Cylinder 1

Flow past a circular cylinder

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Cylinder 2

Flow past a circular cylinder

Frame of reference

A

u1u0

u2

u0

A

u1

Cylinder 3

Flow past a circular cylinder

Reynolds number

A

u1

du0

ρ,µ

νµρ dudu 00Re ==

Cylinder 4

Flow past a circular cylinder

Reynolds number

Viscosity µ (15°C)Air : 1.78x10-5 kg/ms

Water : 1.14x10-3 kg/msdu0

νdu0Re = Kinematic Viscosity ν (15°C)

Air : 0.15 cm2/s Water : 0.01 cm2/s

Example

Air : d=1m, u0=1m/s → Re≈67KWater : d=1m, u0=1m/s → Re≈1M

Air : d=0.1mm, u0=1.5cm/s → Re≈0.1Water : d=1cm, u0=1.5cm/s → Re≈150

Cylinder 5

Flow past a circular cylinder

Flow patterns

Re<<1

Path of fluid elementsSymmetry for upstream and downstream

Effect of a cylinder is large and wide.

Cylinder 6

Flow past a circular cylinder

Flow patterns

Cylinder 7

Flow past a circular cylinder

Flow patterns

Re=10

Cylinder 8

Flow past a circular cylinder

Flow patterns

Re=41

Cylinder 9

Flow past a circular cylinder

Flow patternsIncreasing Re

Re≈30

Re≈40

Re=47

Cylinder 10

Flow past a circular cylinder

Flow patterns

Re=55

Re=67

Re=100

Cylinder 11

Flow past a circular cylinder

Flow patterns

von Karman vortex street

Cylinder 12

Flow past a circular cylinder

Flow patterns

Cylinder 13

Flow past a circular cylinder

Flow patterns

Wake : a flow behind a body

Cylinder 14

Free convection

Density at various temperature

0.997040.999100.99996Water

1.2151.2581.303Air (x10-3)780Torr

25°C15°C5°CDensity

g/cm3

Cylinder 15

Free convection

Internal and external flow

A

u1

Cylinder 16

Free convection external

g

Cylinder 17

Free convection internal

g

Cylinder 18

Free convection internal

g

Cylinder 19

Free convectionRayleigh Number

d

h

Properties of fluidsα : thermal expansion coefficientκ : thermal diffusivityν : kinematic viscosity

T2 T1Rayleigh Number

( )νκ

α 312 dTTgRa −

=g

Cylinder 20

Free convection

Prandtl Number

κν

=PrProperties of fluids

κ : thermal diffusivityν : kinematic viscosity

( ) Pr2

312

να dTTgRa −

=

Grashof Number

( )νκ

α 312 dTTgRa −

=

Cylinder 21

Free convectionConvection in a vertical slot

Control parameter : Ra

When T2>T1 & Ra is small

T2 T1

System parameter : Aspect ratio : h/d

Pr number : Prg

Cylinder 22

Free convectionConvection in a vertical slot

Ra=2.9x105

Ra=1.5x106Ra=1.3x105

h/d=15Pr=480

Cylinder 23

Free convectionConvection in a vertical slot

Distribution of vertical velocity3.1x104

2.95x105

6.6x105

3.6x106

Cylinder 24

Free convectionConvection in a horizontal layer

T1

If T1>T2 → equilibrium state

T2

If T2>T1

T1heaviercolder

hotter lighterT2

Equilibrium is unstable.

Cylinder 25

Free convectionConvection in a horizontal layer

T1

T2

Equilibrium is unstable.

Stabilizing effects :friction by viscosityheat conduction

At larger Ra number

Rayleigh-Benard convection

Critical Ra ≈ 1700

Cylinder 26

Free convectionConvection in a horizontal layer