Verification of Eddy-Viscosity Models in the Planetary Boundary Layer

Investigators

Li-Chuan Chen, William Eichinger, Zhiming Chen (University of Iowa); Fernando Porte-Agel, Markus Pahlow, Charles Meneveau and Marc Parlange

Acknowledgment

We gratefully acknowledge the collaboration of Dr. Gabriel Katul of Duke University who contribute equipment.

Introduction

In the Planetary Boundary Layer (PBL), Large-Eddy Simulation (LES) has become an important tool for the study of transport processes. LES divides the flow into two fields: the resolved-scale field and the subgrid-scale (SGS) field. The resolved-scale field can be solved by the filtered continuity, momentum, and energy equations once the SGS fluxes are determined. The SGS turbulence in the LES model is a local and instantaneous field, and must be parameterized in terms of the resolved-scale field.

There are many different types of SGS models currently in use in LES codes for atmospheric flows. Eddy-viscosity models are one of the most popular models. For many decades, the correctness of eddy-viscosity models have been in question. Speziale (1991) pointed out that the Eddy-viscosity models have two major problems. First, they are purely dissipative and hence cannot account for Reynolds-stress relaxation effects. Second, the models predict isotropy of the normal stresses and, therefore, cannot describe Reynolds-stress driven secondary motion in three-dimensional flows. In the last three decades, most of the verification work has been done in laboratory experiments such as Tucker (1968) and Champagne (1970). Liu, Meneveau and Katz (1994) found that the correlation coefficients of the real stresses and the rates of strain were shown to be very small ( 0.15), implying that locally the eddy-viscosity closures perform poorly.

This research uses arrays of sonic anemometers to verify the eddy-viscosity models in the Planetary Boundary Layer. The data were obtained from 6 sonic anemometers in two-row arrangements in a uniform field of mowed alfalfa in Amana, Iowa. The Reynolds stresses and the rates of strain can be calculated directly from the measurements. Taylor's hypothesis was employed to convert the spatial derivatives from time derivatives for the rates of strain in the dominant wind direction. The results were shown in the following.

Experiment

The experiment was performed in a uniform alfalfa field in Amana, Iowa from Jun. 25. 1998 to Jul. 16. 1998. The experiment field in the upwind direction is shown in the picture above. Eleven setups were arranged for different purposes. The data used in this study were chosen from two of the eleven setups, Setup 7 and Setup 8. Six sonic anemometers were arranged in two rows (three sonic anemometers for each row). The arrangement is shown in the picture on the right-hand-side. The height for the first row from the ground was 3.14m and the second row was 3.65m. The spacing between two adjacent sonic anemometers was 0.4m for Setup 7 and 0.8m for Setup 8. The grass height was 15 - 20 cm while Setup 7 and Setup 8 were performed. The data for all six anemometers were recorded synchronously in order to obtain the instantaneous turbulent field. The sampling rate was 60 Hz. The structure was located 350m from the building to assure sufficient fetch.

The reasons to perform the verification work using field data are that: (1) Atmospheric turbulence is characterized by high Reynold numbers which are difficult to generate in the laboratory; (2) The thermal convection (atmospheric stability) has a strong effect in the atmospheric turbulence which can not be observed in experiments using a wind tunnel.

Eddy-Viscosity Models

For Newtonian fluid with constant properties, the equation of motion can be expressed in the following equation which is the famous Navier-Stokes Equation

Eq. (1)

By substituting the Reynolds Decomposition

and

into Eq. (1) and taking the average of the equation, we can obtain the Reynolds-averaged Navier-Stokes Equation (RANS) for turbulent flow

Eq. (2)

Comparing Eq. (1) and Eq. (2), we can see that there is one additional term in the RANS equation. This term adds six more unknowns () in the turbulence problem and causes it to have more unknowns than equations.

For decades, people have tried to solve the closure problem by many different methods. One of the popular method is turbulence modeling. Smagorinsky (1963) suggested an eddy-viscosity model to simulate the Reynolds stress by rate of strain which is

Eq. (3)

where d _ij is the Kronecker delta.

For the shear Reynolds stress components (i ¹ j),

Eq. (4)

where is the rate of strain and n _t is the eddy viscosity (or diffusivity K_m ) which can be expressed as (Deardorff, 1980)

Eq. (5)

e is the turbulent kinetic energy (TKE), and l is a characteristic length scale. This length scale is often assumed to be (or proportional to) the effective grid spacing l = D s º (D xD yD z)^1/3, where D s is the grid length scale. Deardorff (1980) proposed a stability correction to l that when ¶ q /¶ z > 0

Eq. (6)

if l_s < D s

Eq. (6) was given by Deardorff without any observational or theoretical basis. The coefficient 0.76 is purely empirical.

Results and Discussions

The Reynolds stress and the rate of strain can be calculated directly from the instantaneous field. The diffusivity Km is obtained from the ratio of the Reynolds stress and strain-rate. The figures on the left-hand-side show the scatter plot between the diffusivity and the turbulent kinetic energy in log-scale. The diffusivity increases while the turbulent kinetic energy increases. Also, the data seem to have the similar tendency. However, the data points scatter in a wide range and the correlation coefficients between the real stress and strain-rate in the normal-scale are small which are 0.01, 0.21, and 0.16 for uv-component, uw-component, and vw-component respectively.

From the preliminary results, we have not observed any strong relationship between the real stress and strain-rate as expected from the models. Two possibilities:

(1) The models can not totally represent the relationship between the Reynolds stress and the rate of strain. Most models wide used in different fields have not been tested by the physical experiment data. Some of them are purely empirical.

(2) There are some uncertainties in the data needed to eliminate before the data analysis performed. The sensor array technique is proposed by Tong (1997). Many assumptions, for example, Taylor hypothesis and the homogeneity and isotropy of the atmospheric turbulence, have been employed to the theory because of the physical limitations of the experiment. The highly random nature of the atmosphere also increases the difficulty in the data analysis. Therefore, some data quality control techniques need to be used in the future analysis to assure the results.

Conclusions and Future Work

(1) The preliminary results show that Eddy-viscosity models perform poorly. The average correlation coefficient between the real stress and the strain-rate is 0.13 for different tensor components in normal-scale which consists the result suggested by Liu (1994).

(2) Even the correlation coefficients between the real stress and strain are low. It is early to conclude that there is no relationship between the Reynolds stress and the rate of strain. The first task now is to estimate the uncertainty in the data. Further analysis such as testing the Taylor Hypothesis and the isotropy of turbulence will be performed in the future work.

(3) The long-term objective for this research is to model the SGS flux by using the resolved-scale quantities based on the field data. More data are required to accomplish this work. It is planned to perform a similar experiment in Davis, California in May, 1999. A new setup shown on the right-hand-side will be employed in the experiment.

References

Champagne, F. H., Harris, V. G., and Corrsin, S. 1970: Experiments on Nearly Homogeneous Turbulent Shear Flow. J. Fluid Mech., 41, 81-139.

Deardorff, J. W., 1980: Stratocumulus-capped Mixed Layers Derived From a Three-domensional Model. Bound.-Layer Meteor., 18, 495-527.

Liu, S., Meneveau, C., and Katz, J., 1994: On the Properties of Similarity Subgrid-scale Models as Deduced From Measurements in a Turbulent Jet. J. Fluid Mech., 275, 83-119.

Smagorinsky, J., 1963: General Circulation Experiments with the Primitive Equations, I. The Basic Experiment. Mon. Weath. Rev. 91, 99-164.

Speziale, C. G., 1991: Analytical Methods for the Development of Reynolds-stress Closures in Turbulence. Annu. Rev. Fluid Mech., 23, 107-157.

Tong, C., Wyngaard, J. C., Khanna, S., and Brasseur, J. G., 1997: Resolvable- and Subgrid-scale Measurement in the Atmospheric Surface Layer: Technique and Issues. J. Atmos. Sci., Preprint.

Tucker, H. J., and Reynolds, A. J., 1968: The Distortion of Turbulence by Irrotational Plane Strain. J. Fluid Mech., 32, 657-673.