Is restricted to wavenumbers within the sphere of ra

is restricted to wavenumbers within the sphere of ra-
dius κ
c
(which is within the cube of side 2 κ
c
). Equivalently, the fraction
of the represented coefficients
ˆ
u ( κ , t ) that are non-zero is approximately
4
3
πκ 3
c
/ (2 κ
c
) 3 = π/ 6.
The Navier–Stokes equations in wavenumber space ( Eq. (6.145) on page
214) can be written
d
d t
+ νκ
2 ˆ
u
j
( κ , t ) = − iκ P
jk
( κ )
κ , κ
δ
κ , κ + κ
ˆ
u
k
( κ , t )ˆ u ( κ , t ) , (13.195)
where P
jk
is the projection tensor ( Eq. (6.133) ). The Kronecker delta is unity
for triadic interactions, i.e.,
κ = κ + κ , (13.196)
and is zero otherwise. Equation (13.195) is a coupled set of ordinary differ-
ential equations, one equation for each of the infinite number of modes.

606
13 Large-eddy simulation
The filtered equation obtained from Eq. (13.195) is, for the non-zero
coefficients ( κ < κ
c
),
d
d t
+ νκ
2
ˆ
u
j
( κ , t ) = − iκ P
jk
( κ )
κ , κ
δ
κ , κ + κ
H ( κ
c
− κ )ˆ u
k
( κ , t )ˆ u ( κ , t ) .
(13.197)
This is a finite set of (approximately πN 3 / 6) ordinary differential equations,
but it is unclosed. The closure problem arises because the nonlinear term (on
the right-hand side of Eq. (13.197) ) includes unknown Fourier coefficients,
namely ˆ u ( κ ) and ˆ u ( κ ) for κ ≡ | κ | ≥ κ
c
and κ ≡ | κ | ≥ κ
c
.
EXERCISE
13.34 Consider LES of high-Reynolds-number homogeneous isotropic tur-
bulence using the sharp spectral filter and a pseudo-spectral method
with largest resolved wavenumber κ
max
= κ
c
. As in DNS, adequate
resolution of the large scales requires the size L of the domain to be
eight integral lengthscales, i.e., L = 8 L
11
(see Eq. (9.5) ). So that no
more than 20% of the energy is in the residual motions, the cutoff
wavenumber is taken to be κ
c
L
11
= 15 (see Table 6.2 on page 240).
Show that these requirements determine
κ
max
κ
0
=
15
π/ 4
≈ 19 , (13.198)
as the ratio of the resolved wavenumbers, corresponding to a 38 3
simulation.
13.5.2 Triad interactions
There is, of course, a direct connection between the nonlinear term in
Eq. (13.197) , the velocity product in physical space u
k
u , and various stresses
formed from it (e.g., u
k
u ). From Eq. (13.191) we obtain
u
k
( x , t ) u ( x , t ) =
κ κ
e
i ( κ + κ ) · x
ˆ u
k
( κ , t )ˆ u ( κ , t )
=
κ , κ
δ
κ , κ + κ
e
i κ · x ˆ
u
k
( κ , t )ˆ u ( κ , t ) . (13.199)
Figure 13.10 is a sketch of qualitatively different triad interactions. These
are referred to as types (a)–(d), and are defined in Table 13.4 . In terms of
the modes represented in the LES (i.e., | κ | < κ
c
), only interactions of type
(a) can be represented exactly. Consider the summation in Eq. (13.199) for

13.5 LES in wavenumber space
607
κ '
κ
κ
κ ''
c
(a)
κ '
κ
κ ''
(b)
κ '
κ
κ ''
κ '
κ
κ ''
(c)
κ '
κ
κ ''
(d)
Fig. 13.10. Sketches of the various types of triad interactions defined in Table 13.4:
(a) resolved, (b) Leonard, (c) cross, (d) SGS.
u
k
u restricted to these wavenumbers. The restrictions κ < κ
c
and κ < κ
c
amount to replacing u
k
by u
k
and u by u , and the restriction κ < κ
c
amounts
to filtering the result. Thus, the sum of the interactions of type (a) yields
u
k
u =
κ , κ
δ
κ , κ + κ
H ( κ
c
− κ ) H ( κ
c
− κ ) H ( κ
c
− κ )ˆ u
k
( κ )ˆ u ( κ )
=
κ , κ
δ
κ , κ + κ
H ( κ
c
− κ )
ˆ
u
k
( κ )
ˆ
u ( κ ) . (13.200)

608
13 Large-eddy simulation
Table 13.4. Definitions of the types of triad interactions sketched in Fig. 13.10 ,
and their contributions to u
k
u ( Eq. (13.199) ) (all possible interactions with
κ < κ
c
are included, but there are additional interactions with κ ≥ κ
c
)
Type Defining wavenumber Contribution
designation ranges to u
k
u
Type (a), κ < κ