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, Fluid Dynamics. (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: Laminar Flow 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” Turbulence 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 Reynolds number) the force leading to stability (_________)viscosity 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.