MECENG 106 Notes: Fluid Dynamics

Fall 2024
By Aathreya Kadambi
Expanding on Lectures by Professor Phil Marcus

This fall I’m taking a fluid dynamics class by Professor Phil Marcus at Berkeley! Coincidentally, I actually was really interested in his research on Jupyter’s red spot and these related problems in freshman year, although I ended up going down a different road for a while. Excited to learn fluid dynamics.

Chapter 1: Hydrostatics

Story 1: Dimensions

Relevant lectures: Lecture 1

The lecture notes have anything you need to succeed. You are also strongly recommended to work in teams on the homework. Homework are combination of graded and self-graded. Feel free to go to any discussion section.

When you write down equations, you should keep track of the dimensions of the quanities in the equation. If you are taking the $\sin$ or $\log$ of something, it better be dimensionless.

Dimensional analysis is very important. In fact, people were able to find the energy in the atomic bomb (which was a well-kept secret) based on the speed of the shock wave.

Dimensions are things like length, time, mass, etc.. It is important to distinguish these things from units, which we use to measure these dimensions.

Story 2: Properties of Fluids

Relevant lectures: Lectures 1, 2

Our properties are going to be functions of space and time: $\textbf{x} \in \R^3$ and $t \in \R$. Here are some important properties:

The following properties live in $\text{Maps}(\R^3\times \R, \R)$.

• Temperature: $\text{Temp}$.
• Mass Density: $\rho = \frac{\text{mass}}{\text{Volume}}$. It has dimensions $ML^{-3}$ in MLT units.
• Pressure: $P = \frac{\text{force}}{\text{area}}$. It has dimensions $ML^{-1}T^{-2}$ in MLT units.
• Velocity: $V$. It has dimensions $LT^{-2}$.

In fluid dynamics, we will write the ideal gas law as:

$P = R\rho T$ $R$ is called the gas constant, and the other quantities were discussed above.

In physics we generally use the form $PV = nRT$. So what is the relationship between these two forms? In physics one has to look up the molecular weights and stuff, and in MechE, we have to look up individual gas constants for each gas.

Ideal gas equation of state:

$P = R\rho T$ or in physics, $PV = nR_{\text{physics}}T$ Notice that $\rho = \frac{M}{V} = \frac{\nu_{\text{physics}}n}{V}$ so solving for $n$, $n = \frac{\rho V}{\nu_{\text{physics}}}$. Substituting this into the previous equation, we find that $P = \frac{R_{\text{physics}}}{\nu_{\text{physics}}} \rho T$ so we’ve found that $\boxed{R = \frac{R_{\text{physics}}}{\nu_{\text{physics}}}}$

Story 3: Viscosity

Viscosity is about roughly how well things flow. Physically, it is a diffusion coefficient of momentum.

In fluid dynamics, we have so called boundary conditions. For a Newtonian fluid, the boundary conditions are that the fluid at any surface in the direction that is parallel to the surface must move at the same velocity as the surface. This is called the no-slip boundary condition. The zeroness of momentum diffuses into the middle so that for honey, where that diffusion would happen very quickly for example, the momentum in the middle will also be close to zero. For water on the other hand, the momentum in the middle might be high even if the momentum at the surface is zero.

So far what we have discussed is called the kinematic viscosity. There is another viscosity called dynamic viscosity, which satisfies $\nu_{\text{dyn}} = \rho \nu_{\text{kin}}.$ From now on, $\nu$ will refer to kinematic viscosity. There is a dimensionless quantity called the Reynold’s number, defined: $\text{Re} = \frac{[V][L]}{\nu}$ where we are using brackets here to indicate “characteristic values”. When $\text{Re} \le 10^3$, we call it laminar, when it is $\ge 10^5$ we call it turbulent, and in between this, we call it transitional.

For 3D flows, it is extremely difficult to make calculations in the transitional region without approximations. That’s why you need to take MECENG 106, because that is one of the hardest places to calculate flows.

Story 4: Pressure

Relevant Lectures: Lecture 2

Consider an area. This vector has normal inward vector. We might write: $dF = \hat n P dA$ so that the integral is $\int df = \int \hat n P dA$ A question is, if we change our orientation, then do we get a different force or something? It turns out that the magnitude is independent.

We will show this in the static case, namely where

• $V \equiv 0$
• $\frac{\partial}{\partial t} = 0$

Now let’s imagine a little piece of swiss cheese (proceeds to draw an inclined plane with angle theta).

Chapter 2: Go with the Flow

Story 1: Towards Continuity and Conservation

In the following, we’re going to pretend we know some information: the density $\rho$ (a distribution on $\R^3$ representing mass), $\mathbf{F}_M$ (an idea of flux), and $Q$ (another distribution on $\R^3$ representing creation/destruction of mass).

Consider any fixed region (volume) $V \in \R^3$, and imagine $\R^3$ filled with a fluid. The density of this fluid is given by $\rho(x,y,z,t)$. Then we can obtain a fwe quantities directly:

• Total Mass of Fluid in Container: $\int_V \rho \;dV$ (summing up the density function)
• Time Rate of Change of Total Mass of Fluid in Container: $dM(t)/dt = \int_V \frac{\partial \rho}{\partial t} dV$
• Flux Through The Boundary/Surface (Rate of transport of mass from outside to inside): $\dot{M}_{\text{transport}} = \int_{\partial V} \mathbf{F}_M \cdot \hat{n}\; dA =: \int_{\partial V} \mathbf{F}_M \cdot d\vec{A} = -\int_V \nabla \cdot \mathbf{F}_M\;dV$
  • $\mathbf{F}_M$ is defined so that $(\mathbf{F}_M \cdot \hat{\mathbf{n}}) \; dA$ is the time rate at which mass passes through the surface element

Then we find that the change in the total mass of the region can only change when mass is created or destroyed within the region, or when it flows in through the boundary. This gives us: $\int_V \frac{\partial \rho}{\partial t} dV = -\int_V \nabla \cdot \mathbf{F}_M + Q\;dV$ and since this holds for all regions $V$, we find that

$\boxed{\frac{\partial \rho}{\partial t} = -\nabla \cdot \mathbf{F}_M + Q}.$

Story 2: Mass and Momentum Go with the Flow

We theorize that mass “goes with the flow” in the sense that, as Professor Marcus puts it, “transport of mass in a fluid is due to the fluid velocity”. Mathematically,

$\mathbf{F}_M = \rho \mathbf{v}.$ This sort of makes sense in a dimensional-analysis sort of way, and also intuitively: mass flows at a rate dictated by the velocity. Substituting this into our equation, and considering the case in which mass cannot be created nor destroyed (so $Q = 0$), $\frac{\partial \rho}{\partial t} = -\nabla \cdot (\rho \mathbf{v}) = -\rho \nabla \cdot \mathbf{v} - \nabla \rho \cdot \mathbf{v}.$ which is called the continuity equation or the conservation of mass. We rewrite this as: $\boxed{\frac{\partial \rho}{\partial t} + (\mathbf{v}\cdot \nabla)\rho = -\rho(\nabla \cdot \mathbf{v})}.$ In the special case where $\rho$ is constant, we find that this is the same as $\boxed{\nabla \cdot v = 0}$.

In general, if we make the assumption that a quantity “goes with the flow” and know the density of the quantitiy, we can write an expression for $\mathbf{F}$ and get an appropriate continuity or conservation law. Another important example is momentum. In the case of momentum, the density is something like “mass times velocity”. Since velocity is a vector quantity though, and we want a scalar density field, we instead choose a particular direction called the “accounting direction”, which we will use to study momentum flow in that direction. If we take a certain direction $\hat{d}$, and write $v_d = \vec{v} \cdot \hat{d}$, then we can take the scalar density field $\rho v_d$, and end up with the expression:

$\mathbf{F}_{MOM_d} = \rho v_d \mathbf{v}.$ Plugging this into our conclusion from the last story, $\frac{\partial \rho}{\partial t} = -\nabla \cdot (\rho v_d \mathbf{v}) + Q_{MOM_d}.$ Note that unlike for mass, it is harder to say that momentum cannot be created nor destroyed. In fact, by Newton’s laws, forces are “sources” of momentum. Now for simplicity I’ll take my accounting direction $\hat{d}$ to be $\hat{z}$, but this can be applied in any direction. By definition, $-\nabla P \cdot \hat{z}$ is the force per unit volume due to pressure of our fluid. We may also have external forces, for example due to gravity, which would contribute $-g\rho \hat{z}$. Substituting this in for $Q_{MOM_z}$, we get: $\frac{\partial (\rho v_z)}{\partial t} = -\nabla \cdot (\rho v_z \mathbf{v}) - \frac{\partial P}{\partial z} - g\rho$. which is the $z$-component of Euler’s equation. In the static case, this reduces to our usual hydrostatic equation!

Now if we simplify this equation and also use the conservation of mass equation from before, we can actually simplify this into the form:

$\frac{\partial v_z}{\partial t} + (\mathbf{v} \cdot \nabla)v_z = -\frac{1}{\rho}\frac{\partial P}{\partial z} - g$ And similar equations in the $x$ and $y$ directions can be combined to give the overall Euler’s equation: $\boxed{\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v}\cdot \nabla)\mathbf{v} = -\frac{1}{\rho}\nabla P - g\hat{z}}.$

We actually have a special name for the $\frac{\partial}{\partial t} + \mathbf{v}\cdot \nabla$ operator: it is called the Lagrangian time derivative, or the material or advective or convective derivative. This is kind of like the total time derivative from the perspective of someone who is moving, taking their motion into account via chain rule. A Lagrangian particle is a particle which “goes with the flow”, so the Lagrangian derivative is the rate of change in time as observed by a Lagrangian particle.

To summarize, the Euler equation and continuity equation can be written:

$\frac{D\mathbf{v}}{Dt} = -\frac{1}{\rho}\nabla P - g\hat{z}$ $\frac{D\rho}{Dt} = -\rho(\nabla \cdot \mathbf{v})$

Story 3: Vorticity and Bernoulli’s Theorem

We call the curl of the velocity vector field vorticity: $\mathbf{\omega} = \nabla \times \mathbf{v}$. This satisfies the following identity:

$\mathbf{v} \times \mathbf{\omega} - \frac{1}{2}\nabla\|v\|^2 \equiv -(\mathbf{v} \cdot \nabla)\mathbf{v}$ (I’ll leave this as an exercise to the reader). Then substituting into Euler’s equation gives: $\frac{\partial\mathbf{v}}{\partial t} = \mathbf{v}\times \mathbf{\omega} - \frac{1}{2}\nabla \|v\|^2 - \frac{1}{\rho}\nabla P - \nabla \phi$ where $\phi = gz$ is the gravitational potential.

Now there are a few important definitions:

• Streamline: a continuous curve tangent to $\mathbf{v}$ everywhere
  • Cannot be visualized in an experiment
• Pathline: path of a particle that “goes with the flow”
  • When velocity is independent of time, pathlines and streamlines are the same
• Streakline: path created by a bunch of particles released at every time up to a certain point

Finally, we can discover Bernoulli’s equation by dotting our expression with $\hat{\mathbf{s}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}$ and noting that $(\hat{\mathbf{s}}\cdot \nabla)Q = \frac{\partial Q}{\partial s}$, and considering the case of a steady flow so that $\frac{\partial \mathbf{v}}{\partial t} = 0$:

$\boxed{\frac{\partial}{\partial s}\left[\frac{1}{2}\|v\|^2 + \frac{1}{\rho} P + gz\right] = 0}$ so $\frac{1}{2}\|v\|^2 + \frac{1}{\rho} P + gz$ is constant along streamlines.

Story 4: Energy and the Breakdown of Bernoulli