A smart autonomous swimmer following in a leader's wake. Youtube link here.

### To school, or not to school...

There has been a long-standing debate as to whether schooling fish reduce energy expenditure by adapting their swimming response to unsteady flow. This question has profound evolutionary significance, since any behavior that may lead to energy-savings can give a species an undeniable advantage over others that do not exploit this mechanism.

With the help of unsupervised machine learning algorithms, we have demonstrated that it is feasible to teach an artificial agent (a self-propelled fish-like swimmer) the capability to take adaptive decisions autonomously, so as to exploit energy deposited in the flow by an upstream swimmer^{[1,2]}. The 'smart' agent is able to minimize its own energy expenditure by interacting judiciously with the unsteady wake, while having no a-priori knowledge regarding details of the complex fluid phenomena involved.

Moreover, the agent explicitly chooses to pursue in the leader's wake while attempting to maximize swimming-efficiency, although it is given no direct incentive to do so. This suggests that large groups of fish may indeed resort to schooling as a means of energy-saving. The results lay the groundwork for future robotic applications, where groups of robotic swimmers may attempt to maximize range and endurance by swimming in a coordinated manner, without having to depend upon complex (and potentially sub-optimal) hand-crafted rules.

Optimal burst and coast swimming modes. Youtube link here. Video of recent presentation.

### Discovering the benefits of unsteady swimming

Steady, continuous swimming is rarely observed in most fish species. A large number adopt an intermittent form of locomotion referred to as `burst-and-coast' swimming, where a few quick flicks of the tail are followed by a prolonged unpowered glide. This behavior is believed to confer energetic benefits, in addition to stabilizing the sensory field, and diminishing the wake-signature for predator-avoidance.

Unfortunately, these advantages may be offset by a reduction in average speed. We have coupled high-fidelity simulations with evolutionary-optimization algorithms to discover a range of intermittent-swimming patterns^{[3]}, the most efficient of which resemble the swimming-behavior of live fish. Importantly, the use of multi-objective optimization reveals locomotion patterns that strike the perfect balance between both speed and efficiency. Some of these patterns do not generally occur in nature, but can be invaluable for use in robotic applications. The resulting increase in range, endurance, and average speed can greatly enhance the mission capability of robotic swimmers.

Vorticity (top), pressure (bottom), & force vectors - Youtube link here.

### Numerical techniques for complex deforming geometries

The distribution of forces on the surface of complex, deforming geometries is an invaluable output of flow simulations. One particular example of such geometries involves self-propelled swimmers. Surface forces can provide significant information about the flow field sensed by the swimmers, and are difficult to obtain experimentally. At the same time, simulations of flow around complex, deforming shapes can be computationally prohibitive when body-fitted grids are used.

We have developed numerical techniques that expand the utility of vortex methods in simulations of complex, deforming geometries^{[4]}. The method developed employs a Cartesian grid to avoid the computational overhead associated with remeshing, and provides fast and accurate distribution of flow-induced forces on the surface of temporally evolving geometries. This has allowed us to quantify perturbations sensed by self-propelled swimmers, which is essential for discovering optimal swimming-kinematics, as well as for efficiency-maximization of interacting swimmers.

High Schmidt number scalar mixing in Homogeneous Isotropic Turbulence (HIT). Youtube link here.

### Turbulent scalar transport at high Schmidt numbers

For turbulent simulations at very high Reynolds and Schmidt numbers, it often becomes necessary to use a model to reproduce multiscale dynamics essential to scalar transport. The model accounts for small scales that may be unresolved on the computational grid, as is the case when conducting Large Eddy Simulations (LES). These small scales tend to have an indirect, yet quite significant impact on the dynamics of the larger scales.

Our investigations of scalar transport physics point to certain peculiarities, with regard to the dependence of these multiscale interactions on the flow Schmidt number. With increasing Schmidt number, convective scalar transport at the intermediate and large scales becomes progressively decoupled with the dynamics of the smaller scales. This occurrence of 'spectral de-linking' can be proved mathematically^{[5]} by examining scalar-velocity Fourier mode interactions (commonly referred to as 'triadic interactions').

The occurrence of spectral de-linking has profound consequences in terms of the role that models constructed specifically for very high Schmidt number transport must play. Unlike their low Schmidt number counterparts, these models must make an effort to have minimal impact on the intermediate and larger scales. At the same time, the models must introduce just the right amount of scalar variance dissipation, in a relatively narrow band of wavenumbers close to the filter cutoff. Achieving all of these goals simultaneously is a daunting task, and demands a unison of state of the art tools and techniques in the two disparate domains of physics and numerics.

The Turbulent Mixing Layer (TML) configuration used for testing scalar scheme performance. Youtube link here.

### The Bounded Cubic Hermite (BCH) transport scheme

All numerical schemes used for solving PDEs (Partial Differential Equations) entail some amount of numerical error owing to spatial and temporal discretization. Simulations at high Schmidt numbers become increasingly sensitive to these errors, as numerical diffusion tends to overwhelm the comparatively minuscule molecular diffusion. The detrimental impact on transport characteristics is quite severe, and is discernible even in unity Schmidt number flows.

Data obtained from such simulations cannot be relied upon to furnish an accurate picture of scalar transport physics, unless the numerical errors are minimized. With these issues in mind, we have developed a semi-Lagrangian scalar transport scheme that is both robust and possesses highly accurate numerical characteristics^{[6]}. The scheme is capable of delivering results with fidelity comparable to that of conventional schemes, but with close to a 16X reduction in computational cost. In addition to being beneficial to scientific studies of academic import, the BCH scheme is inherently suitable for use in Large Eddy Simulations involving complicated geometries, where numerical diffusion of conventional schemes may be strong enough to negate the contribution from subgrid models.

[1] Verma, S., Novati, G., Koumoutsakos, P., “Efficient collective swimming by harnessing vortices through deep reinforcement learning”, *Under Review*

[2] Novati, G., Verma, S., Alexeev, D., Rossinelli, D., van Rees, W.M., and Koumoutsakos, P., “Synchronisation through learning for two self-propelled swimmers ”, Bioinspiration & Biomimetics (2017), 12, 036001

[3] Verma, S., Hadjidoukas, P., Wirth, P., Koumoutsakos, P., Multi-objective optimization of artificial swimmers, IEEE Congress on Evolutionary Computation, Donostia - San Sebastian, Spain, June 2017

[4] Verma, S., Abbati, G., Novati, G., Koumoutsakos, P., “Computing the force distribution on the surface of complex, deforming geometries using Vortex methods and Brinkman penalization”, International Journal of Numerical Methods in Fluids (2017), 85, 484--501

[5] Verma, S., Blanquart, G. , “On filtering in the viscous-convective sub-range for turbulent mixing of high Schmidt number passive scalars”, Physics of Fluids (2013), 25, 055104

[6] Verma, S., Xuan, Y., Blanquart, G., “A bounded semi-Lagrangian scheme for the turbulent transport of passive scalars”, Journal of Computational Physics, Journal of Computational Physics (2014), 272, 1--22