Direct numerical simulations of the time evolution of homogeneous stably stratified shear flows have been performed for Richardson numbers from 0 to 1 and for Prandtl numbers between 0.1 and 2. The results indicate that mixing efficiency R(sub f) varies with turbulent Froude number in a manner consistent with laboratory experiments performed with Prandtl numbers of 0.7 and 700. However, unlike the laboratory results, for a particular Froude number, the simulations do not show a clear dependence on the magnitude of R(sub f) on Pr. The observed maximum value of R(sub f) is 0.25. When averaged over vertical length scales of an order of magnitude greater than either the overturning or Ozmidov scales of the flow, the simulations indicate that the dissipation rate epsilon is only weakly lognormally distributed with an intermittency of about 0.01 whereas estimated values in the ocean are 3 to 7

Briggs, D. A.

Ferziger, J. H.

Ivey, G. N.

Koseff, J. R.

English

NASA Technical Reports Server

Center for Turbulence Research
Annual Research Briefs 199_ // _ .__-_-_ _ 7 335
9 4- 1
Mixing in a stratified sh ai" flow:
energetics and sampling
By G. N. Ivey 1, J. R. Kosefl a, D. A. Brlggs 2 AND J. H. Ferziger 2
Direct numerical simulations of the time evolution of homogeneous stably strat-
ified shear flows have been performed by Holt (1990), Holt et al. (1992), and
Itsweire et al. (1992) for Richardson numbers from 0 to 1 and for Prandtl numbers
between 0.1 and 2. The results indicate that mixing efficiency R I varies with turbu-
lent Froude number in a manner consistent with laboratory experiments performed
with Prandtl numbers of 0.7 and 700. However, unlike the laboratory results, for
a particular Froude number, the simulations do not show a clear dependence on
the magnitude of Rf on Pr. The observed maximum value of Rf is 0.25. When
averaged over vertical length scales of an order of magnitude greater than either
the overturning or Ozmidov scales of the flow, the simulations indicate that the
dissipation rate e is only weakly lognormally distributed with an intermittency of
about 0.01 whereas estimated values in the ocean (Baker and Gibson (1987)) are 3
to 7.
1. Introduction
The specification of the buoyancy flux in a fluid with a stable density gradient
and subjected to forcing by either mean shear or a mechanical means of generating
turbulent kinetic energy is crucial to the understanding of mixing processes in a
wide variety of geophysical applications. Quantifying the rate of vertical mixing in
the ocean, for example, is central not only to parameterizing vertical heat and salt
fluxes, but also those of passive tracers such as pollutants and chemicals. Thus,
predicting the mixing efficiency or flux Richardson number R! is of central im-
portance in turbulent mixing in density stratified flows. Ivey and Imberger (1991)
(hereafter II) introduced a generalized definition of Rf based on the full turbulent
kinetic energy equation and demonstrated that available laboratory measurements
had peak values of Rf = 0.20 for fluids with Prandtl number of 700 and 0.15 for
fluids with Prandtl number of 0.7. In the laboratory work, turbulence was gener-
ated with grids and, in the case of Rohr et al. (1988), by both grids and a mean
shear. This work has been extended by the direct numerical simulation work of
Holt (1990), Holt et al. (1992), and Itsweire et al. (1992), who numerically studied
the temporal evolution of homogeneous turbulence subjected to a constant mean
velocity and density gradients using the pseudo-spectral method of Rogallo (1981).
The Boussinesq form of the Navier Stokes equations were solved for the three dl-
mensional velocity and density fields on a 128 × 128 × 128 grid. The velocity fields
1 Centre for Water Research, University of Western Australia, Nedlands, Western Australia
2 Stanford University
https://ntrs.nasa.gov/search.jsp?R=19940007839 2020-06-16T20:55:18+00:00Z
336 G. N. Ivey, J. R. Koseff, D. A. Briggs _ J. H. Ferziger
were initially isotropic with Reynolds numbers based on the integral scale and the
turbulent kinetic energy ranging from 52 to 104, Richardson numbers ranging from
0.058 to 1.0, and Prandtl numbers of either 1 or 2. In the oceanic case, the density
stratification is predominantly due to vertical temperature gradients. Therefore, in
order to investigate the differences in the behavior of R I with Pr observed by II,
we have extended the earlier work of Holt et al. (1992) by performing additional
simulations at values of Ri from 0.075 to 1.0 with Prandtl numbers as low as 0.1;
these results are described in Section 3 below.
Oceanic turbulence is intermittent in character, and there exists considerable con-
troversy about how to correctly sample the flow to determine the volume averaged
dissipation (eg. Baker and Gibson (1987)). We examine this question by averaging
over sub-sections of our 128 × 128 × 128 domain and calculating the intermittency,
defined as the variance of the logarithm of a quantity: the results are summarized
in Section 4. Finally, Section 5 summarizes our conclusions.
2. Dynamics of mixing in stratified flows
The full turbulent kinetic energy equation can be written as:
O_.q 2 0 ujq 2 0 1 cOUi g (1)
m
where q2 = u_ + u_ + u32 is twice the turbulent kinetic energy, po is the background
v °--_ _ is the rate of dissipation of turbulent kinetic energy, anddensity, e = 0., 0.,
velocity component us is in the direction of the gravity vector. For homogeneous
turbulence at steady state, (1) can be written as P = B+e where P is the production
of turbulent kinetic energy (the last term on the left hand side of (1)) Then, II
defined
B 1
R! - p - 1 + (e/B) (2)
and also introduced the concept of a turbulent Froude number associated with the
energy bearing eddies defined by
_ (3)FrT --
where Lc is the scale of the most energetic overturns, Lo = (e/g3) 1/2 represents
the Ozmidov scale, and N is the buoyancy frequency characterizing the stable back-
ground stratification. An alternate form of (2) is
1
RI = 1 + Fr_/R.,_ (4)
where Rp_ is the correlation coefficient betwecn the density and vertical velocity
fluctuations. II found the available laboratory measurements were in accord with
Mizing in a stratified shear flow: energeties and sampling 337
if,
09
0.25
0.20
0.15-
0.10-
0.05-
0.00
. x,_x,kx ",
• x _ ++x x X + + v
' 5'.0 ').0 2.5 7.5 10.0
Froude number, FrT
FIGURE 1. R/ as a function of FrT = (LO/L,) 2/s for simulations with Pr = 0.1
(o), Pr = 0.5 (o) and Pr = 2.0 (o). Experimental data is provided by Stillinger et
al., Itsweire et al., and Rohr et al. for Pr = 700 (×) and Lienhard and Van Atta
for Pr = 0.7 (+).
the simple predictions in (4). Additionally, II introduced two further dimensionless
numbers, the turbulent Reynolds number ReT and small scale Froude number Fr- t
defined by
_ uL____A__ (Lc)4/3ReT L_ (5)
and
T
Fr.r=( e)l/2,vy 2 =(_-_)2/s, (6)
where e/vN 2 is a measure of the range of overturning scales when buoyancy strongly
affects the flow and Lk = (v3/e) 1/a represents the Kolmogorov scale. The dimen-
sionless numbers (3), (5), and (6) are related by
Frr = (_-_)l/_fr,. (7)
One final length scMe is used in discussing the numerical results below. The Ellison
scale Le is defined as
n,= (P_)'/_ (8)
Oz
where _(z) is mean density. As Itsweire et al. (1992) show, except at very high
Richardson numbers (based on mean shear and density gradient), Lc and Le are
the same scale to within a constant of O(1).
338 G. N. Ivey, J. R. Ko_eff, D. A. Briggs _ J. H. Ferziger
0
l0
°1,
10
-2
lO
FIGURE 2.
0 O0 0 0 0 0 0 0
0 @0 _000
o°_
o
o o
o
go
0.5 1. 1.5 2.0 2.5
Froude number, Frr
Row as a function of FrT for Pr = 2.
3. Mixing efficiency
All results in this section were computed by ensemble-averaging data over the
entire computational domain at each time step. In order to eliminate transient
behavior associated with the initial conditions, only the data for St > 2 are con-
sidered, where the shear time, St, is a dimensionless time with S = OU/Oz. R I
was computed according to the definition in (2), as were the dimensionless num-
bers FrT, ReT, and Fr_ from their respective definitions in equations (3), (5), and
(6). Figure 1 summarizes the results for the simulations with Pr = 0.1, 0.5, and
2. Mixing efficiency, RI, is plotted against FrT to facilitate direct comparison with
the laboratory observations (presented by II) which are also shown in Figure 1.
The numerical results show the same general distribution as the laboratory mea-
surements. As FrT becomes very large, the overturning scale is small compared
to the Ozmidov scale, the mixing becomes inefficient because there is much more
turbulent kinetic energy than necessary to mix the scalar gradient, and R! rapidly
decreases. Similarly, for very small values of FrT when buoyancy effects dominate
and reduce the mixing, R I again sharply decreases. At an intermediate value of
I2rT in the range of 1 to 1.5, the mixing is optimal with peak values of R I = 0.25.
The behavior of Rpw as a function of FrT given in Figure 2 for Pr = 2 (as
an example) shows a similar trend. For FrT > 1, the correlation coefficient Rp_,
tends to a constant value of about 0.6; hence, from Equation (4), Rf will rapidly
decrease with increasing FrT. Conversely, for FrT < 1, Rpu, decreases rapidly as
the density and velocity fluctuations become uncorrelated, and R I tends to zero as
a consequence of the denominator in (4) being much greater than unity.
While the overall behavior in the numerical and laboratory results is the same
Mixing in a s_ra_ified _hear flow: energetics and sampling 339
10
16' QO0 ° • •0 •
0 0
0
oo ••
lO _ , , , , ,,u ...... n ...... u ......
101 10° 10! 102 10 3
,/vN 2
FIGURE 3. R I as a function of e/vN 2 for simulations with Pr = 0.1 (o), 0.5 (*),
2.0(×).
in Figure 1, there are differences in detail. Unlike the laboratory data, for a given
FrT, the numerically derived values of R l are independent of Prandtl number and
do not show the experimental tendency of R I to increase with Pr. Nevertheless,
recalling that the laboratory data are derived from time-averaged statistics in a
steady mean flow whereas the numerical value are ensemble-averaged values in an
evolving flow, the differences are remarkably small. The implication is that the
peak mixing efficiency is 0.25 for FrT 1 to 1.5, irrespective of Prandtl number.
In field measurements of oceanic turbulence, the overturning scales Le or Lc are
not usually measured while e invariably is. Substituting (7) into (4) yields the more
practical form
1
RI = 1 + t3(e/vN 2) (9)
where/3 -1= RmoRe T.
In Figure 3, we re-plot the results from the simulations against e/vN2(= Fr_).
The minimum dissipation needed to sustain a vertical buoyancy flux, and hence
positive R$, is clearly a strong function of Pr. For Pr = 0.1, e/vN 2 may be as
little as 0.4 and still sustain a positive buoyancy flux ,whereas for Pr -- 2, the
minimum e/vN 2 for a positive buoyancy flux is about 20. The other interesting
feature of Figure 3 is that R! becomes independent of Pr for large e/vN 2. For
e/vN 2 >> 10, a best fit is/3 = (e/vY2) -°6, hence (9) simplifies to
1
Rs = .0.4 (lO)
1 + (e/vN 2)
340 G. N. Ivey, J. R. Koseff, D. A. Briggs FJ J. H. Ferziger
1.00
0.75
0.50-
0.25"
,00 ' ° ' ' '" * .......
10 -3 10 -2 101
.... , .! ....... i
10-I 10°
dissipation, e cm2s -3
FIGURE 4. Lognormal plot of dissipation estimates with no averaging (Ri = .075,
Pr = 2, St = 6).
and using (2)
B = e°'6(vN2) 0"4 (11)
which provides an approximate but simple means of computing buoyancy flux from
the dissipation for relatively energetic flows.
4. Sampling turbulence in a stratified fluid
Turbulence measurements are made in the ocean with either vertically falling mi-
crostructure instruments or, less commonly, horizontally towed instruments. The
buoyancy flux B is not directly measured but, as indicated above, dissipation esti-
mates are made and then B is computed by estimating R/as outlined in Section 3
(see also Itsweire et al. 1992). For falling probes, dissipation estimates are typically
made by measuring two turbulent velocity components and computing total dissi-
pation using models (see Itsweire et al. 1992). This procedure produces estimates
of dissipation averaged over about 2 meters in the vertical, and these estimates are
further averaged to characterize the dissipation on nmch larger scales such as the
oceanic thermocline (for example see review of Gregg 1987). Gibson and Baker
(1987) and Gibson (1991) have argued that dissipation in oceanic turbulence is log-
normally distributed with an intermittency, a 2, in the range of 3 to 7. Furthermore,
they claim that, due to the large scales and the limited sampling, the dissipation is
greatly undersampled. Given the large intermittency, Baker and Gibson maintain
that to estimate of e to within +10% one would need to average the dissipation
calculated from thousands of independent sampling profiles! Gurvich and Yaglom
Mixin 9 in a s_ratified shear flow: energetics and sampling 341
(1967) (see also Yamazaki and Lueek (1990)) developed three criteria which must
be satisfied in order for dissipation to be a lognormally distributed quantity in the
ocean.
(i) The turbulence must be homogeneous,
(ii) the averaging scale, r, must be small compared to the domain scale, L, and
(iii) the averaging scale must be large compared to the Kolmogorov scale Lk.
Yamazaki and Lueck (1990) suggest that (iii) can be satisfied with r as low as
3Lk, but in all oceanic datasets, there is uncertainty about meeting criterion (i)
given the patchy nature of the turbulence away from regions of highly energetic
forcing such as the near-surface region. We investigated the sampling question by
analyzing the results from one typical simulation. In particular, we examined the
numerical results corresponding to Ri = 0.075, Pr = 2 at St = 6. For this data
set, L -- 25 cm, the grid scale, r = L/128, is 0.195 cm, the ensemble-averaged
dissipation is 0.213 cm2s -3, Lk = 0.0465 cm, Lc = 2.68 cm, and Le = 1.37 cm. For
the full data set of 1283 points without averaging, all three criteria of Yamazaki and
Lueck (1990) are met since r = 4.2Lk. The corresponding distribution of dissipation
should, therefore, be lognormal, and in Figure 4 the strong lognormality of the data
with a 2 __ .75 is evident. However, for comparison with typical oceanic sampling,
of greater interest are the consequences of averaging estimates in the vertical. We
chose to examine the effect of averaging over the full depth of the computational
domain (25 cm), which is equal to 18.2 Le or 9.3 Lo. Even though we are averaging
over the depth, we still have a statistically significant sample of 1282 points. Note
that we do not satisfy criterion (ii) because r = L.
Figure 5 shows the distribution of vertically averaged e for four different Richard-
son numbers. The effect of averaging is quite dramatic and the value of a 2 is reduced
to 0.01 and becomes essentially independent of Ri. Thus, when averaged over ver-
tical scales significantly greater than L_ or Lo, dissipation is only weakly lognormal
with a 2 less than the value of 3 to 7 suggested for the ocean. Using techniques
developed in Baker and Gibson, this translates to about 4 to 8 required profiles to
obtain an estimate of the mean value of e within _10%.
In order to more fully explore the second criterion of Gurvich and Yaglom, we
averaged the data in the vertical but over length scales r which were smaller than
L, e.g. 16 points or about 2.5L_. The distribution of e averaged over this scale is
plotted in Figure 6 which shows that the data is indeed lognormal but with a 2 "_. 1,
much smaller than the 3 - 7 obtained by Baker and Gibson (1987). Again, the
implication is that, if the data are averaged over length scales comparable to the
overturning scale, the intermittency will not be as high as that described by Baker
and Gibson (1987). This implies that much fewer sampling profiles are necessary
to obtain a reasonable estimate of e. Finally, we should note that since oceanic
samples are likely to be drawn from a domain with significant regions of minimal
dissipation the homogeneity criteria may be the most difficult restriction to satisfy.
342 G. IV. lvey, J. R. Koseff, D. A. Briggs CJ J. H. Ferziger
99.99
99.9
99
95
8O
% 5O
20
5
1
.I
.01
-3.5 -3.0 -2.5 -2.0 -i. 5 -i. 0
averaged dissipation, ln(_)
FIGURE 5. Lognormal probability plot of_ for Ri = 0.075 (o), 0.21 (o), 0.37 (×)
and 1.0 (+) with Pr = 2.0, St = 6.
99.999
99.99
99.9
99
95
9O
80
% 50
20
lg
1
.1
.Ol
.001
-3.0 -2.5 -2.0 -1.5 -I.0 -0.5 0.0
averaged dissipation, ln(_)
FIGURE 6. Lognormal probability plot of ln(_) with the averaging scale r _ 2.5Le,
Ri = .075.
Mizin 9 in a stratified shear flow: energetics and sampling 343
5. Conclusions
Direct numerical simulations of homogeneous turbulence in stably stratified shear
flows confirm the variation of flux Richardson number R I with turbulent Froude
number FrT and e/uN 2 observed in laboratory experiments. With R I defined
as the buoyancy flux divided by the production of turbulent kinetic energy, there
appears to be no systematic dependence of R I on Pr in the range of Pr from 0.1
to 2. This result is not consistent with the laboratory observations; however, the
differences in R I between the simulations and the experiments are small, and the
data from all sources indicate that R I has a peak of 0.25, independent of Pr. Sub-
sampling of the computational domain of 1283 points was investigated to examine
the distribution of the dissipation. The results indicate that when dissipation is
estimated by averaging over vertical scales of an order of magnitude greater than
either the Ellison or Ozmidov scales, the distribution is very weakly lognormal with
an intermittency, a 2 _'20.01. This value is considerably smaller than some estimates
in the oceanic literature and suggests sampling restrictions may not be as severe
as previously suggested provided the sampling and averaging are performed over
domains where the turbulence is homogeneous.
Acknowledgements
The authors are very grateful to the CTR for making this work possible. JRK,
DAB and JHF also wish to acknowledge the Office of Naval Research for their
support of t his work through grant number N00014-92- J- 1611.
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Mixing in a stratified shear flow: Energetics and sampling

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