Heat Transfer in High-Speed Flow

In the actual case of a boundary-layer flow problem, the fluid is not brought to rest reversibly because the viscous action is basically an irreversible process in a thermodynamic sense. In addition, not all the free-stream kinetic energy is converted to thermal energy—part is lost as heat, and part is dissipated in the form of viscous work. To take into account the irreversibilities in the boundary-layer flow system, a recovery factor is defined by 

The essential result of the high-speed heat-transfer analysis is that heat-transfer rates may generally be calculated with the same relations used for low-speed incompressible flow when the average heat-transfer coefficient is redefined with the relation 

Notice that the difference between the adiabatic wall temperature and the actual wall temperature is used in the definition so that the expression will yield a value of zero heat flow when the wall is at the adiabatic wall temperature. For gases with Prandtl numbers near unity the following relations for the recovery factor have been derived  

These recovery factors may be used in conjunction with Eq. (5-119) to obtain the adiabatic wall temperature. 

In high-velocity boundary layers substantial temperature gradients may occur, and there will be correspondingly large property variations across the boundary layer. The constant-property heat-transfer equations may still be used if the properties are introduced at a reference temperature T as recommended by Eckert  

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Cary, A.M. and Bertram, M.H., Engineering Prediction of Turbulent Skin Friction and Heat Transfer in High Speed Flow , NASA, TN D-7507, Houston, TX, 1974. 

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