ENERGY CASCADE IN LARGE-EDDY SIMULATIONS
241
A significant portion of the large-scale dissipation is provided by appropriate treatment of this term, In evaluating this term, the variations of
?(fi,i4j),’6xj within an averaging volume defined by G(x) should -. be .. explicitly
accounted for. One obvious possibility is to represent 3(iiiiij)/?xj as a
weighted average of the vitlues of C(ii,Itf)/?.xj at neighboring grid points.
Another possibility is to use the Taylor expansion
(6. I )
iik(x’) = iik(x) + (x’ - X ) VU,(X) + O( I X’ - x 1’ )
in the dtfinition
i i i i i , = G(x - x ’ F ~ ( x ’ ~ ~ ( x ’ )
dx’,
- -
I
(6.2)
with the result
(6.3)
where 7 is the onedimensional second moment of G.
z.
= 1 .x2 dx i ‘ G ( ~ .
?, z ) (iZ.
.-,
‘ - a -
(6.4)
Finally, an expansion of UiUj(x’) including O( I x - x’ 1 )’ terms, gives
c7(li,lij) ~ d (- - I I , (72
( iii ii j ) ) .
u,uj +
(?Xj
c‘S/
2 f?Y, dx,
Using the methods of the previous section, one can show that the approximation (6.3) would yield twice the E~~ determined for curve A, while (6.4)
gives the same value of cRS.
The SGS term ih,,/2xj must produce the remaining dissipation and
probably can be modeled by the eddy viscosity model of (3.1 I ) and (3.12).
The value of the eddy coefficient in (3.12) will be somewhat smaller than that
calculated by Lilly (1967) (c = 0.17). Numerical experiments will probably
be required to obtain a satisfactory value.
Similar considcrations apply to the simulation of large-scale fluctuations
of a passive scalar.
ACKNOWLEDGMENTS
Tlic authot gratefully acknowledges the important comments of Drs. J. W. Deardorff. D. K.
Lilly, and W. C. Reynolds who rcad a draft of the manuscript. Contributions hj Drs. J. H.
Ferziger. C. H. Gibson, and S. Corrsin were also most helpful. Messrs. John Clay and Steve
McConncll kindly made available their experimental measurements prior to publication. This
work was supported by a grant from NASA Ames Research Center.
241
A significant portion of the large-scale dissipation is provided by appropriate treatment of this term, In evaluating this term, the variations of
?(fi,i4j),’6xj within an averaging volume defined by G(x) should -. be .. explicitly
accounted for. One obvious possibility is to represent 3(iiiiij)/?xj as a
weighted average of the vitlues of C(ii,Itf)/?.xj at neighboring grid points.
Another possibility is to use the Taylor expansion
(6. I )
iik(x’) = iik(x) + (x’ - X ) VU,(X) + O( I X’ - x 1’ )
in the dtfinition
i i i i i , = G(x - x ’ F ~ ( x ’ ~ ~ ( x ’ )
dx’,
- -
I
(6.2)
with the result
(6.3)
where 7 is the onedimensional second moment of G.
z.
= 1 .x2 dx i ‘ G ( ~ .
?, z ) (iZ.
.-,
‘ - a -
(6.4)
Finally, an expansion of UiUj(x’) including O( I x - x’ 1 )’ terms, gives
c7(li,lij) ~ d (- - I I , (72
( iii ii j ) ) .
u,uj +
(?Xj
c‘S/
2 f?Y, dx,
Using the methods of the previous section, one can show that the approximation (6.3) would yield twice the E~~ determined for curve A, while (6.4)
gives the same value of cRS.
The SGS term ih,,/2xj must produce the remaining dissipation and
probably can be modeled by the eddy viscosity model of (3.1 I ) and (3.12).
The value of the eddy coefficient in (3.12) will be somewhat smaller than that
calculated by Lilly (1967) (c = 0.17). Numerical experiments will probably
be required to obtain a satisfactory value.
Similar considcrations apply to the simulation of large-scale fluctuations
of a passive scalar.
ACKNOWLEDGMENTS
Tlic authot gratefully acknowledges the important comments of Drs. J. W. Deardorff. D. K.
Lilly, and W. C. Reynolds who rcad a draft of the manuscript. Contributions hj Drs. J. H.
Ferziger. C. H. Gibson, and S. Corrsin were also most helpful. Messrs. John Clay and Steve
McConncll kindly made available their experimental measurements prior to publication. This
work was supported by a grant from NASA Ames Research Center.