## Schedule

International Workshop on Theoretical Aspects of Near-Wall Turbulence Studies

### June 28 (Tuesday)

   13:20-13:30 Opening remarks

issue in sub-critical flows”
Paul Manneville (CNRS, Kyoto University)

14:00-14:30“Dimension reduction method by machine learning
for turbulent plane Couette flow”
Masaki Shimizu and Genta Kawahara (Osaka University)

14:30-15:00 Break

15:00-16:00“A formalised description of quasi-steady quasi-homogeneous
scale interactions in near-wall turbulence”
Sergei Chernyshenko (Imperial College London)

16:00-16:30 Break

16:30-17:00“Subcritical transition in plane Poiseuille flow”
Darren P. Wall (Nippon Bunri University) and
Masato Nagata (Tianjin University)

17:00-17:30“Spatially localized and developed turbulence in 2D Kolmogorov flow”
Toshiki Teramura (Kyoto University)

17:30-18:00 “Drag reduction in spatially developing turbulent boundary layers
by spatially intermittent blowing at constant mass-flux”
Yukinori Kametani (The University of Tokyo)


### June 29 (Wednesday)

    9:00-10:00“The many faces of the Reynolds stresses and other turbulent fluxes”

10:00-10:30“The single-arm spiral roll state in the thermal convection
between concentric double spherical boundaries”
Tomoaki Itano, Takahiro Ninomiya, Kouhei Iida
and Masako Sugihara-Seki (Kansai University)

10:30-11:00 Break

11:00-12:00“New ways to look at turbulence data and
what can be learned (if anything)”
Henrik Alfredsson (KTH)

12:00-12:30“The sustaining mechanism of turbulence in a precessing sphere”
Yasufumi Horimoto and Susumu Goto (Osaka University)

12:30-14:00 Lunch break

14:00-15:00“Geometrical, dynamical and statistical symmetries of turbulence
and its implications for turbulent scaling laws and turbulence
modelling”

15:00-15:30“Application of persistence diagrams to analysis of high
dimensional systems”
Miroslav Kramar (Tohoku University)

15:30-16:00 Break

16:00-16:30“A mathematical aspect of turbulence”
Hayato Nawa (Meiji University)

16:30-17:00“The Navier-Stokes equations and Wiener path integrals:
a probabilistic approach to the regularity problems"
Koji Ohkitani (University of Sheffield, Kyoto University)

17:00-17:30“Lagrangian approach to spectra and energy of Kelvin waves on a vortex core"
Yasuhide Fukumoto and Rong Zou (Kyushu University)

19:00 Dinner


### June 30 (Thursday)

    9:00-10:00“An update on the turbulence open laboratory, finite
time Lyapunov exponents and Lagrangian models of turbulence
fine structure”
Charles Meneveau (Johns Hopkins University)

10:00-10:30“Bifurcation of unstable periodic orbits in plane Couette flow
with the Smagorinsky model”
Eiichi Sasaki, Genta Kawahara (Osaka University)

11:00-11:30“Random reversals of the flow in a precessing sphere”
Shigeo Kida (Doshisha University)

11:30-12:00“Mean-flow reversals in a two-dimensional forced flow
in a square domain”
Takeshi Matsumoto (Kyoto University)

12:00-12:30“Scaling exponents of passive scalars in homogeneous turbulence”
Toshiyuki Gotoh and Takeshi Watanabe
(Nagoya Institute of Technology)

12:30-14:00 Lunch break

14:00-15:00“Optimum transport and exact coherent states:
the Rayleigh-Benard example”

15:00-15:30“The Super Typhoon - its structure and formation process”
Masaki Matsui, Kiyosi Horiuti (Tokyo Institute of Technology)
and   Kazuhisa Tsuboki (Nagoya University)

15:30-16:00“Optimal heat transfer enhancement in wall-bounded shear flow”
Shingo Motoki, Genta Kawahara
and Masaki Shimizu (Osaka University)


Abstracts of Talks

“About the ‘laminar-to-turbulent’ vs. ‘turbulent-to-laminar’ issue in sub-critical flows”

Paul Manneville (CNRS, Kyoto University)

In wall-bounded flows, a transition to/from turbulence can be observed in an intermediate range of Reynolds numbers R where laminar flow is linearly stable so that nontrivial flow regimes can coexist with it, both in phase space and in physical space, at given R within this range. The nature of this coexistence raises interesting questions both theoretical/conceptual and practical.
With respect to the ≪ transition from ≫ at decreasing R, it presents itself as the opening of regions where turbulence intensity is depleted down to the point where the flow is conspicuously locally laminar. Is this the result of some global instability of the Reynolds-averaged flow and related pattern-formation issues, especially in the quasi-2D case of plane wall flows, or of a local and premature interruption of the cascade from large to small scales in the featureless regime, ending in local transient chaos, transient chaotic dynamics being indeed expected in the vicinity of exact solutions of the Navier-Stokes (NS) equation in the Minimal-Flow-Unit framework? Could this be related to Large Deviations of trajectories belonging to the turbulent attractor driving the breakdown according to a nucleation process like in thermodynamic 1st-order transitions, how and with which Extreme Value statistics?
The best studied case of ≪ transition to ≫ is the quasi-1D case of Hagen-Poiseuille flow (HPF) with the puff decay/splitting issue. It clearly hints at what we must understand as an edge state. Trajectories that eventually land on the nontrivial branch are chaotic and highly unstable. Their memoryless character is a consequence of our ignorance of where the trajectory lands in the neighborhood of the chaotic repeller, while the relaxation from the neighborhood of the edge is quasi-deterministic and controlled by viscous effects. Despite being discontinuous at the local stage, the transition can appear globally continuous when the ≪ thermodynamic limit ≫ is taken, the meaning of which has to be well understood in view of large size effects. The Directed Percolation paradigm introduced by Pomeau (1986) seems relevant to most quasi-1D cases such as HPF but the actual reason has to be scrutinized owing to the overall deterministic context,… and what about the quasi-2D cases.
An important path for attacking these problems has been modeling. The most successful attempt is Barkley’s approach to HPF within a reaction-diffusion scheme, giving us a global interpretation of the overall transitional range in terms of excitable vs. bistable local dynamics of only two observables tightly linked to the physics of the problem. A detailed justification from first principles of such an approach ?or other analogical ones? is still lacking and can be searched via truncation of Galerkin approximations to the NS dynamics. Considerable simplifications have however to be faced before the thermodynamic limit alluded to above might be valuably considered.
A possible starting review reference to this general problematics could be §4-5 of ≪ Transition to turbulence in wall-bounded flows: Where do we stand? ≫ to appear in Mechanical Engineering Reviews, Bull. JSME, vol.3, no.2, July 2016 (arXiv:1604.00840).

“Dimension reduction method by machine learning for turbulent plane Couette flow”

Masaki Shimizu and Genta Kawahara (Osaka University)

In a dissipative system, there exists the (global) attractor which has finite fractal dimensions. The flow on the attractor can be parametrized by a finite number of parameters (Temmam 1987). For low-Reynolds-number turbulence in plane Couette flow, we construct precise low-dimensional governing equations on the attractors using machine learning.

“A formalised description of quasi-steady quasi-homogeneous scale interactions in near-wall turbulence”

Sergei Chernyshenko (Imperial College London)

We consider the effect of large-scale structures in a turbulent boundary layer or a channel on the small-scale turbulence near the wall. It might seem obvious that to be able to obtain rigorous results about large and small scales one needs to define rigorously what large and small scales are. However, in our approach we replace this with specifying rigorously the properties of the large-scale filter. This offloads the burden of deciding what large and small scales are on to the end user. Once the end user selects the filter, large scales are the result of filtering the full flow field with this filter, and small scales is the difference between the full field and the large-scale field. We suggested a reasonable way of selecting the filter satisfying the required properties approximately, but we perform the derivations rigorously. We stated a formal hypothesis expressing the idea that the effect of large scales on small scales is quasi-steady and quasi-homogeneous in wall-parallel directions. Expansions for various quantities were found with respect to the amplitude of the large-scale fluctuations, which gave a number of unexpected relationships between seemingly-unrelated statistical characteristics of turbulent flows, and we performed comparisons. Interestingly, the quasi-steady quasi-homogeneous theory implies a dependence of the mean profile log-law constants on the Reynolds number. The main overall result is the demonstration of the relevance of the quasi-steady quasi-homogeneous theory for near-wall turbulent flows.

“Subcritical transition in plane Poiseuille flow”

Darren P. Wall (Nippon Bunri University) and Masato Nagata (Tianjin University)

Although the linear mechanism is bypassed in transition from the laminar state in plane Poiseuille flow, the role of the nonlinear secondary flows that bifurcate from the linear neutral stability points, and the higher-order flows that in turn bifurcate from these flows, remains an open question. We consider secondary and tertiary flows of this kind, thus re-examining and extending the problem previously considered by Ehrenstein & Koch (JFM 1991).

“Spatially localized and developed turbulence in 2D Kolmogorov flow”

Sadayoshi Toh, Yoshiki Hiruta and Toshiki Teramura (Kyoto University)

By controlling the flow rate in the direction perpendicular to the forcing in the 2D Kolmogorov flow, we show that turbulence maintained by inverse energy cascade can be spatially localized. The coarse-grained motion of the localized turbulence is discussed in terms of a large-scale structure such as Karman vortex street.

“Drag reduction in spatially developing turbulent boundary layers by spatially developing turbulent boundary layers by spatially intermittent blowing at constant mass-flux”

Yukinori Kametani (The University of Tokyo)

A series of large-eddy simulations of spatially developing turbulent boundary layers with uniform blowing at moderate Reynolds were performed with the special focus on the effect of intermittent blowing sections; separated in the streamwise direction. How the number of blowing sections, at constant mass-flux, affects on the skin friction drag will be presented.

“The many faces of the Reynolds stresses and other turbulent fluxes”

It is remarked that fluxes in conservation laws, such as the Reynolds stresses in the momentum equation of turbulent shear flows, or the spectral energy flux in isotropic turbulence, are only defined up to an arbitrary gauge transformation. While this is not usually significant for long-time averages, it becomes important when fluxes are modelled locally in large-eddy simulations, or in the analysis of intermittency and cascades. As an example, a numerical procedure is introduced to define fluxes in scalar conservation equations in such a way that their total integrated magnitude is minimised. The result is an irrotational vector field that derives from a potential, thus minimising sterile flux ‘circuits’. The algorithm is generalised to tensor fluxes and applied to the transfer of momentum in a turbulent channel. The resulting instantaneous Reynolds stresses are compared with their traditional expressions, and found to be substantially different.

“The single-arm spiral roll state in the thermal convection between concentric double spherical boundaries”

Tomoaki Itano, Takahiro Ninomiya, Kouhei Iida and Masako Sugihara-Seki (Kansai University) (Kansai University)

The single-arm spiral roll state in Rayleigh-Benard convection confined between non-rotating double concentric spherical boundaries was investigated. It was found that the state exists in a relatively thick gap as an autonomously rotating wave solution with a finite amplitude.

“New ways to look at turbulence data and what can be learned (if anything)”

Henrik Alfredsson (KTH)

Here we discuss the streamwise turbulence fluctuations using the so called diagnostic formulation. The diagnostic formulation was first attempted on the rms of the streamwise velocity fluctuations in wall-bounded flows and it was shown that it could nicely predict the distribution both in the near wall and outer regions, as well as the overlap region. It predicts that a second maximum in the rms profile will appear in the inner part of the logarithmic region at a position in wall units that increase approximately as the square root of the Reynolds number and that for large Reynolds numbers becomes higher than the near-wall peak. A further development has been taken by extending the formulation to rough-wall boundary layers. Also the rms-distribution of jets, wakes and mixing layers, as well as pressure gradient boundary layers seem to follow the diagnostic formulation. A recent study shows that it is also possible to extend it to, both odd and even, higher moments. The possibility to describe the turbulence statistics of various free shear layers and wall-bounded flows in a similar way despite their differences, suggests that the, since long hunted, elusive coherent structures, may have a less significant role in the description/prediction of wall turbulence than previously suggested.

“The sustaining mechanism of turbulence in a precessing sphere”

Yasufumi Horimoto and Susumu Goto (Osaka University)

A weak precession of a container can drive strong turbulence of the confined fluid in it. The sustaining mechanism of this strong turbulence was a long-standing unsolved problem. We have been experimentally tackled this problem by using dilute solutions of surfactants. By examining how the turbulence is modulated by the dilute additives, we conclude the turbulence is sustained by a kind of an energy cascading process in the bulk flow.

“Geometrical, dynamical and statistical symmetries of turbulence and its implications for turbulent scaling laws and turbulence modelling”

In theoretical physics, symmetries are usually differentiated into geometrical, dynamical and statistical symmetries. In statistical turbulence theory, such as in multi-point correlation equations (MPCE), Lundgren’s probability density function or Hopf’s functional approach, all of these symmetries are observed. In this presentation we show that the infinite set of MPCE, which are direct statistical consequences of the Navier-Stokes equations (NSE), admit an infinite set of Lie symmetry groups. In NSE, Lie symmetries constitute axiomatic properties such as Galilean invariance, which may be considered a dynamical symmetry or rotational invariance, which has a geometrical meaning, while the set of these symmetries is finite. In contrast, the set of symmetries of the MPCE is considerable extended compared to the set of groups which are implied from Navier-Stokes equations. Specifically, a new scaling group and translational groups of the correlation functions and all independent variables have been discovered. These new statistical groups have important consequences on our understanding of turbulent scaling laws to be exemplarily revealed employing various examples of wall-bounded shear flow. Beside the classical log-law, which fundamentally relies on one of the new statistical groups, we give various examples of including that of rotating flows, all of which rely on new statistical symmetries. Most important, in all cases both mean velocity and second order statistics scaling laws are presented. Finally, we present symmetry induced constraints to unclosed terms in turbulence equations, which should be obeyed to be consistent with axiomatic symmetry properties of turbulence.

“Application of persistence diagrams to analysis of high dimensional systems”

Miroslav Kramar (Tohoku University)

In the first part of the talk we will introduce the methods of the topological data analysis. Namely, the persistence diagrams which are a relatively new topological tool for describing and quantifying complicated patterns in a simple but meaningful way. We will demonstrate this technique on patterns appearing in Rayleigh-Benard convection. This procedure allows us to transform experimental or numerical data from experiment or simulation into a point cloud in the space of persistence diagrams. There are a variety of metrics that can be imposed on the space of persistence diagrams. By choosing different metrics one can interrogate the pattern locally or globally, which provides deeper insight into the dynamics of the process of pattern formation.

“A mathematical aspect of turbulence”

Hayato Nawa (Meiji University)

This talk is based on a joint work with T. Sakajo (Kyoto Univ) and T. Matsumoto (Kyoto Univ). If we put aside the issue of the model equations governing the turbulent flows, Kolmogorov's statistical laws (appearing in so-called Kolmogorov's Theory of 1941) and Onsager's conjecture (1949) are mathematically relevant in the light of the theory of stochastic processes. This observation invokes the issue of the model equation again.

“The Navier-Stokes equations and Wiener path integrals: a probabilistic approach to the regularity problems"

Koji Ohkitani (University of Sheffield, Kyoto University)

Using probabilistic methods, we study the basic issues of the incompressible Navier-Stokes equations, which are potentially relevant to the problem of turbulence. We compare in detail the 3D Navier-Stokes equations written in the vector potentials and their dynamically-scaled version, Leray equations. On the basis of path integral formulations, we derive and discuss conditions for their global regularity.

Lagrangian approach to spectra and energy of Kelvin waves on a vortex core

Yasuhide Fukumoto and Rong Zou (Kyushu University)

We establish a Lagrangian method, handling the Lagrangian displacement, which rigorously defines wave energy and facilitates its calculation. A steady incompressible Euler flow is characterized as a state of the maximum of the total kinetic energy with respect to perturbations constrained to an isovortical sheet, and the isovortical perturbation is handled conveniently in terms of the Lagrangian variables. The criticality in energy of a steady flow allows us to work out the energy of Kelvin waves only from the disturbance field linear in amplitude. Various energy formulas are obtainable from this approach. This scheme is extended to MHD by considering the isomagnetovortical sheet.

“An update on the turbulence open laboratory, finite time Lyapunov exponents and Lagrangian models of turbulence fine structure”

Charles Meneveau (Johns Hopkins University)

In this presentation we describe and present an update on the open numerical laboratory, the JHTDB which since 2015 contains a channel flow dataset. The benefits of such an effort to bring 3big-data tools2 to turbulence research and to 3democratize2 access to large DNS datasets is illustrated by means of an application to measure the statistics of finite time Lyapunov exponents in channel flow. We also summarize our recent efforts at developing a Lagrangian model for velocity gradients in isotropic turbulence based on the Recent Fluid Deformation closure applied to initially Gaussian random fields. Possible generalizations and applications to wall bounded flows are discussed. This work has been done with Perry Johnson, the JHU Turbulence Database Group, and the group of R. Moser whose code was used to generate the channel flow dataset.

“Bifurcation of unstable periodic orbits in plane Couette flow with the Smagorinsky model”

Eiichi Sasaki, Genta Kawahara (Osaka University) and Javier Jimenez (Universidad Politecnica de Madrid)

The aim of this study is to obtain descriptions of plane Couette turbulence from view points of the theory of dynamical systems. In order to reduce degrees of freedom we consider the Large-Eddy Simulation (LES) introducing the Smagorinsky model and study bifurcation structure of gentle and vigorous unstable periodic orbits (UPOs) found by Kawahara and Kida (2001) for the Navier-Stokes (NS) system. When we take the Smagorinsky constant as a bifurcation parameter, we find the vigorous UPO bifurcates from the gentle UPO. As the Reynolds number increases, the bifurcation diagram has a lot of turning points. We could find similar phenomena in the NS system if we trace bifurcation of UPOs with the lack of the resolution. We conclude that this result implies that the Smagorinsky model could not annihilate errors arising from the low resolution even if we could obtain the correct profile of the mean flow in the LES system.

“Random reversals of the flow in a precessing sphere”

Shigeo Kida (Doshisha University)

The flow in a precessing sphere exhibits a variety of spatio-temporal structures depending on the values of two control parameters, i.e. the Reynolds number Re and the Poincare number Po. Here, we present a series of numerical simulation analyses. For a given Po, starting from small Re for which a steady flow is realized, we increase Re step by step until the flow becomes unsteady. Although a simple periodic flow (of sinusoidal time-dependence) appears for most cases, we happened to find an interesting temporal behaviour for a particular combination of (Re, Po). The flow is still laminar, but the global structure reverses (with respect to the center of the sphere) randomly in time. The mechanism of the random reversals will be examined with the help of a technique of dynamical systems analysis.

“Mean-flow reversals in a two-dimensional forced flow in a square domain”

Takeshi Matsumoto (Kyoto University)

It is known experimentally and numerically that a certain forced flow in a square domain with no-slip walls exhibits reversals of the mean circulating flow. We study the reversals with a stochastic model and also discuss its deterministic aspects

“Scaling exponents of passive scalars in homogeneous turbulence”

Toshiyuki Gotoh and Takeshi Watanabe (Nagoya Institute of Technology)

Local scaling exponents of the structure functions of passive scalar increments convected by homogeneous steady turbulence are examined by using very high resolution direct numerical simulation (DNS). Scalar $\theta$ are excited by Gaussian random fluctuations that are white in time and applied only at low wavenumbers, and scalar $q$ by the mean uniform scalar gradient applied in $z$ direction. It is found that in the inertial range the local scaling exponents of $\theta$ has the logarithmic correction while those of $q$ have a well developed plateau. This finding means that the scaling exponents of passive scalar are not universal. In order to explore the physical explanation, we have conducted DNSs in which the injections of the scalar fluctuations are modified in such a way that the Gaussian random source has a finite correlation time while the scalar injection by the uniform gradient is low pass filtered. It is found that the scaling exponents of the scalar $q$ become closer to those of the scalar $\theta$ and have the logarithmic correction. We argue the physics of passive scalar turbulence and its implication.

“Optimum transport and exact coherent states: the Rayleigh-Benard example”

Among the spectrum of exact coherent solutions, we focus on those that maximize transport. In Rayleigh-Benard thermal convection such solutions are only slightly above the turbulent experimental data suggesting that maximum transport solutions may ‘control’ turbulent flow.

“The Super Typhoon - its structure and formation process”

Masaki Matsui, Kiyosi Horiuti (Tokyo Institute of Technology) and Kazuhisa Tsuboki (Nagoya University)

A characteristic feature of the typhoon is that it possesses the spiraling arms (rain band) which wrap around the low pressure region in the core (eye). Similar spiral vortex was identified in the small scale of turbulence, where existence of two symmetric modes and a third asymmetric mode of configurations with regard to the vorticity alignment along two spiral sheets and the vortex tube in the core was shown (Horiuti et al. 2008, 2011). They are achieved through the interaction of several sheets. Occurrence of transition between the three modes was shown. This talk aims to elucidate the vortical structure and its formation process of the super typhoon using the numerical data generated by the cloud resolving storm simulation method. Similarities and differences between the spiral vortex in turbulence and the typhoon are discussed. It is shown that this typhoon is created by convergence of recirculating flow caused by the dual sheets and concentration of its low-pressure region. The vorticity in the spiral arms is in the direction transversal or azimuthal to the axial vorticity in the core region, and persistence of this configuration is shown. Implication for the energy spectrum generated by this configuration is discussed.

“Optimal heat transfer enhancement in wall-bounded shear flow”

Genta Kawahara, Shingo Motoki and Masaki Shimizu (Osaka University)

An optimal incompressible steady velocity field has been found numerically for heat transfer enhancement with less energy dissipation in plane Couette flow by using a variational method. The functional is defined as wall heat flux (scalar dissipation), from which energy dissipation has been subtracted, to be optimized under the constraints of the continuity equation for velocity and the advection-diffusion equation for temperature. It is shown that at high Reynolds numbers a three-dimensional (streamwise-dependent) velocity field is optimal, while at low Reynolds numbers optimal heat transfer is given by a streamwise-independent field. Theoretical interpretation is proposed for the emergence of the three-dimensional optimal velocity field.