# Random walks and the symplectic representation of the braid groups

Marc Soret and Marina Ville

###### Abstract

We consider the symplectic representation $\rho_{n}$ of a braid group $B(n)$
in $Sp(2l,\mathbb{Z})$, for $l=\Big{[}\dfrac{n-1}{2}\Big{]}$. If $P$ is a
$4l^{2}$ polynomial on the coefficients of the matrices in
$Sp(2l,\mathbb{Z})$, we show that the set $\\{\beta\in
B(n)/P(\rho_{n}(\beta))=0\\}$ is transient for non degenerate random walks on
$B(n)$.
If $n$ is odd, we derive that the $n$-braids $\beta$ verifying
$|\Delta_{\hat{\beta}}(-1)|\leq C$ for some constant $C$ form a transient set:
here $\Delta_{\hat{\beta}}$ denotes the Alexander polynomial of the closure of
$\beta$.
We also derive that for a random $3$-braid, the quasipositive links
$(\beta\sigma_{i}\beta^{-1}\sigma_{j})^{p}$ have zero signature for every
integer $p$ and $1\leq i,j\leq 2$.
As an example of such braids, we investigate the signature of the Lissajous
toric knots with $3$ strands.

##  1 Preliminaries

###  1.1 Representations of the braid group

If $B(n)$ is the group of braids with $n$ strands, we denote by $\mathcal{B}$
its reduced Burau representation which represents braids as $(n-1)\times(n-1)$
matrices with values in $\mathbb{Z}[t,t^{-1}]$. The specialization of
$\mathcal{B}$ for $t=-1$, denoted $\mathcal{B}_{-1}$, is called the integral
reduced Burau representation. The representation $\mathcal{B}_{-1}$ is closely
related to the symplectic representation of $B(n)$ which we now recall, using
the description of A’Campo ([A’C]).
The braid group $B(n+1)$ acts on the homology $H_{1}(X,\mathbb{Z})$ of a
surface $X$ such that $H_{1}(X,\mathbb{Z})$ is freely generated by $n$ loops.
This action is conjugate to the integral reduced Burau representation.
Concretely speaking, this means the existence of a basis of
$H_{1}(X,\mathbb{Z})$ w.r.t. which the matrices representing the elements of
$B(n+1)$ are the ones given by ${\mathcal{B}}_{-1}$.
This action of $B(n+1)$ preserves the intersection form $I$ on
$H_{1}(X,\mathbb{Z})$.

  1. 1.

If $n$ is even, $n=2l$, the form $I$ is symplectic nondegenerate on
$H_{1}(X,\mathbb{Z})$.

  2. 2.

If $n$ is odd, $n=2l+1$, the kernel of $I$ is a line $K$; ${\mathcal{B}}_{-1}$
restricts to the identity on $K$ and acts symplectically on the quotient
$H_{1}(X,\mathbb{Z})/K$

Thus, in both cases, we get a symplectic representation

$\rho_{n}:B(n+1)\longrightarrow Sp(2l,\mathbb{Z})$ (1)

###  1.2 Random walks on groups

If $\mu_{1},\mu_{2}$ are probability measures on a discrete group $G$ with
finite support, their convolution is defined for $g\in G$ as

$\mu_{1}\star\mu_{2}(g)=\sum_{h\in G}\mu_{1}(g^{-1}h)\mu_{2}(h)$ (2)

######  Definition 1.

A probability $\mu$ on a discrete group $G$ is nondegenerate if $supp(\mu)$
generates $G$ as a semigroup.

Malyutin ([Ma]) defines the right random $\mu$-walk as the Markov chain which
starts at the identity of $G$ and with transition probability
$P(g,h)=\mu(gh^{-1})$. If $X$ is a subset of $G$, the probability that this
walk hits $X$ at the $k$-th step is $\mu^{\star k}(X)$.

###  1.3 The signature of a link

The signature is a classical invariant of links and knots; (see for example
[Mu]). If $\gamma$ is a link on $\mathbb{S}^{3}$ which bounds a Seifert
surface $\Sigma$ in $\mathbb{S}^{3}$, we consider the intersection form

$I:H_{1}(\Sigma)\times H_{1}(\Sigma)\longrightarrow\mathbb{Z}$
$(\alpha,\beta)\mapsto lk(\hat{\alpha},\beta)$

where $lk$ is the linking number and $\hat{\alpha}$ is the link obtained by
pushing $\alpha$ in the direction normal to $\Sigma$ in $\mathbb{S}^{3}$.
Given a base for $H_{1}(\Sigma)$, we define the matrix $V$ of $I$: the
signature of $\gamma$ is the signature of the bilinear form defined by the
matrix $V+^{t}V$. We point out that the literature contains two different
definitions of the signature which coincide up to sign.
The signature of a link can be computed on a braid representing it and
Gambodau-Ghys have estimated the signature of a product of two braids in terms
of the signatures of the two braids ([G-G]). We recall (cf. [Mu]) that the for
a knot $K$,

$|sign(K)|\leq 2g_{4}(K)$ (3)

where $g_{4}$ denotes the topological $4$-genus. Thus a slice knot has zero
signature but the converse is not true and we will see more examples of that
here.

###  1.4 Positive links and quasipositive braids

A positive link is a link which has a diagram with all positive crossings.

######  Theorem 1.

([Pr]) A positive link has strictly negative signature.

So it is interesting to look at the signature of quasipositive braids:

######  Definition 2.

([Ru]) A braid $\beta$ is quasipositive if it is a product of conjugates of
positive braid generators

$\beta=\prod_{i=1}^{k}\gamma_{j_{i}}\sigma_{j_{i}}\gamma_{j_{i}}^{-1}$ (4)

If we close the quasipositive $N$-braid (4) in a link $\hat{\beta}$, Rudolph
proved ([Ru]) that

$\chi_{4}(\hat{\beta})=N-k$ (5)

where $\chi_{4}$ denotes the largest Euler characteristic of a smooth surface
in $\mathbb{B}^{4}$ bounded by $\hat{\beta}$.
Tanaka ([Ta]) constructed examples of quasipositive braids with zero
signature.

###  1.5 Transient sets in the braid groups from quasimorphisms

We recall that a quasimorphism on a group $G$ is a function
$\Phi:G\longrightarrow\mathbb{R}$ such that there exists a positive constant
$C$ with

$\forall a,b\in G,\ |\Phi(ab)-\Phi(a)-\Phi(b)|\leq D_{\Phi}$ (6)

where $D_{\Phi}$ is called the defect of $\Psi$.
Quasimorphisms give us exemples of transient sets in the braid groups:

######  Theorem 2.

([Ma]) Let $G$ be a countable group and $\mu$ a nondegerate probability on $G$
(see Definition 1). If a subset $S$ of $G$ has bounded image under an
unbounded quasimorphism $\Phi$, then the probability that the right random
$\mu$-walk hits $S$ at the $k$-th step tends to zero as $k$ tends to infinity.

We will see below how Gambodau-Ghys’s formula for the signature turns it into
a quasimorphism. Several other link invariants such as $\chi_{4}$ can be used
to define quasimorphisms on the braid group ([Br], [B-K]).
Other exemples of transient sets on the braid groups have been constructed by
Ito [It]

## Acknowledgements

We thank Tahl Nowik and Chaim Even-Zohar for reading an earlier draft of this
paper, Misha Brandenbursky for very helpful conversation on braids and
quasimorphisms and Florence Lecomte for necessary advice on algebraic geometry
over finite fields.

##  2 The results

######  Theorem 3.

Let $P$ be a polynomial in $(2l)^{2}$ variables with coefficients in
$\mathbb{Z}$. If $M=(m_{ij})\in Sp(2l,\mathbb{Z})$, we let
$P(M)=P(m_{11},...,m_{12l},m_{21},...,m_{22l},m_{2l1},...,m_{2l2l}).$
Suppose that $P$ does not vanish identically on $Sp(2l,\mathbb{Z})$. Then the
set

$\\{\beta\in B(n)/P(\rho_{n}(\beta))=0\\}$

is transient for the right random $\mu$-walk.

###  2.1 The Alexander polynomial

######  Corollary 1.

Let $C$ be a real number. If $2s+1$ is an odd integer, the following set is
transient for the right random $\mu$-walk on $B(2s+1)$

$\\{\beta\in B(2s+1)/|\Delta_{\hat{\beta}}(-1)|<C\\}$ (7)

where $\Delta_{\hat{\beta}}$ denotes the Alexander polynomial of the closure
of $\beta$.

###### Proof.

The Alexander polynomial can be expressed in terms of the Burau representation
$\mathcal{B}$ ([Bu]),

$\Delta_{\hat{\beta}}(t)=\frac{1-t}{1-t^{n}}det({\mathcal{B}}_{-t}(\beta)-I)$
(8)

If $n$ is odd, $\Delta_{\hat{\beta}}(-1)=det(\rho_{n}(\beta)-I)$ and the set
(7) is a finite union of sets described in Theorem 3. ∎

REMARK. If $n$ is even, $1-(-1)^{n}=0$ and
$det({\mathcal{B}}_{-1}(\beta)-I)=0$ since ${\mathcal{B}}_{-t}$ acts on $K$ as
the identity; thus the formula (8) is not easy to use. Nevertheless, it seems
likely that Theorem 1 is true also for even $n$’s.

###  2.2 The signature of $3$-braids

######  Corollary 2.

Let $\mu$ be a nondegenerate probability on the braid group $B(3)$. For the
right random $\mu$-walks on $B(3)$, the set

$S=\\{\beta\in B(3)/\exists n\in\mathbb{N},\exists i,j\in\\{1,2\\}\ \
\mbox{such that}\ \ \
sign\big{(}(\beta\sigma_{i}\beta^{-1}\sigma_{j})^{n}\big{)}\neq 0\\}$ (9)

is transient.

By contrast, (5) tells us that for every $n$,
$\chi_{4}\big{(}(\beta\sigma_{1}\beta^{-1}\sigma_{1})^{n}\big{)}=3-2n$; in
particular, if $(\beta\sigma_{1}\beta^{-1}\sigma_{1})^{n}$ closes into a knot,
$g_{4}((\beta\sigma_{1}\beta^{-1}\sigma_{1})^{n})=n-1$.

In §5, we recall the Lissajous toric knots with $3$ strands (cf. [S-V]); they
are naturally of the form $(\beta\sigma_{2}\beta^{-1}\sigma_{1}^{\pm 1})^{n}$
and we prove

######  Theorem 4.

A Lissajous toric knot $K(3,p,q)$ has zero signature unless it is isotopic to
a torus knot.

We conclude by statistical estimates, done with SAGE.

These results suggest possibles generalizations.

######  Question 1.

Let $n\in\mathbb{N}$ and let $\mathcal{I}$ be a numerical link invariant which
is unbounded on the closures of the links represented by a $n$-braid. If $C$
is a constant, is the set $\\{{\mathcal{I}}(\hat{\beta})\leq C\\}$ transient
for the right random $\mu$-walks?

######  Question 2.

For an integer $m$, we set

$A_{m}^{(1)}=\prod_{1\leq 2i+1\leq m}\sigma_{2i+1}\ \ \ \
A_{m}^{(2)}=\prod_{1\leq 2i\leq m}\sigma_{2i}$

Is the following set $S_{m}$ transient in $B(m)$ for the right random
$\mu$-walks?

$S_{m}=\\{\beta\in B(m)/\exists n\in\mathbb{N},\exists i,j\in\\{1,2\\}\ \
\mbox{such that}\ \ \ sign\big{(}(\beta
A_{m}^{(i)}\beta^{-1}A_{m}^{(j)})^{n}\big{)}\neq 0\\}$

##  3 Random walks: proof of Theorem 3

Similarly to Rivin in [Ri], we introduce the finite groups
$Sp(2l,\mathbb{Z}_{p})$’s for a prime number $p>2$.
A theorem of A’Campo ([A’C], Theorem 1 (1)) shows that the representation
$B(n)\longrightarrow Sp(2l,\mathbb{F}_{p})$ is surjective. So we consider the
surjective map

$\Pi_{p}:B(n)\overset{\rho_{n}}{\longrightarrow}Sp(2l,\mathbb{Z})\longrightarrow
Sp(2l,\mathbb{F}_{p})\overset{\pi}{\longrightarrow}PSp(2l,\mathbb{F}_{p})$
(10)

We consider the image probability $\Pi_{p}\mu$ of $\mu$ via $\Pi_{p}$ on
$PSp(2l,\mathbb{F}_{p})$. We recall facts on random walks on finite groups.

######  Theorem 5.

(see for exemple [Di]) Let $G$ be a finite group and $\mu$ a probability
measure on $G$ such that

  1. 1.

$G$ is generated as a semigroup by $supp(\mu)$

  2. 2.

$supp(\mu)$ is not included in a coset of $G$ by a normal subgroup $H$ of $G$.

Then $\mu^{\star m}$ converges to the equidistributed measure on $G$ as $m$
tends to infinity.

We know ([As]) that, if $p>5$, the group $PSp(2n,{\mathbb{F}_{p}})$ is simple
so we can apply Theorem 5 and derive

######  Lemma 1.

If $m$ tends to infinity, $(\Pi_{p}\mu)^{\star m}$ converges to the
equidistributed measure on $PSp(2n,{\mathbb{F}_{p}})$.

We now investigate the density of the zero set in $Sp(2l,\mathbb{F}_{p})$ of
the polynomial $P$ appearng in Theorem 3.
We write $P$ in terms of all the multi-indices, $P(M)=\sum a_{I}m^{I}$, where
the $a_{I}$’s belong to $\mathbb{Z}$. If $p$ is an integer, we denote by
$P_{p}$ its reduction modulo $p$. Since $P$ is not identically zero, we derive
an infinite set $\mathcal{P}$ of prime numbers $p$ such that $P_{p}$ is not
identically zero. To prove the theorem, we need to estimate

$\frac{|P_{p}^{-1}(0)|}{|Sp(2l,\mathbb{F}_{p})|}$ (11)

We recall (see for example [O’M] or [wi]) that

$|Sp(2n,\mathbb{Z}_{p})|=\prod_{m=1}^{n}\big{[}(p^{2m}-1)p^{2m-1}\big{]}\geq\frac{1}{2^{2n}}\prod_{s=1}^{2l}p^{s}=\dfrac{p^{2l^{2}+l}}{2^{2n}}$
(12)

To estimate $|P_{p}^{-1}(0)|$, we use the following result by Lachaud and
Rolland

######  Theorem 6.

([L-R]) Let $K$ be the algebraic closure of $\mathbb{F}_{p}$. Let $X$ be an
algebraic variety of $K^{n}$ of dimension $m$ which is the zero set of a
family of polynomials $(f_{1},...,f_{r})$. Let $d_{i}=deg(f_{i})$. Then

$|X\cap\mathbb{F}_{p}^{n}|\leq d_{1}...d_{r}q^{m}$

We view $P_{p}$ as a polynomial with coefficients in $K$ and let
$X=Sp(2l,K)\cap P_{p}^{-1}(0)$. We know that for a field $K$, $Sp(2l,K)$ is
irreducible as an algebraic variety ([Mi]), hence $X$ is of dimension strictly
smaller that $Sp(2l,K)$, i.e. $dim(X)\leq 2l^{2}+l-1$. Thus, for some constant
$C_{1}(2l,P)$ depending on $2l$ and on the degree of $P$,

$|X\cap\mathbb{F}_{p}^{4l^{2}}|\leq C_{1}(2l,P)p^{2l^{2}+l-1}$ (13)

Putting (12) and (13) together, we derive that

$\dfrac{|P_{p}^{-1}(0)|}{|Sp(2l,\mathbb{F}_{p})|}\leq\dfrac{C_{2}(2l,P)}{p}$
(14)

for some constant $C_{2}(2l,P)$.
We conclude the proof of Theorem 3. Given a $\epsilon>0$, we pick a prime
number $p$ in $\mathcal{P}$ such that

$2\dfrac{C_{2}(2l,P)}{p}<\epsilon$ (15)

There exists an integer $k_{0}$ such that for every $k>k_{0}$, we have

$\mu^{\star k}(P^{-1}(0))\leq(\Pi_{p}\mu)^{\star k}(P_{p}^{-1}(0))<\epsilon.$

∎

##  4 The signature of $3$-braids: proof of Corollary 2

###  4.1 Preliminaries: the Gambaudo-Ghys formula for the signature of a
$3$-braid ([G-G])

####  4.1.1 The reduced Burau representation for $t=-1$

We let $B_{3}$ be the $3$-braid group and we denote by

${\mathcal{B}}_{-1}:B(3)\longrightarrow SL(2,\mathbb{Z})$

the reduced Burau representation of the braid group $B(3)$ for $t=-1$. Because
there are different expressions in the literature, we recall the ones [G-G]
uses:

$s_{1}={\mathcal{B}}_{-1}(\sigma_{1})=\begin{pmatrix}1&0\\\ -1&1\end{pmatrix}\
\ \ \ s_{2}={\mathcal{B}}_{-1}(\sigma_{2})=\begin{pmatrix}1&1\\\
0&1\end{pmatrix}$ (16)

We recall (see for example [K-T]):

######  Fact 1.

The center of $B(3)$ is the cyclic group generated by the twist

$\Delta=(\sigma_{1}\sigma_{2}\sigma_{1})^{2}.$

######  Fact 2.

  * •

${\mathcal{B}}_{-1}$ is surjective

  * •

${\mathcal{B}}_{-1}(\Delta)=-Id$

  * •

The kernel of ${\mathcal{B}}_{-1}$ is the cyclic group generated by
$\Delta^{2}$.

####  4.1.2 The Meyer cocycle and the formula for the signature

If $\beta$ is a braid, we denote its closure by $\hat{\beta}$. For two
$3$-braids, $\alpha$ and $\beta$, [G-G] proves

$sign(\widehat{\alpha.\beta})=sign(\hat{\alpha})+sign(\hat{\beta})-Meyer({\mathcal{B}}_{-1}(\alpha),{\mathcal{B}}_{-1}(\beta))$
(17)

where $Meyer\in H^{2}(SL(2,\mathbb{R}))$ is the Meyer cocycle. We recall its
description by [G-G].
If $\gamma_{1},\gamma_{2}\in SL(2,\mathbb{Z})$, we let

$E_{\gamma_{1},\gamma_{2}}=Im(\gamma_{1}^{-1}-Id)\cap Im(\gamma_{2}-Id)$ (18)

For a vector $e$ in $E_{\gamma_{1},\gamma_{2}}$, we take $v_{1},v_{2}$ in
$\mathbb{R}^{2}$ such that

$e=\gamma_{1}^{-1}(v_{1})-v_{1}=v_{2}-\gamma_{2}(v_{2})$

and define the quadratic form

$q_{\gamma_{1},\gamma_{2}}=\Omega(e,v_{1}+v_{2})$ (19)

where $\Omega$ is the standard symplectic form on $\mathbb{R}^{2}$. Then
$Meyer(\gamma_{1},\gamma_{2})$ is the signature of
$q_{\gamma_{1},\gamma_{2}}$.

######  Fact 3.

If $\gamma\in SL(2,\mathbb{Z})$ is hyperbolic (i.e. $|tr(\gamma)|>2$), then
for two positive integers $a,b$,

$Meyer(\gamma^{a},\gamma^{b})=0$

REMARK. Since a generic element of $SL(2,\mathbb{Z})$ is hyperbolic, it
follows from Fact 3 that for almost every braid $\beta\in B(3)$,
$sign(\beta^{n})=nsign(\beta)$.

###  4.2 Proof of Corollary 2

Since $\sigma_{i}$ and $\beta\sigma_{j}\beta^{-1}$ close in trivial knots,
$sign(\sigma_{i})=sign(\beta\sigma_{j}\beta^{-1})=0$; hence

$sign(\sigma_{i}\beta\sigma_{j}\beta^{-1})=Meyer(\sigma_{i},\beta\sigma_{j}\beta^{-1})$
(20)

######  Lemma 2.

Let $i,j\in\\{1,2\\}$ and $\beta\in B(3)$. If
$Meyer(\sigma_{i},\beta\sigma_{j}\beta^{-1})\neq 0$, then one of the
coefficients of the matrix ${\mathcal{B}}_{-1}(\beta)$ is zero.

###### Proof.

Recall that $Meyer(\sigma_{i},\beta\sigma_{j}\beta^{-1})$ is the signature of
a quadratic form on the space $E_{\sigma_{i},\beta\sigma_{j}\beta^{-1}}$,
defined in §4.1.2; thus, if $Meyer(\sigma_{i},\beta\sigma_{j}\beta^{-1})$ is
non zero, the space $E_{\sigma_{i},\beta\sigma_{j}\beta^{-1}}$ contains a non-
zero element, i.e. there exist non zero $v_{1},v_{2}$ such that

$(s_{i}^{-1}-Id)(v_{i})=\beta\circ(Id-s_{j})[\beta^{-1}(v_{2})]$ (21)

If we denote by $e_{1},e_{2}$ the canonical basis of $\mathbb{R}^{2}$, we
notice that

$Im(s_{1}^{\pm 1}-Id)=\mathbb{R}e_{2}\ \ \ \ \ \ \ Im(s_{2}^{\pm
1}-Id)=\mathbb{R}e_{1}$ (22)

It follows that the (LHS) of (21) is parallel to $e_{1}$ or $e_{2}$ and the
(RHS) is parallel to $\beta(e_{1})$ or $\beta(e_{2})$. Thus, for some
$h,k\in\\{1,2\\}$, $\beta(e_{h})$ is parallel to $e_{k}$ which implies that
one of the coefficients of the matrix of $\beta$ is zero. ∎

######  Lemma 3.

Let $M=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in SL(2,\mathbb{Z})$. The matrix
$Ms_{1}M\texttwosuperior^{-1}s_{1}$ (resp.
$Ms_{2}M\texttwosuperior^{-1}s_{2}$, $Ms_{1}M\texttwosuperior^{-1}s_{2}$,
$Ms_{2}M\texttwosuperior^{-1}s_{1}$) is hyperbolic unless

$b^{2}\leq 4\ \ (\mbox{resp}.\ c^{2}\leq 4,\ d^{2}\leq 4,\ a^{2}\leq 4)$ (23)

###### Proof.

Compute the traces, for example
$Trace(Ms_{2}M\texttwosuperior^{-1}s_{1})=2-a^{2}$. ∎

We can now conclude the proof of Corollary 2. We let $\beta$ be a $3$-braid
such that

$\forall i,j\in\\{1,2\\}\ \ \
|\big{(}{\mathcal{B}}_{-1}(\beta)\big{)}_{ij}|>2$ (24)

Then $sign(\sigma_{a}\beta\sigma_{b}\beta^{-1})=0$ (Lemma 2) and
$\sigma_{a}\beta\sigma_{b}\beta^{-1}$ is hyperbolic (Lemma 3). Thus
$Meyer({\mathcal{B}}_{-1}(\sigma_{a}\beta\sigma_{b}\beta^{-1}),{\mathcal{B}}_{-1}((\sigma_{a}\beta\sigma_{b}\beta^{-1})^{n})=0$
(Fact 3 of §4.1.2). It follows that, for every $n$,

$sign((\sigma_{a}\beta\sigma_{b}\beta^{-1})^{n})=0.$

On the other hand, Theorem 3 tells us that the braids not verifying (24) form
a transient set for the right random $\mu$-walks on $B(3)$. ∎

##  5 The Lissajous toric knots

###  5.1 Description

####  5.1.1 The knots $K(N,q,p)$

In [S-V], the authors exhibited Lissajous toric knots as one of the classes of
boundaries of minimal disks in the $4$-ball with a branch point at the origin.
Before that, Lamm investigated these knots in connection with billiards in the
solid torus ([L-O]). If $N,q,p$ are integers, with $(N,q)=(N,p)=1$, the
$K(N,q,p)$ Lissajous toric knot is defined in a $3D$-cylinder as

$F_{N,q,p}:[0,2\pi]\longrightarrow\mathbb{S}^{1}\times\mathbb{R}^{2}$
$F_{N,q,p}:\theta\mapsto(e^{Ni\theta},\sin(q\theta),\cos(p\theta+\alpha))$
(25)

for a phase $\alpha$. Endow $\mathbb{R}^{3}$ with a coordinate system
$(x,y,z)$ and take the Lissajous curve
$C_{p,q}:\theta\mapsto\big{(}0,2+\sin(q\theta),\cos(p\theta+\alpha)\big{)}$;
then the knot $K(N,q,p)$ is described in $\mathbb{R}^{3}$ by a point
travelling along $C_{p,q}$ while $C_{p,q}$ rotates $N$ times along the
vertical axis $Oz$. We recall a few facts

######  Fact 4.

([S-V]) For a finite number of phases $\alpha$’s the expression in (25) gives
us singular crossing points. Otherwise (25) defines a knot and up to mirror
transformation, its knot type does not depend on the phase $\alpha$.

Thus we drop the phase $\alpha$ in (25) and we just talk of a knot $K(N,q,p)$
defined up to mirror symmetry.

####  5.1.2 The braids $B(N,q,p)$

The knot $K(N,q,p)$ has a natural $N$-braid representation $B(N,q,p)$.

######  Fact 5.

([S-V]) Let $\tilde{p},\tilde{q},d$ be three positive integers all coprime
with $N$ and such that $(\tilde{p},\tilde{q})=1$. Moreover assume (without
loss of generality) that $\tilde{q}$ is odd. Then

$B(N,d\tilde{q},d\tilde{p})=B(N,\tilde{q},\tilde{p})^{d}$

and there exists a braid $Q_{N,\tilde{q},\tilde{p}}$ such that

$B(N,\tilde{q},\tilde{p})=Q_{N,\tilde{q},\tilde{p}}\sigma_{2}^{\epsilon(2)}Q_{N,\tilde{q},\tilde{p}}^{-1}\sigma_{1}^{\epsilon(1)}$
(26)

with $\epsilon(1),\epsilon(2)\in\\{-1,1\\}$.
If $2N$ divides $\tilde{p}+\tilde{q}$ or $\tilde{p}-\tilde{q}$,
$\epsilon(1)=\epsilon(2)=1$ so the braid is quasipositive: for $N=3$, this
happens if and only if $\tilde{q}$ and $\tilde{p}$ are both odd.

The braid (26) is a symmetric union as defined by [La] thus:

######  Fact 6.

If $N,q,p$ are all mutually prime, $K(N,q,p)$ is a ribbon knot.

See also [L-O] and [S-V].

###  5.2 The signature of the Lissajous toric knots

For $N=3$, a Lissajous toric knot has zero signature unless it is a torus
knot. More precisely,

######  Theorem 7.

Let $\tilde{q},\tilde{p},n$ be three positive integers, none of them divisible
by $3$ and $\tilde{q},\tilde{p}$ mutually prime. Let
$K(3,n\tilde{q},n\tilde{p})$ be the corresponding Lissajous toric knot.

  1. 1.

If $\tilde{q}$ and $\tilde{p}$ are both odd, up to mirror image,
$K(3,n\tilde{q},n\tilde{p})$ is a quasipositive knot with

$g_{4}(K(3,n\tilde{q},n\tilde{p}))=d-1$ (27)

and verifying one of the following

    1. (a)

the signature of $K(3,n\tilde{q},n\tilde{p})$ is zero

    2. (b)

$B(3,\tilde{q},\tilde{p})=\sigma_{2}\sigma_{1}$ so $K(3,\tilde{q},\tilde{p})$
is a trivial knot and $K(3,n\tilde{q},n\tilde{p})$ is a $(3,n)$-torus knot.

  2. 2.

If $\tilde{q}$ and $\tilde{p}$ have different parities,
$K(3,n\tilde{q},n\tilde{p})$ is isotopic to its mirror image, so it has zero
signature.

REMARK 1. The Lissajous toric knots $K(3,n,n)$ are just the $(3,n)$ torus
knots so clearly they are in the case 1 (b) of Theorem 7, but they are not the
only ones. For example if $\tilde{p}=\tilde{q}+6$ the knot
$K(3,\tilde{q},\tilde{q}+6)$ (e.g. $K(3,7,13)$) is represented by the braid

$B(3,\tilde{q},\tilde{q}+6)=\sigma_{2}\sigma_{1}$

so it is trivial and, for a positive integer $n$,

$B(3,n\tilde{q},n(\tilde{q}+6))=(\sigma_{2}\sigma_{1})^{n}$

so $K(3,n\tilde{q},n(\tilde{q}+6))$ is a $(3,n)$-torus knot.
In §6.2, we see that at least for $\tilde{q}$’s up to $100$, the majority of
knots of 1. in Theorem 7 1\. are in the case 1. (a) of that Theorem.

An example is $K(3,5,7)$ which is, up to mirror image, the knot $10_{155}$ in
the Rolfsen classification and its braid is

$B(3,5,7)=\sigma_{2}\sigma_{1}^{-1}\sigma_{2}\sigma_{1}^{-1}\sigma_{2}\sigma_{1}\sigma_{2}^{-1}\sigma_{1}\sigma_{2}^{-1}\sigma_{1}$
(28)

###  5.3 Proof of Theorem 7

The proof depends on the parities of $\tilde{q}$ and $\tilde{p}$; since these
numbers are mutually prime, either they have the same parity or they are both
odd.

####  5.3.1 1st case: $\tilde{q}$ and $\tilde{p}$ are both odd

Up to mirror symmetry, the braid $B(3,\tilde{q},\tilde{p})$ is of the form
(Fact 5)

$B(3,\tilde{q},\tilde{p})=Q_{3,\tilde{q},\tilde{p}}\sigma_{2}Q_{3,\tilde{q},\tilde{p}}^{-1}\sigma_{1}$

with ([S-V])

$Q_{3,\tilde{q},\tilde{p}}=\sigma_{2}^{\lambda(1)}\sigma_{1}^{\lambda(2)}\sigma_{2}^{\lambda(3)}\sigma_{1}^{\lambda(4)}...\sigma_{2}^{\lambda(\tilde{q}-2)}\sigma_{1}^{\lambda(\tilde{q}-1)}$
(29)

where $\lambda$ is an expression with values in $\\{-1,1\\}$ verifying

$\lambda(k)=-\lambda(\tilde{q}-k)$

so we rewrite $Q_{3,\tilde{q},\tilde{p}}$ and introduce the braid $P$:

$Q_{3,\tilde{q},\tilde{p}}=\underbrace{\sigma_{2}^{\lambda(1)}\sigma_{1}^{\lambda(2)}\sigma_{2}^{\lambda(\frac{\tilde{q}-3}{2})}\sigma_{1}^{\lambda(\frac{\tilde{q}-1}{2})}}_{P}\sigma_{2}^{-\lambda(\frac{\tilde{q}-1}{2})}\sigma_{1}^{-\lambda(\frac{\tilde{q}-3}{2})}...\sigma_{2}^{-\lambda(2)}\sigma_{1}^{-\lambda(1)}$
(30)

We have defined braids and we now turn to their images under the Burau
representation $B_{-1}$. We let

${\mathcal{Q}}={\mathcal{B}}_{-1}(Q_{3,\tilde{q},\tilde{p}})\ \ \ \ \ \ \ \ \
\ \ \ {\mathcal{P}}={\mathcal{B}}_{-1}(P).$

We notice that the $s_{i}$’s of (16) verify

$s_{2}={}^{t}s_{1}^{-1}$ (31)

and derive

${\mathcal{Q}}={\mathcal{P}}^{t}{\mathcal{P}}$ (32)

######  Lemma 4.

${\mathcal{B}}_{-1}(B(3,\tilde{q},\tilde{p}))$ is hyperbolic except if
${\mathcal{P}}$ is of the form

$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$ $\mbox{with}\ \ \ \
a,b\in\\{-1,0,1\\}$ (33)

###### Proof.

Compute $Trace({\mathcal{B}}_{-1}(B(3,\tilde{q},\tilde{p})))=$

$Trace[({\mathcal{B}}_{-1}({\mathcal{P}})^{t}{\mathcal{B}}_{-1}({\mathcal{P}}))s_{2}({\mathcal{B}}_{-1}({\mathcal{P}})^{t}{\mathcal{B}}_{-1}({\mathcal{P}}))^{-1}s_{1}]=2-(a^{2}+b^{2})^{2}.$

hence if it is hyperbolic, $a^{2}+b^{2}=1$ or $a^{2}+b^{2}=2$ ∎

If ${\mathcal{P}}$ verifies (33), it has one of the following forms, for some
integer $h$.

  * •

${\mathcal{P}}=(\pm 1)\begin{pmatrix}1&0\\\ h&1\end{pmatrix}=(\pm
1)s_{1}^{-h}$
Thus ${\mathcal{Q}}={\mathcal{P}}^{t}{\mathcal{P}}=s_{1}^{-h}s_{2}^{h}$ and,
for some integer $k$,

$\beta=(\sigma_{1}^{-h}\sigma_{2}^{h}\Delta^{k})\sigma_{2}(\Delta^{-k}\sigma_{2}^{-h}\sigma_{1}^{h})\sigma_{1}=\sigma_{1}^{-h}(\sigma_{2}\sigma_{1})\sigma_{1}^{h}$

so $\beta$ is conjugate to $\sigma_{2}\sigma_{1}$ and the knot is as in 1. (b)
of Theorem 7.

  * •

${\mathcal{P}}=(\pm 1)\begin{pmatrix}0&1\\\ -1&h\end{pmatrix}=(\pm
1)s_{1}^{1-h}s_{2}s_{1}$
Thus ${\mathcal{Q}}=s_{1}^{-h}s_{2}^{-h}$ and similarly to above,

$\beta=\sigma_{1}^{-h}(\sigma_{2}\sigma_{1})\sigma_{1}^{h}.$

Again we are in Case 1. (b) of Theorem 7.

  * •

${\mathcal{P}}=(\pm 1)\begin{pmatrix}1&1\\\ h&h+1\end{pmatrix}=(\pm
1)s_{1}^{-h}s_{2}$
Then ${\mathcal{Q}}=s_{1}^{-h}s_{2}s_{1}^{-1}s_{2}^{h}$ and

$\beta=\sigma_{1}^{-h}(\sigma_{2}\sigma_{1}^{-1}\sigma_{2}\sigma_{1}\sigma_{2}^{-1}\sigma_{1})\sigma_{1}^{-h}$

This braid $\beta$ closes into a link with $3$ components, thus it cannot be
the braid of a knot.

  * •

${\mathcal{P}}=(\pm 1)\begin{pmatrix}1&-1\\\
h&1-h\end{pmatrix}=s_{1}^{-h}s_{2}^{-1}$ and

$\beta=\sigma_{1}^{-h}(\sigma_{2}^{-1}\sigma_{1}\sigma_{2})(\sigma_{1}^{-1}\sigma_{2}\sigma_{1})\sigma_{1}^{h}$

Again $\beta$ closes in a link with $3$ components. ∎

####  5.3.2 2nd case: $\tilde{q}$ and $\tilde{p}$ have different parities

The knot is preserved by an isometry which is a rotation in one plane and a
symmetry in its orthogonal plane, thus the knot is isotopic to its mirror
image and has zero signature.
To spell this out, we assume that $\tilde{q}$ is odd and $\tilde{p}$ is even.
The knot $K(N,d\tilde{q},d\tilde{p})$ is defined by the function
$F_{N,d\tilde{q},d\tilde{p}}$ (see (25) above), so we can also define it by
the function

$\theta\mapsto
F_{N,d\tilde{q},d\tilde{p}}(\theta+\frac{\pi}{d})=(e^{iN\theta}e^{iN\frac{\pi}{d}},-\sin(d\tilde{q}\theta),\cos(d\tilde{p}\theta+\alpha))$

thus $K(N,d\tilde{q},d\tilde{p})$ is invariant under the transformation

$\begin{pmatrix}\cos(\frac{N\pi}{d})&-\sin(\frac{N\pi}{d})&0&0\\\
\sin(\frac{N\pi}{d})&\cos(\frac{N\pi}{d})&0&0\\\ 0&0&-1&0\\\
0&0&0&1\end{pmatrix}$ (34)

which reverses the orientation on $\mathbb{R}^{4}$ and on $\mathbb{S}^{3}$. ∎

##  6 Statistics

###  6.1 Random quasipositive braids

The following estimates were done with SAGE.

We define

$Z=\\{\beta\in B(3):|\big{(}{\mathcal{B}}_{-1}(\beta)\big{)}_{11}|>2\\}.$

It follows from the previous discussion that, if a $3$-braid $\beta$ belongs
to $Z$, then for all positive integers $n$, we have

$sign\Big{(}(\beta\sigma_{1}\beta^{-1}\sigma_{2})^{n}\Big{)}=0.$

Let $W$ be the $\mu$-random walk on $B(3)$ given by the probability $\mu$ on
$B(3)$ where

$\mu(\sigma_{1})=\mu(\sigma_{2})=\mu(\sigma_{1}^{-1})=\mu(\sigma_{2}^{-1})=\frac{1}{4}$

Here are the probabilities $p(n)$ that $W$ hits $Z$ at the $n$-th step, for
$n\leq 12$.

$\left|\begin{array}[]{ c | c | c | c | c | c | c | c| c | c | c |c|c|c}\hline\cr n&1&2&3&4&5&6&7&8&9&10&11&12\\\ \hline\cr p(n)&0&0&0.06&0.11&0.17&0.22&0.27&0.32&0.36&0.41&0.45&0.48\\\ \hline\cr\end{array}\right|.$

###  6.2 Lissajous toric knots with $3$ strands

We investigate here the proportion of knots which verify (a) or (b) of 1. of
Theorem 7. 1. First we prove

######  Lemma 5.

To get a full list of the $K(3,\tilde{q},\tilde{p})$’s for a given
$\tilde{q}$, it is enough to take

$\tilde{p}\in[\tilde{q}+1,2\tilde{q}]$

###### Proof.

If $\tilde{q}$ and $\tilde{p}$ are odd, mutually prime and not divisible by
$3$, we use the representation of $K(3,\tilde{q},\tilde{p})$ from (5.3.1). The
expression of $\lambda$ is based on the Bezout identity:

$6A+Bq=1$ (35)

for two integers $A$ and $B$. Then, up to mirror transformation

$\lambda(k)=(-1)^{\Big{[}\frac{2Apk}{q}\Big{]}}$ (36)

It follows from (36) that, if we replace $p$ by $p+q$, the expression of
$\lambda$ remains the same. ∎

For a given $\tilde{q}$, we let

$P(\tilde{q})=\\{\tilde{p}\in[\tilde{q}+1,2\tilde{q}],\ \mbox{such that}\
(2,\tilde{p})=(3,\tilde{p})=1\\}$ $S(\tilde{q})=\\{\tilde{p}\in P(\tilde{q}),\
\mbox{such that}\ K(3,\tilde{q},\tilde{p})\ \mbox{verifies 1) (a) of Theorem
\ref{theoreme sur lissajous}}\\}$

The table below gives us, for a given $\tilde{q}$, the percentage of elements
of $P(\tilde{q})$ which belong to $S(\tilde{q})$: i.e. the percentage of
$K(3,\tilde{q},\tilde{p})$’s for which every $K(3,n\tilde{q},n\tilde{p})$,
$n\in\mathbb{N}$, has zero signature. The percentages have been rounded to the
integer part.

$\left|\begin{array}[]{ c | c | c | c| c | c | c |c | c | c | c | c| c | c | c |c | c | c | c | c| c | c | c |c }\hline\cr\tilde{q}&5&7&11&13&17&19&23&25&29&31&35&37&41&43&47&49\\\ \hline\cr\%&40&57&54&61&70&63&69&64&68&77&62&75&78&74&80&69\\\ \hline\cr\end{array}\right.$ $\left|\begin{array}[]{ c | c | c | c| c | c | c |c | c | c | c | c| c | c | c |c | c | c | c | c| c | c | c |c }\hline\cr\tilde{q}&53&55&59&61&65&67&71&73&77&79&83&85&89&91&95&97&101\\\ \hline\cr\%&79&72&77&78&76&80&67&82&75&83&84&77&80&74&80&82&83\\\ \hline\cr\end{array}\right.$

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Marc Soret: Université F. Rabelais, Dép. de Mathématiques, 37000 Tours,
France,
<EMAIL_ADDRESS>

Marina Ville: Univ Paris Est Creteil, CNRS, LAMA, F-94010 Creteil, France
<EMAIL_ADDRESS>


# The emergence of bubble-induced scaling in thermal spectra in turbulence

On-Yu Dung1  Pim Waasdorp1  Chao Sun1,3  Detlef Lohse1,2  Sander G. Huisman1
<EMAIL_ADDRESS>1Physics of Fluids Group, J. M. Burgers Center for
Fluid Dynamics, and Max Planck Center Twente, Faculty of Science and
Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The
Netherlands 2Max Planck Institute for Dynamics and Self-Organization, Am
Fassberg 17, 37077 Göttingen, Germany 3Center for Combustion Energy, Key
Laboratory for Thermal Science and Power Engineering of Ministry of Education,
Department of Energy and Power Engineering, Tsinghua University, 100084
Beijing, China

###### Abstract

We report on the modification of the spectrum of a passive scalar inside a
turbulent flow by the injection of large bubbles. While the spectral
modification through bubbles is well known and well analyzed for the velocity
fluctuations, little is known on how bubbles change the fluctuations of an
approximately passive scalar, in our case temperature. Here we uncover the
thermal spectral scaling behavior of a turbulent multiphase thermal mixing
layer. The development of a $-3$ spectral scaling is triggered. By injecting
large bubbles ($\text{Re}_{\text{bub}}=\mathcal{O}(10^{2})$) with gas volume
fractions $\alpha$ up to 5%. For these bubbly flows, the $-5/3$ scaling is
still observed at intermediate frequencies for low $\alpha$ but becomes less
pronounced when $\alpha$ further increases and it is followed by a steeper
$-3$ slope for larger frequencies. This $-3$ scaling range extends with
increasing gas volume fraction. The $-3$ scaling exponent coincides with the
typical energy spectral scaling for the velocity fluctuations in high Reynolds
number bubbly flows. We identify the frequency scale of the transition from
the $-5/3$ scaling to the $-3$ scaling and show how it depends on the gas
volume fraction.

###### keywords:

multiphase flow, turbulence, bubbles, heat transfer, passive scalar,
temperature

##  1 Introduction

In fully developed turbulent flows, universality is observed for velocity and
temperature fluctuations for the second order structure functions and the
corresponding spectra. According to the Kolmogorov–Obukhov–Corrsin theory
(Kolmogorov (1941); Obukhov (1949); Corrsin (1951)) and verified extensively
in experiment and numerical simulations, the universal scaling of the spectrum
of a scalar exhibits a $-5/3$ scaling ($E_{\theta}(k)\propto k^{-5/3}$) in the
inertial-convective range for high enough Reynolds and Péclet number (see e.g.
Monin & Yaglom (1975)). Here $k$ is the wavenumber and $\theta$ is a passive
scalar (Monin & Yaglom (1975)) such as the concentration of a chemical
substance or the temperature, in case that buoyancy is not relevant. If
additional complexity is added to the system, different scaling phenomena can
appear, such as having an active scalar introducing buoyancy effects (e.g.
Rayleigh-Bénard convection) (Lohse & Xia (2010)), adding visco-elasticity to
the carrier liquid (Steinberg (2020)), or by adding finite particles (Risso
(2018) and Brandt & Coletti (2022)) or bubbles (Balachandar & Eaton (2010);
Risso (2018); Lohse (2018); Mathai et al. (2020)). In this work, we do the
latter and introduce a large number of finite-size rising bubbles to the
liquid flow (Balachandar & Eaton (2010); Risso (2018); Lohse (2018); Mathai et
al. (2020)). Large rising bubbles are defined here as having a bubble Reynolds
number $\text{Re}_{\text{bub}}=V_{r}d/\nu=\mathcal{O}(10^{2})$, where $V_{r}$
is the bubble rise velocity relative to the mean vertical velocity of the
carrier liquid, $d$ is the mean area-equivalent bubble diameter (as compared
to a spherical bubble), and $\nu$ is the kinematic viscosity of the carrier
liquid.

First observed and theoretically addressed in Lance & Bataille (1991), the
introduction of bubbles lead to the emergence of a $k^{-3}$ scaling of the
velocity fluctuation (energy) spectra $E_{u}(k)$. They argued that in a
statistical steady state, the $k^{-3}$ scaling in the spectrum can be achieved
by balancing the viscous dissipation and the energy production due to the
rising bubbles in the spectral space (Lance & Bataille (1991)). The $k^{-3}$
scaling can also result from spatial and temporal velocity fluctuations in
homogeneous bubbly flows (Risso (2018)), and is found to be robust and is
observed for a homogeneous bubble swarm for a wide range of parameters:
$10\leq\text{Re}_{\text{bub}}\leq 1000$ and $1\leq\text{We}\leq 4$ where We is
the Weber number (Risso (2018); Pandey et al. (2020)). Recently, Ma et al.
(2022) employed particle shadow velocimetry in bubble-laden turbulent flow and
obtained a high-resolution two-dimensional velocity field. They revealed
longitudinal and traverse velocity structure functions up to twelfth order, in
which the second order statistics (one-dimensional longitudinal energy
spectra) also show a $-3$ scaling. In contrast, point particle simulations do
not show $k^{-3}$ scaling (Mazzitelli & Lohse (2009)), because the wakes
behind the bubbles which are obviously absent for point particles are crucial
for the emergence of a $-3$ scaling (in either frequency or wavenumber space).

While there are abundant studies on the energy spectra in high-Re bubbly flow,
the study of scalar spectra in such flow is limited. Alméras et al. (2016)
investigated the time-resolved concentration fluctuations of a fluorescent dye
(passive scalar) in a confined bubbly thin cell in which the scalar spectral
scaling $f^{-3}$ is observed, where $f$ is the frequency. For the three-
dimensional case, Gvozdić et al. (2018) investigated the scalar spectra in a
bubbly column in a vertical convection setup, but they were unable to resolve
the frequencies related to the $-3$ scaling of the energy spectra in bubble-
induced turbulence (Gvozdić et al. (2018)). The obvious downside of using dyes
is that the water gets contaminated and only very short runs can be performed
before the water has to be completely replaced. By using temperature as the
approximately passive scalar, and the active cooling of the setup during the
return stage, it is possible to maintain an overall constant temperature of
the setup, allowing for continuous measurements for hours or days if needed.

In the present experimental work, by utilizing a fast-response temperature
probe, we are able to capture the gradual change of the passive scalar
(thermal) spectral scaling for increasing bubble concentration $\alpha$ in a
bubbly turbulent thermal mixing layer. In addition, we will investigate the
transition scale of the $-3$ scaling as it illuminates the physical origin of
the $-3$ subrange of both the scalar and the energy spectra. The transition
frequency to the $-3$ subrange for energy spectra indeed has yet to be exactly
identified (Risso (2018)).

##  2 Experimental setup and methods

###  2.1 Experimental facility

We utilize the Twente Mass and Heat Transfer Tunnel (Gvozdić et al. (2019))
which creates a turbulent vertical channel flow, and allows for injecting
bubbles and heating the liquid in a controlled manner. The turbulence is
actively stirred using an active grid, see Fig. 1a. Using this facility we
create a turbulent thermal mixing layer (de Bruyn Kops & Riley (2000)) in
water with a mean flow velocity of approximately
$0.5\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$, resulting in a Reynolds
number of $\text{Re}=2\times 10^{4}$. The global gas volume fraction $\alpha$
in the measurement section is measured by a differential pressure transducer
of which the two ends are connected to the top and bottom of the measurement
section. A fast-response thermistor (Amphenol Advanced Sensors type FP07, with
a response time of $7\text{\,}\mathrm{ms}$ in water) is placed into the middle
of the measurement section in order to measure the temperature. An AC bridge
with a sinusoidal frequency of $1.3\text{\,}\mathrm{kHz}$ drives the
thermistor in order to reduce electric noise. The bridge potential is measured
by a lock-in amplifier (Zurich Instruments MFLI) with the acquisition
frequency of $13.4\text{\,}\mathrm{kHz}$ and such that the cut-off frequency
that is used is $(13.4/2)\text{\,}\mathrm{kHz}$, which is far away from the
effective range of the thermistor which is limited by the thermal capacity of
the probe. A temperature-controlled refrigerated circulator with water bath
(PolyScience PD15R-30) with a temperature stability of $\pm
0.005\text{\,}\mathrm{K}$ is used for calibrating the thermistor. We employ
the temperature-resistance characteristic equation proposed by Steinhart &
Hart (1968) over a short range of temperature of interest (in which the terms
are kept only up to the first order to avoid over-fitting) in order to convert
the measured resistance of the thermistor to the corresponding temperature.
The thermistor has a relative precision of $1\text{\,}\mathrm{mK}$. With
relative we mean here relative to a baseline temperature, i.e. a temperature
difference ($\Delta T$), as we are mainly interested in temperature changes,
which have a different precision than $T$ itself. The typical large-scale
temperature difference from the heated side to the non-heated side at the mid-
height of the measurement section of height $h=$1\text{\,}\mathrm{m}$$ is
about $0.2\text{\,}\mathrm{K}$. The working liquid is decalcified tap water at
around $22.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$, with a Prandtl number
$\text{Pr}=\nu/\kappa=6.5$, where $\nu$ and $\kappa$ are the kinematic
viscosity and thermal diffusivity of water, respectively.

For the velocity measurements, we employ Laser Doppler Anemometry (LDA)
(Dantec) and constant temperature anemometry (CTA) using a hot film sensor
(Dantec 55R11, $70\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter with a
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ nickel coating,
$1.3\text{\,}\mathrm{mm}$ long and up to $30\text{\,}\mathrm{kHz}$ frequency
response) placed in the middle of the tunnel. CTA is used for calculating the
velocity spectra and the corresponding method employed in this work can be
found in Appendix A. We note that since a CTA measurement requires negligible
temperature variation of the flow, the velocity is measured when the heaters
are switched off, but left in place such as to not alter the incoming flow.
The velocity statistics are not influenced by the temperature because the
fluctuations in the flow are completely dominated by the turbulence generated
by the active grid and the upward mean flow rather than by buoyancy caused by
density differences due to the temperature differences, i.e., it is an
approximately passive scalar. To substantiate this statement, we must compare
the relative importance of the buoyancy force (parallel to the streamwise
direction) to the inertial force due to the advection of a passive scalar. The
ratio of these forces equals the ratio of the Grashof number Gr and the square
of the Reynolds number $\text{Re}_{0.5h}^{2}$ (Schlichting & Gersten (2017)).
Here we base these numbers on the vertical distance between $\pm 0.25h$ from
the middle of the measurement section, the temperature difference across such
height $\Delta_{0.5h}\leq$50\text{\,}\mathrm{mK}$$ and the mean liquid
velocity $U=$0.5\text{\,}\mathrm{m}{\mathrm{s}}^{-1}$$. This gives
$\text{Gr}/\text{Re}_{0.5h}^{2}=0.5hg\beta\Delta_{0.5h}/U^{2}\leq\mathcal{O}(10^{-3})$,
where $g$ and $\beta$ are the gravitational constant and volumetric expansion
coefficient of water, respectively. The small value of this ratio implies that
the temperature can be seen as passive scalar under our flow conditions.

Figure 1: (a) Schematic of the experimental setup used in the current study.
Bubbles are injected by 120 needles (green). The setup is equipped with 12
heaters. A power of $2250\text{\,}\mathrm{W}$ is supplied to the left half of
the heaters (red), while the right half (white) is left unpowered, creating a
turbulent thermal mixing layer. Each heating cartridge (Watlow, Firerod
J5F-15004) has a diameter of $12.7\text{\,}\mathrm{mm}$. The center-to-center
distance between two consecutive heaters is $37.5\text{\,}\mathrm{mm}$. The
turbulence is stirred by an active grid with a total of 15 engines connected
to diamond flaps (yellow) (Gvozdić et al. (2019)). The measurement section
(blue area) has a height of $1.00\text{\,}\mathrm{m}$, a width of
$0.3\text{\,}\mathrm{m}$, and is $0.04\text{\,}\mathrm{m}$ thick. (b,c,d) High
speed snapshots of the bubble flow, as seen from the side for the volume
fractions $\alpha=0.3\%,1.4\%$, and $4.7\%$. Scale bars of
$20\text{\,}\mathrm{mm}$ are added to each figure. (e) Distribution charts of
the bubble diameter obtained using image analysis of high-speed footage for a
variety of gas volume fractions $\alpha$. The resolution of the footage
prohibits us to measure the size of bubbles (dashed line)
$<$1.4\text{\,}\mathrm{mm}$$. Dots indicate the mean bubble diameter $d$. (f)
Histograms of the bubble aspect ratio $\chi$ obtained by fitting the bubbles
on the high-speed footage with ellipses. $\chi$ is defined as the ratio of the
semi-major axis divided by the semi-minor axis. Each black dot represents the
mean aspect ratio $\chi$ and each error bar represent one standard deviation
of the distribution. See also Table 2 for the details of the bubble
properties.

###  2.2 Measurement conditions: single-phase turbulent characterisation

Next, we present the turbulent characteristics in the single-phase turbulent
thermal mixing layer. Firstly, we characterise the velocity field. Laser
Doppler anemometry (LDA) (Dantec) is used for measuring the horizontal
($u_{x}$) and vertical ($u_{z}$) velocity in order to characterise the
velocity fluctuations. Throughout the manuscript we will use a prime (′) to
denote the standard deviation of any quantity, and therefore the standard
deviation of the velocity in the single-phase case is $u_{0}^{\prime}$ and we
approximate $u_{0}^{\prime}\approx u_{z}^{\prime}$. In the single-phase case,
at the middle of the tunnel, $u_{x}^{\prime}/u_{z}^{\prime}=0.69\pm 0.07$,
where the statistical uncertainty is one standard deviation. This is not
ideally isotropic. Therefore, for single-phase case, the estimation of the
relevant scales using the assumption of isotropy below is only a rough
approximation. The viscous dissipation rate $\epsilon$ is estimated by finding
the longitudinal second order structure function $D_{LL}(r)$ in single-phase
flow, where $r$ is the spatial distance between two points (measured by CTA
and used Taylor’s frozen flow hypothesis) and using the velocity two-third law
$D_{LL}=C_{2}(\epsilon r)^{2/3}$ which is valid in the inertial range,
assuming isotropy and where $C_{2}=2.0$ is a universal constant (Pope (2000)).
As the inertial range has limited extension (see Fig. 2a), we first identify
the value of $r$ such that $D_{LL}(r)/r^{2/3}$ becomes locally flat versus
$r$, and then employ the velocity two-third law. We then obtain the Kolmogorov
length scale $\eta=(\nu^{3}/\epsilon)^{1/4}=$0.22\text{\,}\mathrm{mm}$$ and
the dissipation time scale
$\tau_{\eta}=(\nu/\epsilon)^{1/2}=$49\text{\,}\mathrm{ms}$$. Moreover, we
estimate the Taylor-microscale
$\lambda_{u}=\sqrt{15\nu(2k/3)/\epsilon}=$4.9\text{\,}\mathrm{mm}$$ and the
Taylor-Reynolds number $Re_{\lambda_{u}}=(2k/3)^{1/2}\lambda_{u}/\nu=130$,
where $k=3u_{0}^{\prime}/2$ is the average turbulent kinetic energy, by
assuming isotropy. To estimate the velocity integral length scale $L_{u}$, we
use $\epsilon=C_{\epsilon}u_{0}^{\prime 3}/L_{u}$, where $C_{\epsilon}=0.9$
(Valente & Vassilicos (2012); Vassilicos (2015)).

Secondly, for the scalar field characterisation, the temperature fluctuations
are measured by a thermistor. It is well-known that passive scalar fields are
not isotropic at small scales when there is a large-scale mean temperature
gradient (Warhaft (2000)) but for the order of estimation of the quantities
below, we assume isotropy. First, to estimate the scalar dissipation rate
$\epsilon_{\theta}$. To do so, we again assume isotropy and use the
temperature (scalar) two-third law
$D_{\theta\theta}(r)=\tilde{C_{\theta}}\epsilon_{\theta}\epsilon^{-1/3}r^{2/3}$
which is valid in the inertial-convective subrange (Monin & Yaglom (1975)),
where $D_{\theta\theta}$ is the second order structure function of the
temperature (scalar) fluctuation and $\tilde{C_{\theta}}\approx
0.25^{-1}C_{\theta}$ (Monin & Yaglom (1975)) with $C_{\theta}\sim 0.5$ the
Obukhov-Corrsin constant (Mydlarski & Warhaft (1998); Warhaft (2000)). Again
we identify the plateau of the plot $D_{\theta\theta}(r)/r^{2/3}$ versus $r$
in order to locate the position of $r$ at which we employ the temperature two-
third law. Second, we estimate the scalar Taylor microscale
$\lambda_{\theta}=\sqrt{6\kappa\langle T^{\prime
2}\rangle/\epsilon_{\theta}}=$2.1\text{\,}\mathrm{mm}$$ (see e.g. Yasuda et
al. (2020)), again assuming isotropy. We find that our turbulent thermal
mixing layer, for the single phase case, has Péclet numbers (based on both the
velocity and temperature Taylor microscales)
$\text{Pe}_{\lambda_{u}}=u_{0}^{\prime}\lambda_{u}/\kappa=870$ and
$\text{Pe}_{\lambda_{\theta}}=u_{0}^{\prime}\lambda_{\theta}/\kappa=520$
respectively (Yasuda et al. (2020)). The traditional choice of the microscale
is $\lambda_{u}$ (Mydlarski & Warhaft (1998)) but numerical simulations have
uncovered that the small-scale anisotropy of the scalar field depends on the
Péclet number based on $\lambda_{\theta}$ (Yasuda et al. (2020)). So here we
give both values. For the length scale of the scalar dissipation, since
$\text{Pr}=6.5>1$, we use the Batchelor scale to characterise the smallest
scale of the passive scalar field which is given by
$\eta_{\theta}=\eta\text{Pr}^{-1/2}$ (Tennekes & Lumley (1972)). As the mean
temperature profile for the single-phase turbulent mixing layer follows a
self-similar error-function profile (Ma & Warhaft (1986); de Bruyn Kops &
Riley (2000)), we fitted the profile accordingly and obtain the large-scale
characteristic length $L_{\theta}$ of the single-phase thermal mixing layer
from the fitting parameters (see Appendix B for details). The turbulent
characteristics of the single-phase turbulent mixing layer are summarised in
Table 1.

$\eta$ (mm) |  $\tau_{\eta}$ (ms) |  $\eta_{\theta}$ (mm) |  $\lambda_{u}$ (mm) |  $\lambda_{\theta}$ (mm) | $Re_{\lambda_{u}}$ | $Pe_{\lambda_{u}}$ | $Pe_{\lambda_{\theta}}$ |  $L_{u}$ (m) |  $L_{\theta}$ (m)
---|---|---|---|---|---|---|---|---|---
$0.22$ | 49 | $0.085$ | $4.9$ | $2.1$ | 130 | 870 | 520 | 0.04 | 0.15
Table 1: Relevant turbulent flow parameters of the single-phase turbulent
thermal mixing layer in the current study ($\alpha=0\%$, mean liquid flow
speed = 0.5 m/s). Note that isotropy is assumed for the estimation of the
above quantities except $L_{\theta}$. For the definitions of the parameters
and the discussion on the isotropy, see Sec. 2.

###  2.3 Measurement conditions: Bubble properties

Bubbles are created by injecting regular air through needles, see Fig. 1a.
Using high-speed imaging in a backlight configuration we characterize the size
of our bubbles for a variety of volume fractions, see the example photos Figs.
1(b,c,d). We analyze the data by making use of the circular Hough transform to
detect the individual (potentially overlapping) bubbles. We track the bubbles
over time such as to obtain the mean bubble rise velocity, see Table 2. In
Fig. 1(e) we show the PDF of the bubble diameter for several $\alpha$ in the
form of distribution charts. The mean bubble diameter obtained from the image
analysis is approximately constant but there are changes in the bubble
distribution over $\alpha$.

First, we discuss the change of the bubble distribution for different $\alpha$
and the possible contributions that lead to such change. For $\alpha=0.3\%$,
when measuring the bubble diameter through image analysis, there are
unresolved bubbles with sizes smaller than $1.4\text{\,}\mathrm{mm}$, thus
overestimating the mean bubble diameter for that small $\alpha=0.3\%$. The
larger spread of the bubble size distribution for $\alpha=0.3\%$ may be
accounted as follow. In order to produce bubbles in our setup, the array of
needles at the bottom of the setup is fed by pressurised air from only one
side (see Fig. 1a and also Gvozdić et al. (2019)). The case for $\alpha=0.3\%$
is close to the minimum achievable gas volume fraction at which the pressure
that pushes the air out from the array of needles may become more uneven at
different needle positions (the needles that are closer to the inlet valve
have larger pressure) than for the higher $\alpha$ cases. Thus, the bubbles
come out with more diverse sizes than for the cases with higher $\alpha$.
Moreover, higher $\alpha$ also means that there are more bubble-bubble
interactions which include more coalescence of bubbles, thus contributing to
the smaller spread of the bubble sizes than what happens at lower $\alpha$
cases. For $\alpha>0.3\%$, the mean bubble diameter is essentially unchanged,
having the value of $d\approx$2.5\text{\,}\mathrm{mm}$$, see also Table 2
column $d$. However, even for $\alpha>0.3\%$, the standard deviation of the
bubble sizes changes slightly, in particular dropping from
$0.6\text{\,}\mathrm{mm}$ to $0.4\text{\,}\mathrm{mm}$ when $\alpha$ increases
from $2.2\%$ to $3.0\%$. The distributions remain to have similar shape for
higher $\alpha$ except having a slight increase of width at the highest
$\alpha=4.7\%$ (see Table 2). However, this slight change of the width of the
distribution for $\alpha>0.3\%$ can be considered minor if we see the width of
the bubble distribution is around $0.5\text{\,}\mathrm{mm}$ for
$\alpha>0.3\%$.

For the slight decrease of the width of the bubble size distribution when
$\alpha$ increases further from $0.3\%$, we can attribute to (1) the
possibility that more even pressure distribution over the different needles
that yields more evenly distributed sizes of bubbles when the bubbles comes
out, and (2) more bubble-bubble interaction and coalescence of bubbles, which
makes the spread of the sizes lower. There are also (3) the effect of
incidence turbulence and (4) the effect of the cutting of the bubbles due to
active grid (discussed in the next paragraph) which can also affect the bubble
size distribution.

Second, we compare the variation of the mean bubble diameter in our case with
other studies and discuss a possibility that leads to our observation. While
the mean bubble diameter stays roughly constant over $\alpha$, other research,
which also makes use of capillary needles to inject bubbles, indicates that
the bubble diameter increases with $\alpha$. This includes a homogeneous
bubbly flow in quiescent liquid (Colombet et al. (2015)) and a rising bubbly
flow with a background upward liquid flow which is weakly turbulent (Roig &
Larue de Tournemine (2007)). Bubble swarms with a background upward liquid
flow that is strongly turbulent were investigated in Alméras et al. (2017) and
Prakash et al. (2016) by using the Twente Water Tunnel (Poorte & Biesheuvel
(2002)), and they find also a small increase of $d$ with $\alpha$ but the
investigated range of $\alpha$ was limited ($\alpha<1\%$). It is not clear why
the bubble sizes are roughly constant in our case. The active turbulent grid
in our setup has a similar design but with different aspect ratio and has a
smaller overall size than the one used in the Twente Water Tunnel (Alméras et
al. (2017) and Prakash et al. (2016)). It is also reported in Alméras et al.
(2017) and Prakash et al. (2016) that the fragmentation of bubbles caused by
the active grid leads to smaller bubble sizes for high liquid mean flow. In
our case, while there is a possibility that bubble diameter may increase with
$\alpha$ before the bubbles entering the active turbulent grid, possibly the
rotating speed of the active turbulent grid is too high that it cuts the
bigger bubbles into smaller bubbles with sizes similar to those at lower
$\alpha$.

Third, we obtained the aspect ratio $\chi$ of the bubbles by fitting the
bubbles with ellipses in the high-speed footage. The bubble diameter analysis
has a resolution limit of $1.4\text{\,}\mathrm{mm}$, while the aspect-ratio
analysis uses all sizes (though the tiny bubbles of $\mathcal{O}(1)$ pixels
are still not selected for analysis). Bubbles with smaller sizes, in
particular for $\alpha=0.3\%$, are more spherical bubbles because of the
stronger effect of surface tension. For each $\alpha$, we use
$\mathcal{O}(200)$ bubbles for fitting to obtain the aspect ratios statistics.
Fig. 1f shows $\chi$ versus $\alpha$. The trend shows the mean be to almost
monotonically increasing with $\alpha$, saturated at high $\alpha$. When
$\alpha$ increases from $0.3\%$ to $1.0\%$, there is a considerable increase
of aspect ratio $\chi$ from 1.4 to 1.7, see Table 2. Increasing $\alpha$
further, the aspect ratio saturates. In attempt to explain the monotonic
increase of $\chi$ versus $\alpha$, one may check the Weber number and bubble
Reynolds number, which are the dimensionless parameters for a freely rising
single bubble. Although there is a slight increase of $d$ for $\alpha$ being
from $0.3\%$ to $0.6\%$, the relative rise velocity of the bubbles decreased
dramatically when $\alpha$ increases. Therefore, the bubble Reynolds number
drops dramatically, see Table 2. Moreover, a lower Weber number based on the
relative rise velocity $\text{We}_{V_{r}}$ is also resulted, see Table 2.
However, the Weber number based the liquid velocity fluctuations
$\text{We}_{U^{\prime}}$ increases because the liquid velocity agitation
dramatically increases with $\alpha$. There are also higher (1) bubble-bubble
interactions as $\alpha$ increases and we have to take into account of (2)
incident turbulence in our case and its interaction with the bubbles. Then,
apart from the inhomogeneity of the pressure distribution at the needles that
produces gas bubbles for very low $\alpha$, these two contributions may also
affect the value of $\chi$ and subject to further systematic investigation. We
note that in the case of freely rising bubble swarm simulation (see Fig. 5.12
in Roghair (2012)), the bubbles with the same diameter become more spherical
when $\alpha$ increases, as opposed to what we observed here. This simulation
result may limit to the importance of other possible effects such as incidence
turbulence as described above. Finally, we also point out that the change of
shape is linked to $\alpha$ and we can not exclude that the change of aspect
ratio is more important than the change of $\alpha$ itself. We will use these
bubbles to provide additional stirring to the liquid.

For the cases with bubbles, since it is highly anisotropic between vertical
and azimuthal directions, we estimate the velocity fluctuations of the liquid
as $U^{\prime}\equiv\sqrt{2\langle u_{x}^{\prime 2}\rangle+\langle
u_{z}^{\prime 2}\rangle}$, where we assume $\langle u_{x}^{\prime
2}\rangle=\langle u_{y}^{\prime 2}\rangle$. We first explore the parameter
space by increasing $\alpha$ from $0.0\%$ to $5.2\%$ for a fixed flow velocity
of $0.5\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ and with our active grid
turned on. Finally, the bubble Reynolds number
$\text{Re}_{\text{bub}}=V_{r}d/\nu$ as obtained from visualization is in the
range between 480 ($\alpha=0.6\%$) and 220 ($\alpha=4.7\%$). The above
properties of the bubbles are summarised in Table 2.

##  3 The energy and scalar spectra

Figure 2:  (color) $P_{i}(f)$, the power spectra normalised by their
respective variance of (a) the velocity fluctuations ($P_{u}(f)$) and of (b)
the temperature fluctuations ($P_{\theta}(f)$) for $\alpha$ from 0 % (purple)
to 5.2% (red). The compensated spectra $f^{3}P_{i}(f)$ and
$f^{5/3}P_{\theta}(f)$ are shown as insets. The scalar spectra are cut at the
limit of 143 Hz, corresponding to the highest resolved frequency of the
thermistor.

As the scalar in the present study is advected passively by the liquid
velocity, the corresponding energy spectra first need to be examined in order
to understand the passive scalar spectra. In Fig. 2 we show the energy and
scalar power spectra $P_{u}$ and $P_{\theta}$ for different gas concentrations
$\alpha$ normalized by their corresponding variance, corresponding to the
normalized energy and scalar power spectra, respectively. The methods of
calculating the spectra of the velocity and temperature fluctuations for two-
phase flows are discussed in Appendix A. The energy spectra for single-phase
shows a limited inertial range. However, whenever $\alpha>0$, a pronounced
$-3$ scaling is observed with the presence of an energy ‘bump’ just before
this $-3$ scaling when $\alpha$ is large enough, which is consistent with the
results of Alméras et al. (2017). We find that such $-3$ scaling occurs
roughly between $100\text{\,}\mathrm{Hz}$ to $1000\text{\,}\mathrm{Hz}$ and
has a higher energy content than the single-phase in the same frequency range.
The physical mechanism of such small-scale enhanced velocity fluctuations is
attributed to bubble-induced agitations, in which the velocity fluctuations
produced by the bubbles are directly dissipated by viscosity (Lance & Bataille
(1991); Prakash et al. (2016); Risso (2018)).

Next, we examine the passive scalar (thermal) spectra. A scaling of $-5/3$ for
$1\text{\,}\mathrm{Hz}$ $\leq f\leq$ $15\text{\,}\mathrm{Hz}$ is observed for
the single-phase scalar spectrum. The phenomenon that a developed inertial
range is observed in a passive scalar spectrum but not in the corresponding
energy spectrum due to limited Reynolds number was found previously by Jayesh
et al. (1994). As opposed to the relative high frequency (small-scale)
fluctuation enhancement in the energy spectra, the scalar spectra show
relative high frequency (small-scale) diminution when $\alpha$ increases from
zero. The $-5/3$ scaling is followed by a steeper slope when $\alpha$
increases but such change of scaling is gradual and approximately saturated to
a rough scaling of $-3$ for high enough $\alpha$ ($\alpha\approx 4.7\%$)
around $f=\mathcal{O}(10\text{Hz})$. However, as seen also in the inset of
Fig. 2b for $-5/3$ compensated spectra or the spectral local exponent in Fig.
3b, the original $-5/3$ scaling becomes less pronounced and has a steeper
slope also at the frequency range before the developed $-3$ scaling for the
thermal spectra. The main differences between the energy and the passive
scalar spectra are that the rising bubbles produce additional fluctuations in
the former but diminish the fluctuations in the latter. As we will explain
below, this can be understood due to the added small-scale mixing of the
thermal fluctuations by the bubbles which smoothens the temperature field. The
emergence of the $-3$ scaling is abrupt in the former but gradual in the
latter; and the $-3$ scaling occurs at $\mathcal{O}(100\text{Hz})$ for the
former but $\mathcal{O}(10\text{Hz})$ for the latter. Clearly, there are
differences in the physical mechanisms that lead to the same $-3$ scaling for
the energy and passive scalar spectra. We first clarify the physical and
mathematical background before speculating on the emergence of the $-3$
scaling in the scalar spectra.

The main commonalities and differences of the physical settings of the
velocity and temperature fields are as follows. One common aspect is that
along the measurement section, both the viscosity and diffusivity reduce the
velocity and temperature fluctuations, respectively. On the other hand, the
mean velocity field is homogeneous in the bulk of the measurement section
while the temperature field has a large-scale mean temperature gradient. The
high-Re bubbles have viscous boundary layers at their air-water interface but
the thermal boundary layer is expected to be minimal as the heat capacity of
the bubbles is tiny, and their thermal response is fast with at most a time
scale of (conductive case, no thermal convection)
$t=\delta^{2}/D_{\text{air}}=\mathcal{O}($50\text{\,}\mathrm{ms}$)$, where
$\delta=0.4d$ is the penetration depth for a spherical bubble and
$D_{\text{air}}$ the thermal diffusivity of air. If convective flow inside the
bubble is included this typical time-scale is drastically reduced, implying
that the bubble can adjust its temperature very quickly to its surroundings.
Moreover, if we compare the heat capacities of the water and the bubble we
get:
$(95\%C_{p,\text{water}}\rho_{\text{water}})/(5\%C_{p,\text{air}}\rho_{\text{air}})\approx
65000$, meaning that the bubbles can not be very effective carriers of thermal
energy as compared to the water. The viscous boundary layers transport
momentum to the liquid phase at the scale related to bubbles, producing
velocity agitations. The absence of thermal boundary layers implies no heat
transport is present at the scale related to bubbles. Therefore, the only
source of scalar fluctuations is from the mean temperature gradient while the
source of velocity fluctuations is from bubble agitations.

Mathematically, for the spectral behavior of a passive scalar, one can
consider the general spectral equation for the three dimensional scalar
spectra $E_{\theta}(k,t)$ derived from the advection-diffusion equation (Monin
& Yaglom (1975)) which neglects viscous heating,

$\frac{\partial}{\partial t}E_{\theta}(k,t)+2\kappa
k^{2}E_{\theta}(k,t)=T_{\theta}(k,t)+\Pi_{\theta}(k,t),$ (1)

where $t$ is the time and $T_{\theta}$ and $\Pi_{\theta}$ are the local net
transfer and the production at wave-number $k$ respectively. One can employ
Taylor’s frozen-flow hypothesis to transform from the frequency domain to the
wavenumber domain. In the present case, for increasing $\alpha$, the single-
phase $-5/3$ scaling of the spectrum gradually transitions to a $-3$ scaling
at higher $k$ (or $f$).

With the above discussed physical settings and scaling behaviors, we now
speculate on the physical mechanisms on the $-3$ scaling of the scalar spectra
in bubbly flows. The $-5/3$ range is considered to be the inertial range where
there is negligible net local transfer, negligible production and neglected
diffusivity (and viscosity) effect (Pope (2000)). The scalar production due to
the mean temperature gradient occurs at a smaller frequency than the $-5/3$
frequency range (lower than $1\text{\,}\mathrm{Hz}$). Since the production due
to the presence of bubbles is negligible as discussed previously, we speculate
that $\Pi_{\theta}\approx 0$ at the scalar $-3$ subrange. Furthermore, we
speculate that the rising bubbles enhance the mixing of temperature by
homogenizing the temperature with increasing $\alpha$ at the scale related to
the bubbles, and thus we observe a faster drop (steeper slope) in the passive
scalar spectra. When $\alpha$ is large enough, the smoothing of the
temperature inhomogeneity due to the bubbles is so strong that the thermal
fluctuations are directly smoothed out by the molecular thermal diffusivity
which causes the saturation of the scaling. In other words, the thermal
fluctuations that originally have cascading behaviour (as in the single-phase
case) passing them from larger scales to smaller scales and ceasing by the
molecular thermal diffusion. When this situation happens, in this regime, we
speculate that $T_{\theta}=T_{\theta}(\epsilon_{\theta},k)$, where
$\epsilon_{\theta}$ is the scalar fluctuation dissipation rate. From
dimensional analysis, we find that $T_{\theta}\propto\epsilon_{\theta}k^{-1}$.
For a statistically stationary state, we can rearrange the terms in Eq. 1 to
obtain a $E_{\theta}\propto k^{-3}$ scaling. This derivation is similar to
Lance & Bataille (1991) though they neglected the net local transfer term and
performed dimensional analysis on the velocity fluctuation production by the
bubbles instead. Finally, as noted above, the original $-5/3$ plateau found in
Fig. 2b becomes only a peak for increasing $\alpha$, i.e. a pure $-5/3$
scaling disappears. For high $\alpha$, even when $-3$ scaling has developed,
the mixing of the bubbles may lead to an overall more homogeneous temperature
field which leads to a smaller temperature fluctuation at the scales also
before the $-3$ scaling, leading to a steeper slope also before that. As we
will also see in the next section, the onset frequency of $-3$ scaling becomes
lower for higher $\alpha$. If $\alpha$ further increases, this leads us to
expect that such steeper slope (steeper than the originally $-5/3$) before the
developed $-3$ scaling will be further steepened, closer to $-3$.

##  4 Transition frequencies from $-5/3$ to $-3$ scaling

The frequencies corresponding to the onset of the $-3$ scaling of the scalar
spectra are now examined. When $\alpha$ increases from 0% to 5.2%, the $-5/3$
scaling becomes less pronounced and, when $\alpha$ is large enough, it is
followed by a $-3$ scaling, see Fig. 3a for examples. We identify the onset
frequency $f_{t}$ by using the parameterization

$\displaystyle
P_{\theta}(f)=\frac{(f/f_{L})^{\zeta_{b}}}{\left[1+(f/f_{t})^{2}\right]^{\frac{\zeta_{b}-\zeta_{a}}{2}}}$
(2)

which captures a transition from one scaling to another. Here $f_{L}$ is a
fitting parameter which reflects the height of the curve, and $f_{t}$ is the
transition frequency from the scaling $f^{\zeta_{b}}$ (for $f\ll f_{t}$) to
$f^{\zeta_{a}}$ (for $f\gg f_{t}$) with scaling exponents $\zeta_{b}$ and
$\zeta_{a}$, respectively, see Fig. 3. Here we do not fit these exponents, but
set $\zeta_{b}=-5/3$ (Kolmogorov-Obukhov value) (Monin & Yaglom (1975)) and
$\zeta_{a}=-3$, consistent with the exponent of the velocity spectra in high-
Re bubbly flows (Risso (2018)) and reflecting the observed limiting cases. To
determine the range that covers most of the $-5/3$ and $-3$ scaling, we
examine the local logarithmic slope $\xi(f)$ of the scalar power spectra,
which is given by

$\displaystyle\xi(f)=\frac{d\log_{10}P_{\theta}(f)}{d\log_{10}f}.$ (3)

To estimate the local logarithmic slope at different frequency, for each
spectrum, a moving fit of a straight line in logarithmic space (centred at a
frequency with a window size of one-fourth decade of the frequency) is
employed in order to obtain a local logarithmic slope at that frequency. The
results of $\xi(f)$ for different $\alpha$ are shown in Fig. 3b. In the
figure, we see that around $1.2\text{\,}\mathrm{Hz}$, $-5/3$ scaling starts to
develop and the $-3$ scaling ends around $15\text{\,}\mathrm{Hz}$ for
$\alpha=2.2\%$. Therefore, $1.2\text{\,}\mathrm{Hz}$ to
$15\text{\,}\mathrm{Hz}$ is a reasonable range that covers the transition from
$-5/3$ to $-3$ scaling for all cases.

$\alpha$ $\text{\,}\mathrm{(}\%\mathrm{)}$ |  $U^{\prime}$ $\text{\,}\mathrm{(}\mathrm{m}\mathrm{/}\mathrm{s}\mathrm{)}$ |  $d$ $\text{\,}\mathrm{(}\mathrm{m}\mathrm{m}\mathrm{)}$ | $\text{We}_{U^{\prime}}$ | $\chi$ |  $V_{bub}$$\text{\,}\mathrm{(}\mathrm{m}\mathrm{/}\mathrm{s}\mathrm{)}$ |  $\langle V_{r}\rangle$$\text{\,}\mathrm{(}\mathrm{m}\mathrm{/}\mathrm{s}\mathrm{)}$ | $\langle\text{We}_{V_{r}}\rangle$ | $\langle\text{Re}_{\text{bub}}\rangle$ |  $f_{t}$$\text{\,}\mathrm{(}\mathrm{Hz}\mathrm{)}$ |  $f_{L}$$\text{\,}\mathrm{(}\mathrm{Hz}\mathrm{)}$ |  $f_{c,A}$$\text{\,}\mathrm{(}\mathrm{Hz}\mathrm{)}$
---|---|---|---|---|---|---|---|---|---|---|---
$0.3\pm 0.2$ | — |  $2.4\pm 0.5$* | — | $1.4\pm 0.5$ | $0.68\pm 0.14$ | — | — | — | — | — | —
$0.6\pm 0.2$ | $0.042\pm 0.008$ |  $2.5\pm 0.5$* | $0.06\pm 0.03$ | $1.6\pm 0.6$ | $0.70\pm 0.13$ | $0.194\pm 0.03$ | $1.30\pm 0.04$ | $512\pm 9$ | — | — | $11.0\pm 0.4$
$1.0\pm 0.2$ | $0.051\pm 0.007$ |  $2.5\pm 0.5$* | $0.09\pm 0.03$ | $1.7\pm 0.6$ | $0.68\pm 0.11$ | $0.179\pm 0.004$ | $1.13\pm 0.04$ | $480\pm 10$ | $10.6\pm 0.7$ | $0.54\pm 0.01$ | $9.9\pm 0.4$
$1.4\pm 0.2$ | $0.047\pm 0.006$ |  $2.6\pm 0.6$* | $0.08\pm 0.03$ | $1.8\pm 0.7$ | $0.67\pm 0.11$ | $0.164\pm 0.003$ | $0.96\pm 0.03$ | $448\pm 7$ | $8.1\pm 0.5$ | $0.55\pm 0.01$ | $9.0\pm 0.3$
$2.2\pm 0.2$ | $0.064\pm 0.005$ |  $2.6\pm 0.6$* | $0.15\pm 0.04$ | $1.9\pm 0.7$ | $0.64\pm 0.11$ | $0.148\pm 0.003$ | $0.79\pm 0.04$ | $411\pm 9$ | $5.8\pm 0.3$ | $0.58\pm 0.02$ | $7.9\pm 0.4$
$3.0\pm 0.2$ | $0.073\pm 0.006$ |  $2.5\pm 0.4$* | $0.19\pm 0.04$ | $1.9\pm 0.7$ | $0.62\pm 0.11$ | $0.130\pm 0.004$ | $0.59\pm 0.04$ | $350\pm 10$ | $4.8\pm 0.3$ | $0.57\pm 0.02$ | $7.2\pm 0.4$
$3.9\pm 0.2$ | $0.090\pm 0.006$ |  $2.5\pm 0.4$* | $0.29\pm 0.06$ | $1.9\pm 0.7$ | $0.60\pm 0.11$ | $0.117\pm 0.005$ | $0.48\pm 0.04$ | $320\pm 10$ | $4.1\pm 0.2$ | $0.60\pm 0.02$ | $6.4\pm 0.5$
$4.7\pm 0.2$ | — |  $2.6\pm 0.5$* | — | $2.0\pm 0.6$ | $0.59\pm 0.11$ | $0.090\pm 0.004$ | $0.29\pm 0.028$ | $250\pm 10$ | $3.9\pm 0.2$ | $0.59\pm 0.02$ | $4.9\pm 0.5$
$5.2\pm 0.2$ |  |  | — | — | — | — | — | — | $3.5\pm 0.2$ | $0.61\pm 0.02$ | —
Table 2: Experimental parameters as a function of $\alpha$: velocity
fluctuations of the liquid $U^{\prime}\equiv\sqrt{2u_{x}^{\prime
2}+u_{z}^{\prime 2}}$ with primed velocities referring to the standard
deviation of these velocity components, area-equivalent bubble diameter $d$,
the Weber number based on the turbulent velocity $U^{\prime}$ for the bubbles
$\text{We}_{U^{\prime}}=\rho U^{\prime 2}d/\gamma$ (with $\gamma$ the surface
tension), the aspect ratio of the bubble diameter $\chi$, the bubble rise
velocity $V_{bub}$, the bubble relative (to the liquid phase) rise velocity
$V_{r}\equiv V_{\text{bub}}-\langle u_{z}\rangle$, the mean Weber number based
on the bubble relative rise velocity $\text{We}_{V_{r}}=\rho
V_{r}^{2}d/\gamma$, the bubble Reynolds number
$\text{Re}_{\text{bub}}=V_{r}d/\nu$, the fitting parameters $f_{t}$ and
$f_{L}$ (below Eq. 2), and $f_{c,A}$ (Eq. 4) (Alméras et al. (2017)). The
bubble sizes and velocity are obtained by image analysis (see Sec. 2). *Note
that for $\alpha=0.3\%$, we slightly overestimate the mean bubble diameter and
underestimate the width of the bubble size distribution because of the
resolution limit of the image analysis for the bubble diameter. The aspect
ratio of the bubbles are obtained by fitting the bubbles with ellipses. The
uncertainty of $\alpha$ stems from the precision of the differential pressure
gauge (Gvozdić et al. (2019)). For $U^{\prime}$, $d$,
$\text{We}_{U^{\prime}}$, $\chi$, $V_{bub}$, we show the mean values and the
corresponding standard deviations for the distributions. For $V_{r}$,
$\text{We}_{V_{r}}$ and $\text{Re}_{\text{bub}}$, we show the corresponding
mean values and the standard errors of the mean. For $f_{L}$ and $f_{t}$, we
show the mean values and the corresponding confidence intervals for a 95%
confidence level obtained from the fitting. For $f_{c,A}$ , we show the mean
values and the corresponding confidence intervals for a 95% confidence level.

We fit the spectra of Fig. 2b with Eq. 2 with $\zeta_{b}=-5/3$ and
$\zeta_{a}=-3$ only for the $\alpha\geq 1.0\%$ cases, since for $\alpha<1\%$
the transition frequencies resulting from the fits are either outside the
fitting range or very close to the boundary of the fitting range. The fits for
$\alpha=1.0\%$ and $\alpha=5.2\%$ are shown in Fig. 3a. It shows qualitatively
nice fits for $\alpha=1.0\%$, even though this case does not exhibit a fully
developed $-3$ scaling. The figure also shows that for the case of
$\alpha=5.2\%$ the scaling is close to $-3$ after the transition and that the
transition frequency $f_{t}$ is smaller for this higher value of $\alpha$.

Figure 3:  (a) The power spectral densities $P_{\theta}(f)$ (normalized by
their respective variance) of the temperature fluctuation from the
measurements (dots) and the corresponding fits (Eq. 2) in logarithmic scale
for the gas volume fractions $\alpha=1.0\%$ and $\alpha=5.2\%$ at the
frequency range of $1.2\text{\,}\mathrm{Hz}$ to $15\text{\,}\mathrm{Hz}$
(colors same as Fig. 2), plotted together to the purpose of showing the
quality of the fits. The two grey sloped lines labeled with ‘$-5/3$’ and
‘$-3$’ indicate the scaling behaviours of the two limits. The vertical grey
lines labeled by $f_{t}$ indicate the fitted transition frequency for the two
cases. (b) The local scaling exponent $\xi(f)$ (defined by Eq. 3) of
$P_{\theta}(f)$ over an interval of more than three decades for different gas
volume fractions $\alpha$. Two grey horizontal lines indicates the the slopes
$-5/3$ and $-3$. The details of obtaining the local slope is described in
Section 4. (c) Comparison of $f_{c,A}(\alpha)$ (open symbols, Eq. 4) and
$f_{t}(\alpha)$ (solid symbols). The uncertainty of $\alpha$ is related to the
precision of the differential pressure gauge (Gvozdić et al. (2019)). while
the error bars of $f_{d}$ and $f_{c,A}$ are the confidence intervals from the
fitting Eq. 2 for a $95\%$ confidence level. The the fitting results of
$f_{t}$ are also tabulated in Table 2.

Table 2 shows the transition frequencies $f_{t}$ obtained from the
parameterisation (Eq. 2) as function of $\alpha$ for $\alpha\geq 1.0\%$. A
decreasing trend of $f_{t}$ with increasing $\alpha$ is found. Currently,
there is no prediction for the onset frequencies $f_{t}$ of the bubble-induced
subrange for the passive scalar spectra. However, there is a candidate of the
onset frequency of the $-3$ scaling in the energy spectra, which is
proportional to $V_{r}/d_{eq,V}$, where $V_{r}$ is the relative (to the liquid
phase) rise velocity of the bubbles and $d_{eq,V}$ is the mean volume
equivalent diameter of the bubbles (Alméras et al. (2017)). This frequency is
given by $f_{\text{bub}}\equiv V_{r}/\lambda_{\text{bub}}\equiv
C_{D}V_{r}/d_{eq,V}$ (Alméras et al. (2017)), where $C_{D}$ is the drag
coefficient of the bubbles, and $\lambda_{\text{bub}}\equiv d_{eq,V}/C_{D}$ is
the candidate for the onset scale for the $-3$ scaling in the energy spectra
in the wavenumber space (Riboux et al. (2010); Risso (2018)). Apart from
$f_{\text{bub}}$, there are several other frequency scales that are
proportional to $V_{r}/d_{eq,V}$, which roughly locate a ‘bump’ in the energy
spectra just before the $-3$ scaling for high enough $\alpha$, summarised in
Alméras et al. (2017). For example, based on their observations in their
energy spectra, Alméras et al. (2017) gives

$\displaystyle f_{c,A}=0.14\langle V_{r}\rangle/\langle d\rangle,$ (4)

where we stress again that $d$ is the area-equivalent diameter whereas in
Riboux et al. (2013) the mean volume-equivalent diameter is used instead.
Table 2 includes these values of $f_{c,A}$, which we are also shown in Fig.
3c. Since the $-3$ scaling has not developed for low $\alpha$ and thus the
identification of $f_{t}$ for low $\alpha$ is hard to determine, we can only
conclude that both $f_{c}$ and $f_{t}$ identified here have decreasing trends
but of the same order of magnitude in our parameter regime.

Apart from $f_{c,A}$, when $\alpha$ becomes large, the typical time scale for
the effective diffusion for passive scalar in bubble-induced turbulence is the
time scale for successive bubble passages (Alméras et al. (2015)), which is
given by

$\displaystyle T_{\text{2B}}=\frac{d}{\alpha V_{r}\chi^{2/3}}.$ (5)

The numeral values of the quantities on the right hand side of the above
equation can be found in Table 2. For $\alpha=4.7\%$,
$T_{\text{2B}}=$0.39\text{\,}\mathrm{s}$$ which results in a frequency of
$2.6\pm 0.7\text{\,}\mathrm{Hz}$, which is two times smaller than
$f_{c,A}=$4.9\pm 0.5\text{\,}\mathrm{Hz}$$. The transition frequency
$f_{t}=$3.9\pm 0.2\text{\,}\mathrm{Hz}$$ for the corresponding $\alpha$. For
low void fraction ($\alpha=0.6\%$), we find
$T_{\text{2B}}=$1.57\text{\,}\mathrm{s}$$, giving a frequency of $0.7\pm
0.3\text{\,}\mathrm{Hz}$. Therefore, for high $\alpha$, this is comparable to
our $f_{t}$. However, we see that the trend of this prediction is going up
with increasing $\alpha$, while we find that the transition frequency coming
from the spectra goes down with $\alpha$ (see Fig. 3c). This is consistent
with that for large $\alpha$ the time scale for effective diffusion by bubble-
induced turbulence can be dominated by the bubble passage time; while for
small $\alpha$ such effective diffusive time scale can be, for example, the
Lagrangian integral time scale $T_{L}$ (Alméras et al. (2015, 2019)). The
cross-over of the relevant time scale can occur when $T_{\text{2B}}=T_{L}$
(Alméras et al. (2019)). However, the Lagrangian integral time scales for
$\alpha>0$ is not available in our study. We cannot exclude the time scales
mentioned above to be relevant to the onset of the $-3$ scaling in the thermal
spectra in bubbly flows.

##  5 Conclusion

The above speculation on the $-3$ scaling mechanism for a scalar spectrum
requires diffusivity to be important. This means that the inverse of scalar
diffusive time scale needs to be around the onset frequency $f_{t}$ of the
$-3$ subrange. A direct verification of this hypothesis may be accomplished by
investigation of the diffusive and spectral transfer budgets of the scalar
fluctuations at the $-3$ subrange by direct numerical simulations similar to
what Pandey et al. (2020) did. Our hypothesis will be confirmed if within the
–3 subrange, the spectral transfer of the scalar fluctuation (behaving as
$T_{\theta}\propto k^{-1}$) is balanced by the diffusive dissipation ($2\kappa
k^{2}E_{\theta}(k)$).

To conclude, when a swarm of high-Reynolds number bubbles are injected into a
turbulent thermal mixing layer ($0.3\%\leq\alpha\leq 5.2\%$), the scalar
spectra that originally have a $-5/3$ scaling, for large frequencies develop a
steeper $-3$ slope for larger gas concentration $\alpha$. As opposed to the
energy spectra, which abruptly developed a $-3$ scaling with the enhancement
of the small-scale content for non-zero $\alpha$, the change of spectral
scaling for passive scalar spectra is more gradual by smoothing out the small-
scale content when $\alpha$ increases. While the $-3$ scaling at
$\mathcal{O}(100\text{Hz})$ of the energy spectra is attributed to the balance
of the production of velocity fluctuations due to bubbles and the molecular
viscous dissipation, we speculate that the physical mechanism of the steeper
spectral slope in the passive scalar spectra is due to enhanced mixing of
bubbles that promotes the homogenization of small-scale temperature
differences. The scaling saturates to $-3$ around
$\mathcal{O}($10\text{\,}\mathrm{Hz}$)$ when the smoothing of temperature
fluctuations is so strong such that the local net transfer of the spectral
fluctuation is directly diffused by molecular diffusivity. The transition
frequency $f_{t}$ from the $-5/3$ to $-3$ scaling in the scalar spectra are
obtained from the parameterization (Eq. 2) and found to monotonically decrease
with increasing $\alpha$.

In our experiments we had $\text{Pr}=6.5$ at
$22.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$. This Pr could experimentally be
changed in a certain narrow regime by changing the water temperature (e.g.
$\text{Pr}=12.9$ and $\text{Pr}=2.2$). To explore the Pr-dependence, direct
numerical simulations would be most welcome. Note however that the much
smaller Pr numbers of liquid metal (e.g. $\text{Pr}\approx 0.006$ for liquid
sodium at $570\text{\,}\mathrm{K}$, see Cioni et al. (1997)) are again
experimentally accessible and that the fluctuations of the temperature in
turbulent bubbly sodium may be relevant for applications in working facilities
in nuclear power station.

Acknowledgments. We thank Dennis P.M. van Gils, Gert-Wim Bruggert, and Martin
Bos for the technical support. We thank Timothy Chan and Jack Cheung for
preliminary measurements.

Funding statement. This work was supported by The Netherlands Center for
Multiscale Catalytic Energy Conversion (MCEC), an NWO Gravitation Programme
funded by the Ministry of Education, Culture and Science of the government of
The Netherlands. Chao Sun acknowledges the financial support from Natural
Science Foundation of China under Grant nos. 11988102 and 91852202.

Declare of interests. The authors report no conflict of interest.

##  Appendix A Method of calculating the energy and scalar spectra in two-
phase flow

The hot-film voltage $E_{hf}(t)$ is measured over time. We remove parts of the
signal where bubbles interact with the sensor using a threshold method on the
time derivative of the measured voltage across the hot-film (see Rensen et al.
(2005a) and the references therein) or the time derivative of the liquid
velocity (Alméras et al. (2017)). During a bubble collision with a probe,
there are rapid changes of $E_{hf}$, and the typical shapes of $E_{hf}(t)$
when bubbles colliding with a hot-film can be found in, for example, Rensen et
al. (2005a) and Rensen et al. (2005b). In this work, by inspection of the
derivative $dE_{hf}/dt$ over time, a fixed threshold
$|dE_{hf}/dt|\geq$500\text{\,}\mathrm{V}\text{\,}{\mathrm{s}}^{-1}$$ is used
to detect the moments when bubbles are colliding the probe. We find that the
findings and the conclusions of this work are robust against the threshold
values within a reasonable range. Apart from the said threshold, we also set
the minimal time between two successive bubble collisions
($5\text{\,}\mathrm{ms}$) in order to determine the moments when bubbles
impinging and leaving the probe and to capture the entire bubble interaction
events. We check the latter threshold by examining the distributions of the
bubble residence time and any unphysical bumps present in the energy spectra
due to many unphysically short bubble collisions. To calculate the velocity
power spectra, we use the Bartlett Method (Oppenheim & Schafer (2009)) with
linear interpolations between the ‘gaps’ of the liquid phase velocity over
time for each $\alpha$. Each partition segment ($20\text{\,}\mathrm{s}$) cover
at least one turnover time by inspecting the auto-correlation function of the
signal. Linear interpolation between the gaps of the liquid signal to
calculate the power spectra was also used in Alméras et al. (2016, 2017). The
discussion on the effects of using linear interpolation can be found in Rensen
et al. (2005a) and Alméras et al. (2016). For the optical method that
calculated the energy spectra while also having evenly sampled time of the
liquid velocity field, recently Ma et al. (2022) employed high-resolution
particle shadow velocimetry. The advantage of this method is that the two-
dimensional velocity field is captured simultaneously without employing Taylor
hypothesis (Ma et al. (2022)). However, the method in this work only measured
single-point velocity field and requires Taylor hypothesis when the time
coordinate is converted into spatial coordinate (and it is only one-
dimensional).

For the computation of the scalar spectra, we did not create gaps of the
temperature time series to distinguish between the gas and liquid phases. This
is mainly due to that the response time of the thermistor is comparable to the
bubble residence time and the bubbles have similar temperature as the
surrounding liquid, which is discussed and shown in Dung (2021). The Bartlett
method (Oppenheim & Schafer (2009)) is applied to the temperature signal using
partition segments of $10\text{\,}\sec$ which cover at least one turnover time
which was found by inspecting the auto-correlation function of the signal.

##  Appendix B Obtaining the thermal integral length scale

Figure 4: The mean temperature profile $\tilde{T}$ versus
$\frac{x-x_{c}}{L_{\theta}/2}$ for $\alpha=0\%$ and the corresponding fit
using Eq. (6) (solid black line), where $\tilde{T}\equiv\langle
T-T_{\text{inlet}}\rangle-\langle T-T_{\text{inlet}}\rangle|_{x/L=0.9}$,
$T_{\text{inlet}}$ is the temperature measured of the incoming liquid before
being heated up by the heaters, $x_{c}=L/2$ is the x-coordinate of the middle
of the tunnel, and $L_{\theta}$ is the fitting parameter that measures the
width of the thermal mixing layer (Eq. (6)).

For the thermal integral length scale, intuitively it is of the order of the
size of the thermal mixing layer. For a single-phase thermal-mixing layer, the
mean temperature profile follows a self-similar shape that can be
characterised by a complementary error function (Ma & Warhaft (1986); de Bruyn
Kops & Riley (2000)). Therefore, quantitatively we first fit the mean
temperature profile (with an offset of the mean temperature at the coldest
position) $\tilde{T}(x)$ by the function

$\displaystyle\tilde{T}(x)=\frac{\Delta
T}{2}\text{erfc}\left(\frac{x-x_{c}}{L_{\theta}/2}\right),$ (6)

where $x_{c}$ is the $x$-coordinate at the center of the measurement section,
$\text{erfc}(x)$ is the complementary error function; and $\Delta T$ and
$L_{\theta}/2$ are the fitting parameters which describe the asymptotic value
of the mean profile and the characteristic length of the thermal mixing layer,
respectively, see Fig. 4 for the fit. $L_{\theta}$ is then defined as the
integral length scale of the scalar fluctuations. The results are summarised
in Table 1.

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