Eulerian Point Of View Research Articles

Convective rain cell properties and the resulting precipitation scaling in a warm‐temperate climate

AbstractConvective precipitation events have been shown to intensify at rates exceeding the Clausius–Clapeyron rate (CC rate) of ca. 7% K−1 under current climate conditions. In this study, we relate atmospheric variables (low‐level dew point temperature, convective available potential energy, and vertical wind shear), which are regarded as ingredients for severe deep convection, to properties of convective rain cells (cell area, maximum precipitation intensity, lifetime, precipitation sum, and cell speed). The rain cell properties are obtained from a rain gauge‐adjusted radar dataset in a mid‐latitude region, which is characterized by a temperate climate with warm summers (Germany). Different Lagrangian cell properties scale with dew point temperature at varying rates. While the maximum precipitation intensity of cells scales consistently at the CC rate, the area and precipitation sum per cell scale at varying rates above the CC rate. We show that this super‐CC scaling is caused by a covarying increase of convective available potential energy with dew point temperature. Wind shear increases the precipitation sum per cell mainly by increasing the spatial cell extent. From a Eulerian point of view, this increase is partly compensated by a higher cell velocity, which leads to Eulerian precipitation scaling rates close to and slightly above the CC rate. Thus, Eulerian scaling rates of convective precipitation are modulated by convective available potential energy and vertical wind shear, making it unlikely that present scaling rates can be applied to future climate conditions. Furthermore, we show that cells that cause heavy precipitation at fixed locations occur at low vertical wind shear and, thus, move relatively slowly compared to typical cells.

Tamed spaces – Dirichlet spaces with distribution-valued Ricci bounds

We develop the theory of tamed spaces which are Dirichlet spaces with distribution-valued lower bounds on the Ricci curvature and investigate these from an Eulerian point of view. To this end we analyze in detail singular perturbations of Dirichlet form by a broad class of distributions. The distributional Ricci bound is then formulated in terms of an integrated version of the Bochner inequality using the perturbed energy form and generalizing the well-known Bakry-Émery curvature-dimension condition. Among other things we show the equivalence of distributional Ricci bounds to gradient estimates for the heat semigroup in terms of the Feynman-Kac semigroup induced by the taming distribution as well as consequences in terms of functional inequalities. We give many examples of tamed spaces including in particular Riemannian manifolds with interior singularities and singular boundary behavior.

On the mean field limit for Cucker-Smale models

<p style='text-indent:20px;'>In this note, we consider generalizations of the Cucker-Smale dynamical system and we derive rigorously in Wasserstein's type topologies the mean-field limit (and propagation of chaos) to the Vlasov-type equation introduced in [<xref ref-type="bibr" rid="b13">13</xref>]. Unlike previous results on the Cucker-Smale model, our approach is not based on the empirical measures, but, using an Eulerian point of view introduced in [<xref ref-type="bibr" rid="b9">9</xref>] in the Hamiltonian setting, we show the limit providing explicit constants. Moreover, for non strictly Cucker-Smale particles dynamics, we also give an insight on what induces a flocking behavior of the solution to the Vlasov equation to the - unknown a priori - flocking properties of the original particle system.</p>

On the time irreversibility of compressible turbulence reflected by particles of various inertias

Time irreversibility of compressible homogeneous isotropic turbulence (HIT) is investigated from a Lagrangian point of view and single-particle statistics. For this purpose, direct numerical simulation (DNS) is implemented for compressible HIT at Taylor-mircoscale Reynolds number Reλ∼100 and turbulent Mach number Mt up to 1.01, in which tracers and inertial particles in a wide Stokes number (St) range are instantaneously tracked with time. The statistics of instantaneous power of particles corroborates that the violation of detailed balance of turbulence in compressible HIT is much stronger than in incompressible HIT. It turns out that the third moment of dimensionless instantaneous power (Ir) of tracers scales as Mt4. A possible explanation based on Kolmogorov-like argument proves to be plausible due to the lacks of direct verification and generality for other moments of the power. A further analysis from an Eulerian point of view suggests that the underlying mechanism for time irreversibility of highly compressible turbulence is quite different from that for weakly compressible or incompressible turbulence. For inertial particles, the moments of instantaneous power are suggested to scale as St−2n/3 at relatively large St numbers, and be dependent only on Mach number or Reynolds number at the small-St number end, which are manifested by the present numerical data. It is further shown that the empirical Mt4 scaling of Ir also approximately applies to various inertial particles, but only at high Mt numbers due to the diminishing effect of compressibility to the low-Mach number end. The time irreversibility announced by the Lagrangian statistical properties of particles of different inertias is shown to be highly associated with their responses to the vortex and shocklet structures in compressible turbulence.

Eulerian spectrum of finite-time Lyapunov exponents in compound channels

Fluid flows reveal a wealth of structures, such as vortices and barriers to transport. Usually, either an Eulerian or a Lagrangian frame of reference is employed in order to detect such features of the flow. However, the two frameworks detect structures that have different properties. Indeed, common Eulerian diagnostics (Hua-Klein and Okubo-Weiss criterion) employed in order to detect vortices do not always agree with Lagrangian diagnostics such as finite-time Lyapunov exponents. Besides, the former are Galilean-invariant whereas the latter is objective. However, both the Lagrangian and the Eulerian approaches to coherent structure detection must show some links under any inertial-frame. Compound channels flows have been accurately studied in the past, both from a Lagrangian and an Eulerian point of view. The features detected do not superimpose: Eulerian vortices do not coincide with barriers to transport. The missing link between the two approaches is here recovered thanks to a spectral analysis.

Particle clustering and subclustering as a proxy for mixing in geophysical flows

Abstract. The Eulerian point of view is the traditional theoretical and numerical tool to describe fluid mechanics. Some modern computational fluid dynamics codes allow for the efficient simulation of particles, in turn facilitating a Lagrangian description of the flow. The existence and persistence of Lagrangian coherent structures in fluid flow has been a topic of considerable study. Here we focus on the ability of Lagrangian methods to characterize mixing in geophysical flows. We study the instability of a strongly non-linear double-jet flow, initially in geostrophic balance, which forms quasi-coherent vortices when subjected to ageostrophic perturbations. Particle clustering techniques are applied to study the behavior of the particles in the vicinity of coherent vortices. Changes in inter-particle distance play a key role in establishing the patterns in particle trajectories. This paper exploits graph theory in finding particle clusters and regions of dense interactions (also known as subclusters). The methods discussed and results presented in this paper can be used to identify mixing in a flow and extract information about particle behavior in coherent structures from a Lagrangian point of view.

Assessment of the Representation of West African Storm Lifecycles in Convection‐Permitting Simulations

AbstractConvection‐permitting models perform better at representing the diurnal cycle and the intermittency of convective rainfall over land than parameterized‐convection models. However, most of the previous model assessments have been from an Eulerian point of view, while key impacts of the rainfall depend on a storm‐relative perspective of the system lifecycle. Here a storm‐tracking algorithm is used to generate storm‐centered Lagrangian lifecycle statistics of precipitation over West Africa from regional climate model simulations and observations. Two versions of the Met Office Unified Model with and without convection parameterization at 4‐, 12‐, and 25‐km resolution were analyzed. In both of the parameterized‐convection simulations, storm lifetimes are too short compared to observations, and storms have no preferred propagation direction; the diurnal cycle of initiations and dissipations and the spatial distribution of storms are also inaccurate. The storms in the convection‐permitting simulations have more realistic diurnal cycles and lifetimes but are not as large as the largest observed storms. The convection‐permitting model storms propagate in the correct direction, although not as fast as observed storms, and they have a much improved spatial distribution. The rainfall rate of convection‐permitting storms is likely too intense compared to observations. The improved representation of the statistics of organized convective lifecycles shows that convection‐permitting models provide better simulation of a number of aspects of high‐impact weather, which are critical to climate impacts in this important geographic region, providing the high rainfall rates can be taken into account.

Some remarks on recent developments in micropolar continuum theory

This paper considers micropolar media that can undergo structural changes and do not a priori consist of indestructible material particles. Initially the pertinent literature is reviewed. Then the necessary theoretical framework for a continuum of that type is presented. The standard macroscopic equations for mass, linear and angular momentum are complemented by a recently proposed balance for the moment of inertia tensor, which contains a production term. Two examples illustrate the effect of the production. In the first example, we study a continuous stream of matter on a conveyor belt going through a crusher so that the total number of particles will change. In context with this example, it is also clear that the traditional Lagrangian way of describing the motion of solids is no longer adequate and must be replaced by the Eulerian point of view known from fluid mechanics. The second example deals with hollow particles which rotate because of the presence of body couples. Now a transient temperature field is superimposed such that the moment of inertia field changes due to the thermal expansion of particles. This in turn results in rotational motion that is no longer constant but varies in space and time.

Saint-Venant Equations and Friction Law for Modelling Self-Channeling Granular Flows: From Analogue to Numerical Simulation

Rock avalanches are catastrophic events involving important granular rock masses (>106 m3) and traveling long distances. In exceptional cases, the runout can reach up to tens of kilometers. Even if they are highly destructive and uncontrollable events, they give important insights to understand interactions between the displaced masses and landscape conditions. However, those events are not frequent. Therefore, the analogue and numerical modelling gives fundamental inputs to better understand their behavior. The objective of the research is to understand the propagation and spreading of granular mass released at the top of a simple geometry. The flow is unconfined, spreading freely along a 45° slope and deposit on a horizontal surface. The evolution of this analogue rock avalanche was measured from the initiation to its deposition with high speed camera. To simulate the analogue granular flow, a numerical model based on the continuum mechanics approach and the solving of the shallow water equations was used. In this model, the avalanche is described from a eulerian point of view within a continuum framework as single phase of incompressible granular material. The interaction of the flowing layer with the substratum follows a Mohr-Coulomb friction law. Within same initial conditions (slope, volume, basal friction, height of fall and initial velocity), results obtained with the numerical model are similar to those observed in the analogue. In both cases, the runout of the mass is comparable and the size of both deposits matches well. Moreover, both analogue and numerical modeling gave same magnitude of velocities. In this study, we highlighted the importance of the friction on a flowing mass and the influence of the numerical resolution on the propagation. The combination of the fluid dynamic equation with the frictional law enables the self-channelization and the stop of the granular mass.

Simulation of nanoparticle coagulation under Brownian motion and turbulence in a differential–algebraic framework: Developments and applications

This paper is devoted to the modeling of the dynamics of nano-aerosols under Brownian motion and turbulence effects. An Eulerian point of view is considered through a population balance equation which is solved in a differential–algebraic framework (DAE-QMOM). This choice is supported by making a brief comparison with reference numerical methods. In order to evaluate the physical insights of the model, a comparison with five available experiments is carried out. The present results reveal the necessity to take into consideration the Brownian and turbulence effects together in the coagulation kernel and also highlight the need to introduce the fractal dimension of aggregates.

Effect of computer-generated turbulent wind field on trajectory of compact debris: A probabilistic analysis approach

A model and numerical algorithm is developed to simulate wind-borne debris trajectories in a fully-developed atmospheric boundary layer wind. The model works in two dimensions and makes use of synthetically-generated turbulence time histories; it accounts for variable mean velocity field with elevation and turbulence. The simulation of a partially coherent wind field was based on the wave superposition method (Di Paola, 1998) [1]. For the simulation of the turbulence field, a simplified approach is proposed. First, turbulence is generated at discrete points located on the “inlet boundary” of the field; second, turbulence is propagated through the field using either Taylor’s “frozen turbulence” hypothesis or a simplified “Eulerian–Lagrangian” formulation. The latter term is used to emphasize that an expression is employed to approximately replicate the features of the Lagrangian turbulence wind spectrum (for high-speed moving objects), even though turbulence is still synthetically generated on a large portion of the field at all times, from an Eulerian point of view. After generating the wind field, the trajectory of compact objects is estimated by means of a point mass dynamic model, converted to state-space form and integrated by fifth-order Runge–Kutta method.The numerical model is applied to the study of the risk of impact by wind-borne debris against tall building facades, recently investigated by the authors for uniform non-turbulent wind field only. The analysis is conducted by using the computer-generated wind field to estimate “universal probability curves” (probability-of-impact curves) for compact debris, conditional on the initial distance of the object from the building before takeoff. Both qualitative and quantitative variations are noted in comparison with previous results.

An Eulerian Approach to the Analysis of Krause's Consensus Models

In this paper we analyze a class of multiagent consensus dynamical systems inspired by Krause's original model. As in Krause's model, the basic assumption is the so-called bounded confidence: two agents can influence each other only when their state values are below a given distance threshold $R$. We study the system under an Eulerian point of view considering (possibly continuous) probability distributions of agents, and we present original convergence results. The limit distribution is always necessarily a convex combination of delta functions at least $R$ far apart from each other: in other terms these models are locally aggregating. The Eulerian perspective provides the natural framework for designing a numerical algorithm, by which we obtain several simulations in $1$ and $2$ dimensions.

A Eulerian approach to the analysis of rendez-vous algorithms

In this paper we analyze rendez-vous algorithms in the situation when agents can only exchange information below a given distance threshold R. We study the system under an Eulerian point of view considering (possibly continuous) probability distributions of agents and we present convergence results both in discrete and in continuous time. The limit distribution is always necessarily a convex combination of delta functions at least R far apart from each other: in other terms these algorithms are locally aggregating. Numerical simulations seem to show that starting from continuous distributions, in general these algorithms do not converge to a unique delta (rendez-vous) in agreement with previous literature on this subject.

On stress rate and plasticity constitutive equations referred to a body-fixed coordinate system

Investigators have viewed the stress rate in two different ways: the material (body-fixed) point of view and the Eulerian point of view. We discuss the Zaremba–Jaumann rate and Oldroyd’s rate from the material viewpoint and apply them to the material formulation of a theory of plasticity for materials undergoing anisotropic plastic deformation. Significant advantages of the material formulation are that the derivation of equations is straight forward, the distortion of yield surface can be easily accounted for, and the issue of self-consistent elastic equation does not arise. A convected (body-fixed) coordinate system that is imbedded in the material element is used to define a convected material element. Constitutive equations in a theory of plasticity are written with respect to this element referred to the body-fixed coordinate system. A material type yield function is a function of stress components and it is invariant under either a superimposed rigid rotation of the body or an arbitrary change of observer. The kinematic hardening evolves by the co-rotational rate and it is disassociated from the general kinematic of deformation. The latter evolves by the convected rate.

Dispersion measurements using time-of-flight remote detection MRI

Remote detection nuclear magnetic resonance and magnetic resonance imaging can be used to study fluid flow and dispersion in a porous medium from a purely Eulerian point of view (i.e., in a laboratory frame of reference). Information about fluid displacement is obtained on a macroscopic scale in a long-time regime, while local velocity distributions are averaged out. It is shown how these experiments can be described using the common flow propagator formalism and how experimental data can be analyzed to obtain effective porosity, flow velocity inside the porous medium, fluid dispersion and flow tracing of fluid.

Existence of weak solutions for an interaction problem between an elastic structure and a compressible viscous fluid

In this article, we are interested by the three-dimensional motion of an elastic structure immersed in a viscous compressible fluid. The fluid and the structure are contained in a fixed bounded set. To describe the structure motion, we choose an Eulerian point of view and we strongly regularize the equation of the solid motion in order to get additional estimates on the elastic deformations. Our maim result is an existence result of weak solutions defined as long as no collisions occur and as long as conditions of non-interpenetration and of preservation of orientation are satisfied.

RECONSTRUCTION OF COMPLEX AND MULTIPLE SHAPE OBJECT CONTOURS USING A LEVEL SET METHOD

A very accurate inverse scattering method for reconstructing the contours of perfectly conducting cylinders of arbitrary crosssection illuminated by electromagnetic waves under TM polarization with time-harmonic dependence is presented. No assumption is made on the shapes of the contours nor on their number. Inversion strategy consists in computing a special deformation velocity, defined on the contours of the objects, and allowing to decrease a cost function at each iteration. Level set methods, used in as diverse fields as fluid dynamics, image processing and crystal growth, are used to perform the evolution of the contours. These methods allow to handle automatically topological changes by using a Eulerian point of view built on a level set representation of the contours. Hence, several objects can be retrieved from one single initial contour. Moreover, contours regularization is easily performed by adding a curvature term to the velocity. The inverse scattering method is based on a very accurate forward scattering algorithm characterized by the use of well-behaved pseudo-currents which are developed in Fourier series along the contours. The frequency hopping technique turns out to play a key role in the inversion process. The method is proved to be flexible, allowing to reconstruct accurately complex and multiple shape contours from one single initial guess. The deformation velocity is even able to retrieve very sharp corners and is quite robust with respect to high noise levels.

Experimental study of chaotic advection regime in a twisted duct flow

The generation of Lagrangian chaos has been studied experimentally in a twisted duct flow, a configuration representing a three-dimensional steady open flow in which various signatures of Lagrangian chaos are documented. The twisted duct consists of four 90 ∘ bends of square cross-section; the plane of curvature of each bend is at 90 ∘ to that of its neighbors. Dean roll-cells, generated by centrifugal forces and the geometrical perturbation due to the change in curvature plane, are the source of the irregular trajectories of the fluid particles. The Eulerian nature of the flow was investigated using a laser Doppler velocimeter (LDV). From the Eulerian point of view, the flow is completely regular. We show by laser-induced fluorescence (LIF) visualization that many characteristics of a Lagrangian chaotic system are present in this flow: − strong stretching and folding of material lines and surfaces; − sensitivity to initial conditions; − exponential growth of stretching in some flow regions. It is also shown that in some regions of the flow stretchings grow linearly with space, indicating non-chaotic behavior. Due to the chaotic nature of the flow, an equalization of the fly-time of fluid particles was observed during their passage through the twisted duct.

Unified approach to electromagnetic shape-inverse and mechanically coupled problems via a Lagrangian technique

The aim of this paper is to present a Lagrangian unified approach to shape-dependent electromagnetic problems (equilibrium, dynamic coupled, shape inverse, trajectory inverse problems). The main features of this approach are illustrated, and a comparison with the usual Eulerian point of view is made, from both the analytical and the numerical side, showing several advantages of the Lagrangian approach. A simple analytical example is presented in order to help comprehension and to show the effectiveness of the method. In addition, some practical applications are described.

Elastic Finite Element Analysis for a Flexible Beam Structure.

A finite element anlaysis is performed for large deformations of a felxible beam. The total Lagrangian formulation for a general large deformation, which involves finite rotations, is chosen and the exponential map is used to treat finite rotations from the Eulerian point of view. The finite elements results are confirmed for several cases of deformations through comparison to a first order elasticity solution obtained by numerical integration, and the agreement between the two is found to be excellent. For lateral buckling, the point of vanishing determinant of the resulting unsymmetric tangent stiffness is traced to examine its relationship to bifurcation points. It is found that the points of vanishing determinant is not corresponding to bifurcation points for large deformation in general, which suggests that the present unsymmetric tangent stiffness is not an exact first derivative of internal forces with respect to displacement.