TURBULENCE EFFECTS ON DRAG OF SHARP-EDGED BODIESa. The problem of turbulence effects on the drag. coefficient is both interesting and important.

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1 TURBULENCE EFFECTS ON DRAG OF SHARP-EDGED BODESa Discussion by Farid Mansour 1 and Walter Graf 2 The problem of turbulence effects on the drag coefficient is both interesting and important. The authors presented valuable and significant experimental work. They illustrated the effect of turbulence on the drag coefficient by comparing the drag values corresponding to different turbulence intensities with variation of Reynolds number for several body shapes. However, it is also very beneficial to relate the drag coefficient directly to the turbulence parameters, but no attempt was made by the authors towards this end. Based on a dimensional analysis and experimental studies for a two dimensional circular cylinder, Ko and Graf (1972) found a relationship between the drag coefficient and an argument which they called the "turbulence characteristics." This argument is a function of the turbulence intensity, turbulence scale, and Reynolds number and was expressed ahy7, 1972, by J. Roberson, c. Lin, G. Rutherford, M. Stine. 1 Grad. Teaching Assist., Civil Engineering Department, Lehigh University, Bethlehem, Pennsylvania. 2 Associate Professor, Director of Hydraulics Division, Fritz Engineering Lab., Lehigh University, Bethlehem, Pa. -1-

2 , as: u (D) 1/5 CD= 1[ l~gr ] (i) where c = D ul = D = R = s = u = u = drag coefficient, turbulence intensity, ul = characteristic dimension, Reynolds number, turbulence scale, average flow velocity, and velocity fluctuation. 1:2 u-' n a separate study Ko and _Graf (1971) presented another relationship similar to Eq. i, but instead of using the drag coefficient they used the term, (u 1 ) 2 ;cd. Eq. (i) was rewritten as: (u 1) 2 CD U 1 (!2.) 1/5 = 2 [ lo~ R ] (ii) where, the term, (u 1 ) 2 /cd, represents a ratio of the kinetic energy of the turbulent component of the flow per unit 2 volume, 1/2 p u, to the force exerted by the mean flow upon 2 the cylinder per unit area, 1/2 CD p U -2-

3 Among the shapes considered by the authors, the cube received particular attention; drag coefficients were evaluated under turbulence intensities as shown in Fig. 5. The test results in Fig. 5 were obtained by using a screen of mesh size= 1.25 in. (3.17 em); opening dimension, M = 1.0 in. (2.54 em); and dowel diameter, b =.25 in. (.63 em); while the distance, x, between the screen and tested cube was varied from 7.5 in. (19.0 em) to 80.0 in. (2.0 em). Following the same technique used by Ko and Graf (1971, 1972), and referred to above, the data in Fig. 5 have been analyzed to establish a relationship between the drag coefficient and turbulence characteristics for the cube. t should be mentioned that the values of turbulence scale, S, used in both Eqs. (i) and (ii) were calculated according to the expression developed by Naudascher and Farell (1970) which was successfu~~y.applied by Ko and Graf (1972). After the required calculations, the results are presented in Figs. A and B. The curves shown in these figures were fitted by using the least square polynomial, with the aid of a computer, yielding the following relations:' (iii) and -3-

4 u' (Q)l/5 u' (D)l/5 B B 2 = [ log R.lOO]+.l06[ log R loo] (iv} Figure A and Eq. (iii) show that the drag coefficient decreases with an increase in turbulence characteristics up to a value of 2.5. A further increase in turbulence characteristics produces a slight increase in the drag coefficient. From Fig.. Band Eq. (iv) it can be seen that the term (u') 2 /CD increases continuously with an increase of turbulence characteristics. Both Figs. A and B show a similar trend if compared with the studies of Ko and Graf (1971, 1972) for circular cylinders. Accordingly, the drag coefficient of the cube can be evaluated for turbulent flow provided the turbulence characteristics can be determined. References 1. Ko, S. and Graf, W., "Drag Coefficient and Turbulence Characteristics," Proc., XV Congress of ntern. Assoc. Hydr. Res. in Paris, France, Vol. 2, Ko, S. and Graf, w., "Drag Coefficient of Cylinders in Turbulent Flow," Proc. Am. Soc. Civil Engs., Vol. 98, No. HY7, Naudascher, E. and Farell, c., "Unified Analysis of Grid Turbulence," Proc. Am. Soc. Civl Engs., Vol. 96, Nl. EM2,

5 1. 95 i ln E 0 G~ Fig. A % ~-. i oo '0. -.o '. i L u... LJ)f i! L 0, 0 -r r U 1 (Q) 1/5 B logr 100 Drag Coefficient Versus Turbulence Characteristics for a Cube l ' l

6 11. ~ ~ : 1.9~ ~ ~~----~~ ~ ~ ~ 1.7 j ! ~~------~ ~ ~ Fig. B (u 1) 2 s;- Versus Turbulence Characteristics for a U 1 (Q) 1/5 --,,----:.::B..,.,... 1 o o CubelogR