Convergent-Divergent Nozzle (De Laval) Design Calculator
Set exit Mach number, throat area, and stagnation conditions to instantly compute isentropic flow distributions of Mach, pressure, and temperature along the nozzle axis. Optional normal shock visualization included.
What exactly is a De Laval nozzle, and why does it have that funny shape—narrow in the middle and wide at the ends?
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Basically, it's a convergent-divergent nozzle designed to accelerate a gas flow to supersonic speeds. The "funny shape" is key! The flow speeds up in the converging section, reaches the speed of sound (Mach 1) at the narrowest point—the throat—and then becomes supersonic as it expands in the diverging section. Try moving the "Exit Mach number (M)" slider in the simulator above from 0.5 to 3. You'll see the nozzle shape change dramatically to accommodate that supersonic flow.
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Wait, really? So the throat is always where Mach = 1? What happens if I change the gas properties, like the "Specific heat ratio (γ)"?
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In an ideal, isentropic design, yes—the throat is the sonic point. The specific heat ratio (γ) is crucial. For instance, diatomic air has γ ≈ 1.4, while rocket exhaust (like from hydrogen) has γ ≈ 1.2. A lower γ means the gas can expand more easily. Change γ in the simulator and watch how the required area ratio (A/A*) for a given Mach number changes. It directly affects the nozzle's contour.
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Okay, I see the "Shock position" slider. What's that for? I thought the flow was isentropic and smooth.
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Great question! In practice, if the back pressure outside the nozzle is too high, a shock wave forms inside the diverging section. This is a sudden, non-isentropic jump where flow properties change drastically—pressure spikes, temperature rises, and the flow can go from supersonic to subsonic. Slide the "Shock position x/L" control. You'll see a vertical line appear, representing the shock, and the pressure plot will show a sharp discontinuity at that point.
Physical Model & Key Equations
The core of the design relies on isentropic (constant entropy) flow relations. The first key equation relates the local static temperature (T) to the stagnation temperature (T₀), which is the temperature if the flow were brought to rest adiabatically.
Here, M is the local Mach number, and γ is the specific heat ratio (cp/cv). As M increases, T drops significantly. This is why nozzles get very cold internally.
The most critical equation for shape design is the area-Mach number relation. It defines the nozzle's cross-sectional area (A) at any point relative to the throat area (A*), where M=1.
For a given exit Mach number (M_exit) and γ, this equation gives the expansion area ratio. This is the value you are directly calculating with the simulator's input parameters. A/A* is minimum (1) at the throat and grows for both subsonic and supersonic Mach numbers.
Frequently Asked Questions
It depends on the application. If the exit velocity of the nozzle you want to design is determined, input the exit Mach number first. Conversely, if the nozzle shape (area ratio) is determined first, input the area ratio. By inputting either one, the other will be automatically calculated from the isentropic relation.
When the nozzle back pressure (pressure outside the exit) is higher than the design condition, a normal shock wave occurs inside the nozzle. In this tool, the position is calculated from the Mach number and pressure ratio before and after the shock wave based on the back pressure condition, and is displayed as a red dashed line on the graph. The pressure recovery after passing through the shock wave can also be confirmed.
For air, it is 1.4; for combustion gas (rocket engine), it is approximately 1.2 to 1.3; for helium, it is 1.67 as standard. If the type of gas is unknown, 1.4 is recommended as a general diatomic molecular gas. Since the relationship between Mach number and area ratio changes with the specific heat ratio, please enter an accurate value.
This is because the assumption of isentropic flow no longer holds when flow separation occurs inside the nozzle or a normal shock wave is generated. This tool stops the isentropic calculation after the shock wave position. By lowering the back pressure or increasing the nozzle area ratio, a wider range of calculations becomes possible.
Real-World Applications
Rocket Engine Nozzles: This is the most iconic application. The bell-shaped nozzle on a rocket is a De Laval nozzle. It maximizes thrust by efficiently expanding hot, high-pressure combustion gases to supersonic speeds. The simulator's stagnation conditions (P₀, T₀) directly model the conditions in a rocket's combustion chamber.
Supersonic Wind Tunnels: To test aircraft and missile models at supersonic speeds, engineers use wind tunnels with a De Laval nozzle in the test section. By carefully designing the area ratio, they can set a precise Mach number for testing, just like you can set the exit Mach number in this tool.
Steam Turbines (Historical & Modern): Gustaf de Laval originally invented this nozzle for steam turbines. It allowed steam to reach supersonic speeds, dramatically increasing the efficiency and power output of early turbines. The principle is still used in some high-power applications today.
Jet Engine Afterburners & Nozzles: Military fighter jets with afterburners often use variable-geometry convergent-divergent nozzles. During afterburner use, the nozzle opens up to act as a De Laval nozzle, allowing the higher-temperature exhaust to expand to supersonic speeds for extra thrust.
Common Misconceptions and Points to Note
When you start using this tool, there are a few common pitfalls to watch out for. First, make sure you understand the assumption that "the inlet condition comes from an infinitely large tank." The simulator calculates starting from a state where the flow velocity at the inlet is nearly zero (Mach number ≈ 0). However, an actual engine's combustion chamber is finite in size and has its own flow. If the inlet Mach number is 0.1 or higher, be aware that the simple area ratio formula used by this tool alone won't determine the accurate shape.
Next, be careful of errors in setting the "specific heat ratio γ." A common mistake is designing a rocket nozzle for air (γ=1.4). Rocket combustion gases contain a lot of water vapor and carbon dioxide, so γ can be around 1.2. For example, simply changing γ from 1.4 to 1.25 increases the required area ratio to achieve the same exit Mach number of 5.0 by about 40. Always double-check the gas you're using.
Finally, there's the "gap between ideal and reality." This calculation is based on "one-dimensional isentropic flow," which completely ignores wall friction, heat losses, and two-dimensional flow effects. In actual nozzles, especially small ones, the boundary layer can reduce the effective flow area, preventing the design performance from being achieved. Remember, even if you achieve an "optimal design" in simulation, the real design work begins from there.
Enter stagnation pressure (P₀) in Pa and stagnation temperature (T₀) in K for your working fluid (e.g., combustion gases at P₀=2.5 MPa, T₀=3000 K for rocket engines).
Set throat area A* in mm² and target exit Mach number (typically 2.5–4.5 for supersonic operation).
Click Calculate to generate isentropic flow properties: exit velocity, pressure ratio Pe/P₀, area expansion ratio Ae/A*, mass flow rate, thrust, and specific impulse at sea-level conditions.
Worked Example
Solid rocket motor with nitrogen tetroxide/hydrazine: P₀=3.0 MPa, T₀=2800 K, throat A*=45 mm², exit Mach=3.2, gamma=1.25. Calculator yields exit velocity=3650 m/s, Pe/P₀=0.028, Ae/A*=8.6, mass flow=12.8 kg/s, thrust=46.7 kN, specific impulse=374 seconds. Expanding from subsonic throat to Mach 3.2 requires divergent section area ratio of 8.6:1.
Practical Notes
Verify convergence before divergence: subsonic flow (M<1) in converging section, sonic at throat (M=1), supersonic expansion in diverging section—mixing occurs if back-pressure exceeds design Pe.
For hypergonic propellants, gamma varies with temperature; use gamma=1.20–1.22 above 2500 K to avoid underestimating expansion.
Thrust peaks when nozzle operates at design altitude (typically 1 atm); lower back-pressure causes overexpansion and side-load forces on divergent walls.