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# Pole vaulting of long fibres in a turbulent wall-bounded flow

Christophe Brouzet, J Bec, Christophe Henry

► To cite this version:

Christophe Brouzet, J Bec, Christophe Henry. Pole vaulting of long fibres in a turbulent wall-bounded flow. Complex Days, Rudy Valette; Claire Michel, Feb 2024, NICE, France. hal-04806528

HAL Id: hal-04806528

https://hal.science/hal-04806528v1

Submitted on 27 Nov 2024

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Distributed under a Creative Commons Attribution 4.0 International License
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Pole-vaulting of long fibres
in a turbulent wall-bounded flow

C. Brouzet

in collaboration with J. Bec & C. Henry
## ● Fibres in turbulent flows:

### - environmental applications:

<figure><img src="image_4.png" /><figcaption>* plastic pollution</figcaption></figure>

Allen et al. (2019)

<figure><img src="image_5.png" /></figure>

Chubarenko et al. (2018)

transport ?
deposition ?
fragmentation ?

<figure><img src="image_6.png" /><figcaption>* natural fibre balls</figcaption></figure>

<figure><img src="image_7.png" /></figure>

Verhille et al. (2017)

aggregation ?
## ● Fibres in turbulent flows:

- industrial applications:

* paper making &
material fabrication

<figure><img src="image_8.png" /></figure>

Lundell et al. (2011)

orientation ?
aggregation ?

* water intakes

<figure><img src="image_9.png" /></figure>

Credits: EDF

<figure><img src="image_10.png" /></figure>

Redlinger-Pohn et al. (2020)

deposition ?
clogging ?
● Fibres in turbulent flows:

- homogeneous & isotropic turbulence:“idealised case”

* rigid fibres (mainly)

* flexible fibres
## Fibres in turbulent flows:

- homogeneous & isotropic turbulence: “idealised case”

* rigid fibres (mainly)

* flexible fibres

- wall-bounded turbulent flows: inhomogeneous & anisotropic

* rigid & short fibres (only)

long flexible fibres in wall-bounded turbulence

=> experiments & simulations
● Fluid flow:

- open-source spectral solver Channelflow 2.0

$$ * L_x \times L_y \times L_z = 4\pi h \times 2h \times 4\pi h/3 $$

* periodic boundary conditions in x and z

* constant bulk velocity imposed

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>
● Fluid flow:

- open-source spectral solver Channelflow 2.0

$$ * L_x \times L_y \times L_z = 4\pi h \times 2h \times 4\pi h/3 $$

* periodic boundary conditions in x and z

* constant bulk velocity imposed

performed by Jérémie Bec

$$ Re = 5600 $$

<figure><img src="image_13.png" /></figure>
● Fluid flow:

- open-source spectral solver Channelflow 2.0

$$ * L_x \times L_y \times L_z = 4\pi h \times 2h \times 4\pi h/3 $$

* periodic boundary conditions in x and z

* constant bulk velocity imposed

- mean velocity profile:

performed by Jérémie Bec

$$ Re = 5600 $$

<figure><img src="image_14.png" /></figure>
● Fluid flow:

- open-source spectral solver Channelflow 2.0

$$ * L_x \times L_y \times L_z = 4\pi h \times 2h \times 4\pi h/3 $$

* periodic boundary conditions in x and z

* constant bulk velocity imposed

- mean velocity profile:

* wall shear stress: $\tau_w$

performed by Jérémie Bec

$$ \text{Re} = 5600 $$

<figure><img src="image_15.png" /></figure>
● Fluid flow:

- open-source spectral solver Channelflow 2.0

$$ * L_x \times L_y \times L_z = 4\pi h \times 2h \times 4\pi h/3 $$

* periodic boundary conditions in x and z

* constant bulk velocity imposed

- mean velocity profile:

* wall shear stress: $\tau_w$

* wall units rescaling: $Re_\tau = 180$

performed by Jérémie Bec

$$ Re = 5600 $$

<figure><img src="image_16.png" /></figure>
● Fluid flow:

- open-source spectral solver Channelflow 2.0

$$ * L_x \times L_y \times L_z = 4\pi h \times 2h \times 4\pi h/3 $$

* periodic boundary conditions in x and z

* constant bulk velocity imposed

- mean velocity profile:

* wall shear stress: $\tau_w$

* wall units rescaling: $Re_τ = 180$

* the law of the wall

performed by Jérémie Bec

$$ Re = 5600 $$

<figure><img src="image_17.png" /></figure>
● **Fibres:**

- inextensible and inertialess

- one-way coupling

performed by Jérémie Bec
● **Fibres:**

performed by Jérémie Bec

- inextensible and inertialess

- one-way coupling

- over-damped slender-body theory: Cosserat equation for $X(s, t)$

$$ \partial_t X = u(X, t) + \frac{c}{8\pi\eta} [\mathbb{I} + \partial_s X \partial_s X^\top] [\partial_s (T \partial_s X) - EI \partial_s^4 X] $$

with $|\partial_s X|^2 = 1$

<figure><img src="image_18.png" /></figure>
● **Fibres:**

performed by Jérémie Bec

- inextensible and inertialess

- one-way coupling

- over-damped slender-body theory: Cosserat equation for $X(s, t)$

$$ \partial_t X = u(X, t) + \frac{c}{8\pi\eta} [\mathbb{I} + \partial_s X \partial_s X^\top] [\partial_s(T\partial_s X) - EI\partial_s^4 X] $$

with $|\partial_s X|^2 = 1$

- elastic length: $\ell_E \equiv (cEI\tau_v/(8\pi\eta))^{1/4} \approx 7.1\delta_v$

<figure><img src="image_19.png" /></figure>
● **Fibres:**

- inextensible and inertialess

- one-way coupling

- over-damped slender-body theory: Cosserat equation for $X(s, t)$

$$ \partial_t X = u(X, t) + \frac{c}{8\pi\eta} [\mathbb{I} + \partial_s X \partial_s X^\top] [\partial_s(T\partial_s X) - EI\partial_s^4 X] $$

$$ \text{with} \quad |\partial_s X|^2 = 1 $$

- elastic length: $l_E \equiv (cEI\tau_v/(8\pi\eta))^{1/4} \approx 7.1\delta_v$

- wall contact: soft boundary approach

performed by Jérémie Bec

<figure><img src="image_20.png" /></figure>
● Fibre orientation:

<figure><img src="image_21.png" /></figure>

- in the centre: relatively isotropic

- close to the wall: aligned with the flow direction

- small variations with length
* Fibre tumbling rate:

<figure><img src="image_22.png" /></figure>

- in the centre:
  * mainly independent of length
  * consistent with HIT

- close to the wall:
  * strong variations with length
  * the longer, the higher the rate
* Fibre tumbling rate:

<figure><img src="image_23.png" /></figure>

- in the centre:
* mainly independent of length
* consistent with HIT

- close to the wall:
* strong variations with length
* the longer, the higher the rate

<figure><img src="image_24.png" /></figure>

Baker & Coletti (2022)
## ● Fibre-wall interactions:

- rebound on the wall

* high tumbling rate

* “pole vaulting”

<figure><img src="image_25.png" /></figure>

$$ \ell^+ = 45 $$

$$ \ell/h = 0.25 $$
● Fibre-wall interactions:

- rebound on the wall

* high tumbling rate

* “pole vaulting”

$$ \ell^+ = 45 $$

$$ \ell/h = 0.25 $$

<figure><img src="image_26.png" /></figure>

- in the literature:

* laminar case:

Stover & Cohen (1990)

Moses et al. (2001)

<figure><img src="image_27.png" /></figure>

Stover & Cohen (1990)

* turbulent case:

Capone et al. (2017)

Alipour et al. (2022)

Baker & Coletti (2022)

<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /></figure>

Stover & Cohen (1990)
## ● Fibre density distribution:

<figure><img src="image_31.png" /></figure>

$$ \rho(y) = \langle \delta(X_2 - y) \rangle $$

## ■ inhomogeneous distribution:

* higher concentration in the center

* depleted region close to the wall

* increases with fibre length

due to wall collisions ?
● Fibre density distribution:

<figure><img src="image_32.png" /></figure>

$$ \rho(y) = \langle \delta(X_2 - y) \rangle $$

inhomogeneous distribution:

* higher concentration in the center

* depleted region close to the wall

* increases with fibre length

high collision probability when tumbling if $y_{CoM}^{+} < \ell^{+}/2$

due to wall collisions ?
● Fibre density distribution:

<figure><img src="image_33.png" /></figure>

- inhomogeneous distribution:

* higher concentration in the center

* depleted region close to the wall

* increases with fibre length

- high collision probability when tumbling if $y_{CoM}^{+} < l^{+}/2$

due to wall collisions !
● Fibre net flux:

<figure><img src="image_34.png" /></figure>

- fibres net flux $\Phi$:

* velocity X concentration

* compared to fluid flux $\Phi_0$

* increases with fibre length

- enhancement:

* for fibre length within the log-layer

* log law scaling?

<figure><img src="image_35.png" /></figure>
● Main results:

- first study on long flexible fibres in turbulent channel flow

* numerical simulations

* $\ell/h \in [0.08 - 1.5]$

- fibre-wall collisions: “pole vaulting”

* high tumbling rates

* depleted regions

increase with fibre length !

- effect on the fibre net flux:

* higher fibre concentration in the centre

* enhancement of the flux up to 6%
## Experiments at INPHYNI:

<figure><img src="image_36.png" /></figure>

$L \approx 4.5 \text{ m}$

- turbulent channel flow:

* rectangular cross-section

* gravity driven

- 3D reconstruction:

* orientation / deformation

* 4 high-speed cameras
  (20 000 fps at 1 MPixels)

<figure><img src="image_37.png" /></figure>

<figure><img src="image_38.png" /></figure>
## Experiments at INPHYNI:

<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /></figure>

- turbulent channel flow:

* rectangular cross-section

* gravity driven

- 3D reconstruction:

* orientation / deformation

* 4 high-speed cameras
  (20 000 fps at 1 MPixels)

- Fundings:

* IDEX (Académie 2)

* CNRS

* INPHYNI

* Région Sud
## Experiments at INPHYNI:

<figure><img src="image_41.png" /></figure>

- turbulent channel flow:

* rectangular cross-section

* gravity driven

- 3D reconstruction:

* orientation / deformation

* 4 high-speed cameras
  (20 000 fps at 1 MPixels)

- Design & manufacturing:

* Sandra Bosio

* Cyrille Claudet

* Christophe Pitiot

* Florian Zumbo

* Magali Varlet-Dusaucy
# Questions ?
● **Fibres:**

- inextensible and inertialess

- one-way coupling

- over-damped slender-body theory:

* discretisation in $N_s$ points of length $\delta s^+ = l^+/N_s \approx 0.9$

* Cosserat equation for $X(s, t)$

$$ \partial_t X = u(X, t) + \frac{c}{8\pi\eta} [\mathbb{I} + \partial_s X \partial_s X^\top] [\partial_s(T\partial_s X) - EI\partial_s^4 X] $$

$$ \text{with } |\partial_s X|^2 = 1 $$

- elastic length: $l_E \equiv (cEI\tau_v/(8\pi\eta))^{1/4} \approx 7.1\delta_v$

- wall contact: soft boundary approach
● **Fibres:**

- wall contact: soft boundary approach

- fibres are:

* allowed to penetrate the wall

* strongly repulsed by an effective force

- fluid velocity outside the domain $u = (0, v, 0)$

$$ \text{bottom} \quad \begin{cases} v = -\gamma(y + h) \\ \text{for } y < -h \end{cases} $$

with $\gamma = 0.5/\Delta t$

$$ \text{top} \quad \begin{cases} v = -\gamma(y - h) \\ \text{for } y > h \end{cases} $$