1. MEKANIKA FLUIDA II Nazaruddin Sinaga
KULIAH VIII - IX
MEKANIKA FLUIDA II
Nazaruddin Sinaga

2. Entrance Length

3. Shear stress and velocity distribution in pipe for laminar flow

4. Typical velocity and shear distributions in turbulent flow near a wall: (a) shear; (b) velocity.

5. Solution of Pipe Flow Problems
Single Path
Find Dp for a given L, D, and Q
Use energy equation directly
Find L for a given Dp, D, and Q

6. Solution of Pipe Flow Problems
Single Path (Continued)
Find Q for a given Dp, L, and D
Manually iterate energy equation and friction factor formula to find V (or Q), or
Directly solve, simultaneously, energy equation and friction factor formula using (for example) Excel
Find D for a given Dp, L, and Q
Manually iterate energy equation and friction factor formula to find D, or

7. Example 1
Water at 10C is flowing at a rate of 0.03 m3/s through a pipe. The pipe has 150-mm diameter, 500 m long, and the surface roughness is estimated at 0.06 mm. Find the head loss and the pressure drop throughout the length of the pipe.
Solution:
From Table 1.3 (for water):  = 1000 kg/m3 and  =1.30x10-3 N.s/m2
V = Q/A and A=R2
A = (0.15/2)2 = m2
V = Q/A =0.03/ =1.7 m/s
Re = (1000x1.7x0.15)/(1.30x10-3) = 1.96x105 >  turbulent flow
To find , use Moody Diagram with Re and relative roughness (k/D).
k/D = 0.06x10-3/0.15 = 4x10-4
From Moody diagram,   0.018
The head loss may be computed using the Darcy-Weisbach equation.
The pressure drop along the pipe can be calculated using the relationship:
ΔP=ghf = 1000 x 9.81 x 8.84
ΔP = 8.67 x 104 Pa

8. Example 2
Determine the energy loss that will occur as 0.06 m3/s water flows from a 40-mm pipe diameter into a 100-mm pipe diameter through a sudden expansion.
Solution:
The head loss through a sudden enlargement is given by;
Da/Db = 40/100 = 0.4
From Table 6.3: K = 0.70
Thus, the head loss is

9. Example 3
Calculate the head added by the pump when the water system shown below carries a discharge of 0.27 m3/s. If the efficiency of the pump is 80%, calculate the power input required by the pump to maintain the flow.

10. Solution: Applying Bernoulli equation between section 1 and 2
(1)
P1 = P2 = Patm = 0 (atm) and V1=V2 Thus equation (1) reduces to:
(2)
HL1-2 = hf + hentrance + hbend + hexit
From (2):

11. The velocity can be calculated using the continuity equation:
Thus, the head added by the pump: Hp = 39.3 m
Pin = Watt ≈ 130 kW.

12. EGL & HGL for a Pipe System
Energy equation
All terms are in dimension of length (head, or energy per unit weight)
HGL – Hydraulic Grade Line
EGL – Energy Grade Line
EGL=HGL when V=0 (reservoir surface, etc.)
EGL slopes in the direction of flow

13. EGL & HGL for a Pipe System
A pump causes an abrupt rise in EGL (and HGL) since energy is introduced here

14. EGL & HGL for a Pipe System
A turbine causes an abrupt drop in EGL (and HGL) as energy is taken out
Gradual expansion increases turbine efficiency

15. EGL & HGL for a Pipe System
When the flow passage changes diameter, the velocity changes so that the distance between the EGL and HGL changes
When the pressure becomes 0, the HGL coincides with the system

16. EGL & HGL for a Pipe System
Abrupt expansion into reservoir causes a complete loss of kinetic energy there

17. EGL & HGL for a Pipe System
When HGL falls below the pipe the pressure is below atmospheric pressure

18. FLOW MEASUREMENT Direct Methods
Examples: Accumulation in a Container; Positive Displacement Flowmeter
Restriction Flow Meters for Internal Flows
Examples: Orifice Plate; Flow Nozzle; Venturi; Laminar Flow Element

19. Definisi tekanan pada aliran di sekitar sayap

24. Flow Measurement Linear Flow Meters
Examples: Float Meter (Rotameter); Turbine; Vortex; Electromagnetic; Magnetic; Ultrasonic
Float-type variable-area flow meter

25. Flow Measurement Linear Flow Meters Turbine flow meter
Examples: Float Meter (Rotameter); Turbine; Vortex; Electromagnetic; Magnetic; Ultrasonic
Turbine flow meter

26. Flow Measurement Traversing Methods
Examples: Pitot (or Pitot Static) Tube; Laser Doppler Anemometer

30. The measured stagnation pressure cannot of itself be used to determine the fluid velocity (airspeed in aviation).
However, Bernoulli's equation states:
Stagnation pressure = static pressure + dynamic pressure
Which can also be written

31. Solving that for velocity we get:
Note: The above equation applies only to incompressible fluid.
where:
V is fluid velocity;
pt is stagnation or total pressure;
ps is static pressure;
and ρ is fluid density.

32. The value for the pressure drop p2 – p1 or Δp to Δh, the reading on the manometer:
Δp = Δh(ρA-ρ)g
Where:
ρA is the density of the fluid in the manometer
Δh is the manometer reading

33. EXTERNAL INCOMPRESSIBLE VISCOUS FLOW

36. Main Topics The Boundary-Layer Concept Boundary-Layer Thickness
Laminar Flat-Plate Boundary Layer: Exact Solution
Momentum Integral Equation
Use of the Momentum Equation for Flow with Zero Pressure Gradient
Pressure Gradients in Boundary-Layer Flow
Drag
Lift

37. The Boundary-Layer Concept

38. The Boundary-Layer Concept

39. Boundary Layer Thickness

40. Boundary Layer Thickness
Disturbance Thickness, d where
Displacement Thickness, d*
Momentum Thickness, q

45. Boundary Layer Laws

46. Laminar Flat-Plate Boundary Layer: Exact Solution
Governing Equations

47. Laminar Flat-Plate Boundary Layer: Exact Solution
Boundary Conditions

48. Laminar Flat-Plate Boundary Layer: Exact Solution
Equations are Coupled, Nonlinear, Partial Differential Equations
Blassius Solution:
Transform to single, higher-order, nonlinear, ordinary differential equation

49. Laminar Flat-Plate Boundary Layer: Exact Solution
Results of Numerical Analysis

50. Momentum Integral Equation
Provides Approximate Alternative to Exact (Blassius) Solution

51. Momentum Integral Equation
Equation is used to estimate the boundary-layer thickness as a function of x:
Obtain a first approximation to the freestream velocity distribution, U(x). The pressure in the boundary layer is related to the freestream velocity, U(x), using the Bernoulli equation
Assume a reasonable velocity-profile shape inside the boundary layer
Derive an expression for tw using the results obtained from item 2

52. Use of the Momentum Equation for Flow with Zero Pressure Gradient
Simplify Momentum Integral Equation (Item 1)
The Momentum Integral Equation becomes

53. Use of the Momentum Equation for Flow with Zero Pressure Gradient
Laminar Flow
Example: Assume a Polynomial Velocity Profile (Item 2)
The wall shear stress tw is then (Item 3)

54. Use of the Momentum Equation for Flow with Zero Pressure Gradient
Laminar Flow Results (Polynomial Velocity Profile)
Compare to Exact (Blassius) results!

55. Use of the Momentum Equation for Flow with Zero Pressure Gradient
Turbulent Flow
Example: 1/7-Power Law Profile (Item 2)

56. Use of the Momentum Equation for Flow with Zero Pressure Gradient
Turbulent Flow Results (1/7-Power Law Profile)

57. Pressure Gradients in Boundary-Layer Flow

58. Drag
Drag Coefficient
with or

59. Drag Pure Friction Drag: Flat Plate Parallel to the Flow
Pure Pressure Drag: Flat Plate Perpendicular to the Flow
Friction and Pressure Drag: Flow over a Sphere and Cylinder
Streamlining

60. Drag Flow over a Flat Plate Parallel to the Flow: Friction Drag
Boundary Layer can be 100% laminar, partly laminar and partly turbulent, or essentially 100% turbulent; hence several different drag coefficients are available

61. Drag
Flow over a Flat Plate Parallel to the Flow: Friction Drag (Continued)
Laminar BL:
Turbulent BL:
… plus others for transitional flow

62. Drag Coefficient

63. Drag Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag
Drag coefficients are usually obtained empirically

64. Drag
Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag (Continued)

65. Drag
Flow over a Sphere and Cylinder: Friction and Pressure Drag

66. Drag
Flow over a Sphere and Cylinder: Friction and Pressure Drag (Continued)

67. Streamlining
Used to Reduce Wake and Pressure Drag

68. Lift
Mostly applies to Airfoils
Note: Based on planform area Ap

69. Lift
Examples: NACA 23015; NACA

70. Lift
Induced Drag

71. Lift Induced Drag (Continued) Reduction in Effective Angle of Attack:
Finite Wing Drag Coefficient:

72. Lift
Induced Drag (Continued)

73. The End
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