June 2000 NASA/CR-2000-210298 An Estimation of Turbulent Kinetic Energy and Energy Dissipation Rate Based on Atmospheric Boundary Layer Similarity Theory Jongil Han, S. Pal Arya, Shaohua Shen, and Yuh-Lang Lin North Carolina State University, Raleigh, North Carolina
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An Estimation of Turbulent Kinetic Energy and Energy Dissipation
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June 2000
NASA/CR-2000-210298
An Estimation of Turbulent Kinetic Energyand Energy Dissipation Rate Based onAtmospheric Boundary Layer SimilarityTheory
Jongil Han, S. Pal Arya, Shaohua Shen, and Yuh-Lang LinNorth Carolina State University, Raleigh, North Carolina
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June 2000
NASA/CR-2000-210298
An Estimation of Turbulent Kinetic Energyand Energy Dissipation Rate Based onAtmospheric Boundary Layer SimilarityTheory
Jongil Han, S. Pal Arya, Shaohua Shen, and Yuh-Lang LinNorth Carolina State University, Raleigh, North Carolina
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NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS)7121 Standard Drive 5285 Port Royal RoadHanover, MD 21076-1320 Springfield, VA 22161-2171(301) 621-0390 (703) 605-6000
Abstract
Algorithms are developed to extract atmospheric boundary layer pro�les for turbulence
kinetic energy (TKE) and energy dissipation rate (EDR), with data from a meteorological
tower as input. The pro�les are based on similarity theory and scalings for the atmospheric
boundary layer. The calculated pro�les of EDR and TKE are required to match the observed
values at 5 and 40m. The algorithms are coded for operational use and yield plausible pro�les
over the diurnal variation of the atmospheric boundary layer.
i
Contents
1 Introduction 1
2 Similarity Relations for the TKE and EDR Pro�les 2
2 Same as Fig. 1 but for TKE at z = 5m. . . . . . . . . . . . . . . . . . . . . 20
3 Same as Fig. 1 but for TKE at z = 40m. . . . . . . . . . . . . . . . . . . . . 21
4 Same as Fig. 1 but for EDR at z = 5m. . . . . . . . . . . . . . . . . . . . . 22
5 Same as Fig. 1 but for EDR at z = 40m. . . . . . . . . . . . . . . . . . . . . 23
6 Diurnal variation of TKE pro�les obtained from the ABL similarity theory
and the observed values of TKE and EDR at z = 5m and 40m. . . . . . . . 24
7 Same as Fig. 8 but for EDR pro�les. . . . . . . . . . . . . . . . . . . . . . . 25
iv
Nomenclature
e = turbulence kinetic energyf = Coriolis parameterg = gravitational accelerationh = height of atmospheric boundary layerL = Obukhov lengthk = von Karman constant, 0.4Ri = gradient Richardson numberT0 = reference temperature�u = mean velocity in the direction of the windu� = friction velocityu0; v0 = horizontal velocity uctuationsw� = convective velocity scalew0 = vertical velocity uctuationz = elevation above the ground� = turbulent kinetic energy dissipation rate� = Monin-Obukhov stability parameter, z=L��v = mean virtual potential temperature�v
0 = virtual potential temperature uctuation�h = dimensionless potential temperature gradient in the surface layer�m = dimensionless wind shear in the surface layer
v
1 Introduction
To safely reduce aircraft spacing and increase airport capacity, NASA is developing a
predictor system, called the Aircraft VOrtex Spacing System (AVOSS: Hinton, 1995; Hinton
et al., 1999; Perry et al., 1997). This system includes prediction algorithms for aircraft
wake vortex transport and decay. Semi-empirical vortex prediction algorithms have been
developed and incorporated within the AVOSS (Robins et al., 1998; Sarpkaya et al., 2000).
One of the key input elements for the AVOSS prediction algorithms is the atmospheric
boundary layer (ABL) turbulence of which the intensity can be represented by turbulent
kinetic energy (TKE) or eddy energy dissipation rate (EDR). While the prediction algorithms
require the vertical pro�les for the TKE and EDR at least up to the vortex generation height,
the observational data for the TKE and EDR are available only from a meteorological tower
at the heights of 5 and 40m above the ground.
In supporting the NASA AVOSS project, the wake vortex research group at North Car-
olina State University (NCSU) has developed algorithms and software which can generate
the vertical pro�les of TKE and EDR. These algorithms are based on the ABL similarity
relations (Arya, 1988, 1995, 2000; Caughey et al., 1979; Rao and Nappo, 1998; Sorbjan,
1989; Stull, 1988) and available experimental data.
Section 2 describes the ABL similarity relations with respect to the vertical pro�les of
the TKE and EDR that are dependent upon the ABL stability. Section 3 contains a detailed
description of the software. In Section 4, estimates from the similarity relations are compared
with the observed data. Section 5 provides information on running the software. Finally,
the summary of this study is given in Section 6.
1
2 Similarity Relations for the TKE and EDR Pro�les
ABL observations frequently show consistent and repeatable characteristics from which
empirical similarity relationships have been obtained for the variables of interest such as
TKE and EDR. Similarity theory is based on the organization of variables into dimensionless
groups that come out of the dimensional analysis. The dimensional analysis is a technique
used in science and engineering to establish a relationship between di�erent quantities. The
functional relationships between dimensionless groups are referred to as similarity relations,
because they express the conditions under which two or more ow regimes would be similar.
Similarity relationships that have certain universal properties are usually designed to apply
to equilibrium (steady-state) situations. One of the well-known similarity relations is the
logarithmic velocity pro�le law observed in surface or wall layers under neutral strati�cation.
The proposed similarity relationships for TKE and EDR are fundamentally based on the
Monin-Obukhov similarity (Monin and Obukhov, 1954) and mixed-layer similarity (Dear-
dor�, 1972) theories. The former is applied to a strati�ed surface layer and is sometimes
called the surface-layer similarity theory, whereas the latter is applied to mixed layers that
often develops during daytime convective conditions. These similarity theories have provided
the most suitable and acceptable framework for organizing and presenting the ABL data, as
well as for extrapolating and predicting certain ABL information where direct measurements
of the same are not available.
Using the framework of these similarity theories, a variety of similarity relationships have
been suggested to describe the vertical pro�les of mean and turbulence �elds as functions of
the dimensionless groups z=L and/or z=h, covering whole ABL including the surface layer
(Arya, 1988, 1995, 2000; Caughey et al., 1979; Hogstrom, 1996; Rao and Nappo, 1998;
Sorbjan, 1989; Stull, 1988). Occasionally, various investigators have suggested di�erent
values for the empirical coe�cients. Based on the similarity scaling in the atmospheric
surface layer and boundary layer under di�erent stability conditions, the expressions and
parameterizations for the vertical pro�les of TKE and EDR and related characteristic scales
2
are suggested in the following subsections; some of the expressions are adopted directly
from those references, whereas the others are derived using the similarity relationships of
turbulence variables other than TKE and EDR.
2.1 Neutral and Stable Boundary Layers (z=L � 0)
The boundary layer may be subdivided into a surface layer (in which stress is nearly
constant with height) and an outer layer. A separate set of algorithms is assigned to each
sublayer as follows.
2.1.1 Surface Layer (z � 0:1h)
In the surface layer, the TKE (e) and EDR (�) are given by (Hogstrom, 1996; Rao and
Nappo, 1998)
e = 6u2�; (1)
� =u3�
kz
�1:24 + 4:3
z
L
�; (2)
where k ' 0:4 is von Karman constant. The friction velocity, u�, is de�ned as
u2�=h(u0w0)
2
s + (v0w0)2
s
i1=2; (3)
where the right hand side of Eq: (3) represents the total vertical momentum ux near the
surface (the subscript s denotes the ground surface). The Obukhov length L depends on
both the momentum and heat uxes near the surface and is de�ned later; the ratio z=L is
the fundamental similarity parameter of the Monin-Obukhov similarity theory.
2.1.2 Outer Layer (z > 0:1h)
Expressions for the outer layer can be assigned according to the level of strati�cation.
(1) Neutral and Stable Boundary Layer
In the neutral and moderately stable boundary layer, the TKE and EDR are given by
(Hogstrom, 1996; Rao and Nappo, 1998)
e = 6u2�
�1�
z
h
�1:75
; (4)
3
� =u3�
kz
�1:24 + 4:3
z
L
��1� 0:85
z
h
�1:5
: (5)
Alternatively, Eqs:(4) and (5) may also be used for the entire boundary layer, including the
surface layer.
(2) Very Stable and Decoupled Layers
In the very stable boundary layer and decoupled layers, the TKE and EDR can be
expressed by extension of Eqs:(1) and (2) as
e = 6u2L; (6)
� = 4:3u3LkLL
; (7)
where uL is the local (friction) velocity scale and LL is the local buoyancy length scale. Under
very stable conditions, the elevated layers of turbulence are decoupled from the surface and
the local uxes and scales cannot be reliably estimated. Perhaps, an empirical relationship
between the overall turbulence intensity (e1=2=�u) and Richardson number should be explored.
It is worthwhile to note that some experimental results show that Eq:(5) can be still used
to estimate � even in a very stable boundary layer.
2.2 Unstable Boundary Layer (z=L < 0)
The unstable ABL such as during daytime surface heating can be divided into three
regimes depending upon the stability parameter, z=L or h=L.
Rao, K. S. and Nappo, C. J., 1998: Turbulence and Dispersion in the Stable Atmospheric
Boundary Layer. Dynamics of the Atmospheric Flows: Atmospheric Transport
and Di�usion Processes, Singh, M. P. and Raman, S., eds., Computational Me-
chanics Publications, 39-91.
Robins, R. E., Delisi, D. P., and Greene, G. C., 1998: Development and Validation of a
Wake Vortex Prediction Algorithm. 36th Aerospace Sciences Meeting & Exhibit,
Reno, NV, AIAA-98-0665, 10 pp.
Sarpkaya, T., Robins, R. E., and Delisi, D. P., 2000: Wake-Vortex Eddy-Dissipation
Model Predictions Compared with Observations. 38th Aerospace Sciences Meet-
ing & Exhibit, Reno, NV, AIAA-2000-0625, 10 pp.
Sorbjan, Z., 1989: Structure of the Atmospheric Boundary Layer. Prentice Hall, 317 pp.
Stull, R. B., 1988: An Introduction to Boundary Layer Meteorology. Kluwer Academic
Publishers, 666 pp.
17
Stability Frequency (%)Stable 54Weakly Unstable 5Moderately Unstable 37Strongly Unstable 4
Table 1: Stability frequency (%) in the Dallas/Ft. Worth (DFW) Airport Field Experimentduring the period of September 15 - October 3, 1997, where data from two rainy days andone day for which data are partly missing have been omitted from the total data set.
18
10−2
10−1
100
10−2
10−1
100 (a) Neutral and stable
Measured u* (m/s)
Est
imat
ed u
* (m
/s)
10−2
10−1
100
10−2
10−1
100 (b) Unstable
Measured u* (m/s)
Est
imat
ed u
* (m
/s)
Figure 1: Comparison plots between measured and estimated u�, where the solid lines rep-
resent lines of estimated u�= measured u
�.
19
10−3
10−2
10−1
100
101
10−3
10−2
10−1
100
101 (a) Neutral and stable (z=5m)
Measured TKE (m2/s2)
Est
imat
ed T
KE
(m
2 /s2 )
10−3
10−2
10−1
100
101
10−3
10−2
10−1
100
101 (b) Unstable (z=5m)
Measured TKE (m2/s2)
Est
imat
ed T
KE
(m
2 /s2 )
Figure 2: Same as Fig. 1 but for TKE at z = 5m.
20
10−3
10−2
10−1
100
101
10−3
10−2
10−1
100
101 Neutral and stable (z=40m)
Measured TKE (m2/s2)
Est
imat
ed T
KE
(m
2 /s2 )
Figure 3: Same as Fig. 1 but for TKE at z = 40m.
21
10−4
10−3
10−2
10−1
10−4
10−3
10−2
10−1 (a) Neutral and stable (z=5m)
Measured EDR (m2/s3)
Est
imat
ed E
DR
(m
2 /s3 )
10−4
10−3
10−2
10−1
10−4
10−3
10−2
10−1 (b) Unstable (z=5m)
Measured EDR (m2/s3)
Est
imat
ed E
DR
(m
2 /s3 )
Figure 4: Same as Fig. 1 but for EDR at z = 5m.
22
10−5
10−4
10−3
10−2
10−1
10−5
10−4
10−3
10−2
10−1 Neutral and stable (z=40m)
Measured EDR (m2/s3)
Est
imat
ed E
DR
(m
2 /s3 )
Figure 5: Same as Fig. 1 but for EDR at z = 40m.
23
0 0.2 0.4 0.6 0.8 1 1.20
100
200
300
400
500
600 (a) 00 GMT (DFW 9/28/97)
TKE (m2/s2)
A
ltitu
de
(m
)
0 0.2 0.4 0.6 0.8 1 1.20
100
200
300
400
500
600 (b) 03 GMT (DFW 9/28/97)
TKE (m2/s2)
A
ltitu
de
(m
)
0 0.2 0.4 0.6 0.8 1 1.20
100
200
300
400
500
600 (c) 06 GMT (DFW 9/28/97)
TKE (m2/s2)
A
ltitu
de
(m
)
0 0.2 0.4 0.6 0.8 1 1.20
100
200
300
400
500
600 (d) 09 GMT (DFW 9/28/97)
TKE (m2/s2)
A
ltitu
de
(m
)
0 0.2 0.4 0.6 0.8 1 1.20
100
200
300
400
500
600 (e) 12 GMT (DFW 9/28/97)
TKE (m2/s2)
A
ltitu
de
(m
)
0 0.2 0.4 0.6 0.8 1 1.20
100
200
300
400
500
600 (f) 15 GMT (DFW 9/28/97)
TKE (m2/s2)
A
ltitu
de
(m
)
0 0.2 0.4 0.6 0.8 1 1.20
100
200
300
400
500
600 (g) 18 GMT (DFW 9/28/97)
TKE (m2/s2)
A
ltitu
de
(m
)
0 0.2 0.4 0.6 0.8 1 1.20
100
200
300
400
500
600 (h) 21 GMT (DFW 9/28/97)
TKE (m2/s2)
A
ltitu
de
(m
)
Figure 6: Diurnal variation of TKE pro�les obtained from the ABL similarity theory andthe observed values of TKE and EDR at z = 5m and 40m.
24
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10−3
0
100
200
300
400
500
600 (a) 00 GMT (DFW 9/28/97)
EDR (m2/s3)
A
ltitu
de
(m
)
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10−3
0
100
200
300
400
500
600 (b) 03 GMT (DFW 9/28/97)
EDR (m2/s3)
A
ltitu
de
(m
)
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10−3
0
100
200
300
400
500
600 (c) 06 GMT (DFW 9/28/97)
EDR (m2/s3)
A
ltitu
de
(m
)
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10−3
0
100
200
300
400
500
600 (d) 09 GMT (DFW 9/28/97)
EDR (m2/s3)
A
ltitu
de
(m
)
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10−3
0
100
200
300
400
500
600 (e) 12 GMT (DFW 9/28/97)
EDR (m2/s3)
A
ltitu
de
(m
)
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10−3
0
100
200
300
400
500
600 (f) 15 GMT (DFW 9/28/97)
EDR (m2/s3)
A
ltitu
de
(m
)
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10−3
0
100
200
300
400
500
600 (g) 18 GMT (DFW 9/28/97)
EDR (m2/s3)
A
ltitu
de
(m
)
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10−3
0
100
200
300
400
500
600 (h) 21 GMT (DFW 9/28/97)
EDR (m2/s3)
A
ltitu
de
(m
)
Figure 7: Same as Fig. 8 but for EDR pro�les.
25
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Contractor Report4. TITLE AND SUBTITLE
An Estimation of Turbulent Kinetic Energy and Energy Dissipation RateBased on Atmospheric Boundary Layer Similarity Theory
5. FUNDING NUMBERS
NCC1-188
6. AUTHOR(S)Jongil Han, S. Pal Arya, Shaohua Shen, and Yuh-Lang Lin
576-02-11-11
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) North Carolina State University Department of Marine, Earth and Atmospheric Sciences Raleigh, NC 27695-8208
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NASA/CR-2000-210298
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12a. DISTRIBUTION/AVAILABILITY STATEMENTUnclassified-UnlimitedSubject Category 03 Distribution: NonstandardAvailability: NASA CASI (301) 621-0390
12b. DISTRIBUTION CODE
13. ABSTRACT (Maximum 200 words)Algorithms are developed to extract atmospheric boundary layer profiles for turbulence kinetic energy (TKE)and energy dissipation rate (EDR), with data from a meteorological tower as input. The profiles are based onsimilarity theory and scalings for the atmospheric boundary layer. The calculated profiles of EDR and TKE arerequired to match the observed values at 5 and 40 m. The algorithms are coded for operational use and yieldplausible profiles over the diurnal variation of the atmospheric boundary layer.
14. SUBJECT TERMSTurbulence Kinetic Energy
15. NUMBER OF PAGES35
16. PRICE CODEA03
17. SEC U RITY CL ASSIF IC AT ION O F REPO R TUnclassified
18. SEC U RITY CL ASSIF IC AT ION O F TH IS PA GEUnclassified
19. SECURITY CLASSIFICATION OF ABSTRACTUnclassified
20. LIMITATION OF ABSTRACT UL
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