Heisenbergs turbulent spectral theory determining thefiltering procedure in LES model

Acevedo, Otávio C.

Degrazia, Gervásio A.

Puhales, Franciano S.

Velho, Haroldo F. de Campos

Heisenbergs turbulent spectral theory determining thefiltering procedure in LES model

Universidade Federal de Santa Maria: Portal de Periódicos Eletrônicos da UFSM

                          V Workshop Brasileiro de Micrometeorologia                                  47Heisenbergs turbulent spectral theory determining thefiltering procedure in LES modelsGervÆsio A. Degrazia1, OtÆvio C. Acevedo1,Haroldo F. de Campos Velho2, Franciano S. Puhales11Departamento de Física/Universidade Federal de Santa Maria Santa Maria, RS, Brazil2LAC, Instituto Nacional de Pesquisas EspaciaisSªo JosØ dos Campos, SP, Brazil1. IntroductionLarge-eddy simulation (LES) models represent a well-establishedtechnique to study the physical behaviour of the planetary boundary-layer(PBL). In general, the governing equations in the LES models are theincompressible Navier-Stokes equations described for a horizontallyhomogeneous boundary-layer. The resolved turbulent flow variables areobtained by the application of a low-pass spatial filter of characteristic width,the turbulence resolution length-scale (Pope, 2004), smaller than the scalesof the resolved turbulent motions. This low-pass spatial filter width hasthe same order of magnitude as the numerical grid dimensions and, basedon Kolmogorov spectral characteristics can be expressed in terms of theinertial subrange scales. The purpose of the present study is to show thatHeisenberg·s classical theory for inertial transfer of turbulent energy(Heisenberg, 1948) can be used to select the filtering in LES models.2. Heisenberg model for the kinematic turbulence viscosityIn his classical work, based on intuitive arguments, Heisenberg(1948) assumed that the process of the energy transfer from the small tothe large wavenumbers in a Kolmogorov turbulent spectrum is similar tothe conversion of mechanical energy into thermal energy through the agencyof molecular viscosity. In other words, the physical picture that forms thebasis of Heisenbergs theory is that, in the energy cascade process withinthe kinetic turbulent spectrum, the mechanism of inertial exchange ofenergy from large to small eddies is controlled by a kinematic turbulence48                                             CiŒncia e Natura Especial, UFSMviscosity (KTV). This introduces a division between scales at any arbitrarywave number in the inertial subrange. This scale separation is indeednaturally relevant in the LES frame, where a cut-off wave number isarbitrarily chosen in the inertial range, introducing then a well determinatedivision, to which Heisenbergs approach seems well suited to be applied.An eddy viscosity is the product of a characteristic turbulent length scaleand a velocity, and thus dimensional analysis yields (Heisenberg, 1948 see also Hinze, 1975):∫∞=νckHT dkkkEC 3)((1)where k is the wavenumber, HC  is Heisenberg·s dimensionless spectraltransfer constant and )(kE  is the three-dimensional (3-D) turbulenceenergy spectrum in the inertial subrange, with the following form(Kolmogorov, 1941):3532)( −εα= kkE K (2)where Kα  is the Kolmogorov constant. Assuming that the small-scaleturbulence (inertial subrange) should act on the large-scale turbulence likean additional eddy viscosity we substitute eq. (2) into eq. (1), where HC ≈0.47 and Kα ≈ 1.52 (Muschinski and Roth, 1993; Corrsin, 1963), to obtain343144.0 −= cT kεν (3)It is important to note that Eq. (3) was firstly derived byMuschinski and Roth (1993). Following the philosophy of Kraichnans eddy viscosity in spectralspace, Lesieur and Metais (1996) present an eddy viscosity expressed by2123 )(32=−kkE cKT αν (4)where )( ckE  is the kinetic energy spectrum at the cutoff ck .Now, substituting (2) with ckk =  in Eq. (4) yields343144.0 −ε=ν cT k (5)suggesting that Heisenbergs eddy viscosity is identical to the one proposedby Lesieur and Metais (1996, Eqs. (3.5) and (4.1)). A remarkable point                          V Workshop Brasileiro de Micrometeorologia                                  49here is the consistency of the theoretical model (4) with the experimentalvalue for the Heisenbergs constant.At this point its important to note that the KTV obtained fromthe Heisenbergs model (eq. (3)) is expressed in terms of a cutoffwavenumber that defines the beginning of the inertial subrange andestablishes the concept of sharp spectral filter. Therefore, Heisenbergsspectral transfer theory becomes the sharp spectral filter a natural filteringprocedure to be applied in LES models. In fact, this type of filter has beenlargely employed in LES methodology.3. AcknowledgementsThis work has been supported by CNPq and CAPES.4. ReferencesCorrsin, S.: Estimates of the Relations between Eulerian and LagrangianScales in Large Reynolds Number Turbulence, J. Atmos. Sci., 20, 115-119, 1963.Heisenberg, W.: Zur Statistischen Theorie der Turbulenz, Z. Phys., 124,628-657, 1948.Hinze, J. O.: Turbulence, McGraw-Hill, New York, 1975.Kolmogorov A. N.: The Local Structure of Turbulence in IncompressibleViscous Fluid for Very Large Reynolds Numbers, Dokl. Akad.  Nauk.SSSR, 30, 301-305, 1941.Lesieur, M., MØtais, O.: New Trends in Large-Eddy Simulations ofTurbulence, Annu. Rev. Fluid. Mech., 28, 45-82, 1996.Muschinski, A., Roth, R.: A Local Interpretation of HeisenbergsTransfer Theory, Beitr. Phys. Atmosph., 66, 335-346, 1993.Pope, S. B.:  Ten Questions Concerning the Large-Eddy Simulation ofTurbulent Flows, New J. of  Phys., 6, 1-24, 2004.50                                             CiŒncia e Natura Especial, UFSM

Heisenberg.s Turbulent Spectral Theory Determining the Filtering Procedure in les Models

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Heisenberg.s Turbulent Spectral Theory Determining the Filtering Procedure in les Models

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