# Research

### Brief descriptions, images, and videos summarizing my major research projects. Further details on these projects can be found in my publications.

Velocity gradient analysis of a head-on vortex ring collision

Rahul Arun and Tim Colonius

Vortex rings are ubiquitous flow phenomena, and they also form fundamental building blocks of more complex turbulent flows. Collisions between vortex rings provide a sandbox for investigating mechanisms (e.g., flow instabilities) that underly transition and the turbulent cascade. We perform and analyze an adaptive, multiresolution simulation of turbulence generated by a head-on vortex ring collision. Transition is excited by the development of the elliptic instability through the generation of antiparallel secondary vortex filaments. During turbulent decay, the partitioning of the velocity gradients approaches an equilibrium that is similar to that observed in forced isotropic turbulence. To analyze these regimes, we introduce new phase spaces based on the velocity gradient tensor and the structure of local streamlines. Our analyses highlight that the interplay between the elliptic instability and other mechanisms, e.g., the Crow instability, becomes more important during turbulent decay. Moving forward, our phase space analyses show promise for modeling velocity gradients and identifying the effects of mechanisms driving turbulence.

Flow configuration: head-on collision between vortex rings of equal strength and opposing circulation. The vortex cores are shaded according to their Gaussian vorticity profiles.

Temporal evolution of the vortex boundaries (left) and vortex cores (right), colored by the projection of the direction of rotation onto the z-axis. Visualizations are shown throughout the initial evolution (rows 1-3), transition (rows 4-6), and turbulent decay (rows 7-9) of the collision.

Temporal evolution of the vortex ring collision, depicted by vortex boundaries colored by the projection of the direction of rotation onto the z-axis.

Towards real-time reconstruction of velocity fluctuations in turbulent channel flow

Rahul Arun, Jane Bae, and Beverley McKeon

Efficiently processing streaming measurements of turbulence is a critical, but challenging requirement in real-time flow estimation and control schemes. We develop a framework for efficient streaming reconstructions of turbulent velocity fluctuations from limited sensor measurements with the goal of enabling real-time applications. The estimation techniques are based on the generalized Wiener filter, and we apply them to linear models related to resolvent analysis and the spectral proper orthogonal decomposition. Linear estimators are computed using training data and evaluated during a subsequent testing period. During training, we introduce blockwise inversion to efficiently compute the resolvent operator. During testing, we enable efficient streaming reconstructions by using a sliding discrete Fourier transform to recursively update measurements in the frequency domain. Our techniques capture dominant features of the velocity fluctuations in a turbulent channel using relatively few measurements and they can reconstruct each snapshot in a fraction of a second on a laptop. Promising future directions include experimental validation, nonlinear modeling, and real-time control.

Methods overview: computing linear estimators for each reconstruction method during training (top); reconstructing velocity fluctuations from sparse, streaming measurements during testing (bottom). Discrete Fourier transforms (DFTs) and their inverses (IFTs) are labeled by their dimensionality and the sliding discrete Fourier transform (SDFT) is applied every time step.

Temporal evolution of isosurfaces of the reconstructed streamwise velocity fluctuations (left), the true streamwise fluctuations (middle), and their difference (right).

Temporal evolution of isosurfaces of the reconstructed wall-normal velocity fluctuations (left), the true wall-normal fluctuations (middle), and their difference (right).

Temporal evolution of isosurfaces of the reconstructed spanwise velocity fluctuations (left), the true spanwise fluctuations (middle), and their difference (right).

Control of instability by injection rate oscillations in a radial Hele-Shaw cell

Rahul Arun, Scott Dawson, Peter Schmid, Angeliki Laskari, and Beverley McKeon

The Saffman-Taylor instability leads to the growth of perturbations that resemble "fingers" at the interface of a fluid displacing a more viscous fluid in a porous medium or, analogously, in a Hele-Shaw cell. Controlling the growth of this instability is useful for applications including enhanced oil recovery, microfluidic mixing, and CO2 sequestration. We analyze the effect of injecting the driving fluid with an oscillatory flow rate on instability growth in a radial Hele-Shaw cell using a combined experimental and analytical approach. The experimental results show that instability growth can be mitigated using relatively low- and high- frequency oscillations, but it is amplified at an intermediate frequency. In these settings, we bolster our predictions of instability growth by accounting for wetting effects and by using experimental observations to model a time-varying effective surface tension at the interface. Moving forward, further characterizing unsteady and nonlinear effects will help assess the efficacy of the present control scheme.

Experimental apparatus: air is pumped into a radial Hele-Shaw cell with a more viscous silicone oil as the working fluid. A high-speed camera captures the evolution of the air-oil interface, which is identified using Sobel edge detection.

Comparisons of instability growth throughout typical experiments using various oscillation frequencies. The perturbations (black) are depicted about the mean radii (red) with respect to the injection point (filled circles). The experiments are shown from left to right in order of increasing instability growth: low-frequency, high-frequency, constant flow rate (CFR), and intermediate-frequency.

Temporal evolution of the air-oil interfaces during typical experiments using various oscillation frequencies. The experiments are shown from left to right in order of increasing instability growth.

A mechanical model of bacteriophage DNA ejection

Rahul Arun and Sandip Ghosal

Bacteriophages are viruses that replicate by injecting their DNA into the gene transcription mechanism of host bacteria cells. Absent external complexities, this process is largely driven by the release of elastic and electrostatic energy of the DNA, which is initially coiled within the capsid. Experiments with Î»-phages suggest that the DNA mobility decreases exponentially with the length that remains in the capsid. We perform experiments to assess the proposal that this behavior is attributed to the capstan effect, which amplifies frictional forces as a flexible line is wound around a cylinder. Absent electrostatic forces, we mechanically model the capsid and DNA using a film canister and fishing line, respectively. Though our model is a million times larger than typical phages, its ejections in various working fluids mimic the behavior of real phages and collapse onto the capstan frictional scaling law.

Schematic drawing (left) of the mechanical model of a bacteriophage and a top view (right) showing the spiral arrangement of the nylon fiber within the film canister. The drag on a hypothetical particle (filled circle) represents the fluid resistance on the trailing end (SE).

Normalized evolution of the model ejections in air (stars), water (circles) and glycerin (crosses). The solid line represents the capstan frictional scaling law.

Video (slowed 8x) of a typical ejection using the model in air. The fishing line has 5cm color bands for enhanced visibility during the ejection.