fluid using the standard Oldroyd-B model [14] together with the incompressibility condition ∇ · u = 0. In (1) T ( x , t ) is the temperature field, proportional to the density via the thermal expansion coefficient β as ρ = ρ 0 [1 − β ( T − T 0 )] ( ρ 0 and T 0 are reference values), σ$_{ij}$ ( x , t ) is the positive symmetric conformation tensor of polymer molecules, g = (0 , 0 , − g ) is gravity acceleration, ν is the kinematic viscosity, κ is the thermal diffusiv- ity, η is the zero-shear polymer contribution to viscosity (proportional to polymer concentration) and τ$_{p}$ is the (longest) polymer relaxation time [14]. The initial condition for the RT problem is an unstable temperature jump T ( x , 0) = − ( θ 0 / 2) sgn ( z ) in a fluid at rest u ( x , 0) = 0 and coiled polymers σ ( x , 0) = I . The physical assumptions under which the set of equations (1) is valid are of small Atwood number A = (1 / 2) βθ 0 (dimensionless density fluctuations) and dilute polymer solution. Experimentally, density fluctuations can also be obtained by some additives (e.g., salts) instead of temper- ature fluctuations: within the validity of Boussinesq ap- proximation, these situations are described by the same set of equations (1). In the following, all physical quanti- ties are made dimensionless using the vertical side, L$_{z}$ , of the computational domain, the temperature jump θ 0 and the characteristic time τ = ( L$_{z}$/Ag ) 1 / 2 as fundamental units. Elasticity of the polymer solution is measured by the Deborah number De , the ratio of polymer relaxation time to a characteristic time of the flow. In our unsteady case De grows in time starting from De = 0, therefore viscoelastic effects are initially absent. An estimate of the largest Deborah number achievable is based on the large scale convective time as De = τ$_{p}$/τ . case (of about 40% at t = 2 . 5 τ ). This is the fingerprint of a “drag reduction” as defined for homogeneous-isotropic turbulence in the absence of a mean flow [10, 11]. ∂$_{t}$ u + u · ∇ u = −∇ p + ν ∇ $^{2}$u − β g T + 2 νη τ$_{p}$ ∇ · σ ∂T + u · ∇ T = κ ∇ $^{2}$T ∂$_{t}$σ + u · ∇ σ = ( ∇ u ) T · σ + σ · ( ∇ u ) − 2 $_{τ$_{p}$}$( σ − I ) (1) 2 0 3 6 9 12 15  0  1  2  3 E (x 10 -3 ) t 0.1 0.12 0 1 2 3 P t 0 3 6 9 0 1 2 3 K (x 10 -3 ) t FIG. 2: Time evolution of total energy E = K + Σ, kinetic energy K (upper inset) and potential energy P (lower inset) for the Newtonian run ( De = 0, black continuous line) and the viscoelastic run ( De = 0 . 2, red dashed line). FIG. 3: Time evolution of velocity anisotropy w rms /u rms where w and u are the vertical and one horizontal veloc- ity components, respectively. Black line is the Newtonian run at De = 0, red dashed line is the viscoelastic run at De = 0 . 2. Inset: the evolution of the ratio of velocity gradi- ents ( ∂ z w ) rms / ( ∂ x u ) rms .  1.5  2  2.5  3  3.5  4  0  0.5  1  1.5  2 w rms /u rms t 1 2 3 0 1 2 $_{(}$∂ z  w) rms /( x  u) rms t The most important effect of polymers on turbulent velocity is to generate more coherent thermal plume with respect the Newtonian case, as it is evident in Fig. 1. This reflects in larger vertical component of the velocity with respect the horizontal one, i.e. , an increased anisotropy of the velocity field. This effect is evident in Fig. 3 where we plot the ratio of vertical rms velocity w$_{rms}$ to horizontal one u$_{rms}$ . The anisotropy ratio, which is around 1 . 8 for the Newtonian case [15], becomes larger than 2 . 5 for the viscoelastic run. More important, in the viscoelastic case where P = − βg 〈 zT 〉 is the potential energy and ε$_{ν}$ = ν 〈 ( ∂$_{α}$u$_{β}$ ) $^{2}$〉 is the viscous dissipation and the last term represents elastic dissipation. Because this last term in (2) is not negative, one might expect that the presence of polymers accelerates the consumption of potential energy with respect to the Newtonian case ( η = 0), as it is indeed observed in Fig. 2. Of course, the speed-up of potential energy consump- tion due to polymers does not automatically imply the in- crease of kinetic energy growth. Part of potential energy is indeed converted to elastic energy Σ by polymers elon- gation. The inset of Fig. 2 shows indeed that kinetic en- ergy for viscoelastic runs is larger than in the Newtonian Total energy of the solution has an additional elastic contribution to kinetic energy E = K + Σ = (1 / 2) 〈 u $^{2}$〉 + ( νη/τ$_{p}$ ) 〈 trσ 〉 and the energy balance for (1) reads − dP dt = βg 〈 wT 〉 = dE dt + ε$_{ν}$ + 2 νη τ 2 p [ 〈 trσ 〉 − 3] (2)