Fluid Dynamics (Overview): Basics, Terminology & Equations

The study of fluid dynamics might seem like a narrow topic in physics. In day-to-day speech, for one, you say "fluids" when you mean liquids, in particular something like the flow of water. And why would you want to spend so much time just looking at the motion of something so mundane?

But this way of thinking misunderstands the nature of the study of fluids and ignores the many different applications of fluid dynamics. As well as being useful for understanding things like ocean currents, fluid dynamics has applications in areas like plate tectonics, stellar evolution, blood circulation and meteorology.

The key concepts are also crucial for engineering and design, and mastery of fluid dynamics opens doors to working with things like aerospace engineering, wind turbines, air conditioning systems, rocket engines and pipe networks.

The first step to unlocking the understanding you need to work on projects like these, though, is to understand the basics of fluid dynamics, the terms physicists use when talking about it and the most important equations governing it.

The meaning of fluid dynamics can be understood if you break down the individual words in the phrase. "Fluid" refers to a liquid or an incompressible fluid, but it can technically also refer to a gas, which substantially broadens the scope of the topic. The "dynamics" part of the name tells you it involves studying moving fluids or fluid motion, rather than fluid statics, which is the study of fluids not in motion.

There is a close relationship between fluid dynamics, fluid mechanics and aerodynamics. Fluid mechanics is the broad term covering both the study of ​fluid motion​ and static fluids, and so fluid dynamics really comprises half of fluid mechanics (and it's the part with the most ongoing research).

Aerodynamics, on the other hand, deals ​exclusively​ with gases, while fluid dynamics covers both gases and liquids. While there is a benefit in specializing if you know you'd rather work in aerodynamics, fluid dynamics is the broadest-ranging and most active field in the area.

The key focus of fluid dynamics is ​how fluids flow​, and so understanding the basics is crucial for any student. However, the key points are intuitively simple: Fluids flow downhill and as a result of pressure differences. The downhill flow is driven by gravitational potential energy, and the flow due to pressure differences is essentially driven by the imbalance between the forces at one location and another, in line with Newton's second law.

The continuity equation is a fairly complicated-looking expression but it really just conveys a very simple point: Matter is conserved during fluid flow. So the amount of fluid flowing past point 1 must match the point flowing past point 2, in other words, the ​mass flow rate​ is constant. The equation makes it easy to see specifically what this means:

\(ρ_1A_1v_1= ρ_2A_2v_2\)

Where ​ρ​ is the density, ​A​ is the cross-sectional area, and ​v​ is the velocity, and the subscripts 1 and 2 refer to point 1 and point 2, respectively. Think about the terms in the equation carefully while considering fluid flow: The cross-sectional area takes a single, two-dimensional "slice" of the fluid flow at a given point, and the velocity tells you how rapidly any single cross-section of the fluid is moving.

The remaining piece of the puzzle, the density, ensures that this is balanced against the amount of compression of the fluid at different points. This is so that if a gas is compressed between point 1 and point 2, the greater amount of matter per unit volume at point 2 is accounted for in the equation.

If you combine the units for the three terms on each side, you'll see that the resulting unit for the expression is a value in mass/time, i.e. kg / s. The equation explicitly matches the rate of flow of matter at two different points on its journey.

Bernoulli's principle is one of the most important results in fluid dynamics, and in words, it states that the pressure is lower in regions where a fluid flows more quickly. However, when this is expressed in the form of Bernoulli's equation, it becomes clear that this is a statement of the ​conservation of energy​ applied to fluid dynamics.

It essentially states that the energy density (i.e. the energy in a unit of volume) is equal to a constant, or (equivalently) that before and after a given point, the sum of these three terms remains the same. In symbols:

\(P_1 + \frac{1}{2} ρv_1^2 + ρgh_1 = P_2 + \frac{1}{2} ρv_2^2 + ρgh_2\)

The first term gives the pressure energy (with pressure = ​P​), the second term gives the kinetic energy per unit volume, and the third gives the potential energy (with ​g​ = 9.81 m/s² and ​h​ = height of tube). If you're familiar with conservation of energy or momentum equations in physics, you'll already have a good idea of how to use this equation.

If you know the initial values and at least some details of the pipe and fluid after the chosen point, you can find out the remaining value by re-arranging the equation.

It's important to note some caveats about Bernoulli's equation. It assumes that both points lie on a streamline, that the flow is steady, that there is no friction and that the fluid has a constant density.

These are restrictive limitations on the formula, and if you were being ​strictly​ accurate, no moving fluids would meet these requirements. However, as is often the case in physics, many cases can be approximately described this way, and to make the calculation much simpler, it's worth making these approximations.

Bernoulli's equation actually applies to what is called laminar flow, and essentially describes moving fluids with a smooth or streamline flow. It can help to think about it as the opposite to turbulent flow, where there are fluctuations, vortices and other irregular behavior.

In this steady flow, the important quantities like velocity and pressure used to characterize the flow remain constant, and the fluid flow can be thought of as taking place in layers. For example, on a horizontal surface, the flow could be modeled as a series of parallel, horizontal layers of water, or through a tube it could be thought of as a series of increasingly small concentric cylinders.

Some examples of laminar flow should help you understand what it is, and one everyday example is the water emerging from the bottom of a tap. At first, it dribbles, but if you open the tap a little more, you get a smooth, perfect stream of water out of it – this is laminar flow – and at higher levels still it becomes ​turbulent​. The smoke emerging from the tip of a cigarette also shows laminar flow, a smooth stream at first, but then becomes turbulent as it gets farther away from the tip.

Laminar flow is more common when the fluid is moving slowly, when it has high viscosity or when it only has a small amount of space to flow through. This was demonstrated in a famous experiment by Osborne Reynolds (known for the Reynolds number, which will be discussed more in the next section), in which he injected dye into a fluid flow through a glass tube.

When the flow was slower, the dye moved in a straight line path, at higher speeds it moves to a transitional pattern, while at much higher speeds it becomes turbulent.

Turbulent flow is the chaotic flow motion that tends to happen at higher speeds, where the fluid has a larger space to flow through and where the viscosity is low. This is characterized by vortices, eddies and wakes, which makes it very difficult to predict the precise motions in the flow because of the chaotic behavior. In turbulent flow, the speed and direction (i.e. the velocity) of the fluid changes continuously.

There are many more examples of turbulent flow in day-to-day life, including wind, river flow, the water in the wake of a boat's travel, the air flow around the tips of an aircraft's wing and the flow of blood through arteries. The reason for this is that laminar flow really only happens under special circumstances. For example, you have to open a faucet a specific amount to get a laminar flow, but if you just open it to an arbitrary level, the flow will likely be turbulent.

The Reynolds number of a system can give you information about the ​point of transition​ between laminar and turbulent flow, as well as more general information about situations in fluid dynamics. The formula for the Reynolds number is:

\(Re = \frac{ρvL}{μ}\)

Where ​ρ​ is the density, ​v​ is the velocity, ​L​ is the characteristic length (e.g. the diameter for a pipe), and ​μ​ is the dynamic viscosity of the fluid. The result is a dimensionless number that characterizes the fluid flow, and it can be used to distinguish between laminar flow and turbulent flow when you know the characteristics of the flow. A flow will be laminar when the Reynolds number is less than 2,300 and turbulent when it's a high Reynolds number over 4,000, with the intermediate stages being turbulent flow.

Fluid dynamics has tons of real-world applications, from the obvious to the not-so-obvious. One of the more expected applications is to the design of plumbing systems, which need to take into account how the fluid will flow through the pipes in order to ensure everything works as intended. In practice, a plumber can go through their tasks without an understanding of fluid dynamics, but it's essential to the design of pipes, corners and plumbing systems in general.

Ocean currents (and atmospheric currents) are another area where fluid dynamics plays an integral role, and there are many specific areas physicists are researching and working with. The ocean and atmosphere are both rotating, stratified systems and both have a multitude of complexities affecting their behavior.

However, understanding what drives the different oceanic and atmospheric currents is a crucial task in the modern age, especially with the additional challenges posed by global climate change and other anthropogenic impacts. The systems are generally complex, though, and so computational fluid dynamics is often used to model and understand these systems.

A more familiar example shows the smaller-scale ways that fluid dynamics can contribute to understanding physical systems: a curveball in baseball. When spin is imparted onto the throw, it has the effect of slowing down part of the air moving against the spin, and speeding up the part moving with the spin.

This creates a pressure differential across different sides of the ball, according to Bernoulli's equation, which propels the ball toward the low pressure region (the side of the ball spinning into the direction of motion).