Nonlinear Drag Research Articles

A multi-scale model for simulating liquid-fabric interactions

We propose a method for simulating the complex dynamics of partially and fully saturated woven and knit fabrics interacting with liquid, including the effects of buoyancy, nonlinear drag, pore (capillary) pressure, dripping, and convection-diffusion. Our model evolves the velocity fields of both the liquid and solid relying on mixture theory, as well as tracking a scalar saturation variable that affects the pore pressure forces in the fluid. We consider the porous microstructure implied by the fibers composing individual threads, and use it to derive homogenized drag and pore pressure models that faithfully reflect the anisotropy of fabrics. In addition to the bulk liquid and fabric motion, we derive a quasi-static flow model that accounts for liquid spreading within the fabric itself. Our implementation significantly extends standard numerical cloth and fluid models to support the diverse behaviors of wet fabric, and includes a numerical method tailored to cope with the challenging nonlinearities of the problem. We explore a range of fabric-water interactions to validate our model, including challenging animation scenarios involving splashing, wringing, and collisions with obstacles, along with qualitative comparisons against simple physical experiments.

Nonlinear drag forces and the thermostatistics of overdamped motion.

Diverse processes in statistical physics are usually analyzed on the assumption that the drag force acting on a test particle moving in a resisting medium is linear on the velocity of the particle. However, nonlinear drag forces do appear in relevant situations that are currently the focus of experimental and theoretical work. Motivated by these developments, we explore the consequences of nonlinear drag forces for the thermostatistics of systems of interacting particles performing overdamped motion. We derive a family of nonlinear Fokker-Planck equations for these systems, taking into account the effects of nonlinear drag forces. We investigate the main properties of these evolution equations, including an H-theorem, and obtain exact solutions of the stretched q-exponential form.

Mixed finite elements for global tide models with nonlinear damping

We study mixed finite element methods for the rotating shallow water equations with linearized momentum terms but nonlinear drag. By means of an equivalent second-order formulation, we prove long-time stability of the system without energy accumulation. We also give rates of damping in unforced systems and various continuous dependence results on initial conditions and forcing terms. A priori error estimates for the momentum and free surface elevation are given in $$L^2$$ as well as for the time derivative and divergence of the momentum. Numerical results confirm the theoretical results regarding both energy damping and convergence rates.

Exact solutions of the Navier-Stokes equations generalized for flow in porous media

Flow of Newtonian fluids in porous media is often modelled using a generalized version of the full non-linear Navier-Stokes equations that include additional terms describing the resistance to flow due to the porous matrix. Because this formulation is becoming increasingly popular in numerical models, exact solutions are required as a benchmark of numerical codes. The contribution of this study is to provide a number of non-trivial exact solutions of the generalized form of the Navier-Stokes equations for parallel flow in porous media. Steady-state solutions are derived in the case of flows in a medium with constant permeability along the main direction of flow and a constant cross-stream velocity in the case of both linear and non-linear drag. Solutions are also presented for cases in which the permeability changes in the direction normal to the main flow. An unsteady solution for a flow with velocity driven by a time-periodic pressure gradient is also derived. These solutions form a basis for validating computational models across a wide range of Reynolds and Darcy numbers.

A linearization coefficient for Morison force considering the intermittent effect due to free surface fluctuation

For members located near the free surface, an intermittent effect caused by the free surface fluctuation makes statistical properties of fluid particle kinematics different. Therefore, a linearization coefficient of the drag force derived under the assumption that the particle velocity and acceleration follow the Gaussian process are not valid above the still water level. The present paper derives the linearization coefficient for the drag force considering the intermittent effect. Force spectra and mean values of force maxima at each location are compared between nonlinear and linearized drag force to verify the validity of the proposed coefficient. The linearization coefficients derived from fully submerged section are constant irrespective of the observation position, and it tends to underestimate both the mean values of peak forces and force spectra as the observation height increases near the free surface. The proposed linearization coefficient reflects the effect of the free surface fluctuation and improves the error between the nonlinear and linear drag with increasing observation position.

Directed motion of spheres induced by unbiased driving forces in viscous fluids beyond the Stokes' law regime.

The emergence of directed motion is investigated in a system consisting of a sphere immersed in a viscous fluid and subjected to time-periodic forces of zero average. The directed motion arises from the combined action of a nonlinear drag force and the applied driving forces, in the absence of any periodic substrate potential. Necessary conditions for the existence of such directed motion are obtained and an analytical expression for the average terminal velocity is derived within the adiabatic approximation. Special attention is paid to the case of two mutually perpendicular forces with sinusoidal time dependence, one with twice the period of the other. It is shown that, although neither of these two forces induces directed motion when acting separately, when added together, the resultant force generates directed motion along the direction of the force with the shortest period. The dependence of the average terminal velocity on the system parameters is analyzed numerically and compared with that obtained using the adiabatic approximation. Among other results, it is found that, for appropriate parameter values, the direction of the average terminal velocity can be reversed by varying the forcing strength. Furthermore, certain aspects of the observed phenomenology are explained by means of symmetry arguments.

Effects of polymer stresses on analogy between momentum and heat transfer in drag-reduced turbulent channel flow

The effects of polymer stresses on the analogy between momentum and heat transfer are examined by using a direct numerical simulation (DNS) of viscoelastic turbulent channel flows using a constant heat flux boundary condition. The Reynolds number based on the friction velocity and channel half height is 125, and the Prandtl number is 5. The polymer stress is modeled using the finitely extensible nonlinear elastic-Peterlin constitutive model, and low (15%), intermediate (34%), and high drag reduction (DR) (52%) cases are examined. The Colburn analogy is found to be inapplicable for viscoelastic turbulent flows, suggesting dissimilarity between the momentum and heat transfer at the macroscopic coefficient level. The mean temperature profile also shows behaviour different from the mean velocity profile in drag-reduced flows. In contrast to the dissimilarity in the mean profiles, the turbulent Prandtl number Prt predicted by the DNS is near unity. This implies that turbulent heat transfer is still analogous t...

Correction scheme for point-particle models applied to a nonlinear drag law in simulations of particle-fluid interaction

Drag laws for particles in fluids are often expressed in terms of the undisturbed fluid velocity, defined as the fluid velocity a particle sees before its disturbance develops in the fluid. In two-way coupled point-particle simulations the information from the undisturbed state is not available and must be approximated using the disturbed velocity field. ([Horwitz, J. A. K. & Mani, A. 2016 Accurate calculation of stokes drag for point-particle tracking in two-way coupled flows. Journal of Computational Physics318, 85–109]) recently developed a procedure to estimate the undisturbed velocity for particles moving at low Reynolds number and obeying the linear Stokes drag law. Using this correction, convergence of numerical simulations was demonstrated to match the expected physical behavior for a range of canonical settings. In this paper we examine this correction scheme for particles moving at finite Reynolds number, by considering the nonlinear Schiller–Naumann drag law. Our results indicate that a linear correction can significantly improve prediction of drag force up to particle Reynolds number of about 10. Additionally, we present an investigation of the impact of this correction when applied to simulations of forced homogeneous turbulent particle-laden flows to further demonstrate the importance of modelling the undisturbed fluid velocity. While the shapes of particle acceleration and Reynolds number pdfs are not sensitive to correcting for the undisturbed fluid velocity, we show that an uncorrected scheme can result in significant under-prediction of the mean particle Reynolds number and standard deviation of particle acceleration. Furthermore, examination of particle radial distribution function reveals modest enhancements in preferential concentration predicted by the correction scheme (for St > 1). Our investigations of turbulent flows indicate the undisturbed fluid velocity correction to be more important for particles with larger size, while the Stokes number dependence is more complicated. The correction procedure shows a greater difference in particle slip velocity at higher Stokes numbers but more enhancement in preferential concentration at lower Stokes numbers. Finally, we propose a regime diagram to guide scheme selection for point-particle modelling.

LBM simulation of unsteady flow and heat transfer from a diamond-shaped porous cylinder

Incompressible, two-dimensional unsteady flow and heat transfer around a permeable diamond-shaped cylinder placed in an infinite stream of fluid have been numerically analyzed employing D2Q9 lattice model of the lattice Boltzmann method. The variations in hydrodynamic and thermal behaviour of the permeable diamond cylinder have been studied for different values of Darcy numbers (10-6⩽Da⩽10-2) and Reynolds numbers (50⩽Re⩽150). The force term which comprises the effects of linear and nonlinear drag forces of the porous medium (the Darcy-Forchheimer term) is directly coupled with collision equation for flow through the porous zone. Single relaxation parameter (SRT-BGK collision operator) is used to relax the particles towards equilibrium. A comprehensive analysis of the effects of Re and Da values on vortex suppression and wake depletion is presented. In addition, the influence of permeability on thermal enhancement ratio at different surfaces of the porous diamond-shaped cylinder is elucidated. A substantial reduction in vortex strength is witnessed for the same Re with higher permeability. Reduction in drag, suppression of vortex shedding and heat transfer augmentation are seen in the permeably rich cylinder. Correlations for time-averaged mean Nusselt number, valid for the range of parameters considered in the present study, are also provided. Furthermore, a comparative study on thermal dissipation from the permeable square and diamond shaped cylinders is carried out at Re = 50, 100 and 150 at different values of Da. This analogy sheds light on understanding the effect of porous body orientation on heat transfer enrichment, which may be handy while modeling porous medium for different engineering applications.

Seismic analysis of nonlinear offshore moored floating structures

This article presents a numerical model for solving the nonlinear random vibrations of offshore moored floating structures under seismic excitation. The offshore moored floating structure consists of the floating platform and mooring cables. The floating platform is considered as a rigid body with 3 degrees of freedom. The nonlinear equations of motions of the mooring cables are established using the nonlinear cable elements that are formulated based on the extended Hamilton principle. The nonlinear hydrodynamic drag forces that act on both the floating platform and cables are considered. In order to carry out the random vibrational analysis, the connection conditions between the floating structure and mooring cables are given to formulate the equations of motions of the whole system. Finally, the moored floating structure under horizontal seismic ground accelerations with Kanai–Tajimi model are analyzed using Monte Carlo simulation method. The probability density functions of the displacements of the moored floating structure and the maximum tensile force in cables are presented. The influences of different sag-to-span ratios or inclined angles of the mooring cables on the mean value and standard deviation of the displacements of the floating structure and the maximum tensile force in cables are analyzed.

Nonlinear Froude-Krylov and viscous drag representations for wave energy converters in the computation/fidelity continuum

Designing, optimizing and controlling a wave energy converter requires the construction of a mathematical model in order to simulate the behaviour of the device. Given the nonlinear nature of fluid-structure interactions, the definition of the model is not straightforward and should take into account the specific application it is intended for. Two of the most important characteristics of a model are the computational time and the expected accuracy, which usually are mutually conflicting. The inclusion of nonlinearities potentially increases the model accuracy, but at a higher computational price.Considering a heaving wave energy converter with and without the application of latching control, this paper studies and compares nine different modelling options, eight of which are based on potential theory and consider nonlinear Froude-Krylov and viscous drag forces, while one is based on fully-nonlinear computational fluid dynamics. The value of including nonlinearities in the hydrodynamic model is discussed in relation to the computational cost of the eventual accuracy benefits, under a range of scenarios.

2D Observer-Based Control of a Vascular Microrobot

The paper addresses the 2D observer-based control of a magnetic microrobot navigating in a cylindrical blood vessel along a reference trajectory. In particular, this robot faces the nonlinear drag force induced by the pulsatile blood flow, which can hardly be measured. We consequently propose a mean value theorem (MVT) based observer to estimate the blood velocity from the sole measurement of the robot position. We also prove the stability of the observer-based backstepping controller. The resulting estimation and tracking are then illustrated through simulations, as well as robustness to parametric uncertainty, measurement noise, and dynamical errors when the pulsatile blood flow is incorrectly modeled.

Enhanced settling of nonheavy inertial particles in homogeneous isotropic turbulence: The role of the pressure gradient and the Basset history force.

The Stokes drag force and the gravity force are usually sufficient to describe the behavior of sub-Kolmogorov-size (or pointlike) heavy particles in turbulence, in particular when the particle-to-fluid density ratio ρ_{p}/ρ_{f}≳10^{3} (with ρ_{p} and ρ_{f} the particle and fluid density, respectively). This is, in general, not the case for smaller particle-to-fluid density ratios, in particular not for ρ_{p}/ρ_{f}≲10^{2}. In that case the pressure gradient force, added mass effects, and the Basset history force also play important roles. In this study we focus on the understanding of the role of these additional forces, all of hydrodynamic origin, in the settling of particles in turbulence. In order to qualitatively elucidate the complex dynamics of such particles in homogeneous isotropic turbulence, we first focus on the case of settling of such particles in the flow field of a single vortex. After having explored this simplified case we extend our analysis to homogeneous isotropic turbulence. In general, we found that the pressure gradient force leads to a decrease in the settling velocity. This can be qualitatively understood by the fact that this force prevents the particles from sweeping out of vortices, a mechanism known as preferential sweeping which causes enhanced settling. Additionally, we found that the Basset history force can both increase and decrease the enhanced settling, depending on the particle Stokes number. Finally, the role of the nonlinear Stokes drag has been explored, confirming that it affects settling of inertial particles in turbulence, but only in a limited way for the parameter settings used in this investigation.

Dynamic response of articulated towers under correlated wind and waves

This paper deals with the dynamic behavior of a double-hinged articulated tower to wave alone, and correlated wind and waves. The analysis includes the nonlinearities due to nonlinear drag force, fluctuating buoyancy, variable added mass and instantaneous tower orientation. The fluctuating wind load is modeled by Ochi and Shin spectrum, while the wave load is characterized by Pierson–Moskowitz (P–M) spectrum. The nonlinear dynamic equation of motion is derived by Hamilton's principle. The equations of motion are solved in time domain by using Wilson-θ method. Power spectral density function (PSDF) of surge, tilting motion, hinge shear and bending moment are presented under high, moderate and low sea states. Studies of correlated wind and waves are found to be imperative for double hinged articulated towers to serve and survive in the extreme ocean environment. The response PSDF highlights the wind induced dynamic responses of the tower.

Non-linear drag induced entropy generation analysis in a microporous channel: The effect of conjugate heat transfer

The entropy generation analysis in a viscous dissipative flow of a Newtonian fluid through a hyper-porous microchannel formed between two heated parallel plates is conferred. Employing an analytical method, which is consistent with the perturbation analysis, the transport equations governing the thermo-hydrodynamics are studied. The effects of nonlinear Forchheimer drag and conjugate heat transfer on the thermal transport characteristics of heat are considered, while the thermal boundary conditions of third kind have been employed at the outer boundaries of channel for the conjugate heat transfer analysis. The explicit alterations are made in the thermal transport of heat in the system as attributable to the effect of dissipative heat generated due to the non-linear effect Forchheimer drag, Darcy frictional effect and the viscous shearing stress in the flow field. To account these effects, the explicit variation of Forchhiemer constant, Darcy number, porosity, thickness and conductivity of upper wall and Biot number of upper wall are carried out to shows the changes in the available energy of the system. Also, it is shown that the effect of non-linear drag mainly stemming from the presence of complex porous structure in the flow field and its interaction with the conjugate transport of heat alters the heat transfer rate in the system non-trivially, which, in turn, gives rise to the entropy generation in the system. The individual contribution of two different effects viz., the heat transfer and viscous dissipation on the system entropy generation rate for different cases are studied. It is believed that the implications of the present analysis may have direct bearing on the design of micro devices/systems typically used in electronic cooling, micro-heat pipes.

Response spectrum method for extreme wave loading with higher order components of drag force

Response spectra of fixed offshore structures impacted by extreme waves are investigated based on the higher order components of the nonlinear drag force. In this way, steel jacket platforms are simplified as a mass attached to a light cantilever cylinder and their corresponding deformation response spectra are estimated by utilizing a generalized single degree of freedom system. Based on the wave data recorded in the Persian Gulf region, extreme wave loading conditions corresponding to different return periods are exerted on the offshore structures. Accordingly, the effect of the higher order components of the drag force is considered and compared to the linearized state for different sea surface levels. When the fundamental period of the offshore structure is about one third of the main period of wave loading, the results indicate the linearized drag term is not capable of achieving a reliable deformation response spectrum.

Response and stability of two-point mooring system under random waves

The stochastic response and stability of a two-point mooring system are investigated for random sea state represented by a P-M sea spectrum. The two-point mooring system is modelled as an SDOF system having only stiffness nonlinearity; drag nonlinearity is represented by an equivalent linear damping. Since no parametric excitation exists and only the linear damping is assumed to be present in the system, only a local stability analysis is sufficient for the system. This is performed using a perturbation technique and by the application of Infante’s method. The analysis requires the mean square response of the system, which may be obtained in various ways. In the present study, the method using van-der-Pol transformation and F–P–K equation is used to obtain the probability density function of the response under random wave forces. From the moment of the probability density function, the mean square response is obtained. Stability of the system is represented by an inequality condition expressed as a function of some important parameters. A two-point mooring system is analysed as an illustrative example for a water depth of 141.5 m and a sea state represented by P-M spectrum with 16 m significant height. It is shown that for certain combinations of parameter values, stability of two-point mooring system may not be achieved.

Collective properties of defects, multiscale plasticity, and shock induced phenomena in solids

A statistically based approach is developed for the construction of constitutive equations that provides linkages between defect-induced mechanisms of structural relaxation, thermally activated plastic flow, and material response to extreme loading conditions. The collective properties of defects have been studied to establish the interaction of multiscale defect dynamics and plastic flow, and to explain the mechanisms leading to the universal self-similar structure of shock wave fronts. Pn explanation for structural universality of the steady-state plastic shock front (the four power law) and the self-similarity of shock wave profiles under reloading (unloading) is proposed. Structural characterization under transition from thermally activated dislocation glide to nonlinear dislocation drag effects is developed in terms of scaling invariants (effective temperatures) related to mesodefect induced morphology formed during the different stages of plastic deformation.

DYNAMIC MODEL AND SIMULATION OF ONE ACTIVE JOINT ROBOTIC FISH

This study considers the dynamic model of one active joint robotic fish by using Lagrange method and simulation of the robotic fish model in MATLAB/SimMechanics environment. Compared results of these two different models are given in the study. The mathematical model of the system is derived from Lagrange energy equations of the robotic fish inspired from a real carangiform fish. The Computer Aided Design (CAD) model of the robotic fish is designed by using SolidWorks and it is transferred to the SimMechanics environment. The hydrodynamic effects, which are linear and nonlinear drag force, are also adapted and head motion, one active joint, and one passive joint angles found by using MATLAB Simulink environment. Obtained results for joint angles from both dynamic and SimMechanics models are compared and proved with animation video of the robotic fish.

The frequency domain estimate of fatigue damage of combined load effects based on the rain-flow counting

Based on the rain-flow counting technique, a frequency domain method is developed for calculating the fatigue damages caused by the combined loads, which can be linear or nonlinear and Gaussian or non-Gaussian loads. Firstly, the new method is developed for the fatigue damage estimates of the combination of high and low frequency Gaussian loads, by which the ranges of the small rain-flow cycles are obtained as high frequency cycles with considering the reduction effect of low frequency loads on them; the ranges of the large rain-flow cycles are determined by means of Turkstra's rule of load combination. Secondly, the method is extended to calculate the fatigue damages caused by the nonlinear Morison load which is the combination of linear inertial and nonlinear drag forces. The method is benchmarked against the rain-flow damage estimates and compared with the existing ones. The numerical analyses show that the damages estimated by the new method are close to the rain-flow damages.