Laminar Flow: Turbulence and Convection”

PHY131H1S – Class 25 A little pre-class reading quiz…

Today: • Special Topic in Fluid Flow: “Beyond Laminar Flow: Turbulence and Convection”

No Practicals this Turbulent flow in a tube occurs for week! A. Re ≥ 2000 B. Re ≤ 2000

Laminar Flow • Steady, gently varying flow • Everything is smooth and uniform

See Zinke Q-12.15

Laminar to Recall viscosity turbulent transition • It takes a force, F, to maintain a “shear”, • Turbulence is an Δv/Δy. extreme form of • Units of viscosity, η, are N·s/m2. chaotic motion. • 1 N·s/m2 = kg / (m·s) • It is usually treated statistically, as if it was just random.

1 When is viscosity important? Dimensional Analysis • Every quantity in mechanics is either: – a length, L (SI unit = metre) – a mass, M (SI unit = kg) – a time, T (SI unit = second) • Viscosity is always important close enough – some combination of these, to solid boundaries – or “dimensionless” (like an angle measured in radians) • Otherwise, we need to compare the • Equations have to work out dimensionally. viscous forces in the fluid to other kinds of • LMT are the 3 fundamental dimensions. forces. • square brackets are used to indicate “dimensions • We need to do this in a systematic way: of”. • ie, for kinetic energy, K, [K] = M L2 T–2 Reynolds number

Dimensions of Kinetic and Potential Energy Dimensional Analysis

• What are the dimensions of Pressure?

A. M L T–2 B. M L–2 T–2 C. M L2 T–1 D. M–1 L T–1 E. M L–1 T–2

Dimensional analysis, applied to pipe flow Which of the following combinations is dimensionless? • Osborne Reynolds, in 1883, wanted to study flow patterns in a pipe. • He searched for a dimensionless number that A. ρ·η·d / would describe all the important parameters of B. d· / ρ·η the flow. Recall: C. ρ··d / η • His number includes: [ρ] = M / L3 –, the typical speed of the fluid – d, the diameter of the pipe [η] = M / L·T – ρ, the density of the fluid [d] = L – η, the viscosity of the fluid (HINT: Think about M..) [] = L / T

2 The most important number in fluid Dynamic Similarity mechanics ρ ⋅ v ⋅d ρ ⋅ v ⋅d Re = Re = η η • Re controls when viscosity is important • All viscous flows with the same Reynolds number have similar flow patterns. • Small Re Æ viscous effects dominate • In this sense they are just rescaled • Large Re Æ viscous effects weak, except in versions of each other. boundary layers • Double d but take half of : the flow • Large Re fluids tend to behave like ideal fluids, pattern is the same. except that they are likely to be turbulent.

What Reynolds saw NOTE: this is not a frictionless, ideal fluid… in fluid flow in a tube • Small pipe, high viscosity, slow flow: • Poiseuille flow looks very different than ideal fluid flow in a pipe. • Subject to a pressure difference, an ideal fluid flow would accelerate forever! • Poiseuille flow is like pushing a box over a floor • Laminar, parabolic Poiseuille at a constant speed. flow (eq.12.29 of Zinke) • All the work goes into heat, and the box does •0 < Re < 2000 not accelerate. • Seen in blood flow • That is why you need a heart!

Transition to turbulence

•For Re > 2000, but < 100,000, there are puffs of unsteady flow: turbulence. • This is a bad thing if it happens in your aorta. •For Re > 105, turbulence is fully developed. • It takes more pressure to move the same flow rate if the flow is turbulent.

3 that’s all the new material for PHY131… • Final Exam is Friday, May 1, 2:00 PM in EX 300 • The 2 hour final exam will cover the entire course, including all of the assigned reading plus Practicals materials and what was discussed in class • Approximately even spread over Knight Chs 1-14 (21/24 of course) plus Zinke reading (3/24 of course) • I recommend you are familiar with all Masteringphysics homework and Practicals work • Please email me ( jharlow @ physics.utoronto.ca ) with any questions or review suggestions for Wednesday.