1. Incompressible Flow in Pipes and Channels
Shear Stress and skin friction in Pipes
Laminar Flow in Pipes and Channels
Turbulent flow in pipes and channels
Friction from changes in velocity or direction

2. Shear Stress and Skin Friction in pipe with fully developed laminar flow

3. Fluid Flow Fluid flowing in pipes has two primary flow patterns.
It can be either laminar when all of the fluid particles flow in parallel lines at even velocities and it can be turbulent when the fluid particles have a random motion interposed on an average flow in the general direction of flow.  
There is also a critical zone when the flow can be either laminar or turbulent or a mixture.  

4. Reynold Number
It has been proved experimentally by Osborne Reynolds that the nature of flow depends on the mean flow velocity (v), the pipe diameter (D), the density (ρ) and the fluid viscosity Fluid Viscosity( μ).
A dimensionless variable for the called the Reynolds number which is simply a ratio of the fluid dynamic forces and the fluid viscous forces , is used to determine what flow pattern will occur.
The equation for the Reynold Number is

5. For normal engineering calculations , the flow in pipes is considered laminar if the relevant Reynolds number is less than 2000, and it is turbulent if the Reynolds number is greater than 4000.  
Between these two values there is the critical zone in which the flow can be either laminar or turbulent or the flow can change between the patterns...
It is important to know the type of flow in the pipe when assessing friction losses when determining the relevant friction factors

6. Reynolds Experiment Reynolds Number
Laminar flow: Fluid moves in smooth streamlines
Turbulent flow: Violent mixing, fluid velocity at a point varies randomly with time
Transition to turbulence in a 2 in. pipe is at V=2 ft/s, so most pipe flows are turbulent
Laminar
Turbulent

7. Shear stress distribution
Flow
-(P+dP) P rw r y τ
Relation between skin friction and wall shear

8. Relations between skin friction parameters
Fanning friction factor
Fanning Equation
Use to calculate skin friction loss in straight pipe

9. Flow in non-circular channels
For noncircular cross section, the diameter in the Reynolds number is taken as equivalent diameter Deq.
Hydraulic radius, rH = ratio of the cross sectional area of the channel to the wetted perimeter of the channel (S/Lp)
For s circular tube, the hydraulic radius is

10. Laminar Flow (Method 1)

11. Velocity Distribution
du/dr
The relation between the local velocity and position in the stream is

12. Hagen-Poiseuille Equation
Eliminating τw in favor of ΔPs
(Hagen-Poiseuille Equation)

13. Laminar Flow (Method 2)

14. Dimensional Analysis
Darcy Weisbach Equation

15. Assumptions: Fully developed, Low
Approach: Simplify momentum equation, integrate, apply boundary conditions to determine integration constants and use energy equation to calculate head loss
Schematic
Exact solution :
Friction factor:
Head loss:

16. Both laminar sublayer and overlap layer are affected by roughness
Inner layer:
Outer layer: unaffected
Overlap layer:
constant
Three regimes of flow depending on k+
K+<5, hydraulically smooth (no effect of roughness)
5 < K+< 70, transitional roughness (Re dependent)
K+> 70, fully rough (independent Re)
For 3, using EFD data to adjust constants:
Friction factor:

17. Example
A straight stretch of horizontal pipe having a diameter of 5 cm is used in the laboratory to measure the viscosity of crude oil (γ = 0.93 t/m3). During a test run a pressure difference of 1.75 t/m2 is obtained from two pressure gages, which are located 6 m apart on the pipe. Oil is allowed to discharge into a weighing tank, and a total of 550 kg of oil is collected for a duration of 3 min. Determine the viscosity of the oil.

18. Solution

19. Laminar (Non-newtonian liquids)
Difference in the relation between shear stress and velocity gradient
For fluid following the power law model
The pressure difference becomes

20. Turbulent Flow

21. Turbulent flow in pipes and channels
Turbulent flow occurs when the Reynolds number exceeds 4000.
Eddy currents are present within the flow and the ratio of the internal roughness of the pipe to the internal diameter of the pipe needs to be considered to be able to determine the friction factor.
In large diameter pipes the overall effect of the eddy currents is less significant.
In small diameter pipes the internal roughness can have a major influence on the friction factor.
The ‘relative roughness’ of the pipe and the Reynolds number can be used to plot the friction factor on a friction factor chart

22. Turbulent flow
The velocity at the interface between fluid and solid wall is zero
Within a thin volume immediately adjacent to the wall, the velocity gradient is essentially constant viscous sublayer
Within this layer, viscous shear is important, eddy diffusion is minor

23. Turbulent flow
Transition layers exist immediately adjacent to the viscous sublayer is called buffer layer
Both viscous shear and shear due to the eddy diffusion occur
In the turbulent core, viscous shear is negligible in comparison with that from eddy viscosity

24. Velocity distribution
The velocity distribution in turbulent flow can be expressed in terms of dimensionless parameters
u*=friction velocity
u+=velocity quotient, dimensionless
y+=distance, dimensionless
y=distance from wall of tube u+
Universal velocity distribution laws y+

25. Turbulent flow
When fluid flow at higher flowrates, the streamlines are not steady and straight and the flow is not laminar. Generally, the flow field will vary in both space and time with fluctuations that comprise "turbulence
For this case almost all terms in the Navier-Stokes equations are important and there is no simple solution
P = P (D, , , L, U,) uz úz
Uz average ur úr
Ur average p P’
p average
Time

26. Turbulent flow
All previous parameters involved three fundamental dimensions,
Mass, length, and time
From these parameters, three dimensionless groups can be build

27. Turbulent flow in pipes and channels
The friction factor can be used with the Darcy-Weisbach formula to calculate the frictional resistance in the pipe.
Between the Laminar and Turbulent flow conditions (Re 2300 to Re 4000) the flow condition is known as critical. The flow is neither wholly laminar nor wholly turbulent.
It may be considered as a combination of the two flow conditions. The friction factor for turbulent flow can be calculated from the Colebrook-White equation:

28. Internal roughness (e) of common pipe materials.

29. Major Loss (Friction Loss)
In fully developed pipe flow, the pressure drop caused by friction in a horizontal constant-area pipe is known to dependent on pipe diameter, D, pipe length, L, pipe roughness, ε, average flow velocity, V, fluid density, ρ and fluid viscosity, µ .

30. Major Loss (Friction Loss)- dimensional analysis
(Fanning Chart)
Laminar – Analytical Eq.
(Moody Chart)
(Colebrook)
Turbulent-Empirical
(Haaland)

31. Head Loss / Friction loss
The value for the Reynold number is to be used to evaluate if the flow is laminar or turbulent and can be used to obtain the friction factor " f " from a moody chart.
The moody chart plots the friction factor (f) against the Reynold number with a number of different plotted lines for different values of absolute roughness/Diameter .
The head loss along the pipe can now be calculated using the Darcy-Weisbach equation

32. Friction Loss
The result of the calculation is in units of head of the fluid.  . It is based on the pipe being all one dia and the fluid is incompressible
For a single pipe line with a number of fittings the total head loss is calculated as
K p = f (L/D)
for the length of pipe. ( this may be made up of ∑ f(L/D). for a number of different pipe lengths of different diameters ) K 1..n = fT(L/D) equivalent for each fitting

33. Friction Factor for Smooth, Transition, and Rough Turbulent flow
Smooth pipe, Re>3000
Rough pipe, [ (D/)/(Re√ƒ) <0.01]
Transition function for both smooth and rough pipe

34. Fanning Chart

35. Moody Chart

36. hydraulic roughness (ε)
In the moody chart above (ε /D ) is identified with both numerator and denominator in metres (for consistency with all other equations on this page.  

37. Friction from changes in velocity or direcion

38. Values of L/D for Fittings
The losses through fittings are generally evaluated by obtaining K = fT(L/D)
Table of pipe friction values for clean pipe in region of complete turbulence

39. For laminar fluids with low Re numbers ( "<" 500) the K values obtained using the above are probably very innaccurate. The table below illustrates how this affects the K values

40. K values for Sudden Expansion-Contraction & Orifice
The losses through these fitting are generally evaluated by first obtaining β = d2 / d1 Important Note: the resulting K values as tabled below are based on the flow velocity in the larger pipe if the flow velocity in the small pipe is used to evaluate the head loss then the K values tabled below should be multiplied by ( β ) 4 = (d2 / d1) 4

41. K values for Sudden Expansion-Contraction & Orifice

42. Table of Ke,Kc & KO against β = d2 / d1

43. Example
Given: Figure
Find: Estimate the elevation required in the upper reservoir to produce a water discharge of 10 cfs in the system. What is the minimum pressure in the pipeline and what is the pressure there?

44. Example
Water flows from the ground floor to the second level in a three-storey building through a 20 mm diameter pipe (drawn-tubing,  = 0.0015 mm) at a rate of 0.75 liter/s. The layout of the whole system is illustrated in Figure below. The water flows out from the system through a faucet with an opening of diameter 12.5 mm. Calculate the pressure at point (1).

45. Solution: From the modified Bernoulli equation, we can write
In this problem, p2 = 0, z1 = 0. Thus,
The velocities in the pipe and out from the faucet are respectively
The Reynolds number of the flow is

46. Solution
The roughness d = /20 = From the Moody chart,   (or, via the Colebrook formula). The total length of the pipe is
Hence, the friction head loss is
The total minor loss is
Therefore, the pressure at (1) is

47. Summary