The Kolmogorov Constant for the Lagrangian Velocity
Spectrum and Structure Function
Ren-Chieh Lien
Applied Physics Laboratory, University of Washington, Seattle, Washington.
(206)685-1079, [email protected]
Eric A. D’Asaro
Applied Physics Laboratory and School of Oceanography, College of Ocean and Fishery
Sciences, University of Washington, Seattle, Washington.
Abstract
The inertial subrange Kolmogorov constant for the Lagrangian velocity structure
function Co is related to the inertial subrange constant for the Lagrangian acceleration
spectrum β by Co = πβ. However, Rλ must be greater than about 105 for the inertial
subrange of the structure function to be sufficiently wide to accurately determine C o ,
while values of Rλ greater than 102 are sufficient to determine β. Taking these Rλ
limitations into account, the only two known high-quality independent measurements of
Co are 5.5 and 6.4 .
1
1. Introduction
Kolmogorov scaling predicts that for a turbulent flow with sufficiently large Reynolds
number the Lagrangian velocity structure function D(τ ) = h[w(t + τ ) − w(t)] 2 i = Co ετ
in an inertial subrange Tν ¿ τ ¿ T.1 Here ε is the average dissipation rate of kinetic
energy and w is turbulence velocity, hi denotes an average, τ is a time lag, T is the
integral time scale, Tν = (ν/ε)1/2 is the viscous time scale, ν is the viscosity, and Co is the
Kolmogorov constant. Similarly, the Lagrangian spectrum of acceleration ω 2 Φw (ω) = βε
in an inertial subrange ων À ω À ωo . Here Φw (ω) is the spectrum of velocity, ω is the
frequency, ων = Tν−1 is the viscous frequency, ω0 ≈ T −1 is the “Large-Eddy” frequency,
and β is the Kolmogorov constant.2 The two Kolmogorov constants are related by
Co = πβ.3 In this paper, we show that β is often easier to estimate than Co and combine
published measurements of both quantities to yield improved estimates of their values.
2. Estimation of Co and β at finite Reynolds Number
At finite Reynolds number the inertial subrange have finite width. Thus for the
acceleration spectrum (Fig. 1) the value of β is determined from the level of the
plateau of ω 2 Φw (ω)/ε. Similarly, for the structure function (Fig. 2), the value of C o is
determined from the level of the plateau of D(τ )/τ ε. For insufficiently high Reynolds
numbers the plateaus may be short or exist only as bumps. The maximum values
of these curves will be denoted by β ∗ and Co∗ and are biased estimates of β and Co ,
respectively. The ratios β∗ /β and Co∗ /Co will be functions of Reynolds number.
A Lorentzian form for the Lagrangian velocity spectrum is assumed, equivalent to
an exponential correlation function4
2
Φw (ω) = βε
ω2
1
+ ωo2
(1)
The Reynolds number is varied by truncating the spectrum at ωη which is varied
from 1 to 106 ωo . The resulting acceleration spectra are shown in Fig. 1. The structure
functions computed from these spectra are shown in Fig. 2.
The microscale Reynolds number, Rλ , is computed from the spectrum
Rλ = 1.23(βπ)(ωη /ωo )tan−1 (ωη /ωo )
(2)
and plotted in Fig. 3. For Rλ > 100 a linear relation exists, Rλ = 1.9Co ωη /ωo (dashed
√
line), which is nearly identical to Sawford’s5 result TL (∞)/tη = 2Rλ / 15Co (Fig. 3 solid
dots) assuming TL (∞)/tη = ωη /ωo . Another linear relationship also exists for Rλ < 4.
At Rλ < 10 neither the spectrum (Fig. 1) nor the structure function (Fig. 2) shows
an inertial subrange; the values of β ∗ and Co∗ are both biased low relative to β and Co .
The structure function requires Rλ > 104 for Co∗ to reach 95% of Co , as suggested by
Yeung6 and Sawford.5 However, the spectrum achieves an inertial subrange with β ∗ very
close to β for Rλ > 100. Thus accurate estimates of β can be made at much lower Rλ
than accurate estimates of Co .
The functional dependences of β∗ /β and Co∗ /Co on Rλ are shown in Fig. 4. The
results agree very well with data of Sawford and Yeung7 (circles) and of Sawford5
(squares). For Rλ > 50, our results suggest Co∗ = Co [1 − (0.1Rλ )−1/2 ]. For Rλ < 50,
1/2
1/2
Co∗ ≈ 0.07Co Rλ . The Rλ
dependence has been suggested by Yeung and Pope8 (their
Fig. 4).
The dispersion coefficient9 Ke (τ ) = (1/2)dhz 2 i/dt is not sensitive to the properties
of the inertial subrange, as shown by the dashed curves in Fig. 2. The Ke (τ ) curves
are nearly identical for all plotted values of Rλ . Estimates of Co based on dispersion
3
rates are therefore indirect; they mostly measure the properties of the energy containing
scales and address the properties of the inertial subrange only through assumptions
about the form of the energy spectrum or, equivalently, the structure function.
The sensitivity of these results was investigated by replacing the sharp cutoff
at ωη with a smoother transition from ω 0 to ω −2 such as that in Sawford’s5 second
auto-regressive model. Examples for ωη /ωo = 100 are shown in Figs. 1–4 by the
solid gray curves. The calculations were also redone using other spectral forms Φw (ω)
described by Lien and D’Asaro.10 The results were qualitatively identical: 1) Co∗
1/2
converges to Co for Rλ > 105 , 2) Co∗ is proportional to Rλ
for Rλ < 100, and 3) β ∗
converges to β for Rλ > 100.
3. Estimates of Co and β
Table 1 shows known quality estimates of Co or β based on Lagrangian
measurements or theory. These are grouped by method. All β estimates are reported in
Co units as πβ.
An early theoretical estimate of β is due to Tennekes and Lumley.11 They transform
the Eulerian inertial subrange k −5/3 spectrum to a Lagrangian spectrum using the
transformation ω = a(εk 2 )1/3 and find πβ = 3.8. Lien et. al12 allow for a bandwidth
of frequencies for each wavenumber and find πβ = 5.6. However, neither provides a
rationalization for setting the constant a = 1 and therefore neither estimate of β is
reliable. Rodean13 uses the known diffusivity and ε profiles of a logarithmic boundary
layer, a Langevin model relating ε to the dispersion and the equality of diffusion and
dispersion. Kaneda14 uses Lagrangian renormalization theory. Fung et. al
15
produce
isotropic “kinematic turbulence” from the random superposition of Fourier modes with
a specified -5/3 Eulerian spectrum and, from this, produce Lagrangian trajectories,
4
Lagrangian spectra and estimates of β.
A series of numerical simulations (DNS) with increasingly large Rλ have yielded a
wealth of information on Lagrangian turbulence statistics.6−8 At the largest Rλ = 234,
the above analysis indicates that πβ ∗ has converged to πβ, while Co∗ is still below Co .
The most recent attempt to extrapolate the Co∗ values to infinite Rλ yields Co = 6.4,16
in good agreement with πβ from the same simulations.
Several studies have used measurements of dispersion to estimate Co . Most of
these17−19 have been in laboratory air or water flows at low Rλ and thus yield low
values of Co∗ . Du20 analyzes atmospheric boundary layer with Rλ ≈ 103 and estimates
Co∗ = 2.5 − 3.5. Even after correction for Rλ , this is still well below the DNS results. As
noted above, however, dispersion rate is insensitive to the inertial subrange properties,
depending mostly on velocities at lower frequencies. Furthermore, these measurements
were made within the atmospheric surface layer where the gradients in ε are large and
the flow is anisotropic. The value of Co∗ was determined from the best fit to the data of
a Langevin equation model with variable ε. Due to the complexity of the environment
and analysis and their insensitivity to Co it seems safe to discard these measurements.
Similarly, the analysis of Degrazia and Anfossi21 and Anfonssi et al.22 are discarded
because they are based entirely on Eulerian measurements within the atmospheric
surface layer plus theoretical assumptions.
Measurements of Lagrangian trajectories in high Reynolds number flow with known
ε provide data ideally suited for the direct estimation of β. The particles being tracked
are typically larger than the viscous Kolmogorov scale. The width of the Lagrangian
inertial subrange is therefore not ωη /ω0 as in Fig. 1, but approximately ωL /ω0 , where
ωL = (ε/L2 )1/3 is the highest frequency that a particle of size L can accurately track.12
An effective Reynolds number RL can be computed from Fig. 3 using ωL /ω0 for ωη /ω0 on
5
the vertical axis. Hanna23 uses neutrally buoyant balloons in the atmospheric boundary
layer to generate Lagrangian spectra with an inertial subrange in Φw (ω) nearly two
decades long. Unfortunately, the spectral analysis is crude by modern standards and
the resulting estimates in βπ have wide error bars. Lien et. al12 use neutrally buoyant
floats in wind and convectively driven ocean boundary layers. The floats are quite
large (1m) so the resulting inertial subranges are short. Corrections for the float size
are used to extend the analysis over a wider frequency range, but introduce additional
uncertainties. More importantly, direct measurements of ε, simultaneous with the float
trajectories are not available and estimates of β inferred from the flow properties lead to
significant error bars. Mordant et al4 acoustically track small particles in a laboratory
flow between two counter-rotating disks to generate Lagrangian spectra with an inertial
subrange approximately one decade wide. A value of ε is obtained from the power input
to the system. This yields a value of πβ ∗ = 5.5. The structure function does not show
an inertial subrange; instead a peak with Co∗ = 2.9 is seen, much as predicted in Fig. 2.
The value of πβ ∗ is only slightly below that predicted by the DNS analysis.
4. Conclusions
A compilation of estimates of the Kolmogorov constant Co which describes
the inertial subranges of the Lagrangian velocity structure function and Lagrangian
acceleration spectrum indicates that existing measurements are consistent with
Co = 6 ± 0.5. The spectrum achieves an inertial subrange at much lower Reynolds
number (Rλ > 100) than does the structure function (Rλ > 105 ) and is thus more
suitable for estimation of Co at finite Rλ . Only two independent high quality direct
estimates at sufficiently high Rλ exist: the results of a series of numerical experiments16
yielding Co = 6.4 and a high quality laboratory experiment4 yielding Co = 5.5. Many
6
other estimates in the literature are biased low due to their low Reynolds number.
Various theoretical estimates are close to these two direct estimates. A few more high
quality estimates of Co using independent methods would add considerable confidence
to its value. However, since the present uncertainty is comparable to that between high
quality estimates of the Eulerian one-dimensional longitudinal Kolmogorov constant
measured by many dozen investigators over the last 50 years24 large improvements in
the accuracy of the estimate of Co seem unlikely.
7
References
1. S. B. Pope, ”Lagrangian PDF methods for turbulent flows,” Annu. Rev. Fluid Mech.,
26, 23–63 (1994).
2. S. Corrsin, ” Estimates of the relations between Eulerian and Lagrangian scales in
large Reynolds number turbulence,” J. Atm. Sci., 20, 115–119 (1963).
3. A. S. Monin and A. M. Yaglom, ”Statistical fluid mechanics,” vol II, ed. Lumley, J.
L., Cambridge, Ma: MIT Press (1975).
4. N. Mordant, P. Metz, O. Michel, and J.-F. Pinton, ”Measurement of Lagrangian
velocity in fully developed turbulence”, Phys. Rev. Let., 87, 21, 214501-214504
(2001).
5. B. L. Sawford, ”Reynolds number effects in Lagrangian stochastic models of turbulent
dispersion,” Phys. Fluids A, 3, 1577–1586 (1991).
6. P. K. Yeung, ”Lagrangian characteristics of turbulence and scalar transport in direct
numerical simulations,” J. Fluid Mech., 427, 241–274 (2001).
7. B. L. Sawford and P. K. Yeung, ”Lagrangian statistics in uniform shear flow: direct
numerical simulation and Lagrangian stochastic models,” Phys. Fluid, 13,
2627-2634 (2001).
8. P. K. Yeung and S. B. Pope, ”Lagrangian statistics from direct numerical simulations
of isotropic turbulence,” J. Fluid Mech., 207, 531–586 (1989).
9. G. I. Taylor, ”Diffusion by continuous movements,” Proc. Lon. Math. Soc., 20,
196–211 (1921).
10. R.-C. Lien and E. A. D’Asaro, ”Particle dispersion and turbulent diffusion”,
submitted to Phys. Fluid (2002).
11. H. Tennekes and J. L. Lumley, ”A first course in turbulence,” MIT Press (1972).
12. R.-C. Lien, E. A. D’Asaro and G. Dairiki, ”Lagrangian frequency spectra of vertical
8
velocity and vorticity in high-Reynolds-number oceanic turbulence,” J. Fluid
Mech., 362, 177–198 (1998).
13. H. C. Rodean, ”The universal constant for the Lagrangian structure function,” Phys.
Fluid, 3, 1479–1480 (1991).
14. Y. Kaneda, ”Lagrangian and Eulerian time correlations in turbulence,” Phys. Fluids
A, 5, 2835–2845 (1993).
15. J. C. H. Fung, J. C. R. Hunt, N. A. Malik, R. J. Perkins, ”Kinematic simulation of
homogeneous turbulence by unsteady random Fourier modes,” J. Fluid Mech.,
236, 281–318 (1992).
16. P. K. Yeung, ” Lagrangian investigations of turbulence,” Annu. Rev. Fluid Mech.,
34, 115–142 (2002).
17. M. S. Anand, and S. B. Pope, ”Diffusion behind a line source in grid turbulence,”
In Turbulent Shear Flows 4, ed. L. J. S. Bradbury, F. Durst, B. E. Launder, F.
W. Schmidt, J. J. Witelaw, pp. 46–61. Berlin: Springer-Verlag. (1985).
18. S. B. Pope and Y. L. Chen, ”The velocity-dissipation probability density function
model for turbulent flows,” Phys. Fluids A, 2, 1437–1449 (1990).
19. S. Du, B. L. Sawford, B. L. J. D. Wilson, and D. J. Wilson, ”Estimates of
the Kolmogorov constant (Co ) for the Lagrangian structure function, using a
second–order Lagrangian model of grid turbulence,” Phys. Fluids, 7, 3083–3090
(1995).
20. S. Du, ”Universality of the Lagrangian velocity structure function constant (C o )
across different kinds of turbulence,” Boundary-Layer Meteo., 83, 207–219
(1997).
21. G. Degrazia, and D. Anfossi, ”Estimation of the Kolmogorov constant Co from
classical statistical diffusion theory,” Atm. Env., 32, 3611–3614 (1998).
9
22. D. Anfossi, G. Degrazia, E. Ferrero, S. E. Grying, M. G. Morselli, and S. Trini
Castelli, ”Estimation of the Lagrangian structure function constant Co from
surface-layer wind data,” Boundary-Layer Metero., 95, 249–270 (2000).
23. S. R. Hanna, ”Lagrangian and Eulerian time-scale relations in the daytime boundary
layer,” J. Appl. Meteor., 20, 242–249 (1981).
24. K. Sreenivasan, ”On the universality of the Kolmogorov constant,” Phys. Fluids, 7,
2778–2784 (1995).
Received
10
Table 1. Estimates of Universal Constant Co of Lagrangian Structure Function
Source
Co
Co∗
βπ
Rλ
Remarks
∞
log layer structure
THEORY
Rodean13 (1991)
5.7
Kaneda14 (1993)
5.9
Lagrangian renormalization
Fung et al.15 (1992)
∞
5.0
∞
kinematic simulation
2.6
3.9
38
DNS
4.0
5.5
93
DNS
Yeung & Pope8 (1989)
Sawford & Yeung7 (2001)
4.3
140
DNS
4.8
240
DNS
234
DNS
Yeung6 (2001)
6.4
DNS Extrapolated to Rλ = ∞
Rλ Range
Sawford5 (1991)
7.0
38–93
Pope1 (1994)
6.2
38–93
Sawford & Yeung7 (2001)
6.0
38–240
Yeung16 (2002)
6.4
38–234
DISPERSION DATA fit with Langevin model
C̃o
Rλ
Anand & Pope17 (1985)
2.1
70
lab
Pope & Chen18 (1990)
3.5
70
lab
Du et al.19 (1995)
2.5–3.5
Du20 (1997)
≈ 50
2.5–3.5
≈ 103
lab water and air
atmospheric surface layer
Rλ (RL )
TRAJECTORY DATA
Hanna23 (1981)
2.2–6.1
atmospheric surface layer
Lien et al.12 (1998)
O103 (≈ 103 )
3.1-6.2
2200(≈ 40)
oceanic boundary layers
5.5
740(≈ 200)
laboratory experiment
Mordant et al.4 (2001)
2.9
11
Figure Captions:
Figure 1. Normalized Lagrangian acceleration spectra (thin curves) with different
viscous frequencies ωη . For ωη = 100ωo both sharply (thick black) and smoothly (thick
grey) truncated spectra are shown. Values of microscale Reynolds number Rλ are shown
in parentheses.
Figure 2. Normalized Lagrangian velocity structure function (thin solid curves) and
normalized effective diffusivity (dashed curves) computed from spectra in Fig. 1. Estimates of effective diffusivity computed from spectra of different Rλ are indistinguishable.
Thick black and thick grey curves are computed from corresponding curves in Fig. 1.
Values of microscale Reynolds number Rλ are shown in parentheses.
Figure 3. Relation between ωη /ωo and Rλ . The thick gray curve and thin solid curve are
the exact relation for the sharp truncated and smoothly truncated spectra, respectively.
Two dashed curves represent the approximate forms at high and low Rλ regimes. Solid
dots illustrate Sawford’s5 results for Rλ between 40 and 100.
Figure 4. Co∗ /Co (thick solid curves) and β ∗ /β (thick dashed curves) as a function
of Rλ computed from the sharply (black) and smoothly (grey) truncated spectra
shown in Fig. 1. The thin curve, overlapped mostly with the thick solid curve,
shows Co∗ /Co = [1 − (0.1Rλ )−1/2 ]. The dashed-solid-dotted curve at Rλ < 100 shows
1/2
Co∗ /Co = 0.07Rλ . Symbols show Sawford and Yeung’s7 DNS estimates (circles) and
results of Sawford’s5 second order auto-regressive model (squares). The horizontal
dashed line marks Co∗ /Co = 1 and β ∗ /β = 1.
12
100
10−1
10−2 −2
10
Φw ω2 / β ε
10 (~100)
105 (~106)
6
10
106 (~107)
−2
4
10
104 (~105)
ω
o
103 (~104)
Acceleration Spectrum
2
10
ω/ω
13
102 (~103)
Fig. 1. Lien, Physics of Fluids
0
10
ωη / ωo = 1 (Rλ = 5.5)
Fig. 2. Lien, Physics of Fluids
1.2
1.2
Structure Function
1
106 (~107)
Ke
1
105 (~106)
104 (~105)
103 (~104)
0.8
102 (~103)
0.6
0.6
10(~102)
0.4
0.4
0.2
0 -4
10
ωη/ωo=1 (Rλ=5.5)
10
-2
0
τω
0
14
10
0.2
0
Ke / π ε ωo -2
D / (π β ε τ)
0.8
Fig. 3. Lien, Physics of Fluids
2
10
1
η
ω /ω
o
10
0
10
0
10
R
λ
15
10
2
Fig. 4. Lien, Physics of Fluids
1
0.8
C*0 /(π β)
[1
−
R
.1
(0
)−
λ
1/2
]
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
2
10
4
Rλ
16
10
6
10
0
β*/β
β */β
1