Spectral behavior of low-Prandtl-number temperature fluctuations in isotropic turbulence: Insights from a GOY shell model

The present study investigates the spectral behavior of temperature fluctuations in homogeneous isotropic turbulence at low Prandtl numbers using an extended GOY shell model that incorporates a passive-scalar field. The model enables simultaneous computation of the velocity and temperature dynamics and reveals a unified picture linking the inertial-convective and dissipation ranges. Numerical results demonstrate that the apparent “long-tail” behavior of the temperature fluctuation spectrum at low Prandtl numbers does not arise from delayed diffusion but from a scale effect associated with the reduction of the scalar-dissipation wavenumber kκ=ε/κ31/4. By applying nondimensional variables k∗=k/kκ and Eθ∗k∗=Eθk/χ/ε1/3kκ−5/3, spectra obtained under various Reynolds and Prandtl numbers collapse onto a single curve, confirming a universal transition at k∗≃1 from the inertial-convective to the diffusion-dominated regime. Extending the Hebishima–Inage exponential spectrum, an empirical expression Eθk∝k−53exp−2.37k/kκαθ with the fitted correlation αθ=0.5116Pr3−1.565Pr2+1.6802Pr+0.8692 reproduces the shell-model results quantitatively over the entire wavenumber range. The findings establish a coherent theoretical framework for passive-scalar turbulence and provide a foundation for modeling thermal transport and mixing in low-Prandtl-number fluids such as liquid metals and plasma flows.