*Human-built air vehicles glide, chop, or propel themselves through the air. Nature’s fliers, however, flap their wings. Flapping flight is a complicated phenomenon, and understanding it could help to truly imitate natural flight someday. *

Who would have thought that the pendulum – the staid symbol of predictability and rhythm – could behave so wildly when doubled? The brain searches in vain, thinking it has caught a pattern here, a structure there, when, without warning, everything goes haywire again. Watching a double pendulum swing is in some ways like listening to a piece by Stravinsky, or reading a poem with no meter.

Such unpredictable behaviour is not the exception but the norm in nature, Dr. Sunetra Sarkar tells us as we sit in her office late one Friday evening, talking about how birds and insects fly. After centuries of admiring natural flyers and longing to imitate them, there is a reason why only a handful of our flying machines come anywhere close to the dream of flying *like a bird*, with the flapping of wings and the freedom to perch, hover and take off at will. Natural flight is something that humans have not been able to fully understand despite millennia of effort, even now that we have the equations describing fluid flow at our disposal. Dr. Sarkar and her group at the Department of Aerospace Engineering work at the intersection of fields, combining dynamical systems theory with unsteady aerodynamics to understand these processes.

One might wonder what is unpredictable about classical physical systems on a macroscopic scale with no quantum physical phenomena involved. After all, if the position and velocity of all particles in the system is known at a certain time, Newton’s laws dictate the entire future of the system. How the double pendulum is configured when it starts swinging should dictate how it behaves later. How a bird flaps its wings should determine exactly how the air flows around it.

This is perfectly true in principle. Two kinds of mistakes, then, can keep us from seeing why we are unable to predict the behaviour of some systems. First, we take for granted that it is possible to know the starting configuration of a system exactly. This is all very well in mathematics, but never happens in real life. Second, we naïvely imagine that if we know the starting configuration approximately, then we should be able to predict the future behaviour approximately. Our minds somehow assume that the system is linear, that the result of a slightly wrong input would be a slightly wrong output. However, not all systems are linear. In chaotic systems, a tiny change in input can alter the behaviour beyond recognition. The weather is chaotic, hence the phrase “unpredictable as weather” and the Butterfly Effect description of chaos: a butterfly flapping its wings in a rainforest can cause a tornado weeks later halfway across the world. A double pendulum is chaotic as well, and so is the flow of air around a flapping wing, explains Chandan Bose. As a doctoral research scholar studying the dynamics of natural flight with Dr. Sarkar and Dr. Sayan Gupta (at the Department of Applied Mechanics), Chandan works to understand the physics of chaos, and how birds use it to fly. An enthusiastic member of the institute’s Aero Club at the Centre For Innovation, he also builds flapping wing prototypes, both to study and for fun. As it turns out, these are entirely different from other flying vehicles, such as planes, and single or multi-rotor helicopters.

Blowing air over a strip of paper is a nice demonstration of the principle behind the flight of planes. Since air flows faster above the strip, the air pressure is higher below it than above, and this pushes the paper upward. An aeroplane wing is designed with a shape that forces the air above it to move faster, even when the oncoming stream of air arrives with a uniform speed. The plane first speeds up on the runway to get the air streaming past it to move fast enough. The shape of the wing does the rest by creating a pressure difference, so that the craft receives enough *lift *from the air to overcome its own weight. It is also important that the shape of the plane be streamlined, or else the *drag* from the air would put up too much resistance to any movement.

Things are different when a bird or an insect flies. It needs neither runway nor landing strip. It could have the clumsiest imaginable shape and still manage to move through the air without any trouble. It knows how to avoid crashing into things on its way, how to stop when it needs to, and how to react to changing winds and to other flying objects. To paraphrase what Leonardo da Vinci said half a millenium ago, and what Dr. Sarkar says now, human beings are intelligent and creative, but often the best thing that we can do with these capabilities is to use them to learn from the usually superior designs of nature.

The equations governing the way fluids such as air and water move were figured out a couple of centuries ago, as a culmination of contributions from a series of thinkers. Called the Navier-Stokes equations (NSE), they are a set of equations which tie together fundamental physical laws – the conservation of mass, energy and momentum – with the properties of the fluid in question, such as its density and viscosity. They do this in a way that relates the flow of the fluid to the pressures at different places.

Imagine standing somewhere in the middle of a stream, noting how fast the water flows past your feet and in which direction. Theoretically, if this could be done at every point in the stream at once, then the flow at that instant would have been completely described. This is how Euler, the great mathematician, looked at flow – the way a fluid flows is described by treating it as a set of velocities in space, a “velocity field”, which changes with time. Since the velocity field describes a quantity that changes with both time and location, the NSE do not talk about velocity directly. They are partial differential equations instead: they tell us how the velocity at a certain point in the region changes with time, how the velocity at a frozen instant in time changes as one looks at different locations, and so on. These equations must be solved to find the velocity field, before one can see how the fluid actually behaves. Surprisingly, not only do we not have a general solution for the NSE, we do not know how to say for sure whether a solution even exists.

In such situations, there are two avenues available to engineers, and they choose whichever works best. The first is to make assumptions, treating the system as a simpler one than it really is. This reduces the complexity of the equations and sometimes they become open to an analytical solution. This solution is a mathematically perfect fit for the equations in question; the trouble is that the equations themselves may no longer fit the physical system that they are meant to represent.

The second method is to find an approximate solution to the equations. This is done by dividing space into a grid, treating it as a collection of points – many, but not infinitely many. Time, too, is treated not as continuously varying but as a series of steps, a chain of moments. This allows us to find something that is not quite the exact solution, but is close. Say we were trying to guess at an invisible smooth curve on a sheet of paper, given a string of a hundred closely spaced points on it but not the curve itself. One could get a reasonable picture of reality by joining consecutive points with straight lines, or for that matter tiny parabolic segments. Numerical solution techniques generalise this idea. How one chooses to join the points, how many points are necessary to consider the result reasonably close, how close is “reasonably close” – all these depend on the system in question, the judgment of the researcher, and how powerful a computer is available. The finer the grids and the smaller the time step, the closer we get to a realistic idea of the physics, and the more difficult it is to actually compute the solution. Today, it takes Chandan weeks and often months of processing to run a single simulation on the Virgo Super Cluster at IIT Madras – one of the country’s most powerful supercomputers running his code day and night on 16 processors simultaneously. These are high-fidelity simulations, which means that they are based on a bare minimum of assumptions and therefore work on equations that quite accurately describe the physics of the system. Fifty years ago, Dr. Sarkar remarks, there was no question of solving such equations in any reasonable amount of time without making drastic assumptions to simplify them.

The first reason why the dynamics of flapping are so different from those of a stationary wing is the effect of time. A stationary wing facing a steady wind has the same flow around it as long as the flow speed does not change, but air around a flapping wing flows past a structure whose position and orientation changes all the time. Then there is layer upon layer of complexity. For one thing, even if one assumes that the wing moves in a repeated pattern over regular intervals, this motion is not a simple up-and-down “plunging”, but is more complicated in reality, involving forward-and-backward tilting as well. To add to this, the wing is a flexible body. It not only flaps in complex ways, but changes shape when pushed and pulled by the air around it. The wind too is never steady. One does not want to stop at understanding how flight would happen in an artificially simplified situation, or build a robot or vehicle that cannot handle a gusty environment.

The problem is a daunting one, but as long as it can be broken into stages of increasing complexity, it is a problem that can be attacked. Dr. Sarkar, Chandan and Sandeep Badrinath (then a Dual Degree student) started with a single, repetitive, plunging motion, and found a wealth of interesting results even in this simplified model. First, they assumed that the air would have a constant density throughout the simulation, meaning that their theoretical model would ignore any compression of it. They then observed what happens when flapping becomes more and more rigorous, and tried to understand and explain it.

A low-speed wind facing a small flapping wing would behave the same way as a wind of twice the speed facing a wing either double the size, or one flapping twice as fast. It would be as if the entire system had been scaled up or down (suppose each of us were to shrink to the size of ants, if everything around us shrank by the same scale, no one would ever notice). This is why it is common practice to use the non-dimensional form of the NSE, that is, all lengths are divided by a characteristic length (such as the wing span), all times by a characteristic time (such as the duration of one flap), and so on. This gives us an equation involving numerical parameters which have values but no units. Loosely speaking, these parameters, by occurring in different terms of the NSE, decide which terms dominate and thus how the system behaves. One such parameter is the Strouhal number, important in periodic systems such as these. It is the product of frequency (how often the wing flaps) and amplitude (how much it moves in each flap), divided by the wind speed. So, to study a system with an increased Strouhal number, flapping harder or flapping faster would have the same effect. However, changing the flapping frequency would affect the ability of the numerical solver to find solutions, unless the time step was also altered suitably, so the researchers chose to change the amplitude to see what happened, knowing that all that mattered was the overall change in Strouhal number.

When a wing does not flap, the air flowing past it keeps close to its edges, flowing smoothly along them. Flapping changes this. At first, the wing flaps softly, moving only slightly every time. The air in its wake swirls around, forming what are called vortices. The vortices form just as regularly as the flapping, making the same pattern as each cycle repeats. As the range of motion of the wing grows, however, something interesting begins to happen. The vortices repeat for a while, and move around unpredictably for a while before they begin to repeat again. As the flapping amplitude grows larger and larger, these intervals or “windows” of unpredictable behaviour begin to grow until the entire behaviour becomes unpredictable.

Once this has happened, if one were to take a picture of the flow field at some time, and superpose it on the original flow field, one would get a blurred image, even if the wing was at the same stage in its flapping cycle. This gives researchers in the field a hunch that the system has become chaotic. However, there is a rigorous mathematical definition for chaos. Establishing that this is indeed chaos, deciding how chaotic it is, and being sure that it transitions from well-behaved repetitive behaviour to chaotic behaviour via an intermediate state of “intermittency”, as the growing windows of unpredictability seem to show, is not a trivial task.

Since the equations involved are a set of non-linear partial differential equations that are solved by creating a mesh of tens of thousands of cells, the numbers being different at each point in time and space, some of the tools most widely used to establish chaos turn out to be poorly suited to these studies. To identify the behaviour as chaos, Sandeep used a technique based on the idea that when some property of the system is observed at two instants some time apart, the extent of correlation between the two should describe how periodic the system is. A system that is periodic would keep coming back to its initial state, unlike a chaotic one.

To study how the originally periodic system becomes chaotic over time, the researchers used a different tool, called recurrence quantification analysis, which shows, in a single plot, the tendency of a system to repeat its behaviour. This gives a visual intuition, like the pictures of vortices in the wake of the flapping wing, but also allows quantitative measurements. Since recurrence analysis leads to results that can be used to measure periodicity, it is useful for identifying the moment when a system begins to show irregular behaviour. The brains of flying animals use highly advanced control strategies to deal with all kinds of air flows, but if we were to build a robot to fly like an insect or a bird, and endow it with a small circuit to control its flapping based on how much lift it was receiving from the air, we would very much like to avoid setting up unpredictable flow fields and losing control of the flight.

The next step in Chandan’s work has been to complicate the system even further, to bring it closer to reality. Just as the flapping wing has an effect on the air, the air has an effect on the wing. He started by forcing the wing to move in a way that had already been determined, regardless of the forces from the air. This is called a kinematic model, because it concerns itself with only the motion of the wing; the theoretical wing in the model does not respond to lift, drag and so on. He now moves to a dynamic model, allowing the mass and the stiffness of the wing to affect its behaviour in the flow. Of course, this means he must stop dictating the flapping, and let the forces dictate it instead. He sets off the system with a tap and then watches what happens.

To be able to watch what happens, he needs an enormously complicated piece of code. The physics has now come full circle, from the flapping of the wing determining the flow, to the flow pushing the wing around. To handle these two aspects, he now needs two solvers. At every instant, the wing has a certain configuration. With this information and the wind speed, a Navier Stokes solver finds the pressures at different locations in the flow. This information is given to a Fluid-Structure Interaction solver, which computes the effect of the pressures in the air on the wing, by creating special kinds of meshes on both the fluid and the structure, and carefully ensuring that their interaction is accurately represented by the equations. This tells the NSE solver the new configuration of the wing, so that the latter is ready to move on to the next time step.

This time, he includes both pitch and plunge, making the wing tilt periodically in addition to moving up and down. To start with, the wing is treated as a combination of a block, representing its mass, and a spring, representing its stiffness. This kind of approximation is called a lumped mass model, and is very useful for modelling physical systems in a basic sense. His intention is to move on to three-dimensional motion and use a finite element model, in which the wing would be treated as a large number of tiny connected elements, bringing the theoretical model closer to a real wing.

It is a never-ending search for the truth, but that is what scientists do. It is also a never-ending effort to be creative with whatever glimpses we have of the truth – that is what engineers do. Scientists cannot give up on the hope of understanding how nature works; engineers cannot allow limited understanding of nature to stop them from making things. These researchers are both, using whatever tools are at their disposal to either break problems open, or peel them apart layer by layer. It would be difficult to decide which would be more exciting: the prospect of seeing tiny flying machines with flapping wings doing reconnaissance work for us, collecting data humans cannot collect, from places where humans cannot go – or the prospect of finally, having watched nature tease us for all this time, truly understanding how birds and insects fly. Whether or not we shall ever be able to beat nature at her own game, every step we take to understand these strange and wonderful systems brings us closer to figuring out the rules.