Qian Jian (1939–2018) and his contribution to small-scale turbulence studies

Qian (Tsien) Jian (1939–2018), a Chinese theoretical physicist and fluid dynamicist, devoted the second part of his scientific life to the physical understanding of small-scale turbulence to the exclusion of all else. To place Qian's contribution in an appropriate position in the field of small-scale turbulence, a historical overview and a state-of-the art review are attempted. Qian developed his own statistical theory of small-scale turbulence based on the Liouville [“Sur l'équation aux différences partielles,” J. Math. Pures Appl. 18, 71–72 (1853)] equation and a perturbation variational approach to non-equilibrium statistical mechanics, which is compatible with the Kolmogorov–Oboukhov energy spectrum. Qian's statistical theory of small-scale turbulence, which appears mathematically and physically valid, successfully led to his contributions to (i) the closure problem of turbulence; (ii) one-dimensional turbulence; (iii) two-dimensional turbulence; (iv) the turbulent passive scalar field; (v) the cascade model of turbulence; (vi) the universal equilibrium range of turbulence; (vii) a simple model of the bump phenomenon; (viii) universal constants of turbulence; (ix) the intermittency of turbulence; and perhaps most importantly, and (x) the effect of the Taylor microscale Reynolds number ( R λ) on both the width of the inertial range of finite R λ turbulence and the scaling exponents of velocity structure functions. In particular, Qian found that the inertial range cannot exist when R λ ≪ 2000. In contrast to the prevailing intermittency models, he discovered that normal scaling is valid in the real Kolmogorov inertial range when R λ approaches infinity while the anomalous scaling observed in experiments reflects the finite R λ effect ( Q e). He then made a correction to the famous Kolmogorov [“Dissipation of energy in the locally isotropic turbulence,” Dokl. Akad. Nauk SSSR 32(1), 19–21 (1941c) (in Russian); reprinted in Proc. R. Soc. London A 434, 15–17 (1991)] equation and obtained the finite R λ effect equation or the Kolmogorov–Novikov–Qian equation. He also independently derived the decay law of the finite R λ effect. Qian steered all of us along the right path to an improved understanding of small-scale turbulence and solutions to its problems. Qian is credited with his contribution to enhanced knowledge about the finite R λ effect of turbulence, which has profoundly shaped and stimulated thinking about the K41 turbulence, the K62 turbulence, and t