Energy-Equation Approximations in Fluid Mechanics

Abstract

Fig. 1 is a dimensionless plot of the digital-computer solution of Eq. (1). Each constant-^ curve peaks at a depth Froude Number, c/vgH < 1. A s ^ decreases the peaks become sharper and larger in magnitude, approaching the two-dimensional solution which gives infinite drag at c/vgH = l. The inflection point of each curve, to the right of the peak, corresponds approximately to c/vgH = 1 or Froude 1. Fig. 2 shows the relation of peak speed and resistance to water depth. Values other than those shown in Fig. 1 were obtained from solutions of Eq. (1) in the immediate vicinity of the peak. Peak locations were computed only to within ±0.005, which accounts for the slight scatter in these curves. A comparison of Havelock's original hand-computed curves with Fig. 1 yields interesting results. Except at p = 0.75 and 1.00 and Vl /x greater than about 1.50, the digital-computer resistances are smaller than Havelock's values. The peak values, however, occur at approximately the same x. As an example of this difference, at p = 1, Havelock's peak resistance value is approximately 0.27 while the digital value is 0.217. In addition, the digital curves do not exhibit the secondary shallow-water peak found by Havelock at Vi/ x = 1.25 and p = 2 or his enhanced deep-water resistance values at \/llx = 0.92 and 0.81 for p = 1.43 and 1.00 respectively. As shown in Fig. 2, the transition from deepto shallow-water characteristics follows a single, continuous curve.