Cahn-Hilliard-Navier-Stokes Turbulence: Physics,
Numerical Simulations, and Mathematics
Rahul Pandit
Indian Institute of Science, India
I introduce the Cahn-Hilliard-Navier-Stokes (CHNS) equations and show that they
provide a natural theoretical framework for describing two-phase flows, both laminarand turbulent. I then cover some of our recent results, based on extensive numericalsimulations, of turbulence in the CHNS system and the important dimensionless control
parameters for such turbulence. In particular, I discuss (a) the multifractal fluctuationsof a droplet in a two-phase, turbulent flow and (b) the suppression of phase separation by
turbulence in a two-fluid system. I end with a Beale-Kato-Majda-type theorem for the
regularity of solutions of the three-dimensional CHNS equations. The first two parts ofthis work have been done with Nairita Pal and Prasad Perlekar; the last part has been
done with Nairita Pal, Anupam Gupta, and John Gibbon

Energy dissipation in blood flows related to
cardiovascular problems
Hiroshi Suito1 , Viet Q.H. Huynh1 , Kenji Takizawa2
and Takuya Ueda3
1 Advanced Institute for Materials Research, Tohoku University,

Katahira 2-1-1, Aoba-ku, Sendai, 980-8577, Japan,
Email: hiroshi.suito@tohoku.ac.jp
2 Faculty of Science and Engineering, Waseda University, Japan
3 Department of Radiology, Tohoku University Hospital, Japan
Abstract
We present numerical experiments related to cardiovascular problems suchas the aortic aneurysm, which is one of the life-threatening diseases. A number of patient-specific models of the aorta as constructed from CT scans are
considered. Blood flows are aﬀected by curvature and torsion distributions
as well as branches to smaller arteries. Such geometrical characteristics ofthe vessels bring about diﬀerences in the flow characteristics, which lead todiﬀerent energy dissipation patterns and outcomes.
This work was supported by JST CREST Grant Number JPMJCR15D1,
Japan.

Friction induced instability of surface of velocity
discontinuity of a shallow-water flow
Yasuhide Fukumoto1 , Thi Thai LE2 , and Liangbing JIN3
1
Institute of Mathematics for Industry, Kyushu University, Japan,
2
Graduate School of Mathematics, Kyushu University, Japan
3
Zhejiang Normal University, China
We examine the frictional eﬀect on the stability of velocity discontinuity in a shallowwater stream. The ﬂuid is moving with a uniform velocity U in a half region but is at
rest in the other, and the bottom surface is assumed to exert the drag force, quadratic
in velocity, on the thin ﬂuid layer. In the absence of the drag, the interface of√velocity
discontinuity undergoes√ the instability of the Kelvin-Helmholtz type for U ≤ 8c, but
is stabilized for U > 8c, with c being the propagating speed of the gravity wave. The
bottom drag, however small it may be, destabilizes the interface in the latter range of the
Froude number U/c. For small values of the drag coeﬃcient, the growth rate is written
down in tidy form. We ﬁnd by an asymptotoc analysis that the instability never ceases
as the drag strength is increased.

Numerical implementation of the construction ofdissipative Euler solutions with the Mikado flow
Takeshi Matsumoto
Department of Physics, Kyoto University, Japan
Certain weak solutions of the incompressible Euler equations which dissipate the energy are likely to follow the same laws of high Reynolds number turbulent flows. A
mathematical method to construct such a weak solution has been proposed by De Lellis,
Szekelyhidi and coworkers. In this talk, we explain why these weak solutions are appealing also for physicists and mention our current attempt at numerical implementation of
their mathematical construction.

Study of Biot-Savart models for vortex reconnection
Yoshifumi Kimura
Graduate School of Mathematics, Nagoya University, Japan
Vortex reconnection under Biot-Savart evolution is investigated geometrically and numerically using vortex ﬁlament models. It is shown that the tips of these vortices approacheach other accelerating as they do so to form a ﬁnite-time singularity. The minimum separation of the vortices and the maximum velocity and axial strain rate exhibit nearly selfsimilar Leray scaling. By way of validation of the models, the structure of the eigenvalues
and eigenvectors of the rate-of-strain tensor is investigated. At the tips, it is observedthat the second eigenvalue is positive and the corresponding eigenvector is tangent to theﬁlament, implying persistent stretching of the vortex. This is a joint work with H. K.
Moﬀatt (Cambridge University).

On scalar and vectorial spherical harmonics foranalysing and calculating strongly anisotropic
shear-driven rotating turbulent flows
Claude Cambon
Ecole Centrale Lyon, France
Isotropy is often recovered in shear-driven and/or in thermally-driven turbulent ﬂows
for length scales signiﬁcantly smaller than typical threshold scales (Corrsin 1958, Ozmidov,
Woods-Hopﬁnger-Zeman (1984)); weakly anisotropic theories and models were proposed
for length scales close to, or smaller than, these threshold ones. On the other hand, an
extended scale-by-scale analysis reveals that anisotropy is strong in the largest scales andcannot be represented by angular harmonic expansions at low degree. Spherical harmonics
are used for representing the three-dimensional spectral distribution of kinetic energy, withdirectional anisotropy, and then polarization anisotropy, that ought to be disentangled(e.g. CC and Rubinstein 2006, Rubinstein et al. 2015). On this occasion, a large historicalreview of using both scalar and vectorial harmonics is proposed. Regarding realizations ofincompressible velocity ﬁelds, various toroidal/poloidal decompositions are discussed andreconciled or not: Chandrasekkar 1963, Craya (1957), Herring (1964), Riley (1981), CC
and Teissèdre (1985), Rieutord (1987). Linkage to symmetry groups, as SO(3), is touched
upon, as well as Wiener-Kinchin relationship for passing from spectra or co-spectra to
structure functions in physical space , and conversely. Finally, applications are illustrated
by the development of two-point statistics of homogeneous turbulence in the presence of
uniform mean shear and system rotation. The so-called ‘rapid distortion’ limit ought tobe captured in a large range of scales, even if non-linearity is signiﬁcant. For instance, thislimit is relevant in the infrared domain, in which purely linear dynamics compete withthe backscatter transfer, that is mediated by non-linear non-local interactions from small
scales. Depending on various dynamics, from algebraic growth to exponential growthor decay, spherical harmonics are used either for data processing only, or as a basis
for projecting dynamical equations. In our most elaborate model-equations, exact linear
eﬀects that reﬂect rapid distortion, if considered alone, are solved with maximum accuracy
on spherical shells with many angles. Meanwhile the nonlinear terms are closed by ageneralized EDQNM (Orszag 1970, Mons et al. 2016) procedure. Cross-validation with
DNS in distorted boxes (Rogallo 1981, Lesur 2005, “snoopy” code) are expected.

WEAKENED REGULARITY CONDITIONS TO INCOMPRESSIBLE
NAVIER-STOKES EQUATIONSHI JUN CHOE

1. Introduction
Incompressible viscous fluid is governed by incompressible Navier-Stokes equations:
ut − ν∆u + ∇ · (u ⊗ u) + ∇p = 0
div(u) = 0,
where ν is viscosit constant, u(x, t) is fluid velocity field at (t, x) and p(x, t) is pressure at(x, t). For convenience of presentation we assume viscosity is one.
In 1934, Leray proved that there is a weak solution in L2,∞ for initial value problem when
the initial data u(x, 0) ∈ L2 and weaklly solenoidal although the uniqueness is still unkown.
He constructed weak solution by Galerkin method. The energy inequality
Z ∞Z||∇u|2 dxdt ≤ ||u(x, 0)||2L2
supt ||u||2L2 (t) +02,2
implies strong compactness in L and one obtain strong convergece of nonlinear term u⊗uwhich is convection. We can not mention all important results related to exsitence questions in various circumstances. With perturbation for nonlinear term under boundedness
assumption for the velocity u, we obtain a stability estimate and therefore the uniquenessfollows. Similarly we have a stability estimate if the boundedness of enstrophy holds so thatZ
sup |∇u|2 dx < ∞.
t

We note that from solenodality condition the vorticity ω satisfies
ZZ
2|∇u| dx = |ω|2 dx, ω = ∇ × u.
We call a solution strong if it is weak and has bounded enstrophy.
Among the Galilean transforms, we have a special scaling for Navier-Stokes equations:(uλ (x, t), pλ (x, t)) = (λu(λx, λ2 t), λ2 p(λx, λ2 t)),since (uλ , pλ ) is a solution, too. First we find that the Lebesgue spaces
3 2
Lp,q ,+ = 1, p > 3, q > 2
p q
1

2

HI JUN CHOE

are invariant under the scaling and the condition of parameters is called LadyzhenskayProdi-Serrin condition. There are many authors, including Serrin, Prodi, Kato, Fujita,
Kozono and Kang, who considered regularity property. One notable result in view of scalingis due to Kato.
Theorem 1.1. (Kato) Suppose u(x, 0) is in L3 , then there is a positive time T such thata strong soution exists in (0, T ).
We skip recent activities to extend short time existence of strong solutions for various
scale invariant spaces.
In theories of minimal surface, harmonic map, elliptic system and freeboundary, regularity argument are very successful by scale invariance and perturbation of linear operator. In 1970’s, Scheffer was the first person who considered local regularity condition in thetheory of -regularity. His parabolic dimension estimate of possible singular set in spacetime was two. By the way, Leray already showed the Hausdorff dimension of singular timeis less than 12 and it is still unkown that 21 is best or not. We like to emphasize that there
are singular solutions to minimal surface, harmonic map, elliptic system and freeboundary
although one might need higher space dimension to construct. In Navier-Stokes equations,singular set S is the set of point z in domain such that u is unbounded in any neighborhoodof z and the space dimension is strictly physical dimension 3.
The Galerkin weak solution due to Leray for whole domian and Hoff for bounded domain
is not convenient to consider local properties and Scheffer introduced the suitable weaksolution.
Definition 1.2. We say (u, p) ∈ V × L3/2 (ΩT ) is suitable weak solution to the initial
3 ×R )
boundary value problem if for all φ ∈ C0∞ (R+
+
Z
Z
Z
Z
(1.1)
u · φt dz + ∇u : ∇φdz + u ⊗ u : ∇φdz − pdivφdz = 0

and u is weakly divergence free for almost all time, satisfies the localized energy inequality
for almost all t
Z
Z tZ
2
(1.2)
|u(x, t)| φdx + 2|∇u|2 φdxds
0
Z tZ
Z tZ
2
≤
|u| (φt + ∆φ)dxds +(|u|2 + 2p)u · φdxds
0∞
for all nonnegative φ ∈ C0 (R3 × R+ ) andZ|u(x, t) − u0 (x)|2 dx → 0

0

ast → 0,

COUNTEREXAMPLE

3

3 ).
where the initial data u0 is weakly divergence free in L2 (R+The following existence theorem is due to Scheffer and Caffarelli-Kohn-Nirenberg.
Theorem 1.3. There is a suitable weak solution to the initial boundary value problem.
We define a ball Br (x0 ) = {x : |x − x0 | < r} and a parabolic cylinder Qr (x0 , t0 ) =Br (x0 ) × (t0 − r2 , t0 + r02 ). The following  regularity theorem is due to Caffarelli-KohnNirenberg:
Theorem 1.4. There is an absolute constant 0 such thatZ1|∇u|2 dz ≤ 0
lim supr
r→0
Qr
implies that for an r0
sup |u(z)| ≤
z∈Qr0

c
r0

for an absolute constant c.
As a natural consequence we have an estimate of Hausdorff dimension of singular set S.
Theorem 1.5. We let L(δ) be the family of all coverings {Qri (zi ), ri < δ} of S and h(r) =r|logr|. We defineΨδ (S, h) = inf

X

L(δ)

h(ri )iand setΛ(S, h) = lim Ψδ (E, h).
δ→0

Then
Λ(S, h) = 0.
Recently we observed that the possible oscillation of local energy density is large nearsingular point. To see that we define
Eq (z, r) = r

−5+2q

ZZ

|∇u|q dxdt
Q(z,r)and denote
E q (z) = lim sup Eq (z, r)
r→0

and

E q (z) = lim inf Eq (z, r).
r→0

Theorem 1.6. Let 9/5 ≤ q < 2. There exists a positive number  such that z ∈ Ω × (0, T )is a regular point if
E q (z)(5−q)/(q−1) E q (z) < .

4

HI JUN CHOE

As in the case of Kato, Escauriaza–Sergin–Šverák proved that if u ∈ L3,∞ then it isregular. From the scale invariant property of Navier-Stokes equations, we have a natural
question of type one singularity and the smallest space for the type one singularity is
weak −L3 space in space. We study local regularity properties of a L∞ (0, T ; weak −L3 (R3 ))solution u to the Cauchy problem of the incompressible Navier–Stokes equations, whereu ∈ weak − L3 if
sup h3 |{u > h}| < Mhfor some M .
As an application, we conclude that there are at most a finite number of blowup points at
any singular time t. The condition that the weak Lebesgue space norm of the veclocity field uis bounded in time is encompassing type I singularity and significantly weaker than the end
point case of the so-called Ladyzhenskaya–Prodi–Serrin condition proved by Escauriaza–
Sergin–Šverák.
Theorem 1.7. For each M > 0 there exists a positive number (M ) < 1/4 such that if aweak solution u ∈ V (QT ) to the Cauchy problem satisfies the conditions(1.3)

ess sup ||u||weak−L3 ≤ M
0≤t≤T

and for some z0 = (x0 , t0 ) ∈ QT and 0 < r ≤

√

t0

1

|{x ∈ B(x0 , r) : |u(x, t0 )| > }| ≤ ,
r3
r
where |E| denotes the Lebesgue measure of the set E, then u is bounded in the space-time
cylinder Q(z0 , r).
As an application of this criterion, we are able to estimate the size of possible singularpoints at a singular time t, denoted by

Σ(t) = {x : (x, t) ∈ Σ}.
We know that the Hausdorff dimension of the possible singular time is at most 1/2. Many
researchers have been investigating the size of Σ(t) at the singular time t under various
conditions on u. Seregin obtained a result on estimating Σ(t) under the slightly weakercondition. More recently, Wang–Zhang gave a unifying results on the number of singularpoints under the Ladyzhenskaya–Prodi–Serrin type conditions.
Utilizing Theorem 1.7, we can obtain the following theorem which shows that the numberof possible singular points at any singular time t is at most finite.

COUNTEREXAMPLE

5

Theorem 1.8. Suppose u ∈ Vσ2 (QT ) is a weak solution to the Cauchy problem and satisfies
the condition (1.3) for some M > 0. Then there exist at most finite number N (M ) of
singular points at any singular time t.
At each singular time t, only a few singular points exist, yet we do not know that blowup
points are of type I or not.
References[1] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes
equations, Comm. Pure Appl. Math. 35(1982), 771-831.
[2] D. Chae and J. Wolf. On the liouville theorem for self similar solutions to the Navier-Stokes equations.
Preprint, 2016.
[3] H. J. Choe, Partial Regularity of Leray-Hopf solution for the Navier-Stokes Equations, submitted.
[4] H. J. Choe, Boundary regularity of weak solutions of the Navier-Stokes equations, J. Differential Equations 149 (1998), no. 2, 211-247.
[5] H. J. Choe and J. Lewis, On the singular set in the Navier-Stokes equations, J. Functional Analysis
175(2000), 348-369.
[6] L. Escauriaza, G. Seregin, and V. Šverák. Backward uniqueness for parabolic equations. Arch. Ration.
Mech. Anal., 169(2):147–157, 2003.
[7] G. Galdi, C. Simader and H. Sohr, On the Stokes problem in Lipshitz domain, Annali di Mat. Pura
Appl., (IV), 167 (1994), pp. 147-163.
[8] E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr.
4(1951), 213-231.
[9] S. Gustafson, K. Kang, and T. P. Tsai. Interior regularity criteria for suitable weak solutions of theNavier-Stokes equations. Comm. Math. Phys., 273(1):161–176, 2007.
[10] E. Hopf. Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr.,
4:213–231, 1951.
[11] J. Kinnunen and J. Lewis, Higher integrability for parabolic systems of p-Laplacian type, Duke Math.
J. 102(2000), 253-271.
[12] H. Kozono. Removable singularities of weak solutions to the Navier-Stokes equations. Comm. Partial
Differential Equations, 23:949–966, 1998.
[13] O. A. Ladyženskaja. Uniqueness and smoothness of generalized solutions of Navier-Stokes equations.
Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 5:169–185, 1967.
[14] O. A. Ladyzhenskaya and G. A. Seregin. On partial regularity of suitable weak solutions to the threedimensional Navier-Stokes equations. J. Math. Fluid Mech., 1(4):356–387, 1999.
[15] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta. Math. 63(1934) 193-248.
[16] F. Lin, A new proof of the Caffarelli, Kohn and Nirenberg theorem, Comm. Pure Appl. Math. 51(1998),
241-257.
[17] J. Neustupa. Partial regularity of weak solutions to the Navier-Stokes equations in the class
L∞ (0, T ; L3 (Ω)3 ). J. Math. Fluid Mech., 1(4):309–325, 1999.
[18] G. Prodi. Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. (4), 48:173–
182, 1959.
[19] V. Scheffer, Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys. 55(1977), 97-112[20] V. Scheffer, Boundary regularity for the Navier-Stokes equations. Comm. Math. Phys, 85 (1982), no.
2, 275-299.
[21] G. A. Seregin, Local regularity of suitable weak solutions to the Navier-Stokes equations near theboundary, J. Math. Fluid Mech. 4 (2002), no. 1, 1-29.
[22] J. Serrin. The initial value problem for the Navier-Stokes equations. In Nonlinear Problems (Proc.
Sympos., Madison, Wis., 1962), pages 69–98. Univ. of Wisconsin Press, Madison, Wis., 1963.
[23] V. A. Solonnikov, Estimates for solutions of the nonstationary Stokes problem in anisotropic Sobolev
spaces and estimates for the resolvent of the Stokes operator. (Russian) Uspekhi Mat. Nauk 58 (2003),
no. 2(350), 123–156; translation in Russian Math. Surveys 58 (2003), no. 2, 331-365

6

HI JUN CHOE

[24] E. Stein, ”Singular Integrals and Differentiability Properties of Functions”, Princeton Univ. Press, NJ,
1970.
[25] R. Temam, ”Navier-Stokes Equations, Theory and Numerical Analysis,” North-Holland, Amsterdam/New York, 1977.
Hi Jun Choe: CMAC/Department of Mathematics, Yonsei University, 50 Yonsei-ro, Seodaemungu, Seoul, South Korea 120-749
E-mail address: choe@yonsei.ac.kr

Recent works on the Naiver-Stokes equations withharmonic analysis
Okihiro Sawada
Gifu University, Japan
Bourgain-Pavlovic proved the ill-posedness theory of 3D Navier-Stokes equations in
critical Besov spaces by norm-inflation argument. The proof is based on tools of theharmonic analysis. However, seeking the structure of velocity field, a blow-up does notoccur. The key is the annihilation of gradient of pressure terms. This fact implies thatthe maximum principle is applicable, and then global-in-time smooth unique solutions
exist. The same fact is obtained on the Euler equations. Some recent ill-posedness results
by Bourgain-Li and Tao are also discussed.

Time decay estimate with diﬀusive property forsolution to the compressible Navier-Stokes system
Takayuki Kobayashi
Osaka University, Japan
Time decay estimate of a solution to the compressible Navier-Stokes-Korteweg systemis studied. It is known that the system describes two phase ﬂow with phase transition
between liquid and vapor as a diﬀuse interface model in a compressible ﬂuid. Concerning
time decay estimate, Wang and Tan ([3]) show convergence rates of L2 norm of a solution
for 3 dimensional case under small initial value around constant state. The decay rateis same as that of heat kernel. In this talk concerning the linearized problem, the decay
estimates with diﬀusive wave property (Cf., [1, 2]) for initial date are derived. Furthermore, as an application, we give time decay estimate of a solution to nonlinear system.
In contrast to the compressible Navier-Stokes system, for linear system regularities of
initial dates are lower and independent of the order of derivative of the solution owing
to smoothing eﬀect from the Korteweg tensor. Furthermore, for the nonlinear system
diﬀusive wave properties are obtained with initial dates having lower regularity than thatof studies of the compressible Navier-Stokes system and [3]. The results in this talk wereobtained in a joint work with K. Tsuda (Osaka University, Japan).

References[1] D. Hoﬀ and K. Zumbrun, Multi-dimensional diﬀusion waves for the Navier- Stokes
equations of compressible ﬂow, Indiana Univ. Math. J., 44, (1995), pp.603-676.
[2] T. Kobayashi and Y. Shibata, Remark on the rate of decay of solutions to linearizedcompressible Navier-Stokes equations, Paciﬁc Journal of Mathematics, 207 (2002), pp.
199-234.
[3] Y. Wang and Z. Tan, Optimal decay rates for the compressible ﬂuid models of Korteweg
type, J. Math. Anal. Appl., 379 (2011), pp. 256-271.

GLOBAL WELL-POSEDNESS OF THE TWO-DIMENSIONAL EXTERIORNAVIER-STOKES EQUATIONS FOR NON-DECAYING DATA
K. ABE

Abstract. We consider the two-dimensional Navier-Stokes equations in an exterior domainΩ ⊂ R2 :∂t u − ∆u + u · ∇u + ∇p = 0

in Ω × (0, ∞),div u = 0 in Ω × (0, ∞),u=0

on ∂Ω × (0, ∞),u = u0 onΩ × {t = 0}.

It is well known that the two-dimensional Navier-Stokes equations is globally well-posed
for initial data with finite energy (Leray 1934, Ladyzhenskaya 1959). However, global wellposedness is unknown in general for initial data with infinite energy. In this talk, we report
global well-posedness of the problem for bounded initial data with a finite Dirichlet integral and unique existence of asymptotically constant solutions for arbitrary large Reynolds
numbers.

1

Viscous shock wave and singular limit for hyperbolic
systems with Cattaneo’s law
Tohru Nakamura
Kumamoto University, Japan
In this talk, we consider large time behavior of solutions to scalar conservation laws withan artificial heat flux term. In the case where the heat flux is governed by Fourier’s law,
the equation is scalar viscous conservation laws. In this case, existence and asymptotic
stability of one-dimensional viscous shock waves have been studied. The main concern inthe current talk is a 2 × 2 system of hyperbolic equations with relaxation which is derived
by prescribing Cattaneo’s law for the heat flux. We consider the one-dimensional Cauchy
problem for the system of Cattaneo-type and show existence and asymptotic stability ofviscous shock waves. We also obtain the convergence rate by utilizing the weighted energy
method. By letting the relaxation time zero in the system of Cattaneo-type, the system
is formally deduced to scalar viscous conservation laws of Fourier-type. This is a singular
limit problem which occurs an initial layer. We also consider the singular limit problem
associated with viscous shock waves.

Spectrum of turbulence in nonlinear Schrödinger(Gross-Pitaevskii) equation
Kyo Yoshida1 , Yoshiyuki Tsuji2 , and Hideaki Miura3
1
Faculty of Pure and Applied Sciences, University of Tsukuba,
2
Graduate School of Engineering, Nagoya University,
3
National Institute for Fusion Science
Dynamical properties of quantum fluids such as superfluid 4 He or Bose-Einstein condensates (BECs) of ultracold atoms are described by the order parameter field ψ(x, t) ∈ C,
where x and t are space and time coordinates. Under certain conditions, the dynamics ofψ(x, t) is governed by the nonlinear Schrödinger (NLS) equation,
iℏ

∂
ℏ2 2
ψ=−
∇ ψ − µψ + g|ψ|2 ψ,∂t2mwhich is also called the Gross-Pitaevskii equation. Here, m is the mass of the particles, µ
the chemical potential, g the coupling constant and ℏ is the reduced Planck constant.
Turbulent states of quantum fluids have received considerable attention both in experimental and numerical studies. See, e.g. Ref. [1] for a review for the numericalstudies. Due to the recent progress in the control of ultracold atomic gas, it has become
possible ∫to stir BEC of 87 Rb and generate turbulent states [2,3]. Data of the spectrum
F (k) ∝ dk′ δ(|k′ | − k)⟨|ψk′ |2 ⟩, where ψk is the Fourier transform of ψ(x), δ(k) is the
Dirac delta function and ⟨ · ⟩ denotes an ensemble average, have been accumulated bothin experiments and numerical simulations. However, the synthetic understanding of thespectrum has not been established, especially in the case such that the nonlinearity in the
NLS equation is strong.
In the present study, we performed the numerical simulation of NLS in a threedimensional box with periodic boundary conditions in each coordinate direction. Pumping
and dissipation of the mass are applied in small- and large-wavenumber regions, respectively, in order to maintain a turbulent state. For some cases such that the nonlinearity
is strong, we observed F (k) ∝ k −2 which is consistent with the theoretical prediction for
the energy-transfer region in Ref. [4].

References[1] M. Tsubota, K. Fujimoto, and S. Yui. J. Low Temp. Phys. 188(2017) 119.
[2] N. Navon et al.. Nature 539(2016) 72.
[3] M. C. Tsatsos et al.. Phys. Rep. 622(2016) 1.
[4] K. Yoshida and T. Arimitsu. J. Phys. A: Math. Theor. 46(2013) 335501.

Self-similarity of heavy-particle clusteringin the inertial range of turbulence
Taketo Ariki1 , Kyo Yoshida2 , Keigo Matsuda3 , & Katsunori Yoshimatsu1
1 Institute of Materials and Systems for Sustainability, Nagoya University
2 Division of Physics, Faculty of Pure and Applied Sciences, University of Tsukuba
3 Center for Earth Information Science and Technology, Japan Agency for Marine-Earth Science and Technology

Inertial particles suspended in fluid turbulence represent spatially non-uniform distribution
due to their non-negligible inertia, forming clusters of variable sizes. In the inertial range of
homogeneous isotropic turbulence, we found a power-law behavior in particle’s statistics whichindicates self-similarity of particle clustering.
We consider particle’s number-density field n and its normalized fluctuation, i.e. θ(≡ n/⟨n⟩−1).
A perturbation analysis and Kolmogorov’s inertial-range scaling result in a self-similar solution ofpair-correlation function (PCF):
Θ(r) ≡ ⟨θ(r + x)θ(x)⟩ = C2 τp2 ϵ2/3 r−4/3 ,(1)

where C2 is a universal constant, τp is relaxation timescale of relative velocity between particles
and fluid, and ϵ is the mean dissipation rate of turbulence energy.
A closure theory for PCF further allows us to reach underlying physics behind Eq. (1); scalelocal transfer of the number-density variance is discovered on the basis of dynamical equation of
PCF, which guarantees the universal power law (1) free from both forcing-scale and dissipationrange properties. Our DNS on particle-laden homogeneous isotropic turbulence reasonably supports the predicted −4/3-power law of Eq. (1). Theoretical prediction C2 ≈ 16 ± 2 well agrees
with the DNS data.