Library / Aerospace / AER-37
Aerospace
Velocity gradient in a varying-area duct
Rate of change of speed along a smooth duct in steady quasi-one-dimensional isentropic flow with no mass addition, from the local area slope.
- Ref
- AER-37
- Rev
- v1
- Field
- Aerospace
- Result
- dV/dx [1/s]
Mass, momentum and energy conservation in quasi-one-dimensional isentropic flow give the area–velocity relation dV/V = [1/(M² − 1)] dA/A, which means dV/dx = [V/(M² − 1)](1/A)(dA/dx). In subsonic flow (M < 1) the speed rises as the area falls, and in supersonic flow it rises as the area increases, so a nozzle converges, then diverges. The expression is singular at M = 1, where the area is a minimum.
Rate of change of speed along a smooth duct in steady quasi-one-dimensional isentropic flow with no mass addition, from the local area slope. Subsonic flow speeds up in a converging duct; supersonic flow speeds up in a diverging one. The form is singular at M = 1.
Calculator
- Relative area slope, (1/A)dA/dx0.2 1/m
- Factor 1/(M² − 1)0.3333
- Speed change over 1 m of duct, V(dA/dx)/(A(M² − 1)) × 1 m40 m/s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The flow speeds up by 40 m/s for each metre of duct, a relative change of 6.67 % per metre.
- The flow is supersonic, so widening the duct (positive area slope) accelerates it; the reverse is true below Mach 1.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (1/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| M Mach number | 0.2 to 6 | −2,280 | 332.3 | 2,613 | +10% | −21.9 % |
| dA/dx Area gradient | −0.1 to 0.1 m²/m | −200 | 200 | 400 | +0.01 m²/m | +50 % |
| A Local area | 0.01 to 1 m² | 4 | 400 | 396 | +10% | −9.09 % |
| V Flow speed | 50 to 1,500 m/s | 3.333 | 100 | 96.67 | +10% | +10 % |
Sweep of Mach number
| Mach number | Velocity gradient (1/s) |
|---|---|
| 0.2 | −125 |
| 0.925 | −831.2 |
| 1.65 | 69.67 |
| 2.375 | 25.86 |
| 3.1 | 13.94 |
| 3.825 | 8.804 |
| 4.55 | 6.091 |
| 5.275 | 4.473 |
| 6 | 3.429 |
Sweep one variable
Variables and how each one contributes
Velocity gradient
What the equation gives you. Everything else feeds it.
Flow speed
The local flow speed, in m/s. Scales the gradient in direct proportion. Raising it by 10 % (to 660) moves the result by +10 %, from 40 to 44 1/s.
Area gradient
The local area slope dA/dx, in m²/m. A faster-widening duct gives a larger acceleration (supersonic) or deceleration (subsonic). Raising it by 10 % (to 0.022) moves the result by +10 %, from 40 to 44 1/s.
Local area
The local duct area, in m². A larger area lowers the relative area change and so the gradient. Raising it by 10 % (to 0.11) moves the result by −9.09 %, from 40 to 36.36 1/s.
Mach number
The local Mach number, away from 1. The gradient is largest near Mach 1, and changes sign across it. Raising it by 10 % (to 2.2) moves the result by −21.9 %, from 40 to 31.25 1/s.
Worked example with the values above
- Relative area slope: (1/A)dA/dx = 0.02 / 0.1 = 0.2 1/m.
- Mach factor: M² − 1 = 4 − 1 = 3.
- dV/dx = 600 × 0.2 / 3 = 40 1/s.
Assumptions and limits
- Steady, adiabatic, reversible flow of a calorically perfect gas, with no shocks or friction.
- A constant heat-capacity ratio, and quasi-one-dimensional flow where an area enters.
Common mistakes
- Using the formula at M = 1, where it is singular and the flow is choked.
- Using the wrong sign for the area slope in a converging section.
- Applying it to flows with friction, heat addition or shocks.
Statistics and uncertainty
Isentropic relations are exact for a perfect gas. Real nozzle flow departs from them: boundary layers and non-equilibrium chemistry make mass flow a few per cent lower, and for hot combustion gas γ is a function of temperature, not a constant.
Where it is used
- Sizing nozzles, inlets and diffusers.
- Referring engine data to standard conditions for comparison.
Maths you need
- Stagnation conditions: the state reached if the flow is slowed to rest isentropically
- Choking: the flow is sonic at the minimum area and the mass flow is maximal
- Mach number: M = V/a