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Mécanique des fluides 2014

Mécanique des fluides Appliquée 2014

Session 1 Mass, species balances!1.  Solving problem methodology 2.  Continuum medium 3.  Flow types 4.  Material derivative 5.  Transport theorems 6.  Mass balance 7.  Mixture of gases 8.  Species balances

Lecture 1, 30/09/2014

State !equation!

Constitutive !law! Models!

Approximations! Experimental !data!

Numerical !methods!

Solving problems : methodology!

mass balance!momentum balance!energy balance!species balance!

Continuum medium!

V!

P!δ"V!

δV

ρ

O

δm/δV

δVl

L System characteristic size!

Knudsen !number!Kn = lpm

L<<1

Local ETL!

ρ = lim δmδVδV→δVl

Density!

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v(r, t) =

mivii=1

N

∑

mii=1

N

∑

Velocity!

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v(r,t) = lim δpδmδV→δVl

Fluid particle P!

P!

n

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δF

δAV

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δF = t(n)δA

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t(n) = limδFδAδA→δAl

Stress vector!

Flow types!

Incompressible! Compressible!Steady! Unsteady!Laminar ! Turbulent!Single phase! Multiphase flow!One-dimensional! Multidimensional!Non-reactive! Reactive!Ideal! Viscous!Single species! Multi species!

Low Mach flow approximation!

M =vc

Mach !number!

In absence of heat transfer (adiabatic flows)!

Δρρ

≈O(M 2 )

With heat transfer (non adiabatic flows)!

Iso-volume flow!

Flow velocity!

Speed of sound!

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∇⋅ v ≈ 0

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M < 0.3

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M < 0.3

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v⋅ ∇phρv⋅ ∇h

≈O M 2( )Hydrodynamic pressure gradient can be neglected in energy budget but not in momentum balance!

Laminar !plume!

Heated !cylinder!

Natural convection flow is induced by buoyancy forces (Archimedes)!

Example : buoyancy flow – Natural convection

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ρ = ρ T( )Isobaric flow!

Pressure changes can be neglected compared to temperature changes!

ΔTHeated cylinder!

Boundary layer

Laminar plume

Transition

Turbulent plume

Natural convection due to buoyancy forces!

Laminar Turbulent Motion is smooth Regular Deterministic!!

Motion is fluctuating Irregular !Random!

Laminar

Transition

Turbulent

Cigarette

Flow around a winglet with increasing attack angle!

LAMINAR! TURBULENT!Increase angle of incidence!

Flow separation - Instabilities !

movie!

Reversible flow! Irreversible flow!movie! movie!

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Re =ρvLµ

Re < 1! Re > 1!

Reynolds !number!

Re

turbulent!

Flow regimes in a pipe!

laminar!

2500

transition!

2000 3000

movie!

Re = ρvdµ

Re 2500 2000 3000

turbulent!laminar! transition!

Flow regimes in a pipe! Re = ρvdµ

Re

Turbulent Laminar

2500

transition

2000 3000

movie!

Flow regimes in a pipe! Re = ρvdµ

Steady! Unsteady!Time independent!Permanent!

Time dependent!Non stationary!

A turbulent flow is steady in the mean if mean variables are time independent!

A turbulent flow is said unsteady if mean variables explicitly depend on time!

Two-phase jet flow submitted to harmonic modulations!movie!

Trajectories of fluid particles in waves. Depth is small compared to the wavelength!

Cylinder within a flow !

Steady !! Unsteady!

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Re <<1

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Re =10

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Re =100Steady or unsteadiness depends also on the flow velocity!

movie! movie!

Single phase! Two phase flows!

Liquid injection at high pressure!

Vortices created around a delta wing!

Voir cours « Ecoulements diphasiques »!

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dφdt

=∂φ∂t

+ v ⋅ ∇φ

t

t + δtx

x + δx

Material derivative!Of a fluid particle P at X at t=0 as a function of space x coordinates!

The flow is steady but fluid particles accelerate between the pipe inlet and outlet!

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ve

€

vs

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dv dt = v⋅ ∇v

Inlet flux! Outlet flux!

n

dA €

ddt

f (x,t)dV =∂f∂tVa ( t )

∫Va ( t )∫ dV + fw ⋅ndA

Aa (t )∫

Va (t)

w

Transport Theorems!

Transport Theorems!

Am n

Vm

Material volume!Useful for proof!!

Fixed volume!Useful for applications!!

A n

V

v €

ddt

φdV =Vm

∫ ddt

φdV + φvA∫

V∫ ⋅ ndA

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∂ρ∂t

+∇ ⋅ ρv = 0

Local mass balance!

Global mass balance!

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0 =ddt

ρdV =Vm

∫ ddt

ρdVV∫ + ρv⋅ ndA

A∫

Material volume!

Mass remains constant in a material volume!

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ddt

ρdVV∫ = − ρv⋅ ndA

A∫

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∂m∂t

= ρ1v1 ⋅N1A1 − ρ2v2 ⋅N2A2

Uniform flow at the INLET and OUTLET!

A1

A2

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N1€

N2

Mass balance – 1D inlets/outlets!

N : unit normal vector oriented in the flow direction!

Steady flow!

Mass balance – Steady in/out!

m = ρ1v1 ⋅N1 dA∫ = ρ2v2 ⋅N2 dA∫

A1

A2

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N1€

N2

The mass flowrate !remains constant through the system!

Mixture of species!

Mole fraction! Mass fraction!

Concentration! Density!

Partial pressure (perfect gas)!

Species j!

Mixture of species!

Molar mass!

Concentration! Density!

Pressure (perfect gas)!

N species!

J/kg/K!

∂ρ j

∂t+∇⋅ρ jv j = !ω j

Local species balance!

Species j density in the mixture!

Mass production rate of species j [kg/m3/s]!

Species j velocity!

Mixture velocity! Diffusion velocity of

species j with respect to the mixture velocity!

Diffusion velocity!Stefan-Maxwell equations (in the case when diffusion is only controlled by gradients of molar fractions)!

Dj Diffusion coefficient of !species j in the mixture!

Fick law!Approximate solution of Stefan-Maxwell equations!