# On the fluctuations generated by a dispersed phase

Frederic Risso

► To cite this version:

Frederic Risso. On the fluctuations generated by a dispersed phase. The 11th International Conference on Multiphase Flow, ICMF 2023, Apr 2023, Kobe, Japan. hal-04736962

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<figure><img src="image_1.png" /></figure>
# On the Fluctuations Generated by a Dispersed Phase

Frédéric Risso

<figure><img src="image_2.png" /></figure>
In dispersed two phase-flows...

<figure><img src="image_3.png" /><figcaption>bubbly flow</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>particulate flow</figcaption></figure>

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

... liquid velocity always experiences random fluctuations
Those fluctuations depend on many parameters

$$ \text{Fluid inertia versus Fluid viscosity: } Re = \frac{\rho_c U d}{\mu_c} $$

$$ \text{Body inertia versus Fluid viscosity: } St = \frac{\rho_d U d}{\mu_c} $$

$$ \text{Body-phase volume fraction: } \alpha $$

$$ \text{Body shape: } \chi \text{ (aspect ratio)} $$

$$ \text{Deformability - breakup - coalescence: } We = \frac{\rho_d U^2 d}{\sigma} $$

$$ \text{Nature of the bodies: } \text{solid - liquid - gas} $$

Specific studies seem required for each case
Those fluctuations depend on many parameters

**Fluid inertia versus Fluid viscosity:** Re = $\frac{\rho_c U d}{\mu_c}$

**Body inertia versus Fluid viscosity:** St = $\frac{\rho_d U d}{\mu_c}$

**Body-phase volume fraction:** α

**Body shape:** χ (aspect ratio)

**Deformability - breakup - coalescence:** We = $\frac{\rho_d U^2 d}{\sigma}$

**Nature of the bodies:** solid - liquid - gas

However, similar statistical signatures are observed in contrasted configurations
# Exponential PDFs of liquid velocity

sheared particulate flow

<figure><img src="image_6.png" /></figure>

Re ≪ 1
1% ≤ α ≤ 20%

normalized velocity fluctuations (shear direction)

Abbas et al, Phys. Fluids 2006

liquid-solid fluidized bed

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

Re ≈ 700
14% ≤ α ≤ 42%

normalized velocity fluctuations (horizontal direction)

Alméras et al, Phys. Rev. Fluids 2021

bubble column

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

Re ≈ 700
α ≤ 5%

normalized velocity fluctuations (horizontal direction)

Riboux et al, J. Fluid Mech. 2010
$k^{-3}$ spectral subrange

<figure><img src="image_9.png" /><figcaption>3D bubble column (turbulent)</figcaption></figure>

Riboux et al, J. Fluid Mech. 2010

<figure><img src="image_10.png" /><figcaption>2D bubble column (not turbulent)</figcaption></figure>

Bouche et al, J. Fluid Mech. 2014
* Some properties rely on the exact nature of the flow

* Others may depend on emergent statistical effects

“More is different” Philip Warren Anderson, *Science* 1972

Need to analyze the mechanisms by distinguishing dynamic effects from statistical ones
Various mechanisms causing fluctuations
<figure><img src="image_11.png" /></figure>

Continuous Phase

mean flow

agitation

Dispersed Bodies

mean flow

agitation
Pressure gradient

<figure><img src="image_12.png" /></figure>

Buoyancy

Continuous Phase

mean flow

agitation

Dispersed Bodies

mean flow

agitation
<figure><img src="image_13.png" /><figcaption>Single-phase flow</figcaption></figure>
<figure><img src="image_14.png" /></figure>
<figure><img src="image_15.png" /><figcaption>Turbulent flow
with small bodies</figcaption></figure>
<figure><img src="image_16.png" /></figure>

Wake-Induced
Fluctuations (WIF)
<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>
Heterogeneous swarm

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

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

<figure><img src="image_22.png" /></figure>
<figure><img src="image_23.png" /><figcaption>General case</figcaption></figure>
## Homogeneous BIA

<figure><img src="image_24.png" /></figure>
## Homogeneous BIA

### Occurrences of homogeneous falling or rising body swarm

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

<figure><img src="image_26.png" /></figure>
<figure><img src="image_27.png" /><figcaption>Large-Re Bubble Column</figcaption></figure>

<figure><img src="image_28.png" /></figure>
# Sedimenting suspensions / Fluidized beds

increasing
volume
fraction

isolated
Body

$3.5\% < \alpha_{low} < 45\%$

heterogeneous
regime

homogeneous
regime

jamming

$0.95 \alpha_{pack} \le \alpha_{pack} \le 60\%$

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

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

characteristic
length d, $\Lambda$

drift velocity $V_0$

fluctuating
velocity

$\Lambda$
disturbed-flow length

$U_r < V_0 \Leftrightarrow \alpha$
hindering effect

$u'^2 \propto \alpha V_0^2$
at large Re
Homogeneous BIA

Focus on High-Re Homogeneous Body-Induced Agitation
<figure><img src="image_31.png" /><figcaption>Wave-Number Spectra (loglog)</figcaption></figure>

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

Riboux et al, J. Fluid Mech. 2010

► Asymmetric and anisotropic

► Non Gaussian with exponential tails

► Spectrum independent of α

$$ \frac{E}{\langle u'^2 \rangle \Lambda} \propto (\Lambda k)^{-3} $$

with Λ independent of α
<figure><img src="image_33.png" /></figure>

► How can we study them separately ?
Separation of the two contributions: case of fixed bodies

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

1. Perturbations induced in the body vicinity (LBD): time average $u_{LBD}(\mathbf{x})$

2. Flow instabilities (BIT): $u_{BIT}(t) = u(\mathbf{x}, t) - u_{LBD}(\mathbf{x})$

$$ \langle u^2 \rangle = \langle u_{BIT}^2 \rangle + \langle u_{LBD}^2 \rangle $$
**Random array of spheres**

Amoura et al, J. Fluid Mech. 2017

<figure><img src="image_35.png" /></figure>

Total fluctuations

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

**Coarse-grained simulations**

Riboux et al, J. Fluid Mech. 2013

LBD

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

BIT

<figure><img src="image_38.png" /></figure>

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

► dominated by wakes

► homogeneous
# Separation of the two contributions: case of fixed bodies

## PDFs of *normalized* fluctuations

Total

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

LBD

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

BIT

<figure><img src="image_42.png" /></figure>

black: exp. with bubbles

red: simulations with bubbles

blue: exp. fixed spheres

similar to real bubble swarm

<figure><img src="image_43.png" /></figure>

anisotropic
exponential

<figure><img src="image_44.png" /></figure>

almost isotropic
exponential

<figure><img src="image_45.png" /></figure>
Separation of the two contributions: case of fixed bodies

Spatial spectra ($200 \le \text{Re} \le 1000$)

<figure><img src="image_46.png" /></figure>

thick lines: bubble swarm

dashed lines: simulations with bubbles

thin lines: fixed spheres

* structure independent of Re

* k⁻³ subrange for the two kinds of fluctuations
Two contributions to BIA: LBD and BIT

* LBD: superposition of individual bubble perturbations

* anisotropic, inhomogeneous

* non Gaussian

* k⁻³ spectrum

* dominated by average wake

* BIT: real bubble-induced turbulence

* almost isotropic and homogeneous

* non Gaussian

* k⁻³ spectrum

* generated by instability of randomly distributed interacting wakes
# Statistical modeling of LBD

## Probability density functions and moments

<figure><img src="image_47.png" /></figure>
Liquid velocity induced by a bubble within a swarm ($\alpha > 0.5\%$)

Risso & Ellingsen, J. Fluid Mech. 2002

<figure><img src="image_48.png" /></figure>

► The wake is strongly attenuated
# Body-wake attenuation in a swarm

Risso et al, Phil. Trans. R. Soc. 2008

<figure><img src="image_49.png" /><figcaption>Average wake in a fixed array of spheres</figcaption></figure>

<figure><img src="image_50.png" /></figure>

<figure><img src="image_51.png" /></figure>

► Exponential decay of the wake as $\exp(-z/\Lambda)$
# LBD modeling: Agitation resulting from the sum of individual disturbances

<figure><img src="image_52.png" /></figure>

## Assumptions:

* body locations $x_i$ independent of each other

* velocity disturbances $\tilde{v}_i$ independent of each other

$$ \tilde{u} = \sum_{i=1}^{N} \tilde{v}(x_i) \qquad \begin{array}{l} \tilde{u} = u/V \\ \tilde{v} = v/V \end{array} $$

* Potential flow around an ellipsoidal body (depends only on the aspect ratio)

* The wake behind the body

$$ \left\{ \begin{array}{l} \tilde{v}_x(x, y, z) = 0 \\ \tilde{v}_y(x, y, z) = 0 \\ \tilde{v}_z(x, y, z) = \exp(-z/L) \exp(-(x^2 + y^2)/w^2) \end{array} \right. $$

## Sets of samples of fluctuations generated numerically
<figure><img src="image_53.png" /><figcaption>horizontal PDFs by potential flows</figcaption></figure>

<figure><img src="image_54.png" /></figure>

<figure><img src="image_55.png" /></figure>

## Exponential tails in the way towards the Gaussian
# LBD modeling: Application to bubbly flow ($\alpha \le 10\%$)

Risso, J. Fluid Mech. 2016

<figure><img src="image_56.png" /><figcaption>PDF of vertical fluctuations</figcaption></figure>

► Dominated by LBD (wakes)

<figure><img src="image_57.png" /><figcaption>PDF of horizontal fluctuations</figcaption></figure>

► Dominated by BIT
PDFs of vertical liquid fluctuations

<figure><img src="image_58.png" /></figure>

$$ \text{skewness: } S_k = \frac{\langle u_i^3 \rangle}{\langle u_i^2 \rangle^{3/2}} $$

* Body-Induced Turbulence

* Localized Body Disturbances
LBD modeling: solid-liquid fluidized beds at large Re ≈ 500 (14% ≤ α ≤ 42%)

<figure><img src="image_59.png" /></figure>

<figure><img src="image_60.png" /></figure>

► α < α_c: wakes are rare with more intense fluctuations

► α = α_c: wakes and interstices are equally probable

► α > α_c: interstitial regions are rare with more intense fluctuations
# Statistical modeling of LBD

## Wavenumber Spectra

<figure><img src="image_61.png" /></figure>
# LBD modeling: Agitation resulting from the sum of individual disturbances

Risso, Phys. Fluids 2011

<figure><img src="image_62.png" /></figure>

*Vn random variable of finite variance

*Xcn random variable uniformly distributed within an interval Δx

*λn random variable uniformly distributed in between λmin and λmax

The three random variables are independent

► LBD generates a k⁻³ spectrum
Statistical modeling of BIT

<figure><img src="image_63.png" /></figure>
Power budget in the spectral domain

$$ \begin{gathered} \frac{dE(k)}{dt} = \underbrace{P(k)}_{\text{work of } F_{b \to f}} + \underbrace{D(k)}_{\text{dissipation}} + \underbrace{T(k)}_{\text{nonlinear transfer}} \\ =0 \text{ steady flow} \end{gathered} $$
## Bubble wakes

$$f_w = \frac{V_w}{L_w}$$

<figure><img src="image_64.png" /></figure>

Navier-Stokes eqs. + forcing

<figure><img src="image_65.png" /></figure>

* Reproduce well the wake interactions

* Dissipation rate proportional to the wake shear rate: $\epsilon \propto \nu f_w^2$
Power budget in the spectral domain

<figure><img src="image_66.png" /></figure>

$$ \left. \begin{array}{l} -D(k) = 2\nu k^2 E(k) \\ E(k) = f^2 k^{-3} \end{array} \right\} \Rightarrow -D(k) = \underbrace{2\nu f^2 k^{-1}}_{\epsilon} $$
# Spectral dissipation density in $k^{-1}$ subrange

normalization by production scales: $\sigma$ and $\epsilon$

normalization by dissipation scales: $\eta = (\nu^3/\tilde{\epsilon})^{1/4}$ and $\epsilon$

<figure><img src="image_67.png" /></figure>

► First part of the $k^{-1}$ subrange

<figure><img src="image_68.png" /></figure>

► Second part of the $k^{-1}$ subrange
+ dissipation range
# Spectral dissipation density in $k^{-1}$ subrange
## Asymptotic matching of the shear-rate spectrum

Large scales
Shear-rate spectrum

depends only on $f, \sigma$ and $k$

$$S_{LS} = (f^2 \sigma) F_{LS}(k\sigma)$$

Small scales
Shear-rate spectrum

depends only on $f, \nu$ and $k$

$$S_{SS} = (f^2 \eta) F_{SS}(k\eta)$$

$$2\pi/\sigma \ll k \ll 2\pi\eta$$

$$S_{LS} = S_{SS} \propto f^2 k^{-1}$$

$$D = 2\nu S = 2\nu f^2 k^{-1} = \epsilon k^{-1}$$

$k^{-3}$ subrange:

* transition between production-dominated and dissipation-dominated range

* controlled by the shear-rate of the average wakes

more detail were given in yesterday Zamansky talk
# Implications for BIT modeling

Risso, Ann. Rev. Fluids Mech. 2018

$$ 
\left\{ 
\begin{array}{l}
E = \frac{\epsilon k^{-3}}{2\nu} \\
\\
\frac{E}{\langle u'^2 \rangle \Lambda} \propto (\Lambda k^{-3})
\end{array}
\right.
\qquad \epsilon = \nu \frac{\langle u'^2 \rangle}{\Lambda^2}
$$

$$ 
\left\{ 
\begin{array}{l}
\langle u'^2 \rangle = \alpha V^2 \\
\\
\epsilon = \alpha g V
\end{array}
\right.
\qquad \Lambda = \sqrt{\frac{\nu V}{g}}
$$

in agreement with experiments

$$ 
\left\{ 
\begin{array}{l}
\text{Re} = Vd/\nu \\
\\
C_d = 4gd/3V^2
\end{array}
\right.
\qquad \frac{\Lambda}{d} \propto \sqrt{C_d Re}
$$

<figure><img src="image_69.png" /><figcaption>Riboux et al
J. Fluid Mech. 2013</figcaption></figure>

<figure><img src="image_70.png" /><figcaption>Amoura et al
J. Fluid Mech. 2017</figcaption></figure>

<figure><img src="image_71.png" /></figure>
# RANS modeling of Body-Induced Agitation

du Cluzeau, Bois, Leoni, Toutant, Phys. Rev. Fluids 2022

$$ \text{Total fluctuation} \qquad \langle u'_i u'_j \rangle_{\text{BIA}} = \langle u'_i u'_j \rangle_{\text{LBD}} + \langle u'_i u'_j \rangle_{\text{BIT}} $$

$$ \text{LBD} \qquad \langle u'_i u'_j \rangle_{\text{LBD}} = \alpha u_r^2 \begin{pmatrix} 3/20 & 0 & 0 \\ 0 & 3/20 & 0 \\ 0 & 0 & 1/5 + \text{wake}(\text{Re}) \end{pmatrix} $$

$$ \text{BIT} \qquad \left. \begin{array}{l} \epsilon = \alpha g U_r = \nu \frac{\langle u^2 \rangle}{\Lambda^2} \\ \Lambda \propto \frac{d}{\sqrt{C_d \text{Re}}} \end{array} \right\} \begin{array}{l} \Pi_{ij_{\text{BIT}}} = \alpha g u_r f(\text{Re}) \delta_{ij} \\ \epsilon_{ij_{\text{BIT}}} = 0.35 C_d U_r \frac{\langle u'_i u'_j \rangle_{\text{BIT}}}{d} \end{array} $$

► Validated by DNS
Wall-turbulence
and bubbles

SIT + BIA

<figure><img src="image_72.png" /></figure>

du Cluzeau, Bois, Toutant,
J. Fluid Mech. 2019

Forced isotropic turbulence
and bubbles

SIT + BIA

<figure><img src="image_73.png" /></figure>

Pandey, Mitra, Perlekar,
Fluid Mech. 2022

Forced isotropic turbulence
with neutrally-buoyant droplets

SIT modulated by deformable interfaces

<figure><img src="image_74.png" /></figure>

Crialesi-Esposito, Rosti, Chibbaro, Brandt,
J. Fluid Mech. 2022
Thanks to my co-workers...

<figure><img src="image_75.png" /></figure>

...and to you for your attention!
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