The field of fluid mechanics is rapidly advancing, driven by unprecedented volumes of data from field measurements, experiments and large-scale simulations at multiple spatiotemporal scales. Machine learning offers a wealth of techniques to extract information from data that could be translated into knowledge about the underlying fluid mechanics. Moreover, machine learning algorithms can augment domain knowledge and automate tasks related to flow control and optimization. This article presents an overview of past history, current developments, and emerging opportunities of machine learning for fluid mechanics. It outlines fundamental machine learning methodologies and discusses their uses for understanding, modeling, optimizing, and controlling fluid flows. The strengths and limitations of these methods are addressed from the perspective of scientific inquiry that considers data as an inherent part of modeling, experimentation, and simulation. Machine learning provides a powerful information processing framework that can enrich, and possibly even transform, current lines of fluid mechanics research and industrial applications.
Elastoviscoplastic (EVP) fluid flows are driven by a non-trivial interplay between the elastic, viscous, and plastic properties, which under certain conditions can transition the otherwise laminar flow into complex flow instabilities with rich space-time-dependent dynamics. We discover that under elastic turbulence regimes, EVP fluids undergo dynamic jamming triggered by localised polymer stress deformations that facilitate the formation of solid regions trapped in local low-stress energy wells. Below the jamming transition $φ<φ_J$, the solid volume fraction $φ$ scales with $\sqrt{Bi}$, where $Bi$ is the Bingham number characterizing the ratio of yield to viscous stresses, in direct agreement with theoretical approximations based on the laminar solution. The onset of this new dynamic jamming transition $φ\geqφ_J$ is marked by a clear deviation from the scaling $φ\sim \sqrt{Bi}$, scaling as $φ\sim \exp{Bi}$. We show that this instability-induced jamming transition -- analogous to that in dense suspensions -- leads to slow, minimally diffusive, and rigid-like flows with finite deformability, highlighting a novel phase-change in elastic turbulence regimes of complex fluids.
For high-fidelity predictions of turbulent flows in complex practical engineering problems, the Wall-Modeled (WM) Large-Eddy Simulation (LES) has aroused great interest. In the present study, we prove that the conventional Wall-Stress Models (WSMs) developed for WMLES of Newtonian fluids fail to predict the shear-thinning-induced drag reduction in power-law fluids. Therefore, we propose novel algebraic, integrated, and Ordinary-Differential-Equation (ODE) WSMs, for the first time, for WMLES of power-law Non-Newtonian (NN) fluids and assess their performance against reference Wall-Resolved (WR) LES solutions. In addition, the effects of the key model parameters, including the WSM type, sampling height, sampling cell, and axial grid resolution are explored, and it is revealed that turbulent NN flow predictions have a much higher sensitivity to the choice of WSM, compared to their Newtonian counterparts. It is manifested that, in contrast to WRLES, accurate modeling of the mean apparent and subgrid-scale NN viscosities in the NNODE model can improve the predictions considerably. Therefore, closures with lower uncertainties on coarse WMLES grids are sought for these terms. Finally, the
Displacement flows are common in hydraulic fracturing, as fracking fluids of different composition are injected sequentially in the fracture. The injection of an immiscible fluid at the center of a liquid-filled fracture results in the growth of the fracture and the outward displacement of the interface between the two liquids. We study the dynamics of the fluid-driven fracture which is controlled by the competition between viscous, elastic, and toughness-related stresses. We use a model experiment to characterize the dynamics of the fracture for a range of mechanical properties of the fractured material and fracturing fluids. We form the liquid-filled pre-fracture in an elastic brittle matrix of gelatin. The displacing liquid is then injected. We record the radius and aperture of the fracture, and the position of the interface between the two liquids. In a typical experiment, the axisymmetric radial viscous flow is accommodated by the elastic deformation and fracturing of the matrix. We model the coupling between elastic deformation, viscous dissipation, and fracture propagation and recover the two fracturing regimes identified for single fluid injection. For the viscous-dominated
Microorganism motility often takes place within complex, viscoelastic fluid environments, e.g., sperm in cervicovaginal mucus and bacteria in biofilms. In such complex fluids, strains and stresses generated by the microorganism are stored and relax across a spectrum of length and time scales and the complex fluid can be driven out of its linear response regime. Phenomena not possible in viscous media thereby arise from feedback between the "swimmer" and the complex fluid, making swimming efficiency co-dependent on the propulsion mechanism and fluid properties. Here we parameterize a flagellar motor and filament properties together with elastic relaxation and nonlinear shear-thinning properties of the fluid in a computational immersed boundary model. We then explore swimming efficiency over this parameter space. One exemplary insight is that motor efficiency (measured by the volumetric flow rate) can be boosted vs.\ degraded by moderate vs.\ strong shear-thinning of the viscoelastic environment.
Nanofluidics has firmly established itself as a new field in fluid mechanics, as novel properties have been shown to emerge in fluids at the nanometric scale. Thanks to recent developments in fabrication technology, artificial nanofluidic systems are now being designed at the scale of biological nanopores. This ultimate step in scale reduction has pushed the development of new experimental techniques and new theoretical tools, bridging fluid mechanics, statistical mechanics and condensed matter physics. This review is intended as a toolbox for fluids at the nanometre scale. After presenting the basic equations that govern fluid behaviour in the continuum limit, we will show how these equations break down and new properties emerge in molecular scale confinement.
Big data and machine learning are driving comprehensive economic and social transformations and are rapidly re-shaping the toolbox and the methodologies of applied scientists. Machine learning tools are designed to learn functions from data with little to no need of prior knowledge. As continuous developments in experimental and numerical methods improve our ability to collect high-quality data, machine learning tools become increasingly viable and promising also in disciplines rooted in physical principles. These notes explore how machine learning can be integrated and combined with more classic methods in fluid dynamics. After a brief review of the machine learning landscape, we show how many problems in fluid mechanics can be framed as machine learning problems and we explore challenges and opportunities. We consider several relevant applications: aeroacoustic noise prediction, turbulence modelling, reduced-order modelling and forecasting, meshless integration of (partial) differential equations, super-resolution and flow control. While this list is by no means exhaustive, the presentation will provide enough concrete examples to offer perspectives on how machine learning might
The symmetries of a free incompressible fluid span the Galilei group, augmented with independent dilations of space and time. When the fluid is compressible, the symmetry is enlarged to the expanded Schroedinger group, which also involves, in addition, Schroedinger expansions. While incompressible fluid dynamics can be derived as an appropriate non-relativistic limit of a conformally-invariant relativistic theory, the recently discussed Conformal Galilei group, obtained by contraction from the relativistic conformal group, is not a symmetry. This is explained by the subtleties of the non-relativistic limit.
Analytical expressions for the flow field as well as for the effective slip length of a shear flow over a surface with periodic rectangular grooves are derived. The primary fluid is in the Cassie state with the grooves being filled with a secondary immiscible fluid. The coupling of both fluids is reflected in a locally varying slip distribution along the fluid-fluid interface, which models the effect of the secondary fluid on the outer flow. The obtained closed-form analytical expressions for the flow field and effective slip length of the primary fluid explicitly contain the influence of the viscosities of the two fluids as well as the magnitude of the local slip, which is a function of the surface geometry. They agree well with results from numerical computations of the full geometry. The analytical expressions allow investigating the influence of the viscous stresses inside the secondary fluid for arbitrary geometries of the rectangular grooves. For classic superhydrophobic surfaces, the deviations in the effective slip length compared to the case of inviscid gas flow are are pointed out. Another important finding with respect to an accurate modeling of flow over microstructured
The Hall-Vinen-Bekharevich-Khalatnikov (HVBK) model is widely used to numerically study quantum turbulence in superfluid helium. Based on the two-fluid model of Tisza and Landau, the HVBK model describes the normal (viscous) and superfluid (inviscid) components of the flow using two Navier-Stokes type equations, coupled through a mutual friction force term. This feature makes the HVBK model very appealing in applying statistical tools used in classical turbulence to study properties of quantum turbulence. A large body of literature used low-order statistics (spectra, or second-order structure functions in real space) to unravel exchanges between the two fluids at several levels. The novelty in this study is to use a theoretical approach based on first principles to derive transport equations for the third-order moments for each component of velocity.
Predicting how much mixing occurs when a given amount of energy is injected into a Boussinesq fluid is a longstanding problem in stratified turbulence. The huge number of degrees of freedom involved in those processes renders extremely difficult a deterministic approach to the problem. Here we present a statistical mechanics approach yielding prediction for a cumulative, global mixing efficiency as a function of a global Richardson number and the background buoyancy profile.
In this paper, we present a theoretical, experimental, and numerical study of the dynamics of cavitation bubbles inside a droplet suspended in another host fluid. On the theoretical side, we provided a modified Rayleigh collapse time and natural frequency for spherical bubbles in our particular context, characterized by the density ratio between the two liquids and the bubble-to-droplet size ratio. Regarding the experimental aspect, experiments were carried out for laser-induced cavitation bubbles inside oil-in-water (O/W) or water-in-oil (W/O) droplets. Two distinct fluid-mixing mechanisms were unveiled in the two systems, respectively. In the case of O/W droplets, a liquid jet emerges around the end of the bubble collapse phase, effectively penetrating the droplet interface. We offer a detailed analysis of the criteria governing jet penetration, involving the standoff parameter and impact velocity of the bubble jet on the droplet surface. Conversely, in the scenario involving W/O droplets, the bubble traverses the droplet interior, inducing global motion and eventually leading to droplet pinch-off when the local Weber number exceeds a critical value. This phenomenon is elucidated
Bubbly drinks are surprisingly attractive. There is something about the nature of the these beverages that make them preferable among other choices. In this article we explore the physics involved in this particular kind of two-phase, mass-transfer-driven flows.
Fluid deformable surfaces are ubiquitous in cell and tissue biology, including lipid bilayers, the actomyosin cortex, or epithelial cell sheets. These interfaces exhibit a complex interplay between elasticity, low Reynolds number interfacial hydrodynamics, chemistry, and geometry, and govern important biological processes such as cellular traffic, division, migration, or tissue morphogenesis. To address the modelling challenges posed by this class of problems, in which interfacial phenomena tightly interact with the shape and dynamics of the surface, we develop a general continuum mechanics and computational framework for fluid deformable surfaces. The dual solid-fluid nature of fluid deformable surfaces challenges classical Lagrangian or Eulerian descriptions of deforming bodies. Here, we extend the notion of Arbitrarily Lagrangian-Eulerian (ALE) formulations, well-established for bulk media, to deforming surfaces. To systematically develop models for fluid deformable surfaces, which consistently treat all couplings between fields and geometry, we follow a nonlinear Onsager formalism according to which the dynamics minimize a Rayleighian functional where dissipation, power input a
Viscous fingering is a well-known hydrodynamic instability that sets in when a less viscous fluid displaces a more viscous fluid. When the two fluids are miscible, viscous fingering introduces disorder in the velocity field and exerts a fundamental control on the rate at which the fluids mix. We present a fluid dynamics video of the mixing process in a viscously unstable flow, generated from a high-resolution numerical simulation using a computational strategy that is stable for arbitrary viscosity ratios. We develop a two-equation dynamic model of concentration variance and mean dissipation rate to quantify the degree of mixing in such a displacement process. The model reproduces accurately the evolution of these two quantities as observed in high-resolution numerical simulations and captures the nontrivial interplay between channeling and creation of interfacial area as a result of viscous fingering.
Many biological materials such as cervical mucus and collagen gel possess a fibrous micro-structure. This micro-structure affects the emergent mechanical properties of the material, and hence the functional behaviour of the system. We consider the canonical problem of stretching a thin sheet of transversely-isotropic viscous fluid as a simplified version of the spinnbarkeit test for cervical mucus. We propose a novel solution to the model constructed by Green & Friedman by manipulating the model to a form amenable to arbitrary Lagrangian-Eulerian techniques. The system of equations, reduced by exploiting the slender nature of the sheet, are solved numerically and we discover that the bulk properties of the sheet are controlled by an effective viscosity dependent on the evolving angle of the fibres. In addition, we confirm a previous conjecture by demonstrating that the centre-line of the sheet need not be flat, and perform a short timescale analysis to capture the full behaviour of the centre-line.
Non-modal amplification of disturbances in streamwise-constant channel flows of Oldroyd-B fluids is studied from an input-output point of view by analyzing the responses of the velocity components to spatio-temporal body forces. These inputs into the governing equations are assumed to be harmonic in the spanwise direction and stochastic in the wall-normal direction and in time. An explicit Reynolds number scaling of frequency responses from different forcing to different velocity components is developed, showing the same $Re$-dependence as in Newtonian fluids. It is found that some of the frequency response components peak at non-zero temporal frequencies. This is in contrast to Newtonian fluids, where peaks are always observed at zero frequency, suggesting that viscoelastic effects introduce additional timescales and promote development of flow patterns with smaller time constants than in Newtonian fluids. The temporal frequencies, corresponding to the peaks in the components of frequency response, decrease with an increase in viscosity ratio (ratio of solvent viscosity to total viscosity) and show maxima for non-zero elasticity number. Our analysis of the Reynolds-Orr equation de
A pressure-driven two-layer channel flow of a Newtonian fluid with constant viscosity (top layer) and a fluid with a time-dependent viscosity (bottom layer) is numerically investigated. The bottom layer goes through an aging process in which its viscosity increases due to the formation of internal structure, which is represented by a Coussot-type relationship. The resultant flow dynamics is the consequence of the competition between structuration and destructuration, as characterized by the dimensionless timescale for structuration (tau) and the dimensionless material property (beta) of the bottom fluid. The development of Kelvin-Helmholtz type instabilities (roll-up structures) observed in the Newtonian constant viscosity case was found to be suppressed as the viscosity of the bottom layer increased over time. It is found that, for the set of parameters considered in the present study, the bottom layer almost behaves like a Newtonian fluid with constant viscosity for tau greater than 10 and beta greater than 1. It is also shown that decreasing the value of the Froude number stabilizes the interfacial instabilities. The wavelength of the interfacial wave increases as the capillary
We present a numerical study of convection in a horizontal layer comprising a fluid-saturated porous bed overlain by an unconfined fluid layer. Convection is driven by a vertical, destabilising temperature difference applied across the whole system, as in the canonical Rayleigh-Bénard problem. Numerical simulations are carried out using a single-domain formulation of the two-layer problem based on the Darcy-Brinkman equations. We explore the dynamics and heat flux through the system in the limit of large Rayleigh number, but small Darcy number, such that the flow exhibits vigorous convection in both the porous and the unconfined fluid regions, while the porous flow still remains strongly confined and governed by Darcy's law. We demonstrate that the heat flux and average thermal structure of the system can be predicted using previous results of convection in individual fluid or porous layers. We revisit a controversy about the role of subcritical "penetrative convection" in the porous medium, and confirm that such induced flow does not contribute to the heat flux through the system. Lastly, we briefly study the temporal coupling between the two layers and find that the turbulent flu
The late-stage demixing following spinodal decomposition of a three-dimensional symmetric binary fluid mixture is studied numerically, using a thermodynamicaly consistent lattice Boltzmann method. We combine results from simulations with different numerical parameters to obtain an unprecendented range of length and time scales when expressed in reduced physical units. Using eight large (256^3) runs, the resulting composite graph of reduced domain size l against reduced time t covers 1 < l < 10^5, 10 < t < 10^8. Our data is consistent with the dynamical scaling hypothesis, that l(t) is a universal scaling curve. We give the first detailed statistical analysis of fluid motion, rather than just domain evolution, in simulations of this kind, and introduce scaling plots for several quantities derived from the fluid velocity and velocity gradient fields.