HYDRODYNAMICS IN SlllP Dl-.SKiN 



Sec. -to. 5 



\\\c thion- ami principles of similiUuie is assuredly 

 iiecessan'. 



Fur tlie reader who wishes to study lliis method, 

 refresh his menmry, or look up doubtful points, a 

 partial list of references is given: 



il) HiBbourhinsky, D., "M^-thodc den Variables dc 

 Dimcndion Zero ct son Application en Af rodynam- 

 iquc (DimensionleRR \'arial>los and Their Uso in 

 AiToilynamics\" A6rophilo, 1 Sep 1911 



(2) Rayli-igh, IajmI, Brit. Adv. t'omm. Aero., Ann. Rop., 



1::JS. 1010: 2:20, 1911; 3:30, 1912 



(3) Tolmnn, U. C, "The Principle of Similitude," Phys. 



Rev., 1914, Vol. Ill, p. 244 



(4) BuckinRliam, Iv, "On Phy.sically Similar Syplems," 



Phys. Rev., 1914, Vol. IV, p. :}45 



(5) Tolman, R. C, "The Principle of Similitude and the 



Principle of Dimensional Homogeneity," Phys. 

 Rev., 1915, Vol. VI, p. 219 



(6) Buckingham, E., "Model E.\i)eriment8 and the Forms 



of Empirical Equations," Trans. ASME, 1915, 

 Vol. 37. pp. 263-296 



(7) Tolman, R. C, "Note on the Homogeneity of Physical 



Equations," Phvs. Rev., 1916, Vol. VIII (Ser. II), 

 p. 8 



(8) Bridgman, P. W., "Tolman's Principle of Similitude," 



Phys. Rev., 1910, Vol. VIII, p. 423 



(9) Buckingham, E., "Notes on the Methml of Dimen- 



sions," Phil. Mag., 1921, Vol. 52, p. ti<)6 



(10) Tavlor, D. W., "Propeller Design Based upon Model 



E.\i)eriments," SNAME, 1923, pp. 57-109: csp. 

 pp. 99-106, which include the deduction of the 

 n Theorem on pp. 102-106, from Buckingham's 

 paper, ref. (4) of this series 



(11) Slocum, S. E., Discussion of ref. (10) of this series, 



SNAME, 1923, pp. R0-S7 



(12) Dryden, H. L., Muriiaghan, F. D., Bateman, H., 



"Hydrodynamics," Nat. Res. Council, Wa.shington, 

 19:J2, pp. 4-6 



(13) Buckingham, E., "Dimensional Analysis of M(mI(I 



Proi«-ller Tests," ASNE, May 1936, pp. 147-19S. 

 Pages 107-198 of this paper list 24 references on 

 the subject. 



(14) Bridgman, P. W., "Dimensional Analysis," Yale 



University Pre.ss, rev. edition, 19;J7 



(15) Rouse, H., "Fluiil Mechanics for Hydraulic Engi- 



neers," McGraw-Hill, 1938, Chap. 1 

 (10) Davidson, K. S. M., PNA, 1939, Vol. II, pp. 67-58, 



00-61 

 (17) Hnnkins, O. A., "Experimental Fluid Dynamics 



Applie<l to I'^ngincering Practice," NECI, 1943- 



HM4, Vol. 60, pp. 24-25 

 (IS) Rouw, H., EMF, 10^16, Appx., pp. 'AhX'ATA 



(10) Van Driest, Iv R., "On Dimensional .'Vnalysis and 



Prc!(«,-ntatinn of Data in Fluid Flow Problems," 

 Jour. Appl. Mcch., ASME, 19-16, Vol. 13, No. 1 



(20) Vcnnjird, J. K., "Elementary Fluid Mechanics," 



Wiley, New York, 2nd e«Jitiori, 1947, pp. 142 153 



(21) C1inrp(^nticr, H., "Introduction aiix NK'thodes Di- 



merurionnelloK (Intrmluclion to Dimensional Mcth- 

 O.U)." ATMA, UM7. Vol. 46, p. 156 



(22) Roiwc. H., i;il, IBM), Appx., pp. 1MI5 WS. 1001 lOW 



(23) O. BirkhofT givoo a lint of cightMn references on this 



subject, to accompany a chapter on modeling and 

 dimensional analysis in his book "Hydrodynamics" 

 (Princeton Univ. Press, 1950, pp. 182-183| 



(24) I.,!inghaar, H. I,., "Dimensional Analysis and Theorj' 



of Mmiels," Wiley, New York, 1951 



(25) Huntley, H. E., "Dimensional Analysis," Macdonald, 



Ivondon, 1952 



(26) Duncan, W. J., "Physical Similarity and Dimensional 



.\n:ily»i8," St. Martin's Press, New York, 1953. 



40.5 Dimensions of Physical Quantities. 



Wliellier one is or is not versed in dimensional 

 analysis or the theory of similitude, the concept 

 of the dimensions of a physical ciuantity, in terms 

 of the basic dimensions of length, mass, and time, 

 needs to be clearly visualized. Only in this way 

 can the dimensionless (0-diml) relationships 

 described elsewhere and employed constantly in 

 the book be well understood. The dimensions of 

 the various physical quantities in general use by 

 architects and engineers are derived bj' relatively 

 simple processes and are tabulated in full in 

 Appendix 2 of Volume I. They may he memorized 

 or a list may be kept handy for ready reference. 

 Better still, they may be derived each time they 

 are needed by the procedure described in .Vppen- 

 (lix 2. 



X;itural sines, cosines, and tangents of angles 

 are perhaps the simplest examples of dimensionless 

 ratios. Others are pitch ratio, blade-thickne.ss 

 ratio, and aspect ratio. Unfortunately for the 

 engineer, or so he may think, there are no limits 

 to the coni])iexity or intricacy of other dimension- 

 less combinations. Among the particular 0-diml 

 exprc.'^sions of interest to the na^■al architect are 

 the comijlete set of hull-form and ])ropcller-form 

 coetficients, the Froude luimber /•'„ , the Reynolds 

 number //„ , and the cavitation index (^(sigma). 

 All of these are important, and they arc in constant 

 use in one form or another. 



The opposite sides of all etiviations involving 

 physical (]uantitie.s must have the .sjune dimen- 

 sions (House, II., EH, 10.50, pp. i)'.l.'-)-!)!)8). A 

 formula for a (luantity must have the dimensions 

 of that (luantity. Take the familiar V = 2gh. 

 Here 2 is 0-<linil; g is an acceleration having the 

 dimensions of a length // divided by a time I 

 stiuared, and the height h has the dimensions of 

 a length /.. Hence V = (L/l'){L) = // l' = 

 (/y//)' = v. Another example is the formula for 

 lift force /., which is L = r',.(()..'ip).t f '', where .1 is 

 the |)rojected area of lui airfoil or hydrofoil and f/ 

 is the fluid velocity pa.st it. Since (",. is 0-<liml. for 

 tilt! reasons given in the section following, with 

 the dimensions of unity. 



