Within the framework of the Navier–Stokes equations, the Weissenberg effect of turbulence is investigated. We begin with our investigation on the elastic effect of homogeneous turbulent shear flow. First, in the sense of Truesdell (Physics of Fluids, 1964) on the natural time of materials, we derive the natural time of turbulence, and use it together with the natural viscosity of turbulence derived in the article of Huang et al. (Journal of Turbulence, 2003) to define the natural Weissenberg number of turbulence as a measure of the elastic effect of homogeneous turbulence. Second, we define a primary Weissenberg number of turbulence, which in laminar flow reduces to the Weissenberg number widely applied in rheology to characterize the elasticity of visco-elastic fluids. Our analysis based on the experimental results of Tavoularis and Karnik (Journal of Fluid Mechanics, 1989) indicates that the larger is the Weissenberg number of turbulence, the more elastic becomes the turbulent flow concerned. Furthermore, we put forth a general Weissenberg number of turbulence, which includes the primary Weissenberg number of turbulence as a special case, to measure the overall elastic effects of turbulence. Besides, it is shown that the general Weissenberg number can also be used to characterize the elastic effects of non-Newtonian fluids in laminar flow.

Introduction

It is well known that one of the most famous actions of normal stress differences is the appearance of the Weissenberg effect in a visco-elastic fluid. In his celebrated article entitled “A continuum theory of rheological phenomena”, Weissenberg [1] reported the so-called rod-climbing phenomenon in laminar flow of a visco-elastic fluid, a popular phenomenon that can be observed in daily life when one stirs a bowl of cake batter or a can of paint—though, interestingly, this phenomenon can never be seen in laminar flow of a Newtonian fluid, such as drinking water. In fact, it is the inequality of the normal stresses, a.k.a. the normal stress effect or the Weissenberg effect, that plays a vital role in this rod-climbing phenomenon of a fluid. The normal stress effect of a visco-elastic fluid generated in a viscometric flow, i.e., the Weissenberg effect, is the equivalent of the Poynting effect [2] in a nonlinear elastic material—that is, to produce a state of simple shear in a nonlinear elastic material, normal stresses as well as a shear stress are needed, and the normal stresses are unequal. It was Truesdell [3] who first proposed in 1952 to call this phenomenon the Poynting effect. Interestingly, both well-known effects are a direct consequence of non-zero normal stress differences of the materials under shear or shearing.