Turbulent diffusion

Siegfried Großmann

Fachbereich Physik der Philipps-Universität
Marburg, Germany

Dust, aerosoles, smoke and other air pollutions spread, in still air, by molecular diffusion. In addition there can be advection by atmospheric flow, either well directed in jets or winds, or nondirected in air turbulence. Diffusional spreading by turbulence is extraordinary effective. Instead of the well known linear growth of the mean square particle distance with time $t$

$$\sigma_t^2 = 6 K t ~ , ~ \mbox {regular diffusion} ~ ,$$

$\sigma_t^2$ grows much faster under turbulent advection, according to a cubic $t$ dependence

$$\sigma_t^2 = c \epsilon t^3 ~ , ~ \mbox {turbulent diffusion} ~ .$$

This was discovered by Lewis Fry Richardson (1926, 1929) when he performed his ingenious study of the turbulent diffusivity $K_{turb}$ . He presented data on impressively many scales. A bulk of later measurements confirmed his results, as e.g. balloon campaignes, cf. Lundgren (1981).

The anomalously fast particle distance growth could be explained from fluid dynamics (Navier-Stokes equations) by Grossmann and Procaccia (1984), Effinger and Grossmann (1984), Grossmann (1990) in mean field approximation. New results on dynamical Lagrangian time correlation decay (numerical, Grossmann and Wiele 1997 and analytical, Daems et al. 1999) stimulated efforts to even determine the absolute magnitude as characterized by the prefactor $c$ in addition to the scaling exponent (Grossmann 2002).

In particular the memory effects in the time correlation decay have turned out to be very important. And turbulent intermittency implies additional scale dependence of the turbulent diffusivity (Grossmann 2002). A survey on the past development and on the most recent surprising findings is offered in the lecture.

References
Daems D, Grossmann S, L'vov V and Procaccia I 1999 Phys Rev E60 6656
Effinger H and Grossmann S 1984 Phys Rev Lett 50 442
Grossmann S and Procaccia I 1984 Phys Rev A29 1358
Grossmann S 1990 Annalen der Physik (Leipzig) 47 577
Grossmann S and Wiele C 1997 Z Phys B103 469
Grossmann S 2002 to be published
Lundgren T S 1981 J Fluid Mech 111 27
Richardson L F 1926 Proc Roy Soc (London) A110 709
Richardson L F 1929 Beitr Phys Atm 15 24

Turbulent correlation decay

Siegfried Großmann

Fachbereich Physik der Philipps-Universität
Marburg, Germany

The appropriate tool to study the dynamics of a statistical system is its (stationary) time correlation function. This has also proven to be true for turbulent fluid flow. The observable quantities of interest are here the Lagrangian, scale dependent eddies $v(r)$, i.e., the Eulerian velocity differences ${\bf v}({\bf r};{\bf x},t) = {\bf u}({\bf x} + {\bf r}, t) - {\bf u}({\bf x},t)$. These objects are statistically time $t$ and space ${\bf x}$ independent in stationary and homogeneous turbulence, but depend besides on scale $r$ on the time lapse $\tau$ between two observations of an $r$-eddy at times $t$ and $t + \tau$. By means of a continued fraction expansion the time correlation function can be uniquely expressed in terms of the static, stationary, time independent structure functions; but all orders of those are needed.

Analysis of the dynamical time correlation function $D(r,\tau) = \langle v(r) v(r,\tau) \rangle$ is presented. $D(r,\tau)$ was first studied in 1-pole approximation (Grossmann and Thomae 1982), because no estimate of higher order stationary structure functions was available then. Since Grossmann and Wiele (1997) provided numerical data for large Reynolds number, highly turbulent flow, it became clear that the memory effects surprisingly reduce the decorrelation time. The additional effects of turbulent intermittency were elucidated analytically in Daems et al. (1999). The main, rather unexpected results are:

i. The static multifractality of turbulent flow destroys dynamical scaling despite good scaling of stationary moments, i.e., of power laws in $r$ of all $p^{th}$ order structure functions. ii. The deviations from dynamical scaling are a direct measure of the strength of intermittency. iii. The scale dependence of the correlation decay rate can be expressed approximately in terms of the turbulent structure function of $2^{nd}$ order (Grossmann 2002).

References
Daems D, Grossmann S, L'vov V and Procaccia I 1999 Phys Rev E60 6656
Grossmann S and Thomae S 1982 Z Phys B49 253
Grossmann S and Wiele C 1997 Z Phys B103 469
Grossmann S 2002 to be published

Turbulent heat convection

Siegfried Großmann

Fachbereich Physik der Philipps-Universität
Marburg, Germany

Turbulent heat transport in fluid layers heated from below is one of the most intensely studied fluid flow problems. In 1900 Bénard detected interesting pattern formation in this system, in 1916 Rayleigh calculated the underlying instability. If the heating is increased further, chaotic motion and finally turbulence is observed in the bulk , surrounded by Blasius type boundary layers near the plates and walls. In the 50th and 60th power laws for the heat transport as a function of the thermal driving were suggested for the Rayleigh-Bénard system in accordance with other scaling behavior in turbulence,

$$Nu \propto Ra^{\beta} Pr^{\beta '} ~ .$$

The Nusselt number $Nu$ is the effective heat current including turbulent advection, nondimensionalised with the molecular heat current. The Rayleigh number $Ra$ measures the thermal driving due to the temperature difference $\Delta$ across the fluid layer of height $L$, originating from buoyancy in the gravitational field $g$ by thermal expansion $\alpha$, which is counteracted by the fluid's kinematic viscosity $\nu$ and its thermal diffusivity $\kappa$,

$$Ra = \frac{\alpha g L^3 \Delta}{\nu \kappa} ~ .$$

The Prandtl number $Pr = \nu / \kappa$ weighs the ratio of the space and time scales according to molecular momentum ($\nu$) and energy ($\kappa$) transport. Recent measurements of increasing accuracy (Libchaber group (Castaing et al. 1989), Cioni et al. (1997), Chavanne et al. (1997), Niemela et al. (2000), Xu et al. (2000), Ahlers et al. (2001), Xia et al. (2002)) led to various unexpected results. A unifying theory explains all these data (Grossmann and Lohse 2000, 2001, 2002). In particular the theory's prediction is confirmed that simple power laws are insufficient to describe $Nu$ versus $Ra$ adequately. In the lecture the theory of heat convection in a very extended $Ra-Pr-$parameter space of up to 10 orders of magnitude is presented and discussed.

References
Ahlers G and Xu X 2001 Phys Rev Lett 86 3320
Castaing B, Gunaratne G, Heslot F, Kadanoff L, Libchaber A, Thomae S, Wu X, Zaleski S and Zanetti G 1989 J Fluid Mech 204 1
Chavanne X, Chilla F, Castaing B, Hebral B, Chaboud B and Chaussy J 1997 Phys Rev Lett 79 3648
Cioni S, Ciliberto S and Sommeria J 1997 J Fluid Mech 335 111
Grossmann S and Lohse D 2000 J Fluid Mech 407 27
Grossmann S and Lohse D 2001 Phys Rev Lett 86 3316
Grossmann S and Lohse D 2002 to be published
Niemela J, Skrbek L, Sreenivasan K R and Donnelly R 2000 Nature 404 837
Xia K Q, Lam S and Zhou S Q 2002 Phys Rev Lett 88 064501
Xu X, Bajaj K M S and Ahlers G 2000 Phys Rev Lett 84 4357