Session Q11: Transitional Flows and Non-linear Dynamics I

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The helicity of a laminar vortex ring is prescribed by its geometry in the forms of writhe and twist. In viscous fluids, helicity is not conserved, but nonetheless its evolution is naturally characterized by the geometry and topology of the vorticity field. By generating helical vortices using hydrofoils, we are able to measure their helicity and its evolution over a range of Reynolds numbers. Fully resolved DNS with adaptive mesh refinement provide complementary insight. We present an analytic model for helicity evolution in vortex tubes with a natural geometric interpretation and compare its predictions to experiments and simulations.   

Scaling of the turbulence transition in pipe flow

Despite significant experimental and theoretical advances, a complete physical picture for the turbulence transition in pipe flow remains elusive. Here we present a theoretical model for a flow disturbance’s critical amplitude to trigger sustained turbulent puffs downstream as a function of the pipe’s Reynolds number Re. Current theory (Trefethen 1993) suggests that flow disturbances excite a spectrum of pseudo modes, which grow transiently in time before exponential decay. These modes may in turn excite a sustained travelling wave packet in the pipe, marking the transition to turbulence. We find this yields a simple scaling for the critical disturbance amplitude u ~ g/Re, where g plays the role of an amplitude response function for the travelling wave. We computed g for several pipe disturbance methods and successfully collapsed the data from corresponding experimental papers onto our scaling; to date, flow obstacles (Nishi 2008) and pulsed fluid injections (Hof 2003 & 2005, Peixinho 2007). We find the travelling wave manifests with the same frequency in the direction of flow regardless of the disturbance method, evidencing the same pathway to turbulence.

Grant funding provided by Shell and NSF (CMMI-1346638 to A.J.H., CMMI- 1727565 to J.W.M.B.)   

Dynamic Phase Alignment in Navier-Stokes Turbulence

In Navier-Stokes turbulence, energy and helicity injected at large scales are subject to a joint direct cascade, with both quantities exhibiting a spectral scaling $\sim k^{-5/3}$. A ``na\"ive", dimensional estimate of the spectrum of helicity would, however, yield the scaling $\mathcal{H}(k) \sim k v_{\lambda} \omega_{\lambda} \sim k^{-2/3}$, violating conservation of the helicity flux in the inertial range. We demonstrate via direct numerical simulations that this apparent contradiction is revolved because of the existence of a strong scale-dependent Fourier phase alignment between velocity and vorticity fluctuations, with the phase alignment angle scaling as $\cos\alpha_k\propto k^{-1}$ [L. M. Milanese \textit{et al.}, ``Dynamic Phase Alignment in Navier-Stokes Turbulence", arXiv:2104.13518]. This strong dependence on scale of $\cos\alpha_k$, termed \textit{dynamic phase alignment}, underpins the spectral scaling of helicity, i.e., $\mathcal{H}(k) \sim k v_{\lambda} \omega_{\lambda} \cos\alpha_k \sim k^{-5/3}$. Dynamic phase alignment plays a role in the turbulent dynamics in the presence of two invariants beyond Navier-Stokes, and it has been shown to underpin the joint direct cascade of energy and (generalized) helicity in a variety of turbulent plasma environments [L. M. Milanese \textit{et al.}, ``Dynamic Phase Alignment in Inertial Alfv\'en Turbulence", Physical Review Letters, 2020].    

Measurements of turbulence over a streamwise preferential porous medium

This study experimentally examines the possibility of the turbulent drag reduction by an orthotropic porous medium whose streamwise permeability is larger than the wall-normal permeability.

We made a layered porous medium whose measured streamwise, wall-normal and spanwise permeabilities are K_xx=26.9, K_yy=1.45, and K_zz=4.26 mm², respectively. Hence, ψ_xy = 4.3 and ψ_zy = 1.7, where ψ_xy =√(K_xx/K_yy).These values satisfy the drag-reducing condition suggested by numerical studies in the literature. We carry out particle image velocimetry measurements of turbulent square duct flows over the porous medium at the bulk Reynolds numbers of 3000-15000 and examine the drag reduction probability. Although the present experiments do not show a remarkable turbulence reduction over the porous medium, from the detailed analyses of the obtained data, it is confirmed that the present porous medium is suppressed at least to the same level as that of the flat smooth wall side while the near-wall flow structures are different.   

Odd transport phenomena from broken symmetries

Linear constitutive relations connect fluxes of quantities such as mass, heat, and momentum to corresponding thermodynamic driving forces. Such relations reveal macroscopic consequences of microscopic fluctuations. When time-reversal and parity symmetries are broken at the microscale, as is often the case in chiral active matter, new "odd" transport coefficients can emerge at the continuum scale. We discuss two odd transport coefficients in particular: the odd diffusivity, which generates diffusive fluxes perpendicular to concentration gradients, and the odd viscosity, which generates normal stresses perpendicular to shear flow. In both cases, we show that the behavior under time reversal and parity inversion can be understood through deriving Green-Kubo relations, which additionally provide an avenue for direct numerical verification in model active matter systems using molecular dynamics simulations. Our work provides steps towards extending the framework of non-equilibrium thermodynamics to systems like active matter, which are inherently out of equilibrium, or for which no equilibrium reference state exists.   

Design optimization for Richtmyer—Meshkov instability suppression

The Richtmyer—Meshkov instability (RMI) is a phenomenon that occurs at the interface of two substances of different densities due to an impulsive acceleration, such as a shock wave passing through this interface. Under these conditions, the instability can be seen as interface perturbations begin to grow into jets or spikes of one substance that propagate into the other. The control of RMI growth is one major limiting factor for technological challenges such as inertial confinement fusion, which involves using high-pressure shock waves to implode a fuel target. RMI growth can lead to asymmetry in the implosion process that significantly reduces the energy yield. This work is at the forefront of understanding RMI and designing for the suppression and control of perturbation growth. We use hydrodynamic simulations of impactor shock-wave experiments and design optimization to suppress RMI growth by altering the geometry and other properties of a shocked material target. Our results show that RMI suppression can be achieved by intentionally creating a secondary instability to counteract RMI growth at a perturbed interface.   

Linear stability of the hydromagnetic pipe flow subject to a transverse magnetic field

Despite decades of research, the problem of developping a fully consistent framework to explain the instability of Poiseuille pipe flow remains challenging. Surprisingly, little attention has been devoted to the linear stability analysis of the circular pipe in the hydromagnetic case and the only thorough numerical study available considers a parallel magnetic field acting on the flow.

We analyse here the linear stability of an electrically conducting circular pipe flow subject to a uniform transverse magnetic field. Both classical stability and transient growth theories are used to analyze the evolution of disturbances. We show that the elongation of the velocity profile in the direction of the magnetic field has a destabilizing effect on the flow. The situation is similar when considering the hydrodynamic flow through an elliptic pipe, which has been shown to be linearly unstable even when the aspect ratio of the pipe is just slightly larger than 1. In the hydromagnetic case, this destabilizing effect is however counterbalanced by the Joule dissipation resulting from the occurrence of induced currents. A thorough parametric study of the stability of the flow will be presented for Reynolds numbers up to Re=10^8 in the cases of perfectly conducting and insulating pipe wall.   

Particle capture in a model chaotic flow with a moving capture unit

To investigate the capture of passive scalars in geophysical flows, we computationally study capture in the simpler, but still chaotic, time-dependent 2d double-gyre flow. For a range of model parameters, the double-gyre flow consists of a rapidly mixing chaotic region interspersed with non-mixing islands in which particle trajectories are regular. Here we consider a moving capture unit (plant) rather than a fixed one, because the former can potentially capture more material. We restrict plant motion to be along a straight line connecting the centers of the flow’s two gyres. Three prescriptions of plant motion are studied for flow conditions ranging from non-chaotic to fully chaotic: motion based on maximum capture for non-chaotic flow but applied to chaotic flow; a method based on Lagrangian coherent structures (LCS) in which the plant follows a maximum in the finite-time Lyapunov exponent field; and a biologically-inspired approach that tracks concentration gradients. The capture efficiency depends on flow conditions. The maximum capture approach based on non-chaotic flow is effective when the flow is less chaotic, while the LCS method is better for fully chaotic flow. The biologically-inspired approach works well when plant motion is not restricted to a straight line.   

Physics-constrained data-driven subgrid-scale parameterization of 2D turbulence in the small-data regime

In this work, we develop a data-driven subgrid-scale (SGS) model for large eddy simulation (LES) of 2D turbulence using a fully convolutional neural network (CNN). In the small-data regime, the LES-CNN generates artificial instabilities and thus leads to unphysical results. We propose four remedies for the CNN to work in the small-data regime: (1) data augmentation (DA), (2) group equivariant convolution neural network (GCNN), leveraging the rotational equivariance of the SGS term, (3) incorporating a physical constraint on the SGS enstrophy transfer, and (4) variational autoencoder providing stochasticity and uncertainty quantification. The rotational equivariance of SGS terms can be accounted for by either including rotated snapshots in the training data set (DA) or by a GCNN that enforces rotational equivariance as a hard constraint. Additionally, The SGS enstrophy transfer constraint can be implemented in the loss function of the CNN. Stochasticity can be crucial in modeling backscattering (energy transferred from subgrid scales to resolved scales) of SGS terms. A priori and a posteriori analyses show that the proposed approaches enhance the SGS model and allow the data-driven model to work stably and accurately in a small-data regime.