Fluid flow lecture to be studied, Lecture notes of Civil Engineering

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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

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 18th century and derived a relationship between velocity, height and pressure

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

d 1 v 1

1 2 d 2 v 2

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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 - 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

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

Reynolds Number

He combined these observations into a dimensionless number which now carries his name

ρvd

Re =

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

Conservation of Energy

Bernoulli reasoned that the sum of pressure and kinetic energy is the same for any two points in a pipe

v^2 + P = 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

ρ v^2 + P + ρgh = C

This is often written as follows: 2 2 2 2

2

P + ρ gh + ρv = P + ρgh + ρv

Points 1 and 2 could be at two places in a pipe:

d 1 v 1 P 1

1 2 d 2 v 2 P 2

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