Subsonic and transonic wind tunnel

Schematic of an open wind tunnel with a closed test section

The working principle is based on the continuity and Bernoulli's equation:

The continuity equation is given by:

${\displaystyle AV=constant\Rightarrow {\frac {dA}{A}}=-{\frac {dV}{V}}}$

The Bernoulli equation states:-

${\displaystyle P_{total}=P_{static}+P_{dynamic}=P_{s}+{\frac {1}{2}}\rho V^{2}}$

Putting Bernoulli into the continuity equation gives:

${\displaystyle V_{m}^{2}=2{\frac {C^{2}}{C^{2}-1}}{\frac {P_{settl}-p_{m}}{\rho }}\approx 2{\frac {\Delta p}{\rho }}}$

The contraction ratio of a windtunnel can now be calculated by: ${\displaystyle C={\frac {A_{settl}}{A_{m}}}}$

Schematic of a closed (return-flow) wind tunnel

In a return-flow wind tunnel the return duct must be properly designed to reduce the pressure losses and to ensure smooth flow in the test section. The compressible flow regime: Again with the continuity law, but now for isentropic flow gives:

${\displaystyle -{\frac {d\rho }{\rho }}=-{\frac {1}{a^{2}}}{\frac {dp}{\rho }}=-{\frac {1}{a^{2}}}{\frac {-\rho VdV}{\rho }}={\frac {V}{a^{2}}}dV}$

The 1-D area-velocity is known as:

${\displaystyle {\frac {dA}{A}}=(M^{2}-1){\frac {dV}{V}}}$

The minimal area A where M=1, also known as the sonic throat area is than given for a perfect gas:

${\displaystyle \left({\frac {A}{A_{throat}}}\right)^{2}={\frac {1}{M^{2}}}\left({\frac {2}{\gamma +1}}\left(1+{\frac {\gamma -1}{2}}M^{2}\right)\right)^{\frac {\gamma +1}{\gamma -1}}}$