## Projects

1) **High-Rey****nolds number stratified turbulent wakes:** Submerged stratified turbulent wakes are a canonical turbulent flow where one can examine the competition between stratification and shear in suppressing and enhancing turbulence, respectively. Moreover, stratified wakes are of interest to researchers in geophysical fluid dynamics and naval hydrodynamics. Our existing spectral multidomain penalty method (SMPM) flow solver has allowed us to conduct implicit large eddy simulations (LES) of the intermediate-to-late wake of a towed sphere at Reynolds number values O(10) larger than those of the laboratory. The unprecedented high resolution of the wake core, enabled by the SMPM, has revealed novel physics inside within it in the form of secondary Kelvin-Helmholtz instabilities and turbulence. These secondary events drive a significant prolongation of the commonly perceived life-cycle of the original turbulence with important implications for the parameterization of vertical transport and mixing by stratified turbulence.

We have also analyzed the high-frequency internal gravity wave field radiated by stratified turbulent wakes. The wave-emission process by localized stratified turbulence becomes increasingly inviscid at high Reynolds numbers. As a result, the radiated wave-field extracts non-negligible momentum from the wake and the waves show an increasing likelihood for near-field breaking (leading to additional sources of dissipation and mixing).

Ongoing efforts are focused towards characterizing the (sub)surface manifestation of the wake-radiated wave-field, namely in terms of the induced surface strain field and mean flows. The associated results can be used by remote sensing researchers. Very recently, we have begun pursuing the formulation of universal scaling laws for high-Reynolds number stratified wakes. We will also soon explore the development of subgrid scale parameterizations informed by the state-of-the-art homogeneous stratified turbulence simulations of Prof. Steve de Bruyn Kops at the University of Massachussetts. The PhD student working on this project is Mr. Qi Zhou.

2) **Nonlinear effects in reflecting and refracting internal wave beams:** When finite amplitude internal gravity wave beams impact the ocean bottom/surface or variations in the background stratification/shear nonlinear effects emerge. Beyond wave breaking, such effects include the formation of time-averaged mean currents and higher-frequency harmonics. So far, we have investigated such phenomena in two fundamental configurations: the reflection of an internal wave beam off a free-slip surface (a surrogate for the ocean surface on a calm day) and the interaction of such a beam with a model localized ocean pycnocline. Harmonic generation is particularly complex in the latter case, where the spectral multidomain penalty method solver has been invaluable in providing very high-resolution of the pycnocline. For thin pynoclines, whose thickness is 10% of the incident IWB’s horizontal wavelength, harmonics trapped within the pycnocline have maximum amplitude when their frequency and wavenumber match those of the natural pycnocline interfacial wave mode. Results in this case are compared with weakly nonlinear theory for harmonic generation by plane wave refraction. For thicker pycnoclines, whose thickness is equal to the incident IWB’s horizontal wavelength, IWB refraction results in harmonic generation at multiple locations in addition to pycnocline entry, giving rise to complex flow structure inside the pycnocline.

Ongoing work is focused on exploring a number of outstanding questions and also determining the role of not only beam angle and amplitude but also background shear in harmonic formation. We work in close collaboration with Dr. Scott Wunsch at the Applied Physics Laboratory of Johns Hopkins University. The PhD student working on this project is Mr. Anil Aksu.

3) **Bottom turbulence and resuspension under internal solitary waves:** Internal solitary waves (ISWs) are long, high-amplitude, waves propagating along the interfacial layer of the lakes and coastal ocean known as a pycnocline (or thermocline). ISWs are strongly non-linear and non-hydrostatic, and maybe thought of as a type of internal tsunami. They impose a pressure gradient on the ocean/lake bottom that is variable in the along-wave-direction and adverse (increasing pressure) in the rear end of the wave. Consequently, boundary layer separation (and possible reattachment) occurs along the bed in the rear of the wave. The resulting streamline configuration is highly prone to a particular form of shear instability, known as a global instability ; its properties are defined by the along-bed variation of the wave-induced current. This instability gives rise to near-bed vortex shedding in analogy to the vortices shed by separating flow over an airfoil. In previous work, we have extensively characterized this instability under mode-1 waves of depression (the pycnocline is above mid-depth and the wave pushes it downward). In collaboration with Profs. Leon Boegman (Queens U., Canada) and Kevin Lamb (U. Waterloo, Canada) we have also proposed a stability boundary for such waves in terms of quantities that are directly measurable in the field.

Current work is focused on direct numerical simulation of the 3-D turbulent boundary layer under an ISW. This is a non-trivial effort on account of the very large number of grid-points (250 million) entailed by the very long domain encompassing the entire wave and also the significant challenges involved in establishing a numerically simulated self-sustained near-bed turbulent wake in the ISW footprint. Dr. Takahiro Sakai, a recently-departed postdoc, continues to work on this problem. Mr. Gustavo Rivera is the PhD student actively involved in this research. Our work is in close collaboration with Prof. Gustaaf Jabos at San Diego State University. We plan to use his custom-designed Lagrangian particle tracking techniques to obtain a better understanding of near-bed turbulence-driven resuspension and the formation of benthic nepheloid layers.

4) **Instabilities in the bottom boundary layer under a transient surface wave (collaboration with Prof. Phil Liu):** Tsunami waves are long highly nonlinear transient surface waves whose destructive effect is well-known. U-tube laboratory experiments serve as a model to investigate the boundary layer dynamics under such a long wave over a region limited in the along-wave direction. We are using direct numerical simulations of the U-tube configuration and hydrodynamic instability analysis tools to study the transition to turbulence under a surface solitary wave. The transient nature of such a wave, as sensed by a stationary observer, gives rise to a near-bed base flow whose stability properties might not always be described by classical stability analysis. In close collaboration with Prof. Luis Parras at the University of Malaga in Spain, we are applying alternative stability analysis tools to more effectively characterize and interpret the above transition, and its associated coherent structures, in both two and three dimensions. The PhD student conducting this research is Mr. Mahmoud Sadek, whose primary advisor is Prof. Liu.

5) **Development of a deformed quadrilateral spectral multidomain penalty method model for the incompressible Navier-Stokes equations:** Our existing incompressible Navier-Stokes equation (NSE) solver used in all the above projects employs a spectral multidomain penalty method (SMPM) model only in the vertical direction. By virtue of using a Fourier discretization, the computational domain is always restricted to being periodic in the horizontal with a uniform grid. For the purpose of allowing for more complex boundary conditions and flexibility in localized resolution in one horizontal direction and complex bathymetry, we have been developing a quadrilateral SMPM incompressible NSE solver. The quadrilateral SMPM was first applied to the simpler inviscid shallow water equations. Its implementation in a incompressible NSE framework has been completed for the case of undeformed rectangular subdomains. To this end, in collaboration with Prof. Charles Van Loan in Cornell Comp. Sci., we developed a novel scheme for the iterative solution of the SMPM-discretized pressure Poisson equation (PPE). The associated linear system is non-symmetric, inconsistent and singular giving rise to numerous challenges from a numerical linear algebra standpoint.

We are now in the process of extending the quadrilateral SMPM solver to deformed subdomains, as we progressively overcome challenges arising from the modifications this deformation brings to the highly regular structure of the PPE matrix for the undeformed subdomain case. Following parallelization of the new solver, it will be applied to the study of internal solitary waves shoaling over gentle slopes for the purpose of identifying the associated 2-D and 3-D structural transformations of the waves. In addition, an underwater acoustics model will be coupled to the flow solver to study the propagation of sound through such waves. The PhD student working on this project is Mr. Sumedh Joshi.