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Homogeneous, Isotropic TurbulencePhenomenology, Renormalization and Statistical Closures$

W. David McComb

Print publication date: 2014

Print ISBN-13: 9780199689385

Published to Oxford Scholarship Online: April 2014

DOI: 10.1093/acprof:oso/9780199689385.001.0001

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(p.399) Appendix C Evaluation of the L(k,\,j) coefficient

(p.399) Appendix C Evaluation of the L(k,\,j) coefficient

Source:
Homogeneous, Isotropic Turbulence
Publisher:
Oxford University Press

As the mathematical treatment of statistical theory in this book is deliberately light, I thought that it would be helpful to put in a complete calculation as an example. This calculation has the virtue that it can be be applied to all Eulerian closure theories, ranging from quasi-normality to renormalized perturbation theories, both fundamental and ad hoc, provided only that we work with Edwards’ (k,j,μ) formalism, rather than Kraichnan’s (k,j,l) formalism. The work presented draws on Section 2.8.2 and Appendix E of the book PFT [1], and is to some extent a unified treatment of these. Reference can also be made to (1.32) and (1.35) in the present book, although these are in a very much simplified form.

As well as the coefficient L(k,j), there is also L(k,kj), but we only define this latter coefficient in passing. A fuller treatment can be found in Appendix E of PFT. The relationship to Kraichnan’s formalism is also discussed in PFT, while a more extensive treatment of that formalism can be found in the book by Leslie [2].

C.1 Derivation of the closed covariance equation

For the sake of simplicity, we consider the single-time covariance equation as derived by the quasi-normality hypothesis. In our usual notation (see Section 3.1), this can be written as

(C.1)
t+2ν0k2C(k;t)=W(k,t)4πk2+H(k;t),

where W(k,t) is the injection spectrum and

(C.2)
H(k;t)=0tdseν0(k2+j2+|kj|2)(ts)Mαβγ(k)d3jXβγα(j,kj,k;s).

As is well known, the expression for Xβγα(j, kj,k;s) is obtained by solving its evolution equation in terms of the fourth-order moment, which is then factorized according to the rules for a Gaussian distribution:

(C.3)
Xβγα(j,kj,k;s)=2Mβρδ(j)Pργ(kj)Pδα(k)C(|kj|;s)C(k;s)+2Mγρδ(kj)Pρβ(j)Pδα(k)C(j;s)C(k;s)+2Mαρδ(k)Pρβ(j)Pδγ(kj)C(j;s)C(|kj|;s),

(p.400) or see equation (2.139) in [1]. Next, we define the coefficients

(C.4)
A(k,j,kj)=2Mαβγ(k)Mαρδ(k)Pρβ(j)Pδγ(kj),
(C.5)
B(j,k,kj)=2Mαβγ(k)Mβρδ(j)Pργ(kj)Pδα(k)=2Mαβγ(k)Mβαρ(j)Pργ(kj),
(C.6)
B(kj,k,j)=2Mαβγ(k)Mγρδ(kj)Pρβ(j)Pδα(k)=2Mαβγ(k)Mβαρ(kj)Pργ(j).

Then we can express Mαβγ(k)Xβγα(j,kj,k;s), as contained in (C.2), in the form

(C.7)
Mαβγ(k)Xβγα(j,kj,k;s)=B(j,k,kj)C(|kj|;s)C(k;s)B(kj,k,j)C(j;s)C(k;s)+A(k,j,kj)C(j;s)C(|kj|;s).

An identity that originates from the energy-conserving nature of the nonlinear term is

(C.8)
Mαβγ(k)+Mβαγ(j)+Mγαβ(l)Pασ(k)Pβρ(j)Pγδ(l)=0,

provided that kjl=0. From this, it is simple to show that the coefficients satisfy the condition

(C.9)
A(k,j,kj)B(j,k,kj)B(kj,k,j)=0.

Using this identity to replace A(k,j,kj), we have

(C.10)
Mαβγ(k)Xβγα(j,kj,k;s)=B(j,k,kj)C(|kj|;s)C(j;s)C(k;s)+B(kj,k,j)C(j;s)C|kj|;sC(k;s),

with which we may write H(k;t) from (C.2) as

(C.11)
H(k;t)=0tdseν0(k2+j2+|kj|2)(ts)×d3j{B(j,k,kj)C(|kj|;s)[C(j;s)C(k;s)]+B(kj,k,j)C(j;s)[C(|kj|;s)C(k;s)]}.

(p.401) Since j is integrated over all space, we can make the change of variables jkj in either term. Doing so to the second term on the right-hand side, we find

(C.12)
H(k;t)=0tdseν0(k2+j2+|kj|2)(ts)×d3j2B(j,k,kj)C(|kj|;s)[C(j;s)C(k;s)]=0tdseν0(k2+j2+|kj|2)(ts)×2d3jL(k,j)C(|kj|;s)[C(j;s)C(k;s)],

where the L(k,j) coefficient is defined as

(C.13)
L(k,j)=B(j,k,kj).

As an aside, we note that, alternatively, making the change of variable in the first term instead would yield

(C.14)
H(k;t)=2d3jL(k,kj)0tdseν[j2+|kj|2+k2](ts)×C(j;s)[C(|kj|;s)C(k;s)].

Multiplying by 4πk2 on both sides of (C.1), and using (C.11) for H(k;t), we find the energy equation written as

(C.15)
t+2ν0k2C(k;t)=W(k,t)+T(k,t),

where the transfer spectrum is given by

(C.16)
T(k;t)=4πk2H(k;t)=8πk2d3jL(k,j)0tdseν0(k2+j2+|kj|2)(ts)×C(|kj|;s)[C(j;s)C(k;s)].

C.2 Evaluation of L(k,j)

We start from the definition of the L(k,j) coefficient, as given by (C.13) and (C.5):

(C.17)
L(k,j)=2Mαβγ(k)Mβαδ(j)Pγδ(kj).

(p.402) Expanding out the projection and vertex operators Pαβ(k) and Mαβγ(k), respectively, using their definitions as given by (2.40) and (2.44), we have

(C.18)
L(k,j)=2{12i[kβPαγ(k)+kγPαβ(k)]12i[jαPβδ(j)+jδPαβ(j)]Pγδ(kj)}=12[(kβδαγ2kαkβkγk2+kγδαβ)(jαδβδ2jαjβjδj2+jδδαβ)×(δγδ(kγjγ)(kδjδ)|kj|2)]=12[kαjα2kβjβ+kαjαkβ(kβjβ)jα(kαjα)|kj|2+2kβjβj2jα(kαjα)jγ(kγjγ)|kj|24kαjαjβ(kβjβ)kα(kαjα)|kj|24kαjαkβjβk2j2kγ(kγjγ)jδ(kδjδ)|kj|2+2jαkαk2kβ(kβjβ)kγ(kγjγ)|kj|2+4kαjαkβjβkγjγk2j2+2kα(kαjα)jβ(kβjβ)|kj|2+2kαjαkα(kαjα)jβ(kβjβ)|kj|2+2kα(kαjα)jβ(kβjβ)|kj|23kα(kαjα)jβ(kβjβ)|kj|2],

where the factors of 3 come from δαα=3, since we are using the summation convention. Collecting similar terms, this can be simplified to

(C.19)
L(k,j)=12[2kαjα(jβ(kβjβ)jγ(kγjγ)j2|kj|2+kβ(kβjβ)kγ(kγjγ)k2|kj|21)+4kαjαkβjβkγjγk2j22(1+2kαjαkβjβk2j2)(kγ(kγjγ)jδ(kδjδ)|kj|2)],

which is equivalent to

(C.20)
L(k,j)=[k·j((k·jj2)2j2|kj|2+(k2k·j)2k2|kj|2|kj|2|kj|2)+2(k·j)3k2j2|kj|2|kj|2(1+2(k·j)2k2j2)(k2k·j)(k·jj2)|kj|2].

We can simplify this by writing kj=kjcosθkjμ, where θ is the angle between k and j in the plane spanned by the two vectors. Thus,

(C.21)
L(k,j)=1|kj|2{kjμ[1j2(kjμj2)2+1k2(k2kjμ)2(k2+j22kjμ)]+2kjμ3(k2+j22kjμ)(1+2μ2)(k2kjμ)(kjμj2)}.

(p.403) Expanding the brackets and collecting terms, we arrive at

(C.22)
L(k,j,μ)=kjk2+j22μkj{μ[3μ2(k2+j2)2kjμ(1+2μ2)](1+2μ2)[μ(k2+j2)kj(1+μ2)]}=kjk2+j22μkj[μ(μ21)(k2+j2)+kj(1+2μ2)(1μ2)],

which finally yields our expression for the L(k,j) coefficient in terms of the absolute magnitudes k,j and the cosine of the enclosed angle, μ:

(C.23)
L(k,j,μ)=kj(1μ2)k2+j22μkj[kj(1+2μ2)μ(k2+j2)].

C.2.1 A note on numerical evaluation in closures

It should be noted that, in the evaluation of closures such as LET, it is common to evaluate the momentum integral in spherical polar coordinates (due to isotropy) as

(C.24)
d3jL(k,j)f(k,j,|kj|)=2πdjj20πdθsinθL(k,j,cosθ)f(k,j,cosθ)=2πdjj211dμL(k,j,μ)f(k,j,μ),

where μ=cosθ and so dμ=sinθdθ. This negative sign is often absorbed into the definition of the L(k,j,μ), but this is not consistent with the original definition. The form above should be used, using the negative sign to switch the limits of the μ integral.