Kinetic energy spectrum of low-Reynolds-number turbulence with

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Kinetic energy spectrum of low-Reynolds-numberturbulence with polymer additivesTo cite this article Takeshi Watanabe and Toshiyuki Gotoh 2013 J Phys Conf Ser 454 012007

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Kinetic energy spectrum of low-Reynolds-number

turbulence with polymer additives

Takeshi Watanabe and Toshiyuki Gotoh

Graduate School of Engineering Department of Scientific and Engineering SimulationsNagoya Institute of Technology Gokiso Showa-ku Nagoya 466-8555 Japan

E-mail watanabenitechacjp gotohtoshiyukinitechacjp

Abstract The effects of polymer additives on the behavior of the kinetic energy spectrumin isotropic decaying turbulence were numerically investigated by hybrid Eulerian-Lagrangiansimulations making full use of large-scale parallel computation The kinetic energy spectrum wasfound to obey the power-law E(k) sim kminusα in the range below the Kolmogorov length lK whenthe turbulence decayed The exponent α satisfied 4 lt α lt 5 and decreased with the increasein the Weissenberg number Wi The value of α obtained by the largest Wi run was close tothe values obtained in previous experimental and numerical studies on elastic turbulence whichis characterized by a larger Wi and a Reynolds number of less than unity The relationshipbetween the results of the present study and the results obtained for elastic turbulence is alsodiscussed

Small amounts of polymers added to a fluid significantly affect large-scale flow structuresTurbulence drag reduction [1 2] is one of the most important phenomena related to polymersolution flows because this phenomenon has a broad range of industrial applications involvingturbulent transport On the other hand the polymer solution flow characterized by both asmaller Reynolds number and higher elasticity indicates chaotic fluctuations in space and timeThis phenomenon is referred to as elastic turbulence [3] It is vital to examine the meso-scaledynamics of polymers in order to clarify the peculiar nature of polymer solution flows becausethe origin of the turbulence modifications or the origin of the appearance of random motions forlow-Reynolds-number flow are due to the strong elasticity of polymers in flows

Dilute polymer solution flows have been extensively investigated using constitutive equationssuch as the Oldroyd-B or FENE-P models [4] These are the evolution equations for the tensorfield representing the polymer conformation and are constructed based on a simple polymermodel The constitutive equations are widely used in numerical studies of the above-mentionedphenomena [2] because of the ease of handling the polymer effects on fluid motions for areasonable computational cost

Another method by which to numerically investigate the polymer solution flow is a hybridapproacheg [5] the fluid motion is computed using the NavierndashStokes (NS) equations whilethe polymer dynamics are determined by molecular or Brownian dynamics simulations (BDSs)using an appropriate polymer model [6] In a previous study [7] we developed a two-way coupledsimulation method using this approach in which a BDS for a dumbbell model coupled with adirect numerical simulation (DNS) of turbulent flow was performed using large-scale parallelcomputations A number of dumbbells on the order of ten billion (O(1010)) were dispersed ina turbulent flow and their advection and deformation were tracked during the time evolution of

24th IUPAP Conference on Computational Physics (IUPAP-CCP 2012) IOP PublishingJournal of Physics Conference Series 454 (2013) 012007 doi1010881742-65964541012007

Content from this work may be used under the terms of the Creative Commons Attribution 30 licence Any further distributionof this work must maintain attribution to the author(s) and the title of the work journal citation and DOI

Published under licence by IOP Publishing Ltd 1

the system We then examined the modification of the decaying turbulence for various polymerconcentration and Weissenberg number Wi = ττK which is the ratio of the polymer relaxationtime τ to the Kolmogorov time τK equiv (νsε)

12 for the smallest eddy in turbulence (νs and εare respectively the kinematic viscosity of the solvent fluid and the average rate of the energydissipation of turbulence)

Fundamental quantity characterizing the fluctuations of the solvent fluid is the kinetic energyspectrum which is defined by

E(k t) =

primesumk

1

2|u(k t)|2 (1)

in terms of the Fourier amplitude u(k t) wheresumprimek is the summation taken over the spherical

shell within kminus∆k2 lt |k| le k+ ∆k2 in the wavenumber space One of the most remarkableresults reported in [7] is the fact that when turbulence sufficiently decays and Wi gt 1 the kineticenergy spectrum obeys the power-law decay of E(k t) sim kminusα with α = 47 at wavenumbershigher than the Kolmogorov wavenumber 1lK equiv (εν3

s )14 On the other hand previousstudies on elastic turbulence revealed experimentally that E(k) sim kminus35 [3] and by the DNSof the Oldroyd-B model under a two-dimensional Kolmogorov flow that E(k) sim kminus38 [8] Thusthe power-law exponent obtained in a previous study (approximately 47) is larger than theexponents obtained for elastic turbulence although this result is consistent with the theoreticalprediction using the simplified viscoelastic model where the scaling exponent α must be greaterthan 3 [9]

The intrinsic difference between the previous study [7] and studies on elastic turbulence [3 8]is that the former was performed with a smaller Wi and a larger Reynolds number on the orderof Rλ 35 than in [3 8] where Rλ is the Taylor microscale Reynolds number defined by (9)This raises the question of how the scaling behavior of the kinetic energy spectrum varies withthe variation of parameters Wi and Rλ in the hybrid simulation

The purpose of the present study is to examine the scaling behavior of the kinetic energyspectrum E(k t) in isotropic decaying turbulence with polymer additives for a wider parameterspace We investigate the power-law behavior of E(k t) for cases of larger Wi values than thosein our previous study [7] This means that the system more closely resembles elastic turbulence

The dumbbell model used in the present study is given by

dR(n)

dt= u

(n)1 minus u

(n)2 minus 1

2τf

(|R(n)|Lmax

)R(n) +

reqradic2τ

(W

(n)1 minusW

(n)2

) (2)

dr(n)g

dt=

1

2

(u

(n)1 + u

(n)2

)+

reqradic8τ

(W

(n)1 + W

(n)2

) u(n)

α equiv u(x(n)α (t) t) (3)

where R(n)(t) and r(n)g (t) are respectively the end-to-end vector and the center-of-mass vector

of the n-th dumbbell We adopt the finitely extensible nonlinear elastic (FENE) modelf(z) = 1(1 minus z2) for the elastic force of a dumbbell In equation (2) Lmax is the maximum

extension length of the dumbbell because f(z rarr 1) = infin The term W(n)12 (t) indicates a

random force representing the Brownian motion of particles in the solvent fluid which obeys

Gaussian statistics with a white-in-time correlation of 〈W (n)αi (t)〉 = 0 and 〈W (m)

αi (t)W(n)βj (s)〉 =

δαβδijδmnδ(t minus s) where 〈middot middot middot〉 denotes the ensemble average The subscripts α β i j n andm take the values (α β) = 1 or 2 (i j) = 1 2 and 3 and (nm) = 1 2middot middot middot Nt respectivelyMoreover δij denotes the Kronecker delta and δ(t) is the Dirac delta function The constantsτ equiv ζ4k and req equiv

radickBTk are respectively the relaxation time and the equilibrium length


2

of the dumbbell under u(x t) = 0 Here k is the spring constant and ζ equiv 6πνsρsa (ρs is thedensity of the solvent fluid) Also kB and T are the Boltzmann constant and temperaturerespectively

The turbulent velocity field obeys the continuity equation for an incompressible fluid and theNS equations

nabla middot u = 0partu

partt+ u middot nablau = minusnablap+ νsnabla2u +nabla T p (4)

where p(x t) is the pressure field Here ρs is set to unity and is equal to the density of bead ρprepresenting the polymer and T p(x t) is the polymer stress tensor due to the force acting onthe fluid from the dispersed dumbbells and is defined by

T pij(x t) =νsη

τ

(L3box

Nt

)Ntsumn=1

R(n)i R

(n)j

r2eq

f

(|R(n)|Lmax

)minus δij

δ(xminus r(n)g ) (5)

where Nt is the total number of polymers required to realize the experimental situationη equiv (3req4a)2ΦV is the zero-shear viscosity ratio of the polymer νp to the solution viscosity(η = νpνs) and ΦV equiv (8πNt3)(aLbox)3 represents the volume fraction of the ensemble ofdumbbells

The numerical simulations of equations (2) through (5) are performed in a periodic box withperiodicity Lbox = 2π using the pseudo-spectral method in space and the second-order RungendashKutta method in time A total of 1283 grid points are set to solve the NS equations The totalnumber of dumbbells in computation is Ncomp = Ntb = 504 times 108 where b is the artificialparameter representing the number of replica dumbbells This reduces the computational costof the computation of Nt = O(1013) dumbbells [7] The initial velocity field is given by therandom solenoidal field obeying Gaussian statistics with an energy spectrum

E(k 0) = 16

radic2

π

(u2

0

k0

)(k

k0

)4

exp

(minus2

(k

k0

)2)

(u0 = 1 k0 = 2) (6)

In this setting the initial value of Rλ is Rλ(0) = 52 The dumbbells are uniformly and randomlydistributed over the computational domain The initial configuration of each dumbbell is setaccording to R(m)(0) =

radic3reqn

(m) where n(m) is a random unit vector which is isotropicallydistributed

Parallel computations are performed in order to evaluate the convection and deformationof the dispersed dumbbells The program is parallelized using Message Passing Interface(MPI) The maximum number of MPI processors is 64 and the total number of dumbbellsin computation Ncomp is divided into 63 groups with each group assigned one processor thatcomputes the temporal evolution of the group One processor is also assigned to perform theDNS of the solvent fluid Figure 1 shows the schematic diagram of the parallel computationin the hybrid simulations The field data of the fluid velocity u(x t) and u(x + ∆x2 t) fromthe process dedicated to the time integration of the turbulence is transferred to all of the other63 processes for the computation of the dumbbells The fluid velocity at the bead positions isthen interpolated using the TS13 scheme [10] whereas the polymer stress term is evaluated asfollows

(i) The polymer stress field T pm(x) (m = 1 middot middot middot 63) is computed according to (5) for each

processor

(ii) The results are gathered by the processor (m = 0) for the turbulence DNS and are summedas T p(x) =

sum63m=1 T

pm(x)

(iii) The obtained T p(x) is incorporated into the DNS of the NS equations


3

Figure 1 Schematic diagram of theparallelized method used in the hybridsimulation The total number of processM + 1 is set to be M + 1 = 64 in the presentstudy

In step (i) since the center-of-mass vector r(n)g for each of the dumbbells is not on the grid points

for the DNS computation we need an approximate expression instead of (5) The delta function

in (5) is approximated by the weight function δ∆(xminus r(n)g ) used for the tri-linear interpolation

scheme [11] The other parameter settings are described in detail in [7] We examine three casesof decaying turbulence for three values of Wi while fixing the other parameters The values ofthe numerical parameters are listed in Table 1

Table 1 Parameters for the hybrid simulations of decaying turbulence and Brownian dynamicsfor dispersed dumbbells The values of τ listed below are determined using the values of Wi

given in the table and the minimum value of τK(t) during the simulation without the polymeradditives (one-way coupling simulation)

Ncomp (times109) b (times105) ΦV (times10minus4) η Wi τ

Run D1 0504 09 101 01045 25 4903Run D2 0504 09 101 01045 50 9806Run D3 0504 09 101 01045 100 19612

Figure 2 shows the temporal evolutions of the kinetic energy of fluid motion E(t) and of thepotential energy for the ensemble of dumbbells U(t) which are defined respectively as follows

E(t) =1

2〈u(x t)2〉V (7)

and

U(t) = minusνsη2τ

(Lmaxreq

)21

Ncomp

Ncompsumn=1

ln

1minus(|R(n)(t)|Lmax

)2 (8)

for all of the runs where 〈middot middot middot〉V represents the volume average over the computational domainV = L3

box Hereinafter figures are plotted using non-dimensional time tlowast equiv k0u0t In figure 2 (a)E(t) decays monotonically with time and the curves collapse onto a single curve irrespectiveof Wi The values of E(t) at the end of the simulations are much smaller than their initialvalues meaning that the turbulence adequately decayed at tlowast = 20 On the other hand U(t)increases with time for tlowast lt 7 which indicates that the turbulence kinetic energy is transferred


4

10-4

10-3

10-2

10-1

100

0 5 10 15 20

E(t

) U

(t)

t

(a)

E(t)U(t)

Run D1Run D2Run D3

10-1

100

0 5 10 15 20

E(t

) +

U(t

)

t

(b)

Run D1Run D2Run D3

Figure 2 Comparison of the temporal evolutions of (a) the kinetic energy of fluid motionE(t) and the potential energy for the ensemble of dumbbells U(t) and of (b) the total energyE(t) + U(t) for Runs D1 D2 and D3

100

101

0 5 10 15 20

Rλ (

t)

t

Run D1Run D2Run D3

100

101

102

0 5 10 15 20

Wi (

t)

t

Run D1Run D2Run D3

Figure 3 Comparison of the temporalevolutions of the Taylor microscale Reynoldsnumber Rλ(t) for Runs D1 D2 and D3

Figure 4 Comparison of the temporalevolutions of the local Weissenberg numberWi(t) for Runs D1 D2 and D3

to the potential energy of the ensemble of dumbbells in addition to the heat Moreover U(t)starts to decay at around tlowast = 7 indicating that the portion of restored potential energy is alsoback-transferred to turbulence and is dissipated due to the viscosity Note that U(t) is largerthan E(t) in the range of tlowast gt 10 for all of the runs Figure 2(b) shows the temporal variationof the total energy E(t) + U(t) based on the data of figure 2(a) We confirmed that the totalenergy also decays monotonically with time

The temporal evolutions of the Taylor microscale Reynolds number Rλ(t) which is definedby using E(t) (7) and ε(t) = νs〈(nablau)2〉V as follows

Rλ(t) =

radic20

3νsε(t)E(t) (9)

are shown in figure 3 The curves almost collapse onto a single curve for tlowast lt 10 and the curvefor Run D1 slightly deviates upward from the other curves for tlowast gt 10 Thus Rλ is almostinsensitive to the variation of Wi and the values at tlowast = 20 are Rλ = 35 32 and 31 for RunsD1 D2 and D3 respectively

Figure 4 compares the temporal evolutions of the local Weissenberg number Wi(t) = ττK(t)


5

10-5

10-4

10-3

10-2

10-1

100

101

10-1

100

101

ε (

t)-2

3 l

K(t

)-53

E

(kt

)

k lK(t)

t = 20

k-47

Run D1Run D2Run D3

10-3

10-2

10-1

100

10-1

100

101

kα E

(kt

)

k lK(t)

t = 20

Run D1 (α=458)Run D2 (α=433)Run D3 (α=420)

Figure 5 Comparison of the kinetic energyspectra normalized using the Kolmogorovscaling of (10) obtained at tlowast = 20 for RunsD1 D2 and D3 The reference line shows thepower law scaling of kminus47

Figure 6 Compensated kinetic energyspectra kαE(k t) obtained at tlowast = 20 forRuns D1 D2 and D3 The value of α isevaluated by a least squares fit in the rangeof 2 le klK(t) le 5 Plateaus are clearlyconfirmed in the range klK(t) gt 1

for all runs Based on figure 4 we confirm that the curves reach their maximum values atapproximately tlowast = 2 and Wi(t) decays with time for tlowast gt 2 The value obtained for RunD3 is the greatest among the three values in the entire time region At the end of simulation(tlowast = 20) we have Wi = 17 38 and 75 for Runs D1 D2 and D3 respectively which indicatesthat numerous dumbbells remain in the stretched configuration for Runs D2 and D3 becausethe values of Wi are just above the critical value W c

i = 3minus 4 at which the coil-stretch transitionoccurs [12]

Since the turbulence adequately decays for tlowast = 20 where Wi is larger than unity for all ofthe runs we investigate the scaling behavior of E(k t) obtained at tlowast = 20 Figure 5 comparesthe kinetic energy spectra obtained for all of the runs where we plot the dimensionless form ofspectra E(klK(t) t) defined using the Kolmogorov scaling as

E(klK(t) t) =E(k t)

ε(t)23lK(t)53 (10)

As shown in figure 5 the normalized spectra generally collapse in the range of klK(t) lt 1and exhibit a power-law decay similar to kminus47 over one decade for klK(t) gt 1 However thespectrum becomes less steep with the increase of Wi To see this point in more details we makethe compensated plot kαE(k t) in figure 6 where the exponent α is evaluated by using a leastsquares fit in the range 2 le klk(t) le 5 This figure clearly indicates that the value of α decreasesfrom 458 to 420 when Wi is increased For the greatest values of Wi α = 420 which is closeto the values obtained for elastic turbulence [3 8]

Next we will comment on the difference between the hybrid DNS using FENE dumbbellsand the DNS of the elastic turbulence using the Ordroyd-B model [8] The FENE model hasthe upper limit of the dumbbell extension length Lmax This means that many dumbbells tendto remain at around Lmax when Wi is larger Figure 7 shows the probability density function(PDF) for the end-to-end distance of dumbbell r = |R(n)|Lmax at tlowast = 20 for all of the runsThis figure indicates that there are more stretched dumbbells for larger Wi even when Rλ issufficiently small This also implies that the contribution of each dumbbell to the polymer stresstensor T p becomes larger as Wi increases because f(z)rarrinfin as z rarr 1 In contrast the Oldroyd-B model is a linear equation for the conformation tensor of the polymer which corresponds to


6

10-6

10-5

10-4

10-3

10-2

10-1

100

101

0 02 04 06 08 1

P(r

t)

r

t = 20

Run D1Run D2Run D3 Figure 7 Probability density function for

the end-to-end distance of dumbbells P (r t)normalized by the maximum extension lengthLmax obtained at tlowast = 20

the case in which the linear spring model is adapted to (2) by setting f(z) = 1 In this casethe stretched dumbbells can exceed Lmax leading to a greater contribution to the increase inT p due to an increase in |R(n)| Thus the mechanism for increasing the amplitude of T p whichstrongly affects the power-law behavior of E(k t) through the NS equations differs significantlybetween the FENE model and the Oldroyd-B model

Another important point is the nature of the isotropy of the flow field In the present case theflow field is regarded as nearly isotropic so that the effect of anisotropy on the scaling behaviorof E(k t) is negligible In contrast the flow field is strongly anisotropic for the DNS study underthe Kolmogorov flow [8] and for the experimental viscoelastic flow within the rotating disks [3]The persistent anisotropy leads to the modification of the spectral dynamics at small scales sothat we must carefully compare the present results to the previous results for elastic turbulence

The above considerations raise the question of how the power-law form of the kinetic energyspectrum is universal for the changes in the mechanical properties of the chain polymer model orthe degree of anisotropy Moreover since the flow field is nearly isotropic in the present study itis appropriate to explore whether the power-law exponent is universal for Wi 1 and Rλ 1We may need more computational resources to examine this problem because the case in whichWi 1 requires much finer spatial resolution for computing the NS equation and much finertime steps for computing the dumbbell deformations in the Lagrangian frame with reasonablecomputation accuracy This is perhaps the greatest computational challenge in examining theultimate state of elastic turbulence by large-scale simulations

We investigated the scaling behavior of the kinetic energy spectrum in decaying isotropicturbulence with polymer additives for various values of Wi We obtained the power-law formof E(k) sim kminusα in the range klK gt 1 and α was approximately 42ndash46 and decreased with theincrease in Wi The value of 420 obtained for Run D3 which has the greatest value of Wi inthe present study was similar to the values obtained in the study on elastic turbulence (α = 35experimentally [3] and α = 38 by DNS of the constitutive equation [8])

T W and T G were supported in part by Grants-in-Aid for Scientific Research Nos23760156 and 24360068 respectively from the Ministry of Education Culture Sports Scienceand Technology of Japan T W would like to thank S Uno and D Sugimoto for their assistancein writing the parallelized code The authors would like to thank the Theory and ComputerSimulation Center of the National Institute for Fusion Science and JHPCN and HPC at theInformation Technology Center of Nagoya University for providing the computational resources

References[1] Lumley J L 1973 J Polymer Sci Macromolecular Reviews 7 263[2] Proccacia I Lrsquovov V S and Benzi R 2008 Rev Mod Phys 80 225[3] Groisman A and Steinberg V 2000 Nature 405 53


7

[4] Bird R B Curtiss C F Amstrong R C and Hassager O 1987 Dynamics of Polymetric Liquids Vol2 KineticTheory 2nd ed Wiley New York

[5] Laso M and Ottinger H C 1993 J Non-Newtonian Fluid Mech 47 1[6] Doi M and Edwards S F 1986 The Theory of Polymer Dynamics Oxford Univ Press New York[7] Watanabe T and Gotoh T 2013 J Fluid Mech 717 535[8] Berti S Bistagnino A Boffetta G Celani A and Musacchio S 2008 Phys Rev E 77 055306[9] Fouxon A and Lebedev V 2003 Phys Fluids 15 2060[10] Yeung P K and Pope S B 1988 J Comp Phys 79 373[11] Prosperetti A and Tryggvason G 2007 Computational Methods for Multiphase Flow Cambridge University

Press Cambridge[12] Watanabe T and Gotoh T 2010 Phys Rev E 81 066301


8

Kinetic energy spectrum of low-Reynolds-number

turbulence with polymer additives

Takeshi Watanabe and Toshiyuki Gotoh

Graduate School of Engineering Department of Scientific and Engineering SimulationsNagoya Institute of Technology Gokiso Showa-ku Nagoya 466-8555 Japan

E-mail watanabenitechacjp gotohtoshiyukinitechacjp

Abstract The effects of polymer additives on the behavior of the kinetic energy spectrumin isotropic decaying turbulence were numerically investigated by hybrid Eulerian-Lagrangiansimulations making full use of large-scale parallel computation The kinetic energy spectrum wasfound to obey the power-law E(k) sim kminusα in the range below the Kolmogorov length lK whenthe turbulence decayed The exponent α satisfied 4 lt α lt 5 and decreased with the increasein the Weissenberg number Wi The value of α obtained by the largest Wi run was close tothe values obtained in previous experimental and numerical studies on elastic turbulence whichis characterized by a larger Wi and a Reynolds number of less than unity The relationshipbetween the results of the present study and the results obtained for elastic turbulence is alsodiscussed

Small amounts of polymers added to a fluid significantly affect large-scale flow structuresTurbulence drag reduction [1 2] is one of the most important phenomena related to polymersolution flows because this phenomenon has a broad range of industrial applications involvingturbulent transport On the other hand the polymer solution flow characterized by both asmaller Reynolds number and higher elasticity indicates chaotic fluctuations in space and timeThis phenomenon is referred to as elastic turbulence [3] It is vital to examine the meso-scaledynamics of polymers in order to clarify the peculiar nature of polymer solution flows becausethe origin of the turbulence modifications or the origin of the appearance of random motions forlow-Reynolds-number flow are due to the strong elasticity of polymers in flows

Dilute polymer solution flows have been extensively investigated using constitutive equationssuch as the Oldroyd-B or FENE-P models [4] These are the evolution equations for the tensorfield representing the polymer conformation and are constructed based on a simple polymermodel The constitutive equations are widely used in numerical studies of the above-mentionedphenomena [2] because of the ease of handling the polymer effects on fluid motions for areasonable computational cost

Another method by which to numerically investigate the polymer solution flow is a hybridapproacheg [5] the fluid motion is computed using the NavierndashStokes (NS) equations whilethe polymer dynamics are determined by molecular or Brownian dynamics simulations (BDSs)using an appropriate polymer model [6] In a previous study [7] we developed a two-way coupledsimulation method using this approach in which a BDS for a dumbbell model coupled with adirect numerical simulation (DNS) of turbulent flow was performed using large-scale parallelcomputations A number of dumbbells on the order of ten billion (O(1010)) were dispersed ina turbulent flow and their advection and deformation were tracked during the time evolution of


Content from this work may be used under the terms of the Creative Commons Attribution 30 licence Any further distributionof this work must maintain attribution to the author(s) and the title of the work journal citation and DOI

Published under licence by IOP Publishing Ltd 1




E(k t) =

primesumk

1

2|u(k t)|2 (1)







dR(n)

dt= u

(n)1 minus u

(n)2 minus 1

2τf

(|R(n)|Lmax

)R(n) +

reqradic2τ

(W

(n)1 minusW

(n)2

) (2)

dr(n)g

dt=

1

2

(u

(n)1 + u

(n)2

)+

reqradic8τ

(W

(n)1 + W

(n)2

) u(n)











2






T pij(x t) =νsη

τ

(L3box

Nt

)Ntsumn=1

R(n)i R

(n)j

r2eq

f

(|R(n)|Lmax

)minus δij




E(k 0) = 16

radic2

π

(u2

0

k0

)(k

k0

)4

exp

(minus2

(k

k0

)2)

(u0 = 1 k0 = 2) (6)


radic3reqn




processor


sum63m=1 T

pm(x)



3









Run D1 0504 09 101 01045 25 4903Run D2 0504 09 101 01045 50 9806Run D3 0504 09 101 01045 100 19612


E(t) =1

2〈u(x t)2〉V (7)

and


(Lmaxreq

)21

Ncomp

Ncompsumn=1

ln


)2 (8)




4

10-4

10-3

10-2

10-1

100

0 5 10 15 20

E(t

) U

(t)

t

(a)

E(t)U(t)

Run D1Run D2Run D3

10-1

100

0 5 10 15 20

E(t

) +

U(t

)

t

(b)

Run D1Run D2Run D3


100

101

0 5 10 15 20

Rλ (

t)

t

Run D1Run D2Run D3

100

101

102

0 5 10 15 20

Wi (

t)

t

Run D1Run D2Run D3





Rλ(t) =

radic20

3νsε(t)E(t) (9)




5

10-5

10-4

10-3

10-2

10-1

100

101

10-1

100

101

ε (

t)-2

3 l

K(t

)-53

E

(kt

)

k lK(t)

t = 20

k-47

Run D1Run D2Run D3

10-3

10-2

10-1

100

10-1

100

101

kα E

(kt

)

k lK(t)

t = 20







E(klK(t) t) =E(k t)

ε(t)23lK(t)53 (10)




6

10-6

10-5

10-4

10-3

10-2

10-1

100

101

0 02 04 06 08 1

P(r

t)

r

t = 20










7





8




E(k t) =

primesumk

1

2|u(k t)|2 (1)







dR(n)

dt= u

(n)1 minus u

(n)2 minus 1

2τf

(|R(n)|Lmax

)R(n) +

reqradic2τ

(W

(n)1 minusW

(n)2

) (2)

dr(n)g

dt=

1

2

(u

(n)1 + u

(n)2

)+

reqradic8τ

(W

(n)1 + W

(n)2

) u(n)











2






T pij(x t) =νsη

τ

(L3box

Nt

)Ntsumn=1

R(n)i R

(n)j

r2eq

f

(|R(n)|Lmax

)minus δij




E(k 0) = 16

radic2

π

(u2

0

k0

)(k

k0

)4

exp

(minus2

(k

k0

)2)

(u0 = 1 k0 = 2) (6)


radic3reqn




processor


sum63m=1 T

pm(x)



3









Run D1 0504 09 101 01045 25 4903Run D2 0504 09 101 01045 50 9806Run D3 0504 09 101 01045 100 19612


E(t) =1

2〈u(x t)2〉V (7)

and


(Lmaxreq

)21

Ncomp

Ncompsumn=1

ln


)2 (8)




4

10-4

10-3

10-2

10-1

100

0 5 10 15 20

E(t

) U

(t)

t

(a)

E(t)U(t)

Run D1Run D2Run D3

10-1

100

0 5 10 15 20

E(t

) +

U(t

)

t

(b)

Run D1Run D2Run D3


100

101

0 5 10 15 20

Rλ (

t)

t

Run D1Run D2Run D3

100

101

102

0 5 10 15 20

Wi (

t)

t

Run D1Run D2Run D3





Rλ(t) =

radic20

3νsε(t)E(t) (9)




5

10-5

10-4

10-3

10-2

10-1

100

101

10-1

100

101

ε (

t)-2

3 l

K(t

)-53

E

(kt

)

k lK(t)

t = 20

k-47

Run D1Run D2Run D3

10-3

10-2

10-1

100

10-1

100

101

kα E

(kt

)

k lK(t)

t = 20







E(klK(t) t) =E(k t)

ε(t)23lK(t)53 (10)




6

10-6

10-5

10-4

10-3

10-2

10-1

100

101

0 02 04 06 08 1

P(r

t)

r

t = 20










7





8






T pij(x t) =νsη

τ

(L3box

Nt

)Ntsumn=1

R(n)i R

(n)j

r2eq

f

(|R(n)|Lmax

)minus δij




E(k 0) = 16

radic2

π

(u2

0

k0

)(k

k0

)4

exp

(minus2

(k

k0

)2)

(u0 = 1 k0 = 2) (6)


radic3reqn




processor


sum63m=1 T

pm(x)



3









Run D1 0504 09 101 01045 25 4903Run D2 0504 09 101 01045 50 9806Run D3 0504 09 101 01045 100 19612


E(t) =1

2〈u(x t)2〉V (7)

and


(Lmaxreq

)21

Ncomp

Ncompsumn=1

ln


)2 (8)




4

10-4

10-3

10-2

10-1

100

0 5 10 15 20

E(t

) U

(t)

t

(a)

E(t)U(t)

Run D1Run D2Run D3

10-1

100

0 5 10 15 20

E(t

) +

U(t

)

t

(b)

Run D1Run D2Run D3


100

101

0 5 10 15 20

Rλ (

t)

t

Run D1Run D2Run D3

100

101

102

0 5 10 15 20

Wi (

t)

t

Run D1Run D2Run D3





Rλ(t) =

radic20

3νsε(t)E(t) (9)




5

10-5

10-4

10-3

10-2

10-1

100

101

10-1

100

101

ε (

t)-2

3 l

K(t

)-53

E

(kt

)

k lK(t)

t = 20

k-47

Run D1Run D2Run D3

10-3

10-2

10-1

100

10-1

100

101

kα E

(kt

)

k lK(t)

t = 20







E(klK(t) t) =E(k t)

ε(t)23lK(t)53 (10)




6

10-6

10-5

10-4

10-3

10-2

10-1

100

101

0 02 04 06 08 1

P(r

t)

r

t = 20










7





8









Run D1 0504 09 101 01045 25 4903Run D2 0504 09 101 01045 50 9806Run D3 0504 09 101 01045 100 19612


E(t) =1

2〈u(x t)2〉V (7)

and


(Lmaxreq

)21

Ncomp

Ncompsumn=1

ln


)2 (8)




4

10-4

10-3

10-2

10-1

100

0 5 10 15 20

E(t

) U

(t)

t

(a)

E(t)U(t)

Run D1Run D2Run D3

10-1

100

0 5 10 15 20

E(t

) +

U(t

)

t

(b)

Run D1Run D2Run D3


100

101

0 5 10 15 20

Rλ (

t)

t

Run D1Run D2Run D3

100

101

102

0 5 10 15 20

Wi (

t)

t

Run D1Run D2Run D3





Rλ(t) =

radic20

3νsε(t)E(t) (9)




5

10-5

10-4

10-3

10-2

10-1

100

101

10-1

100

101

ε (

t)-2

3 l

K(t

)-53

E

(kt

)

k lK(t)

t = 20

k-47

Run D1Run D2Run D3

10-3

10-2

10-1

100

10-1

100

101

kα E

(kt

)

k lK(t)

t = 20







E(klK(t) t) =E(k t)

ε(t)23lK(t)53 (10)




6

10-6

10-5

10-4

10-3

10-2

10-1

100

101

0 02 04 06 08 1

P(r

t)

r

t = 20










7





8

10-4

10-3

10-2

10-1

100

0 5 10 15 20

E(t

) U

(t)

t

(a)

E(t)U(t)

Run D1Run D2Run D3

10-1

100

0 5 10 15 20

E(t

) +

U(t

)

t

(b)

Run D1Run D2Run D3


100

101

0 5 10 15 20

Rλ (

t)

t

Run D1Run D2Run D3

100

101

102

0 5 10 15 20

Wi (

t)

t

Run D1Run D2Run D3





Rλ(t) =

radic20

3νsε(t)E(t) (9)




5

10-5

10-4

10-3

10-2

10-1

100

101

10-1

100

101

ε (

t)-2

3 l

K(t

)-53

E

(kt

)

k lK(t)

t = 20

k-47

Run D1Run D2Run D3

10-3

10-2

10-1

100

10-1

100

101

kα E

(kt

)

k lK(t)

t = 20







E(klK(t) t) =E(k t)

ε(t)23lK(t)53 (10)




6

10-6

10-5

10-4

10-3

10-2

10-1

100

101

0 02 04 06 08 1

P(r

t)

r

t = 20










7





8

10-5

10-4

10-3

10-2

10-1

100

101

10-1

100

101

ε (

t)-2

3 l

K(t

)-53

E

(kt

)

k lK(t)

t = 20

k-47

Run D1Run D2Run D3

10-3

10-2

10-1

100

10-1

100

101

kα E

(kt

)

k lK(t)

t = 20







E(klK(t) t) =E(k t)

ε(t)23lK(t)53 (10)




6

10-6

10-5

10-4

10-3

10-2

10-1

100

101

0 02 04 06 08 1

P(r

t)

r

t = 20










7





8

10-6

10-5

10-4

10-3

10-2

10-1

100

101

0 02 04 06 08 1

P(r

t)

r

t = 20










7





8





8

Documents

Kinetic energy spectrum of low-Reynolds-number turbulence with