Mass Transfer in Turbulent Flow

ChEn 6603

References:• S. B. Pope. Turbulent Flows. Cambridge University Press, New York, 2000.

• D. C. Wilcox. Turbulence Modeling for CFD. DCW Industries, La Caada CA, 2000.

• H. Tennekes and J. L. Lumley. A First Course in Turbulence. MIT Press, Cambridge, MA, 1972.

• R. O. Fox. Computational Models for Turbulent Reacting Flows. Cambridge University Press, 2003.

1Friday, April 15, 2011

William H. Cabot and Andrew W. CookNature Physics 2, 562 - 568 (2006)

Rayleigh-Taylor instability (DNS calculation) Photograph of Jupiter from Voyager

~15,000 Miles(2 Earth diameters)

Mixing in reacting flow (DNS)

~ 6 cm

Methane pool fire

2Friday, April 15, 2011

Energy balance perspective.• Consider steady, isothermal, fully developed turbulent flow in a horizontal

pipe‣ Increasing pressure drop does not increase flow rate proportionally. Why? Where is

the energy going? How?

‣Work done by pressure forces balanced by work done by viscous forces

• Energy provided at “large” scales, dissipated at “small” scales.‣ Length scales reduce to meet demand of energy balance.

‣ Smaller length scales ⇒ steeper gradients ⇒ more dissipation.

Origins of Turbulence

pressure work viscous dissipation

∂ρk

∂t+∇ · (ρkv) = −v ·∇ · τ − v ·∇p + ρ

n�

i=1

ωiv · fi

0 = v ·∇p + v ·∇ · τ

∆pKinetic energy

equation:What assumptions?

3Friday, April 15, 2011

Velocity Length ScalesL - largest length scale (m)η - smallest length scale (m)U - velocity at L-scale (m/s)

ν - kinematic viscosity (m2/s) = µ/ρε - kinetic energy dissipation rate (m2/s2· s-1)

Most kinetic energy is contained in “large” length scales (L). It is dissipated primarily at “smallest” (Kolmogorov) length

scales (η) by molecular viscosity (ν).

Can we form a length scale from ε and ν?

η ∝�

ν3

�

�1/4

� ∝ UU

L/U

kinetic energy

“integral” or “large” time scale

Note: ε doesn’t depend on ν. ν just determines the smallest

length scale in the flow.

η ∝�

ν3

�

�1/4

∝ L1/4� ν

U

�3/4

L

η∝

�LU

ν

�3/4

= Re3/4 Key result! Tells us how length-scales separate!

4Friday, April 15, 2011

Scalar Length ScalesℓB - smallest scalar length scale (Batchelor scale)

L

�B=

L

η

η

�B∝ Re3/4Sc3/4L

�B=

L

η

η

�B∝ Re3/4Sc1/2

Sc ≡ ν

DSc > 1“mixing paint” - ℓB <η - scalar only feels straining

from smallest velocity scales.(mass diffuses slower than momentum)

Sc < 1ℓB>η - at ℓB, there are still velocity fluctuations,

but the scalar field is uniform (mass diffuses faster than momentum)

�2B ∝ Dt

�B ∝ D1/2�ν

�

�1/4

�B

η∝ D1/2

�ν

�

�1/4�

ν3

�

�−1/4

∝�

D

ν

�1/2

= Sc−1/2

Form a time scale from the “Kolmogorov” time scale

(i.e. from ν and ε).

Relevant parameters are D, ε. (ν only dominant near η).�B ∝

�D3

�

�1/4

�B

η∝

�D

�

�1/4 �ν3

�

�−1/4

∝�

D

ν

�3/4

= Sc−3/4

Gases: Sc ≈ 1, Liquids: Sc ~ 103.5Friday, April 15, 2011

Sc→∞

http://sensitivelight.com/smoke2

6Friday, April 15, 2011

Solution OptionsDirect Numerical Simulation (DNS)• Resolve all time/length scales by solving the governing equations directly.• Restricted to small problems.• Cost scales as Re3 for turbulence alone! (Species with Sc>1, and/or

complex chemistry could further increase cost)‣(L/η∼Re3/4, 3D, time)

Large Eddy Simulation (LES)• Resolve “large” spatial & temporal scales• Model “small” (unresolved) time/space scales

Reynolds-Averaged Navier Stokes (RANS)• Time-averaged.• Describes only mean features of the flow.

Model all effects of the flow field• Useful only for some classes of problems (usually interfaces like walls)• Commonly done in heat transfer & mass transfer (also for some problems

involving aerodynamics )

Incr

ease

d M

odel

ing

7Friday, April 15, 2011

Time-Averaging (RANS)Constant density, viscosity:

� ∞

0∇ · vdt = 0

∇ ·�� ∞

0vdt

�= 0

∇ · v = 0

Continuity:

Momentum:

Definition of time-average:

index (Einstein) notation: − ∂

∂xj

� ∞

0

vivjdt = − ∂

∂xjvivj .

∇ · v = 0∂v∂t

= −∇ · (v ⊗ v)− 1ρ∇p + ν∇2v

� ∞

0

∂v

∂tdt = −∇ ·

� ∞

0vv dt− 1

ρ∇p+ ν∇2v

0 = −∇ ·� ∞

0vv dt− 1

ρ∇p+ ν∇2v

∂

∂xjvivj +

1

ρ

∂p

∂xi+ ν

∂

∂xj

∂vi∂xj

φ ≡ limT→∞

1

T

� t0+T

t0

φ(t) dt

8Friday, April 15, 2011

The Closure Problem

vivj = (vi + v�i)(vj + v�

j)

= vivj + viv�j + v�

ivj + v�iv

�j

vivj = vivj + v�iv

�j

For large Re, µt≫µ (molecular viscosity is negligible).

“Fluctuating” component

φ� = 0 ¯φ = φφϕ� = 0

φ� ≡ φ− φ

Model this term using a “gradient diffusion” model.

∂

∂xj

�v�iv

�j

�≈ −µt

ρ

∂

∂xi

∂

∂xjvj

∂

∂xj(vivj) +

1

ρ

∂p

∂xi+ ν

∂

∂xj

∂vi∂xj

+∂

∂xj

�v�iv

�j

�= 0

∂

∂xj(vivj) +

1

ρ

∂p

∂xi+ 1

ρ (µ+ µt)∂

∂xj

∂vj∂xj

= 0

9Friday, April 15, 2011

Time-Averaged Species Equations

@ Large Re, ji,turb ≫ji

(molecular diffusion is negligible)

A very difficult problem...∇ · ωiv +1ρ∇ · ji = si/ρ

∇ · (ωiv) +1ρ∇ · ji +∇ ·

�ω�

iv��

= si/ρ

∇ · (ωiv) +1ρ∇ ·

�ji + ji,turb

�= si/ρ

• D°turb - turbulent diffusivity (for mass flux relative to mass avg. velocity)

• µturb - eddy viscosity• Typically, Scturb is specified.

MODEL for turbulent species diffusive flux:

∇ · (ωiv) +1ρ∇ · ji,turb = si/ρ

Multicomponent effects are irrelevant at sufficiently high Re.

constant properties & density...

ji,turb = −ρD◦turb∇ωi

Scturb =νturbD◦

turb

=µturb

ρD◦turb

10Friday, April 15, 2011

Spatial Averaging (LES)

G(x) - filter kernel functionremoves “high wavenumber” components of ϕ.

• Filter governing equations. (similar procedure as for RANS, but a little more complicated).

• Write models for unclosed terms.• Solve filtered equations (for filtered variables).• Provides time-varying solutions at a “coarse”

level.

Courtesy R.J. McDermott

Courtesy R. J. McDermott

φ ≡� ∞

−∞φ(x)G(x)dx

¯φ �= φ

φ� �= = 0

φϕ� �= 0

11Friday, April 15, 2011

Variable Density∂ρωi

∂t= −∇ · ρωiv −∇ · ji + si.

Favre-averaging (RANS)Favre-filtering (LES) φ ≡ ρφ

ρ−→ ρφ = ρφ

Leads to many additional complications, most of which are typically ignored...

example: ji =n�

k=1

ρDoik∇ωk

?≈ ρn�

k=1

Doik∇ωk

LES: ifΔ = “filter width”

∆� �B , ∆� η then ji,turb � ji

∂ρωi

∂t= −∇ · ρ�ωiv −∇ · ji + si

= −∇ · ρωiv −∇ ·�ji + ji,turb

�+ si

12Friday, April 15, 2011