Fluid Mechanics I Chapter 1 3rd semester, autumn

Shinichiro YANO Department of Urban and Environmental Engineering, Kyushu University

17 October, 2012

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1. 4 Description of fluid flows (1) Eulerian & Lagrangian description systems

– Lagrangian system --- trace particular fluid particle X(x0, y0, z0, t) • Inconvenient for describing overall flow field

– Eulerian system --- observe the flow at fixed points V(x, y, z, t)

Flow properties (§1.7-1.8) – Kinematic properties

• Velocity (vector field) V(x, y, z, t) m/s {L/T}

– Major thermodynamic properties

• Pressure (scalar field) p(x, y, z, t) Pa (N/m2=kg/ms2) {ML-1T-2} • Density (scalar field) ρ (x, y, z, t) kg/m3 {M/L3} • Temperature (scalar field)T(x, y, z, t) K {Θ}

unknown variables

),,,(),,,(),,,(),,,( tzyxwtzyxvtzyxutzyx kjiV ++=

1 atm =101300 Pa =101.3 kPa =1013 hPa

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1. 4 Description of fluid flows (2) – Minor thermodynamic properties (§1.8)

• Internal energy m2/s2 {L2T-2} • Enthalpy m2/s2 {L2T-2} • Entropy J/K (=Kgm/s2K){MLT-2Θ-1} • …..

– Constitutive equation (state law)

• Gas ---- State equation of gas – Perfect gas law

• Liquid ---- incompressible fluid

– approximately constant ρ for wide range of p and T » ex. ρwater=998 kg/m3, ρmercury=13,580 kg/m3

Reduces 3 thermodynamic properties into 2 unknowns

uρ/ˆ puh +=

sFunctions of P, T and ρ

#) Refer to P.18-24. Detail will be introduced in the course “Thermodynamics” for Mech. and Aero. Eng. course.

RTp ρ= vp ccR −= =gas constant {L2T-2Θ-1} cp & cv = specific heat under constant pressure/volume {L2T-2Θ-1}

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1. 4 Description of fluid flows (3) Unknown flow properties

– Velocity – Pressure p(x, y, z, t) – Density ρ (x, y, z, t) – Temperature T(x, y, z, t)

Fundamental Equations of Fluid Mechanics --- 6 eqs. – Conservation law of mass (continuity equation) – Conservation law of momentum (Newton’s second law) 3 comp. – Conservation law of energy (First law of thermodynamics) – Constitutive equation

),,,(),,,(),,,(),,,( tzyxwtzyxvtzyxutzyx kjiV ++=

provided appropriate boundary conditions

6 unknowns

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1. 5 Other properties of fluids --- viscosity (1) (Molecular) Viscosity µ kg/ms {ML-1T-1}

– Shear stress due to friction between fluid molecules is

• proportional to velocity gradient • Constant µ Newtonian fluid

» ex. Water, air, etc. » Dependent of T (Fig.1.7)

– Kinematic viscosity ν =µ/ρ m2/s2 {L2T-2}

Reynolds number

– Inertial force/viscous force – Re<Rec laminar flow – Re>Rec turbulent flow

dydu

dtd µθµτ ==

Fig. 1.6

Table. 1.4

Non-slip condition

νµρ VLVL

==Re

Rec: critical Re number

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1. 5 Other properties of fluids --- viscosity (2) Flow between plates (Fig. 1.8)

– Steady (zero acceleration) – One plate moves and another at rest – Called “Couette flow” – No pressure variation in flow direction – Shear stress is constant

– Boundary conditions • u=0 at y=0 • u=V at y=h

– Shear stress

.constdydu

== µτ

byayu +=)(

hyVyu =)(

hVµτ =

Fig. 1.8

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1. 5 Other properties of fluids --- surface tension Coefficient of surface tension Υ N/m {MT-2}

– Tension force per unit length – Surface of liquid-air boundary/ liquid-liquid boundary – Typical values of Υ (upsilon)

• 0.073 N/m for air-water • 0.48 N/m for air-mercury

Pressure jump across curved surface

capillary effect

meniscus effect

Rp

LpRLΥ

=∆

Υ=∆ 22

Rp

RpRΥ

=∆

Υ=∆222 ππ

+Υ=∆

21

11RR

p

Fig. 1.11

contact angle ・between water and glass: 0-9° ・between mercury and glass: 130-140°

θ

solid: glass

θ

water mercury

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capillary phenomenon

θ

cv water mercury

T

D h θ

T

gDThDThDgρ

θθππρ cos4cos4

2 =∴=

8

capillary phenomenon water mercury

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1. 5 Other properties of fluids --- speed of sound Speed of sound a m/s {LT-1}

– Propagating speed of pressure waves

under entropy s is constant • Ideal gas aideal gas = (κRT)1/2 where κ=cp/cv

• Air aair=340 m/s • Water awater>1400 m/s

Mach number – Ma<1 Subsonic flow – Ma~1 Transonic flow – Ma>1 Supersonic flow – Ma>0.3 compressibility effect becomes important

s

pa

∂∂

=ρ

2

==aVMa Representative flow velocity

Speed of sound

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1.6 Flow patterns (§1.11) Flow patterns: streamlines, streaklines and pathlines

– Streamline tangent to velocity vector at a moment

– Pathline actual path of a particular fluid particle

– Streakline locus of particles passing through a point – They are identical in steady flow

Streamline and streamtube

Vdr

wdz

vdy

udx

===

∫∫∫ === wdtzvdtyudtx ,, Fig. 1.17

Fig. 1.16

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1.7 Flow conditions Time-dependency

– Steady flow ---- flow pattern is independent of time – Unsteady flow

Viscosity – Viscous flow – Inviscid flow (ideal flow)

Compressibility – Compressible flow --- high speed flows (M>0.3) – Incompressible flow --- low speed flows, most liquid flows

Turbulence – Laminar flow --- Low Re number – Turbulent flow --- High Re number