Introduction to Fluid Flow

Introduction to Fluid Flow

Learning Outcomes After this lecture you should be able to Explain viscosity and how it changes with temperature Write the continuity equation Define laminar and turbulent flow by using the Reynolds number Determine if a flowrate is laminar or turbulent Write and Explain the Bernoulli equation Apply the Bernoulli equation

Basics of Fluid Flow A fluid is a substance that flows When subjected to a shearing stress layers of the fluid slide relative to each other Both gases and liquids are defined as fluids Fluid mechanics is the study of the flow of gases and liquids The degree of resistance to shear stress is represented by the term viscosity High viscosity means high resistance to shear stress does not flow easily

Viscosity Dynamic Viscosity or Viscosity is a measure of resistance to shearing motion The unit is Ns/m 2.but it has no name! The poise or centipoise is the SI cgs unit 1 centipoise = 1 x 10-3 Ns/m 2 Typical values for viscosity Water at 20 C = 1 cp Air at 20 C = 1.8 x 10-2 cp Crude Oil = 7.2 cp Petrol = 0.29 cp You may hear the term kinematic viscosity This is dynamic viscosity divided by fluid density Its SI cgs unit is the Stoke (= 1 cm 2 /s) NB Viscosity is a function of temperature. For liquids, viscosity decreases as temperature increases

Basics Equations for Fluid Flow The continuity equation Q = v.a where v is the velocity (m/s) and a the area available for flow (m 2 e.g. cross sectional area of a pipe) and Q is the flowrate (m 3 /s) The Reynolds number is used to define laminar and turbulent flow Laminar flow is defined by slow moving, uniform, even, smooth flow (e.g. a canal) Turbulent flow is uneven and rough (e.g. a white water river) Bernoulli equation. Daniel Bernoulli lived in the 18 th century and derived a relationship between velocity, height and pressure

The Continuity equation Q=va Q flowrate, m 3 /s v fluid velocity, m/s a area available for flow, m 2 What is the flowrate from your kitchen tap? (What is the volume of your kettle and how long does it take to fill it?) The pipe feeding the tap is 15mm. What is the cross sectional area? Use the continuity equation to determine the velocity

Continuity Equation contd. Imagine a long pipe of varying diameter. The flowrate is constant Where the diameter is large, the velocity is small Where the diameter is small, the velocity is large 1 2 d 1 v 1 < > d 2 v 2

Osborne Reynolds 1842-1912 A pioneer in Fluid Mechanics He discovered the nature of flow depends on Velocity Fluid physical properties Geometry of the channel/pipe Sometimes flow is even and smooth Sometimes it is uneven and rough He asked Why?

Reynolds Experiment He investigated fluid flow using this apparatus Dye

Reynolds Experiment - Velocity His first discovery At very low water flowrates, dye did not break up Implies no mixing between dye and water! Dye

Reynolds Experiment - Velocity.. And at high water flowrates, dye did break up Dye mixed with water Dye

Reynolds Concluded that At low flowrates we get streamline or laminar flow Flow is characterised by streams that don t mix At high flowrates we get turbulent flow and a lot of mixing Increase Velocity

Further Experiments - Viscosity Reynolds heated the water When heated the change from laminar to turbulent occurred sooner (at a lower velocity) This is explained by viscosity Viscosity decreases as temperature increases Decrease Viscosity

Further Experiments - Density Reynolds replaced water with liquids of different density The change from laminar to turbulent occurred sooner for high density liquids Increase Density

Further Experiments Tube diameter Reynolds used tubes of different diameter He discovered that as the diameter increased the change to turbulent occurred sooner Increase Diameter

Reynolds Number He combined these observations into a dimensionless number which now carries his name Re = ρvd µ Re = Reynolds number ρ = density (kg/m 3 ) v = velocity (m/s) d = pipe diameter (m) µ = viscosity (kg/ms)

Activity Laminar or Turbulent? Is the flow from your kitchen tap laminar or turbulent? Determine the Reynolds No. and then use the table below 0 < Re <2000 Laminar flow 2000 < Re < 4000 Transition region Re > 4000 Turbulent flow

Daniel Bernoulli (1700 1782) Bernoulli was a pioneer in Science. His interests were medicine and engineering Bernoulli, with Leonard Euler, investigated the relationship between pressure and velocity They punctured a pipe with a straw and observed that the height of liquid in the straw is related to the pressure in the pipe This was used to measure blood pressure where patients arms were punctured with glass capillaries

Conservation of Energy Bernoulli reasoned that the sum of pressure and kinetic energy is the same for any two points in a pipe 1 ρ 2 v + P = 2 C This implies that if the velocity increases, pressure decreases. This is true for a horizontal pipe only.

Bernoulli Equation Include a term for gravity, ρgh, to get the Bernoulli Equation as follows 1 2 ρ 2 v + P + ρgh = This is often written as follows: P 1 + C 1 2 ρ gh1 + ρv1 = P2 + ρgh2 + 2 1 2 ρv Points 1 and 2 could be at two places in a pipe: 1 2 2 2 d 1 v 1 P 1 < > < d 2 v 2 P 2

Activity Bernoulli Eqn Units Determine the units of each term in the Bernoulli equation 1 2 ρ 2 v + P + ρgh = C

Bernoulli Eqn Rearranged Instead of expressing each term in units of Pressure, rearrange to give units of height v 2 2 g P + ρ g + h = C

How a chimney works Point 1 is at the top of the chimney where the velocity is the same as the wind speed Point 2 is in the fireplace where the velocity is almost zero

Activity Flow in a pipe A water mains supply enters a house at ground level (point 1) and rises vertically to the attic tank at an elevation of 10 m (point 2). No change in diamter. What is the P? Point 2 V = 2 m/s 10m Point 1

Activity Bernoulli Eqn 2 Same as before except the pipe changes from 40mm diameter to 20mm. What is the P? Point 2 20mm V = 2 m/s 10m Point 1 40mm V =? m/s

Conversion Table Litre/s Litre/min m 3 /hr m 3 /s Ft 3 /hr Ft 3 /min gpm 1 Litre/s 1 60 3.6 0.001 127.1 2.119 15.85 1 litre/min 0.0167 1 0.06 1.66x10-5 2.12 0.035 0.264 1 m 3 /hr 0.278 16.67 1 0.00028 35.3 0.588 4.438 1 m 3 /s 1,000 60000 3,600 1 127,133 2,119 15,850 1 Ft 3 /hr 0.0078 0.472 0.0283 7.87x10-6 1 0.0167 0.124 1 Ft 3 /min 0.472 28.3 1.699 0.00047 60 1 7.481