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:
- [math]\displaystyle{ F = \int \left(\rho \mathbf{V} \cdot d \mathbf{A} \right) \mathbf{V} \cdot \mathbf{f} +\int pd \mathbf{A} \cdot \mathbf{f}. }[/math]
Mass flow:
- [math]\displaystyle{ \dot m = \int \rho \mathbf{V} \cdot d \mathbf{A}. }[/math]
Stagnation enthalpy:
- [math]\displaystyle{ H = {1 \over \dot m} \int \left({\rho \mathbf{V} \cdot d \mathbf{A}} \right) \left( h+ {|\mathbf{V}|^2 \over 2} \right), }[/math]
- [math]\displaystyle{ \overline{U}^2 \left({1- {R \over 2C_p}}\right) -\overline{U}{F\over \dot m} +{HR \over C_p}=0. }[/math]
Solutions
Solving for [math]\displaystyle{ \overline{U} }[/math] 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.
- [math]\displaystyle{ \overline{\rho} = {\dot m \over \overline{U}A}, }[/math]
- [math]\displaystyle{ \overline{p} = {F \over A} -{\overline{\rho} \overline{U}^2}, }[/math]
- [math]\displaystyle{ \overline{h} = {\overline{p} C_p \over \overline{\rho} R}. }[/math]
Second law of thermodynamics:
- [math]\displaystyle{ \nabla s = C_p \ln({\overline{T}\over T_1}) +R \ln({\overline{p} \over p_1}). }[/math]
The values [math]\displaystyle{ T_1 }[/math] and [math]\displaystyle{ p_1 }[/math] are unknown and may be dropped from the formulation. The value of entropy is not necessary, only that the value is positive.
- [math]\displaystyle{ \nabla s = C_p \ln(\overline{T}) +R \ln(\overline{p}). }[/math]
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". NASA/TM—1999-209279. NASA. https://ntrs.nasa.gov/api/citations/19990062664/downloads/19990062664.pdf. Retrieved 18 February 2013.
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