## Hydrodynamics: The Reynolds Number – Li Kewei

The Reynolds number is an important quantity in hydrodynamics. According to Wikipedia, the Reynolds number is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces. I guess that doesn’t tell you much about what the Reynolds number actually is, so I’m going to show you how such a number was obtained.

First things first, the Reynolds number is defined as

where ρ is the density of the fluid, v is the velocity of the fluid, η is the viscosity of the fluid and L is a quantity called the characteristic length. Unless you have some foundation in hydrodynamics, such a definition of Reynolds number probably didn’t make much sense to you. To see how the Reynolds number came about, we have to start from something more fundamental: the equations of hydrodynamics. But before that, since you might not be familiar with the concept of viscosity, we’ll take a short detour to define viscosity.

Consider the situation pictured below. There are two plates with some fluid between them. The bottom plate is stationary while the top plate is moving with velocity u. Both plates have area A and the separation between them is d.

Now let us take a look at what is happening inside the fluid. We know that the fluid must “stick” to the plates at the boundaries between the fluid and plate. Such a condition is called the “no-slip condition”. So fluid next to the bottom plate has zero velocity while fluid next to the top plate has velocity u. What about the fluid in between? In the simplest kind of fluids (called Newtonian fluids), the velocity just increases linearly from 0 to u as we move up from the bottom plate to the top plate, as shown by the arrows in the diagram. In other words, a velocity gradient with magnitude u/d exists within the fluid.

Because the fluid is between them, you’d expect that the top plate will experience a drag force due to the fluid. Due to the drag force from the fluid, a force F must be applied to the top plate to keep it moving at velocity u. And since the top plate is not accelerating, the force F is numerically equal to the drag force from the fluid. Intuitively, this force is proportional to the area of the plates A, and the velocity gradient u/d. The constant of proportionality is defined as the viscosity of the fluid. In other words:

In general, the velocity gradient may not be linear, so u/d should be replaced by the more general derivative :

Don’t be too bother with the partial derivative, just treat it as a normal derivative. The difference only becomes important when we are dealing with more than one dimension.

The study of hydrodynamics is an extremely complicated and diverse field. The most fundamental equations of hydrodynamics are the Navier-Stokes equations. These equations come about by applying Newton’s Second law on a fluid element (that may sound simple enough, but it is actually very complicated even in the simplest of cases!). Due to the great range of forces that become important in different cases, it’s almost impossible to put down a general form of the Navier-Stokes equations. To get the essence of the idea behind the Reynolds number, we will consider a simplified scenario: the flow of a Newtonian, non-compressible fluid past a stationary cylinder, in the absence of gravity. For such a situation, the Navier-Stokes equation looks something like this:

What the hell does this equation mean? Let me tell you a piece of good news: you don’t really have to know unless you intend to do some fluid dynamics in the future! Anyway, to be complete, I’ll just define the terms in the equation. v is the velocity field, which is a vector field (since velocity is a vector). It tells you how the velocity v varies with the space coordinates and time, so v is a function of x,y,z and t. This the thing that we want to find when we solve this equation, since it contains all the information about the flow patterns, etc. The left hand side of the equation is actually the ma term in F=ma (I told you it’s complicated…), and the right hand side is the F term. The two terms on the right hand side represents the force due to pressure and viscosity respectively.

It turns out that this form of the equation does not best demonstrate the Reynolds number is. A better form is:

I’m not going to attempt to explain what this equation means. It’s not that important to the current discussion. Just know that it is equivalent to the Navier-Stokes equation, but written in some new variables that allowed to get rid of some terms. However, do remember how this equation looks like, for we will return to it later on.

Now you might be thinking, so what if the equations are complicated? If they are equations, can’t we solve them for v(x,y,z,t) and be done with the whole problem? Well, it turns out that the solutions to these equations vary with the lengths, velocities, densities and viscosities involved! If we obtain a solution to the problem for a low fluid velocity, and we subsequently increase the velocity, the flow pattern does not remain the same! Experimentally, the various flow patterns for various flow velocities are illustrated below. As you can see, many different types of flow patterns develop as the velocity increases.

Now the interesting thing is, if you believe that the equations are right, that same equation must give rise to all these flow patterns as certain parameters are changed. Now it is our job to find out what these parameters are.

Since the equations are so complicated, “normal” techniques used in analyzing mathematical and physics problems don’t quite work here. The method that gives the right answer actually looks something like a farfetched attempt at dimensional analysis. Now suppose we write all our dimensions in a length that is characteristic of the problem, the characteristic length L, then our new dimensions will be:

This characteristic length is normally taken to be the diameter of the cylinder in the context of this problem.

We can also write our time in terms of a “characteristic time” T. However since no such quantity is readily available in the problem, we will take the T to be the characteristic length divided by the velocity of the fluid, V.

If we diligently make these substitutions into the original equation, we will get:

Since now all the dimensions and time are standardized, the equation will always appear the same except for the coefficient in front of the term. If you are observant, you would have identified this as one over the Reynolds number:

Since the Reynolds number is the only part of the equation that can change, it alone determines which kind of flow pattern develops. So there we have it, all the characteristics of flow patterns captured in one dimensionless number: the Reynolds number.