12th Workshop on Synthetic Turbulence Models 3rd-4th July 2017, Université Paris-Ouest, Nanterre La Défense, France

Synthetic ows for heat and mass transfer b

c

F. C. G. A. Nicolleau , D. Queiros Conde , A. F. Nowakowski

b

and T. Michelitsch

d

http://www.sig42.group.shef.ac.uk/SIG42-12.htm

Introduction

to investigate a wide range of two-component ow

The workshop was the 12th of the ERCOFTAC Special Interest Group on Synthetic Turbulence Models (SIG42).

It took place at Université Paris-Ouest,

Nanterre La Défense, France.

It was co-organised

with the SIG 14 dedicated to Multipoint Turbulent Structure and Modelling. The workshop particular theme was `Synthetic ows for heat and mass transfer'

problems. The robustness of the numerical approach was demonstrated using the cases of shock-bubble interaction representing media of non-uniform thermodynamic properties for which experimental data are available. The governing equations include non-conservative terms and therefore their numerical solution require a non-conventional approach. The numerical framework required a scheme ensuring conservativeness in

About 20 participants attended from dierent Eu-

a predictor-corrector and Godunov-type nite vol-

ropean countries (France, Germany, Russia, Spain,

ume environment.

United Kingdom) and beyond the European Commu-

methodology is the diuse interface treatment of the

nity (Columbia, Irak, Saudi Arabia) and 10 dierent

discontinuities and boundaries between ow compo-

institutions. It was an opportunity for the KS com-

nents. An ecient pressure relaxation procedure has

munity to strengthen the links with the energy en-

been implemented within the algorithm, to achieve

gineering community. Abstracts from the main con-

the equilibrium state, in the case of the pressure non-

tributions are reported below. An Ercoftac Book Se-

equilibrium model. [36]

An integral part of the applied

ries that will publish more detailed contributions is in preparation (to appear early 2018)

Abstracts of Talks

Instabilities and Pattern Formation in Mechanically Vibrating Thin Films

M. Bestehorn

a

Helicity in the atmospheric dynamics and turbulence theory

O. G. Chkhetiani and M.V. Kurgansky [email protected] n

n

An overview of the helicity concept and of its role

In a liquid layer with a free surface several hydrody-

in the atmospheric dynamics (including Nambu me-

namic instabilities may occur and may lead to self-

chanics) and the turbulence theory is given, also in

organized pattern formation.

relation to recent direct eld measurements of tur-

The patterns can be

seen as macroscopic motion of the uid, surface defor-

bulent helicity (Koprov,

mations, temperature or concentration distributions

Chkhetiani.

etc.

Physics, 2015,

In this contribution, the Faraday instability for thin lms was studied in detail.

Izvestiya,

Koprov,

Kurgansky,

and

Atmospheric and Oceanic

51, 565-575).

By considering a super-

position of the classic Ekman spiral solution and a jet-

The liquid layer is

like wind prole that mimics a shallow breeze circula-

thereby vibrated with one or more frequencies in nor-

tion over a non-uniformly heated Earth surface, a pos-

mal and tangential direction(s) with respect to its sur-

sible explanation is provided, why the measured mean

face. For a partially wetting substrate (small contact

turbulent helicity sign is negative. Whereas, the pos-

angle) a reduced theoretical description based on the

itive helicity is injected into the Ekman boundary

lubrication approximation is possible [2, 3].

Linear

layer in the northern hemisphere. A simple turbulent

stability as well as nonlinear results were presented

relaxation model is discussed that explains the mea-

and compared to ndings of the full set of hydrody-

sured positive values of the contribution to turbulent

namic basic equations. Recent results of a two-layer

helicity from the vertical components of velocity and

system consisting of two immiscible uids separated

vorticity.

by a deformable interface were discussed [4, 5].

A computationally ecient method for multicomponent ows

A. F. Nowakowski , F.C.G.A. Nicolleau and T. M. Michelitsch b

b

d

Large-eddy simulations of a wind turbine wake above a forest

J. Schroettle , Z. Piotrowski q,s

r

Since the pioneering large-eddy simulations of Shaw and Schumann (1992), the forest stands have been

This contribution focused on the developed numeri-

treated as a porous body of horizontally uniform

cal methods that constitute in house software used

(leaf ) area density with constant drag coecient.

This approach is sometimes called eld-scale ap-

A kinematic simulation was used to reconstruct the

proach.

Current ner scale applications and eld

missing-scale motions in isotropic turbulence [15] and

campaigns consider the heterogeneity of the canopies

turbulent shear ows [16]. Yao and He [19] proposed

at the plant-scale (Schrottle and Dornbrack 2013).

to use a kinematic simulation (KS) to explicitly con-

Results from plant-scale simulations of ow through

struct the unresolved velocity elds with the required

resolved tree structures originating from laser scans

statistic properties in both space and time. The kine-

were presented in this contribution.

Simulating the

matic SGS model was used to calculate sound power

wake of one wind turbine, e.g., above a forest requires

spectra from isotropic turbulence and yielded an im-

a new computational approach.

For this purpose,

proved result: the missing portion of the sound power

2008) is modied to accom-

spectra was approximately recovered in the LES. We

modate two independent hydrodynamic solvers. The

discuss the procedure of implementation a kinematic

two solvers are integrated simultaneously.

SGS model in SOWFA code. The computational do-

EULAG (Prusa et al.

The tur-

2π

bulence structure above a forest is simulated in the

main as a box of length

rst solver at the eld scale and acts as inow for the

periodic boundary conditions were applied, was used

wind turbine wake ow.

for verication of dierent SGS models on dierent

LES and KS Applications of Wind Energy in the Atmospheric Boundary Layer

J.M. Redondo , J. Tellez-Alvarez , S. Strijhak f

i

m

The Large-Eddy Simulation (LES) became a powerful tool to simulate physical processes in Atmospheric Boundary Layer and aerodynamics of wind farm. The dierent sub-grid scale (SGS) models are widely used for simulation by dierent authors. The mathematical model was realized in SOWFA (Simulator for On/Oshore Wind Farm Application) code. SOWFA is an OpenFOAM-based incompressible atmospheric/wind farm LES solver, which is based on nite-volume method, that models wind turbine as actuator lines and actuator disk. The results of velocity magnitude's elds at dierent time steps are shown on Fig. 1.

on each side, where the

grids.

A non-local theory of turbulent pair diusion

N. A. Malik [email protected], [email protected] g

In 1926, Richardson (1926) [44] proposed that the

pair diusivity

K(l)

is scale dependent with a single

power law over all turbulent scales, and that the diffusion process is local. He collected available data at

K(l) ∼ l4/3 , (which is hl2 i ∼ t3 ) although this has never been unambiguously veried. l is the pair separation, and h·i is the ensemble average. (It is usual p to evaluate K at typical values of, l, namely at σl = hl2 i, so this 4/3 scaling is replaced by, K ∼ σl .) the time, and assumed the t,

equivlent to

A re-examination of the original data set actually shows that

K(l) ∼ l1.564

is a much better t to the

data Malik (2016b). To date, experiments and DNS have not reached the high Reynolds numbers need to generate a large enough inertial subrange to test these ideas fully. [52] remark that, ...

there has not been an experiment

that has unequivocally conrmed R-O scaling over a broad-enough range of time and with sucient accuracy. A non-local theory for generalised power law spectra of the form

E(k) ∼ k −p , 1 < p ≤ 3,

[26],

features a novel calculation for the pair diusivity, Figure 1:

U0

magnitude value at dierent time a) 150

s, b) 300 s, c) 500 s, d) 1000 s

K ∼ hl · vi, in terms of a Fourier decomposition of the relative velocity eld, v(l) = u(x + l) − u(x), where R u(x) = A(k) exp(ik · x)d3 k; then

The analysis of fractal dimension for turbulent wakes was performed with the results of numerical simulation by LES and Lagrangian-averaged scale-

Z K(l) ∼

independent dynamic Smagorinsky, One Eddy Equa-

(1)

exp(ik · x)id3 k

tion ABL turbulence models for a horizontal-axil wind single turbine and two turbines.

h(l · A)[exp(ik · l) − 1]

It is known

that such SGS models don't reproduce correctly the

Partitioning the integral in to local and non-local

energy spectrum for maximum wave number, the

ranges, and assuming a number of closures, leads to,

normalized time correlation coecients for veloc-

K(l) ∼ O(σl p ) + O(σl2 ),

ity modes and sound power spectra comparing with Direct Numerical Simulation (DNS) results.

The

stochastic SGS models, such as random forcing models,

represent alternatives that could correct the

γl

where the

nl is the locality scaling and γp scaling.

=2

γpl = (1 + p)/2

is the non-locality

Now assuming a single power law t to

K(l) over p, and a

timescales of resolved velocity elds. Kinematic sim-

the inertial subrange of scales for any given

ulations that are limited to SGS motion (or kinematic

smooth and uniform transition between the asypm-

SGS models) can be used to represent unresolved velocity elds with imposed spatial-temporal statistics.

totic limits

K ∼ σl2 ,

p = 1

where

K ∼ σl ,

to

p = 3

where

the observable scaling is intermediate be-

At the current time, we are carrying out a sys-

tween local and non-local scalings,

tematic parametric study of turbulent inertial parti-

γ

K(l) ∼ σl p , with γpnl > γp > γpl

cle pair diusion for a wide range of Stokes' num-

for 1 < p < 3

(2)

p = 1.74, we obtain K ∼ σlγI , with γI > 1.37, and σl2 ∼ tχI , with χI > 3.2. The exact values of γI and χI can only be

For intermittent turbulence,

established through detailed experiments and DNS with extended inertial subranges of more than

105 .

This is not possible at the current time and probably not for many decades to come. However, Lagrangian models such as Kinematic Simulations (KS), [25], can indicate the scalings. KS with spectra

E ∼ k −p ,

over inertial subranges

γp of size 10 , yielded power law scalings for K ∼ σl . For intermittent turbulence, p = 1.74, KS produced K ∼ σl1.57 , which is within 2% of the revised data. 2 4.7 This is equivalent to σl ∼ t . 6

It has been shown that a non-local theory of pair diusion is at least as good an explanation of the observed and simulated results, and perhaps even a better one, than locality. However, only detailed experiments or observations in homogeneous turbulence with extened inertial subranges can give a decisive answer to this century old problem. The 1926 dataset is subject to signicant uncertainty in the length scales and bouyancies under dierent conditions, and although the overall sense is now leaning towards non-locality, the matter is far from certain.

In the

new non-local theory, a number of assumptions and closures have been made, but all such theorie contain similar assumptions; locality makes the extra ad hoc assumption that the non-local term in equation is zero. The idea of non-locality could lead to important revision of our view of turbulence. The implications for modeling are equally important.

A theory of turbulent inertial particle pair diffusion

S. M. Usama and N. A. Malik [email protected] [email protected] g

g

0 < St < 100, 102 ≤ kη /k1 ≤ 106 ,

ber

and size of the inertial range and also for

1 < p ≤ 3.

The

method of in vestigation is numerical using Kinemataic Simulations (KS), Malik (2016a). Theoretical considerations suggest that inertial particle pair diusivity should follow a triple regime of diusion. For inertial pairs initially close together the pair diusion follows a ballistic regimes for short times, ity is

t < τp , such that the inertia pair diusivK(l) ∼ σl1 . At these short times, the energy

in the smallest turbulence scales is very small and therefore inertia dominates over the turbulence until the particles response time is comparable to the local tubulence time scale at the spearation

t∗ = t(l∗ ) ≈ τp .

l∗ ,

i.e. when

After this time, the inertial particles will begin to feel more and more of the turbulent energies in the bigger scales as their speration increases.

The pair

diusion will therefore asyptote towards the non-local passive particle pair diusion regime in the limit that

t/τp → ∞

or

σl /η → ∞,

such that

1.57 K(l) → σL

as

predicted in the non-local theory of Malik (2016b). This regime diers from the classical

4/3 − t3

law of

Richardson (1926) and Obukhov (1941). In between these two asyptotic regimes, there may be a long period of transition between the ballistic and passive pair diusion regimes, long enough to constitute a third, transition regime. This is what we are observing in the nuermical results from KS. In the near future, we will complete the parametric study of inertial particle pair diusion. After that, in a second part of the study, we will investigate the inerial pair diusion process under gravity.

Eect of non Kolmogorov spectra on particle clustering

F.C.G.A. Nicolleau and N.M. Sangtani Lakhwani b

b

Clustering could be dened as the propensity of an initially uniformly distributed cloud of particles to accumulate in some regions of the physical space. This

It is natural to extend the ideas of non-locality,

is an important phenomenon to understand in or-

Malik (2016b), to inertial particles because most real

der to explore, identify and possibly monitor some

world applications concern inertial particle transport.

natural or hand-made mixing processes such as those

Although innumerabe studies have been carried out

causing rain formation sediments transportation, fuel

on single inertial paticle transport,

mixing and combustion.

surpising our

literature survey found just one previous study of

In this contribution we study the clustering of

turbulent inertial particle pair diusion, the DNS

inertial particles using a periodic kinematic simula-

study by [6].

tion.

The systematic Lagrangian tracking of parti-

The parameters governing turbulent inertial parti-

cles makes it possible to identify the particles' clus-

cle pair diusion are, the size of the inertial subrange

tering patterns for dierent values of particle's inertia

(which is related to the Reynolds number), and the

and drift velocity. The main focus is to identify and

Stokes' number which is given by,

then quantify the clustering attractor - when it ex-

St =

ists - that is the set of points in the physical space

τp tη

(3)

where the particles settle when time goes to innity. Depending on gravity or drift eect and inertia values, the Lagrangian attractor can have dierent di-

tη

τp

is the

mensions varying from the initial three-dimensional

particle relaxation time. The energy spectrum power

space to two-dimensional layers and one-dimensional

p, 1 < p ≤ 3,

attractors that can be shifted from an horizontal to

where

is the Kolmogorov time scale, and where

E(k) ∼ k −p ,

is an additional

parameter. In this part of the study, we will ignore gravity.

a vertical position. The particles initially uniformly distributed in the

Figure 2: Dierent Lagrangian attractors

ow eld are allowed to evolve until an asymptotic

ever passage probabilities fulll Riesz potential power

clustering pattern - also referred to as Lagrangian at-

law decay asymptotic behavior for nodes far from the

tractor - is achieved.

departure node [32, 41].

The shape of this cluster varies from clear onedimensional structures to three-dimension distributed structures or two-dimensional layer-like structures (Fig. 2). We quantify the variations in particle clustering in the presence of gravity with modied spectral power laws, from very steep (p

→ 2.5) to very at (p → 1.4)

energy distributions. Our main ndings are:

•

The spectral law can have a signicant eect on the Lagrangian attractor shape but for limited ranges of Froude or Stokes numbers,

•

ing. Without gravity (F r

random walk (α

= 2).

Random Walks on Networks: Lévy Flights and Fractional Transport

A. Pérez Riascos

e,d

We study fractional random walks on networks deequation in graphs and their relation with Lévy walk navigation strategies on networks. We explore these processes for dierent structures; in particular, rings, circulant networks, n-dimensional lattices, complete graphs and random networks. From the spectra and

grangian attractor (1D ou 2D) whatever the

the eigenvectors of the Laplacian matrix, we deduce

∈ [1.4, 2]).

explicit results for dierent quantities that character-

With gravity for Stokes numbers large enough there is a `2D layered' structure whatever the spectral power.

•

is faster than a search strategy based on the normal

there is no La-

spectral power (p

•

= ∞)

a search strategy based on a fractional random walk

ned from the equivalent of the fractional diusion

in particular when a 2D Layer is achieved the spectral law has little or no eect on the cluster-

•

Our results conrm our recent ndings [32, 41] that

ize this dynamical process and their capacity to explore networks. We obtain analytical expressions for the fractional transition matrix, the fractional degree and the average probability of return of the random

If gravity is large

F r < 0.5,

there is no `quasi-

horizontal attractors'. Gravity tends to deform the attractor towards the vertical direction.

T.M. Michelitsch , B A Collet , A P Riascos , A.F. Nowakowski and F.C.G.A. Nicolleau d

Also, we discuss the Kemeny constant that

gives the average number of steps necessary to reach any site of the network.

Through all this work, we

analyze the mechanisms behind the fractional trans-

Fractional Random Walks on regular networks and lattices d

walker.

port on networks and how this long-range process induces dynamically the small-world property in dierent structures. [40, 41, 42, 43, 31, 32]

d,e

b

b

We establish a generalization of Polya's recurrence

Performance assessment of numerical modelling for hydraulic eciency of a grated inlet

0

Synthetic ows for heat and mass transfer b

c

F. C. G. A. Nicolleau , D. Queiros Conde , A. F. Nowakowski

b

and T. Michelitsch

d

http://www.sig42.group.shef.ac.uk/SIG42-12.htm

Introduction

to investigate a wide range of two-component ow

The workshop was the 12th of the ERCOFTAC Special Interest Group on Synthetic Turbulence Models (SIG42).

It took place at Université Paris-Ouest,

Nanterre La Défense, France.

It was co-organised

with the SIG 14 dedicated to Multipoint Turbulent Structure and Modelling. The workshop particular theme was `Synthetic ows for heat and mass transfer'

problems. The robustness of the numerical approach was demonstrated using the cases of shock-bubble interaction representing media of non-uniform thermodynamic properties for which experimental data are available. The governing equations include non-conservative terms and therefore their numerical solution require a non-conventional approach. The numerical framework required a scheme ensuring conservativeness in

About 20 participants attended from dierent Eu-

a predictor-corrector and Godunov-type nite vol-

ropean countries (France, Germany, Russia, Spain,

ume environment.

United Kingdom) and beyond the European Commu-

methodology is the diuse interface treatment of the

nity (Columbia, Irak, Saudi Arabia) and 10 dierent

discontinuities and boundaries between ow compo-

institutions. It was an opportunity for the KS com-

nents. An ecient pressure relaxation procedure has

munity to strengthen the links with the energy en-

been implemented within the algorithm, to achieve

gineering community. Abstracts from the main con-

the equilibrium state, in the case of the pressure non-

tributions are reported below. An Ercoftac Book Se-

equilibrium model. [36]

An integral part of the applied

ries that will publish more detailed contributions is in preparation (to appear early 2018)

Abstracts of Talks

Instabilities and Pattern Formation in Mechanically Vibrating Thin Films

M. Bestehorn

a

Helicity in the atmospheric dynamics and turbulence theory

O. G. Chkhetiani and M.V. Kurgansky [email protected] n

n

An overview of the helicity concept and of its role

In a liquid layer with a free surface several hydrody-

in the atmospheric dynamics (including Nambu me-

namic instabilities may occur and may lead to self-

chanics) and the turbulence theory is given, also in

organized pattern formation.

relation to recent direct eld measurements of tur-

The patterns can be

seen as macroscopic motion of the uid, surface defor-

bulent helicity (Koprov,

mations, temperature or concentration distributions

Chkhetiani.

etc.

Physics, 2015,

In this contribution, the Faraday instability for thin lms was studied in detail.

Izvestiya,

Koprov,

Kurgansky,

and

Atmospheric and Oceanic

51, 565-575).

By considering a super-

position of the classic Ekman spiral solution and a jet-

The liquid layer is

like wind prole that mimics a shallow breeze circula-

thereby vibrated with one or more frequencies in nor-

tion over a non-uniformly heated Earth surface, a pos-

mal and tangential direction(s) with respect to its sur-

sible explanation is provided, why the measured mean

face. For a partially wetting substrate (small contact

turbulent helicity sign is negative. Whereas, the pos-

angle) a reduced theoretical description based on the

itive helicity is injected into the Ekman boundary

lubrication approximation is possible [2, 3].

Linear

layer in the northern hemisphere. A simple turbulent

stability as well as nonlinear results were presented

relaxation model is discussed that explains the mea-

and compared to ndings of the full set of hydrody-

sured positive values of the contribution to turbulent

namic basic equations. Recent results of a two-layer

helicity from the vertical components of velocity and

system consisting of two immiscible uids separated

vorticity.

by a deformable interface were discussed [4, 5].

A computationally ecient method for multicomponent ows

A. F. Nowakowski , F.C.G.A. Nicolleau and T. M. Michelitsch b

b

d

Large-eddy simulations of a wind turbine wake above a forest

J. Schroettle , Z. Piotrowski q,s

r

Since the pioneering large-eddy simulations of Shaw and Schumann (1992), the forest stands have been

This contribution focused on the developed numeri-

treated as a porous body of horizontally uniform

cal methods that constitute in house software used

(leaf ) area density with constant drag coecient.

This approach is sometimes called eld-scale ap-

A kinematic simulation was used to reconstruct the

proach.

Current ner scale applications and eld

missing-scale motions in isotropic turbulence [15] and

campaigns consider the heterogeneity of the canopies

turbulent shear ows [16]. Yao and He [19] proposed

at the plant-scale (Schrottle and Dornbrack 2013).

to use a kinematic simulation (KS) to explicitly con-

Results from plant-scale simulations of ow through

struct the unresolved velocity elds with the required

resolved tree structures originating from laser scans

statistic properties in both space and time. The kine-

were presented in this contribution.

Simulating the

matic SGS model was used to calculate sound power

wake of one wind turbine, e.g., above a forest requires

spectra from isotropic turbulence and yielded an im-

a new computational approach.

For this purpose,

proved result: the missing portion of the sound power

2008) is modied to accom-

spectra was approximately recovered in the LES. We

modate two independent hydrodynamic solvers. The

discuss the procedure of implementation a kinematic

two solvers are integrated simultaneously.

SGS model in SOWFA code. The computational do-

EULAG (Prusa et al.

The tur-

2π

bulence structure above a forest is simulated in the

main as a box of length

rst solver at the eld scale and acts as inow for the

periodic boundary conditions were applied, was used

wind turbine wake ow.

for verication of dierent SGS models on dierent

LES and KS Applications of Wind Energy in the Atmospheric Boundary Layer

J.M. Redondo , J. Tellez-Alvarez , S. Strijhak f

i

m

The Large-Eddy Simulation (LES) became a powerful tool to simulate physical processes in Atmospheric Boundary Layer and aerodynamics of wind farm. The dierent sub-grid scale (SGS) models are widely used for simulation by dierent authors. The mathematical model was realized in SOWFA (Simulator for On/Oshore Wind Farm Application) code. SOWFA is an OpenFOAM-based incompressible atmospheric/wind farm LES solver, which is based on nite-volume method, that models wind turbine as actuator lines and actuator disk. The results of velocity magnitude's elds at dierent time steps are shown on Fig. 1.

on each side, where the

grids.

A non-local theory of turbulent pair diusion

N. A. Malik [email protected], [email protected] g

In 1926, Richardson (1926) [44] proposed that the

pair diusivity

K(l)

is scale dependent with a single

power law over all turbulent scales, and that the diffusion process is local. He collected available data at

K(l) ∼ l4/3 , (which is hl2 i ∼ t3 ) although this has never been unambiguously veried. l is the pair separation, and h·i is the ensemble average. (It is usual p to evaluate K at typical values of, l, namely at σl = hl2 i, so this 4/3 scaling is replaced by, K ∼ σl .) the time, and assumed the t,

equivlent to

A re-examination of the original data set actually shows that

K(l) ∼ l1.564

is a much better t to the

data Malik (2016b). To date, experiments and DNS have not reached the high Reynolds numbers need to generate a large enough inertial subrange to test these ideas fully. [52] remark that, ...

there has not been an experiment

that has unequivocally conrmed R-O scaling over a broad-enough range of time and with sucient accuracy. A non-local theory for generalised power law spectra of the form

E(k) ∼ k −p , 1 < p ≤ 3,

[26],

features a novel calculation for the pair diusivity, Figure 1:

U0

magnitude value at dierent time a) 150

s, b) 300 s, c) 500 s, d) 1000 s

K ∼ hl · vi, in terms of a Fourier decomposition of the relative velocity eld, v(l) = u(x + l) − u(x), where R u(x) = A(k) exp(ik · x)d3 k; then

The analysis of fractal dimension for turbulent wakes was performed with the results of numerical simulation by LES and Lagrangian-averaged scale-

Z K(l) ∼

independent dynamic Smagorinsky, One Eddy Equa-

(1)

exp(ik · x)id3 k

tion ABL turbulence models for a horizontal-axil wind single turbine and two turbines.

h(l · A)[exp(ik · l) − 1]

It is known

that such SGS models don't reproduce correctly the

Partitioning the integral in to local and non-local

energy spectrum for maximum wave number, the

ranges, and assuming a number of closures, leads to,

normalized time correlation coecients for veloc-

K(l) ∼ O(σl p ) + O(σl2 ),

ity modes and sound power spectra comparing with Direct Numerical Simulation (DNS) results.

The

stochastic SGS models, such as random forcing models,

represent alternatives that could correct the

γl

where the

nl is the locality scaling and γp scaling.

=2

γpl = (1 + p)/2

is the non-locality

Now assuming a single power law t to

K(l) over p, and a

timescales of resolved velocity elds. Kinematic sim-

the inertial subrange of scales for any given

ulations that are limited to SGS motion (or kinematic

smooth and uniform transition between the asypm-

SGS models) can be used to represent unresolved velocity elds with imposed spatial-temporal statistics.

totic limits

K ∼ σl2 ,

p = 1

where

K ∼ σl ,

to

p = 3

where

the observable scaling is intermediate be-

At the current time, we are carrying out a sys-

tween local and non-local scalings,

tematic parametric study of turbulent inertial parti-

γ

K(l) ∼ σl p , with γpnl > γp > γpl

cle pair diusion for a wide range of Stokes' num-

for 1 < p < 3

(2)

p = 1.74, we obtain K ∼ σlγI , with γI > 1.37, and σl2 ∼ tχI , with χI > 3.2. The exact values of γI and χI can only be

For intermittent turbulence,

established through detailed experiments and DNS with extended inertial subranges of more than

105 .

This is not possible at the current time and probably not for many decades to come. However, Lagrangian models such as Kinematic Simulations (KS), [25], can indicate the scalings. KS with spectra

E ∼ k −p ,

over inertial subranges

γp of size 10 , yielded power law scalings for K ∼ σl . For intermittent turbulence, p = 1.74, KS produced K ∼ σl1.57 , which is within 2% of the revised data. 2 4.7 This is equivalent to σl ∼ t . 6

It has been shown that a non-local theory of pair diusion is at least as good an explanation of the observed and simulated results, and perhaps even a better one, than locality. However, only detailed experiments or observations in homogeneous turbulence with extened inertial subranges can give a decisive answer to this century old problem. The 1926 dataset is subject to signicant uncertainty in the length scales and bouyancies under dierent conditions, and although the overall sense is now leaning towards non-locality, the matter is far from certain.

In the

new non-local theory, a number of assumptions and closures have been made, but all such theorie contain similar assumptions; locality makes the extra ad hoc assumption that the non-local term in equation is zero. The idea of non-locality could lead to important revision of our view of turbulence. The implications for modeling are equally important.

A theory of turbulent inertial particle pair diffusion

S. M. Usama and N. A. Malik [email protected] [email protected] g

g

0 < St < 100, 102 ≤ kη /k1 ≤ 106 ,

ber

and size of the inertial range and also for

1 < p ≤ 3.

The

method of in vestigation is numerical using Kinemataic Simulations (KS), Malik (2016a). Theoretical considerations suggest that inertial particle pair diusivity should follow a triple regime of diusion. For inertial pairs initially close together the pair diusion follows a ballistic regimes for short times, ity is

t < τp , such that the inertia pair diusivK(l) ∼ σl1 . At these short times, the energy

in the smallest turbulence scales is very small and therefore inertia dominates over the turbulence until the particles response time is comparable to the local tubulence time scale at the spearation

t∗ = t(l∗ ) ≈ τp .

l∗ ,

i.e. when

After this time, the inertial particles will begin to feel more and more of the turbulent energies in the bigger scales as their speration increases.

The pair

diusion will therefore asyptote towards the non-local passive particle pair diusion regime in the limit that

t/τp → ∞

or

σl /η → ∞,

such that

1.57 K(l) → σL

as

predicted in the non-local theory of Malik (2016b). This regime diers from the classical

4/3 − t3

law of

Richardson (1926) and Obukhov (1941). In between these two asyptotic regimes, there may be a long period of transition between the ballistic and passive pair diusion regimes, long enough to constitute a third, transition regime. This is what we are observing in the nuermical results from KS. In the near future, we will complete the parametric study of inertial particle pair diusion. After that, in a second part of the study, we will investigate the inerial pair diusion process under gravity.

Eect of non Kolmogorov spectra on particle clustering

F.C.G.A. Nicolleau and N.M. Sangtani Lakhwani b

b

Clustering could be dened as the propensity of an initially uniformly distributed cloud of particles to accumulate in some regions of the physical space. This

It is natural to extend the ideas of non-locality,

is an important phenomenon to understand in or-

Malik (2016b), to inertial particles because most real

der to explore, identify and possibly monitor some

world applications concern inertial particle transport.

natural or hand-made mixing processes such as those

Although innumerabe studies have been carried out

causing rain formation sediments transportation, fuel

on single inertial paticle transport,

mixing and combustion.

surpising our

literature survey found just one previous study of

In this contribution we study the clustering of

turbulent inertial particle pair diusion, the DNS

inertial particles using a periodic kinematic simula-

study by [6].

tion.

The systematic Lagrangian tracking of parti-

The parameters governing turbulent inertial parti-

cles makes it possible to identify the particles' clus-

cle pair diusion are, the size of the inertial subrange

tering patterns for dierent values of particle's inertia

(which is related to the Reynolds number), and the

and drift velocity. The main focus is to identify and

Stokes' number which is given by,

then quantify the clustering attractor - when it ex-

St =

ists - that is the set of points in the physical space

τp tη

(3)

where the particles settle when time goes to innity. Depending on gravity or drift eect and inertia values, the Lagrangian attractor can have dierent di-

tη

τp

is the

mensions varying from the initial three-dimensional

particle relaxation time. The energy spectrum power

space to two-dimensional layers and one-dimensional

p, 1 < p ≤ 3,

attractors that can be shifted from an horizontal to

where

is the Kolmogorov time scale, and where

E(k) ∼ k −p ,

is an additional

parameter. In this part of the study, we will ignore gravity.

a vertical position. The particles initially uniformly distributed in the

Figure 2: Dierent Lagrangian attractors

ow eld are allowed to evolve until an asymptotic

ever passage probabilities fulll Riesz potential power

clustering pattern - also referred to as Lagrangian at-

law decay asymptotic behavior for nodes far from the

tractor - is achieved.

departure node [32, 41].

The shape of this cluster varies from clear onedimensional structures to three-dimension distributed structures or two-dimensional layer-like structures (Fig. 2). We quantify the variations in particle clustering in the presence of gravity with modied spectral power laws, from very steep (p

→ 2.5) to very at (p → 1.4)

energy distributions. Our main ndings are:

•

The spectral law can have a signicant eect on the Lagrangian attractor shape but for limited ranges of Froude or Stokes numbers,

•

ing. Without gravity (F r

random walk (α

= 2).

Random Walks on Networks: Lévy Flights and Fractional Transport

A. Pérez Riascos

e,d

We study fractional random walks on networks deequation in graphs and their relation with Lévy walk navigation strategies on networks. We explore these processes for dierent structures; in particular, rings, circulant networks, n-dimensional lattices, complete graphs and random networks. From the spectra and

grangian attractor (1D ou 2D) whatever the

the eigenvectors of the Laplacian matrix, we deduce

∈ [1.4, 2]).

explicit results for dierent quantities that character-

With gravity for Stokes numbers large enough there is a `2D layered' structure whatever the spectral power.

•

is faster than a search strategy based on the normal

there is no La-

spectral power (p

•

= ∞)

a search strategy based on a fractional random walk

ned from the equivalent of the fractional diusion

in particular when a 2D Layer is achieved the spectral law has little or no eect on the cluster-

•

Our results conrm our recent ndings [32, 41] that

ize this dynamical process and their capacity to explore networks. We obtain analytical expressions for the fractional transition matrix, the fractional degree and the average probability of return of the random

If gravity is large

F r < 0.5,

there is no `quasi-

horizontal attractors'. Gravity tends to deform the attractor towards the vertical direction.

T.M. Michelitsch , B A Collet , A P Riascos , A.F. Nowakowski and F.C.G.A. Nicolleau d

Also, we discuss the Kemeny constant that

gives the average number of steps necessary to reach any site of the network.

Through all this work, we

analyze the mechanisms behind the fractional trans-

Fractional Random Walks on regular networks and lattices d

walker.

port on networks and how this long-range process induces dynamically the small-world property in dierent structures. [40, 41, 42, 43, 31, 32]

d,e

b

b

We establish a generalization of Polya's recurrence

Performance assessment of numerical modelling for hydraulic eciency of a grated inlet

0