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**MEKANIKA FLUIDA II Nazaruddin Sinaga**

KULIAH VIII - IX MEKANIKA FLUIDA II Nazaruddin Sinaga

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Entrance Length

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**Shear stress and velocity distribution in pipe for laminar flow**

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

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**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

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**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

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Example 1 Water at 10C 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

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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

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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.

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**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):

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**The velocity can be calculated using the continuity equation:**

Thus, the head added by the pump: Hp = 39.3 m Pin = Watt ≈ 130 kW.

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**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

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**EGL & HGL for a Pipe System**

A pump causes an abrupt rise in EGL (and HGL) since energy is introduced here

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**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

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**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

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**EGL & HGL for a Pipe System**

Abrupt expansion into reservoir causes a complete loss of kinetic energy there

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**EGL & HGL for a Pipe System**

When HGL falls below the pipe the pressure is below atmospheric pressure

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**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

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**Definisi tekanan pada aliran di sekitar sayap**

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**Flow Measurement Linear Flow Meters**

Examples: Float Meter (Rotameter); Turbine; Vortex; Electromagnetic; Magnetic; Ultrasonic Float-type variable-area flow meter

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**Flow Measurement Linear Flow Meters Turbine flow meter**

Examples: Float Meter (Rotameter); Turbine; Vortex; Electromagnetic; Magnetic; Ultrasonic Turbine flow meter

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**Flow Measurement Traversing Methods**

Examples: Pitot (or Pitot Static) Tube; Laser Doppler Anemometer

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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

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**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.

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**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

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**EXTERNAL INCOMPRESSIBLE VISCOUS FLOW**

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**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

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**The Boundary-Layer Concept**

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**The Boundary-Layer Concept**

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**Boundary Layer Thickness**

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**Boundary Layer Thickness**

Disturbance Thickness, d where Displacement Thickness, d* Momentum Thickness, q

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Boundary Layer Laws

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**Laminar Flat-Plate Boundary Layer: Exact Solution**

Governing Equations

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**Laminar Flat-Plate Boundary Layer: Exact Solution**

Boundary Conditions

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**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

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**Laminar Flat-Plate Boundary Layer: Exact Solution**

Results of Numerical Analysis

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**Momentum Integral Equation**

Provides Approximate Alternative to Exact (Blassius) Solution

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**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

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**Use of the Momentum Equation for Flow with Zero Pressure Gradient**

Simplify Momentum Integral Equation (Item 1) The Momentum Integral Equation becomes

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**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)

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**Use of the Momentum Equation for Flow with Zero Pressure Gradient**

Laminar Flow Results (Polynomial Velocity Profile) Compare to Exact (Blassius) results!

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**Use of the Momentum Equation for Flow with Zero Pressure Gradient**

Turbulent Flow Example: 1/7-Power Law Profile (Item 2)

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**Use of the Momentum Equation for Flow with Zero Pressure Gradient**

Turbulent Flow Results (1/7-Power Law Profile)

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**Pressure Gradients in Boundary-Layer Flow**

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Drag Drag Coefficient with or

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**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

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**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

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

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Drag Coefficient

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**Drag Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag**

Drag coefficients are usually obtained empirically

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

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

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

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Streamlining Used to Reduce Wake and Pressure Drag

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

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Lift Examples: NACA 23015; NACA

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Lift Induced Drag

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**Lift Induced Drag (Continued) Reduction in Effective Angle of Attack:**

Finite Wing Drag Coefficient:

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Lift Induced Drag (Continued)

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The End Terima kasih

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Pharos University ME 352 Fluid Mechanics II

Pharos University ME 352 Fluid Mechanics II

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