From Hydrodynamic instability to chaotic mixing - Nektar++ · 2019. 6. 13. · WARSAW UNIVERSITY OF...

WARSAW UNIVERSITY OF TECHNOLOGY

Nektar++ Workshop 2019Nektar++ Workshop 2019

From Hydrodynamic instability to chaotic

mixingwith NEKTAR++

Stanis law Gepner, Nikesh, Jacek Szumbarski

Institute of Aeronautics and Applied Mechanics, Warsaw University of Technology



Plan of the Presentation

Introduction

Mixing and mixing quantification

Hydrodynamic stability in a corrugated channel

Nonlinear saturation

Enhancement of transport

Experiment



IntroductionRationale

• Pattern based flow control

• Mass and heat transfer intensification

• Cooling of microelectronics

• Blood oxygenations, DNA screening microarray

• Drag reduction and roughness modelling



What is mixing?’If you have to ask what jazz is, you’ll never know.’ Louis Armstrong

• In fluids it is a two-stage process (Eckart, 1948)1

• mechanical stirring• inter-material diffusion

• Stirring produces small scales (layers) in a stretching and folding action muchlike the horseshoe transformation, that can be rapidly smoothed by diffusion

• Turbulization is effective, but not always applicable

1Eckart, C. 1948. An Analysis of the Stirring and Mixing Processes in Incompressible Fluids. J.Mar. Res., 7, 265–275.



Chaotic advection and mixing

• Simple, low Reynolds number flow lead to the onset of Lagrangian chaos (Aref1984)1

• Motion described by ẋ = u(x, t)

• Or ẋ = ∂Ψ/∂y , ẏ = −∂Ψ/∂x , equivalent to a single-degree-of-freedomsystem

1 Two dimensional, steady flows result in integrable advection equations meaningthat particle trajectories are regular.

2 Unsteady flows in two dimensions and steady or unsteady flows in threedimensions may result in non-integrable advection equations leading to chaoticparticle trajectories.

1Aref, Hassan. 1984. Stirring by chaotic advection. Journal of Fluid Mechanics, 143, 1–21.



Double gyre example



Quantifying mixingIs not well agreed upon

• Strange eigenmodes - eigen function to the advection-diffusion process.Leading AD operator mode corresponds to the slowest decaying initialdistribution of the advected scalar. This mode is a representation of structuresthat are persistent under the flow, with the corresponding eigen valuedetermining the exponential decay rate. Consequently, it imposes an upperlimit on the efficiency of the stirring

• Dynamical system analogy (Ottino 1989)1

• strives to quantify the stirring action by the amount of stretching experienced byindividual fluid parcels traced in the advection field

• based on the notion of the specific rate of stretch

• Norms2: L2 and H−1

• L2: Variance Varθ = 1

|Ω|

∫Ωθ2dΩ)

• Mix-norm based on negative index Sobolev norms

1Ottino, Julio M. 1989. The kinematics of mixing: stretching, chaos, and transport. Vol. 3.Cambridge university press.

2Thiffeault, Jean-Luc. “Using multiscale norms to quantify mixing and transport.” (2012).



How to invoke chaotic advection without turbulization?

• Baffles, obstacles, bends

• External actuation

• Why not hydrodynamic stability?

• Optimal initial perturbation in Poiseuille flow (Vermach 20181, Foures 20142)

Mixers for the Plastics Processing Industry, Sulzer Chemtech, https://sulzer.com/

1Foures, D., Caulfield, C., & Schmid, P. (2014). Optimal mixing in two-dimensional planePoiseuille flow at finite Péclet number. Journal of Fluid Mechanics, 748, 241-277.

2Vermach, L., & Caulfield, C. (2018). Optimal mixing in three-dimensional plane Poiseuille flowat high Péclet number. Journal of Fluid Mechanics, 850, 875-923.



What everybody knowsabout hydrodynamic stability

• We look for eigenfunctions of the linearised NS operator

• Those could be either attenuated or amplified, stationary or travelling

• In the smooth channel case the critical perturbation is the 2D TS wave thatbecomes unstable at Recr = 5772, δ = 1.02 and travels downstream withfrequency σr ≈ 0.27 and phase speed vp = σr/βcr ≈ 0.26

u: v:



Longitudinal grooves

• S - corrugation amplitude

• Re = ULν

- reference flow, Qr =43

• n - number of corrugations in computations

• α - spanwise wave number → λα =2πα

• β - streamwise wave number

(α, S ,Re, n, β)



2D base flow and stability

0.5 1 1.5 2 2.5 3

0.2

0.4

0.6

0.8

60 6570

80 Re cr=100

160

200

350500

0.720

.93

QQr

= 1

1.17

0.88

0.6

vp = 0.4

α

S



Nonlinear saturation

α = 1, S = 0.4,Re = 70, α = 1, β = 0.4

0 1 2 3 410−20

10−14

10−8

10−2∼

e2∗σ it

k = 0 1

2

4

k

t · 10−3

Ek

20 40 6010−20

10−14

10−8

10−2

Re=

60

65

70

80

90, 100, 110, 120

150

200 250

300

Re

k

Ek



Nonlinear Saturationα = 1, S = 0.4, α = 1, β = 0.4

0.22 0.24 0.26 0.28 0.3 0.320

1

2

3

4

·10−2

80

90

100

110

120

150

200

250Re = 300

Base flow

Re

E0

N∑

k=1

Ek

Re = 70

−0.1

0

0.1

u

0.8 1 1.2

−0.2

0

0.2

w

v·

102



Nonlinear SaturationFlow pattern α = 1, S = 0.4,Re = 80, α = 1, β = 0.4

z =λβ4

z =λβ2

z =3λβ4

z = λβ



Nonlinear Saturation

Mq =1Ω

∫Ω

(qs − qb)2dΩ

0 100 200 300

0.9

0.95

1

1.05QQr

Re

QQr

50

100

150

200

Reb

base flow

100 200 300

10−4

10−3

10−2

10−1

Mu

Mv

Mw

Re

MqE0




Re = 60




Re = 70




Re = 100




Re = 250



Quantification of mixingHomogenization of a passive scalar




Re = 100, Sc = 10, θ = 0



Interface areaHomogenization of a passive scalar

0 50 1000

2

4

6

Re = 60

200

Re

Sc = ν/D = 1

Aθ=0(t)

Aθ=0(t=0)

0 50 100

Re = 60

6570

80

90

100

120

150200 Sc = 5

t

0 50 1000

2

4

6

Re = 60

65

70

8090

100

120

150

200Sc = 10

Aθ=0(t)

Aθ=0(t=0)




0 50 1000

0.2

0.4

0.6

0.8

1

Re = 60

200

Re

Sc = ν/D = 1

Var

(θ)

0 50 100

Re = 60

200

Re

Sc = 5

t

0 50 1000

0.2

0.4

0.6

0.8

1

Re = 60

200

Re

Sc = 10

Var

(θ)

0 50 1000

0.2

0.4

0.6

0.8

1Re = 60

65

7080

90

Re ≥ 100

Var(θ)

Var(θ) b

ase

0 50 100

60

65

70

Re ≥ 80

t

0 50 1000

0.2

0.4

0.6

0.8

160

65

70

Re ≥ 80

Var(θ)

Var(θ) b

ase




100 150 2000

0.2

0.4

0.6

0.8

1

Sc = 15

10

Re

Var(θ,t=120)

Var(θ,t=120) b

ase



Poincaré sectionsL = 320h, intersections every 10h



Poincaré sectionsL = 320h, intersections every 10h

Re = 60 Re = 65 Re = 80

Re = 100 Re = 150 Re = 200



Experiment

L = 0.3mm, t = 0.12s, Re = 120

0Szumbarski, Jacek, Blonski, Slawomir, & Kowalewski, Tomasz. 2011. Impact oftransversely-oriented wallcorrugation on hydraulic resistance of a channel flow. Archive ofMechanical Engineering, 58(4), 441.

OutlineIntroductionMixing and mixing quantificationHydrodynamic stability in a corrugated channelNonlinear saturationEnhancement of transportExperiment

From Hydrodynamic instability to chaotic mixing - Nektar++ · 2019. 6. 13. · WARSAW UNIVERSITY OF...

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