SpaceX put another batch of Starlink satellites up this week on a Falcon 9, which by my count was the hundredth Falcon 9 launch of the year. Watching the booster come back and land has become almost routine, which is an astonishing sentence to be able to write. It sent me back to my engineering degree at UCL, where rocket propulsion and fluid dynamics took up a good part of my time, and I thought it might be worth setting down the handful of principles that make a launch like that work. None of it is secret. All of it is beautiful once it clicks.
Why a rocket moves
A rocket engine is Newton’s third law with plumbing. Throw mass out of the back fast enough and the vehicle goes forward. In a chemical rocket like the Falcon 9’s Merlin engines, kerosene and liquid oxygen burn in a combustion chamber at very high pressure and temperature, and the hot gas is expelled through a nozzle. The thrust is essentially the mass flow rate of that gas multiplied by how fast it leaves:
\[F = \dot{m}\, v_e + (p_e – p_a)\, A_e\]where ṁ is the mass flow rate, ve the exhaust velocity, and the second term is a correction for the difference between the pressure of the exhaust at the nozzle exit, pe, and the surrounding atmosphere, pa, acting over the exit area Ae. At sea level that pressure term matters; in vacuum pa is zero and it becomes a small bonus. Most of the thrust, in any case, is the first term, which is why everything about engine design comes down to pushing more mass out faster.
The efficiency of an engine is measured by specific impulse, which is thrust per unit of propellant consumed per second:
\[I_{sp} = \frac{F}{\dot{m}\, g_0}\]with g0 the standard gravity, included only so that the answer comes out in seconds whatever units you started in. A higher specific impulse means more thrust from each kilogram of propellant, which on a rocket, where propellant is most of the take-off mass, is the number everyone is trying to raise.
The mass flow rate itself is the density of the gas times the area it passes through times its velocity:
\[\dot{m} = \rho\, A\, v\]and because mass is conserved, that product has to be the same at every point along the nozzle. That one constraint is the key to everything that follows.
The garden hose, and where it stops working
Everyone has done the experiment. Put your thumb over the end of a hose and the water comes out faster. The area went down, the flow rate stayed the same, so the velocity had to go up. That is the continuity equation, A1v1 = A2v2, and it works because water is effectively incompressible, so the density term drops out. Bernoulli’s principle goes with it: where the flow is faster the pressure is lower, which for an incompressible fluid along a streamline is
\[P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant}\]So far so intuitive. Narrow the pipe, speed up the flow. The surprise comes when the fluid is a gas moving near or above the speed of sound, because then the density can change, and the hose intuition reverses.
The quantity that decides which regime you are in is the Mach number, the ratio of the flow speed to the local speed of sound. For a gas moving through a duct of changing area, the relationship between a fractional change in area and the resulting fractional change in velocity is
\[\frac{dA}{A} = (M^2 – 1)\,\frac{dv}{v}\]Look at the sign. When M is less than 1 the bracket is negative, so shrinking the area speeds the flow up, exactly as with the hose. When M is greater than 1 the bracket is positive, and the behaviour flips: to make a supersonic gas go faster you have to widen the duct, not narrow it. And at M equal to 1 the bracket is zero, which says that the only place the flow can pass through the speed of sound is where the area is not changing, which means a minimum. That minimum is the throat.
The nozzle
This is why every rocket nozzle is the shape it is. The gas leaves the combustion chamber subsonic, is squeezed through a narrowing section that accelerates it, reaches exactly Mach 1 at the throat, and is then expanded through a widening bell that, because the flow is now supersonic, accelerates it further still. The design is called a converging-diverging or de Laval nozzle, and the condition at the throat is called choked flow: once the throat is at Mach 1, the mass flow rate through the engine is fixed by the chamber pressure and temperature and cannot be increased by lowering the pressure downstream. The engine, in effect, no longer knows what is happening outside.
What the diverging section is doing, in energy terms, is converting the pressure and heat of the gas into velocity. For a compressible gas the energy balance along a streamline, with no heat added and no losses, takes the form
\[\frac{v^2}{2} + \int \frac{dP}{\rho} + g z = \text{constant}\]and for a perfect gas expanding isentropically the pressure integral can be done, giving
\[\frac{v^2}{2} + \frac{\gamma}{\gamma – 1}\,\frac{P}{\rho} + g z = \text{constant}\]where γ is the ratio of specific heats of the gas. In a rocket nozzle the height term is negligible and drops out. What remains says that as the pressure and density fall along the bell, the velocity has to rise to keep the total constant, and the whole art of nozzle design is choosing the expansion ratio so that the gas reaches the exit at roughly the pressure of the surrounding air. Too little expansion and you are throwing away pressure that could have been velocity. Too much, and the exhaust is below ambient pressure at the exit, which costs thrust and, in extreme cases, pulls the flow off the nozzle wall. That is why a vacuum-optimised upper-stage engine has a much larger bell than a sea-level engine, and why the Falcon 9’s second-stage Merlin looks so different from the nine on the first stage even though the combustion chamber is nearly identical.
You can see the mismatch in the exhaust itself. When the exit pressure does not quite match the atmosphere the plume forms a repeating pattern of shock diamonds as it alternately over- and under-expands on the way out. At sea level that pattern is visible in almost every launch photograph.
Why I still think about this
Fluid dynamics was the subject I enjoyed most at UCL. My tutor, Professor Ian Eames, supervised my third-year thesis on the hydrodynamic forces a tsunami exerts on a building, a project funded by the UK government that later became part of a published paper, and working with him was the best part of my degree. What that work taught me, more than any specific equation, was how to take a system that will not sit still, write down the conservation laws that have to hold whatever else is going on, and let the mathematics tell you what the system is allowed to do. I have used that habit in markets far more often than I have used the Mach number. But every time a Falcon 9 lands on a barge, I am reminded that the equations above are not approximations of the world. They are the world, and somebody built a machine that trusts them.