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A Variable Mesh Spacing for Large-Eddy Simulation Models in the Convective Boundary Layer

A Variable Mesh Spacing for Large-Eddy Simulation Models in the Convective Boundary Layer
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  Boundary-Layer Meteorol (2009) 131:277–292DOI 10.1007/s10546-009-9360-zARTICLE A Variable Mesh Spacing for Large-Eddy SimulationModels in the Convective Boundary Layer Gervásio Annes Degrazia  ·  Umberto Rizza  ·  Franciano Scremin Puhales  · Antônio Gledson Goulart  ·  Jonas Carvalho  ·  Guilherme Sausen Welter  · Edson Pereira Marques Filho Received: 27 March 2008 / Accepted: 5 February 2009 / Published online: 24 February 2009© Springer Science+Business Media B.V. 2009 Abstract  A variable vertical mesh spacing for large-eddy simulation (LES) models in aconvective boundary layer (CBL) is proposed. The argument is based on the fact that in thevertical direction the turbulence near the surface in a CBL is inhomogeneous and thereforethe subfilter-scale effects depend on the relative location between the spectral peak of thevertical velocity and the filter cut-off wavelength. From the physical point of view, this lack of homogeneity makes the vertical mesh spacing the principal length scale and, as a conse-quence, the LES filter cut-off wavenumber is expressed in terms of this characteristic lengthscale. Assuming that the inertial subrange initial frequency is equal to the LES filter cut-off frequency and employing fitting expressions that describe the observed convective turbulentenergy one-dimensional spectra, it is feasible to derive a relation to calculate the variablevertical mesh spacing. The incorporation of this variable vertical grid within a LES modelshowsthatboththemeanquantities(andtheirgradients)andtheturbulentstatisticsquantitiesare well described near to the ground level, where the LES predictions are known to be achallenging task. Keywords  Convective boundary layer · Large-eddy simulation · Variable mesh spacing G. A. Degrazia ( B ) · U. Rizza · F. S. Puhales · G. S. WelterLaboratório de Fisica da Atmosfera, Universidade Federal de Santa Maria, Santa Maria, RS, Brazile-mail: Degrazia@ccne.ufsm.brU. RizzaIstituto di Scienze dell’Atmosfera e del Clima, Consiglio Nazionale delle Ricerche, Lecce, Italye-mail: u.rizza@isac.cnr.itA. G. GoulartCentro de Tecnologia de Alegrete, Universidade Federal do Pampa, Alegrete, RS, BrazilJ. CarvalhoUniversidade Federal de Pelotas, Pelotas, RS, BrazilE. P. Marques FilhoUniversidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil  1 3  278 G. A. Degrazia et al. 1 Introduction In large-eddy simulation (LES), the energy-containing eddies of the turbulent motion areexplicitly resolved and the effect of the smaller, more isotropic, eddies (typical eddies in theinertial subrange) needs to be parameterized (Muschinski 1996). Modelling these residual turbulent motions, which are also termed subfilter-scale (SFS) motions (Pope 2004), is in largepartaphenomenologicalprocedurebasedonheuristicarguments(Sullivanetal.1994).The resolved turbulent flow time-dependent random variables are obtained by the applica-tion of a low-pass spatial filter of characteristic width   , the turbulent resolution length scale(Pope 2004), smaller than the scales of the resolved turbulent motions. According to Mason (1994) “The term large-eddy simulation has indeed been usually reserved for applicationsin which the scale of the filter operation is within or at least close to an inertial subrangeof turbulence”. Therefore, this low-pass spatial filter width that plays a decisive role in theconstruction of the resolved turbulence variables can be expressed in terms of the inertialsubrange parameters. Thus, there is a clear expectation that an adequate physical deriva-tion of the parameter    provides robust fluxes of heat and momentum from resolved-scalemotions and that the contributions of the unresolved subfilter motions are responsible for thedissipation of the resolved turbulence. However, most geophysical flows present boundaryregions where the turbulence scale reduces and becomes smaller than the spatial filter scalewidth. Indeed, in the atmospheric convective boundary layer (CBL), the wavelength of thepeak in the vertical velocity spectrum  (λ m ) w  decreases in proximity to the ground (Kaimalet al. 1976; Caughey and Palmer 1979) and this surface closeness constrains our capacity to accomplish high-Reynolds-number LES (Sullivan et al. 2003). Near the ground, the fil- ter cut-off wavelength  λ  f   employed in LES modelling is of the same order as  (λ m ) w  andconsequently the SFS fluxes in LES become important and their contribution to the totalflux (resolved and filtered eddies) grows in proximity to the surface. Recent testing of SFSmodels using observations show that the contribution of the SFS turbulent energy modes tothe total turbulence is dominant when  (λ m ) w /λ  f   ≤  1, while the resolved motions play apreponderant role for  (λ m ) w /λ  f   ≫ 1 (Sullivan et al. 2003). Based on the universal form of the energy spectrum for fully developed turbulence therelative proportion of resolved and SFS motions is established by the location of the filtercut-off wavenumber  k   f   in the turbulent energy spectrum. If the energy-containing eddiesare described by the peak wavenumber  k   p  of the turbulent spectrum, for cut-off and peak wavenumbers, then the proportion of the SFS turbulent motions to the total turbulence islarge when  k   f   ≤  k   p , while the resolved turbulent motions are dominant for  k   f   ≫  k   p . As aconsequence,thedeterminationofaphysicalcriteriontoobtain k   f   indistinctverticalregionsof a CBL is a fundamental procedure in LES methodology. A commonly used filter in LES isthe sharp Fourier cut-off  (Moeng and Wyngaard 1988). The sharp spectral filter annihilates all Fourier modes of wavenumber | k  | greater than the cut-off wavenumber k   f   ≡ π (1)whereas it has no effect upon lower wavenumber modes. In practical applications of LES,the parameter    is expressed in terms of     x  , and normally it is replaced by this horizontalmesh spacing (Degrazia et al. 2007). Following Moeng and Wyngaard (1988), the low-pass LES filter width is given by  =  32  2   x    y   z  1 / 3 (2)  1 3  A Variable Mesh Spacing 279 where    x  ,  y  and    z  are the computational mesh sizes in the coordinate directions  x  ,  y ,  z and the constant  ( 3 / 2 ) 2 accounts for the de-aliasing. Furthermore, for a LES model thatresolves the most energetic turbulent eddies in a CBL, the filter characteristic width canbe chosen in the inertial subrange, closer to the integral scale, and far from Kolmogorov’sdissipative scales (Mason 1994; Degrazia et al. 2007). Observations show that a CBL presents a homogeneous structure in the horizontal direc-tions. Therefore, throughout the CBL, both  u  (longitudinal velocity) and  v  (lateral velocity)spectradisplayawell-establishedfilterz-lesslimitingwavelength λ  f   fortheinertialsubrange(Degrazia et al. 2007). This horizontal homogeneity imposes large values of     x   =    y  ≈ 0 . 05  z i  (Degrazia et al. 2007, where  z i  is defined as the height of the lowest inversion base atthetopoftheCBL)inthesurfaceregionsofaCBL,andasaconsequencedecreasesthemag-nitude of   k   f   and favours the condition  k   f   ≤ k   p . This experimental evidence indicates that afilter wavenumber, such as  k   f   =  π/  x  , is physically not adequate for LES applications toreproduceturbulentfluxesinproximitytothesurface.Thus,toobtainapossiblesolutionthatavoids this condition and ensures that  k   f   ≫  k   p , it is necessary to define the sharp spectralfilter as k   f   = π  z .  (3)However, in comparison to  u  and  v , the vertical velocity component  w  exhibits the larg-est height dependence among the three velocity components. This vertical inhomogeneitymakes the inertial subrange limiting wavelength vary strongly with height in the lower CBL.Therefore, for the turbulent vertical velocity, observational values of   λ  f   or  k   f   valid nearthe surface in the CBL cannot be clearly determined. This lack of definition of   λ  f   for thevertical turbulent velocity does not allow the magnitudes of     z  and consequently of     to beestablished for the vertical region near the ground in a CBL.Motivatedbytheselectionofafiltercut-offwavenumberprovidedbyEq.3thepurposeof thepresentinvestigationistodevelopamethodologythatprovidesvaluesfor   z  inthelowerCBL.Theapproachisbasedonfittedmathematicalexpressionsdescribingtheobservedcon-vective turbulent energy spectra. Therefore, the idea is to determine, for distinct levels in thelower CBL, a limiting or filter wavelength associated with the vertical turbulent velocityspectrum. This procedure that allows derivation of a variable and non-uniform grid spacing   z  is based on the physical criterion that constrains  (λ m ) w  to  λ  f   and requires the condition (λ m ) w /λ  f   ≫ 1 in the proximity of the surface in a CBL. 2 Methodology to Obtain a Vertical Mesh Spacing Near the Ground in a CBL A review of the literature from the past 20 years shows that many observed turbulent velocityspectra present a simple shape in the planetary boundary layer (Kaimal and Finningan 1994;Panofsky and Dutton 1984; Foken 2006). Thus, in the frequency range associated with the turbulent energy, spectra can often be approximated by a smooth curve containing one max-imum only (when plotted as log ( nS  ( n ))  versus log ( n )) . This statement is valid for the  w spectrum under convective conditions (Kaimal et al. 1976; Caughey 1982; Olesen et al. 1984).The equation for Eulerian velocity spectra under unstable conditions can be expressed asa function of convective scales as follows (Degrazia et al. 2001): nS  i w 2 ∗ = 1 . 06 c i  f   ψ 2 / 3 (  z /  z i ) 2 / 3 (  f  m ) 5 / 3 i  ( 1 + 1 . 5 (  f  /  f  m ) i ) 5 / 3 (4)  1 3  280 G. A. Degrazia et al. where  i  = u ,v,w  are the turbulent wind components,  c i  = α i ( 0 . 5 ± 0 . 05 )( 2 πκ) − 2 / 3 ,α i  = ( 1 ; 4 / 3 ; 4 / 3 )  (Champagne et al. 1977),  κ  =  0 . 4 is the von Karman constant,  f   =  nz / U   isthenondimensionalfrequency, n  isthefrequencyinHz, U   isthehorizontalmeanwindspeed, (  f  m ) i  is the nondimensional frequency of the spectral peak,  w ∗  is the convective velocityscale,andthenondimensionalmoleculardissipationratefunctionisdefinedby ψ  = ε  z i /w 3 ∗ ,where  ε  is the mean dissipation of turbulent kinetic energy per unit time per unit mass of fluid, with the order of magnitude of   ε  determined only by those parameters that characterizethe energy-containing eddies. The magnitude of the above defined  α i  parameters is derivedfrom the turbulence isotropic properties in the inertial subrange.From Eq.4, the vertical turbulent velocity spectrum can be written as nS  w w 2 ∗ ψ 2 / 3  = 1 . 06 c w nzU   ψ 2 / 3 (  z /  z i ) 2 / 3 (  f  m ) 5 / 3 w  1 + 1 . 5 ( nz / U  )(  f  m ) w  5 / 3  (5)where  c w  = 0 . 36.The peak frequency,  (  f  m ) w , is an important parameter for studies on turbulent dispersionin the CBL. The knowledge of this quantity in Eq.5 allows reproduction of the observedspectral curves and captures the temporal and spatial characteristics associated with theenergy-containing eddies in distinct vertical regions of a CBL. From the Ashchurch andMinnesota experiments (Caughey 1982), for  z  <  0 . 1  z i  (surface layer, Weil et al. 2004), therelation for  (  f  m ) w  is linear and can be approximated by (  f  m ) w  =  z (λ m ) w = 15 . 9 = 0 . 17 ,  (6)which is precisely the free convection limit observed in the Kansas observations (Kaimalet al. 1972; Caughey 1982). Substituting the relation (6) into Eq.5, one obtains the following vertical turbulent spectrum in the lower CBL: nS  w w 2 ∗ ψ 2 / 3  = 7 . 32  f  i  (  z /  z i ) 5 / 3 1 + 8 . 85  f  i  (  z /  z i ) 5 / 3 .  (7)Furthermore, the asymptotic behaviour of Eq.7 for large values of   f  i  in the inertial subrangeof the spectrum, can be described as nS  w w 2 ∗ ψ 2 / 3  = 0 . 19  f  − 2 / 3 i  .  (8)Theideainthisanalysisistodeterminealimitingorinitialcut-offwavelength(orfrequency)for the inertial subrange at heights  z /  z i  <  0 . 1 in the CBL, and consider it identical to theLES filter cut-off wavelength (or frequency). We are interested in obtaining the initial fre-quencies (or wavelengths) of the inertial subrange for distinct vertical levels in the proximityof the surface. From these initial frequencies the turbulent energy will be filtered and theireffects parameterized in LES models. Therefore, to obtain mathematically these filter cut-off frequencies or limits, we equate Eq.7 with its asymptotic behaviour given by (8) and solve this equality to obtain the fundamental frequencies,  f   f   , that represent the limiting or filtercut-off initial frequency of the inertial subrange,0 . 026   f   f   5 / 3  = (  z /  z i ) 5 / 3  1 + 8 . 85  f   f   (  z /  z i )  5 / 3 .  (9)  1 3  A Variable Mesh Spacing 281 After some simple algebraic manipulations, we obtain (  z /  z i )  f   f   × 0 . 03 = 0 . 11 ,  (10)and  f   f   = 3 . 66   z z i  − 1 .  (11)Now employing the relation  f   f   =  z i λ  f  one obtains  λ  f   =  0 . 27   z z i   z i . Furthermore,  k   f  can be written as  k   f   =  2 πλ  f  and a comparison of this ratio with  k   f   = π/  z  leads to   z  = λ  f  2 ∼= 0 . 14   z z i   z i .  (12)Equation12 imparts a physical spatial constraint (obtained from observations), that helps usdetermine the correct choice of the dimension of the vertical mesh spacing in LES modelsin the lower CBL; this discussion shows that the choice of the vertical length scale,    z , asgiven by relation (12), is not an arbitrary assumption, but is imposed by observations in the CBL.On the other hand, the determination of     z , as given by Eq.12, makes the relation  k   f   = π/  z  well-defined since the height dependency of   λ  f   is now known in the lower CBL.Therefore, the inhomogeneous character of the turbulence in the vertical direction in a CBLis responsible for the distinct values of the filter cut-off wavenumber  k   f   that need to beselected in a LES model. For  z /  z i <  0.1, the vertical inhomogeneity naturally imposes    z as a characteristic length scale of the flow and selects the ratio  k   f   =  π/  z  as having thequalities needed for a LES filter cut-off wavenumber.The above development enables calculation of the magnitude of the ratio between thewavelength of the peak in the vertical velocity spectrum and the LES filter cut-off wave-length. From our derivation the value for  (λ m ) w /λ  f   is of the order of 20. As the magnitudefor this ratio is much larger than 1.0 one expects that the employment of   k   f   = π/  z  in LESmodels, with    z  given by Eq.12 in the lower CBL, reproduces SFS fluxes that are a small fraction of the resolved turbulent fluxes. 3 Numerical Experiment Using a Variable Vertical Grid Spacing  z In order to provide numerical experiments using the approach previously presented in thisstudy, the LES model, firstly developed by Moeng (1984) and later modified by Sullivan et al. (1994), is used with a variable vertical grid spacing    z  for  z  <  0 . 1  z i . Regarding thedependence of the SGS closure on the filter scale, we would like to point out that in order toaccomplish our simulations the filter has not been modified, as well as the  (λ m ) w  /λ  f   ratio.Thus, the subfilter turbulence parameterization aspects in this study are based exclusively onSullivan et al. (1994). A (5, 5, 2) km box domain with 128 grid points in each direction  (  x  ,  y ,  z )  has beenused. In the simulation we held the kinematic turbulent heat flux constant with a magnitude  w ′ θ  ′  0 =  0 . 24K ms − 1 , and the geostrophic wind was set to  U  g  =  10ms − 1 . The initialvalues for the CBL height and the surface potential temperature were set respectively equalto  (  z i ) 0  =  1 , 000m and  θ   =  300K. Furthermore, to establish a comparison, two simula-  1 3
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