) for a system of two weakly interacting Bose gases in the semiclassical approximation. Using the coupled defocusing nonlinear Schrodinger (NLS) equations as a representative model, we focus on condensation in the phase mixing regime. We employ weak turbulence theory to provide a complete thermodynamic description of the classical condensation process. We show that the temperature and the condensate mass fractions are fully determined by the total number of particles in each component and the initial total energy. Moreover, we find that, at higher energies, condensation can occur in only one component. We derive an analytic result for the variation of the critical energy where this transition occurs. The theory presented provides excellent agreement with results of numerical simulations obtained by directly integrating the dynamical model.

Prediction of Lobed Mixer Vortical Structures with a k-≤ Turbulence Model

Numerical simulations of an incompressible planar shear layer and a lobed mixer e owe eld are presented and validated against the detailed experimental measurements of McCormick and Bennett (McCormick, D. C., and Bennett, J. C., Jr., “ Vortical and Turbulent Structure of a Lobed Mixer Free Shear Layer,” AIAA Journal, Vol. 32, No. 9, 1994, pp. 1852 ‐1859). The study focused on quantifying the predictability of these e ows using the standard k‐≤ turbulence model. Simulations for the planar shear layer showed that, whereas the self-similar behavior can be captured by the model, the measured near-e eld development of the shear layer could not be reproduced in the simulations. Inconsistencies between simulations and experiments arise as a result of the Kelvin ‐Helmholtz instability that is not captured in the simulations. Predictions for the lobed mixer shear layer revealed a lag of 1 :75 lobe heights in the shear-layer development with respect to the measured data. Global parameters such as momentum thickness and streamwise circulation generally showed an underprediction of 20% and an overprediction of 65% with respect to measured values, respectively. Good prediction of the primary Reynolds shear stresses that control the variation of the momentum thickness was obtained. An analysis for the equation of the streamwise component ofvorticity revealedthattheimportantcontribution tothestreamwisecirculation decay rateisthesecondary shear stress and the normal stress anisotropy. Neither is well predicted by the k‐≤ model, leading to poorer predictions of the streamwise circulation decay rate.

Helicity within the vortex filament model

Kinetic helicity is one of the invariants of the Euler equations that is associated with the topology of vortex lines within the fluid. In superfluids, the vorticity is concentrated along vortex filaments. In this setting, helicity would be expected to acquire its simplest form. However, the lack of a core structure for vortex filaments appears to result in a helicity that does not retain its key attribute as a quadratic invariant. By defining a spanwise vector to the vortex through the use of a Seifert framing, we are able to introduce twist and henceforth recover the key properties of helicity. We present several examples for calculating internal twist to illustrate why the centreline helicity alone will lead to ambiguous results if a twist contribution is not introduced. Our choice of the spanwise vector can be expressed in terms of the tangential component of velocity along the filament. Since the tangential velocity does not alter the configuration of the vortex at later times, we are able to recover a similar equation for the internal twist angle to that of classical vortex tubes. Our results allow us to explain how a quasi-classical limit of helicity emerges from helicity considerations for individual superfluid vortex filaments.

Linear and non-linear turbulence model predictions of vortical flows in lobed mixers

Lobed mixers are widely used in gas turbine engines to increase mixing between hot and cold streams and consequently reduce jet noise. CFD predictions are presented for a simplified experimental configuration of a planar, three lobe geometry. Results are shown for a standard linear k–e turbulence model, the same model with a time scale limitation invoked and a non-linear quadratic model also employing a time scale limitation. Comparisons are presented between the three models for axial velocity, velocity vectors, shear stress and turbulence kinetic energy at a selected plane in the mixing region. The non-linear model was found to have little influence on the mean flow but some effect on the turbulence structure was observed. Comparison with measurements showed that all major features were reproduced but detail differences were evident. The use of a time scale limit reduced peak values of predicted turbulence quantities by 20-30%. As compared to the standard linear model, the time scale limited non-linear model moved the position of the zero streamwise circulation location by about one lobe wavelength upstream so giving better agreement with experiment.