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 φ. The main message of the investigation is that the variation of φ 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 Wic of the elastic instability threshold on φ. 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/Wic characterizes the ET regime with the exponent independent of φ. The torque Γ and pressure p power spectra show power-law decays with well-defined exponents, which has values independent of Wi and φ separately at 100≤φ≤900 ppm and 1600≤φ≤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 φ differs drastically. In the case of the vertical boundary layer near the driving disk, wvv is independent of Wi/Wic and linearly decreases with φ/φ∗, while in the case of the horizontal boundary layer wvh its width is independent of φ/φ∗, linearly decreases with Wi/Wic, and is about five times smaller than wvv. Moreover, these Wi and φ dependencies of the vertical and horizontal boundary layer widths are found in accordance with the inverse turbulent intensity calculated inside the boundary layers Vθh/Vθhrms and Vθv/Vθvrms, respectively. Specifically, the dependence of Vθv/Vθvrms in the vertical boundary layer on Wi and φ agrees with a recent theoretical prediction [S. Belan, A. Chernych, and V. Lebedev, Boundary layer of elastic turbulence (unpublished)].