Comment on "Evolution of the Laminar Wake behind a Flat Plate and Its Upstream Influence"

Abstract

D URING last few years considerable effort has been expended in an attempt to understand character of boundary-layer flow at a sharp trailing edge. The Introduction to Ref. 1 summarizes most of published investigations of steady incompressible laminar flow past trailing edge of a finite plate at zero incidence and high Reynolds number. In particular, asymptotic descriptions considered to be valid in limit as R -> oo (R = UL/v; U = external flow speed, L = plate length, v = kinematic viscosity) were given independently by Stewartson and by Messiter. For most part, Ref. 1 correctly points out deficiencies of earlier work, but several comments about theory derived by Stewartson and Messiter are either unjustified or incorrect. First of all, Ref. 1 states incorrectly that work of Ref. 4 assumed isobaric flow. The assumptions actually made were that: a) effect of trailing edge is confined to a distance O(R~L); and b) Navier-Stokes equations can be linearized about .the Blasius solution in this neighborhood. The serious error lay in neglecting discontinuity of streamline slopes in boundary layer at' x = 1; this was corrected in Refs. 2 and 3. Imai treated same problem as in Ref. 4 but with isobaric assumption. Incidentally, Ref. 4 is not without value for it still provides a useful guide to structure of region within a distance O(R ~ L) of trailing edge provided one adjusts constant 1 as explained in Ref. 2. Hakkinen and O'Neil, not mentioned in Ref. 1, have also discussed this region, in terms of expansions for Rr/L '-> oo. Second, it is stated in Ref. 1 that use of linearized airfoil theory is not justified because of large slopes of (effective) slender body. It is certainly true that slopes are large compared with streamline slopes predicted by ordinary boundary-layer theory, but slopes do remain small compared to 1. As explained in Refs. 2 and 3, u = 1 + 0(#~), v = 0(R~), du/dx = 0(R + ), etc. In fact, Eq. (7) of Ref. 1 essentially incorporates notion of linearization because term vfiu/dy is omitted. Also, as Fig. 6 of Ref. 1 makes clear, ~ 0.01 and therefore cp = -2(u1) with error ~ 0.0001. Third, main problem studied in Eqs. (4-7) of Ref. 1 contains formulation of Refs. 2 and 3 and reduces to it as R -> oo. Thus there is agreement in this limit. However, this does not provide justification for neglecting terms when R = 10. A number of terms are omitted in Ref. 1, but perhaps most important is pressure variation across boundary layer. It is established in Refs. 2 and 3 that this is 0(R~), just a fraction smaller than longitudinal pressure variation O(R~). Neglecting this term is justified in limit as R -* oo, but may well lead to numerical inaccuracy at R = 10. Fourth, pressure plotted in Fig. 6 of Ref. 1 shows a singularity at x = 1, and so dp/dx is seriously in error if x is very close to 1. The solutions to Eqs. (1-3) of Ref. 1 presumably require that pressure be known at the edge of Region I; however, pertinent details are not discussed. Furthermore, numerical work of Ref. 1 would have benefited from making use of an asymptotic analysis of outer region of influence of trailing edge. Conditions at infinity are notoriously difficult to handle numerically, and discrepancy between results of Ref. 1 and those of Goldstein, when x = 2, should be regarded with caution. Finally, Fig. 6 of Ref. 1 is in qualitative agreement with predictions of Refs. 2 and 3, in that a favorable pressure gradient is established for x < 1, an adverse pressure gradient at x = 1 + , and a favorable pressure gradient as (x— 1)#-+ oo. Fig. 6 shows that displacement effect leads to large dcp/dx for Ax « 0.2 (say) whereas Fig. 10 shows significant changes in skin friction for Ax « 0.0005. While dependence on R is not clearly established by these results, curves do show unequivocally existence of two length scales, again qualitatively in agreement with results of Refs. 2 and 3. The latter references do not contend that region O(jR~L) is irrelevant to structure of flow, but rather that dominant correction occurs on larger scale, and that even for R = 10 (say) largest effects are found in a qualitatively correct way by asymptotic analysis given. For example, correction to drag coefficient on larger scale is O(R ~ ), whereas effects on smaller scale lead to a change O(R ~). The correction may be calculated either by integrating skin friction along plate or by calculating a momentum deficit in wake. For consistency eigenfunction proposed in Eq. 5.10 of Ref. 2 is required.