Mixing in modulated turbulence. Analytical results

Abstract

Recent numerical results show that if a scalar is mixed by periodically forced turbulence, the average mixing rate is directly affected for forcing frequencies small compared to the integral turbulence frequency. We elucidate this by an analytical study using simple turbulence models for spectral transfer. Adding a large amplitude modulation to the forcing of the velocity field enhances the energy transfer and simultaneously diminishes the scalar transfer. Adding a modulation to a random stirring protocol will thus negatively influence the mixing rate. We further derive the asymptotic behaviour of the response of the passive scalar quantities in the same flow for low and high forcing frequencies.

Passive scalar mixing in modulated turbulence

“…In this manuscript, we have answered to the question whether a periodic modulation of a turbulent flow changes its mixing properties. It was clearly shown that for large modulation amplitudes, the modulation affects the energy transfer positively, whereas it diminishes the mixing rate, as was predicted in a recent analytical study (Bos and Rubinstein 2017). The average mixing rate was shown to decrease by approximately 20%, for two distinct Reynolds numbers, whereas the energy transfer rate increases by approximately 7%.…”

“…In this summary we will only consider the case where the kinetic energy injection is modulated ¹ p 0, = q p 0, and where the modulation frequency is small. For the high frequency analytical results we refer to references (Bos et al 2007, Bos andRubinstein 2017).…”

“…More importantly, retaining second-order contributions in the perturbation analysis, we obtained estimates of the influence of the modulation on the time-averaged quantities, χ and χ θ (Bos and Rubinstein 2017). We will here give a simplified derivation of the results in (Bos and Rubinstein 2017). For this analysis we will use the Taylor expression for the dissipation (or injection) rate In our setting the forcing-scale is fixed, so that we find immediately…”

“…In a recent paper [16], we carried out an analytical study of such a case: mixing in periodically forced turbulence. We showed that the second-order contributions of a perturbation analysis of the nonlinear transfer with respect to a modulation of the velocity forcing will lead to (1) an enhanced energy transfer (2) a reduced scalar transfer.…”

Periodically Modulated Thermal Convection

“…In periodically driven turbulence in shear flows, a meanfield theory has been used to analyze the resonance maxima of the Reynolds number [3,4]. Periodic forcing in other turbulent systems, for example, in the homogeneous isotropic turbulence [5][6][7][8], pipe flow [9][10][11][12], channel flow [13,14], Taylor-Couette flow [15,16], and Rayleigh-Bénard (RB) convection [17][18][19], is also shown to have highly nontrivial response properties.…”