The Flow Field

There are two ways to watch a river. You can pick a floating leaf and follow it downstream with your eyes, or you can stand on a bridge and watch the water stream past a pier. The first way follows one bit of water from start to finish; the second records whatever passes one spot. Two observers of the same river write down different numbers. This chapter shows that one term of the equation comes from that difference.

An arrow at every point

Fluid dynamics mostly takes the second view. It gives every point in space, at every moment, the velocity of the water passing there. This is the flow field, or velocity field, written $\vel{\mathbf{u}(\mathbf{x},t)}$. Give it a position $\mathbf{x}$ and a time $t$, and it returns the velocity there and then as one arrow. The flow field is the part at the far left of the equation bar at the top of the page.

Side note

A particle in this book is not a molecule. It is a blob of fluid too small to see but holding a great many molecules, so each blob has a velocity and a pressure of its own. This is the continuum assumption.

Once you know the flow field, you also know how every particle moves: at any moment, a particle’s velocity is the field’s velocity where the particle is. So the two ways describe the same flow in different terms. Following a particle is the Lagrangian view; watching a fixed point is the Eulerian view.

Following a particle and watching a point

Picture water flowing through a channel that narrows and then widens again. The same amount of water passes every cross-section each second, so the water must move faster through the narrow throat. Chapter 2 works out this rule of “what flows in must flow out”. The flow itself never changes. In the experiment below, a sky-blue ring stays at one point while a yellow particle rides with the water. The plot at the top left draws both speeds over time.

Press Restart and watch the first few seconds. The speed at the ring never changes, as you would expect from a flow that stays the same. The yellow particle, though, speeds up as it nears the throat and slows down again after the throat. The flow did not change at all, yet the particle accelerated. It moved to where the water flows at a different speed.

Side note

This flow is not a simulation. This book draws it from formulas chosen to conserve the water’s volume, and it solves no equation of motion. Where the channel’s half-width is $h(x)$, the speed along the centre line is inversely proportional to $h$, and the crosswise velocity keeps any water from appearing or disappearing. There is no viscosity, so the water slides along the walls; real water sticks to them (chapter 4). The particle moves with these velocities, one fourth-order Runge–Kutta step every 1/60 s.

Watch the moment the particle passes the ring. They are at the same place, so they have the same velocity. Their lines on the plot still differ in slope: the ring’s line is flat and the particle’s line climbs. Same place, same velocity, a different rate of change. That difference is what this chapter is about.

The Squeeze slider sets how many times wider the inlet is than the throat. Lower it to 1 and the channel straightens, and the particle keeps its speed too. Raise the Pulsing slider instead, and the whole flow speeds up and slows down in time, so the speed at the ring swings as well. Pulse a straight channel and the particle’s line and the ring’s line become one.

The material derivative

How much does a particle’s velocity change over a short time $\Delta t$? Write the two components of the velocity as $\mathbf{u} = (u, v)$. The velocity changes for two reasons. First, the flow at the particle’s place may change with time even if the particle stayed put: that part is $\partial\mathbf{u}/\partial t$ times $\Delta t$. Second, in that time the particle moves $u\,\Delta t$ across and $v\,\Delta t$ up, to a place where the velocity is different: that part is the distance moved times the change in velocity per unit distance. Add the two:

$$\Delta\mathbf{u} \approx \frac{\partial \mathbf{u}}{\partial t}\,\Delta t + u\,\Delta t\,\frac{\partial \mathbf{u}}{\partial x} + v\,\Delta t\,\frac{\partial \mathbf{u}}{\partial y}$$

Divide both sides by $\Delta t$ and let $\Delta t$ shrink to zero, and you get the acceleration the particle feels. This is the material derivative, written $D/Dt$:

$$\mat{\frac{D\mathbf{u}}{Dt}} = \frac{\partial \mathbf{u}}{\partial t} + \adv{(\mathbf{u}\cdot\nabla)\mathbf{u}}$$

Here $\mathbf{u}\cdot\nabla$ is short for $u\,\partial/\partial x + v\,\partial/\partial y$. Apply it to any quantity and you get the change in that quantity per unit distance along the flow, times the speed. The first term on the right, $\partial\mathbf{u}/\partial t$, is the change the ring sees. The second, $\adv{(\mathbf{u}\cdot\nabla)\mathbf{u}}$, is the change the particle meets because it moves. It is called the advection term. In the channel as the experiment opens, nothing changes in time, so the first term is zero and all of the particle’s acceleration comes from the second. Straighten the channel and the second term vanishes; add pulsing and the first term comes alive.

Side note

$D/Dt$ means the rate of change as you follow a particle. It works for any quantity, not only velocity. For a temperature $T$:

$$\frac{DT}{Dt} = \frac{\partial T}{\partial t} + (\mathbf{u}\cdot\nabla)T$$

In Newton’s second law, $\mathbf{F} = m\mathbf{a}$, you measure the acceleration $\mathbf{a}$ by following the object. Fluids are no different: to apply Newton’s law to a fluid particle, you need $\mat{D\mathbf{u}/Dt}$. That is why the left side of the Navier–Stokes equation is $\mat{D\mathbf{u}/Dt}$, and the right side holds the forces, per unit mass, that push the particle: pressure, viscosity and forces from outside.

$$\mat{\frac{D\mathbf{u}}{Dt}} = \prs{-\frac{\nabla p}{\rho}} + \vis{\nu\nabla^2\mathbf{u}} + \frc{\mathbf{f}}$$

This book fills in the pressure force in chapter 3 and the viscous force in chapter 4. You already used the outside force in chapter 0, when you stirred the water.

Notice that $\adv{(\mathbf{u}\cdot\nabla)\mathbf{u}}$ holds the velocity twice: the flow carries itself along. That makes the term nonlinear. A pair of swirls that pushes itself forward, and a flow that breaks up into turbulence, both need this term. Chapter 5 switches it on alone.

Streamlines, pathlines and streaklines

There are also three ways to draw a flow as lines.

• A streamline joins up the arrows of the flow field at one instant. At every point on it, the velocity there is tangent to the line. It is a snapshot.
• A pathline is the path one particle actually took. Fix a small light to a particle, leave a camera’s shutter open, and the photo shows its pathline.
• A streakline joins every particle that passed one point. Smoke from a chimney, or ink let out at one spot, draws a streakline.

Side note

In this flow the velocity is the same at every point, and its direction swings in time, so the streamline is a straight line. The swing is a sine wave with a period of 3 s plus a smaller wave at twice the frequency. With a pure sine wave, the pathline and the streakline would land exactly on each other once every swing and look like one line. A new yellow particle leaves the source every 6 s.

When the flow does not change in time, the three are the same line. At the start of the experiment, the sky-blue band, the yellow line and the pink dots lie on top of each other. Once the flow’s direction starts to swing up and down, they part. The streamline points along this instant’s velocity, so it stays straight and swings with the flow. The pathline is the way the particle went, so once drawn, its shape stays put. The streakline behaves like smoke: the dots already out drift away together, while new dots leave the source in the direction of the moment, so its shape keeps changing. The larger you set Swing, the further apart the three lines go.

People often describe streamlines as the paths that particles follow. In a flow that does not change in time, that is true. Once the flow’s direction changes, it is wrong. A streamline joins up the velocities of one instant; a particle follows it only for a moment before it moves on to the next instant’s streamline. In a flow whose speed changes but whose direction does not, like the pulsing channel above, the streamlines stay put, so the particles still follow them. Even the smoke trail in a photo of a flow that changes direction is a streakline, not a streamline. Keep this in mind when you read the streamlines of a simulation.

In one sentence

The rate of change of velocity you see following a particle, $\mat{D\mathbf{u}/Dt}$, is the rate you see at a fixed point, $\partial\mathbf{u}/\partial t$, plus the rate the particle meets as it moves, $\adv{(\mathbf{u}\cdot\nabla)\mathbf{u}}$.