CHAPTER 41 



General Formulas Relating to Liquid Flow 



41.3 

 41.4 

 41.5 

 41.6 

 41.7 



41.8 



The Use of Pure Formulas 



The Quantitative Use of Dimensionless Num- 

 bers; The Mach and Cauchy Numbers 



The Euler and the Cavitation Numbers . . 



The Froude Number and the Taylor Quotient 



Calculation of the Reynolds Numbers . . . 



Application of the Strouhal Number . . . 



The Planing, Boussinesq, and Weber 

 Numbers 



Derivation of Stream-Function and Velocity- 



7 Potential Formulas for Typical Two- 



Dimensional Flows 17 



7 41.9 Stream-Function and Velocity-Potential For- 



8 mulas for Three-Dimenaional Flows ... 20 

 11 41.10 The Determination of Liquid Velocity 



15 Around Any Body 24 



16 41.11 Conformal Transformation 25 



41 . 12 Quantitative Relationship Between Velocity 



16 and Pressure in Irrotational Potential Flow 25 



41 . 13 Tables of Velocity Ratios, Pressure Coeffi- 

 cients, Ram Pressures and Heads ... 30 



41.1 The Use of Pure Formulas. The ahnost 

 exclusive employment of pure mathematical 

 formulas in this book, described in Sec. 1.7 of the 

 Introduction to Volume I, is emphasized here. 

 Unless specific exceptions are mentioned, these 

 formulas contain only symbols representing 

 physical concepts and dimensionless ratios, and 

 they are dimensionally consistent. They may be 

 used as given, with consistent units belonging 

 to any system of meas urement. 



Examples are c = y/gLw/^Tz for the celerity or 

 velocity of a trochoidal surface wave, and 

 h = kw{Bx/LE){Vy2g) for the predicted height 

 of the bow-wave crest on a ship. Substituting the 

 dimensions of the physical quantities in the first, 



f=[Hr= 



For the second, 



There is no restriction whatever upon the units 

 used in these and other formulas like them, 

 provided they are consistent. The wave celerity, for 

 example, may be in mi per hr, kt, ft per sec, 

 meters per sec, or what not, so long as g and Lw 

 are in the same units. The examples in the sec- 

 tions following illustrate the use of pure formulas. 



41.2 The Quantitative Use of Dimensionless 

 Numbers; The Mach and Cauchy Numbers. 

 The ten dimensionless numbers or relationships 

 of hydrodynamics, previously described in Sec. 

 2.22 and listed in Table 2.a, are expressed briefly 

 here in quantitative terms to illustrate the manner 



in which they are generally used. Table 2. a is 

 repeated as Table 4 La in this volume for the 

 convenience of the reader. 



The Mach number ilf „ in liquids is of primary 

 interest in the analysis of underwater-explosion 

 phenomena or high-order impact studies, espe- 

 cially in the immediate vicinity of the impact or 

 explosion. The Cauchy number C„ , related to it, 

 may eventually be found of interest in studies of 

 cavitation erosion on propeller blades and similar 

 objects. Taking account of the elasticity and mass 

 density of the material, it may be useful in 

 analyzing shock-wave erosion on different pro- 

 peller materials. 



As a study of high-order impact and of the 

 details of shock-wave erosion is somewhat beyond 

 the scope of this book, the treatment of the Mach 

 or Cauchy numbers in this section is limited to 

 examples giving the derivation of the shock-wave 

 velocities in salt water and propeller bronze. For 

 the first of these examples it is assumed that salt 

 ocean water at an average temperature of 60 

 deg F possesses an elastic modulus K at that 

 temperature, from Table X3.m of Sec. X3.7 of 

 Appendix 3, of 340,000 psi, or 144(340,000) lb per 

 ft^. The mass density of salt ocean water, for the 

 examples quoted here, may be taken as 1.9903 

 slugs per ft^. The celerity c of an elastic shock 

 wave in ocean water is then 



[" 144(340,000) 1°" 

 L 1.9903 J 



= 4,960 ft per sec, approx. 



This is the speed of sound in the ocean water 

 under the conditions described. 



