Polymer concentration and properties of elastic turbulence in a von Karman swirling flow

We report detailed experimental studies of statistical, scaling, and spectral properties of elastic turbulence (ET) in a von Karman swirling flow between rotating and stationary disks of polymer solutions in a wide, from dilute to semidilute entangled, range of polymer concentrations phi. The main message of the investigation is that the variation of phi just weakly modifies statistical, scaling, and spectral properties of ET in a swirling flow. The qualitative difference between dilute and semidilute unentangled versus semidilute entangled polymer solutions is found in the dependence of the critical Weissenberg number Wi(c) of the elastic instability threshold on phi. The control parameter of the problem, the Weissenberg number Wi, is defined as the ratio of the nonlinear elastic stress to dissipation via linear stress relaxation and quantifies the degree of polymer stretching. The power-law scaling of the friction coefficient on Wi/Wi(c) characterizes the ET regime with the exponent independent of phi. The torque Gamma and pressure p power spectra show power-law decays with well-defined exponents, which has values independent of Wi and phi separately at 100 <= phi <= 900 ppm and 1600 <= phi <= 2300 ppm ranges. Another unexpected observation is the presence of two types of the boundary layers, horizontal and vertical, distinguished by their role in the energy pumping and dissipation, which has width dependence on Wi and phi differs drastically. In the case of the vertical boundary layer near the driving disk, w(v)(v) is independent of Wi/Wi(c) and linearly decreases with phi/phi*, while in the case of the horizontal boundary layer w(v)(h) its width is independent of phi/phi*, linearly decreases with Wi/Wi(c), and is about five times smaller than w(v)(v). Moreover, these Wi and phi dependencies of the vertical and horizontal boundary layer widths are found in accordance with the inverse turbulent intensity calculated inside the boundary layers V-theta(h)/V-theta(hrms) and V-theta(v)/V-theta(vrms), respectively. Specifically, the dependence of V-theta(v)/V-theta(vrms) in the vertical boundary layer on Wi and phi agrees with a recent theoretical prediction [S. Belan, A. Chernych, and V. Lebedev, Boundary layer of elastic turbulence (unpublished)].