Boundary Layer Equations

In the attached or slightly detached external flow cases, we can obtain the surface pressure distribution using the methods described in Sect. 2.1 and further simplify set of Eqs. 2.49 and 2.54. In these simplifications we again resort to the order of magnitude analysis. Assuming again that the viscous effects are only in the vicinity of the surface of the body, we can consider the gradients and the diffusion normal to the surface we obtain

Here, x is the direction parallel to the surface, z is the normal direction and h, in Eq. 2.62 is the enthalpy of species i.

The real gas effect in an external flow can be measured with the change caused in the stagnation enthalpy. If we neglect the effect of vertical velocity component, the stagnation enthalpy of the boundary layer flow reads ; ho = h + u2/2. The normal gradient of the stagnation enthalpy at a point then reads

dho dh du

dz dz + dz

Hence the new form of the energy equation becomes

(2.63)

During the non dimensionalization process of the boundary layer equations, we introduce the Lewis number to represent the magnitude of diffusion in terms of heat conduction as a non dimensional number; Le = pD12cp/k. The non dimen­sional form of Eq. 2.63 reads as

0ho 0ho 0ho dp 0 PoT + Pu0X + pw~0Z = at+ 0Z
(2.64)
l 0h0 – + 0Z Pr 0Z
Pr	Ц
1

In Eq. 2.64 the local value 1 for the Lewis number makes the contribution of diffusion vanish and as the Lewis number gets higher the diffusion gets stronger. The cp value in the Lewis number is obtained from the average cpi values of the species involved in the boundary layer under the frozen flow assumption.