VIII. Viscous Flow and Head Loss

Contents 1. Introduction 2. Laminar and Turbulent Flows 3. Friction and Head Losses 4. Head Loss in Laminar Flows 5. Head Loss in Turbulent Flows 6. Head Loss of Steady Pipe Flows 7. Minor Losses 8. Examples

1. Introduction

Shear stress due to fluid viscosity

D’Alembert Paradox

For real fluid flows Head Loss

Head Loss:  Losses due to friction  Minor Losses entrance and exit sudden change of cross sections valves and gates bends and elbows ……

2. Laminar and Turbulent Flows

Reynolds’ Experiment

Laminar Flows: Movement of any fluid particle is regular Path lines of fluid particles are smooth Turbulent Flows: Movement of any fluid particle is random Path lines of fluid particles are affected by mixing

Transition from Laminar to Turbulent Flow:  for different fluid  for different diameter of pipe

Head Loss due to laminar and turbulent flows

Turbulent Flows: Laminar Flows:

Critical Condition Reynolds Number

3. Friction and Head Losses

Momentum Equation A : area of the cross-section P: wetted perimeter

Hydraulic radius

Darcy-Weisbach equation

4. Head Loss in Laminar Flows

5. Head Loss in Turbulent Flows

Mean flow and fluctuation

Basic Equations of Turbulent Flows:

Reynolds’ Average

Reynolds Stresses

Mean flux of horizontal momentum: Equivalent Shear Stress:

Reynolds Equations:

Theory of Mixing Length

Logarithmic Velocity Distribution

6. Head Loss of Steady Pipe Flows

Logarithmic Velocity Distribution

Logarithmic Overlap Layer

Logarithmic Velocity Distribution in a Pipe

ViscousTurbulent

Viscous sublayer: Turbulent zone: Transition zone:

Velocity Distribution in Viscous Sublayer

Velocity Distribution in a Pipe

Blasius’ 7th-root law Valid for R = 3000  10 5

Wall Roughness

Hydraulically smooth wall: Roughness height is smaller than the thickness of the viscous sublayer Hydraulically rough wall: Roughness height is larger than the lower boundary of the turbulent zone

Hydraulically smooth pipe: Hydraulically rough pipe:

Velocity Distribution in a Pipe

Mean velocity in hydraulically smooth pipe:

Mean velocity in hydraulically rough pipe:

Relation among mean velocity, friction velocity and friction factor:

Friction factor in hydraulically smooth pipe:

Friction factor in hydraulically rough pipe:

Experiment of Nikuradse

Modified friction factor in hydraulically smooth pipe:

Modified friction factor in hydraulically rough pipe:

Colebrook Equation:

Head loss in hydraulically smooth pipe:

Practical pipe: equivalent roughness

7. Minor Losses

Head Loss due to Sudden Expansion

12

Head Loss due to Sudden Contraction

Head Loss at Entrance

Head Loss at Bell-Mouthed Entrance

Head Loss in Bend

8. An Example