%0 Book Section %1 statphys23_0005 %A Shibata, H. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K boltzmann coefficient coefficients equation homogeneous lattice method navier-stokes statphys23 topic-5 transport turbulence turbulent viscosity %T Turbulent Viscosity Coefficient in Low-Reynolds-Number Turbulence %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=5 %X The turbulent transport coefficients have been studied extensively for long years1. Recently, the turbulent transport coefficients have been calculated on the basis of statistical mechanics2. Above all, the divergence of the turbulent viscosity coefficient was found in the Rayleigh-B\' enard convection by numerical calculation. These results are going to be confirmed. E. Helfand represented the statistical mechanical expression of the transport coefficients3. He assumed that fluids satisfied the Navier-Stokes equation with the molecular viscosity coefficient. In addition, the velocity components were assumed to be subject to the Maxwell distribution. His expression holds for the quiescent fluids, i.e., the fluids in a microscale. However, the concepts of the turbulent viscosity coefficients are necessary for us to characterize fluids in the state of turbulence. It should be noticed that the turbulent transport coefficients are quantities in the scales much larger than the microscale. So, we assume the Navier-Stokes equation with the turbulent viscosity coefficient for a fluid parcel. In addition, the velocity components of the fluid parcel are assumed to satisfy the Gaussian statistics. Then it is possible for us to describe the same formulas for the turbulent transport coefficients as the ones by Helfand for the transport coefficients. Here we take up the turbulent viscosity coefficient. The formula for the turbulent viscosity coefficient is expressed as $$ =V kTınt_0^ınfty dt C(t), \eqno(1) $$ $$ C(t) <J_xy(t)J_xy(0)> -<J_xy(t)><J_xy(0)>, \eqno(2) $$ and $$ J_xy(t) ınt_V d r (r, t) v_x (r, t) v_y(r,t), \eqno(3) $$ where $V$ is the volume where the turbulent viscosity coefficient is calculated. $k$ is a constant and $T$ is the effective temperature. $<>$ means the long time average along the long trajectory of a solution. The behavior of the turbulent viscosity coefficient in 3-dimensional turbulence is shown here. The lattice Boltzmann method is used in order to simulate the homogeneous turbulence and to calculate the turbulent viscosity coefficient of it. It is clearly shown that the turbulent viscosity coefficient becomes large as the Reynolds number of the homogeneous turbulence becomes large. The dependency of the turbulent viscosity coefficient on the Reynolds number will be shown.\\ 1) S.B. Pope, Turbulent Flows, Cambridge Univ. Press, Cambridge, 2000.\\ 2) H. Shibata, Physica A 333, 71(2004).\\ 3) E. Helfand, Phys. Rev. 119, 1(1960).

@incollection{statphys23_0005, abstract = {The turbulent transport coefficients have been studied extensively for long years[1]. Recently, the turbulent transport coefficients have been calculated on the basis of statistical mechanics[2]. Above all, the divergence of the turbulent viscosity coefficient was found in the Rayleigh-B{\' e}nard convection by numerical calculation. These results are going to be confirmed. E. Helfand represented the statistical mechanical expression of the transport coefficients[3]. He assumed that fluids satisfied the Navier-Stokes equation with the molecular viscosity coefficient. In addition, the velocity components were assumed to be subject to the Maxwell distribution. His expression holds for the quiescent fluids, i.e., the fluids in a microscale. However, the concepts of the turbulent viscosity coefficients are necessary for us to characterize fluids in the state of turbulence. It should be noticed that the turbulent transport coefficients are quantities in the scales much larger than the microscale. So, we assume the Navier-Stokes equation with the turbulent viscosity coefficient for a fluid parcel. In addition, the velocity components of the fluid parcel are assumed to satisfy the Gaussian statistics. Then it is possible for us to describe the same formulas for the turbulent transport coefficients as the ones by Helfand for the transport coefficients. Here we take up the turbulent viscosity coefficient. The formula for the turbulent viscosity coefficient is expressed as $$ \nu ={V \over kT}\int_0^{\infty} dt C(t), \eqno(1) $$ $$ C(t) \equiv <J_{xy}(t)J_{xy}(0)> -<J_{xy}(t)><J_{xy}(0)>, \eqno(2) $$ and $$ J_{xy}(t) \equiv \int_V d \vec r \rho (\vec r, t) v_x (\vec r, t) v_y(\vec r,t), \eqno(3) $$ where $V$ is the volume where the turbulent viscosity coefficient is calculated. $k$ is a constant and $T$ is the effective temperature. $<\cdots >$ means the long time average along the long trajectory of a solution. The behavior of the turbulent viscosity coefficient in 3-dimensional turbulence is shown here. The lattice Boltzmann method is used in order to simulate the homogeneous turbulence and to calculate the turbulent viscosity coefficient of it. It is clearly shown that the turbulent viscosity coefficient becomes large as the Reynolds number of the homogeneous turbulence becomes large. The dependency of the turbulent viscosity coefficient on the Reynolds number will be shown.\\ 1) S.B. Pope, Turbulent Flows, Cambridge Univ. Press, Cambridge, 2000.\\ 2) H. Shibata, Physica A 333, 71(2004).\\ 3) E. Helfand, Phys. Rev. 119, 1(1960).}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Shibata, H.}, biburl = {https://www.bibsonomy.org/bibtex/2dcc458340eee940a21f7dbd98b4e0906/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {5b2449b1f2d4d051612bec6fc460f696}, intrahash = {dcc458340eee940a21f7dbd98b4e0906}, keywords = {boltzmann coefficient coefficients equation homogeneous lattice method navier-stokes statphys23 topic-5 transport turbulence turbulent viscosity}, month = {9-13 July}, timestamp = {2007-06-20T10:16:09.000+0200}, title = {Turbulent Viscosity Coefficient in Low-Reynolds-Number Turbulence}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=5}, year = 2007 }