# Jet Discharge Propulsion

## Propulsive force or thrust induced by an incompressible jet flow

The velocity out of a free jet can be expressed as

*v _{2} = (2 (p_{1} - p_{2}) / ρ)^{1/2} (1)*

*where *

*v _{2} = velocity out of the jet (m/s)*

*p _{1} = pressure before the jet (N/m^{2}, Pa) *

*p _{2} = ambient pressure after the jet (N/m^{2}, Pa)*

*ρ = density of the fluid (kg/m ^{3}) *

**Note!** - the density *ρ* is constant in incompressible flows and the equations are valid for liquids (like water) but not for gases (like air).

The flow volume out of a jet can be expressed as

*q = A _{} v_{2} (2)*

*where*

*q =flow volume (m ^{3}/s)*

*A _{} = orifice area of the jet (m^{2})*

The propulsive force or thrust induced by the jet can be expressed as

*F = ρ q (v _{2} - v_{1}) (3)*

*where *

*v _{1} = jet velocity (m/s)*

If the jet is not moving

*v _{1} = 0*

and (3) can be expressed as

*F = ρ q v _{2} (3b)*

- or alternatively with *(2)*

*F = ρ A v _{2}^{2} (3c)*

Substituting *v _{2}* with eq.

*(1)*in eq.

*(3c)*- the propulsive force or thrust induced by a jet not moving

*(v*be expressed as

_{1}= 0)*F = 2 A (p _{1} - p_{2}) *

*(4)*

### Example - Propulsive Force

Water flows through a garden hose with diameter *15 mm*. The water pressure just before the outlet is *4 10 ^{5} Pa* (abs) and the atmospheric pressure is

*1 10*(abs).

^{5}PaThe outlet area of the hose can be calculated as

A = *π ((15 mm) (0.001 m/mm) / 2) ^{2}*

* = 1.77 10 ^{-4} (m^{2})*

The propulsive force can be calculated as

*F = 2 ( 1.77 10^{-4} m^{2}) ((4 10^{5} Pa) - (1 10^{5} Pa))*

* = 106 N** *

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