Fernando T. Pinho

Direct numerical simulations employing the finitely extensible nonlinear elastic constitutive model closed with Peterlin's approximation (FENE-P) are used to investigate the far-field region of turbulent planar wakes of viscoelastic fluids and to develop the theory describing these flows. The theoretical results display excellent agreement with the simulations and provide new scaling laws for the evolution of the shear layer thickness $\delta (x) \sim x^{1/2}$ , mean velocity deficit ${\rm \Delta} U(x) \sim x^{-1/2}$ and, for very high Deborah numbers, of the maximum polymer shear stresses $\sigma ^{[p]}_c(x) \sim x^{-2}$ and averaged polymer chain extension $\textrm {tr}(\bar {C}(x) - \boldsymbol{\mathsf{I}}) \sim x^{-2}$ , where $x$ is the streamwise distance from the solid body generating the wake. The theory is able to show the existence of self-similarity for the profiles of mean velocity, mean polymer shear stress, averaged polymer chain extension and the conditions for similarity of the turbulent shear stress, and is very well supported by the numerical simulations. Similarly to the case of viscoelastic turbulent planar jets (Guimarães et al., J. Fluid Mech., vol. 899, 2020, p. A11), when the inlet Weissenberg and Deborah numbers are sufficiently large, turbulent viscoelastic wakes exhibit a considerable reduction of the spreading rate and of the normalised Reynolds stresses. However, for very large downstream locations the turbulent viscoelastic wake recovers the classical evolution laws observed for Newtonian turbulent planar wakes.

Direct numerical simulations (DNS) of turbulent fronts spreading into an irrotational flow region are used to analyse the turbulent entrainment mechanism for viscoelastic fluids. The simulations use the FENE-P fluid model and are initiated from DNS of isotropic turbulence with Weissenberg and turbulence Reynolds numbers varying in the ranges $1.30 \le Wi \le 3.46$ and $206 \le Re_{\lambda }^{0} \le 404$ , respectively. The enstrophy dynamics near the turbulent/non-turbulent interface (TNTI) layer, that separates regions of turbulent and irrotational flow, includes a new mechanism – the viscoelastic production – caused by the interaction between the vorticity field and the polymer stresses. This term can be a sink or a source of enstrophy in the turbulent core region of the flow, depending on the Weissenberg number, and contributes to the initial growth of the enstrophy in the viscous superlayer, together with the viscous diffusion, which is the only mechanism present for Newtonian fluids. For low and moderate Weissenberg numbers the scaling of the TNTI layer is similar to the scaling of TNTI layers for Newtonian fluids, but this is no longer the case at high Weissenberg numbers where the enstrophy tends to be concentrated into thin vortex sheets instead of vortex tubes. Finally, it is shown that the substantial decrease of the entrainment rates observed in turbulent flows of viscoelastic fluids, compared with Newtonian fluids, is caused by a reduction of the surface area and fractal dimension of the irrotational boundary, originated by the depletion of ‘active’ scales of motion in the fluid solvent caused by the viscoelasticity.

This work presents a detailed numerical investigation on the required development length ($\mathcal{L}=L/B$) in laminar Newtonian fluid flow in microchannels with rectangular cross section and different aspect ratios ($AR$). The advent of new microfluidic technologies shifted the practical Reynolds numbers ($Re$) to the range of unitary (and even lower) orders of magnitude, i.e., creeping flow conditions. Therefore, accurate estimations of $\mathcal{L}$ at $Re\le \mathcal{O}\left(1\right)$ are important for microsystem design. At such low Reynolds numbers, in which inertial forces are less dominant than viscous forces, flow characteristics become necessarily different from those at the macroscale where $Re$ is typically much larger. A judicious choice of mesh refinement and adequate numerical methods allowed obtaining accurate results and a general correlation for estimating $\mathcal{L}$, valid in the ranges $0\le Re\le 2000$ and $0.1\le AR\le 1$, thus covering applications in both macro and microfluidics.