Nonlinear Viscosity Research Articles

Generalized field inversion strategies for data-driven turbulence closure modeling

Most data-driven turbulence closures are based on the general structure of nonlinear eddy viscosity models. Although this structure can be embedded into the machine learning algorithm and the Reynolds stress tensor itself can be fit as a function of scalar- and tensor-valued inputs, there exists an alternative two-step approach. First, the spatial distributions of the optimal closure coefficients are computed by solving an inverse problem. Subsequently, these are expressed as functions of solely scalar-valued invariants of the flow field by virtue of an arbitrary regression algorithm. In this paper, we present two general inversion strategies that overcome the limitation of being applicable only when all closure tensors are linearly independent. We propose to either cast the inversion into a constrained and regularized optimization problem or project the anisotropy tensor onto a set of previously orthogonalized closure tensors. Using the two-step approach together with either of these strategies then enables us to quantify the model-form error associated with the closure structure independent of a particular regression algorithm. Eventually, this allows for the selection of the a priori optimal set of closure tensors for a given, arbitrary complex test case.

Constraining Earth’s nonlinear mantle viscosity using plate-boundary resolving global inversions

Variable viscosity in Earth's mantle exerts a fundamental control on mantle convection and plate tectonics, yet rigorously constraining the underlying parameters has remained a challenge. Inverse methods have not been sufficiently robust to handle the severe viscosity gradients and nonlinearities (arising from dislocation creep and plastic failure) while simultaneously resolving the megathrust and bending slabs globally. Using global plate motions as constraints, we overcome these challenges by combining a scalable nonlinear Stokes solver that resolves the key tectonic features with an adjoint-based Bayesian approach. Assuming plate cooling, variations in the thickness of continental lithosphere, slabs, and broad scale lower mantle structure as well as a constant grain size through the bulk of the upper mantle, a good fit to global plate motions is found with a nonlinear upper mantle stress exponent of 2.43 [Formula: see text] 0.25 (mean [Formula: see text] SD). A relatively low yield stress of 151 [Formula: see text] 19 MPa is required for slabs to bend during subduction and transmit a slab pull that generates asymmetrical subduction. The recovered long-term strength of megathrusts (plate interfaces) varies between different subduction zones, with South America having a larger strength and Vanuatu and Central America having lower values with important implications for the stresses driving megathrust earthquakes.

Integrating NSGA-II and CFD for enhanced urban airflow prediction: Recalibration of closure coefficients for a nonlinear eddy viscosity model

Accurately predicting urban environmental airflows using the Reynolds-averaged Navier‒Stokes (RANS) approach presents significant challenges due to unsolved closure issues. Previous attempts to address these limitations through coefficient adjustments were constrained by isotropic assumptions, resulting in limited generalizability and reliability, especially for complex building configurations. In this study, we proposed a recalibration approach within a multi-objective optimization framework. Nondominant sorting genetic algorithm-II (NSGA-II) and computational fluid dynamics (CFD) numerical computations were utilized to optimize the closure coefficients of a nonlinear eddy viscosity (NLEV) model, with a focus on improving the accuracy of the mean velocity, turbulent kinetic energy, and airflow characteristic predictions. To ensure the robustness of the model, appropriate objective functions were carefully defined. The proposed approach was evaluated using three-dimensional (3D) building cases related to urban prototypes, and the results demonstrated the necessity of multi-objective optimization. Our findings indicated that the trade-off solution on the Pareto front consistently outperformed the baseline and single-objective optimal versions across different types of urban airflow, demonstrating superior generalizability. The simulation results of this solution exhibited closer agreement with the wind tunnel experimental data and provided more accurate distributions of the time-averaged values and airflow characteristics. The normalized root mean square errors are reduced by about 33 %, 52 %, 35 % and 9 % in velocity, and 31 %, 25 %, 24 % and 11 % in turbulent kinetic energy, respectively. These results underscore the effectiveness of the multi-objective optimization framework in addressing closure issues and improving the prediction accuracy of complex urban environmental airflows.

A tensor basis neural network-based turbulence model for transonic axial compressor flows

Traditional turbulence models encounter limitations when simulating intricate flows within transonic axial compressors. In contrast, recent advancements in machine learning turbulence models have demonstrated enhanced potential in refining the precision of turbulence modeling. Notably, the tensor basis neural network (TBNN) methodology has successfully developed non-linear eddy viscosity turbulence models. These models possess the capability to capture the anisotropic characteristic of Reynolds stress. However, applying machine learning non-linear eddy viscosity models to aircraft turbomachinery remains relatively infrequent. Current research mainly focuses on linear eddy viscosity turbulence models based on the Boussinesq hypothesis that presupposes isotropy in Reynolds stress. In this work, we introduce a non-linear eddy viscosity turbulence model, denoted as the k-ω-SST-TBNN model based on the TBNN framework. This model has been employed to simulate the transonic axial compressor NASA Rotor 37. The TBNN is trained using large eddy simulation (LES) results datasets. The importance of input scalar features is analyzed using the random forest method, from which significant and low-degree variables are selected as inputs for the TBNN. Furthermore, this study proposes including the turbulent Mach number as one of the extra features, representing fluid compressibility, thereby extending the computational mass flow rate range of compressor. Additionally, this paper proposes a method involving the weighted average of Reynolds stress, combining the high-precision but less robust TBNN predictions with results based on the Boussinesq assumption to enhance the turbulence model's robustness. The trained k-ω-SST-TBNN model undergoes validation on Rotor 37, where it exhibits a marked improvement in the prediction of overall performance, tip-gap vortex, and radial distribution of flow parameters. The model also displays a commendable capacity for generalization.

Intraoperative tumor mapping using pyridine-carbazole based multifunctional fluorescent probes for precise resection and photodynamic therapeutics

Precise intraoperative tumor localization plays a pivotal role in the quest for refined cancer therapeutics. Addressing this, we have developed a series of novel pyridine-carbazole-based probes (Mito-Pyca1 ∼ Mito-Pyca4) featuring viscosity-enhanced fluorescence and multiphoton absorption properties. Among them, Mito-Pyca4 showcases significant promise for two-photon activated photodynamic therapy. When subjected to near-infrared irradiation, it reliably produces copious singlet oxygen, crucial for targeted tumor treatment. In vitro and in vivo experiments validated the capacity of Mito-Pyca4 to effectively induce cell apoptosis and impede tumor growth with minimal toxicity, positioning it as a potential candidate for photodynamic therapy. Furthermore, the specificity of Mito-Pyca4 for cancer cells was confirmed in co-culture experiments, and the probe was utilized for in vivo fluorescence-guided surgery. Successfully distinguishing tumor tissue from normal tissue, Mito-Pyca4 facilitates precise tumor resection. In summary, this comprehensive study establishes Mito-Pyca4 as a versatile tool for nonlinear optical imaging, viscosity sensing, and fluorescence-guided cancer surgery and treatment, providing surgeons an unparalleled advantage in discerning tumor margins. Through this research, we underscore the transformative potential of integrating chemical innovation with surgical and therapeutic techniques.

Overlapping Grid-Based Spectral Collocation Technique for Bioconvective Flow of MHD Williamson Nanofluid over a Radiative Circular Cylindrical Body with Activation Energy

The amalgamation of motile microbes in nanofluid (NF) is important in upsurging the thermal conductivity of various systems, including micro-fluid devices, chip-shaped micro-devices, and enzyme biosensors. The current scrutiny focuses on the bioconvective flow of magneto-Williamson NFs containing motile microbes through a horizontal circular cylinder placed in a porous medium with nonlinear mixed convection and thermal radiation, heat sink/source, variable fluid properties, activation energy with chemical and microbial reactions, and Brownian motion for both nanoparticles and microbes. The flow analysis has also been considered subject to velocity slips, suction/injection, and heat convective and zero mass flux constraints at the boundary. The governing equations have been converted to a non-dimensional form using similarity variables, and the overlapping grid-based spectral collocation technique has been executed to procure solutions numerically. The graphical interpretation of various pertinent variables in the flow profiles and physical quantities of engineering attentiveness is provided and discussed. The results reveal that NF flow is accelerated by nonlinear thermal convection, velocity slip, magnetic fields, and variable viscosity parameters but decelerated by the Williamson fluid and suction parameters. The inclusion of nonlinear thermal radiation and variable thermal conductivity helps to enhance the fluid temperature and heat transfer rate. The concentration of both nanoparticles and motile microbes is promoted by the incorporation of activation energy in the flow system. The contribution of microbial Brownian motion along with microbial reactions on flow quantities justifies the importance of these features in the dynamics of motile microbes.

Riemann problem for a general variable coefficient Burgers equation with time-dependent damping

In this paper, we study the Riemann problem for a general variable coefficient Burgers equation with time-dependent damping. We use a nonlinear time-dependent viscosity equation with a similarity variable. Thus, when the viscosity goes to zero, we obtain Riemann solutions to the general variable coefficient Burgers equation with time-dependent damping. Moreover, we use the Lax–Friedrichs scheme to obtain numerical evidence of the Riemann solutions.

Ensemble variational method with adaptive covariance inflation for learning neural network-based turbulence models

This work introduces an ensemble variational method with adaptive covariance inflation for learning nonlinear eddy viscosity turbulence models where the Reynolds stress anisotropy is represented with tensor-basis neural networks. The ensemble-based method has emerged as an important alternative to data-driven turbulence modeling due to its merit of non-derivativeness. However, the training accuracy of the ensemble method can be affected by the linearization assumption and sample collapse issue. Given these difficulties, we introduce the hybrid ensemble variational method, which inherits the merits of the ensemble method in non-derivativeness and the variational method in nonlinear analysis. Moreover, a covariance inflation scheme is proposed based on convergence states to alleviate the detrimental effects of sample collapse. The capability of the ensemble variational method in model learning is tested for flows in a square duct, flows over periodic hills, and flows around the S809 airfoil, with increasing complexity in the training data from direct observation to sparse indirect observation. Our results show that the ensemble variational method can learn relatively accurate neural network-based turbulence models in scenarios of small ensemble size and sample variances, compared to the ensemble Kalman method. It highlights the superiority of the ensemble variational method in practical applications, since small ensemble sizes can reduce computational costs, and small sample variance can ensure the training robustness by avoiding nonphysical samples of Reynolds stresses.

Abaqus implementation of a large family of finite viscoelasticity models

In this paper, we introduce an Abaqus UMAT subroutine for a family of constitutive models for the viscoelastic response of isotropic elastomers of any compressibility – including fully incompressible elastomers – undergoing finite deformations. The models can be chosen to account for a wide range of non-Gaussian elasticities, as well as for a wide range of nonlinear viscosities. From a mathematical point of view, the structure of the models is such that the viscous dissipation is characterized by an internal variable Cv, subject to the physically-based constraint detCv=1, that is solution of a nonlinear first-order ODE in time. This ODE is solved by means of an explicit Runge–Kutta scheme of high order capable of preserving the constraint detCv=1 identically. The accuracy and convergence of the code is demonstrated numerically by comparison with an exact solution for several of the Abaqus built-in hybrid finite elements, including the simplicial elements C3D4H and C3D10H and the hexahedral elements C3D8H and C3D20H. The last part of this paper is devoted to showcasing the capabilities of the code by deploying it to compute the homogenized response of a bicontinuous rubber blend.

Numerical spectral analysis of standing waves in quantum hydrodynamics with viscosity

<abstract><p>We study the spectrum of the linearization around standing wave profiles for two quantum hydrodynamics systems with linear and nonlinear viscosity. The essential spectrum for such profiles is stable; we investigate the point spectrum using an Evans function technique. For both systems we show numerically that there exists a real unstable eigenvalue, thus providing numerical evidence for spectral instability.</p></abstract>

INFLUENCE OF EXTERNAL CUBIC VISCOSITY ON THE DYNAMICS OF A CANTILEVER ROD LOADED WITH A TRACKING FORCE

The analysis of the influence of external nonlinear cubic viscosity on the behavior of a cantilever rod loaded with a compressive tracking force is performed. The Bernoulli – Euler beam model is used for the describing of the rod bending deformations. After the transition to dimensionless variables, we obtained the form of notation in which the coefficients of the equation do not depend on the coefficient of cubic viscous friction. This indicates the independence of the critical load and system frequency characteristics from the level of nonlinear attenuation. Using the standard procedure of the Bubnov – Galerkin method, taking into account the first two forms of eigen oscillations, the problem is reduced to the analysis of a fourth order nonlinear system of ordinary differential equations with respect to generalized coordinates and generalized velocities. For the linear version of the equations with low attenuation, the values of the critical load are obtained in dependence on the ratio of the damping coefficients of the eigenforms. The behavior of the system is analyzed taking into account the considered cubic nonlinearity using the method of nonlinear normal forms. The problem is reduced to the analysis of a nonlinear dynamical system with one degree of freedom. There are constructive analytical methods of qualitative and approximate quantitative research for this class of problems. The critical value of the compressive force was determined by the analysis of the dependence of the first Lyapunov magnitude on the load parameter. It turned out to be less than the value corresponding to the system without damping (the destabilizing effect of external nonlinear friction). A bifurcation occurs in the system at the critical load value. Therefore, the zero equilibrium becomes an unstable complex focus of the first order and a limit cycle is born. The function of the self-oscillations amplitude of the load magnitude is constructed using the harmonic linearization method.

Modeling viscoelasticity and cyclic creep of PEEK by parallel rheological framework (PRF)

Although creep is usually regarded as a quasistatic phenomenon, in multiple engineering applications the load is not constant, hence creep under cyclic loading conditions should be considered. The current work presents the theoretical and experimental study on the viscoelasticity and cyclic creep of PEEK (Polyether ether ketone) – the polymer with good chemical and thermal stability. Among other demanding applications, PEEK is used in bearings industry for the fabrication of cages, which main function is to isolate and to guide rolling elements. In this application the polymer cage is subjected to the long-term load alternating in time, meaning that classical creep (under constant load) cannot be presumed. In the current study, the non-linear viscosity is included into the generalized Maxwell model, employing the Eyring equations which postulate that the viscosity coefficient is not constant but is dependent on stress. The model is implemented with the Parallel Rheological Framework (PRF) (the numerical technique recently implemented in the commercial software ABAQUS), used to construct the generalized Maxwell model with hyper-elastic springs and dashpots of non-linear viscosity. Initially, the viscoelastic material parameters are identified from the uniaxial test, and later these parameters are used to simulate the PEEK response under cyclic loading, to predict the time to creep/fatigue failure. Eventually, the numerical predictions are compared to the test results. The comparison indicates that the developed method and its numerical implementation by PRF can be successfully used to model the non-linear viscoelastic behavior of PEEK under uniaxial load. It is also found, that the simulation of cyclic creep can be reasonably performed by using the current material model and PRF.

A novel refined Caputo kernel and constitutive concepts for semi-exact nonlinear dynamic and creep analyses of suddenly pressurized hollow fractional-order visco-hyperelastic cylinders

Some reports have confirmed that discretizing the mass, damping, and stiffness characteristics of a continuum can result in remarkable deviations from the experimental results; as all these properties are intertwined and coupled within a single material. The viscoelastic constitutive laws with fractional-order time derivatives can model this integrity much more accurately, due to their capability of using a set of an infinite number of orders. Here, an innovative constitutive model with a nonlinear hyperelastic memory element that treats simultaneously the fractional-order viscoelasticity and Mooney-Rivlin hyperelasticity of the continuum is proposed to track the suppression of the short-term dynamic stresses and displacements and the redistribution of the long-term creep stress and deformations of the suddenly pressurized thick cylinders. While the spatial variations of the quantities of the incompressible cylinder are treated by an exact analytical approach, the resulting time-dependent system of equations is converted by a novel numerical solution scheme to an integrodifferential system that contains integer-order partial derivatives and a time-growing number of integral terms. An enhancement to Caputo's singular-kernel integral definition of the fractional-order derivatives is proposed here to replace these derivatives with equivalent integrals whose integer derivatives are computed based on second-order Taylor's expansion. The integer-order and fractional-order time derivatives are then computed numerically using the second-order Runge-Kutta and accumulation-based trapezoidal techniques, respectively. Finally, comparisons are made among the dissipative pattern and time variations of the dynamic/vibration stresses and large-deformations of the hyperelastic and fractional-order visco-hyperelastic cylinders in addition to the assessment of the effects of the constitutive parameters of the fractional-order model. Results show that while the larger magnitudes of both the nonlinear viscosity coefficients and fractional-order time derivatives diminish the resulting deformations and stresses, they respectively lead to smaller and larger frequencies of oscillation.

CFD simulation of drag-reducing fluids in a non-Newtonian turbulent pipe flow

The conveyance of fluids from one terminal to another incurs considerable expenses primarily due to the significant costs associated with addressing pressure losses arising from shear stresses and friction along the inner surface during the pumping operations within pipelines. This study was conducted with the objective of simulating turbulent flow in pipes containing drag-reducing fluids. This was achieved through the utilization of both RKE and RNG versions of a non-Newtonian model for low Reynolds numbers, based on the k-ε model, along with a non-Newtonian damping function. The outcomes were subsequently compared to simulations carried out by Pinho and experimental data sourced from the existing literature. To simulate the flow in pipelines, computational fluid dynamics (CFD) software was employed, which incorporated non-linear molecular viscosity and a damping function to accurately account for near-wall effects. In addition, this research entailed the optimization of C and C0 parameters in the damping function and the stress term, which quantifies the cross-correlation between fluctuating viscosity and the fluctuating rate of strain. Furthermore, non-Newtonian terms were incorporated into the equations for turbulent kinetic energy (k), dissipation rate (ε), and momentum transfer. The study involved a comprehensive comparison of model predictions with experimental data across various flow parameters, including friction factor, axial velocity, turbulent kinetic energy, and Reynolds shear stresses. The results revealed a significant enhancement in the model's ability to predict critical flow parameters, such as friction factor, mean axial velocity, and Reynolds stress profiles. Nevertheless, the model displayed some limitations in predicting turbulent kinetic energy. Notably, the average error in calculating the friction factor for the fluids under investigation was found to be 5.45%, marking a substantial improvement compared to the Pinho model, which exhibited an average error of 32.49%.

Effect of Elasticity and Nonlinear Viscosity on Fingering Instability in Heterogeneous Porous Media

AbstractThe Saffman‐Taylor instability of polymeric fluids flooding through saturated heterogeneous porous media is studied numerically. The displacement of a Newtonian fluid by a viscoelastic liquid is examined using a Hartley transformation and a fourth‐order Adams‐Bashforth technique. The flow is assumed to be miscible, and the Carreau‐Yasuda equation is used to model the shear‐thinning viscometric functions. The effects of different rheological properties and domain parameters on flow instability are studied in detail. It is found that a channeling regime is possible as the main difference between displacements through homogeneous and heterogeneous media.

A unified Maxwell model with time-varying viscosity via [formula omitted]-Caputo fractional derivative coined

In order to describe the mechanical behaviors of viscoelastic materials that couple memory effects and time-varying viscosity properties toggled between thixotropy and rheopexy, this paper explores and establishes the constitutive equation of a unified Maxwell model with a variable kernel in terms of the ψ-Caputo fractional derivative. By virtue of a Volterra integral equation of the second kind and the ψ-Laplace transform technique, the closed-form expressions of creep and relaxation responses for the proposed model are derived and compared in detail. Furthermore, to enhance practicality, the exponential and power-law time-varying viscosity candidates are embedded into the proposed model, along with exhibiting the corresponding creep and relaxation behaviors in light of illustration and reasoning, respectively. The results may provide a fresh perspective for detecting the relationship between the viscoelastic system with nonlinear time-varying viscosity and generalized fractional calculus.

Effect of Blade Trimming Length on the Performance of Marine Centrifugal Pump

To study the hydrodynamic characteristics of blade trimming length in centrifugal pumps, Delayed Detached Eddy Simulation (DDES) with nonlinear eddy viscosity was utilized to conduct unsteady calculations on the centrifugal pump. A comprehensive examination of the fluid dynamic properties of the centrifugal pump, including external features, flow conditions, and pressure fluctuations, was carried out. By applying the theory of entropy production, the areas of high energy loss within the centrifugal pump were identified, and the correlations between local entropy production, energy loss, and unsteady flow in different areas with varying blade trimming lengths were analyzed. The results indicate that with the increase in blade trimming length, under rated flow conditions, the head decreases by 1.8%, 3.2%, and 5.7% for different blade trimming lengths, respectively, compared to normal impellers. Similarly, the efficiency decreases by 0.5%, 0.8%, and 1.0% for different blade trimming lengths, respectively. Similar trends were observed under other working conditions as well. As the degree of blade trimming increases, the irreversible losses after the failure of the centrifugal pump also increase significantly, indicating that the flow inside the centrifugal pump becomes disorder. Blade trimming leads to a disorderly fluid flow inside the centrifugal pump, causing an increase in the radial force during operation, which in turn leads to an increase in vibration amplitude and affects its operational stability. Blade trimming failure has a significant impact on the frequency and amplitude of pressure pulsation, resulting in abnormal pressure pulsation and abnormal vibration of the centrifugal pump. Therefore, early warning and diagnosis of blade trimming can be achieved through pressure pulsation monitoring.

Insight into the Eyring–Powell fluid flow model using degenerate operator: geometric perturbation

This work provides a formulation of a fluid flow under a nonlinear diffusion based on a viscosity of Eyring–Powell type along with a degenerate semi-parabolic operator. The introduction of such a degenerate operator is significant as it allows us to explore a further general model not previously considered in the literature. Our aims are hence to provide analytical insights and numerical assessments to the mentioned flow model: firstly, some results are provided in connection with the regularity and uniqueness of weak solutions. The problem is converted into the travelling wave domain where solutions are obtained within an asymptotic expansion supported by the geometric perturbation theory. Finally, a numerical process is considered as the basis to ensure the validity of the analytical assessments presented. Such numerical process is performed for low Reynolds numbers given in classical porous media. As a main finding to highlight: we show that there exist exponential profiles of solutions for the velocity component. This result is not trivial for the non-linear viscosity terms considered.

Viscosity database for ternary Cu–Cr–X (X=Ni, Si, Zr) alloys based on CALPHAD-type modeling

A viscosity model with composition and temperature dependence for the liquid alloys of the Cu–Cr–X (X = Ni, Si, Zr) systems was developed using the CALPHAD (CALculation of PHAse Diagrams) approach. Based on the critical review of the available experimental data for pure metal viscosity, the viscosities of pure liquid Cu, Cr, Zr, Ni and Si were simulated with the Arrhenius formula. The viscosities of the binary liquid alloys of the Cu–Ni, Cu–Si, Cu–Zr, Cr–Ni, Cr–Si and Ni–Zr systems at different temperatures were evaluated using the CALPHAD-type formalism. The non-linear viscosity behavior resulting from chemical short-range order was described using temperature-dependent binary interaction parameters. The viscosities of the Cu–Cr and Cr–Zr liquid alloys were predicted employing the Hirai equation. The viscosities of the Cu–Cr–Zr, Cu–Cr–Ni and Cu–Cr–Si ternary liquids were directly extrapolated using the Redlish-Kister-Muggianu equation from the parameters of the binary systems. A comprehensive understanding of the temperature-composition-solidification phase-viscosity relationship was established for ternary melts. The satisfactory agreement observed between calculations and experimental data validates the use of CALPHAD-type viscosity modeling, supporting the suitability of the constructed viscosity database for predicting the viscosity of high-performance Cu-based alloys.

Physical interpretation of neural network-based nonlinear eddy viscosity models

Neural network-based turbulence modeling has gained significant success in improving turbulence predictions by incorporating high–fidelity data. However, the interpretability of the learned model is often not fully analyzed, which has been one of the main criticisms of neural network-based turbulence modeling. Therefore, it is increasingly demanding to provide physical interpretation of the trained model, which is of significant interest for guiding the development of interpretable and unified turbulence models. The present work aims to interpret the predictive improvement of turbulence flows based on the behavior of the learned model, represented with tensor basis neural networks. The ensemble Kalman method is used for model learning from sparse observation data due to its ease of implementation and high training efficiency. Two cases, i.e., flow over the S809 airfoil and flow in a square duct, are used to demonstrate the physical interpretation of the ensemble-based turbulence modeling. For the flow over the S809 airfoil, our results show that the ensemble Kalman method learns an optimal linear eddy viscosity model, which improves the prediction of the aerodynamic lift by reducing the eddy viscosity in the upstream boundary layer and promoting the early onset of flow separation. For the square duct case, the method provides a nonlinear eddy viscosity model, which predicts well secondary flows by capturing the imbalance of the Reynolds normal stresses. The flexibility of the ensemble-based method is highlighted to capture characteristics of the flow separation and secondary flow by adjusting the nonlinearity of the turbulence model.