Eulerian Kinematics Research Articles

Flow Structures With High Lagrangian Coherence Rate Promote Diatom Blooms in Oligotrophic Waters

AbstractDiatoms are among the most efficient autotrophic organisms for oceanic primary production and carbon sequestration. Yet, the spatial distributions of these planktonic organisms remain puzzling and the underlying physical processes poorly known, especially in oligotrophic open waters. Here we investigate what dynamical conditions are conducive to episodic diatom blooms in oceanic deserts based on Lagrangian diagnosis and satellite‐derived phytoplankton functional types and currents. The coherence of the flow is diagnosed in space and time simultaneously through the Lagrangian coherence rate (LCR) to identify which dynamical structures favor diatom growth. Observations evidence that flow structures with high LCR (40 days or longer) in areas with elevated eddy kinetic energy and vorticity sustain high diatom concentrations in the sunlit layers. Our findings show that the integration of Eulerian kinematic variables into a Lagrangian frame reveals new dynamical aspects of geophysical turbulence and unveil their biological impacts.

Prescribed Performance Adaptive Balance Control for Reaction Wheel-Based Inverted Pendulum-Type Cubli Rovers in Asteroid

This paper investigates the issue of balance control for reaction-wheeled inverted pendulum-type Cubli Rovers on asteroids, and an adaptive control scheme is proposed via the prescribed performance control technique. The main feature lies in the fact that the transient behavior is satisfied which is required critically in the environment of asteroids. The attitude model of reaction-wheeled inverted pendulum-type Cubli Rovers is first constructed by virtue of the momentum moment theorem and Eulerian kinematics. Based on that, the gravitational field in the asteroid is described and the avoiding jumping condition is analyzed. Then, an adaptive prescribed performance control (APPC) method is proposed to obtain the fine tracking performance of the equilibrium error such that the inverted pendulum-type Cubli Rovers achieve the self-balancing motion. The proposed method is capable of ensuring the tracking errors inside the preset boundary functions, and the asymptotic stability of all states in the closed-loop system is guaranteed via the Lyapunov stability theory. The simulation and comparison results on the environment of asteroids verify the effectiveness and superiority of the presented control law.

A Newtonian mechanics formulation for the vibration of translating and rotating elastic continua

In this paper, we present a Newtonian mechanics formulation for modeling the vibration of a convecting elastic continuum, i.e., a system characterized by mean kinematic translation or rotation with small superposed vibrations. The proposed Newtonian approach complements customary energy-based techniques and serves as a convenient means to validate and physically interpret their results. We develop the equations of motion and matching conditions in a continuum mechanics setting with respect to a stationary inertial reference frame. Interaction of the convecting continuum with discrete space-fixed elements (e.g., springs and dampers) is enabled, without introducing time-dependent coefficients, through the use of Eulerian kinematics and kinetics. Kinematic discontinuities inherent in these interactions are accommodated by employing a global (or integral) form of balance of linear momentum applied to a space-fixed control volume. A generalized form of Navier's equation of elastic wave propagation is derived, with unsteady, Coriolis, centripetal, and convective contributions to the inertia. The resulting formulation is applied to a broad class of translating and rotating systems – including spinning rings, axially moving strings and beams, and general three-dimensional elastic structures – and shown to successfully reconcile with existing energy-based derivations in the literature.

Eulerian description of non-stationary motion of an idealized belt-pulley system with dry friction

We develop and validate a mathematical model of a belt drive with dry friction between the belt and the pulleys. Considering an idealized setting, we make use of the Eulerian (spatial) kinematic description for the belt, which moves quasistatically along a given contour. The focus of the paper lies on modelling the friction law in a non-material finite element framework. We also obtained a novel analytical result for the time evolution of the zones of sliding friction and stick. The correspondence of the analytical solution to both the non-material and conventional material finite element simulations has been demonstrated.

A Lagrangian–Eulerian formulation for reeling analysis of history-dependent multilayered beams

This paper presents a novel Lagrangian–Eulerian finite element formulation for reeling analysis of multilayered beams with gross interlayer slippage. In contrast to the conventional Lagrangian approach, the mesh becomes practically fixed in space, which yields significant benefits for the performance of the contact algorithms and the overall computational efficiency. The needed Lagrangian–Eulerian kinematic relations are derived, special attention is given to the convective transport term for the constitutive variables and an implicit update scheme for the elasto-plastic bending model is formulated. The proposed formulation is shown to predict responses with the same accuracy as offered by the conventional Lagrangian formulation.

Numerical Simulation of Fluid Flow Past an Oscillating Triangular Cylinder in a Channel

The structure of laminar incompressible flow in a two-dimensional horizontal channel past a triangular cylinder undergoing vertical oscillating motion is simulated using the finite element method. The oscillating motion is carried out for a fixed Reynolds number equal to 100 and dimensionless oscillating amplitudes of 0.125, 0.25, 0.5, and 1, while the dimensionless forced oscillation frequencies are chosen from St/4 to 4 St, where St is the natural Strouhal number of a stationary triangular cylinder. The arbitrary Lagrangian–Eulerian kinematics is utilized to simulate the oscillating motion. The present problem is first solved for a stationary triangular cylinder to obtain the natural Strouhal number. Then, the fluid dynamics for an oscillating triangular cylinder are solved in terms of instantaneous drag and lift, mean of drag, and root mean square (RMS) of lift coefficients. Detailed illustrations of flow streamlines and vortices contours are presented. The results indicate that when the oscillating frequency is equal to the natural Strouhal number of the stationary cylinder, the RMS of lift coefficient reaches its maximum and the oscillating amplitude has no effect at high oscillating frequency.

Finite element analysis of natural heat transfer from an isothermal array of cylinders in presence of vertical oscillations

A numerical simulation is performed to study the effects of vertical vibrations on natural convection heat transfer in an array of isothermal circular cylinders. Governing equations consist of continuity, momentum and energy equations were solved numerically by finite element method and Arbitrary Lagrangian–Eulerian kinematics description. The results showed that vertical oscillations of cylinders do not necessarily results in increment of average Nusselt number of each cylinder.

An investigation of a mixed convection in a ⊔ shape channel moving with a reciprocating motion

This study focuses on utilizing numerical calculation to investigate the heat transfer mechanisms in a ⊔ shape reciprocating channel system comprised of a horizontal channel at the bottom and vertical channels on both left and right sides. The issue is considered one kind of moving boundary problems and the finite element and Arbitrary Lagrangian–Eulerian (ALE) kinematic methods can be applied to this study. Due to the high temperature at the bottom surface of the horizontal channel and the direction of inlet cooling fluids in the same direction of the gravity, the heat transfer mechanisms induced by the mixed convection flow become extremely complex. The results show that thermal layers near the heat surface are disturbed drastically and the effect of reciprocating motion upon the heat transfer mechanisms strongly depends on a relationship between Reynolds and Grashof numbers.

A mixed convection in a reciprocating $$\Uppi$$ shape channel with opposite direction of gravity and inlet cooling fluids

A numerical study is executed to investigate heat transfer mechanisms of a reciprocating channel cooled by fluids. The problem of the reciprocating piston is a moving boundary problem, and the Finite Element Method and arbitrary Largragian–Eulerian Kinematics method are then utilized. Due to the high temperature of the channel, a mixed convection is taken into consideration for simulating a more realistic situation. Directions of inlet cooling fluids and gravity are opposite. Concerning important parameters of the Reynolds number and Grashof number, reciprocating frequency and amplitude are examined. Thermal stratification layers which are found firmly on the top surface are deeply dependent on the magnitude of Gr/Re 2 ratio which directly affects the heat transfer mechanisms.

Natural convection flow in a horizontal annulus with an oscillating inner cylinder using Lagrangian–Eulerian kinematics

In this article, the natural convection flow in a horizontal annulus enclosure with a transversely oscillating inner cylinder is studied numerically. The arbitrary Lagrangian–Eulerian kinematics description method is utilized to simulate flow and temperature fields. The finite element method is used to solve the governing equations. The effect of Rayleigh number, oscillating eccentricity, and oscillating frequency of the inner cylinder on the flow structure and heat transfer is presented. The results are validated with published results for a stationary inner cylinder case with different eccentricities. The results indicate that the oscillating inner cylinder highly affects the average Nusselt number at the inner cylinder. However, the oscillating eccentricity and frequency of the inner cylinder have a negative effect on the Nusselt number at the outer cylinder.

Finite element analysis of rubber extrusion forming process for automobile weather strip

The liquid-state melted rubber flow near the die region, during the extrusion forming process of automobile weather strips, is numerically investigated by the finite element thermal flow analysis. The shear viscosity of the melted rubber flow, which shows the significant dependence on both temperature and shear strain rate, is described by the power-law and Arrhenious-law models, and the associated parameters are interpolated by the least-square fitting of experimental values. The steady-state temperature and velocity fields are approximated by the coupled finite element method in the Eulerian kinematic description. The parametric characteristics of the melted rubber flow are examined through the illustrative numerical experiments.

Motion of a spherical particle in a cylindrical channel using arbitrary Lagrangian–Eulerian method

A finite element particle transport model, consisting of Navier–Stokes and continuity equations defined in arbitrary Lagrangian–Eulerian (ALE) kinematics, is employed to describe the motion of a rigid uncharged spherical particle in a cylindrical channel of uniform cross-section. The wall correction factors for the spherical particle moving with a fluid confined in an infinitely long cylindrical channel, as well as in finite length channels are presented. Two finite channel effects are considered, namely, motion of the particle at the entrance and exit of an open channel, and the motion of a particle toward the capped end of the channel. The numerical model demonstrates good agreement with many existing analytical results for infinite channels in the Stokes flow regime. Simple correlations for the hindrance factors are presented.

Limitation enhancement of a mixed convection by a vibrational heat surface in a vertical channel

A numerical simulation is performed to study effects of a vibrational heat surface on a mixed convection in a vertical channel flow. This subject is a kind of moving boundary problems, and the finite element method and arbitrary Lagrangian–Eulerian kinematics description method is then utilized. The main parameters of Grashof number, amplitude and frequency are taken into consideration, and the Reynolds number is limited and equals 100. According to the results, the equations of the critical vibration frequency could be derived and expressed in terms of Grashof number and amplitude. In the forced convection, the critical vibration frequency is only dependent on the amplitude, and in the mixed convection the critical vibration frequency becomes larger as the Grashof number increases. The variation of the critical vibration frequency of the study could league with the critical vibration frequency of the natural convection shown in the previous study consistently.

Solution of moving-boundary problems by the spectral element method

This paper describes a novel numerical model aiming at solving moving-boundary problems such as free-surface flows or fluid–structure interaction. This model uses a moving-grid technique to solve the Navier–Stokes equations expressed in the arbitrary Lagrangian–Eulerian kinematics. The discretization in space is based on the spectral element method. The coupling of the fluid equations and the moving-grid equations is essentially done through the conditions on the moving boundaries. Two- and three-dimensional simulations are presented: translation and rotation of a cylinder in a fluid, and large-amplitude sloshing in a rectangular tank. The accuracy and robustness of the present numerical model is studied and discussed.

An investigation of heat transfer of a reciprocating piston

A numerical simulation of a reciprocating piston cooled by fluid is performed to investigate effects of frequency, amplitude and Reynolds number on heat transfer rate of the piston. For facilitating the analyses, the model of the reciprocating piston is replaced by a “⨅” shape cooling channel. This subject belongs to a kind of moving boundary problems, and the Finite Element Method and arbitrary Lagragian–Eulerian Kinematics description method are then utilized. The heat transfer mechanisms affected by the main parameters of frequency, amplitude and Reynolds number are discussed in detail. The results show that the heat transfer mechanisms influenced by the frequency are more remarkable than those influenced by the amplitude and Reynolds number. Besides, the larger the above three parameters are, the apparent enhancements of heat transfer are usually obtained.

Effects of a vibrational heat surface on natural convection in a vertical channel flow

A numerical simulation is performed to study effects of a vibrational heat surface on natural convection in a vertical channel flow. This subject is a kind of moving boundary problems, and the finite element method and arbitrary Lagrangian–Eulerian kinematics description method is then utilized. The main parameters of Rayleigh number, amplitude and frequency are taken into consideration, and the mechanisms are examined in detail. The results show that for the same Rayleigh number, natural convection of a vibration heat plate with a certain combination of frequency and amplitude is possibly smaller than that of a stationary state. According to the results, the critical frequency could be derived and expressed in terms of Rayleigh number and amplitude.

Mesh Update Techniques for Free-Surface Flow Solvers Using Spectral Element Method

This paper presents a novel mesh-update technique for unsteady free-surface Newtonian flows using spectral element method and relying on the arbitrary Lagrangian–Eulerian kinematic description for moving the grid. Selected results showing compatibility of this mesh-update technique with spectral element method are given

Numerical investigation of heat transfer characteristics of the heated blocks in the channel with a transversely oscillating cylinder

A numerical simulation is performed to study the influence of an oscillating cylinder on the heat transfer from heated blocks in a channel flow. An arbitrary Lagrangian–Eulerian kinematics description method is adopted to describe the flow and thermal fields. A penalty consistent finite element formulation is applied to solve the governing equations. The effects of Reynolds number, oscillating amplitude and oscillating frequency on the heat transfer characteristics of the heated wall are examined. The results show that the heat transfer from heated blocks is enhanced remarkably as the oscillating frequency of the cylinder is in lock-in region.

Numerical study of heat transfer enhancement in a channel flow using an oscillating vortex generator

A numerical simulation is performed to study the unsteady flow and heat transfer in a channel with an oscillating bar, which is called an oscillating vortex generator. The behavior of the oscillating bar and flow is coupled, and the variation of the flow and thermal fields is classified as a moving boundary problem. An arbitrary Lagrangian–Eulerian kinematic description method is adopted to describe the flow and thermal fields, and a Galerkin finite element formulation with moving meshes is applied to solve the governing equations. The effects of Reynolds number, maximum oscillating speed, oscillating amplitude, and oscillating frequency of the bar on the flow and heat transfer are examined in details. The results show that the unsteady flow of transverse vortices is actively and largely formed behind the bar as the bar oscillates in the channel. The transverse vortices transport the low-temperature and high-speed core flow toward the heated regions of the channel. Also, the high-temperature wall flow is carried away from the heated regions of the channel to mix with the low-temperature core flow by transverse vortices. Based upon the fixed heat transfer area, the efficiency index is greater than 1.0 for almost all cases.

A numerical investigation of effects of a moving operator on airflow patterns in a cleanroom

The variations of the airflow induced by a moving operator in a cleanroom installed with a curtain were studied numerically. This situation is cataloged to a class of the moving boundary problems. An arbitrary Lagrangian–Eulerian kinematic description method is utilized to describe the flow field and a penalty finite element formulation with moving meshes is adopted to solve this problem. The effects of the moving operator and curtain on the airflow patterns under different distances from the workbench to the curtain with the moving speed of the operator equal to 2.0 and Reynolds number Re=500 are taken into account. The results show that recirculation zones are formed around the operator and workbench due to the movement of the operator. The recirculation zones are not favorable to the cleanroom because they may induce a local turbulent flow and entrain and trap contaminants. These phenomena are remarkably different from those of the moving operator assumed as stationary in the cleanroom. Based on the length of the curtain, it is useful for protecting the operator from the hazardous gases.