Fluctuations in laminar flow Jos e M. Ortiz de Z arate - - PowerPoint PPT Presentation

Fluctuations in laminar flow

Jos´ e M. Ortiz de Z´ arate∗ Universidad Complutense. Madrid, Spain IWNET2012, Røros, August 20th, 2012

∗In collaboration with Jan V. Sengers

Hydrodynamics

Balance of momentum: ∂ ∂t(ρv) = −∇[(ρv)v] +∇Π − ∇p + fV Entropy production: ˙ S = 1 T Π : (∇v) Linear phenomenological laws (incompressible flow): Πij =

• kl

η(δikδjl + δilδjk)∂vk ∂xl

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Fluctuating hydrodynamics I

• Dissipation is due to molecular collisions (interaction)
• To reflect the random nature of these collisions,

in fluctuating hydrodynamics dissipative fluxes are supplemented with stochastic

• contributions. For incompressible flow:

Πij = η(δikδjl + δilδjk)∂vk ∂xl + δΠij = η ∂vi ∂xj + ∂vj ∂xi

• + δΠij
• Stochastic properties: δΠij = 0, and fluctuation-dissipation theorem:

δΠij(r, t) δΠkl(r′, t′) = 2kBT η(δikδjl + δilδjk) δ(r − r′) δ(t − t′)

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Fluctuating hydrodynamics II

• Substituting into momentum balance, we obtain an stochastic Navier-

Stokes equation, with a δΠ forcing term. For incompressible flow and no volume forces: ρ ∂v ∂t + (v · ∇)v

• = −∇p + η∇2v + ∇δΠ

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Plane Couette flow I

Solution of the deterministic equation

wall-normal (z) spanwise (y) L L

0,x

v z γ = ɺ

streamwise (x)

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Plane Couette flow II

• Studying velocity fluctuations around the plane Couette flow, in the usual

way, we obtain stochastic Orr-Sommerfeld and Squire equations for wall normal velocity δvz and vorticity δωz fluctuations: ∂t(∇2δvz) + z∂x(∇2δvz) − 1 Re∇4δvz = {∇ × ∇ × ∇δΠ}z ∂t(δωz) + z∂x(δωz) − ∂yδvz − 1 Re∇2δωz = {∇ × ∇δΠ}z

• Problem:

deduce the fluctuations of fields δvz(r, t)δvz(r′, t) and δωz(r, t)δωz(r′, t) from the stochastic properties of the thermal noise.

• Non-equilibrium enhancement due to mode-coupling

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Nonequilibrium enhancement of thermal noise

In nonequilibrium systems thermal noise is generically amplified due to mode

• coupling. Illustration with a temperature gradient.

T0

• Fluctuations in δvz “mix” regions with

different (local) temperature.

• Advective term ∇T0 δvz in hydrodynamic

equations.

• Local version of FDT.
• Problem:

What are the fluctuations maximally enhanced?

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Amplification of thermal fluctuations in plane Couette

Most important effect: vorticity fluctuations enhancement due to coupling with Orr-Sommerfeld.

0.25 0.50 1.0 1.5 2.0 2.5 3.0 0.50 1.0 2.5 5.0 7.5 10 15 20 25 30 35

1 2 3 4 5 1 2 3 4 5

C(NE)

z

(q||)

Re=100

qx qy

1 2 3 4 5 1 2 3 4 5

streamwise W(NE)

z

(q||) qx

• Wall-normal

vorticity fluctuations with spanwise modulation

• When

towed by the flow: streaks

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Voticity fluctuations with spanwise wave vector

0.1 1 10 100 10

• 5

10

• 4

10

• 3

10

• 2

10

• 1

10 10

1

∆Wzz

(NE)(q)

q

Spanwise wave vector (qx = 0), great simplification. Relatively simple analytical expressions are possible.

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Thermal noise in real space

Generation of streaks

• Thermal noise adopts a

streak form

• Streaks are maximally

amplified

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Thank you for your attention!!

To learn more. . .

• J. M. Ortiz de Z´

arate, J. V. Sengers Hydrodynamic Fluctuations in fluids and fluid mixtures Elsevier, 2006

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