Comparison of dynamic subgrid-scale models for simulations of ...

Environ Fluid Mech DOI 10.1007/s10652-007-9023-x ORIGINAL ARTICLE

Comparison of dynamic subgrid-scale models for simulations of neutrally buoyant shear-driven atmospheric boundary layer flows William C. Anderson · Sukanta Basu · Chris W. Letchford

Received: 15 August 2006 / Accepted: 11 April 2007 © Springer Science+Business Media B.V. 2007

Abstract Several non-dynamic, scale-invariant, and scale-dependent dynamic subgridscale (SGS) models are utilized in large-eddy simulations of shear-driven neutral atmospheric boundary layer (ABL) flows. The popular Smagorinsky closure and an alternative closure based on Kolmogorov’s scaling hypothesis are used as SGS base models. Our results show that, in the context of neutral ABL regime, the dynamic modeling approach is extremely useful, and reproduces several establised results (e.g., the surface layer similarity theory) with fidelity. The scale-dependence framework, in general, improves the near-surface statistics from the Smagorinsky model-based simulations. We also note that the local averaging-based dynamic SGS models perform significantly better than their planar averaging-based counterparts. Lastly, we find more or less consistent superiority of the Smagorinsky-based SGS models (over the corresponding Kolmogorov’s scaling hypothesis-based SGS models) for predicting the inertial range scaling of spectra. Keywords Atmospheric boundary layer · Large-eddy simulation · Neutral · Subgrid-scale · Turbulence Abbreviations ABL—Atmospheric boundary layer LAD—Locally averaged (scale-invariant) dynamic LASDD—Locally averaged scale-dependent dynamic LES—Large-eddy simulation

W. C. Anderson · C. W. Letchford Wind Science and Engineering Research Center, Texas Tech University, Lubbock, TX 79409, USA S. Basu (B) Atmospheric Science Group, Department of Geosciences, Texas Tech University, Lubbock, TX 79409, USA e-mail: [email protected]

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NBL—Neutral boundary layer PAD—Planar averaged (scale-invariant) dynamic PASDD—Planar averaged scale-dependent dynamic SGS—Subgrid-scale TKE—Turbulence kinetic energy

1 Introduction The dynamic subgrid-scale modeling approach of Germano et al. [1] has been quite successful in large-eddy simulations (LESs) of various engineering flows [2]. In this approach, one dynamically computes the values of the unknown SGS coefficients at every time and position in the flow. By looking at the dynamics of the flow at two different resolved scales, and assuming scale similarity as well as scale invariance of the SGS coefficients, one can optimize their values. Thus, the dynamic modeling approach avoids the need for a priori specification and tuning of the SGS coefficients. A recent study [3] based on extensive database analysis further suggests that the dynamic modeling approach closely reproduces the minimal simulation error strategy (termed as optimal refinement strategy), which is highly desirable in turbulence modeling. In atmospheric boundary layer turbulence, where shear and stratification and associated flow anisotropies are (almost) ubiquitous, the inherent scale-invariance assumption of the original dynamic modeling approach breaks down. Porté-Agel et al. [4] relaxed this assumption and introduced a scale-dependent dynamic modeling approach in which the SGS coefficients are assumed to vary as powers of the LES filter width ( f ). The unknown power-law exponents, and subsequently the SGS coefficients, can be determined in a self-consistent manner by filtering at three levels [4,5]. In the simulations of neutral boundary layers (NBLs), the scale-dependent dynamic SGS model was found to exhibit appropriate dissipation behavior and more accurate spectra in comparison to the original (scale-invariant) dynamic model [4,5]. Recently the scale-dependent dynamic modeling approach was modified and extended by incorporating a localized averaging technique in order to simulate intermittent, patchy turbulence in the stably stratified flows [6,7]. In parallel, scale-dependent dynamic SGS models based on Lagrangian averaging over fluid flow path lines were developed by Bou-zeid et al. [8] and Stoll and Porté-Agel [9] to simulate neutrally stratified flows over heterogeneous surfaces. The dynamic modeling approach and its variants so far mostly used the popular eddyviscosity formulation of Smagorinsky [10] as the SGS base model. According to this base model, the i j component of the SGS stress tensor (the deviatoric part) can be written as follows: 1 ˜ S˜i j τi j − τkk δi j = −2C S2 2f | S| (1) 3 ˜ denote the resolved strain rate tensor and its magnitude, respectively. C S where S˜i j and | S| is the so-called Smagorinsky coefficient, which is adjusted empirically or dynamically to account for shear, stratification, and grid resolution. Similar to the SGS stresses, the SGS scalar fluxes for a generic scalar (c) could be written as: qi = −

C S2 ˜ ∂ c˜ 2 | S| PrSGS f ∂ xi

where PrSGS is the SGS Prandtl number.

123

(2)

Environ Fluid Mech Table 1 Summary of LES runs

ID

Base model

Dynamic

SMAG_p17

ScaleAveraging dependent

Smagorinsky No No (C S = 0.17) SMAG_p24 Smagorinsky No No (C S = 0.24) PAD-SM Smagorinsky Yes No PAD-WL Wong–Lilly Yes No LAD-SM Smagorinsky Yes No LAD-WL Wong–Lilly Yes No PASDD-SM Smagorinsky Yes Yes PASDD-WL Wong–Lilly Yes Yes LASDD-SM Smagorinsky Yes Yes LASDD-WL Wong–Lilly Yes Yes

Not applicable Not applicable Planar Planar Local (3 × 3) Local (3 × 3) Planar Planar Local (3 × 3) Local (3 × 3)

The Smagorinsky SGS model assumes that the energy dissipation rate locally equals the energy production rate. In order to avoid this strong assumption, Wong and Lilly [11] proposed a new SGS model based on Kolmogorov’s scaling hypothesis: 1 4/3 τi j − τkk δi j = −2C W L f S˜i j 3

(3)

and qi = −

C W L 4/3 ∂ c˜ PrSGS f ∂ xi

(4)

C W L is a model coefficient to be specified or determined dynamically. In contrast to C S , C W L is dimensional in nature and is proportional to the energy dissipation rate () as follows: C W L ∼ 1/3 . In simulations of the buoyancy-driven Rayleigh–Bénard convection [11], a dynamic version of the Wong–Lilly SGS model to some extent outperformed the dynamic Smagorinsky model. In a more recent study, Chow et al. [12] utilized a dynamic version of this SGS model in conjunction with the approximate deconvolution model (ADM) for resolvable subfilterscale (RSFS) components in order to simulate neutral ABL flow. To account for the smaller underresolved eddies in the surface layer, they used a near-wall stress model in addition to the dynamic Wong–Lilly SGS and ADM-RSFS models. The dynamic Wong–Lilly SGS model is computationally less expensive in comparison to the dynamic Smagorinsky SGS model. The combination of lesser assumptions and cheaper computational cost certainly make the Wong–Lilly model an attractive SGS base model for LES. Therefore it is of interest to explore if the Wong–Lilly SGS model or its variants are capable of simulating different flow regimes of the ABL. It is generally agreed upon that in comparison to buoyancy-driven flows, large-eddy simulations of shear-driven boundary layer flows are far more challenging. Thus, in the present study, we focus on neutrally buoyant shear-driven ABL flow. We comprehensively compare several scale-invariant and scale-dependent variations (see Appendix for derivation) of the Wong–Lilly SGS model with their Smagorinsky SGS model-based counterparts (see Table 1). Non-dynamic (constant SGS coefficients) Smagorinsky SGS models (SMAG_p17 and SMAG_p24 in Table 1) with a wall damping function have also been utilized as baseline simulations in order to establish the superiority of the dynamic modeling approach.

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The structure of this paper is as follows. In Sect. 2, we briefly provide the technical details of a case study. Extensive comparisons (in terms of the similarity theory, spectra, and flow visualizations) among several SGS models are performed in Sect. 3. Finally, concluding remarks are made in Sect. 4.

2 Description of simulations In this work, we perform large-eddy simulations of a turbulent Ekman layer (i.e., pure shear flow with a neutrally stratified environment in a rotating system) utilizing several non-dynamic, scale-invariant, and scale-dependent dynamic versions of the Smagorinsky and Wong–Lilly SGS models (see Table 1). All these simulations are identical in terms of initial conditions, forcings, and numerical specifications (e.g., grid spacing). In dynamic −1 scale-invariant and scale-dependent runs, the SGS coefficients (C S2 , C S2 PrSGS , C W L , and −1 C W L PrSGS ) are either obtained by (non-weighted) averaging locally on the horizontal plane with a stencil of 3 × 3 grid points or by planar averaging over the entire horizontal plane. The non-dynamic simulations SMAG_p17 and SMAG_p24 use C S = 0.17, and C S = 0.24, respectively, in conjunction with PrSGS = 0.7. The selection of these C S values is entirely based on past literature usage. In a pioneering study, Lilly [13] theoretically derived that for homogeneous, isotropic turbulence C S is approximately equal to 0.17 (assuming spectral cutoff filtering operation). In the neutral boundary layer intercomparison study [14], Mason’s code used C S = 0.17. In the same study, the derived C S coefficient from Moeng’s simulation equaled to 0.24 [14]. The non-dynamic runs also involve wall-damping function of the form [4,15]: 1 1 1 = 2 + λ2 λ◦ {κ(z + z ◦ )}2

(5)

where κ (= 0.4) is the von Karman constant, z ◦ is surface roughness length, z is height above gound, and λ is the SGS mixing length. λ◦ = C S f is the asymptotic SGS mixing length far from the wall. We have used a modified version of the LES code described in [6]. The salient features of this code are as follows: (i) it solves the filtered incompressible Navier–Stokes equations written in rotational form; (ii) derivatives in the horizontal directions are computed using the Fourier Collocation method, while vertical derivatives are approximated with second-order central differences; (iii) dealiasing of the non-linear terms in Fourier space is done using the 3/2 rule; (iv) explicit second-order Adams-Bashforth time advancement scheme is used; (v) spectral cutoff filtering is used in the horizontal directions (no explicit filtering is performed in the vertical direction); (vi) only Coriolis terms involving horizontal wind are considered; (vii) forcing is imposed by geostrophic wind; and (viii) staggered vertical grid is used. The selected case study is similar to that of the LES intercomparison study by Andrén et al. [14]. The simulated boundary layer is driven by an imposed geostrophic wind of Ug , Vg = (10, 0) ms−1 . The Coriolis parameter is equal to f c = 10−4 s−1 , corresponding to latitude 45◦ N. The computational domain size is: L x = L y = 4,000 m and L z = 1,500 m. This domain is divided into N x × N y × Nz = 40 × 40 × 40 nodes (i.e., x = y = 100 m, and z = 38.5 m). The motivation behind the selection of this coarse grid-resolution is two-fold. Primarily it allows us to perform a direct comparison with the results from [14], which used almost the same grid-resolutions. More importantly, coarse grid-resolution enables us to identify the strengths and/or weaknesses of different SGS models, as well as, to underscore their impacts on large-eddy simulations. The simulations are run for a period

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of 10 × f c−1 (i.e., 100,000 s), with time steps of 1.6–2 s. The last 3 × f c−1 interval is used to compute statistics. A passive scalar is introduced in the flows by imposing a constant flux (wc0 ) of 10−3 kg m−2 s−1 at the surface. The lower boundary condition is based on the Monin-Obukhov similarity theory with a surface roughness length of z ◦ = 0.1 m. 3 Results and discussions In this section, we report the results of the 10 diverse SGS models-based simulations and compare them with results from the intercomparison study [14], wherever possible. This particular case (without the inclusion of passive scalars) was also simulated by Kosović [16] using a nonlinear SGS model, and recently by Chow et al. [12], who utilized a sophisticated hybrid RSFS/SGS model. Our simulations show that all the dynamic SGS models perform very well, and the results are comparable to the past studies. Temporal evolution of the surface friction velocity (u ∗ ) is very similar in all the simulations (not shown). The average values of u ∗ during the last 3 × f c−1 interval are approximately in the range of 0.435–0.460 m s−1 . The corresponding values found in [14] are: 0.425 ms−1 (Moeng), 0.448 ms−1 (Mason—backscatter), 0.402 ms−1 (Mason—nonbackscatter), 0.402 ms−1 (Nieuwstadt), and 0.425 ms−1 (Schumann). In Fig. 1, we present the nonstationary parameters Cu and Cv (see [14] for definitions). Under steady state conditions, these parameters should approach unity. Although none of the past [14,12] and present simulations are quite close to steady state conditions, they are more or less in phase with each other. All these simulations clearly portray the inertial oscillation of period 2π/ f c , as anticipated. Accurately simulating the non-dimensional velocity gradient (φ M ), and the scalar gradient (φC ) in the neutrally stratified surface layer has proven to be a very challenging task for many atmospheric LES models. It is well known that the traditional Smagorinsky model is overdissipative in the near-surface region and gives rise to excessive mean gradients in velocity and scalar fields (cf. [14]). Fortunately, state-of-the-art LES–SGS modeling approaches of Mason and Thomson [17], Sullivan et al. [18], Kosović [16], Porté-Agel et al. [4], PortéAgel [5], Esau [19], Chow et al. [12], Bou-zeid et al. [8], and Stoll and Porté-Agel [9] offer major improvements over traditional Smagorinsky-type SGS models, and reproduce the non-dimensional gradients reasonably well. In the framework of Monin-Obukhov similarity theory, the non-dimensional velocity gradient (φ M ) should be equal to one (the dotted line in Fig. 2-left). However, in the literature there is no consensus on the “true” magnitude of the non-dimensional scalar gradient (φC ). Businger et al. [20], based on the Kansas field experiment, proposed a value of 0.74. Recent field observations, however, suggest values close to 0.9 (for a review, see [21]). From the present coarse-resolution simulations, it is difficult to favor either of these values. To facilitate quantitative comparison, in Table 2 the mean absolute errors in simulated surface-layer nondimensional velocity and scalar gradients are reported. The errors are computed as follows: (φ M )err

1 = Ni

z≤0.15z i

|1 − φ M (z)|

(6)

|0.9 − φC (z)|

(7)

z>0

and (φC )err =

1 Ni

z≤0.15z i z>0

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3

SMAG_p17 SMAG_p24

1.8

SMAG_p17 SMAG_p24 2.5

1.6 1.4

2

v

1

C

C

u

1.2 1.5

0.8 1

0.6 0.4

0.5

0.2 0

0

2

4

6

8

10

12

0

14

0

2

4

6

8

10

12

14

tf

tf c

c

2

3 PAD−SM PAD−WL LAD−SM LAD−WL

1.8 1.6

PAD−SM PAD−WL LAD−SM LAD−WL

2.5

1.4

2

v

1

C

C

u

1.2 1.5

0.8 1

0.6 0.4

0.5

0.2 0

0 0

2

4

6

8

10

12

14

0

2

4

6

tfc

8

10

12

14

tfc 3

2

PASDD−SM PASDD−WL LASDD−SM LASDD−WL

1.8 1.6


2.5

1.4

2

v

1

C

C

u

1.2 1.5

0.8 1

0.6 0.4

0.5

0.2 0

0 0

2

4

6

8

tf

c

10

12

14

0

2

4

6

8

10

12

14

tfc

Fig. 1 Temporal evolution of the non-stationarity parameters Cu (left) and Cv (right). Non-dynamic, scale-invariant dynamic, and scale-dependent dynamic results are shown in top, middle, and bottom panels, respectively.

where z i denotes the inversion height. In the present study, z i is assumed to be equal to L z = 1,500 m. Ni represents the numbers of vertical model levels in the surface layer (0 < z ≤ 0.15z i ). From Fig. 2 and Table 2, it is clear that almost all the dynamic SGS models behave satisfactorily, albeit, the performances of the local averaging-based SGS models are superior. Results from the non-dynamic simulations (i.e., SMAG_p17 and SMAG_p24) convincingly portray the sensitivity of the non-dimensional gradients on the chosen value of C S . In other words, the need for dynamic estimation of C S is clearly justified. We would like to stress

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0.1 SMAG_p17 SMAG_p24

0.09

0.08

0.07

0.07

0.06

0.06

zfc/u*

zfc/u*

0.08

0.05

0.05

0.04

0.04

0.03

0.03

0.02

0.02

0.01

0.01 0 0

0 0

0.5

1

1.5

SMAG_p17 SMAG_p24

0.09

2

0.5


0.08

2


0.09 0.08 0.07

0.06

0.06 *

0.07

zf /u

0.05

c

zfc/u*

1.5

0.1

0.1 0.09

0.05

0.04

0.04

0.03

0.03

0.02

0.02

0.01

0.01 0

0 0

0.5

1

1.5

0

2

0.5

Φ

1

1.5

2

ΦC

M

0.1

0.1 PASDD−SM PASDD−WL LASDD−SM LASDD−WL

0.09 0.08


0.09 0.08 0.07

0.06

0.06 *

0.07

zf /u

0.05

c

zfc/u*

1

ΦC

ΦM

0.05

0.04

0.04

0.03

0.03

0.02

0.02

0.01

0.01 0

0 0

0.5

1

Φ

M

1.5

2

0

0.5

1

Φ

1.5

2

C

Fig. 2 Simulated non-dimensional velocity (left) and scalar (right) gradients. The dashed lines correspond to the values of 1 (left) and 0.9 (right). These values are expected to hold in the surface layer under neutral conditions according to the similarity theory. Non-dynamic, scale-invariant dynamic, and scale-dependent dynamic results are shown in top, middle, and bottom panels, respectively.

that all the dynamic SGS modeling approaches employed here do not require any additional stochastic term or supplementary near-wall stress models for reliable performance in an LES. However, all the present simulations (including the non-dynamic ones) utilize the more accurate treatment of the advection term near ground as described in the Appendix of [4]. This treatment marginally reduces the error incurred by the finite difference approximation of the vertical derivatives. The normalized resolved velocity (σu2 /u 2∗ , σv2 /u 2∗ , σw2 /u 2∗ ) and scalar variances (σc2 /c∗2 ) are shown in Figs. 3–5. Here, c∗ is the surface scalar scale (= −wc0 /u ∗ ). Overall, the

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Environ Fluid Mech Table 2 Mean absolute errors in simulated non-dimensional velocity (φ M ) and scalar gradients (φC ) in the surface layer

D

(φ M )err

(φC )err

SMAG_p17 SMAG_p24 PAD-SM PAD-WL LAD-SM LAD-WL PASDD-SM PASDD-WL LASDD-SM LASDD-WL

0.12 0.19 0.17 0.13 0.09 0.09 0.13 0.17 0.12 0.11

0.16 0.17 0.15 0.14 0.15 0.13 0.10 0.22 0.10 0.14

SMAG_p17 SMAG_p24

SMAG_p17 SMAG_p24

0.3 0.25

0.2

0.2

*

0.25

c

zf /u

zfc/u*

0.3

0.15

0.15

0.1

0.1

0.05

0.05 0

0 0

1

2

3

4

5

6

7

8

0

1

2

2 2 σu/u*

SMAG_p17 SMAG_p24

0.25

0.2

0.2

zfc/u*

zf /u

4

5

0.15

SMAG_p17 SMAG_p24

0.3

0.25

c

*

3

v

0.3

0.15

0.1

0.1

0.05

0.05

0 0

σ2/u2*

0 0.5

1

1.5 2 2 σ /u w *

2

2.5

3

0

1

2

3

4 2 c

5

6

7

8

2 *

σ /c

Fig. 3 Simulated normalized longitudinal (top-left), transverse (top-right), vertical (bottom-left) resolved velocity variances. Simulated normalized resolved scalar variances are shown in the bottom-right plot. These results are from the non-dynamic runs.

consensus among most of the runs are very good. The exceptions being the SMAG_p24 and PASDD-WL runs. In the case of the σu2 /u 2∗ profile, these runs produce unusually large elevated peaks. This could be attributed to excessive shear in the simulated mean flow near the surface, as confirmed from Table 2 and Fig. 2. In [14], it was found that the consensus among different SGS models is poorer in the case of passive scalar in comparison to the momentum case. The disagreements between different SGS models could be partially attributed to different a priori prescriptions for the SGS Prandtl (PrSGS ) number, and underscore the need for the determination of PrSGS in a

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Environ Fluid Mech PAD−SM PAD−WL LAD−SM LAD−WL

0.3

0.25

0.2

0.2

zfc/u*

0.25

zfc/u*


0.3

0.15

0.15

0.1

0.1

0.05

0.05

0

0 0

1

2

3

4

5

6

7

0

8

1

2


0.25

0.25

0.2

0.2

0.15

0.1

0.05

0.05

0.5

1

1.5 2

2 2

σw/u*

5

2.5


0.15

0.1

0

4

0.3

zfc/u*

zfc/u*

0.3

0

3

2 2 σ /u v *

2 2 σ /u u *

3

0 0

1

2

3

4 2

5

6

7

8

2

σc /c*

Fig. 4 Simulated normalized longitudinal (top-left), transverse (top-right), vertical (bottom-left) resolved velocity variances. Simulated normalized resolved scalar variances are shown in the bottom-right plot. These results are from the scale-invariant runs.

self-consistent manner, as is done in the present dynamic runs. One must also acknowledge the facts that the passive scalars exhibit complex spatio-temporal structure, and the statistical and dynamical properties of passive scalars are remarkably different from the underlying velocity fields [22,23]. We point out that the individual plots in Figs. 3–5 represent only the normalized resolved variances. The SGS contributions to the total variances have not been added here. In the employed SGS modeling approaches, one does not solve additional prognostic equations for the SGS turbulence kinetic energy (TKE) and the SGS scalar variances. However, the SGS variances can be roughly diagnosed using the approach of Mason [24]. Notwithstanding the fact that the SGS estimation approach is prone to substantial errors near the surface, we report in Fig. 6 the resolved, SGS (estimated), and total (resolved + SGS) variances from the LASDD-SM run. In neutrally stratified ABL flows, the observed peak normalized total velocity variances occur near the surface and are of the magnitude: σu2 /u 2∗ ∼ 5 − 7, σv2 /u 2∗ ∼ 3 − 4, and σw2 /u 2∗ ∼ 1 − 2 [25]. The corresponding values found in our simulation (Fig. 6) approximately fall in these ranges. The simulated results also concur with the outer boundary layer observations. For example, the KONTUR data [26] give σu2 /u 2∗ ∼ σv2 /u 2∗ ∼ 1 and σw2 /u 2∗ ∼ 0.5 at z = 0.75z i . The mean profiles of vertical momentum and scalar fluxes from the LASDD-SM and LASDD-WL simulations averaged over the last 3 × f c−1 interval are given in Fig. 7. As would be anticipated, near the ground the SGS contribution is much larger than its resolved

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Environ Fluid Mech PASDD−SM PASDD−WL LASDD−SM LASDD−WL


0.3 0.25

0.2

0.2

*

0.25

c

zf /u

zfc/u*

0.3

0.15

0.15

0.1

0.1

0.05

0.05

0

0 0

1

2

3

4

5

6

7

8

0

1

2


0.3

4

5


0.3 0.25

0.2

0.2

*

0.25

c

zf /u

zfc/u*

3

σ2v /u2*

2 2 σ /u u *

0.15

0.15

0.1

0.1

0.05

0.05

0

0 0

0.5

1

1.5

σ2w/u2*

2

2.5

3

0

1

2

3

4

5

6

7

8

σ2c /c2*

Fig. 5 Simulated normalized longitudinal (top-left), transverse (top-right), vertical (bottom-left) resolved velocity variances. Simulated normalized resolved scalar variances are shown in the bottom-right plot. These results are from the scale-dependent runs.

counterpart. The total vertical momentum and scalar fluxes are qualitatively very similar for all other runs (not shown). The average values of the SGS coefficients C S and PrSGS , and the scale-dependent parameters β and βc obtained by the dynamic model runs are shown as functions of normalized height in Figs. 8 and 9. The Smagorinsky coefficient (C S ) smoothly increases with height and becomes scale-invariant outside the surface layer (Fig. 8-left). The Wong–Lilly coefficient C W L behaves very similarly (not shown). The SGS Prandtl number (PrSGS ) decreases from its surface layer values and becomes more or less height independent. The higher value of PrSGS signifies that the relative efficiency of SGS momentum transfer (with respect to SGS scalar transport) increases near the surface. This behavior is in agreement with the a priori field study by [27]. Near the surface the scale-dependent parameters are much smaller than 1 and become scale-invariant in the interior of the flow (Fig. 9). The one-dimensional longitudinal velocity and passive scalar spectra are computed at heights z = 0.1z i , and z = 0.5z i , and presented in Figs. 10–12. The spectra highlight the most important difference between the SGS models. Traditional non-dynamic SGS models seem to be over-dissipative as indicated by steeper spectral slopes at higher wavenumbers (cf. [14] and Fig. 10). On the other hand, in the case of the dynamic SGS models, the longitudinal velocity and scalar spectra clearly show extended inertial range (approximately −5/3 k1 scaling) at z = 0.5z i . Near the surface (z = 0.1z i ), the longitudinal velocity spectra show the anticipated production range (approximately k1−1 ), as well as a short inertial range.

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Environ Fluid Mech LASDD−SM (Resolved+SGS) LASDD−SM (Resolved) LASDD−SM (SGS)

LASDD−SM (Resolved+SGS) LASDD−SM (Resolved) LASDD−SM (SGS)

0.3 0.25

0.2

0.2

*

0.25

c

zf /u

zfc/u*

0.3

0.15

0.15

0.1

0.1

0.05

0.05 0

0 0

1

2

3

4 2 2

5

6

7

8

0

1

2


2

3

4

5


0.3 0.25

0.2

0.2

zfc/u*

0.25

c

zf /u

*

0.3

2

σv /u*

σu /u *

0.15

0.15 0.1

0.1

0.05

0.05 0

0 0

0.5

1

1.5 2 2

σw/u*

2

2.5

3

0

1

2

3

4 2 2

5

6

7

8

σc /c*

Fig. 6 Simulated normalized longitudinal (top-left), transverse (top-right), vertical (bottom-left) velocity variances. Simulated normalized resolved scalar variances are shown in the bottom-right plot. These results are from the LASDD-SM run.

Recent research suggests that the production range is (likely) related to elongated streaky velocity structures (see below). From the spectral scaling perspective, we might conclude that all the dynamic models perform quite successfully. However, there are subtle differences between the performance of local and planar averaging-based dynamic models, as well as, between Smagorinsky and Wong–Lilly-based dynamic SGS models (discussed below). To facilitate quantitative comparison, in Table 3 estimated velocity and passive scalar spectral slopes are reported. It is quite difficult to manually yet objectively select scaling ranges from all the simulations. So, we have used an automated inertial scaling range of 2/z ≤ k1 < k N yquist . Overall errors in estimated spectral exponents are computed as follows: 5 5 5 5 z=0.1z i z=0.5z i z=0.1z i z=0.5z i αerr = − − αu + − 3 − αu + − 3 − αc + − 3 − αc (8) 3 Here − 53 − α z refers to the absolution deviation of the simulated inertial range scaling exponent from the theoretical value at height z. Based on Table 3 and Figs. 11 and 12, we can claim that overall the LASDD-SM and LAD-WL models outperform other dynamic SGS models. We note that the plane averaged [4] and the Lagrangian-averaged [8,9] scaledependent dynamic SGS models also reproduced the characteristics of the one-dimensional longitudinal velocity spectra remarkably well. However, near the surface, the passive scalar spectra predicted by these SGS models showed unphysical pile up of scalar variances [5,9]. This was possibly due to small dynamically determined eddy-diffusion coefficients near the

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Environ Fluid Mech 0.35

0.35 LASDD−SM (Resolved+SGS) LASDD−WL (Resolved+SGS) LASDD−SM (SGS) LASDD−WL (SGS)

0.25

0.2

0.2

zf /u c *

0.25

0.15

0.15

0.1

0.1

0.05

0.05

0 −1

LASDD−SM (Resolved+SGS) LASDD−WL (Resolved+SGS) LASDD−SM (SGS) LASDD−WL (SGS)

0.3

c

zf /u

*

0.3

0 −0.8

−0.6

−0.4

/u2

−0.2

0

0.2

−0.6 −0.5 −0.4 −0.3 −0.2 −0.1 /u2 *

*

0

0.1

0.2

0.3

0.35 LASDD−SM (Resolved+SGS) LASDD−WL (Resolved+SGS) LASDD−SM (SGS) LASDD−WL (SGS)

0.3

zfc/u*

0.25 0.2 0.15 0.1 0.05 0 0

0.2

0.4

0.6

0.8

1

−/u c

* *

Fig. 7 Simulated normalized vertical fluxes of x-component momentum (top-left), y-component momentum (top-right), and scalar (bottom). These results are from the LASDD-SM and LASDD-WL runs. PAD−SM LAD−SM PASDD−SM LASDD−SM

0.3

0.25 *

0.2

0.2

c

c

zf /u

*

0.25

zf /u

PAD−SM LAD−SM PASDD−SM LASDD−SM

0.3

0.15

0.15

0.1

0.1

0.05

0.05

0

0 0

0.05

0.1

CS

0.15

0.2

0

0.5

1

Pr

1.5

SGS

Fig. 8 Vertical profiles of the SGS coefficients C S and PrSGS dynamically obtained by the Smagorinsky model-based scale-invariant and scale-dependent runs

surface [5,9]. In the present study we encounter this issue in the case of PAD-SM and PASDDSM runs (Figs. 11, 12, and Table 3). From these figures and table, we can draw another conclusion: the Wong–Lilly-based SGS models are found to be slightly over-dissipative in comparison to their Smagorinksy-based counterparts, especially at z = 0.5z i . A few previous LES studies have reported the existence of elongated streaky structures in the neutral surface layers [16,18,28–31]. Evidences of these structures in various laboratory experiments and field observations are undeniable (for example, see Hutchins and Marusic

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0.3 0.25

0.2

0.2

c

zf /u

*

0.25

c

zf /u

*

0.3

0.15

0.15

0.1

0.1

0.05

0.05 0

0 0

0.5

1

1.5

0

2

0.5

1

1.5

2

β

β

c

Fig. 9 Vertical profiles of the scale-dependent parameters β and βc dynamically obtained by the scale-dependent model runs 0

0

10

10

−1

10

−1

10

−2

1

E (k )

10

u

Eu(k1)

10 −2

−3

10 −3

10

−4

10

−4

SMAG_p17 SMAG_p24 −5/3 −1 −3

10

−2

10

10

k

SMAG_p17 SMAG_p24 −5/3 −3

−1

−2

10

10

10

−1

10

k

1

1 −5

−5

10

10

−6

−6

10

c

1

Ec(k1)

E (k )

10

−7

−7

10

10

SMAG_p17 SMAG_p24 −5/3 −1

−8

10

−3

10

SMAG_p17 SMAG_p24 −5/3

−8

−2

10

k1

−1

10

10

−3

10

−2

10

−1

10

k1

Fig. 10 Spectra of longitudinal velocity (top), and passive scalar (bottom). The spectra are plotted for z = −5/3 0.1z i (left), and z = 0.5z i (right) levels. The dashed and dotted lines depict the inertial range (k1 ) and production range (k1−1 ) scalings, respectively. These results are from the non-dynamic runs

[32] and the references therein). The link between experimentally observed long production range (k −1 scaling) in the streamwise spectra of the longitudinal velocity and the elongated streaky structures has recently been discussed in depth by Carlotti [31]. Moreover, strong correlations between these streaky structures and large negative momentum flux were earlier reported by [29]. From Figs. 13-top and 14, it is clear that all the SGS models in the present study show streaky structures, roughly parallel to the mean wind direction, in the

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0

10

10

−1

10

−1

10

−2

1

E (k )

u

u

1

E (k )

10 −2

10

−3

10 PAD−SM PAD−WL LAD−SM LAD−WL −5/3 −1

−3

10

−4

10

−3

−2

10

PAD−SM PAD−WL LAD−SM LAD−WL −5/3

−4

10

10

k

−1

−3

10

−2

10

10

k

1

−5

−1

10

1

−5

10

10

−6

−6

1 c

c

1

E (k )

10

E (k )

10

−7

10

−8

10

−7

10

PAD−SM PAD−WL LAD−SM LAD−WL −5/3 −1 −3

10

PAD−SM PAD−WL LAD−SM LAD−WL −5/3

−8

−2

10

k

1

−1

10

10

−3

10

−2

10

k

−1

10

1

Fig. 11 Spectra of longitudinal velocity (top), and passive scalar (bottom). The spectra are plotted for z = −5/3 0.1z i (left), and z = 0.5z i (right) levels. The dashed and dotted lines depict the inertial range (k1 ) and production range (k1−1 ) scalings, respectively. These results are from the scale-invariant runs.

surface layer (at z = 0.1z i ). However, significant morphologic differences are noticeable. For example, the SMAG_p17 run produces very long streaky structures and inadequate smallscale structures (Fig. 14-top left). This can be directly associated with the over-dissipative nature of the SMAG_p17 SGS model, as discussed before. In other words, the existence and morphological characterisitcs of coherent structures in NBL flows are strongly dependent on SGS parameterizations, especially for coarse-resolution simulations. A few previous studies somewhat support this inference. For instance, the non-linear SGS model [16], and the modified Smagorinsky SGS model [30] barely produced any elongated streaky structures. Streamwise and spanwise two-point correlations of the streamwise velocity fluctuations Ruu (Fig. 15) can shed more light on the morphological characteristics of the streaky structures. Recently, Hutchins and Marusic [32] found that for a large range of Reynolds numbers, the correlation profiles collapse in a quasi-universal fashion. Moreover, the streamwise length of the correlations shortens significantly going from z/z i 0.06 to z/z i = 0.50 [32]. On the other hand, the spanwise width of the correlation region increases with height [32]. Interestingly, some (not all) of the present simulations do conform with these observational facts (see Fig. 15). In Table 4, we also report the correlation lengths (L 0.1 ) associated with Ruu = 0.1. Clearly, SMAG_p17, LAD-WL, and LASDD-WL runs stand out against the observations z=0.5z i z=0.1z i by [32], as for these runs L 0.1 > L 0.1 . This could be partially associated with the over-dissipative nature of the Wong–Lilly-based SGS models in the mid-ABL flows.

123


0

10

10

−1

10

−1

10

−2

1

E (k )

u

u

1

E (k )

10 −2

10

−3

10 PASDD−SM PASDD−WL LASDD−SM LASDD−WL −5/3 −1

−3

10

−4

10

−3

−2

10

PASDD−SM PASDD−WL LASDD−SM LASDD−WL −5/3

−4

10

−1

10

k

−3

−2

10

10

−1

10

10

k

1

1

−5

−5

10

10

−6

−6

1 c

c

1

E (k )

10

E (k )

10

−7

10

−7

10

PASDD−SM PASDD−WL LASDD−SM LASDD−WL −5/3 −1

−8

−8

10

10 −3

10

PASDD−SM PASDD−WL LASDD−SM LASDD−WL −5/3

−2

−1

10

−3

−2

10

10

−1

10

10

k1

k1

Fig. 12 Spectra of longitudinal velocity (top), and passive scalar (bottom). The spectra are plotted for z = −5/3 0.1z i (left), and z = 0.5z i (right) levels. The dashed and dotted lines depict the inertial range (k1 ) and production range (k1−1 ) scalings, respectively. These results are from the scale-dependent runs.

Table 3 Inertial range spectral exponents

z=0.1z i

z=0.5z i

z=0.1z i

z=0.5z i

ID

αu

αu

αc

αc

αerr


−3.53 −5.80 −1.61 −1.70 −1.93 −1.88 −1.97 −2.18 −2.09 −2.30

−2.25 −3.02 −1.63 −1.86 −1.92 −1.97 −1.60 −2.20 −1.86 −2.01

−2.39 −4.16 −0.72 −0.85 −1.29 −1.37 −1.01 −1.52 −1.64 −1.94

−1.48 −2.30 −1.34 −1.58 −1.52 −1.70 −1.44 −1.86 −1.57 −1.82

3.36 8.61 1.37 1.13 1.04 0.85 1.25 1.39 0.74 1.40

4 Concluding remarks Several non-dynamic and dynamic SGS closures based on the Smagorinsky and Wong–Lilly models have been used to simulate a neutral ABL case. Although the theoretical foundations of the dynamic SGS models are fundamentally different, results presented in Figs. 1–15 illustrate strong congruence between their results, and with firmly established results (i.e., the Monin-Obukhov similarity theory and the inertial range scaling of spectra). The scaledependence framework, in general, improves the near-surface statistics from the Smagorinsky

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4000

ms−1 10

4000

ms−1 10

3500

9.5

3500

9.5

3000

9

3000

9

2500

8.5

2500

8.5

2000

8

2000

8

1500

7.5

1000

7

y (m)

y (m)

Environ Fluid Mech

7

1000

6.5

500

7.5

1500

6.5

500

6

6 500

1000

1500

2000

2500

3000

3500

4000

500

1000

1500

2000

ms−1 12

4000

3000

3500

4000

ms−1 12

4000

3500

11.5

3500

11.5

3000

11

3000

11

2500

10.5

2500

10.5

2000

10

2000

10

1500

9.5

1500

9.5

1000

9

1000

9

8.5

500

y (m)

y (m)

2500

x (m)

x (m)

8.5

500

8 500

1000

1500

2000

2500

3000

3500

4000

8 500

1000

1500

2000

2500

3000

3500

4000

x (m)

x (m)

Fig. 13 Visualization of longitudinal velocity fields simulated by LASDD-SM (left), and LASDD-WL (right) SGS models. The horizontal cross-sections are taken at z = 0.1z i (top), and z = 0.5z i (bottom). Table 4 Correlation length (in m)

z=0.1z i

z=0.5z i

ID

L 0.1

L 0.1


1,398 >2,000 1,558 1,762 >2,000 1,524 1,936 >2,000 972 1,172

1,409 1,487 1,187 1,507 1,587 1,924 1,591 >2,000 921 1,388

model-based simulations. We also note that the local averaging-based dynamic SGS models perform significantly better than their planar averaging-based counterparts. The normalized variances computed in our simulations also closely follow the ones calculated from field measurements. The major noticeable and consistent difference between the results are shown in Figs. 11, 12, 15 and Table 3: several Wong–Lilly model-based dynamic SGS models appear to be over-dissipative at the higher wavenumbers, in comparison to the corresponding Smagorinsky-based SGS models. In Figs. 13 and 14, we see that all the SGS models predict elongated streaky structures in the near-wall region (z = 0.1z i ). These coherent structures are no longer evident at higher locations in the domain for some runs, as would be anticipated. At

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Environ Fluid Mech

4000

10

4000

ms−1 10

3500

9.5

3500

9.5

3000

9

3000

9

2500

8.5

2500

8.5

2000

8

2000

8

1500

7.5

1500

7.5

1000

7

1000

7

y (m)

y (m)

ms−1

6.5

500 1000

1500

2000 2500 x (m)

3000

3500

500

4000

1000

1500

2000

2500

3000

3500

4000

6

x (m)

4000

ms−1 10

4000

ms−1 10

3500

9.5

3500

9.5

3000

9

3000

9

2500

8.5

2500

8.5

2000

8

2000

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1500

7.5

1500

7.5

1000

7

1000

7

6.5

500

y (m)

y (m)

500

6.5

500

6

6.5

500

6 500

1000

1500

2000

x (m)

2500

3000

3500

4000

6 500

1000

1500

2000

2500

3000

3500

4000

x (m)

Fig. 14 Visualization of longitudinal velocity fields simulated by SMAG_p17 (top-left), PAD-SM (top-right), PASDD-SM (bottom-left), and LAD-SM (bottom-right) SGS models. The horizontal cross-sections are taken at z = 0.1z i .

z = 0.5z i , several Wong–Lilly-based dynamic models (including LAD-WL) create spurious enhanced two-point correlations of the streamwise velocity fluctuations. Quantitatively, based on the overall error statistics [(φ M )err , (φC )err , and αerr ], the LASDD-SM and LAD-WL SGS models are found to be the most competitive. Unfortunately, the two-point correlation results from the LAD-WL run do not conform with the laboratory experimental results of [32]. The Wong–Lilly SGS base model requires fewer assumptions and comes at slightly less computational cost in comparison to the commonly used Smagorinsky SGS base model. Unfortunately, these advantages seem to be offset by its over-dissipative tendency at higher wavenumbers. Some inherent assumptions of the Smagorinsky base model can also be eliminated by solving a prognostic equation for the TKE. However, when using this TKE SGS approach, the SGS model coefficients are often tuned for different ABL flow conditions [18,33]. An alternative approach would be to formulate a dynamic version of the TKE SGS model, which will also account for energy backscatter (e.g., [34,35]). We are currently working on amending these dynamic TKE SGS approaches to better represent the physics of atmospheric boundary layer flows. Acknowledgements This work was partially funded by the National Institute of Standards and Technology (3H5003), the National Science Foundation (ANT-0538453) and the Texas Advanced Research Program (003644-0003-2006) grants. All the computational resources were kindly provided by the High Performance Computing Center at Texas Tech University.

123


1 LASDD−SM LASDD−WL SMAG_p17 SMAG_p24

0.9

0.8 0.7

0.6

0.6

uu

0.7

0.5

R

R

uu

0.8

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0 0

500

1000

∆x (m)

1500

0

2000

LASDD−SM LASDD−WL SMAG_p17 SMAG_p24

0.8

500

1000

∆x (m)

1500

2000


0.8

0.6

uu

0.6

0.4

R

Ruu


0.9

0.4

0.2

0.2

0

0

−0.2

−0.2 0

500

1000

∆y (m)

1500

2000

0

500

1000

1500

2000

∆y (m)

Fig. 15 Streamwise (top) and spanwise (bottom) two-point correlations of the streamwise velocity fluctuations Ruu . The computations are performed at z = 0.1z i (left), and z = 0.5z i (right).

Appendix In this appendix, we formulate a locally averaged scale-dependent dynamic version of Eq. 3 (named LASDD-WL). The SGS stress tensor (τi j ) at the filter scale ( f ) is defined as: τi j = u i uj . In a seminal work, Germano et al. [1] proposed to invoke an additional iu j − u explicit test filter of width α f in order to dynamically compute the SGS coefficients. Consecutive filtering at scales f and at α f leads to a SGS turbulent stress tensor (Ti j ) at the test filter scale α f : Ti j = u ui u j, i uj −

(A1)

where an overline (· · · ) denotes filtering at a scale of α f . From the definitions of τi j and Ti j an algebraic relation can be formed, known in the literature as the Germano identity: Li j = ui uj − ui u j = Ti j − τi j .

(A2)

This identity is then effectively used to dynamically obtain unknown SGS model coefficients. In the case of the Wong–Lilly model (Eq. 3), this identity yields: 1 L kk δi j = (C W L ) f Mi j , (A3) 3 (C W L )α 4/3 where Mi j =2 f 1−α 4/3 (C W L ) f S i j . If one assumes scale invariance, i.e., (C W L )α f = Li j −

f

(C W L ) f , then the unknown coefficient (C W L ) f can be easily determined following the

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Environ Fluid Mech

error minimization approach of Lilly [36]: L i j Mi j . Mi j Mi j

(C W L ) f =

(A4)

In the context of the present study, the angular brackets · · · either denote localized spatial averaging on horizontal planes with a stencil of three by three grid points [6,7] or planar averaging over the entire horizontal plane. Recent studies have shown that the assumption of scale invariance is seriously flawed for sheared and stratified boundary layer flows [4–9]. In other words, the ratio of (C W L )α f to (C W L ) f should not be assumed equal to one for most of these ABL flow scenarios. Rather, this scale-dependence ratio should be determined dynamically. In order to implement the scale-dependent dynamic procedure, one needs to employ a second test filtering operation · · )]. Invoking the Germano identity for the second time at a scale of α 2 f [denoted by (· leads to: Qi j −

1 Q kk δi j = (C W L ) f Ni j , 3

(A5)

where i uj − u i u j Q i j = u and

Ni j =

4/3 2 f

1−α

8/3

(C W L )α 2 f

(C W L ) f

S ij.

This results in: (C W L ) f =

Q i j Ni j . Ni j Ni j

(A6)

Following [4], the following scale-dependence assumption can be made: β=

(C W L )α f (C W L ) f

=

(C W L )α 2 f (C W L )α f

.

(A7)

This is a much weaker assumption than the scale-invariance modeling assumption of β = 1. Now, from Eqs. A4 and A6, using Eq. A7, one solves for the unknown parameter β, which in turn is used to compute the Wong–Lilly SGS model coefficient, (C W L ) f , utilizing Eq. A4. Solving for β essentially involves finding the roots of a fifth-order polynomial [4]: A0 + A1 β + A2 β 2 + A3 β 3 + A4 β 4 + A5 β 5 = 0

(A8)

where A0 = a1 a3 − a6 a8 , A1 = a1 a4 − a7 a8 , A2 = a2 a3 + a1 a5 − a6 a9 , A3 = a2 a4 − a7 a9 , A4 = a2 a5 − a6 a10 , and A5 = −a7 a10 . In the case of Wong–Lilly SGS base model, we , a = S 2 , a = −2α 4/3 S 2 , a = 8/3 Q S derive: a1 = Q i j S i j , a2 = −α ij ij 3 ij 4 ij 5 4/3 L S 8/3 S α 8/3 S i j , a6 = L i j S i j , a7 = −α ij i j , a8 = S i j , a9 = −2α i j , and 2 16/3 Si j . Please note that the coefficients (a1 to a10 ) involve significantly fewer a10 = α tensor terms in comparison to the ones derived by Porté-Agel et al. [4] using the Smagorinsky SGS base model. Lesser number of calculations (specifically the tensor multiplications) undoubtedly lead to cheaper computational costs. 2

2

2

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Environ Fluid Mech

Scale-dependent formulation for scalars can be derived in a similar manner [5]. In the dynamic or scale-dependent dynamic modeling approaches, typically the lumped SGS co−1 efficient (C W L PrSGS ) is determined in a self-consistent manner. This procedure not only eliminates the need for any ad hoc assumption about the SGS Prandtl number (PrSGS ), it also completely decouples the SGS scalar flux estimation from SGS stress computation. In the scale-dependent approach [5], one further defines a scale-dependent parameter for scalars (βc ), analogs to Eq. A7. For the Wong–Lilly SGS base model, it could be written as: −1 −1 C W L PrSGS C W L PrSGS α α2 f = f. (A9) βc = −1 −1 C W L PrSGS C W L PrSGS f

α f

As before, βc could be determined by solving the fifth-order polynomial: A0 + A1 βc + A2 βc2 + A3 βc3 + A4 βc4 + A5 βc5 = 0

(A10)

where A0 = a1 a3 − a6 a8 , A1 = a1 a4 − a7 a8 , A2 = a2 a3 + a1 a5 − a6 a9 , A3 = a2 a4 − a7 a9 , A4 = a2 a5 − a6 a10 , and A5 = −a7 a10 . For the Wong–Lilly SGS base model for scalars, 2 2 ∂ c ∂ c we get: a = K , a = −α 8/3 K , a = ∂c , a = −2α 4/3 ∂c , a = 1

2

i ∂ xi

2

i ∂ xi

3

4 2 ∂ c ∂ xi , a9

∂ xi

c ∂ c c 4/3 K ∂ α 8/3 ∂∂ = i ∂ xi , a8 = xi , a6 = K i ∂ xi , a7 = −α 2 ∂ c 16/3 a10 = α c . i c − ui c , and K i = u c − u i i ∂ xi . Here, K i = u

∂ xi

2 ∂ c −2α 8/3 ∂ xi ,

5

and

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