Boundary layer flow over EPRs

The problems considered so far were one-dimensional in essence. Here, a two-dimensional problem will be considered. It should be also noted that no reduction of any term of the complete Navier—Stokes equations was made yet. This means that no groundless assumptions were employed apart from the representation of EPR by a distributed force. In contrast, the Navier—Stokes equations reduced to the boundary-layer approach (1.4), (1.5) were employed by meteorologists. 

A boundary layer flow is one of the simplest two-dimensional formulations in fluid mechanics. This approach suggested by Prandtl assumes that the flow properties change very slowly in the longitudinal direction with respect to the normal direction, Jj. This allows one to drop some terms in the equations, thus having simplified them. 

The boundary-layer approach neglects some terms of the momentum equations (3.29) and yields two simplified equations instead of three  

The initial region extends over an entrance length Xo and consists of boundary layer 2 over the wall with width d0(x), internal boundary layer 3 with depth dx x), and initial region core 1 between them. External boundary layer 4 with width d2(x) transports perturbations caused by the EPR far away over it. 

The calculations reveal that r 0 and dU/dz 0 within the initial region core, and thus the drag force (1.6) dominates. This allows us to reduce the governing equation to the following one  

---

Consider now the boundary layer flow over the surface with a droplet EPR of the height h over it Fig. 3.8 illustrates the physical situation. Both the carying and carried media are governed by the system of equations (3.74) determining the two-dimensional fields U(x,z), V(x,z) and u(x,z) in the boundary-layer approach (with r(x,z) = P vt1U )-The non-slip velocity on the surface z = 0, and the asymptotics lim U = along... 

Figure 3.19 Boundary layer flow over a droplet EPR (A) longitudinal velocities of air and droplets, (B) transversal air velocities, (C) shear stresses in the carrying flow, (D) integrated characteristics of the boundary layer.

The results of the numerical solution of the conjugation boundary-value problem for the mass exchange (3.106), (3.107) are shown in Fig. 3.20. The initially homogeneously cold air gets the latent heat from the droplet layer and becomes warmer and warmer from cross section 1 to 5, etc. (family (I) of curves in Fig. 3.20,A). In turn, the droplets become cooler their temperature is reduced especially intensively at the EPR entrance (the dashed curve 0) but less and less intensively in subsequent cross sections (the family of curves (II)). The curves TE(z) and t(z) attract each other and meet at the theoretical infinity z —> -oo. The complex relation between the two profiles is testified by the local maxima on TE(z) at early cross sections (curves 1 and 2) and by negative values of the mass flow (3.97) presented in Fig. 3.20,B. A dilative thermal boundary layer grows over the droplet EPR. Its width dE(x) grows theoretically to infinity over the EPR, but the internal portions of all the variables tend to a certain final position within the EPR, 0 < z < 1. 

The temperature distributions in the thermal boundary layer over an EPR with t0 = -0.5 (i.e. obstructions are colder than the wall) have been shown in Fig. 3.17,B. The thermal distributions in such a flow are much complicated. However, analytical estimations are possible if the structure of the boundary layer is taken into account. 

The turbulent shear (3.130) is often taken constant in the approximate performances of boundary layers in fact, Fig. 3.31,C highlights that it is constant over a significant depth H of the main EPR region. The flow obeys the following equation ... 

---