Welcome to the second lecture in aerodynamics. Last time, we discussed how we look differently at aerodynamics than we did in fluid mechanics, despite them largely being related. Specifically, we looked at the body forces generated by a flow, and the balancing of these forces, the looking up of empirical quantities, and what different flow properties we encounter in aerodynamics.
Today, we'll dive deeply into these flow variables. We will do our best to understand what they physically mean and how they generate a force on an object. So, let's get started. Throughout our study of aerodynamics, we will regularly come across flow variables that exist in the conservation equations and contribute to object forcing.
The main five we will see are the velocity, the pressure, the viscosity, the density, and temperature. While the flow velocity is kind of a bulk fluid quantity, the other four are molecular quantities and can be explained at the molecular level. Now, depending on your situation, you won't always need all of these variables.
Different sets of assumptions can remove our need for them. If we're dealing with an incompressible and inviscid flow, meaning the density is constant and the viscosity is small compared to the other forces, we will likely only utilize equations where velocity and pressure are dominant. A good example here would be the Bernoulli equation, one of the most common relations in aerodynamics.
Next, we can assume the flow is incompressible, but viscosity matters. This will be true, in at least some part, of all the flows with surfaces. Surfaces make boundary layers. and in the boundary layer the viscosity is important.
We will see viscosity in the typical form of the Navier-Stokes equations, or the conservation of momentum, where there is a term associated with viscous shear along with the flow momentum and pressure. And lastly, there will be times we can't make any assumptions, and the flow is considered compressible and viscous. These are difficult to solve directly.
and often require a number of simulations and measurements to understand completely. All of these variables represent a physical source of force on an object in a flow. Naturally, you might ask, sure, these are just variables that appear in equations, but what physically are they? What do they mean? How do they work?
So, here we will explore the definitions of each variable and what they are. Let's consider a sample flow. here represented by some parallel streamlines. Often, we consider fluid motion macroscopically, where we follow an infinitesimally small element, but never zoom in enough to see individual molecules. For us, it's just a blob of moving fluid.
If we ended up seeing molecules, this would break our continuum assumption, the foundation of our conservation laws. However, the forces are generated at the molecular level. and we need to consider this to clearly understand force sources. Our first variable is a macroscopic one, and that's the flow velocity. This is represented by u, v, and w for the x, y, and z directions respectively, and has the units of distance over time.
For velocity, we consider a small fluid blob, but not individual molecules, and we observe how fast these blobs pass through our observation window. Remember, we take the Eulerian point of view, not the Lagrangian, which means we fix an observation window in space and we watch things that pass through it. Note, this is specifically not a molecular quantity, but more of a statistical average of a group of molecules moving.
Individual molecules each have their own erratic speeds, which we will cover later. Next up we have pressure. Usually represented by capital P. has the units of force per area.
Pressure can be explained at the molecular level. Pressure is the force distributed over a surface from fluid molecules bashing into that surface. As molecules are close to a solid surface, sometimes they hit it and bounce off.
This change in velocity and direction requires a force. If you get enough molecules together near a surface and you zoom out, they produce a continuously distributed force over that surface, which is a stress or a pressure. Now, if there is a difference in pressure across the surface, for example, more molecules are partying on the left side than the right, then that generates a net force on the surface.
This is why pressure difference shows up in the momentum equation. Next, we have fluid density, represented by the Greek letter ρ. and has units of mass over volume. Put quite simply, density is how many molecules are packed into a fixed volume. The more molecules in a space, the more it weighs and the denser it is.
Density often represents mass in our conservation equations because it's more convenient to think of fluids from a per-unit volume perspective. Now... We can see in our drawing how density gradients may lead to a force. More molecules on one side of a surface than the other will lead to more wall collisions and a net force.
The last static fluid property, meaning the property of a fluid even if it's not moving, is temperature, represented by T, or sometimes theta, and is technically dimensionless but represented by Kelvin or Celsius. Temperature is how much kinetic energy each molecule has. Molecules each have their own velocity, and their motion can be erratic. If a group of these molecules moves, on average, a distance over a time, This is the flow velocity that we discussed first.
And temperature is the average kinetic energy of individual molecules for a group of molecules. So here in our sketch, we can also see how temperature changes across a surface might result in a force. If there are the same number of molecules on the left and the right, but one has a lot more kinetic energy, they'll bash into the side with a lot greater force.
And this reveals to us that, in reality, for a gas like air, the pressure, density, and temperature are related through the ideal gas law. If, in a region of flow, density or temperature go up, so does the pressure, or vice versa. If you've ever used a compressed air canister to clean your keyboard, you've noticed that releasing the air can cause the can to get very cold.
This is the ideal gas law at work, where the rapid release of pressure causes the air temperature to plummet. And finally, we have the flow dynamic viscosity, represented by the Greek letter μ, with units of force multiplied by time over area. Sometimes you might see the kinematic viscosity, which is the Greek letter ν, which is the dynamic viscosity normalized by density, but they both represent viscosity forcing.
Essentially, viscosity is the fluid form of friction. Consider a fluid flow with parallel streamlines, but each streamline has a different velocity. In this fluid are molecules, and these molecules move with the general flow, but also jump around a bit.
As a molecule jumps from one streamline to the other, it needs to change velocity to keep up with its new flow. This change in velocity is an acceleration and requires a force. This force is the viscous force.
Note that, although this is a molecular property of fluid, it does require a flow. Specifically, a flow gradient where two neighboring streams have different velocity. This is why it's most dominant near surfaces, where a solid boundary forces the velocity to rapidly approach zero. And that's it for the flow variables.
Let's quickly review. Here, we started with introducing the five main variables of air flow, and all these variables represent a force source. We end up having to consider the flow both macroscopically and microscopically to properly understand the variable roots.
We covered the fluid velocity, or the general motion of a group of molecules, then the pressure, density, and temperature, which represent the force on a surface due to neighboring molecules, which through the ideal gas law is a function of how many molecules there are in the region, and how fast they're moving. Lastly, viscosity, which is a molecular property of fluid, though a flow is needed to generate force. Next time, we'll explore how the general body forces produced by these variables, and how they are calculated and represented in the context of aerodynamics.