Insights from SpaceX’s Falcon 9 and Fluid Dynamics

After the recent SpaceX launch, where a Falcon 9 rocket successfully carried 23 Starlink internet satellites into orbit, marking its 100th rocket launch of 2024, I found myself reflecting on the fundamental principles that make these incredible achievements possible. This event stirred memories from my time studying Rocket Science at University College London (UCL), a topic I extensively covered during my engineering degree.

Understanding Propulsion Engines

In essence, propulsion engines work by converting energy (typically from a fuel source) into motion. This is beautifully explained by Newton’s third law of motion:

“For every action, there is an equal and opposite reaction.”

Propulsion engines range from jet engines, which propel aircraft, to rocket engines, used to launch spacecraft and missiles. In the case of rocket propulsion, the principle remains the same: by ejecting mass (in the form of high-speed gases), an equal and opposite reaction occurs, pushing the rocket forward.

For the Falcon 9, we are dealing with a chemical rocket, where fuel and oxidiser are combined in a combustion chamber. The resulting hot gases are expelled through a nozzle, generating thrust. This is the foundation of modern rocket science, combining thermodynamics, fluid dynamics, and engineering precision.

A Personal Journey into Fluid Dynamics

One of the most fascinating subjects I studied at UCL was Fluid Dynamics, a subdiscipline of Fluid Mechanics. This field deals with the flow of liquids and gases, analysing properties such as velocity, pressure, density, and temperature as they change across space and time. My tutor, Professor Ian Eames, a renowned expert in the field, supervised my third-year thesis on tsunami hydrodynamic forces acting on building structures—a project that was funded by the UK government and later published in a scientific paper.

Working with Professor Eames was one of the most intellectually enriching experiences of my academic journey. His vast knowledge and innovative approach to fluid dynamics and mathematics made him an inspiring mentor. This background has been invaluable later in my finance career, helping me succeed in the field of algorithmic trading.

Key Principles of Rocket Propulsion

Thrust Equation:

The thrust generated by a rocket engine can be calculated using the equation:

F = ṁ ⋅ (v_exit – v_inlet)

Where:
– F is the thrust,
– ṁ is the mass flow rate of the propellant,
– v_exit and v_inlet are the velocities of the exhaust and intake air, respectively.

Specific Impulse (Isp):

Specific Impulse (Isp) is a crucial measure of a rocket engine’s efficiency. It’s defined as the thrust per unit of propellant flow rate.

I_sp = F / (ṁ ⋅ g₀)

Where g₀ is the gravitational constant. A higher I_sp indicates more efficient fuel use, which is vital for space exploration.

Mass Flow Rate:

The mass flow rate (ṁ) mentioned above is the rate at which mass flows through a given point in a fluid flow system. It is described by the equation:

ṁ = ρ ⋅ A ⋅ v

Where:
– ρ is the fluid density (kg/m³),
– A is the cross-sectional area (m²),
– v is the velocity of the fluid (m/s).

Fluid Flow and Mach Number

When discussing fluid dynamics in the context of rocket engines, understanding Mach number—the ratio of flow velocity to the speed of sound—is key. This helps categorise the flow as either subsonic or supersonic.

1. Subsonic Flow (Mach < 1):

In subsonic flow, the fluid velocity is less than the speed of sound. The Continuity Equation describes the relationship between area and velocity in such flows:

A₁v₁ = A₂v₂

In subsonic flows, decreasing the area (a converging section) increases velocity, while increasing the area (a diverging section) decreases velocity.

Additionally, Bernoulli’s Principle states that as velocity increases, pressure decreases, and vice versa:

P + ½ ρ v² = constant

2. Supersonic Flow (Mach > 1):

In supersonic flow, where the velocity exceeds the speed of sound, the behaviour reverses. Here, increasing the cross-sectional area causes the fluid velocity to increase, unlike in subsonic flow.

The Continuity Equation still applies:

A₁v₁ = A₂v₂

3. Isentropic Flow Relations:

In supersonic flow, the relationship between pressure, velocity, and cross-sectional area is described by the following equation:

dA / A = (M² – 1) ⋅ (dv / v)

Where M is the Mach number, and dA/A and dv/v represent fractional changes in area and velocity, respectively.

Choked Flow and Mach = 1

At the throat of a converging-diverging nozzle, the flow reaches Mach 1—the speed of sound. This phenomenon is known as choked flow, where:

– The velocity at the throat is exactly the speed of sound.
– The cross-sectional area at the throat is the smallest in the nozzle.

Real-World Analogy: The Garden Hose Example
To make these concepts easier to visualise, consider a simple garden hose. Imagine holding the hose without placing anything at the end: the water flows smoothly at a certain speed. Now, if you place your thumb partially over the hose opening, you reduce the cross-sectional area available for the water to escape. As a result, the water shoots out much faster. This is a real-world example of how reducing the area increases velocity to satisfy the Continuity Equation. This holds true because the fluid in this case is subsonic and considered incompressible.

Rocket Science: The Fluid at the End of a Nozzle
A similar principle is observed at the end of a rocket nozzle, where the fluid undergoes a dramatic transformation governed by the principles of compressible flow dynamics. Initially, as the fluid (gas) moves through the nozzle, it experiences changes in pressure, density, and velocity. These changes are described by the Continuity Equation, Bernoulli’s Principle, and the Isentropic Relations for compressible fluids.

In the converging-diverging design of most rocket nozzles (such as a de Laval nozzle), the gas accelerates through the throat (the narrowest part of the nozzle) as the cross-sectional area decreases. The kinetic energy of the gas results from the change in pressure downstream, which creates a strong pressure gradient. This gradient not only accelerates the flow but also prevents a build-up of pressure upstream. The reduction in pressure downstream allows the conversion of pressure and thermal energy into velocity, driving the flow forward.

Bernoulli’s Principle

Bernoulli’s equation describes how the total mechanical energy of a fluid remains constant along a streamline, assuming no losses due to friction or heat transfer. For an incompressible fluid, Bernoulli’s equation is written as:

$P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant}$

For compressible flow, this equation is adjusted to account for density variations, but the principle remains the same: a reduction in pressure leads to an increase in velocity, and vice versa.

For compressible flow, Bernoulli’s equation is modified to account for changes in density due to variations in pressure and temperature. The compressible flow version of Bernoulli’s equation can be derived from the conservation of energy and is often expressed for isentropic flow (no heat transfer and no losses):

$\frac{v^2}{2} + \int \frac{\mathrm{d}P}{\rho} + gz = \text{constant}$

Where:

$P$ is the static pressure of the fluid,
$ρ$ is the fluid density,
$v$ is the flow velocity,
$g$ is the acceleration due to gravity,
$h$ is the height (potential energy per unit mass).

Special Case for Isentropic Compressible Flow

For isentropic flow of a perfect gas, we use the relation between pressure and density:

$\frac{\mathrm{d}P}{\rho} = a^2 \, \mathrm{d}\ln{\rho}$

$v$ : velocity of the fluid
$P$ : pressure of the fluid
$ρ$ : density of the fluid (variable for compressible flow)
$g$ : acceleration due to gravity
$z$ : elevation (potential energy term)

$\frac{v^2}{2} + \frac{\gamma}{\gamma – 1} \frac{P}{\rho} + gz = \text{constant}$

Here, $γ$ is the ratio of specific heats ( $γ = c p / c v $ ).

Simplifications in Rocket Nozzle Flow

In rocket nozzles, elevation changes ( $g z$ ) are often negligible compared to the dynamic and pressure energy terms. In this case, the compressible Bernoulli equation simplifies to:

$\frac{v^2}{2} + \frac{\gamma}{\gamma – 1} \frac{P}{\rho} + gz = \text{constant}$

Additionally, using the isentropic relations, the pressure, density, and temperature can all be related to velocity and specific heat ratios to further describe the compressible flow in the nozzle.

High-Velocity Transition: The gas reaches or exceeds the speed of sound (supersonic flow) before or at the nozzle exit, depending on its design. The density of the fluid continues to drop, but its velocity increases dramatically due to the area increase in the diverging section of the nozzle. This satisfies the conservation laws and ensures maximum conversion of thermal energy into kinetic energy.
From Compressible to Incompressible Behaviour: As the gas exits into the atmosphere, its density and pressure stabilise, and its flow begins to approximate incompressible behaviour. This occurs because the velocity becomes so high that density changes become negligible compared to the inertial forces of the fluid. Simply put, once the gas is fully expanded and outside the nozzle, its behaviour is dominated more by momentum than compressibility.

Analogy to the Hose Example

Just as placing a thumb over the garden hose forces water to shoot out faster by reducing the area, the nozzle constrains and then expands the gas flow to achieve maximum velocity. However, in the case of a rocket nozzle, the flow transitions from compressible to incompressible in the final phase. Essentially, the nozzle not only accelerates the fluid but also prepares it for a high-momentum, near-constant-density state that enables effective thrust in open-air or vacuum conditions.

Thus, the end of the rocket nozzle represents a crucial phase where the fluid transitions from compressible to incompressible-like behaviour. The downstream drop in pressure and corresponding acceleration ensure that the maximum possible kinetic energy has been imparted to the exhaust for optimal propulsion.

Caio Marchesani