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the anisotropy persists also at small scales ( i.e. , in the ratio of velocity gradients), while it is almost absent in the Newtonian case (see inset of Fig. 3). 

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Despite the fact that RT turbulence has vanishing mean flow, a natural mean velocity is provided by the growth of the width h ( t ) of the turbulent mixing layer where heavy and light fluids are well mixed. For ordi- nary fluids at small viscosity, as a consequence of con- stant acceleration, one expects h ( t ) = αAgt 2 where α is a dimensionless parameter to be determined empirically [16–18]. Several definitions of h ( t ) have been proposed, based on either local or global properties of the mean temperature profile T ( z, t ) (the overbar indicates aver- age over the horizontal directions) [19–22]. The simplest measure h$_{r}$ is based on the threshold value of z at which T ( z, t ) reaches a fraction r of the maximum value i.e. T ( ± h$_{r}$ ( t ) / 2 , t ) = ∓ rθ 0 / 2. 

Figure 4 shows the growth of the mixing layer thick- ness for both Newtonian and viscoelastic RT turbulence. As already suggested by Fig. 1, in the viscoelastic solu- tion the growth of the mixing layer is faster than in the Newtonian case ( i.e. larger h ( t ), up to 30% at t = 2 . 5 τ ), therefore we have an effect of polymer drag reduction , i.e. polymer addiction makes the transfer of mass more efficient. The inset of Fig. 4 shows that the increased efficiency is a global property of the mixing layer and the temperature profile of the viscoelastic solution cor- responds to the profile of the Newtonian case at a later time. 

Also in Fig. 4 we plot the variance profiles of temper- ature field computed at different times for both Newto- nian and viscoelastic turbulence. In both cases, σ 2 $_{T}$( z ) 

at different times collapse when plotted as a function of rescaled variable z/h ( t ). Therefore as turbulence de- velops in the domain, the level of temperature fluctua- tions within the mixing layer remains constant as a con- sequence of new fluctuations introduced by plumes enter- ing from unmixed regions. As Fig. 4 indicates, the level of fluctuations is larger in the viscoelastic case, as a con- sequence of the reduced mixing at small scales (already observed in Fig. 1). We remark that all together these results are consistent with the accepted phenomenology of viscoelastic homogeneous-isotropic turbulence where polymers simultaneously reduce energy at small scales and enhance energy contain at large scales [10–12]. 

The turbulent mixing layer is responsible for the huge enhancement of the heat exchange with respect to the steady conductive case. The dimensionless measure of the heat transport efficiency is usually given by the Nus- selt number Nu = 〈 wT 〉 h/ ( κθ 0 ), the ratio between con- vective and conductive heat transport. For a convective flow in the fully developed turbulent regime, the Nusselt number is expected to behave as a simple scaling law with respect to the dimensionless temperature jump which de- fines the Rayleigh number Ra = Agh $^{3}$/ ( νκ ) [23]. For a flow in which boundary layers are irrelevant, as in our case, Kraichnan predicted many years ago the so-called ultimate state of thermal convection for which (a part logarithmic corrections) [23, 24] 

where C and D are numerical coefficients. The ultimate state regime has indeed recently been observed in numer- ical simulations of RT turbulence both in two and three dimensions [15, 25, 26]. 

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