26 



HYDRODYNAMICS IN SI1I1» HI SIGN 



Sec. -11.12 



field of rvlnlive liquid motion aroiiiui ii luKiy. the 

 next practical step is usually to determine the 

 pressures at thtvse points. For irrotntional poten- 

 tial flow in an ideal liquid without viscosity, at 

 a depth where the hydrostatic pressure remains 

 sensibly constant, this is a rather simple, straight- 

 fonvard process. 



From the basic assumption, explained in Sec. 

 2.7, that in any stream tube in which the flow is 

 steady and continuous the total pressure remains 

 constant along the tube, the relationship between 

 the dynamic pressure q and the pumping pressure 

 p at two reference points 1 and 2 is expressed bj- 



Pi + I V\ = Pi + I V\ or p, + ?, = p, + 9, 



If the point 1 is taken at a great distance in the 

 undisturbed li(inid from the point 2, say at 

 infinity, then the preceding equation i.s written as 



p. + I VI = P, + 1 c/^ 

 Transposing, 



v.-v^ = livl-v^ = l(x-^ 



whence, omitting the subscripts "2," 



= ^P = ^r 



<'-^) 



This gives the relationship derived in Eq. (2.xvi) of 

 Sec. 2.20, namely 



-^ = :^= 1 --^ 



(2.xvi) 



The equalities expressed in Eq. (2.xvi) are 

 mast important and useful. They can with profit 

 be memorized by everyone who works with this 

 subject. The left-hand and middle terms in this 

 equation are expressions for the pressure coeffi- 

 cient or Euler number 7?, , expressing the difTer- 

 encc in pressures between that at any select<>d 

 point 2 and that at infinity, as a proportion of 

 the ram pres.sure O.TypUl which could be set uj) 

 in the unlimited, undisturbed stream. 



The ratio U/U.. is exactly that given l)y the 

 ratio ^n./^n in the 2-diml How net described in 

 Sec. 2.20. Hence sf|unring the fraction An. /An 

 and subtracting it from unity gives directly the 

 pressure c<H'(ficient for any selected portion of t In- 

 flow pattern. 'I'his ratio is larger than I.O when 

 U IM larger in magnitude than l\ . The pressure 

 coefficient ^p/q is then negative. 



Assume that for the 2-diml flow net around the 

 blunt-ended 2-<liml .section of Fig. 41. J (adapted 

 from Rou.se, H., E.MF, 1940, Fig. 21), p. 50), the 

 stream-tube width An. is 0.25 in. .\nd that at the 

 point A on the fonvard shoulder the width An is 

 narrowed to 0.20 in. Then An,/An is 0.25/0.20 = 

 1.25, whence (An^/An)^ = 1.5G25. The pressure 

 coefficient or Euler number E, at this point is 

 thus 1.000 - 1.5()25 or -0.5625. 



The values thus derived for any selected number 

 of points are laid down graphically in several 

 ways, depending upon what is to be represented 

 by the plot. If the positive and negative diiTerential 

 pressures + Ap and — Ap are to be emphasized the 

 scheme followed in diagnini 2 of Fig. 41. J is 



F"io. 41.J Uki.ation Bbtween Veixjcitv and Pressure 

 IN Fixjw OK AN Ideal LiguiD .\koi:nd a Body 



preferable. Here the normal vectors representing 

 the combination of atmospheric and diiTerential 

 l)re.ssures are laid o)T from the solid-surface contour 

 as a reference line, with the -f Ap vectors directed 

 toward the surface and the — Ap ones away from it. 

 However, the practice of drawing vectors inside 

 the .solid is not recommended for general u.sjige 

 becau.se they are crowded together in n-gions of 

 sharp cur\'ature and the solid may be too narrow 

 to lay them down at a conveniently large scale. 



