Viscous Flow in Pipes Excerpted from supplemental materials of Prof. Kuang-An Chang, Dept. of Civil Engin., Texas A&M Univ., for his spring 2008 course CVEN 311, . (Various edits made by JRR, July 2008) Types of Engineering Problems

 How big does the pipe have to be to carry a flow of x m3/s?  What will the pressure in the water distribution system be when a fire hydrant is open? Example Pipe Flow Problem

cs1

Find the discharge, Q. D=20 cm 100 m L=500 m valve

cs2 Describe the process in terms of energy! loss

V1 << V2 ⇒ Viscous Flow: Dimensional Analysis

γ hl D ⎛ ε ⎞ = f = f ⎜ , ReD ⎟ ρV 2 /2 L ⎝ D ⎠ ρVD where ReD = and ε = r.m.s. wall roughness μ  Two important parameters!

 ReD - Laminar or Turbulent  ε / D - Rough or Smooth  Flow geometry  internal ______in a bounded region (pipes, rivers)  external ______flow around an immersed object Laminar and Turbulent Flows

 Reynolds apparatus

inertial force viscous force

Transition at ReD of ~2000 Boundary layer growth: Transition length

What does the water near the pipeline wall experience? ______Drag or shear Why does the water in the center of the pipeline speed up? ______Conservation of mass

Pipe Entrance

v v v Non-Uniform Flow Need equation for entrance length here Incompressible, Steady, Uniform Flow

 Between parallel walls  Through circular tubes  Approach  Shear stress τ can be easily determined from a force balance (and is same for laminar or turbulent cases)  When flow is laminar, velocity profiles can be determined by integrating du / dr = τ / μ , leads to Hagen-Poiseuille equation  Analysis provides fuller details of solution that are not given by dimensional analysis Flow through Circular Tubes

 Similar analysis to that for flow between parallel walls  Apply equation of equilibrium in x-direction to cylindrical fluid volume of radius r and length dx: x

τ z r el

Note: Applying equations of equilibrium in directions perpendicular p to x shows that h ≡ + z el = h ( x ) , i.e., h is constant in a cross section. γ Flow through Circular Tubes: Diagram

Velocity

⎛ p ⎞ ⎛ p ⎞ Shear stress hl ≡ + zel − + zel ⎝⎜ γ ⎠⎟ ⎝⎜ γ ⎠⎟ τ expressions x=0 x=L same for Shear stress laminar or at the wall: turbulent flow Flow velocity, laminar case: through Circular Tubes: Hagen-Poiseuille flow rate equations

Q = volumetric flow rate VA = Vπa2 Turbulent Pipe and Channel Flow: Overview

 Velocity distributions  Energy losses  Steady flow through circular tubes  Steady flow in “open channels”, like natural streams, can be treated in the same way, generalizing the tube results by use of the concept of “hydraulic radius”

 A characteristic of the flow.  How can we characterize turbulence?  intensity of the velocity fluctuations  size of the fluctuations (length scale)

instantaneous mean velocity  velocity velocity fluctuation Turbulence: Flow Instability

 In turbulent flow (high ) the force leading to stability (______) is small relative to the force leading to instability (______).inertia  Any disturbance in the flow results in large scale motions superimposed on the mean flow.  Some of the kinetic energy of the flow is transferred to these large scale motions (eddies).  Large scale instabilities gradually lose kinetic energy to smaller scale motions.  The kinetic energy of the smallest eddies is dissipated by viscous resistance and turned into heat. (=______)head loss Velocity Distributions

 Turbulence causes transfer of momentum from center of pipe to fluid closer to the pipe wall.  Mixing of fluid (transfer of momentum) causes the central region of the pipe to have relatively ______velocityconstant (compared to laminar flow)  Close to the pipe wall eddies are smaller (size proportional to distance to the boundary) Turbulent Flow Velocity Profile

Turbulent shear is from momentum transfer

η = eddy viscosity Length scale and velocity fluctuation of “large” eddies

Dimensional reasoning

y Turbulent Flow Velocity Profile

Size of the eddies ______increases as we move further from the wall. κ = 0.4 (from fitting to experiments) Log Law for Turbulent, Established Flow, Velocity Profiles near Wall (Law of the Wall)

Integration and empirical results (κ =0.4)

Laminar Turbulent Shear velocity y μ ν = ρ

x Turbulent Pipe Flow: The Problem

 Need to be able to predict the head loss term.  Will use the results obtained using dimensional analysis and law of the wall concepts, with some parameters fitted to experimental observations. Pipe Flow Energy Losses

Dimensional Analysis: Darcy-Weisbach equation

f τ = ρV 2 wall 8 Friction Factor : Major losses

 Laminar flow  Turbulent (Smooth, Transition, Rough)  Colebrook Formula  Moody diagram Laminar Flow Friction Factor

Hagen-Poiseuille

Darcy-Weisbach

Slope of ___-1 on log-log plot Turbulent Pipe Flow Head Loss

 Proportional______to the length of the pipe  ______Proportional to the square of the velocity (almost)  ______Inversely with the diameter (almost)  ______Increase with surface roughness  Is a function of density and viscosity  Is ______independent of pressure Smooth, Transitional, and Rough Turbulent Flow

 Hydraulically smooth pipe law (von Karman, 1930)

 Rough pipe law (von Karman, 1930)

 Transition function for both smooth and rough pipe laws (Colebrook, 1939) (used to draw the Moody, 1944, diagram) Moody (1944) Diagram

0.10 0.08 0.05 0.04 0.06 0.03 f 0.05 0.02 0.015 0.04 0.01 0.008 0.006 0.03 0.004 laminar 0.002 friction factor, 0.001 0.02 0.0008 0.0004 0.0002 0.0001 0.00005 0.01 smooth 1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 ReD Pipe roughness

pipe material pipe roughness ε (mm) glass, drawn brass, copper 0.0015 commercial steel or wrought iron 0.045 asphalted cast iron 0.12 galvanized iron 0.15 cast iron 0.26 concrete 0.18-0.6 rivet steel 0.9-9.0 corrugated metal 45 PVC 0.12 Columbia Basin Irrigation Project

The Feeder Canal is a concrete lined canal which runs from the outlet of the pumping plant discharge tubes to the north end of Banks Lake (see below). The original canal was completed in 1951 but has since been widened to accommodate the extra water available from the six new pump/generators added to the pumping plant. The canal is 1.8 miles in length, 25 feet deep and 80 feet wide at the base. It has the capacity to carry 16,000 cubic feet of water per second. Pipes are Everywhere!

Owner: City of Hammond, IN Project: Water Main Relocation Pipe Size: 54" Pipes Pipes are Everywhere! Water Mains