A million dollars in cash (£640,000) awaits anyone who can develop a rigorous mathematical model for how fluids flow – this week's Millennium Prize Problem.
Fluids are extremely difficult to analyse because they can flow in such complicated ways. The next time you're bored in the kitchen, take a glass of water and let it stand until it's completely still (which takes longer than you might expect). Then use a straw to release a drop of food colouring from a height into the glass and watch how it disperses. Even better: try imagining how you think it would look.
Most people imagine the dye forming a cloud of colour that gradually disperses. In reality, the colouring will spread out, apparently without mixing, into surprisingly aesthetically pleasing but random patterns. More often than not, the drop will magically form a ring and travel down to the bottom of the glass without spreading out.
That a drop of liquid could form a ring and move undiluted through water goes against all our intuition. But it shouldn't come as a surprise: fluids are always doing this. The same process lies behind the formation of smoke rings.
Our world is awash with fluid. From the blood that courses through our veins to the cytoplasm in every cell, our bodies are dependent on liquids. And we spend our lives at the bottom of the atmospheric ocean that is fluid air. You can't move without stirring this sea of gases, so ubiquitous they almost go unnoticed.
Given how inextricably linked we are to fluids, it is disconcerting to discover that we do not have a precise mathematical understanding of how they move. Well, we have a working model, but there is no guarantee that it will not one day go horribly wrong.
Any mathematical description of how an object moves when you hit it starts with Newton's equation F=ma (the force applied equals the mass of the object multiplied by the resulting acceleration). Fluids, however, are a collection of countless particles all interacting, so we need a variation on F=ma that takes into account the sum effect across all the objects that comprise a fluid. Then we need to consider its "viscosity", which is a measure of how much the particles resist flowing over each other. Water is a low viscosity fluid while honey is high viscosity.
Put all of this together and you get:
Where u is the velocity of the fluid at position x and this changes over time t. The symbol v is the viscosity of the fluid and p represents pressure. Both i and j range from 1 to n, where n is the number of dimensions that the fluid is moving in.
This still isn't the whole story. As you quickly discover while washing up, if you force water down in one place it will burst out somewhere else (normally all over your legs). If you squeeze one part of a water balloon, a different part will expand. If you hit water hard enough, it will not move out of the way and you will hurt yourself – the same resistance that allows a fast-moving stone to bounce across the surface of water.
To incorporate this into our mathematical model, we have a second equation that effectively takes into account fluids' incompressibility:
These equations were developed simultaneously in the early 1800s by George Stokes in England and Claude-Louis Navier in France. But if they have been around so long, what's the million-dollar prize for? The problem is that for most situations the Navier-Stokes equations are too hard to solve; they tend to result in partial differential equations that are simply too complicated. For the most part, mathematicians have developed numerical workarounds to squeeze out solutions to the equations, but it's a dark art.
Even when the equations can be forced to spit out a solution, in some situations these answers predict that the fluid will accelerate away at infinite speeds, an outcome that is known – rather appropriately – as "blowing up". Predicting fluids with infinite velocity is a sure sign that the mathematical model no longer matches reality.
Worse still, we do not even know whether a solution exists for all fluids. So while we use the Navier-Stokes equations for everything from aeronautical engineering to medical research, there's no guarantee the answers they give will be sensible or even if there will be an answer at all.
To collect the Clay Institute of Mathematics' $1m you need to show mathematically that either the Navier-Stokes equations can always be made to give realistic "not blowing up" answers, or that there is a case where they definitely cannot give such a solution. This has to be done for all fluids in three dimensions – many of these problems are solvable in two dimensions or for low velocities, but fluids get immensely more complicated in three dimensions and when things speed up.
That said, if you succeed in putting fluid dynamics onto solid mathematical foundations, you can use hand-rolled, cognac-infused cigars to conduct all the smoke-ring experiments you want.
The very fundamental equations when you do computational fluid dynamics are the so
called Navier-Stokes equations.
Today we will work with one version of these equations called the Navier-Strokes
equations for incompressible flow.
They express the conservation of momentum in the fluid and
the incompressibility of the flow.
They are very simple, and they neglect many effects,
many physics, which you can observe in actual fair fluids.
They don't account for the compressibility of gases, for example.
They don't account for temperature or thermal transition effects.
They don't account for chemical reactions and many more phenomena, yet
despite their simplicity, they are very much used on the most common versions of
Navier-Stokes equations in computational fluid dynamics.
In particular, they are very frequently used to simulate both gases and liquids.
Although gasses are compressible, neglecting compressibility,
in many cases, is not so important.
Therefore, the incompressible version of Navier-Stokes equation
will be the baseline of our study today.
It is really three equations, because it consists of
vector quantities like the velocity u.
The velocity u is the central quantity for which we are solving.
Once we know the value of the velocity in every point in space and
at every moment in time, we consider the Navier-Stokes equations solved.
The velocity is a three-component vector in 3D space or
a two-component vector in 2D space.
Therefore the Navier-Stokes equations consist of three scalar equations.
The first term in the Navier-Stokes equations
expressed the acceleration of velocity fluid.
The second term is the convective acceleration,
it has some importance if the velocity changes over space.
For example, if a fluid speeds up when it enters a narrow channel,
it is the only term of Navier-Stokes equations which is non-linear.
And it is responsible for all non-linear effects in the fluid for example,
The pressure gradient is a term which is responsible to maintain
the incompressibility of the fluid.
I will not talk in more details about the mathematics or the detailed significations
of the mathematical operators in the Navier-Stokes equations.
Because today, when we derive a numerical method,
we will not derive it from the Navier-Stokes equations.
Instead, we will take the lattice gas automata and
develop them further into numerical scheme to solve Navier-Stokes equation.
Navier-Stokes equations today are more theoretical background framework,
which gives us some help to understand the physics of fluid flow.
If you've modeled an airplane, then the length could be the diameter of
the airplane and u could be the cruising speed of the airplane.
Both of which are characteristic properties of your system.
Then we can make the velocity in the Navier-Stokes equations dimensionless
by dividing them by the characteristic velocity capital U.
We can get the dimensionless velocity, U star.
In the same wa, the pressure is made dimensionless by dividing it
by the characteristic velocity squared.
It is conventional to take the inverse of this constant and
call it the Reynolds number.
The Reynolds number is equal to the characteristic velocity times length
divided by the viscosity.
The fact that we were able to reduce the Navier-Stocks equations to a dimensionless
form governed by a single parameter means that these equations are scale invariant.
This is a property which is exploited, for example, when you explore
the aerodynamics of an airplane on a model airplane which is placed in a wind tunnel.
The model airplane is smaller than the real airplane.
But you can get the same Reynolds number if you increase
the velocity of the wind around the airplane, or
if you replace the air by another gas which has a lower viscosity.
The dimensionless formulation of the Navier-Stokes equations is also exploited
when you do numerical simulation.
Because the units of your actual physical system is most often not so
convenient a numerical simulation where we will use a different system of units.
I would just make sure that we have the same Reynolds number in the simulation as
we have in the experiment, so you can match the two together.
There are different types of boundaries in the system.
Around the obstacle, where here is a circle obstacle, we have boundary
condition which you can consider to be a physical boundary condition.
It is conditions by the physics by the surface properties of the obstacle
which is placed inside the flow.
But we have also virtual boundary conditions which have
no physical mean which are just placed there to satisfy some condition,
some constraint of our numerical system.
For example, at the inlet, well we'll just cut the domain, the real physical domain
is bigger but we have to cut it somewhere so we need to find the boundary condition.
Which is there to pretend that the system was really bigger, to minimize
the influence of the reduced domain size on the solution of the flow equation.
The same we do on the outflow to have the fluid exiting the system into
nothing in an appropriate way, and also on the lower and on the upper boundary.
This ends our module Equation and
Challenges in Computational Fluid Dynamics.
Stay tuned for the next module.