Adiabatic Flows

For a typical aerodynamic flow we have

• d()/dt — 0 , (steady flow)

• qV — 0 , (no volume heating)

• f V ~ 0 , (volume work negligible)

• T ~ 0 , (negligible viscous stress outside of viscous layers)

•
q ~ 0 , (negligible heat conduction outside of viscous layers)

so that wherever the above conditions are met, then ho is constant and equal to its upstream value.

The requirements t ~ 0 and q ~ 0 seem to preclude viscous regions from having a constant ho. This is true, but somewhat overly restrictive. Consider applying only the first three adiabatic-flow assumptions above to the integral enthalpy equation (1.30), and retaining the surface viscous and heat conduction terms.

p(ho – Л-ож )V ■ n dS — © V ■ t ■ n dS – © q ■ n dS (1.58)

In addition, ho was replaced by ho — hoK) as permitted by the steady mass equation◦ pV ■ П dS — 0. The first viscous shear integral on the right vanishes if either V — 0 as on a solid wall, or T — 0 as on the outer

boundary. The second conduction integral vanishes if q = 0 as on an insulated (not heated or cooled) wall, and also on the outer boundaries. These conditions are met in most typical steady aerodynamic flows whose walls have come to temperature equilibrium with the fluid. For these flows we then have:

© p(ho — ho)V ■ n dS = 0 (steady flows with insulated walls) (1.59)

Hence, viscous stresses and heat conduction cannot change the net flux of total enthalpy out of the flow – field, but can only redistribute it within the flow, and in particular within the thin viscous layers, as shown in Figure 1.10. Therefore, steady viscous aerodynamic flows with insulated walls and no volume heat or work addition do have ho = hOTO in a mass-flow averaged sense.

A real aircraft flow-field which includes the propulsive elements will have heat addition via the qv or q ■ П terms as in a turbine combustor, and will also have work addition via the dp/dt and V ■ r ■ П terms due to a moving propeller or fan. In that case the mass-averaged hO leaving any control volume enclosing the aircraft will exceed hOxi, but this excess is confined to the engine exhaust and propulsive jet.