"When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first." — Werner Heisenberg
Fluid mechanics was one of the hardest subjects during my undergraduate days. It got me one of the lowest grades I have ever received in a subject. Yet, it turned out to be the one I pursued through the last 7 years of research.
It is surprising that not a lot of Physics curricula in the world include fluid mechanics as a compulsory topic. In fact, I feel that the reason I got interested in astrophysical and geophysical fluid dynamics was because of a strong exposure to fluids through my aerospace engineering background. If I had studied physics instead, I would surely be doing something else. Through my undergraduate and masters, I had amazing professors who breathed life into this complex subject, making it shake hands with intuition, which is rather difficult to develop otherwise. Despite the complications, the way the subject revealed itself captivated me strongly. Thereafter, I worked on a variety of systems, from natural to technical applications.
The main point of this post is going to be the breadth of systems that my work on the subject covers.
The beginning
Starting off in aerospace engineering in 2014, most of my early projects were technical in nature. My first project (actually a mini-project) was on hypersonic gas dynamics for reacting flows. My job was to solve a system of coupled nonlinear partial differential equations representing a high-speed flow of reacting gases by modifying a pre-written finite volume code for a simpler system. At that time, I understood little to nothing of any of those heavy words and the project reached no tangible end, but it introduced several fundamental concepts about mathematical modelling to me. For example, beyond the textbook examples of plane Poiseuille flows, pipe flows, or Stokes flows, which started with the complex Navier-Stokes equations but ended up being reduced to simple linear ODEs with analytical solutions, lay the real world of nonlinear turbulent flows. These could not be solved on pen and paper and needed then-cryptic computational codes to solve. Also, it was only rarely that fluid equations were sufficient to model phenomena on their own. Rather, there existed other important aspects, such as chemistry and thermodynamics, which needed to be coupled to the existing equations to study a wider range of systems.
Back to pen and paper
In my second project, I took a step back and built up my analytical skills before moving into computations. This work was about solving compressible Euler equations using the method of characteristics for a rocket nozzle. The project showed me how the same set of equations, under varying assumptions, can change their fundamental nature: dropping the viscous terms transforms the fluid equations from being parabolic to hyperbolic, leading to different behaviour and solution techniques. The other aspect of this project was design. Fluid engineering problems are often about solving complex equations for a known initial system configuration and studying the output to optimize the design based on some control parameters. Here, it was a simple analytical system: I designed the optimum nozzle length for maximum rocket thrust. The practice problem can be found in Modern Compressible Flow by J. D. Anderson.
Summer at the national wind tunnel facility and CFD lab, IITK
During the summer of 2017, I visited one of India's top wind tunnel facilities at IIT Kanpur, which has a long history of studying low-speed aerodynamics, including the flight of cricket balls and badminton shuttles. I was on the computational side of the lab and worked on two problems in low-speed incompressible aerodynamics. The first was vortex-induced vibrations of a cylinder placed in a crossflow. As the flow sheds vortices periodically, they excite the cylinder, and the cylinder's motion feeds back into the flow, making it a fluid-structure interaction problem. Using the finite element method to solve the incompressible Navier-Stokes equations, we studied the vortex shedding frequency and cylinfer response under different configurations identifying resonance scenarios. The second was flow past cylinders with surface trips to prepone turbulence and reduce drag. Both projects revealed the importance of boundaries, their shape and motion, and how they couple with on-coming flow generating myriad responses. Recently, vortex induced vibrations have been utilized for energy generation from wind as an alternative to wind turbines by a company called VortexBladeless. Aside from this, I was amazed by the extent to which fluid structure interactions could diversify in terms of applications - from pulsatile flows in deformable channels resembling blood vessels and tubomachinery, the backbone of several energy production systems, to fish/sandworm locomotion and the flapping of a flag in the wind!
Grains in space
For my masters at the same institute, I worked on granular flows on rubble-pile asteroids. This is where I started moving from technical to natural applications. This project introduced yet another aspect of flow physics: not everything flows like water or honey. Some flows have different types of internal interactions between their microscopic components. I became familiar with what is called rheology, or constitutive relations for flowing material in a post on constitutive laws. When grains flow, they interact with one another, which leads to new physical phenomena such as segregation and sieving, giving rise to patterns, instabilities and waves. Even more interesting is when a fluid is combined with grains, which gives rise to the field of Stokesian dynamics or particulate flows, with many industrial applications. In my masters work, I modelled the flow of grains on rotating, gravitating, undulating topographies using finite volume schemes and discrete element simulations on high-performance computing systems, and published in journals such as the Journal of Fluid Mechanics and the Proceedings of the Royal Society A. Through this experience, I realized how broad the applications of mass, momentum and tracer conservation laws can be.
Stars, planets, and reduced-order models
Finally, for my PhD, I studied two systems, both about hydrodynamics on rotating spherical bodies: stars and planets. Having written finite volume codes for hyperbolic conservation laws from scratch during my masters, I switched to using a prebuilt pseudospectral code called Dedalus, which allows for symbolic equation entry. This meant I could write custom equations that include multiple physical components for the system I was studying, adding and dropping terms at ease to measure their individual effects. For example, rotation and radiation impact these astrophysical and geophysical objects strongly, and I could capture their effect using additional terms in the equations without having to rewrite the entire code. Here, I delved into foundational questions and connected pen-and-paper analytical theory with high-performance pseudospectral simulations, establishing the efficacy of reduced-order modelling in understanding stellar interiors and exoplanetary atmospheres. One of these papers, on the atmospheric circulation of variably irradiated planets, I wrote entirely on my own, which gave me a particular appreciation for what a single well-posed reduced-order model can reveal. Through this work, I developed a strong intuition for the contributions of individual physical components to overall system behaviour, something that is often lost in simulations of highly complex multiphysical systems. This is, in a sense, a central point of fluid mechanics: identify which terms are significant, treat the rest as negligible, and obtain simplified solutions. In this way, my PhD tied back to the very beginning, when I was first being introduced to this beautiful subject.
Looking ahead
Despite this breadth, the world of fluid mechanics remains an ocean compared to the droplet of knowledge and experience I have gathered. However, these experiences have also given me the tools to independently navigate that ocean in the future. Being the most abundant form of matter in the universe, I believe this subject deserves far more attention than it receives in foundational science curricula. The range of problems that can be addressed using this elegant suite of equations is unending, and I look forward to studying diverse engineering and scientific applications in the future. Finally, another reason I love this subject is the similarity it shares with life: it is beautifully chaotic and yet elegantly simple.
Why do you seek water, when you are the stream? — Rumi