Direct Numerical Simulation of the Turbulent Ekman ... - AMS Journals

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Direct Numerical Simulation of the Turbulent Ekman Layer: Evaluation of Closure Models STUART MARLATT,* SCOTT WAGGY, AND SEDAT BIRINGEN Department of Aerospace Engineering Sciences, University of Colorado, Boulder, Colorado (Manuscript received 31 March 2011, in final form 7 October 2011) ABSTRACT A direct numerical simulation (DNS) at a Reynolds number of 1000 was performed for the neutral atmospheric boundary layer (ABL) using the Ekman layer approximation. The DNS results were used to evaluate several closure approximations that model the turbulent stresses in the Reynolds averaged momentum equations. Two first-order closure equations proposed by O’Brien and by Large, McWilliams, and Doney were tested; both models approximate the eddy diffusivity as a function of height using cubic polynomials. Of these two models, the O’Brien model, which requires data both at the surface layer and at the top of the boundary layer, proved superior. The higher-order k–« model also agreed well with DNS results and more accurately represented the eddy diffusivity in this rotational flow.

1. Introduction The boundary layer formed by pressure gradients in a rotating system is called the Ekman layer and is often used as a simpler model for the atmospheric boundary layer (ABL). The purpose of this research is to perform a direct numerical simulation (DNS) of the Ekman layer and analyze the three-dimensional, time-dependent database to test the applicability of known turbulence closure models in time-averaged equations in order to assess their relative performance. A significant volume of previous theoretical and numerical research has been conducted on this topic. Coleman presents an analysis of surface friction velocity and shear angle scaling for the neutrally stratified flow up to a Reynolds number (Re) of 1000 (Coleman 1999). His results demonstrate that the surface friction velocity behaves according to the laboratory-scale Ekman layer experimental measurements of Caldwell et al. (1972); however, the surface shear angle deviates from the measured values. In another study, Hess and Garratt (2002a) compare several closure models used in predicting the

* Current affiliation: United Launch Alliance, Littleton, Colorado.

Corresponding author address: Sedat Biringen, Aerospace Engineering Sciences, CB 429, University of Colorado, Boulder, CO 80309. E-mail: [email protected] DOI: 10.1175/JAS-D-11-0107.1 Ó 2012 American Meteorological Society

surface shear stress and shear angle. They found that lower-order models performed better than the higherorder closure models over a larger range of Rossby numbers (Ro), partly because of free parameters that allow a certain level of tuning of the model. The follow-up paper by Hess and Garratt (2002b) continues this work by analyzing several high-order closure models and comparing the results with atmospheric data. The results agree with their previous finding that lower-order models fare just as well as (if not better than) higher-order parameterizations. Zilitinkevich and Esau (2002) demonstrate that the classical similarity theory for the dimensionless coefficients ! kUg u* Ro 2 cosa0 A 5 ln Ug u* and

B5

kUg u*

sina0

depends on mN 5 N/j f j, the ratio of the free-flow Brunt– Vaïsa¨la¨ frequency N and the magnitude of the Coriolis parameter j f j. Miyashita et al. (2006) provide an analysis of the neutral Ekman layer up to Re 5 1393. The mixing length model of Blackadar (1962) is compared with DNS results and performs poorly because of the

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FIG. 2. Mean velocity profile in terms of wall units. The solid line represents DNS data; the dashed line is given by Q1 5 z1 for z1 , 10 and Q1 5 (1/k) ln(z1) 1 B for z1 . 10, where k 5 0.41 and B 5 5.2.

FIG. 1. Normalized energy spectrum at zf/u* 5 0.014: kxh (solid), kyh (dashed), and }k25/3 (dot-dashed).

absence of viscosity in the mixing length definition. They propose a modified mixing length that incorporates near-wall viscous effects to better fit DNS data. Taylor and Sarkar (2008) compared results for a DNS and large-eddy simulation (LES) of the neutrally and stably stratified Ekman layer. Their work demonstrates that the LES does not sufficiently account for the smallscale motion responsible for entrainment at the top of the mixed layer. They recommend modifying the subgrid-scale model or increasing the LES resolution such that the Ellison scale is resolved. Spalart et al. (2008) pushed the Reynolds number for the neutrally stratified Ekman DNS to Re 5 2828 in an attempt to quantify the Karman constant and to further the work of Coleman (1999) concerning similarity scaling. However, in a later paper, Spalart et al. (2009) concluded that the maximum Reynolds number they could achieve with meshindependent results was around Re 5 2000. We performed our simulation for Re 5 1000, a factor of 2 lower than the limit of DNS capabilities established by Spalart et al. (2009). This Reynolds number is much less than typical ABL conditions, and turbulence statistics may exhibit some Reynolds number sensitivity. Nevertheless, the spectral distribution of turbulent energy suggests an inertial subrange (Fig. 1), and the mean velocity profile displays a well-developed logarithmic region (Fig. 2). Thus, the results that we present here should be relevant to fully developed turbulent flows. The focus of this paper concerns evaluating the ability of various closure models to accurately predict the

Reynolds stresses for the turbulent Ekman layer. Direct simulation of the Navier–Stokes equations enables these closure models to be scrutinized against a high-resolution dataset where all relevant scales are resolved. The governing equations, written in conservative form, for a rotating flow are given by ›ui 5 0, ›xi

(1)

›ui uj ›ui 1 1 2 1 (2u2 di1 1 u1 di2 ) 5 2$P 1 = ui , 1 Ro Re ›t ›xj (2) where dij is the Kronecker delta. The Reynolds number dE) are (Re 5 UgdE/n) and Rossby number (Ro 5 U pg/f ffiffiffiffiffiffiffiffiffi defined in terms of the Ekman depth dE 5 2n/f , the geostrophic velocity Ug, and the Coriolis parameter f 5 2V, where V is the rotation rate. The velocity components u1, u2, and u3 (or u, y, and w) correspond to the Cartesian directions x1, x2, and x3 (or x, y, and z), and P is the total pressure. In Eq. (2), x3 (or z) refers to the vertical (wall-normal) direction and the horizontal component of rotation is neglected. The flow field variables can be written as the sum of a mean Ui and a fluctuating u9i component: ui 5 Ui 1 u9i . Substituting this into Eq. (2) and averaging yields the Reynolds-averaged form of the Navier–Stokes equations: ›Ui Uj ›Ui 1 1 (2U2 di1 1 U1 di2 ) 1 Ro ›t ›xj 5 2$P 1

›u9i u9j 1 2 = Ui 2 , Re ›xj

(3)

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where the overbar denotes spatial and temporal averaging. The mean flow governing equations given by Eq. (3) are not closed—they include terms that are not represented in the equation set, specifically the velocity correlations u9i u9j , which are interpreted as turbulent stresses. Prognostic equations may be derived for these terms; however, the resulting equations are also unclosed as they contain the triple correlation products u9i u9j u9k . In fact, attempts to solve this problem through the use of additional prognostic equations represent an infinite regression: for each new layer of equations added, the number of unknowns increases geometrically. First recognized by Keller and Friedmann (1924), this difficulty is termed the ‘‘closure problem’’ and remains an outstanding unsolved question in fluid dynamics: how are the turbulent stresses related to the mean flow quantities? Thus, the crux of the problem for integration of the Reynolds-averaged equations to predict the behavior of the ABL is the estimation and parameterization of the turbulent diffusivity u9i u9j (i.e., the Reynolds stresses). Since the introduction of the mean flow equation by Reynolds (1895), a number of closure models have been advanced. Because the question of how the turbulent scales are related to the mean flow remains fundamentally unanswered, most of these techniques represent a combination of theory and empiricism. Although current meteorological closures typically utilize high-order models (e.g., Lappen et al. 2010), many methods of relating the mean flow to the Reynolds stresses are based on an analogy with molecular viscosity and assume a form of downgradient diffusion by the turbulent stresses: ! ›Uj ›Ui 2 , u9i u9j 2 kdij 5 2Km 1 3 ›xj ›xi

(4)

where Km is known by a variety of names, such as ‘‘eddy viscosity,’’ ‘‘eddy diffusivity,’’ or ‘‘turbulent diffusivity,’’ and k 5 0.5u9i u9i is the turbulent kinetic energy. For the horizontally homogeneous Ekman problem under consideration, Eq. (4) is identically zero for all normal stresses. In the present work, the governing equations given by Eqs. (1) and (2) were integrated directly without any closure approximations. Rotation was parallel to the vertical axis and periodicity was assumed in both these directions. The wall-normal z direction incorporates an exponentially stretched and staggered mesh that clusters points near the wall boundary (where length scales are smallest), resulting in 18 mesh points for z1 # 10. The plus-sign superscript is used to indicate a variable written in terms of wall units: z1 5 zu*/n and u1 5 u/u*. The no-slip condition was applied at the wall; zero-stress

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TABLE 1. Domain parameters.

Re Ro Lx f /u* 5 Ly f /u* Nx 5 Ny Dx1 5 Dy1 Lz f /u* Nz Dz1 min

NEW

Miyashita et al. (2006)

1000 500 2.08 512 5.48 2.31 256 0.477

1140 570 1.92 512 6.60 1.92 512 0.147

(du/dz 5 dy/dz 5 0) and impermeability (w 5 0) were assumed at the top boundary. Spatial derivatives were computed using fourth-order central finite differences in order to facilitate parallelization. The implicit Crank– Nicholson time integration scheme is used for the vertical diffusion terms; all other terms are advanced using the fully explicit Adams–Bashforth method. Both methods are O(Dt2 ) accurate. As previously mentioned, a Reynolds number of Re 5 1000 was used for the simulation described in this paper. As a consequence of the scales used in defining the Reynolds and Rossby numbers, Ro 5 Re/2; thus, Ro 5 500. Specifications of the computational domain are provided in Table 1. For comparison, the domain parameters used by Miyashita et al. (2006) are also presented as they use a similar numerical method. Domain height dependence was tested by continuing the converged simulation with a height of 1.5Lz, where Lz is the vertical domain height in the original simulation. The results showed no statistical differences between the two test cases for any mean velocities, Reynolds stresses, or energy budgets. Concerning the horizontal extent, Waggy et al. (2011) demonstrated that for an Re 5 400 unstratified turbulent Ekman layer simulation, the largest velocity and pressure autocorrelations have an extent of approximately 2u*/f. Thus, based on these observations the lateral domain extent of 2.08u*/f should be sufficient to capture the relevant large-scale motions. We also note that the horizontal (and vertical) domain sizes are in agreement with other neutrally stratified turbulent Ekman layer simulations (Miyashita et al. 2006; Coleman 1999).

2. Mean quantities The governing equations were integrated until a quasisteady state was reached. Statistics were then computed by temporally and spatially (horizontal plane only) averaging 200 realizations over a dimensionless time period of tf 5 0.8. The mean velocity profiles plotted in Fig. 3 display the three-dimensional mean profile

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TABLE 2. Friction velocity and shear angle comparison. Case

Re

u*/Ug

a0 (8)

NEW Coleman (1999) Miyashita et al. (2006) Miyashita et al. (2006) Spalart et al. (2008)

1000 1000 775 1140 1000

0.0520 0.0539 0.0561 0.0520 0.0535

18.56 19.00 21.2 19.4 19.36

Figure 4 shows the ‘‘constant’’ over the logarithmic region where k exhibits nearly constant behavior over approximately a half decade of z1, which agrees with previously published results for comparable Reynolds numbers (Spalart et al. 2008; Miyashita et al. 2006). Time-averaged surface shear stress and direction are presented in Table 2. The slight deviations from the results of Spalart et al. (2008) are negligible when considering the different computational domains and numerical schemes; hence the agreement lends confidence to both results.

The normal and shear Reynolds stresses are shown in Figs. 5 and 6, respectively. The normal Reynolds stress profiles have extrema at slightly different distances from the wall. The streamwise Reynolds stress u9u9 reaches a maximum below the start of the log-law region at approximately zf /u* ’ 0.011 (z1 ’ 15). The spanwise stress y9y9 peaks at a magnitude less than 1/ 3 that of u9u9 at zf/u* ’ 0.023 (z1 ’ 29). For 0.25 , zf/u* , 0.7 (350 , z1 , 800), y9y9 is the dominant stress. This region corresponds to the location where U reaches a maximum (and dU/dz is nearly zero). The effect of the wall is very pronounced in the w9w9 profile as growth of vertical Reynolds stresses is heavily suppressed. Only when z1 . 50 is the magnitude of w9w9 within a factor of 2 of either horizontal energy component. The Reynolds shear stresses (Fig. 6) are defined by 2u9i u9j when i 6¼ j. The largest contribution in the nearwall region comes from 2 u9y9. The minimum at zf/u* 5 0.009 (z1 ’ 12) is comparable to the magnitudes of the normal Reynolds stresses presented in Fig. 5. Contribution by 2 u9y9 Reynolds shear stress is caused by rotation and is not observed in nonrotating flows (Mansour et al. 1988); even at Re 5 400, Marlatt et al. (2010) demonstrated the significant contribution of 2 u9y9 to the energy budgets for the Ekman layer. Accordingly, 2u9y9 does not contribute turbulent kinetic energy production and simply provides a means of redistributing energy. The 2u9w9 term remains positive for the majority of the boundary layer indicative of turbulence

FIG. 4. Calculated value of von Ka´rmań ‘‘constant.’’ Dashed line indicates k 5 0.41.

FIG. 5. Normal Reynolds stresses: u9u9 (solid), y9y9 (dashed), and w9w9 (dotted).

FIG. 3. Mean velocity profile: Re 5 1000 (solid) and laminar (dashed).

characteristic of the Ekman layer. We also observe that the total velocity plotted in terms of wall units (Fig. 2) accurately represents the near-wall region by Q1 5 z1 for z1 # 5. At this Reynolds number a log-law region develops and appears to be well represented by Q1 5 (1/k) ln(z1) 1 B, where the Ka´rmań constant is taken to be k 5 0.41 and B 5 5.2. A more precise measure of the Ka´rmań constant is evaluated using k5

d(lnz1 ) . dQ1

(5)

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FIG. 6. Shear Reynolds stresses: 2u9w9 (solid), 2y9w9 (dashed), and 2u9y9 (dotted).

kinetic energy production by means of sweeps and ejections. Similar behavior was noted for the Re 5 400 turbulent Ekman layer (Marlatt et al. 2010). The 2y9w9 term shows a small positive region near the wall before switching signs at z1 ’ 58. The point of crossover roughly correlates with the sign change of 2u9y9. The mean dissipation in the near-wall region of the simulation is shown in Fig. 7. Above z1 ’ 13 the dissipation decreases rapidly with increased height. The inflection point in the buffer layer at z1 5 10 is particularly interesting as it appears to correlate with the atmospheric data presented by Balsley et al. (2003). In their work they demonstrate that the dissipation exhibits a small region of constant dissipation before decreasing with altitude. However, no measurements were reported in the region nearest the wall. From Fig. 7 we can speculate that there would be a slight increase in dissipation through the viscous sublayer of the Ekman layer. Comparing the computed dissipation with low Reynolds number turbulent energy budgets, the small inflection may be due to a local minimum in the 2u9y9 dissipation; at this point, the 2u9y9 term acts as an energy source (Marlatt et al. 2010). The agreement between the atmospheric data of Balsley et al. (2003) and our unstratified DNS suggests that the effect of buoyancy on dissipation in the near-wall region is minor. The channel flow analysis by Mansour et al. (1988) demonstrated a similar inflection in « at z1 ’ 10, indicating that in the wall region where viscous effects dominate, small-scale motions are not affected by Coriolis forces.

3. Evaluation of closure models Turbulence models may be generally classified according to the order of the highest level prognostic equations retained. Thus, models that integrate only the mean flow equations and parameterize Km directly are termed firstorder models, while those which also integrate the six

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FIG. 7. Mean dissipation.

variance and covariance equations and parameterize the 10 triple correlation products are second-order schemes (e.g., Umlauf and Burchard 2005; Kantha 2006). Variations on first-order closure methods, termed half-order models or bulk methods, assume a profile for the mean flow but modify this profile depending on bulkaveraged wind or temperatures across either the entire boundary layer or across multiple sublayers. Likewise, variations on the second-order closure, denoted 1.5order models, avoid the requirement of integrating all six variance and covariance equations and retain only equations for the turbulent kinetic energy k and either the turbulent mixing length scale l or the dissipation rate «. In addition to classification by order, turbulence models may also be defined as local or nonlocal methods. Local methods estimate the parameterized unknowns on the basis of the known variables at the same location, based on gradient transport. Nonlocal methods may still retain local downgradient diffusion assumptions, or they may allow countergradient diffusion (Deardorff 1966). Most atmospheric turbulence models are local, although nonlocal schemes have been suggested for half- and firstorder methods.

a. First-order models First-order models integrate the mean flow equations and seek to express the Reynolds stresses in terms of the mean flow quantities. Such parameterization schemes are generally constructed by estimating the eddy diffusivity K from global quantities, such as the friction velocity or the boundary layer depth. These models are often termed K-profile or K-theory methods. In this work two first-order models are considered, both based on a cubic polynomial representation of the vertical eddy diffusivity profiles where height is the independent variable. In both cases, the models are nonlocal in the sense that the eddy diffusivity is a function of

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FIG. 8. Eddy diffusivities for the O’Brien model: zh 5 z1% [closed dots (DNS)], zh 5 z1% [solid line (model)], zh 5 zdK /dz50 m [open dots (DNS)], and zh 5 zdK /dz50 [dashed line (model)].

1111

FIG. 9. As in Fig. 8, but for the Reynolds stress.

m

parameters at the bottom and top of the boundary layer rather than local values across the layer. Both models assume local downgradient diffusion, however. To avoid the uncertainties regarding coordinate system rotation, the data presented here consider the parameterization of the total Reynolds vertical momentum flux s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ›U 2 ›V u9w9 1 y9w9 5 2Km 1 , ›z ›z

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

2

(6)

which follows directly from u9w9 5 2Km1 (›U/›z) and y9w9 5 2Km2 (›V/›z). Note that this assumes isotropic turbulent diffusion (i.e., Km 5 Km1 5 Km2). The definition in Eq. (6) will be used to compute the eddy diffusivity from DNS data for comparison with model predictions.

1) O’BRIEN MODEL The K-profile model of O’Brien (1970) is based on a smooth, continuous transition from the surface layer profile, in which Km is assumed to increase linearly with height, to an asymptotically decaying profile for Km at the top of the boundary layer. This is accomplished by means of the cubic representation

zh 2 z 2 Km (z) 5 Km (zh ) 1 Dz dKm DKm 3 DKm 1 (z 2 zsl ) 12 , (7) dz sl Dz where zsl is the height of the surface layer, zh is the height of the boundary layer, Dz 5 zh 2 zsl, and DKm 5 Km(zsl) 2 Km(zh). Obviously, use of O’Brien’s profile for the eddy diffusivity requires relatively detailed knowledge of the boundary layer velocity field and

turbulent field at both the top of the surface layer and at the top of the boundary layer. Fortunately, the surface layer is generally shallow enough that tower measurements of the ABL are often available. Information concerning the turbulent fluxes at the top of the boundary layer is harder to obtain, and assumptions of the magnitude of the turbulent flux at zh may be required (e.g., 4%, 5%, of the surface flux, etc.). The surface layer height was assumed to be the location where the velocity profile most closely matches the log-law profile. The exact choice of location is somewhat arbitrary as the log law roughly applies over nearly a half decade on the z1 scale. Nevertheless, a surface layer height of z1 sl ’ 110 was chosen based on the results presented in Fig. 4. Small changes of z1 sl were found to have little impact on the accuracy of the model. Two values were considered for the boundary layer depth. The first, z1%, is defined as the point at which the total velocity is within 1% of the free stream velocity. This is analogous to the typical 99% boundary layer thickness with a slight modification since the Ekman layer exhibits a small region of velocity where U/Ug . 1. Except for the very near-wall region, the results shown in Figs. 8 and 9 show good agreement of the model with DNS results in the bottom portion of the boundary layer. However, above z/zh 5 0.5 the eddy diffusivity is underpredicted when zh 5 z1%; consequently, the Reynolds stresses exhibit the same trend. Since the magnitude of the eddy diffusivity remains quite large at zh 5 z1%, it is reasonable to question the effect of specifying larger values for zh. Also shown in Fig. 8 are results when zh is the location of a local minimum in the eddy diffusivity profile (zhf/u* 5 1.112). Surprisingly, the first specification of zh proves superior upon calculating the relative difference between the model and DNS results. However, the results for the latter definition appear more qualitatively correct as the

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location of the diffusivity maximum is more closely predicted. For both cases, the relatively good agreement in the turbulent momentum flux noted in the upper portion of the boundary layer, where poor comparison with Km is noted, is primarily due to the small magnitude of the mean velocity gradient in this region. Hence, the momentum flux profiles are insensitive to the modeled eddy diffusivity near the top of the boundary layer.

2) LARGE, MCWILLIAMS, AND DONEY MODEL A modified K-profile model was proposed for the oceanic boundary layer by Large et al. (1994, hereafter LMD94). This model represents a variation on the Troen and Mahrt (1986) model and the O’Brien (1970) formulation presented above. While the model includes a variety of modifications to account for the effects of stable and unstable stratification, the application here to the neutral layer avoids most of these. Like the O’Brien model, the LMD94 model is a nonlocal model; unlike the O’Brien model, the LMD94 model includes the possibility for countergradient transport, although this feature is not present in the parameterization of the momentum fluxes. In the LMD94 model, the turbulent vertical flux for any given flow field quantity c is parameterized by an eddy diffusivity and a prescribed function that allows nonlocal advective transport: dc 2 gc , w9c9 5 2Kc dz

(8)

where Kc is the eddy diffusivity associated with the quantity c under consideration, and gc is a specified function. Assuming horizontal homogeneity, the eddy diffusivity is described by Kc

z z z 5 zh wc G , zh zh zh

(9)

where zh is the depth of the boundary layer, wc is a height-dependent vertical velocity scale, and G is a cubic polynomial function of z. While the vertical velocity scale is defined using a flexible equation that allows proper scaling with a range of thermal stratification classifications, for the neutral layer, wc in Eq. (9) reduces to a simple form given by wm: wm 5 ku* ,

(10)

where the subscript m denotes ‘‘momentum’’ and k is the von Ka´rmań constant (taken to be 0.41). The vertical profile of the eddy diffusivity is given by the cubic polynomial:

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z z z 2 z 3 1 a3 . 5 a0 1 a1 1 a2 G zh zh zh zh

(11)

The coefficients of the polynomial are selected to enforce the boundary conditions Km 5 0 at z 5 0, Km 5 Km,‘ at z 5 zh, and dKm /dz 5 dKm /dzj‘ at z 5 zh; this assumes a linear decay of the turbulent flux through the surface layer. For the neutral turbulent Ekman layer, assuming zero far-field diffusivity, the momentum eddy diffusivity is simply described by Km

z z 2 5 zh ku* 12 . zh zh

(12)

The countergradient transport function for momentum gm is identically zero given the lack of empirical data regarding the proper form of this parameter. The model is therefore nonlocal only in the sense that the eddy diffusivity is a function of the surface shear stress and the boundary layer height; turbulent diffusion of momentum remains strictly downgradient. This is not of concern here, however, as no countergradient momentum transport is observed in the neutral layer simulation data. The LMD94 model, as formulated by Eq. (12), allows the selection of two parameters with which to attempt to fit the model to the simulation data, u* and zh. This is in contrast to the O’Brien model, which uses information about the eddy diffusivity at the bottom and top of the interior region to fit the cubic equation. The lack of this information in the LMD94 model suggests that less accurate agreement between the model and simulation data may be expected, especially as the model includes implicit a priori assumptions regarding the structure of the boundary layer [zsl ’ 0.1zh, Km(zh) 5 0, etc.]. However, in application to the atmospheric or oceanic boundary layer, it also requires less information; if the eddy diffusivity profiles described by the model are reasonable, this is an advantage, since such data are often difficult to obtain. Presented in Figs. 10 and 11 are the eddy diffusivity and Reynolds stress profiles respectively for zh 5 z1%. Although the term ‘‘friction velocity’’ is typically used to refer to the surface friction velocity, in this instance we wish to use u* as a parameter with which the model can be tuned. Two values for the friction velocity are used: u*, the computed value given in Table 2, and u*,zsl , a velocity scale computed from the Reynolds stresses at the surface layer height (z1 sl ’ 110):

u*,zsl

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # u" u ›U 2 ›V 2 5 tn 1 ›z ›z

z5zsl

.

(13)

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FIG. 10. Eddy diffusivity for the LMD94 model, zh 5 z1%: u* (solid), u*,zsl (dashed), and DNS (dots).

FIG. 11. As in Fig. 10, but for the Reynolds stress. Stresses are normalized by surface friction velocity in all cases.

Since the LMD94 model assumes Km(zh) 5 0 and dKm /dzjz 5 0, this choice of zh proves inadequate for sl both friction velocities. The actual diffusivity is neither zero nor a local minimum at the top of the boundary layer, and it results in large errors in the computed Reynolds stresses. Moreover, the height of the maximum eddy diffusivity is not accurately captured, leading to marginal qualitative agreement between the model and DNS results. Since Km from the DNS is nonzero everywhere in the flow, a better choice for the boundary layer height should correlate with a minimum value of the eddy diffusivity (dKm/dz 5 0). The eddy diffusivity for zh f/u* 5 1.112 is presented in Fig. 12. Qualitative agreement is drastically improved for the new value of zh. The location of the maximum (although overpredicted using the surface value of u*) is accurately captured and the decay at the top of the boundary layer is closer to DNS values. Using a friction velocity based on the Reynolds stress (dashed line) at the surface layer improves the model but the diffusivity is still overpredicted. Since the two parameters u* and zh allow for fine tuning of the model, we attempt to improve agreement between DNS data and LMD94 results by applying a scaling to the surface friction velocity based on flow parameters. Reducing the eddy diffusivity is accomplished by scaling the friction velocity by zsl/zh. Using (zsl/zh)u* instead of u* improves the estimate in the outer region and the Reynolds stress is well modeled for z/zh . 0.2 (Fig. 13). However, this leads to a significant underprediction for z/zh , 0.2. We find that scaling the surface friction by 0.7 leads to the best agreement between DNS and the LMD94 model.

solutions since its introduction by Jones and Launder (1972, 1973). A number of studies have also applied the k–« parameterization to atmospheric boundary layer flows, including Detering and Etling (1985) for a onedimensional boundary layer profile, by Kitada (1987) for two-dimensional calculations of simulated sea-breeze fronts, and Beljaars et al. (1987), for three-dimensional computations of neutrally stratified boundary layer flows over inhomogeneous surfaces (with spatial variations in roughness and topography). The k–« model is a simplification of the second-order closure model, replacing the six prognostic equations for the velocity variances and covariances with equations for the turbulent kinetic energy k and the kinetic energy dissipation rate. As presented for the model, these equations have the form Dk dU dV d(w9p9 1 w9k9) 5 2u9w9 2 y9w9 2 2 «, Dt dz dz dz (14a)

b. 1.5-order models: k–« formulation A variant of the second-order closure models for the Reynolds-averaged equations, the k–« model, has enjoyed wide application in the context of engineering flow

FIG. 12. Eddy diffusivity for the LMD94 model, zh defined by dKm /dzjz 5 0: u* (solid); u*,zsl (dashed), (zsl/zh)u* (dash-dotted), h 0.7u* (dotted), and DNS (large dots).

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FIG. 13. As in Fig. 12, but for the Reynolds stress. Stresses are normalized by surface friction velocity in all cases.

« D« dU dV 5 C«1 2u9w9 2 y9w9 Dt k dz dz 2 dw9«9 « 2 2 C«2 , dz k

(14b)

where D/Dt 5 ›/›t 1 u $ is the total derivative and the effects of buoyancy are at present neglected. For the steady-state, horizontally homogeneous simulation presented here, D/Dt 5 0. The terms k9 and «9 are used to denote instantaneous turbulent kinetic energy and kinetic energy dissipation, respectively. The equations are then closed using the assumptions (Beljaars et al. 1987; Stull 1988)

Km 5

(am k)2 , «

(15a)

dk , dz

(15b)

K d« w9«9 5 2 m . C«3 dz

(15c)

w9p9 1 w9k9 5 2Km

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FIG. 14. Eddy diffusivity for the k–« model: am 5 0.3 (solid), am 5 0.18 (dashed), am 5 f(ReT) (dotted), and DNS (large dots).

where ReT is the turbulent Reynolds number defined «n (the ‘‘hat’’ denotes a dimensional quanReT 5 k^2 /^ tity), or, in terms of the dimensionless kinetic energy and dissipation, ReT 5 Re(k2/«). The terms k and « are ^ 2 and « 5 ^ «dE /Ug3 , renondimensionalized by k 5 k/U g spectively. In the present simulation this value varies across the boundary layer and peaks at ReT 5 624.6 near zf/u* 5 0.305. Comparisons between the modeled eddy diffusivities and those from the DNS are presented in Fig. 14, while the resulting Reynolds stress profiles are shown in Fig. 15. Qualitatively the profiles are correct; however, the location of the maximum diffusivity is off by approximately 0.1u*/f regardless of the value of am. For am 5 0.3, the model overpredicts the Reynolds stress in the lower half of the boundary layer and underpredicts in the upper half. Reducing am to 0.18 results in poor model performance across the entire flow field. The variable coefficient where am 5 f(ReT) yields the best results with great accuracy for zf/u* , 0.3 and zf/u* . 0.6. The middle region of the boundary layer is underpredicted by all three values of am, and performance of the models in the near-wall region is poor.

The constants are defined, primarily on the basis of empirical evidence, by C«1 5 1.44, C«2 5 1.92, C«3 5 1.30, and am 5 0.3. The value am 5 0.3 was originally recommended by Jones and Launder; Panofsky and Dutton (1984) recommend a value of 0.18 as more appropriate for the ABL. For low Reynolds number turbulence, Jones and Launder (1973) recommend several modifications (see White 1991). Without revisiting the prognostic equations, the most interesting is the modification to am: am 5 0:09 exp

22:5 1 1 ReT /50

1/2 ,

(16) FIG. 15. As in Fig. 14, but for the Reynolds stress.

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FIG. 16. As in Fig. 14, but for the production term.

FIG. 17. As in Fig. 14, but for the transport terms.

To better assess the k–« model with respect to the simulation data, comparisons were made of the kinetic energy production and transport terms in Eq. (14a) and the transport term in Eq. (14b). The kinetic energy production is parameterized, using Eq. (15a), as dU dV 2u9w9 2 y9w9 ’ Km dz dz

"

dU dz

2

dV 1 dz

1115

2 # (17)

and the transfer terms are modeled by Eqs. (15b) and (15c), respectively. The vertical profiles of kinetic energy production (Fig. 16) demonstrate that this term is overestimated by all three forms of the model in the near-wall region. However, the location of maximum production is well captured and relatively good agreement is achieved above zf/u* ’ 0.1, especially when am is a function of ReT. Apart from the near-wall region (where the best model is off by over a factor of 4), the variable value of am performs quite well. The transport model is qualitatively correct throughout the entire boundary layer. The crossover is well captured (Fig. 17, bottom) for all three values of am. The location of the minimum (zf/u* ’ 0.01) is accurately predicted; however, each model overshoots the actual value. The maximum at zf/u* 5 0.3 is modeled slightly

nearer the wall than DNS results, but the am 5 f(ReT) curve is quite accurate in predicting the actual maximum value. However, all models underpredict the transport terms above zf/u* 5 0.4. The vertical profiles of the dissipation are qualitatively well represented by the models except in the nearwall region where the models behave poorly because of an inflection in the mean dissipation profile. None of the models captures the small negative region below zf/u* 5 0.01. However, the curve for am 5 f(ReT) shows good agreement above zf/u* 5 0.05. All models underpredict the dissipation in the middle parts of the boundary layer (0.35 , zf/u* , 0.7).

4. Discussion and concluding remarks Evaluation of the present DNS results with previous simulations (Coleman 1999; Miyashita et al. 2006; Spalart et al. 2008) shows good agreement, providing support for the validity of all simulations. The two firstorder, nonlocal models compared in this study, O’Brien (1970) and LMD94, both approximate the vertical distribution of the eddy diffusivity as a function of height by means of a cubic polynomial. This assumption appears to be good given the agreement of the models with DNS results. The fit is better in the O’Brien model simply

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FIG. 18. As in Fig. 14, but for the dissipation term.

because the cubic is defined by observed data at both the top of the surface layer and the top of the boundary layer. Conversely, the LMD94 model reduces the freedom to fit available data. However, the latter model surface layer profiles are physically more realistic than those from O’Brien’s model—the magnitude of the turbulent momentum flux vanishes at the wall, whereas nonzero values are computed at z 5 0 by O’Brien’s model. The results presented here support the use of higherorder k–« turbulent models in rotating flows. Specifically, the parameterization of am as a function of the turbulent Reynolds number (ReT) dramatically improved the results over the models utilizing a constant value for am. The underprediction of the Reynolds stresses in the outer region of the boundary layer appears to be a consequence of the underestimation of the transport terms coupled with a slight difference in the modeled and actual dissipation. Comparing the magnitude of the pressure diffusion term w9p9 and the kinetic energy flux w9k9 demonstrates that both terms make considerable contributions to the transport term. As the pressure strain terms are not included in the turbulent kinetic energy budget for the k–« model, it is possible that neglected intercomponent energy redistribution is responsible for model errors. A more likely source of error, however, is the underestimation of the dissipation

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rate. As production and dissipation are nearly in balance, a low estimate of the dissipation will directly influence the modeled production through Eqs. (15a) and (15c). Figure 18 demonstrates the underprediction of the dissipation in the top portion of the boundary layer. This evaluation of several turbulence closure models provided not only an assessment of the relative accuracy of the strengths and weaknesses of the models but also furthers our understanding of the physics involved with the neutrally stratified turbulent Ekman layer. These results are intended to assist climatologists in developing new closure models by highlighting the shortcomings and successes of common low-order models. The results demonstrated that the vertical profile of the eddy diffusivity is well represented by a cubic polynomial. Firstorder models that describe the diffusivity based on this form provide good estimates of the turbulent momentum fluxes. Also, for a variable value of am, the 1.5-order k–« model yields good estimates of the Reynolds stresses apart from a discrepancy found in the middle of the boundary layer. It is suggested that the influence of rotation leads to an underprediction of the Reynolds stresses in this region as system rotation acts to suppress dissipation and (consequently) the production of turbulence. Acknowledgments. The authors gratefully acknowledge the constructive comments provided by Dr. Hamlington and three anonymous reviewers of the manuscript. Support for Scott Waggy was provided by the Graduate Assistantship in Areas of National Need Fellowship through the Aerospace Engineering Sciences department at the University of Colorado, Boulder. This research was supported by an allocation of advanced computing resources provided by the National Science Foundation. The computations were performed on Kraken, Athena, or Nautilus at the National Institute for Computational Sciences (http://www.nics.tennessee.edu/). The authors also thank the Department of Aerospace Engineering Sciences at the University of Colorado, Boulder, for financial support toward the publication costs of this article.

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