Spectrum of Supersaturation Fluctuations and Lagrangian-Eulerian Computation

Variance spectrum of supersaturation fluctuations is theoretically and numerically studied. Interaction between the water droplets and the water vapor mixing ratio is expressed as the condensation rate which is microscopically corpuscular but treated as continuum by the coarse graining. With the assumption of small amplitudes of the supersaturation and the liquid water mixing ratio, the condensation term appears as the linear damping term in the supersaturation equation with the phase relaxation time τ_θ. And the supersaturation is treated as passive scalar (no reaction to the fluid motion). The theory predicts that the variance spectrum of the supersaturation fluctuations has three scaling ranges depending on the wavenumber dependent Damköhler number D(k)=τ(k)/τ_θ where τ(k) is the eddy turn over time at the wavenumber k, the two k^-5/3 ranges in the inertial convective range and a modified Batchelor spectrum k^-1-γ with γ=2C_BD_K in the so called viscous-convective range, where C_B is the Batchelor constant and D_K=τ(k_d)/τ_θ is Damköhler number at Kolmogorov length η=1/k_d. The spectrum k^-1-γ corresponds to that of the scalar at infinite Schmidt number Sc=ν/κ (zero diffusivity). For the verification, we have developed a new method using particles of zero inertia but carrying scalar quantity with relaxation time. In the numerical simulations, the particles are assumed to be convected by the isotropic turbulence and the evolution of scalar on the particle is computed along their Lagrangian trajectory. The scalar carried by the particles is linearly projected on the grid points for the fluid (PIC, particle in cell) when the Eulerian statistics is computed. The shot noise due to the discreteness of particles and the filtering effects due to the linear projection are removed from the raw spectrum, thus we obtain the spectrum of the supersaturation. The direct numerical simulation of the system has been made with the grid points of 2048³ and 2³⁶ particles (about 8 particles in a grid cell on average). We found that the computed spectrum at R_λ∼550 has one Obukhov-Corrsin spectrum k^-5/3 and the Batchelor spectrum k^-1 with the clear transition at kη∼0.04. An extension of the method to the particles with finite inertia, to understand the spectrum of the liquid water contents, is also discussed.