Viscous Contribution Research Articles

Tailor-Designed Models for the Turbulent Velocity Gradient through Normalizing Flow

Small-scale turbulence can be comprehensively described in terms of velocity gradients, which makes them an appealing starting point for low-dimensional modeling. Typical models consist of stochastic equations based on closures for nonlocal pressure and viscous contributions. The fidelity of the resulting models depends on the accuracy of the underlying modeling assumptions. Here, we discuss an alternative data-driven approach leveraging machine learning to derive a velocity gradient model which captures its statistics by construction. We use a normalizing flow to learn the velocity gradient probability density function (PDF) from direct numerical simulation (DNS) of incompressible turbulence. Then, by using the equation for the single-time PDF of the velocity gradient, we construct a deterministic, yet chaotic, dynamical system featuring the learned steady-state PDF by design. Finally, utilizing gauge terms for the velocity gradient single-time statistics, we optimize the time correlations as obtained from our model against the DNS data. As a result, the model time realizations resemble the time series from DNS statistically closely. Published by the American Physical Society 2024

Modelling of wall-bounded cavitating flow and spray mixing in multi-component environments using the PC-SAFT equation of state

This work introduces a numerical multiphase model for multi-component mixtures, utilizing tabulated data for physical and transport properties across a spectrum of conditions from near-vacuum pressures to supercritical states. The property data are derived using Perturbed Chain Statistical Associating Fluid Theory (PC-SAFT), vapor-liquid equilibrium (VLE) calculations, entropy scaling methodologies, and Group Contribution (GC) methods. These techniques accurately reflect the thermodynamic behaviors of real fluids, avoiding the empirical estimation of Equation of State (EoS) input parameters. Implemented in OpenFOAM, the fluid dynamics solver is designed to address the three-dimensional Navier-Stokes equations for multi-component mixtures. The methodology integrates operator splitting to manage hyperbolic and parabolic steps distinctively. Hyperbolic terms are solved using the HLLC (Harten-Lax-van Leer-Contact) solver with temporal integration performed via a third-order Strong-Stability-Preserving Runge–Kutta (SSP-RK3) method. Viscous stress tensor contributions in the momentum equation are handled through an implicit velocity correction equation, while parabolic terms in the energy equation are explicitly solved. The simulation efficiency is further enhanced by adaptive Local Time Stepping and the Immersed Boundary (IB) method, which addresses interactions between the fluid and solid boundaries. Turbulence is resolved using the Wall Adaptive Large Eddy (WALE) model. Applied to high-pressure diesel fuel spray injections into non-reacting (nitrogen) gas environments, the model has been validated against Engine Combustion Network (ECN) data for the Spray-C configuration, featuring a fully cavitating multi-hole orifice. Results demonstrate that the model achieves accurate predictions across a broad range of tested conditions without the need for tuning or calibration parameters.

Exploring the influence of acetylenic acetogenin unsaturation patterns on lipid interface interactions: Insights from surface chemistry, rheology, and spectroscopy

This study explores how the unsaturation patterns of three acetogenins from Porcelia macrocarpa (2S,3R,4R)-3-hydroxy-4-methyl-2-(eicos-11′-yn-19′-enyl)-butanolide (1), (2S,3R,4R)-3-hydroxy-4-methyl-2-(eicos-11′-ynyl)-butanolide (2), and (2S,3R,4R)-3-hydroxy-4-methyl-2-eicosyl-butanolide (3)—affect their interactions with lipid interfaces, using Langmuir monolayers of dipalmitoyl-phosphoethanolamine (DPPE) as a model for protozoal membranes. The hypothesis posits that unsaturated acetogenins (1 and 2) will exhibit distinct behaviors compared to the saturated acetogenin (3) due to structural differences. Experiments incorporated each acetogenin into DPPE monolayers, employing techniques like tensiometry, dilatational rheology, infrared spectroscopy, and Brewster angle microscopy (BAM) to assess interactions and changes in film properties. Results showed that unsaturated acetogenin 1, active against Leishmania infantum and Trypanosoma cruzi, destabilized the film and induced rapid molecular reorganization, while acetogenin 2, inactive against these parasites, uniquely increased the viscous contribution to viscoelasticity. Saturated acetogenin 3 did not integrate into the film. All acetogenins enhanced fluidity, reducing elasticity and viscosity, with BAM revealing phase separation only in acetogenin 2. Infrared spectroscopy underscored the impact of drug-lipid interactions, leading to the conclusion that unsaturated acetogenins 1 and 2 have a more significant influence on thermodynamic and structural properties than acetogenin 3, offering insights into their potential therapeutic mechanisms.

The hydraulically smooth limit of flow over surfaces with spanwise heterogeneity

Wall slip is conventionally obtained through riblets or grooves. As riblets are macroscopic objects, it is usually not practical to discuss slip patterns at the hydraulically smooth limit. Per Nikuradse, the smooth limit is when the characteristic length of the surface pattern is comparable to the viscous length scale. Recent studies show the possibility of slip at the microscopic scale, making slip patterns at the hydraulically smooth limit relevant. In this study, we leverage a high-fidelity pseudo-spectral code and study flow over surfaces featuring alternating strips of slip and no-slip wall conditions. The wavelength of the surface pattern, denoted as l, varies from 1.5 times the half channel height to 2 viscous units (plus units), eventually approaching the anticipated hydraulically smooth limit. The presence of surface slip gives rise to a slip velocity at the wall, denoted as Us, which contributes to drag reduction. The surface spanwise heterogeneity leads to secondary flows and intensifies turbulent mixing, consequently leading to drag increase. This drag increase effect can be parameterized using the “roughness function” ΔU+. The sum of Us+ and −ΔU+ determines whether the surface increases or reduces drag. Here, the superscript ＋ denotes normalization by the wall units. In most cases, the slip velocity at the wall Us+ predominates over −ΔU+, resulting in drag reduction. However, when l is a few viscous units, the roughness function ΔU+ does not vanish and overwhelms the slip velocity, giving rise to a net drag increase. Considering that the wall is a mixture of slip and no-slip conditions, this drag increase at the anticipated hydraulically smooth limit is unexpected. To gain an insight into the mechanism responsible for this drag increase, we derive a Navier–Stokes-based decomposition of the roughness function. Here, we generalize the definition of the roughness function such that it is a function of the wall-normal coordinate, thereby overcoming the difficulty of measuring the roughness function when the log region is narrow and hard to define. Analysis shows that when l is a few plus units, secondary flows contribute to a slightly positive ΔU+, while turbulent and viscous contributions, by and large, cancel out, ultimately leading to an overall drag increase at the anticipated hydraulically smooth limit. The evidence in the paper suggests that the hydraulically smooth limit does not exist for certain surfaces.

A Wave Drift Force Model for Semi-Submersible Types of Floating Wind Turbines in Large Waves and Current

The correct prediction of slowly varying wave drift loads is important for the mooring analysis of floating wind turbines (FWTs). However, present design analysis tools fail to correctly predict these loads in conditions with current and moderate and large waves. This paper presents a semi-empirical method to correct zero-current potential-flow quadratic transfer functions (QTFs) of horizontal wave drift loads in conditions with current and moderate and large waves. The method is applicable to column-stabilized types of substructures or semi-submersibles. In the first step, the potential-flow QTF is corrected for potential-flow wave–current effects by applying a heuristic method. Second, the generalized Exwave formula corrects for viscous drift effects. Viscous drift effects become important for moderate and large waves. Conditions with current in the same direction as the waves increase the viscous drift contribution further. The method is validated by comparing QTF predictions with empirical QTFs identified from model test data for the INO Windmoor semi. While potential-flow QTFs agree well with the empirical data for small seastates without current, they underestimate the wave drift loads for moderate and large seastates. Conditions with current increase the underestimation. The semi-empirical correction method significantly improves predictions.

Numerical study of the interaction between cylindrical particles and shear-thinning fluids in a linear shear flow

In this paper, the interaction between cylindrical particles and shear-thinning non-Newtonian fluids in a linear shear flow is investigated using particle-resolved direct numerical simulation. The Carreau model is used to represent the rheological properties of shear-thinning fluids, and the numerical method is validated against previously published data. Then, the effects of Reynolds number (Re), aspect ratio (Ar), power-law index (n), Carreau number (Cu), and incident angle (α) on drag coefficient (CD), lift coefficient (CL), and torque coefficient (CT) of cylindrical particles are investigated. The numerical results show that the flow field structure and pressure distribution around the cylindrical particle in a shear flow are different from those in a uniform flow, and the particles in a shear flow generate extra CL and CT. Furthermore, comparing with Newtonian fluids, the shear-thinning properties of the non-Newtonian fluid change the viscosity distribution and significantly decrease the CD, CL, and CT of the particles. The variation laws and influencing mechanisms of CD, CL, and CT under different working conditions are discussed by dividing the total coefficients into pressure and viscous shear contributions. Predictive correlations of CD, CL, and CT are established by considering the effects of Re, Ar, n, Cu, and α. The findings indicate that both the shear flow mode and shear-thinning properties must be considered when evaluating relevant particle–fluid interactions, which provides important guidance for predicting and controlling the orientation and distribution of cylindrical particles in shear-thinning fluids. Meanwhile, the predictive correlations can be used for large-scale simulations of multiphase coupling.

Asphalt binder characterization using waveform-based viscoelastic measures and time-temperature superposition principle under large strains

In the United States, currently peak-to-peak oscillatory shear stress-strain data is used in the asphalt binder characterization protocols. This practice necessitates the measurement, usage, and reporting of only peak responses for material characterization and specification even though a sinusoidal waveform is applied. However, in this paper it is argued that with the proliferation of modified binding systems, it is critical to capture and examine the complete waveform data when subjected to both small and large strains. This approach will help properly differentiating between materials. In this study, four modified binders were tested at small and large strain levels while recording the complete response waveform. The recorded waveform data was then used to examine the linear and nonlinear viscoelastic characteristics of these binders. The orthogonal stress decomposition technique was used to delineate elastic and viscous contributions in the nonlinear regime. This study examined the time-temperature dependency under both linear and nonlinear regimes. The time-temperature shift factors obtained from linear regime were effectively applied in the nonlinear regime. Further, all the tested binders showed strain stiffening and shear thinning behavior under most of the testing conditions. This study provides useful insights about the material behavior under nonlinear regime and can serve as a benchmark for further investigation of nonlinear viscoelastic behavior of asphalt binders.

Turbulence driven goal-oriented anisotropic mesh adaptation for RANS simulations in aerodynamics

Anisotropic mesh adaptation is starting to play a role of great importance within the CFD community. In fact, such a tool allows to accurately capture the physics of the problem in an automatic way, by reducing at the same time the computational resources, in contrast to the best-practice meshing guidelines, which require an a priori knowledge of the flow solution and, for this reason, are approximate and error-prone. In the present work, we consider anisotropic mesh adaptation for steady flows to get mesh-independent solutions with a small number of vertices, that is, we achieve the so-called early capturing property. Two classes of methods exist: feature-based mesh adaptation where the minimization of the interpolation error associated with sensor field is targeted and goal-oriented mesh adaptation where the minimization of the discretization error associated with a goal functional is carried out, such as the lift or the drag coefficients. The goal-oriented mesh adaptation requires the adjoint state computation, thus a crucial part of the treatment deals with this topic. The novelty of the present approach is the taking into account of the turbulent primal and adjoint variables, and the corresponding fluxes (including turbulence sources) inside the mesh adaptation process and the adjoint computation, as up to now only the inviscid and the viscous contributions have been considered in this kind of processes. We show how such additions are extremely beneficial for goal-oriented mesh adaptation, as the resulting meshes are able to capture the physics of the problem with a higher accuracy, in contrast to the exclusively inviscid- or viscous-driven adaptation. Eventually, as the mesh adaptation process tends to produce highly unstructured anisotropic meshes, a relevant part of the present work is devoted to the description of a robust solver capable of solving the RANS equations on such meshes.

Physics-informed data-driven discovery of constitutive models with application to strain-rate-sensitive soft materials

AbstractA novel data-driven constitutive modeling approach is proposed, which combines the physics-informed nature of modeling based on continuum thermodynamics with the benefits of machine learning. This approach is demonstrated on strain-rate-sensitive soft materials. This model is based on the viscous dissipation-based visco-hyperelasticity framework where the total stress is decomposed into volumetric, isochoric hyperelastic, and isochoric viscous overstress contributions. It is shown that each of these stress components can be written as linear combinations of the components of an irreducible integrity basis. Three Gaussian process regression-based surrogate models are trained (one per stress component) between principal invariants of strain and strain rate tensors and the corresponding coefficients of the integrity basis components. It is demonstrated that this type of model construction enforces key physics-based constraints on the predicted responses: the second law of thermodynamics, the principles of local action and determinism, objectivity, the balance of angular momentum, an assumed reference state, isotropy, and limited memory. The three surrogate models that constitute our constitutive model are evaluated by training them on small-size numerically generated data sets corresponding to a single deformation mode and then analyzing their predictions over a much wider testing regime comprising multiple deformation modes. Our physics-informed data-driven constitutive model predictions are compared with the corresponding predictions of classical continuum thermodynamics-based and purely data-driven models. It is shown that our surrogate models can reasonably capture the stress–strain-strain rate responses in both training and testing regimes and improve prediction accuracy, generalizability to multiple deformation modes, and compatibility with limited data.

A correction model for the effect of spanwise flow on the viscous force contribution in BEM and Lifting Line methods

Design optimization and aeroelastic load evaluation for wind turbines are still infeasible for CFD due to computational cost. These tasks are carried out using 3D engineering aerodynamic methods, where the local aerodynamic loads are obtained from tabulated 2D aerodynamic polars, which makes the models fast. With recent developments such as highly flexible blades, coned/prebent/swept blades, the relative inflow direction to the “inner” 2D airfoil section systems can have a significant component in the spanwise direction. The Crossflow Principle (CP) is used to treat the effects of this in a physically consistent manner. Cases dominated by pressure forces are described well with the CP method, but it is shown in the paper that CP fails to describe the part of the forces stemming from the friction forces correctly. It is shown in the paper that the power loss due to friction forces will be underestimated with the crossflow principle for rotors with significantly swept blades or other designs with a significant amount of local crossflow on the blades. The present work presents a simple model to correct the baseline crossflow principle method to take into account also the friction force part in a consistent manner. The model is validated with 3D CFD results on a 2D airfoil section. It is shown that the new model successfully corrects for the addition of the viscous forces due to spanwise flow component. The paper includes examples of the effect of using the model on rotor designs with different amounts of blade sweep simulated using a blade element momentum (BEM) and blade element vortex cylinder (BEVC) methods.

Stability of a dispersion of elongated particles embedded in a viscous membrane

We develop a mean-field model to examine the stability of a ‘quasi-2-D suspension’ of elongated particles embedded within a viscous membrane. This geometry represents several biological and synthetic settings, and we reveal mechanisms by which the anisotropic mobility of particles interacts with long-ranged viscous membrane hydrodynamics. We first show that a system of slender rod-like particles driven by a constant force is unstable to perturbations in concentration – much like sedimentation in analogous 3-D suspensions – so long as membrane viscous stresses dominate. However, increasing the contribution of viscous stresses from the surrounding 3-D fluid(s) suppresses such an instability. We then tie this result to the hydrodynamic disturbances generated by each particle in the plane of the membrane and show that enhancing subphase viscous contributions generates extensional fields that orient neighbouring particles in a manner that draws them apart. The balance of flux of particles aggregating versus separating then leads to a wave number selection in the mean-field model.

Planing Hull Hydrodynamic Performance Prediction Using LincoSim Virtual Towing Tank

This work shows the performance of LincoSim, a web-based virtual towing tank enabling automated and standardized calm water computational fluid dynamics (CFD) data sampling, extending previous published applications to the case of a high-speed hull. The calculations are performed for a 1:10 scale model of a 43 ft powerboat hull form in the Froude number range from 0.3 to 2.0. The counterpart physical model is the experimental fluid dynamics (EFD) campaign performed at the University of Naples Federico II, where the resistance, sinkage and trim data have been measured. The EFD/CFD data comparison is performed and shown with a discussion of the spotted differences. The average percentage differences between the EFD and CFD data for the whole speed range are 1.84, 6.87 and 6.94 for the resistance, dynamic trim, and sinkage, respectively. These results confirm the maturity of the standardized and automated CFD modeling for calm water hydrodynamic analysis included in LincoSim, even at very high Froude numbers. The wetted length of the keel and chine and the wetted surface are calculated from numerical data using the advanced post-processing. Finally, as a work in progress, we test a first comparison for the same hull of the EFD and CFD data, considering two seakeeping conditions for head waves at a given wavelength for two velocity conditions. Also, this kind of analysis confirms the tight correlation between the measured and computed outcomes. This synergic interplay of EFD and CFD can link the advantages of both methods to support hull design but also requires experiment planning and final data analysis to obtain physical parameters not easily measurable in laboratory, such as the wetted surface, wetted lengths, proper viscous contribution, and pressure distribution both in calm water and in waves.

Peeling fingers in an elastic Hele-Shaw channel

Using experiments and a depth-averaged numerical model, we study instabilities of two-phase flows in a Hele-Shaw channel with an elastic upper boundary and a non-uniform cross-section prescribed by initial collapse. Experimentally, we find increasingly complex and unsteady modes of air-finger propagation as the dimensionless bubble speed $Ca$ and level of collapse are increased, including pointed fingers, indented fingers and the feathered modes first identified by Cuttle et al. (J. Fluid Mech., vol. 886, 2020, A20). By introducing a measure of the viscous contribution to finger propagation, we identify a $Ca$ threshold beyond which viscous forces are superseded by elastic effects. Quantitative prediction of this transition between ‘viscous’ and ‘elastic’ reopening regimes across levels of collapse establishes the fidelity of the numerical model. In the viscous regime, we recover the non-monotonic dependence on $Ca$ of the finger pressure, which is characteristic of benchtop models of airway reopening. To explore the elastic regime numerically, we extend the depth-averaged model introduced by Fontana et al. (J. Fluid Mech., vol. 916, 2021, A27) to include an artificial disjoining pressure that prevents the unphysical self-intersection of the interface. Using time simulations, we capture for the first time the majority of experimental finger dynamics, including feathered modes. We show that these disordered states evolve continually, with no evidence of convergence to steady or periodic states. We find that the steady bifurcation structure satisfactorily predicts the bubble pressure as a function of $Ca$ , but that it does not provide sufficient information to predict the transition to unsteady dynamics that appears strongly nonlinear.

Critical review of single-cell mechanotyping approaches for biomedical applications.

Accurate mechanical measurements of cells has the potential to improve diagnostics, therapeutics and advance understanding of disease mechanisms, where high-resolution mechanical information can be measured by deforming individual cells. Here we evaluate recently developed techniques for measuring cell-scale stiffness properties; while many such techniques have been developed, much of the work examining single-cell stiffness is impacted by difficulties in standardization and comparability, giving rise to large variations in reported mechanical moduli. We highlight the role of underlying mechanical theories driving this variability, and note opportunities to develop novel mechanotyping devices and theoretical models that facilitate convenient and accurate mechanical characterisation. Moreover, many high-throughput approaches are confounded by factors including cell size, surface friction, natural population heterogeneity and convolution of elastic and viscous contributions to cell deformability. We nevertheless identify key approaches based on deformability cytometry as a promising direction for further development, where both high-throughput and accurate single-cell resolutions can be realized.

Titin governs myocardial passive stiffness with major support from microtubules and actin and the extracellular matrix.

Myocardial passive stiffness is crucial for the heart's pump function and is determined by mechanical elements, including the extracellular matrix and cytoskeletal filaments; however, their individual contributions are controversially discussed and difficult to quantify. In this study, we targeted the cytoskeletal filaments in a mouse model, which enables the specific, acute and complete cleavage of the sarcomeric titin springs. We show in vitro that each cytoskeletal filament's stiffness contribution varies depending on whether the elastic or the viscous forces are considered and on strain level. Titin governs myocardial elastic forces, with the largest contribution provided at both low and high strain. Viscous force contributions are more uniformly distributed among the microtubules, titin and actin. The extracellular matrix contributes at high strain. The remaining forces after total target element disruption are likely derived from desmin filaments. Our findings answer longstanding questions about cardiac mechanical architecture and allow better targeting of passive myocardial stiffness in heart failure.

On the spurious effects in Lamb-vector-based force decomposition methods

Recent methods based on Lamb vector integration allow for a far-field formulation and decomposition of the whole aerodynamic force, with identification of local flow structures associated with the force generation. Different exact Lamb-vector-based force formulations were proposed in the past and the decomposition of total drag into lift-induced and parasite contributions was subject of very recent studies. However, inconsistencies were also reported, raising questions on the physical role of a non-zero induced drag found in two-dimensional flows. Moreover, the effect of the arbitrary pole (appearing in Lamb-vector-based formulas) on the numerical accuracy of such methods was never investigated. Additionally, the separation of parasite drag into viscous and wave drag contributions by Lamb vector field analysis is still questioned. In this paper, the accuracy of different force formulas and of the drag decomposition in lift-induced and parasite contributions is first analyzed, for steady flows. Two-dimensional numerical solutions around RAE2822 and NACA0012 airfoils are considered, the latter in both inviscid and viscous flows (at low and high Reynolds number), to analyze consistency between lift-induced/parasite drag definitions and the drag generated in isentropic flow conditions (reversible drag) or due to entropy production (irreversible drag). A spurious induced drag contribution (related to the streamwise momentum perturbation) is identified, resulting in negative induced drag values in two-dimensional flows. The effect of a shift in the pole location is also analysed, with reference to its impact on the total force and on drag decomposition accuracy, then a new approach to parasite drag breakdown into physical constituents (viscous and wave contributions) is proposed and results are compared against predictions obtained using consolidated thermodynamic-based methods and against available results from recent literature.

Particle Adsorption Using a Quartz Crystal Microbalance with Dissipation by Applying a Kelvin-Voigt-Based Viscoelastic Model and the Gauss-Newton Method.

The use of a quartz crystal microbalance with dissipation (QCM-D) to study the adsorption of particles larger than 100 nm, such as liposomes, viruses, and nano/micro-plastics, remains challenging owing to the lack of appropriate models for data evaluation. This study presents a method for quantifying the adsorption of negatively charged polystyrene latex (100 nm-1 μm) at the solid-liquid interface. The validity of a viscoelastic model based on Kelvin-Voigt theory was assessed, and the model was used to evaluate particle adsorption data obtained from QCM-D measurements. The Gauss-Newton method was used to fit the data; the values obtained were larger than results from atomic force microscopy, indicating that the viscoelastic model combined with the Gauss-Newton method can quantify the adsorption of large polystyrene particles and the surrounding water around them. We suggested that QCM-D, in combination with an appropriate viscoelastic model, is applicable to estimate adsorption at the solid-liquid interface even for soft particles larger than 1 μm, which are out of the range of applications to the hydrodynamics model. Furthermore, we successfully showed that the recorded dissipation reflects the viscoelastic properties of the layer. The viscoelastic model allowed quantification of the rheological properties of the layer. The ratio of the viscous and elastic contributions was characterized by using loss tangent (tan δ) values that were extracted from the experimental data by applying the viscoelastic model. These values were lower for the adsorption of the negatively charged polystyrene particles on a positive surface than on a negative surface. This suggests that tan δ reflects the strength of the contact between the particle and substrate.

Thermomechanical Stress Analysis of Hydrated Vital Gluten with Large Amplitude Oscillatory Shear Rheology.

Vital gluten is increasingly researched as a non-food product for biodegradable materials. During processing, the protein network is confronted with increased thermal and mechanical stress, altering the network characteristics. With the prospect of using the protein for materials beyond food, it is important to understand the mechanical properties at various processing temperatures. To achieve this, the study investigates hydrated vital gluten under thermomechanical stress based on large amplitude oscillatory shear (LAOS) rheology. LAOS rheology was conducted at increasing shear strains (0.01-100%), various frequencies (5-20 rad/s) and temperatures of 25, 45, 55, 65, 70 and 85 °C. With elevating temperatures up to 55 °C, the linear viscoelastic moduli decrease, indicating material softening. Then, protein polymerization and the formation of new cross-links due to thermal denaturation cause more network connectivity, resulting in significantly higher elastic moduli. Beyond the linear viscoelastic regime, the strain-stiffening ratio rises disproportionately. This effect becomes even more evident at higher temperatures. Lacking a viscous contribution, the highly elastic but also stiff network shows less mechanical resilience. Additionally, at these elevated temperatures, structural changes during the protein's denaturation and network shrinkage due to water evaporation could be visualized with confocal laser scanning microscopy (CLSM).

Methods for Calibrating the Electrochemical Quartz Crystal Microbalance: Frequency to Mass and Compensation for Viscous Load

The main output from an Electrochemical Quartz Crystal Microbalance is a frequency shift. This note describes how to separate the mass- and viscous load contributions to this shift by a calibration procedure. The mass calibration is made by electroplating from a copper sulfate solution in ethanol/water with 100% current efficiency. An estimate of viscous load is obtained by measuring the energy dissipation and is related to frequency change using the Kanazawa–Gordon equation. Two approaches are discussed: either by performing calibration experiments in a series of water–glycerol mixtures or by following oscillations in frequency and dissipation by collecting data during the stabilization phase of the experiment.

Forecasting small-scale dynamics of fluid turbulence using deep neural networks

Turbulence in fluid flows is characterized by a wide range of interacting scales. Since the scale range increases as some power of the flow Reynolds number, a faithful simulation of the entire scale range is prohibitively expensive at high Reynolds numbers. The most expensive aspect concerns the small-scale motions; thus, major emphasis is placed on understanding and modeling them, taking advantage of their putative universality. In this work, using physics-informed deep learning methods, we present a modeling framework to capture and predict the small-scale dynamics of turbulence, via the velocity gradient tensor. The model is based on obtaining functional closures for the pressure Hessian and viscous Laplacian contributions as functions of velocity gradient tensor. This task is accomplished using deep neural networks that are consistent with physical constraints and explicitly incorporate Reynolds number dependence to account for small-scale intermittency. We then utilize a massive direct numerical simulation database, spanning two orders of magnitude in the large-scale Reynolds number, for training and validation. The model learns from low to moderate Reynolds numbers and successfully predicts velocity gradient statistics at both seen and higher (unseen) Reynolds numbers. The success of our present approach demonstrates the viability of deep learning over traditional modeling approaches in capturing and predicting small-scale features of turbulence.