Research Article Refinement of a Previous Hypothesis of the ...downloads.hindawi.com/journals/je/2013/653027.pdf[] N. de Divitiis, Self-similarity in fully developed homogeneous isotropic

Hindawi Publishing CorporationJournal of EngineeringVolume 2013 Article ID 653027 4 pageshttpdxdoiorg1011552013653027

Research ArticleRefinement of a Previous Hypothesis of theLyapunov Analysis of Isotropic Turbulence

Nicola de Divitiis

Dipartimento di Ingegneria Meccanica e Aerospaziale ldquoLa Sapienzardquo University Via Eudossiana 18 00184 Rome Italy

Correspondence should be addressed to Nicola de Divitiis ndedivitiisgmailcom

Received 18 August 2012 Accepted 15 March 2013

Academic Editor Mickael Lallart

Copyright copy 2013 Nicola de Divitiis This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The purpose of this paper is to improve a hypothesis of the previous work of N de Divitiis (2011) dealing with the finite-scaleLyapunov analysis of isotropic turbulence There the analytical expression of the structure function of the longitudinal velocitydifference Δ119906

119903is derived through a statistical analysis of the Fourier transformed Navier-Stokes equations and by means of

considerations regarding the scales of the velocity fluctuations which arise from the Kolmogorov theory Due to these latterconsiderations this Lyapunov analysis seems to need some of the results of the Kolmogorov theory This work proposes a morerigorous demonstration which leads to the same structure function without using the Kolmogorov scale This proof assumesthat pair and triple longitudinal correlations are sufficient to determine the statistics of Δ119906

119903and adopts a reasonable canonical

decomposition of the velocity difference in terms of proper stochastic variables which are adequate to describe the mechanism ofkinetic energy cascade

1 Introduction

In the previous work [1] the author applies the finite-scale Lyapunov theory to analyse the homogeneous isotropicturbulence In particular this theory leads to the analyticalclosure of the von Karman-Howarth equation giving thelongitudinal triple velocity correlation 119896 in terms of thelongitudinal velocity correlation 119891 and 120597119891120597119903 (see also theappendix) and shows that the structure function of thelongitudinal velocity difference is expressed by

Δ119906119903

radic⟨(Δ119906119903)2

⟩

=

120585 + 120595 (120594 (1205782

minus 1) minus (1205772

minus 1))

radic1 + 21205952 (1 + 1205942)

(1)

whereΔ119906119903= (u(x+r)minusu(x)) sdotr119903 120585 120578 and 120577 are uncorrelated

centered Gaussian random variables with ⟨1205852⟩ = ⟨1205782⟩ = ⟨1205772⟩= 1 and 120595 is a function of the Taylor scale Reynolds number119877120582= 119906120582119879] and of the separation distance 119903 equiv |r| according

to

120595 (119903 119877120582) = radic

119877120582

15radic15 (119903) (2)

where 119906 = radic⟨1199062119903⟩ is the longitudinal velocity standard

deviation 120582119879is the Taylor microscale and 120594 = 120594(119877

120582) = 1 is

a proper function of 119877120582which provides nonzero skewness of

Δ119906119903[1 2]In [1] the demonstration of (1) is carried out through sta-

tistical elements regarding the Fourier transformed Navier-Stokes equations whereas the proof of (2) is based on thefact that according to theKolmogorov theory the ratio (smallscale velocity)ndash(large scale velocity) depends on120582

119879ℓ asymp radic119877

120582

where ℓ is the KolmogorovmicroscaleTherefore the analysisof [1] seems to require the adoption of ℓ whose definition isbased on another theory

Here instead of using the Kolmogorov scale we obtain(1) and (2) starting from the canonical decomposition of thefluid velocity in terms of proper centered random variables120585119896 In order to describe the mechanism of kinetic energy

cascade the variables 120585119896are properly chosen in such a

way that each of them exhibits a nonsymmetric distributionfunction Moreover due to the isotropy we postulate thatthe knowledge of 119891 and 119896 represents a sufficient condition todetermine the statistics of Δ119906

119903

2 Journal of Engineering

2 The Lyapunov Analysis of theVelocity Fluctuations

This section renews the procedure for calculating the velocityfluctuations which is based on the Lyapunov analysis ofthe fluid strain [1] and on the momentum Navier-Stokesequations

120597119906119896

120597119905= minus

120597119906119896

120597119909ℎ

119906ℎ+1

120588

120597119879119896ℎ

120597119909ℎ

(3)

where u equiv (1199061 1199062 1199063) 119879119896ℎ and 120588 are the fluid velocity stress

tensor and density respectivelyIn order to obtain the velocity fluctuation consider

now the relative motion between two contiguous particlesexpressed by the infinitesimal separation vector 119889x whichobeys

119889 = nablau 119889x (4)

where 119889x varies according to the velocity gradient whichin turn follows the Navier-Stokes equations As observed in[1] 119889x is much faster than the fluid state variables and theLyapunov analysis of (4) provides the expression of the localdeformation in terms of the maximal Lyapunov exponentΛ gt 0

120597x120597x0

asymp 119890Λ(119905minus1199050) (5)

The map 120594 x0rarr x is the function which gives the

current position x of a fluid particle located at the referentialposition x

0at 119905 = 119905

0[3] Equation (3) can be written in terms

of the referential position x0[3]

120597119906119896

120597119905= (minus

120597119906119896

1205971199090119901

119906ℎ+1

120588

120597119879119896ℎ

1205971199090119901

)

1205971199090119901

120597119909ℎ

(6)

The adoption of the referential coordinates allows tofactorize the velocity fluctuation and to express it in theLyapunov exponential form of the local deformation Asthis deformation is assumed to be much more rapid thanminus1205971199061198961205971199090119901119906ℎ+ 1120588120597119879

119896ℎ1205971199090119901 the velocity fluctuation can be

obtained by integrating (6) with respect to the time whereminus1205971199061198961205971199090119901119906ℎ+ 1120588120597119879

119896ℎ1205971199090119901

is considered to be constant

119906119896asymp1

Λ(minus

120597119906119896

1205971199090119901

119906ℎ+1

120588

120597119879119896ℎ

1205971199090119901

)

119905=1199050

(7)

This assumption is justified by the fact that accordingto the classical formulation of motion of continuum media[3] the terms into the circular brackets of (6) are consideredto be smooth functions of 119905mdashat least during the periodof a fluctuationmdashwhereas the fluid deformation varies veryrapidly according to (4)-(5)

3 Statistical Analysis of Velocity Difference

As explained in this section the Lyapunov analysis of thelocal deformation and some plausible assumptions about the

statistics of u lead to determination of the structure functionof Δ119906119903and its PDF

The statistical properties of Δ119906119903are here investigated

expressing the fluid velocity through the following canonicaldecomposition [4]

u = sum119896

U119896120585119896 (8)

where U119896(119896 = 1 2 ) are proper coordinate functions

of 119905 and x and 120585119896(119896 = 1 2 ) are certain dimensionless

independent stochastic variables which satisfy

⟨120585119896⟩ = 0 ⟨120585

119894120585119895⟩ = 120575119894119895

⟨120585119894120585119895120585119896⟩ = 120603

119894119895119896119901

10038161003816100381610038161199011003816100381610038161003816 ⋙ 1

10038161003816100381610038161199011003816100381610038161003816 ⋙ ⟨120585

4

119896⟩

(9)

where120603119894119895119896

= 1 for 119894 = 119895 = 119896 else120603119894119895119896= 0 It is worth to remark

that the variables 120585119896are adequately chosen in such a way that

they can describe properly the mechanism of energy cascadeSpecifically the adoption of 120585

119896with |119901| ⋙ 1 is justified by

the fact that the evolution equation of the velocity correlation(see eg the von Karman-Howarth equation in appendix)includes also the third-order velocity correlation 119896(119903) whichis responsible for the intensive mechanism of energy cascadeand ⟨(Δ119906

119903)3

⟩⟨(Δ119906119903)2

⟩32

= 0 As a result it is reasonable thatthe canonical decomposition (8) (Δ119906

119903= (r119903) sdotsum

119896(U119896(x+r)minus

U119896(x))120585119896) includes variables 120585

119896with |⟨1205853

119896⟩| ⋙ 1 [5] This has

very important implications for what concerns the statisticsof the fluctiations of Δ119906

119903 In order to analyze this question

consider now the dimensionless velocity fluctuation u Thisis obtained in terms of 120585

119896by substituting (8) into (7)

ℎ= sum

119894119895

119860ℎ119894119895120585119894120585119895+1

119877120582

sum

119896

119887ℎ119896120585119896 (10)

where 119903 = 120582119879and 119906

ℎ= ℎ119906 Therefore sum

119894119895119860ℎ119894119895120585119894120585119895arises

from the inertia and pressure terms whereas 1119877120582sum119896119887ℎ119896120585119896is

due to the fluid viscosity Now thanks to the local isotropy119906ℎis a Gaussian stochastic variable [4 5] accordingly 120585

119896

satisfy into (10) the Lindeberg condition a very generalnecessary and sufficient condition for satisfying the centrallimit theorem [5] This condition does not apply to thevelocity difference In fact as Δu is the difference betweentwo correlated Gaussian variables its PDF could be a non-Gaussian distribution function To study this the fluctuationΔ119906119903is first expressed in terms of 120585

119896

Δ119903(r) = sum

119894119895

Δ119860119903119894119895(r) 120585119894120585119895+1

119877120582

sum

119896

Δ119887119903119896(r) 120585119896

equiv 119871 + 119878 + 119866+

+ 119866minus

(11)

This fluctuation can be reduced to the contributions 119871 119878119866+ and119866minus appearing into (11) [6] in particular 119871 is the sum

of all linear terms due to the fluid viscosity and 119878 equiv 119878119894119895120585119894120585119895

is the sum of all bilinear forms arising from the inertia andpressure terms whereas 119866+ and 119866minus are respectively definitepositive and negative quadratic forms of centered Gaussian

Journal of Engineering 3

variables which are derived from the inertia and pressuretermsThe quantity 119871+119878 tends to a Gaussian random variablebeing the sum of statistically orthogonal terms [5 6] while119866+ and 119866minus are determined by means of the hypotheses of

isotropy and of fully developed flow

119866minus

= minus (1205772

minus 1)1205952(119903)

119866+

= (1205782

minus 1)1205953(119903)

(12)

Observe that due to these hypotheses 119866+ and 119866minus

are uncorrelated thus 120578 120577 are two independent centeredGaussian variables with ⟨1205782⟩ = ⟨1205772⟩ = 1 Furthermore asthe knowledge of 119891 and 119896 is considered to be a sufficientcondition for determining the statistics of Δ119906

119903 1205952and 120595

3

are assumed to be proportional with each other through aconstant which depends only on 119877

120582

120594 (119877120582) =

1205953(119903)

1205952(119903) (13)

Therefore the longitudinal velocity difference can bewritten as

Δ119906119903= 1205951120585 + 1205952(120594 (1205782


minus 1)) (14)

where 120585 is a centered Gaussian random variable with ⟨1205852⟩ = 1that thanks to the hypotheses of fully developed flow and ofisotropy is considered to be statistically independent from 120578and 120577

Comparing the terms of (14) and (11) we obtain that 1205951

and 1205952are related with each other and that their ratio 120595 equiv

12059511205952depends on 119877

120582and 119903

2 (1 + 1205942

)1205952

2

1205952

1

=

⟨(119866+

+ 119866minus

)2

⟩

⟨(119878119894119895120585119894120585119895+ (1119877

120582Δ119887119903119896120585119896))2

⟩

(15)

Now the divisor at the R H S of (15) is the sum of thefollowing three terms

119860 = 119878119894119895119878119901119902⟨120585119894120585119895120585119901120585119902⟩ 119861 =

2

119877120582

119878119894119895⟨120585119894120585119895120585119896⟩Δ119887119903119896

119862 =1

1198772

120582

Δ1198872

119903119896⟨1205852

119896⟩

(16)

Hence taking into account the properties (9) of 120585119896 |119861| ⋙

|119860| |119862| thus120595 tends to a quantity arising only from the terms⟨1205853

119896⟩ which appear in (15)

120595 equiv1205952

1205951

= 120593 (119903)radic119877120582 (17)

This expression corresponds to that obtained in [1]

120595 (119903 119877120582) = radic

119877120582

15radic15 (119903) (0) = 119874 (1) (18)

and the dimensionless longitudinal velocity difference isgiven by (1)

Δ119906119903

radic⟨(Δ119906119903)2

⟩

=

120585 + 120595 (120594 (1205782


minus 1))

radic1 + 21205952 (1 + 1205942)

(19)

It is worth to remark that 120595 expresses the fluctuationsratio (large scale velocity)ndash(small scale velocity) that is 120595 asymp119906119906119904asymp (119906

2

120582119879)(1198971199041199062

119904) where 119897

119904and 119906

119904are respectively

the characteristic small scale and the corresponding velocityThis implies that 119906119906

119904≃ 120582119879119897119904asymp radic119877

120582 thus 119897

119904identifies

the Kolmogorov scale and 119906119904119897119904] asymp 1 is the corresponding

Reynolds numberThe distribution function of Δ119906

119903is then expressed

through the Frobenius-Perron equation [7] taking intoaccount that 120585 120578 and 120577 are independent stochastic variables

119865 (Δ1199061015840

119903)

= int

120585

int

120578

int

120577

119901 (120585) 119901 (120578) 119901 (120577) 120575 (Δ1199061015840

119903minusΔ119906119903(120585 120578 120577)) 119889120585 119889120578 119889120577

(20)

where Δ119906119903(120585 120578 120577) is determined by (19) 120575 is the Dirac delta

and 119901 is a centered Gaussian PDF with standard deviationequal to the unity The dimensionless statistical moments ofΔ119906119903are easily calculated

119867119899equiv

⟨(Δ119906119903)119899

⟩

⟨(Δ119906119903)2

⟩1198992

=1

(1 + 2 (1 + 1205942) 1205952)1198992

times

119899

sum

119896=0

(119899

119896)120595119896

⟨120585119899minus119896

⟩⟨(120594 (1205782


minus 1))119896

⟩

(21)In particular 119867

3 related to the mechanism of energy

cascade is

1198673(119903) =

81205953

(1205943

minus 1)

(1 + 21205952 (1 + 1205942))32

(22)

In conclusion 120594 = 120594(119877120582) is implicitly calculated in

function of 120595(0) taking into account that this Lyapunovtheory gives119867

3(0) = minus37 (see appendix) [1]

1198673(0) =

81205953

(0) (1205943

minus 1)

(1 + 21205952 (0) (1 + 1205942))32

= minus3

7 (23)

where (0) ≃ 1075 is estimated as in [1] andradic⟨(Δ119906119903)2

⟩ and(119903) are calculated in function of 119891(119903) and 119896(119903)

We conclude this paper by observing that the mecha-nism of energy cascade acts on Δ119906

119903whose expression here

calculated with the finite-scale Lyapunov theory and (9)provides a nonsymmetric PDF where the absolute values ofthe dimensionless moments |119867

119899(0)| rise with the Taylor scale

Reynolds number for 119899 gt 3


Appendix

For the sake of convenience this section reports some of theresults dealing with the closure of the von Karman-Howarthequation obtained in [1 2]

For fully developed isotropic homogeneous turbulencethe pair correlation function

119891 (119903) =⟨119906119903(x) 119906119903(x + r)⟩

1199062(A1)

satisfies the von Karman-Howarth equation [8]

120597119891

120597119905=119870 (119903)

1199062+ 2](

1205972

119891

1205971199032+4

119903

120597119891

120597119903) +

10]

1205822

119879

119891 (A2)

the boundary conditions of which are

119891 (0) = 1 lim119903rarrinfin

119891 (119903) = 0 (A3)

where 119906 equiv radic⟨1199062119903(x)⟩ satisfies the equation of the turbulent

kinetic energy [8]

1198891199062

119889119905= minus

10]

1205822

119879

1199062

(A4)

and 120582119879equiv radicminus111989110158401015840(0) is the Taylor scale The function 119870(119903)

giving the mechanism of energy cascade is related to thelongitudinal triple velocity correlation function 119896

119870 (119903) = 1199063

(120597

120597119903+4

119903) 119896 (119903)

where 119896 (119903) =⟨1199062

119903(x) 119906119903(x + r)⟩

1199063

(A5)

Thus the von Karman-Howarth equation provides therelationship between the statistical moments ⟨(Δ119906

119903)2

⟩ and⟨(Δ119906119903)3

⟩ in function of 119903The Lyapunov theory proposed in [1] gives the closure

of the von Karman-Howarth equation and expresses 119870(119903) interms of 119891 and 120597119891120597119903

119870 (119903) = 1199063radic1 minus 119891

2

120597119891

120597119903 (A6)

Accordingly the skewness of Δ119906119903is [9]

1198673(119903) equiv

⟨(Δ119906119903)3

⟩

⟨(Δ119906119903)2

⟩32

=6119896 (119903)

(2 (1 minus 119891 (119903)))32

(A7)

Therefore the skewness of 120597119906119903120597119903 is

1198673(0) = minus

3

7 (A8)

Acknowledgments

This work was partially supported by the Italian Ministry forthe Universities and Scientific and Technological Research(MIUR)

References

[1] N de Divitiis ldquoLyapunov analysis for fully developed homo-geneous isotropic turbulencerdquo Theoretical and ComputationalFluid Dynamics vol 25 no 6 pp 421ndash445 2011

[2] N de Divitiis ldquoSelf-similarity in fully developed homogeneousisotropic turbulence using the lyapunov analysisrdquo Theoreticaland Computational Fluid Dynamics vol 26 no 1ndash4 pp 81ndash922012

[3] C Truesdell A First Course in Rational Continuum MechanicsAcademic Press New York NY USA 1977

[4] E S Ventsel Theorie Des Probabilites Editions Mir CCCPMoskow 1973

[5] E L LehmannElements of Large-SampleTheory Springer 1999[6] W G Madow ldquoLimiting distributions of quadratic and bilinear

formsrdquo The Annals of Mathematical Statistics vol 11 no 2 pp125ndash146 1940

[7] G Nicolis Introduction to Nonlinear Science Cambridge Uni-versity Press 1995

[8] T de Karman and L Howarth ldquoOn the statistical theory ofisotropic turbulencerdquoProceedings of the Royal SocietyA vol 164no 917 pp 192ndash215 1938

[9] G K BatchelorTheTheory of Homogeneous Turbulence Cam-bridge University Press Cambridge Mass USA 1953

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2 The Lyapunov Analysis of theVelocity Fluctuations

This section renews the procedure for calculating the velocityfluctuations which is based on the Lyapunov analysis ofthe fluid strain [1] and on the momentum Navier-Stokesequations

120597119906119896

120597119905= minus

120597119906119896

120597119909ℎ

119906ℎ+1

120588

120597119879119896ℎ

120597119909ℎ

(3)

where u equiv (1199061 1199062 1199063) 119879119896ℎ and 120588 are the fluid velocity stress

tensor and density respectivelyIn order to obtain the velocity fluctuation consider

now the relative motion between two contiguous particlesexpressed by the infinitesimal separation vector 119889x whichobeys

119889 = nablau 119889x (4)

where 119889x varies according to the velocity gradient whichin turn follows the Navier-Stokes equations As observed in[1] 119889x is much faster than the fluid state variables and theLyapunov analysis of (4) provides the expression of the localdeformation in terms of the maximal Lyapunov exponentΛ gt 0

120597x120597x0

asymp 119890Λ(119905minus1199050) (5)

The map 120594 x0rarr x is the function which gives the

current position x of a fluid particle located at the referentialposition x

0at 119905 = 119905

0[3] Equation (3) can be written in terms

of the referential position x0[3]

120597119906119896

120597119905= (minus

120597119906119896

1205971199090119901

119906ℎ+1

120588

120597119879119896ℎ

1205971199090119901

)

1205971199090119901

120597119909ℎ

(6)

The adoption of the referential coordinates allows tofactorize the velocity fluctuation and to express it in theLyapunov exponential form of the local deformation Asthis deformation is assumed to be much more rapid thanminus1205971199061198961205971199090119901119906ℎ+ 1120588120597119879

119896ℎ1205971199090119901 the velocity fluctuation can be

obtained by integrating (6) with respect to the time whereminus1205971199061198961205971199090119901119906ℎ+ 1120588120597119879

119896ℎ1205971199090119901

is considered to be constant

119906119896asymp1

Λ(minus

120597119906119896

1205971199090119901

119906ℎ+1

120588

120597119879119896ℎ

1205971199090119901

)

119905=1199050

(7)

This assumption is justified by the fact that accordingto the classical formulation of motion of continuum media[3] the terms into the circular brackets of (6) are consideredto be smooth functions of 119905mdashat least during the periodof a fluctuationmdashwhereas the fluid deformation varies veryrapidly according to (4)-(5)

3 Statistical Analysis of Velocity Difference

As explained in this section the Lyapunov analysis of thelocal deformation and some plausible assumptions about the

statistics of u lead to determination of the structure functionof Δ119906119903and its PDF

The statistical properties of Δ119906119903are here investigated

expressing the fluid velocity through the following canonicaldecomposition [4]

u = sum119896

U119896120585119896 (8)

where U119896(119896 = 1 2 ) are proper coordinate functions

of 119905 and x and 120585119896(119896 = 1 2 ) are certain dimensionless

independent stochastic variables which satisfy

⟨120585119896⟩ = 0 ⟨120585

119894120585119895⟩ = 120575119894119895

⟨120585119894120585119895120585119896⟩ = 120603

119894119895119896119901

10038161003816100381610038161199011003816100381610038161003816 ⋙ 1

10038161003816100381610038161199011003816100381610038161003816 ⋙ ⟨120585

4

119896⟩

(9)

where120603119894119895119896

= 1 for 119894 = 119895 = 119896 else120603119894119895119896= 0 It is worth to remark

that the variables 120585119896are adequately chosen in such a way that

they can describe properly the mechanism of energy cascadeSpecifically the adoption of 120585

119896with |119901| ⋙ 1 is justified by

the fact that the evolution equation of the velocity correlation(see eg the von Karman-Howarth equation in appendix)includes also the third-order velocity correlation 119896(119903) whichis responsible for the intensive mechanism of energy cascadeand ⟨(Δ119906

119903)3

⟩⟨(Δ119906119903)2

⟩32

= 0 As a result it is reasonable thatthe canonical decomposition (8) (Δ119906

119903= (r119903) sdotsum

119896(U119896(x+r)minus

U119896(x))120585119896) includes variables 120585

119896with |⟨1205853

119896⟩| ⋙ 1 [5] This has

very important implications for what concerns the statisticsof the fluctiations of Δ119906

119903 In order to analyze this question

consider now the dimensionless velocity fluctuation u Thisis obtained in terms of 120585

119896by substituting (8) into (7)

ℎ= sum

119894119895

119860ℎ119894119895120585119894120585119895+1

119877120582

sum

119896

119887ℎ119896120585119896 (10)

where 119903 = 120582119879and 119906

ℎ= ℎ119906 Therefore sum

119894119895119860ℎ119894119895120585119894120585119895arises

from the inertia and pressure terms whereas 1119877120582sum119896119887ℎ119896120585119896is

due to the fluid viscosity Now thanks to the local isotropy119906ℎis a Gaussian stochastic variable [4 5] accordingly 120585

119896

satisfy into (10) the Lindeberg condition a very generalnecessary and sufficient condition for satisfying the centrallimit theorem [5] This condition does not apply to thevelocity difference In fact as Δu is the difference betweentwo correlated Gaussian variables its PDF could be a non-Gaussian distribution function To study this the fluctuationΔ119906119903is first expressed in terms of 120585

119896

Δ119903(r) = sum

119894119895

Δ119860119903119894119895(r) 120585119894120585119895+1

119877120582

sum

119896

Δ119887119903119896(r) 120585119896

equiv 119871 + 119878 + 119866+

+ 119866minus

(11)

This fluctuation can be reduced to the contributions 119871 119878119866+ and119866minus appearing into (11) [6] in particular 119871 is the sum

of all linear terms due to the fluid viscosity and 119878 equiv 119878119894119895120585119894120585119895

is the sum of all bilinear forms arising from the inertia andpressure terms whereas 119866+ and 119866minus are respectively definitepositive and negative quadratic forms of centered Gaussian




119866minus

= minus (1205772

minus 1)1205952(119903)

119866+

= (1205782

minus 1)1205953(119903)

(12)



119903 1205952and 120595

3


120582

120594 (119877120582) =

1205953(119903)

1205952(119903) (13)


Δ119906119903= 1205951120585 + 1205952(120594 (1205782


minus 1)) (14)




12059511205952depends on 119877

120582and 119903

2 (1 + 1205942

)1205952

2

1205952

1

=

⟨(119866+

+ 119866minus

)2

⟩

⟨(119878119894119895120585119894120585119895+ (1119877

120582Δ119887119903119896120585119896))2

⟩

(15)


119860 = 119878119894119895119878119901119902⟨120585119894120585119895120585119901120585119902⟩ 119861 =

2

119877120582

119878119894119895⟨120585119894120585119895120585119896⟩Δ119887119903119896

119862 =1

1198772

120582

Δ1198872

119903119896⟨1205852

119896⟩

(16)




120595 equiv1205952

1205951

= 120593 (119903)radic119877120582 (17)


120595 (119903 119877120582) = radic

119877120582

15radic15 (119903) (0) = 119874 (1) (18)


Δ119906119903

radic⟨(Δ119906119903)2

⟩

=

120585 + 120595 (120594 (1205782


minus 1))

radic1 + 21205952 (1 + 1205942)

(19)


2

120582119879)(1198971199041199062

119904) where 119897

119904and 119906



119904≃ 120582119879119897119904asymp radic119877

120582 thus 119897

119904identifies





119865 (Δ1199061015840

119903)

= int

120585

int

120578

int

120577

119901 (120585) 119901 (120578) 119901 (120577) 120575 (Δ1199061015840

119903minusΔ119906119903(120585 120578 120577)) 119889120585 119889120578 119889120577

(20)



119867119899equiv

⟨(Δ119906119903)119899

⟩

⟨(Δ119906119903)2

⟩1198992

=1

(1 + 2 (1 + 1205942) 1205952)1198992

times

119899

sum

119896=0

(119899

119896)120595119896

⟨120585119899minus119896

⟩⟨(120594 (1205782


minus 1))119896

⟩



cascade is

1198673(119903) =

81205953

(1205943

minus 1)

(1 + 21205952 (1 + 1205942))32

(22)




1198673(0) =

81205953

(0) (1205943

minus 1)

(1 + 21205952 (0) (1 + 1205942))32

= minus3

7 (23)









Appendix



119891 (119903) =⟨119906119903(x) 119906119903(x + r)⟩

1199062(A1)


120597119891

120597119905=119870 (119903)

1199062+ 2](

1205972

119891

1205971199032+4

119903

120597119891

120597119903) +

10]

1205822

119879

119891 (A2)


119891 (0) = 1 lim119903rarrinfin

119891 (119903) = 0 (A3)


kinetic energy [8]

1198891199062

119889119905= minus

10]

1205822

119879

1199062

(A4)



119870 (119903) = 1199063

(120597

120597119903+4

119903) 119896 (119903)

where 119896 (119903) =⟨1199062

119903(x) 119906119903(x + r)⟩

1199063

(A5)


119903)2

⟩ and⟨(Δ119906119903)3



119870 (119903) = 1199063radic1 minus 119891

2

120597119891

120597119903 (A6)


1198673(119903) equiv

⟨(Δ119906119903)3

⟩

⟨(Δ119906119903)2

⟩32

=6119896 (119903)

(2 (1 minus 119891 (119903)))32

(A7)


1198673(0) = minus

3

7 (A8)

Acknowledgments


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119866minus

= minus (1205772

minus 1)1205952(119903)

119866+

= (1205782

minus 1)1205953(119903)

(12)



119903 1205952and 120595

3


120582

120594 (119877120582) =

1205953(119903)

1205952(119903) (13)


Δ119906119903= 1205951120585 + 1205952(120594 (1205782


minus 1)) (14)




12059511205952depends on 119877

120582and 119903

2 (1 + 1205942

)1205952

2

1205952

1

=

⟨(119866+

+ 119866minus

)2

⟩

⟨(119878119894119895120585119894120585119895+ (1119877

120582Δ119887119903119896120585119896))2

⟩

(15)


119860 = 119878119894119895119878119901119902⟨120585119894120585119895120585119901120585119902⟩ 119861 =

2

119877120582

119878119894119895⟨120585119894120585119895120585119896⟩Δ119887119903119896

119862 =1

1198772

120582

Δ1198872

119903119896⟨1205852

119896⟩

(16)




120595 equiv1205952

1205951

= 120593 (119903)radic119877120582 (17)


120595 (119903 119877120582) = radic

119877120582

15radic15 (119903) (0) = 119874 (1) (18)


Δ119906119903

radic⟨(Δ119906119903)2

⟩

=

120585 + 120595 (120594 (1205782


minus 1))

radic1 + 21205952 (1 + 1205942)

(19)


2

120582119879)(1198971199041199062

119904) where 119897

119904and 119906



119904≃ 120582119879119897119904asymp radic119877

120582 thus 119897

119904identifies





119865 (Δ1199061015840

119903)

= int

120585

int

120578

int

120577

119901 (120585) 119901 (120578) 119901 (120577) 120575 (Δ1199061015840

119903minusΔ119906119903(120585 120578 120577)) 119889120585 119889120578 119889120577

(20)



119867119899equiv

⟨(Δ119906119903)119899

⟩

⟨(Δ119906119903)2

⟩1198992

=1

(1 + 2 (1 + 1205942) 1205952)1198992

times

119899

sum

119896=0

(119899

119896)120595119896

⟨120585119899minus119896

⟩⟨(120594 (1205782


minus 1))119896

⟩



cascade is

1198673(119903) =

81205953

(1205943

minus 1)

(1 + 21205952 (1 + 1205942))32

(22)




1198673(0) =

81205953

(0) (1205943

minus 1)

(1 + 21205952 (0) (1 + 1205942))32

= minus3

7 (23)









Appendix



119891 (119903) =⟨119906119903(x) 119906119903(x + r)⟩

1199062(A1)


120597119891

120597119905=119870 (119903)

1199062+ 2](

1205972

119891

1205971199032+4

119903

120597119891

120597119903) +

10]

1205822

119879

119891 (A2)


119891 (0) = 1 lim119903rarrinfin

119891 (119903) = 0 (A3)


kinetic energy [8]

1198891199062

119889119905= minus

10]

1205822

119879

1199062

(A4)



119870 (119903) = 1199063

(120597

120597119903+4

119903) 119896 (119903)

where 119896 (119903) =⟨1199062

119903(x) 119906119903(x + r)⟩

1199063

(A5)


119903)2

⟩ and⟨(Δ119906119903)3



119870 (119903) = 1199063radic1 minus 119891

2

120597119891

120597119903 (A6)


1198673(119903) equiv

⟨(Δ119906119903)3

⟩

⟨(Δ119906119903)2

⟩32

=6119896 (119903)

(2 (1 minus 119891 (119903)))32

(A7)


1198673(0) = minus

3

7 (A8)

Acknowledgments


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Appendix



119891 (119903) =⟨119906119903(x) 119906119903(x + r)⟩

1199062(A1)


120597119891

120597119905=119870 (119903)

1199062+ 2](

1205972

119891

1205971199032+4

119903

120597119891

120597119903) +

10]

1205822

119879

119891 (A2)


119891 (0) = 1 lim119903rarrinfin

119891 (119903) = 0 (A3)


kinetic energy [8]

1198891199062

119889119905= minus

10]

1205822

119879

1199062

(A4)



119870 (119903) = 1199063

(120597

120597119903+4

119903) 119896 (119903)

where 119896 (119903) =⟨1199062

119903(x) 119906119903(x + r)⟩

1199063

(A5)


119903)2

⟩ and⟨(Δ119906119903)3



119870 (119903) = 1199063radic1 minus 119891

2

120597119891

120597119903 (A6)


1198673(119903) equiv

⟨(Δ119906119903)3

⟩

⟨(Δ119906119903)2

⟩32

=6119896 (119903)

(2 (1 minus 119891 (119903)))32

(A7)


1198673(0) = minus

3

7 (A8)

Acknowledgments


References












RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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