Critical points in two-dimensional stationary homogeneous isotropic turbulence

Abstract

Basic properties of critical points in two dimensions are reviewed and related to the velocity and acceleration field of two-dimensional turbulence. A direct numerical simulation (DNS) of two-dimensional homogeneous isotropic turbulence with an inverse energy cascade and a k−5/3 power law is used to study critical points of these fields. The velocity stagnation point based pair separation model of Goto and Vassilicos (S Goto and J C Vassilicos, 2004, New J.Phys., 6, p.65) is revisited and placed on a sound mathematical foundation. The DNS is used to study the time-asymmetry observed between forward and backward separation. A new method has been employed to obtain values for the Richardson constants and the ratio of them for the backwards and forwards case, which is gb/gf = (0.92±0.03) and hence, exhibits a qualitatively different behaviour from pair separation in three-dimensional turbulence, where gb > gf (J Berg et al. , 2006, Phys.Rev.E, 74(1), p.016304). An explanation for this behaviour based on the timeasymmetry related to the inverse versus forward energy cascade is suggested. Zero Acceleration Points (ZAPs) and flow structures around them are studied using the same DNS. A well-defined classification of ZAPs in terms of the acceleration gradient tensor’s (∇a) invariants is presented. About half of all ZAPs are Anti-ZAPs (with det[∇a] < 0) and the number of vortical and straining ZAPs (with det[∇a] > 0) is about the same. Vortical and straining ZAPs are swept by the local fluid velocity to a good statistical approximation whereas Anti-ZAPs are not. The average life-time of ZAPs seems to scale with the time-scale of the smallest eddies in the turbulence, though ZAPs (in particular vortical ones) are able to survive up to a few integral time scales. The new ZAP classification can also be applied to extended flow regions and a discussion of the length-scales and sizes characterising these regions and the distances between ZAPs is given.