Stream thrust averaging

In fluid dynamics, stream thrust averaging is a process used to convert three-dimensional flow through a duct into one-dimensional uniform flow. It makes the assumptions that the flow is mixed adiabatically and without friction. However, due to the mixing process, there is a net increase in the entropy of the system. Although there is an increase in entropy, the stream thrust averaged values are more representative of the flow than a simple average as a simple average would violate the second Law of Thermodynamics.

Equations for a perfect gas

Stream thrust:

F=\int \left(\rho {\mathbf {V}}\cdot d{\mathbf {A}}\right){\mathbf {V}}\cdot {\mathbf {f}}+\int pd{\mathbf {A}}\cdot {\mathbf {f}}.

Mass flow:

{\dot m}=\int \rho {\mathbf {V}}\cdot d{\mathbf {A}}.

Stagnation enthalpy:

H={1 \over {\dot m}}\int \left({\rho {\mathbf {V}}\cdot d{\mathbf {A}}}\right)\left(h+{|{\mathbf {V}}|^{2} \over 2}\right),

{\overline {U}}^{2}\left({1-{R \over 2C_{p}}}\right)-{\overline {U}}{F \over {\dot {m}}}+{HR \over C_{p}}=0.

Solutions

Solving for $\overline {U}$ yields two solutions. They must both be analyzed to determine which is the physical solution. One will usually be a subsonic root and the other a supersonic root. If it is not clear which value of velocity is correct, the second law of thermodynamics may be applied.

\overline {\rho }={{\dot m} \over \overline {U}A},

{\overline {p}}={F \over A}-{{\overline {\rho }}{\overline {U}}^{2}},

\overline {h}={\overline {p}C_{p} \over \overline {\rho }R}.

Second law of thermodynamics:

\nabla s=C_{p}\ln({\overline {T} \over T_{1}})+R\ln({\overline {p} \over p_{1}}).

The values $T_{1}$ and $p_{1}$ are unknown and may be dropped from the formulation. The value of entropy is not necessary, only that the value is positive.

\nabla s=C_{p}\ln(\overline {T})+R\ln(\overline {p}).

One possible unreal solution for the stream thrust averaged velocity yields a negative entropy. Another method of determining the proper solution is to take a simple average of the velocity and determining which value is closer to the stream thrust averaged velocity.

References

DeBonis, J.R.; Trefny, C.J.; Steffen, Jr., C.J. (1999). "Inlet Development for a Rocket Based Combined Cycle, Single Stage to Orbit Vehicle Using Computational Fluid Dynamics" (PDF). NASA/TM—1999-209279. NASA. Retrieved 18 February 2013.

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