When long-chain flexible polymers are dissolved in a turbulent flow, the flow properties can be changed drastically by reducing the drag and enhancing the mixing. A fundamental riddle in materials science is to understand how these polymer additives interact with different spatial scales in turbulent flow to alter the turbulence energy transfer. In a new report now on Science Advances, Yi-Bao Zhang and a research team showed how turbulent kinetic energy could be transferred across different scales in the presence of polymer additives. The team noted the emergence of a previously unidentified scaling range known as the elastic range, where an increased amount of energy could be transferred by the elasticity of the polymers. The findings have important applications across many turbulent systems, including turbulence in plasmas or superfluids.
Flow properties and velocity structure-function (VSF)
Materials scientists have shown how dissolving a tiny amount of long-chain flexible polymer in a fluid could change the flow properties. Reynolds number helps predict flow patterns under different fluid flow situations. At low Reynolds, normal fluid flow is stable and laminar, and the addition of polymers can induce strong fluctuations to create elastic turbulence. High-Reynolds number turbulent flows can result in substantial reduction of drag and the enhancement or reduction of convective heat transfer. Researchers aim to understand the interaction between polymers and the turbulence cascade for theoretical reasons and practical applications. It is presently critical to comprehensively measure the energy spectra or the velocity structure-function (VSF) in turbulent flows with polymer additives. In this report, Zhang et al. detailed experimental observation of the new elastic range in a laboratory turbulent flow setup and measured the scaling of the velocity structure-function in the new elastic range, which diverged from any existing theory.
The experimental setup
The scientists generated the turbulent flow in a von Kármán swirling apparatus containing two counter-rotating disks enclosed in a cylindrical tank filled with 100 L of water or polymer solutions. They measured the three components of fluid velocity in a central planar passing through the axis of the tank using a stereoscopic particle image velocimetry (PIV) system. According to the measurements, the flow near the center of the tank was nearly homogeneous and isotropic for both flows with water and with dilute solutions of long-chain polymers in water. The scientists used polyacrylamide (PAM) for the polymers during the experiments. The team noted a Reynolds number for the pure water to range from 340 to 350, indicating a fully developed inertial range in turbulence. At equilibrium, the polymers remained in the coiled state. During weak flow in the solution, the polymer remained in the coiled state with negligible effect on the flow. Comparatively, during intense flow, the polymers stretched to store elastic energy for release into the fluid. The fluid then displayed viscoelastic behavior. During turbulent flows, they characterized the transition using the Weissenberg number to measure the polymer relaxation time relative to the turbulence time scale. For the polymers to be stretched by the flow, the Weissenberg number had to be larger than unity. During the measurements, Zhang et al. only considered the interaction between the fluid and single polymer, while neglecting direct polymer-polymer interactions.
Measuring the elastic range
To then quantify the boundary of the elastic range, the researchers adapted an analytical form of the second-order longitudinal velocity structure-function (VSF) for Newtonian turbulence proposed by Batchelor et al. As the polymer concentration in the sample increased, the measured mean square of the derivative of the longitudinal velocity decreased, indicating that the energy dissipated by viscosity at very small scales—consistent with previous experiments and numerical simulations. The decreased viscous dissipation with polymer concentration alongside the independence of the turbulence energy transfer rate at larger scales indicated the energy transfer rate in the elastic range to vary non-trivially. The team therefore next investigated methods to obtain the energy transfer rate with a setup that drew incrementally more energy into the elastic energy of the polymer due to interactions between turbulent eddies and polymer elasticity.
The crossover scales
Zhang et al. then identified the elastic range and examined the crossover scale between the elastic range and the dissipation range (termed a1), followed by the crossover scale between the elastic range and inertial range (termed a2). They then studied how the two crossover scales varied with control parameters. The crossover scale between the elastic range and the dissipation range appeared to decrease slightly with polymer concentration; however, the team credited this to likely contamination due to poor spatial resolution of the particle image velocimetry measurements. The scientists then corrected the observed inaccuracy as a function of polymer-concentration and showed that for small polymer concentrations, the crossover scale between the elastic range and inertial range was very small.
Scaling of the high-order velocity structure-function
The team also investigated the problem of turbulence flow to scale the high-order velocity structure-function (VSF) on the inertial range with water and polymer additives. The resulting similarities in behavior showed how the elastic range of the energy transfer through scales was altered by polymers. The team expects to observe common features between Newtonian turbulence and polymeric turbulence. The results showed excellent agreement between the data and prediction to show how the energy transfer was substantially altered by polymers in the elastic range. Meanwhile, the fluctuating local energy transfer followed similar statistical descriptions as that of Newtonian turbulence.
In this way, Yi-Bao Zhang and colleagues experimentally observed the scaling of the elastic range in the turbulent flow with polymer additives. They measured the turbulent kinetic energy transfer in the presence of polymer additives. As the energy flux through the turbulent flow decreased, the energy flux through the elastic degree of freedom of polymers increased. The study shed new light to conduct further theoretical and numerical investigations on the interaction between the elasticity of polymer additives and turbulent eddies. These experimental processes can be noted in practice within physical mechanisms such as electromagnetic interactions in plasmas and Alfvén waves in superfluids.