1. FULLY DEVELOPED TURBULENT PIPE FLOW CLASS 2 - REVIEW

2. = 0 F Sx + F Bx =  /  t (  cv u  dVol )+  cs u  V  dA Eq. (4.17) FULLY DEVELOPED, STEADY, NO BODY FORCES, LAMINAR PIPE FLOW  = (r/2)(dp/dx) Eq 8.13a

3.  yx =  (du/dy) u = - (R 2 /4  )(dp/dx) x [1 – (r/R) 2 ] = U C/L [1-(r/R) 2 ] Q =  0 uldy;V = Q/A V/U C/L = 1 / 2 a  = (r/2)(dp/dx) laminar  yx =  (du/dy)+  u’v’ u avg = U C/L (1-r/R) 1/n Q =  0 uldy;V = Q/A V/U C/L = 2n 2 /(2n+1)(n+1) a turbulent (empirical)

4. u(r)/U c/l = (y/R) 1/n = ([R-r]/R) 1/ n = (1-r/R) 1/ n Laminar Flow u/U c/l = 1-(r/R) 2 n=6-10 u(r)/U c/l = (y/R) 1/n

5. Eq. 8.30 “one of the most important and useful equations in fluid mechanics ” Fox et al. ENTER ENERGY EQUATION

6. Allows calculations of capacity of an oil pipe line, what diameter water main to install or pressure drop in an air duct, ……  V 1 2 / (2) + p 1 /(  ) + gz 1 =  V 2 2 / (2) + p 2 /(  ) + gz 2 + h lT h lT has units of enery per unit mass [V 2 ] “one of the most important and useful equations in fluid mechanics ” Fox et al.  V 1 2 / (2g) + p 1 /(  g) + z 1 =  V 2 2 / (2g) + p 2 /(  g) + z 2 + H lT H lT has units of enery per unit weight [L] from hydraulics during 1800’s

7.  A ½ V(r) 2  V(r)dA =  (dm/dt) ½ V 2  = [  A  V(r) 3 dA ]/ {(dm/dt) V 2 }  V 1 2 / (2g) + p 1 /(  g) + z 1 =  V 2 2 / (2g) + p 2 /(  g) + z 2 + H lT Turbulent Flow: V(r)/U c/l = (1-r/R) 1/n Laminar Flow: V(r)/U c/l = 1 – (r/R) 2

8.  = [  A  V(r) 3 dA ]/ {(dm/dt) V 2 }  = 1 for potential flow  = 2 for laminar flow   1 for turbulent flow  V 1 2 / (2g) + p 1 /(  g) + z 1 =  V 2 2 / (2g) + p 2 /(  g) + z 2 + H lT  = (U c/l /V) 3 2n 2 / (3 + n)(3 + 2n)*  = 1.08 for n = 6;  = 1.03 for n = 10 V(r)/U c/l = (y/R) 1/n

9. + HlHl  V 1avg 2 / (2g) + p 1 /(  g) + z 1 =  V 2avg 2 / (2g) + p 2 /(  g) + z 2 + H l (Eq. 8.30) (Eq. 8.34) HlHl V = Q/Area

10. BREATH (early 20 th Century turbulent pipe flow experiments)

11. f F =  wall /{(1/2)  V 2 } Similarity of Motion in Relation to the Surface Friction of Fluids Stanton & Pannell –Phil. Trans. Royal Soc., (A) 1914

12. ~1914 f F =  wall /{(1/2)  V 2 } f D = (  p/L)D/{(1/2)  V 2 } = (  p/L)2R2/2{ ½  V 2 } = 4  wall /{(1/2)  V 2 } = 4 f F

13. BREATH (rough pipe turbulent flow experiments)

14. Original Data of Nikuradze Stromungsgesetze in Rauhen Rohren, V.D.I. Forsch. H, 1933, Nikuradze  p  U ?

15.  p  u avg 2 Newton believed that drag  u avg 2 arguing that each fluid particle would lose all their momentum normal to the body. Drag = Mass Flow x Change in Momentum Drag = dp/dt  (  U  A)  U   U  2 A Drag/Area   U  2 Sir Isaac Newton (1642 – 1727) aside

16. Fully rough zone where have flow separation over roughness elements and  p ~ V 2 k* =  u * / ; k * < 4: hydraulically smooth 4 60 fully rough (no  effect) White 1991 – Viscous Fluid Flow

17. Curves are from average values good to +/- 10%

18. BREATH (Moody Diagram)

19. H l = f (L/D) V 2 /(2g) f = 64/Re and is proportional to  in laminar flow f is not a function of  /D in laminar flow f = const. and is not a function of  at high enough Re turbulent flows in a rough pipe f is usually a function of  /D in turbulent flows laminar t u r b u l e n t

20. f D = (  p/L)D/{(1/2)  V 2 } Darcy friction factor Re D = UD/ For new pipes, corrosion may cause e/D for old pipes to be 5 to 10 times greater. Curves are from average values good to +/- 10%

21. f F = -2.0log([e/D]/3.7 + 2.51/(Ref F 0.5 )] If first guess is: f o = 0.25[log([e/D]/3.7 + 5.74/Re 0.9 ] -2 should be within 1% after 1 iteration

22. For turbulent flow in a smooth pipe and Re D < 10 5, can use Blasius correlation: f = 0.316/Re D 0.25 which can be rewritten as:  wall = 0.0332  V 2 ( /[RV]) 1/4 )

23. For turbulent flow and Re < 10 5 can use Blasius correlation: f D = 0.316/Re 0.25 Which can be rewritten as:  wall =0.0332  V 2 ( /[RV]) PROOF f D = 4 f F 0.316 1/4 / (V 1/4 D 1/4 ) = 4  wall /( 1 / 2  V 2 )  wall = (0.0395  V 2 ) [ 1/4 / (V 1/4 (2R) 1/4 )  wall = (0.0332  V 2 ) [ / (VR)] 1/4 QED

24. Question? Looking at graph – imagine that pipe diameter, length, viscosity and density is fixed. Is there any region where an increase in V results in an increase in pressure drop?

25. Question? Looking at graph – imagine that pie diameter and kinematic viscosity and density is fixed. Is there any region where an increase in V results in an increase in pressure drop? Instead of non-dimensionalizing  p by ½  V 2 ; use D 3  /(  2 L) Laminar flow Turbulent flow transition From Tritton  pD 3  /(  2 L) Everywhere!!!!!!!

26. Some history ~ “Moody Diagram”

27. f = function of V, D, roughness and viscosity f is dimensionless Antoine Chezy ~ 1770: for channels: V 2 P = AS extrapolate this for pipe: H l = (4/C 2 )(L/D)V 2 Gaspard Riche de Prony (1800) H l = (L/D)(aV + bV 2 ) C; a and b are not dimensionless C; a and b are not a function of roughness HlHl Antoine Chezy

28. f = function of V, D and roughness f is dimensionless HlHl

29. HlHl f is a function of  and D better estimates of f Could be dropped for rough pipes Traditional to call f the Darcy friction factor although Darcy never proposed it in that form

30. HlHl Combined Weisbach’s equation with Darcy and other data, compiled table for f but used hydraulic radius. =  w /( ½  V avg 2 ) prob 8.83

31. HlHl Eq. 8.34

32. 4000< Re R < 80000 Full range of turbulent Reynolds numbers

34. “ These equations are obviously too complex to be of practical use. On the other hand, if the function which they embody is even approximately valid for commercial surfaces in general, such extremely important information could be made readily available in diagrams or tables.” Re f 1/  f Re/  f

35. “The author does not claim to offer anything particularly new or original, his aim merely being to embody the now accepted conclusion in convenient form for engineering use.” HlHl f = [  p/(  g)]D2g/(LV 2 ) f = {[  p/L]D}/{ 1 / 2  V 2 }

38. HlHl

40. THETHE ENDEND