Kai Schneider

• Dipl.-Math. techn., Dr. rer. nat., Dr. habil.
• Professor (Full) at Aix-Marseille University

We investigate the behavior of heavy impurities in edge plasma turbulence by analyzing their trajectories using the Hasegawa-Wakatani model. Through direct numerical simulations, we track ensembles of charged impurity particles over hundreds of eddy turnover times within statistically steady turbulent flows. Assuming that heavy impurities lag behin...

We investigate the behavior of heavy impurities in edge plasma turbulence by analyzing their trajectories using the Hasegawa–Wakatani model. Through direct numerical simulations, we track ensembles of charged impurity particles over hundreds of eddy turnover times within statistically steady turbulent flows. Assuming that heavy impurities lag behin...

We present a new methodology for decomposing flows with multiple transports that further extends the shifted proper orthogonal decomposition (sPOD). The sPOD tries to approximate transport-dominated flows by a sum of co-moving data fields. The proposed methods stem from sPOD but optimize the co-moving fields directly and penalize their nuclear norm...

Confinement quality in fusion plasma is influenced significantly by the presence of heavy impurities, which can lead to radiative heat loss and reduced confinement. This study explores the clustering of heavy impurity, i.e. tungsten in edge plasma, using high-resolution direct numerical simulations of the Hasegawa–Wakatani equations. We use the Sto...

The goal of this numerical study is to get insight into singular solutions of the two-dimensional (2D) Euler equations for nonsmooth initial data, in particular for vortex sheets. To this end, high resolution computations of vortex layers in two-dimensional incompressible Euler flows are performed using the characteristic mapping method (CMM). This...

This work introduces a generalized characteristic mapping method designed to handle non-linear advection with source terms. The semi-Lagrangian approach advances the flow map, incorporating the source term via the Duhamel integral. We derive a recursive formula for the time decomposition of the map and the source term integral, enhancing computatio...

This paper presents a neural network-based methodology for the decomposition of transport-dominated fields using the shifted proper orthogonal decomposition (sPOD). Classical sPOD methods typically require an a priori knowledge of the transport operators to determine the co-moving fields. However, in many real-life problems, such knowledge is diffi...

Confinement quality in fusion plasma is significantly influenced by the presence of heavy impurities, which can lead to radiative heat loss and reduced confinement. This study explores the clustering of heavy impurity, \textit{i.e.}, Tungsten in edge plasma, using high-resolution direct numerical simulations of the Hasegawa--Wakatani equations. We...

Cluster and void formations are key processes in the dynamics of particle-laden turbulence. In this work, we assess the performance of various neural network models for synthesizing preferential concentration fields of particles in turbulence. A database of direct numerical simulations of homogeneous isotropic two-dimensional turbulence with one-wa...

This paper presents a neural network-based methodology for the decomposition of transport-dominated fields using the shifted proper orthogonal decomposition (sPOD). Classical sPOD methods typically require an a priori knowledge of the transport operators to determine the co-moving fields. However, in many real-life problems, such knowledge is diffi...

Direct numerical simulation of homogeneous isotropic turbulence shows pronounced clustering of inertial particles in the inertial subrange at high Reynolds number, in addition to the clustering typically observed in the near dissipation range. The clustering in the inertial subrange is characterized by the bump in the particle number density spectr...

Direct numerical simulation of homogeneous isotropic turbulence shows pronounced clustering of inertial particles in the inertial subrange at high Reynolds number, in addition to the clustering typically observed in the near dissipation range. The clustering in the inertial subrange is characterized by the bump in the particle number density spectr...

We propose an efficient semi-Lagrangian characteristic mapping method for solving the one+one-dimensional Vlasov-Poisson equations with high precision on a coarse grid. The flow map is evolved numerically and exponential resolution in linear time is obtained. Global third-order convergence in space and time is shown and conservation properties are...

Desert ants stand out as some of the most intriguing insect navigators, having captured the attention of scientists for decades. This includes the structure of walking trajectories during goal approach and search behaviour for the nest and familiar feeding sites. We analyse such trajectories with regard to changes in walking direction. The directio...

Taylor's hypothesis of frozen flow is revisited in homogeneous turbulent shear flow by examining the cancellation properties of Eulerian and convective accelerations at different flow scales. Using results of direct numerical simulations, vector-valued flow quantities, including the Lagrangian, Eulerian, and convective accelerations, are decomposed...

We present a new methodology for decomposing flows with multiple transports that further extends the shifted proper orthogonal decomposition (sPOD). The sPOD tries to approximate transport-dominated flows by a sum of co-moving data fields. The proposed methods stem from sPOD but optimize the co-moving fields directly and penalize their nuclear norm...

This work investigates the behavior of impurities in edge plasma of tokamaks using high-resolution numerical simulations based on Hasegawa–Wakatani equations. Specifically, it focuses on the behavior of inertial particles, which has not been extensively studied in the field of plasma physics. Our simulations utilize one-way coupling of a large numb...

A wavelet-based machine learning method is proposed for predicting the time evolution of homogeneous isotropic turbulence where vortex tubes are preserved. Three-dimensional convolutional neural networks and long short-term memory are trained with a time series of direct numerical simulation (DNS) data of homogeneous isotropic turbulence at the Tay...

Clustering dynamics of inertial particles in turbulent channel flow are studied via tessellation-based analysis of high-fidelity simulation data at $Re_\tau \ approx 230$ with various values of mass loading (10% - 100%) and the Stokes number ($St^+ = [1 - 60]$). We then characterise the solenoidal, rotational, and swirling motions of clusters by co...

Adaptive Galerkin numerical schemes integrate time-dependent partial differential equations with a finite number of basis functions, and a subset of them is selected at each time step. This subset changes over time discontinuously according to the evolution of the solution; therefore the corresponding projection operator is time-dependent and nondi...

We propose finite-time measures to compute the divergence, the curl and the velocity gradient tensor of the point particle velocity for two- and three-dimensional moving particle clouds. For this purpose, a tessellation of the particle positions is performed to assign a volume to each particle. We introduce a modified Voronoi tessellation which ove...

We propose an efficient semi-Lagrangian Characteristic Mapping (CM) method for solving the three-dimensional (3D) incompressible Euler equations. This method evolves advected quantities by discretizing the flow map associated with the velocity field. Using the properties of the Lie group of volume preserving diffeomorphisms SDiff, long-time deforma...

We develop and analyze error estimators and mesh adaptation strategies within a discontinuous Galerkin formulation. The basic idea of the study is to reduce the computational cost of the simulation by employing mesh adaptation as a better alternative to the use of uniform grids. The novelty of the study resides in the use of multiwavelets and how t...

We propose finite-time measures to compute the divergence, the curl and the velocity gradient tensor of the point particle velocity for two- and three-dimensional moving particle clouds. To this end, tessellation of the particle positions is applied to associate a volume to each particle. Considering then two subsequent time instants, the dynamics...

The state-of-the-art of insect flight research using advanced computational fluid dynamics techniques on supercomputers is reviewed, focusing mostly on the work of the present authors. We present a brief historical overview, discuss numerical challenges and introduce the governing model equations. Two open source codes, one based on Fourier, the ot...

Lagrangian statistics and particle transport in edge plasma turbulence are investigated using the Hasegawa–Wakatani model and its modified version. The latter shows the emergence of pronounced zonal flows. Different values of the adiabaticity parameter are considered. The main goal is to characterize the role of coherent structures, i.e., vortices...

Adaptive Galerkin methods for time-dependent partial differential equations are studied and shown to be dissipative. The adaptation implies that the subset of the selected basis function changes over time according to the evolution of the solution. The corresponding projection operator is thus time-dependent and non differentiable. We therefore pro...

Investigation of electrodynamic effects considering South American features is essential to extend understanding of middle- to low-latitude space weather phenomena. For retrieving magnetic contributions related to geomagnetically induced currents (GIC), a wavelet-based filtering method is verified and applied to magnetic records on the ground. The...

Lagrangian statistics and particle transport in edge plasma turbulence are investigated using the Hasegawa-Wakatani model and its modified version. The latter shows the emergence of pronounced zonal flows. Different values of the adiabaticity parameter are considered. The main goal is to characterize the role of coherent structures, i.e., vortices...

Direct numerical simulation is used to investigate effects of turbulent flow in the confined geometry of a face-centred cubic porous unit cell on the transport, clustering and deposition of fine particles at different Stokes numbers ( $St = 0.01, 0.1, 0.5, 1, 2$ ) and at a pore Reynolds number of 500. Particles are advanced using one-way coupling a...

The proper orthogonal decomposition (POD) is a powerful classical tool in fluid mechanics used, for instance, for model reduction and extraction of coherent flow features. However, its applicability to high-resolution data, as produced by three-dimensional direct numerical simulations, is limited owing to its computational complexity. Here, we prop...

Insect wings can undergo significant deformation during flapping motion owing to inertial, elastic and aerodynamic forces. Changes in shape then alter aerodynamic forces, resulting in a fully coupled Fluid-Structure Interaction (FSI) problem. Here, we present detailed three-dimensional FSI simulations of deformable blowfly (Calliphora vomitoria) wi...

Insect wings can undergo significant deformation during flapping motion owing to inertial, elastic and aerodynamic forces. Changes in shape then alter aerodynamic forces, resulting in a fully coupled fluid–structure interaction (FSI) problem. Here, we present detailed three-dimensional FSI simulations of deformable blowfly (Calliphora vomitoria) wi...

We develop a wavelet-based three-dimensional convolutional neural network (WCNN3d) for superresolution of coarse-grained data of homogeneous isotropic turbulence. The turbulent flow data are computed by high resolution direct numerical simulation (DNS), while the coarse-grained data are obtained by applying a Gaussian filter to the DNS data. The CN...

We study the influence of the shape of the plasma container on the dynamics of the reversed-field pinch (RFP). The geometries we consider are periodic cylinders with elliptical and circular-shaped cross-sections. Numerical simulations of fully nonlinear viscoresistive magnetohydrodynamics are carried out to illustrate how the plasma dynamics is aff...

We study the influence of the shape of the plasma container on the dynamics of the Reversed Field Pinch (RFP). The geometries we consider are periodic cylinders with elliptical and circular-shaped cross-sections. Numerical simulations of fully nonlinear visco-resistive magnetohydrodynamics are carried out to illustrate how the plasma dynamics are a...

We study dynamical Galerkin schemes for evolutionary partial differential equations (PDEs), where the projection operator changes over time. When selecting a subset of basis functions, the projection operator is non-differentiable in time and an integral formulation has to be used. We analyze the projected equations with respect to existence and un...

Direct numerical simulation is used to investigate effects of turbulent flow in the confined geometry of a face-centered cubic porous unit cell on the transport, clustering, and deposition of fine particles at different Stokes numbers ($St = 0.01, 0.1, 0.5, 1, 2$) and at a pore Reynolds number of 500. Particles are advanced using one-way coupling a...

The Lagrangian (LA) and Eulerian Acceleration (EA) properties of fluid particles in homogeneous turbulence with uniform shear and uniform stable stratification are studied using direct numerical simulations. The Richardson number is varied from $Ri=0$, corresponding to unstratified shear flow, to $Ri=1$, corresponding to strongly stratified shear f...

The Lagrangian and Eulerian acceleration properties of fluid particles in homogeneous turbulence with uniform shear and uniform stable stratification are studied using direct numerical simulations. The Richardson number is varied from Ri=0, corresponding to unstratified shear flow, to Ri=1, corresponding to strongly stratified shear flow. The proba...

We propose an efficient semi-Lagrangian Characteristic Mapping (CM) method for solving the three-dimensional (3D) incompressible Euler equations. This method evolves advected quantities by discretizing the flow map associated with the velocity field. Using the properties of the Lie group of volume preserving diffeomorphisms SDiff, long-time deforma...

This corrigendum contains corrections to some figures and the corresponding text in the article “Volume penalization for inhomogeneous Neumann boundary conditions modeling scalar flux in complicated geometry” [J. Comput. Phys. 390 (2019) 452–469] [1] due to some errors in the implementation of our code. The corrected results yield improved converge...

Multiscale statistical analyses of inertial particle distributions are presented to investigate the statistical signature of clustering and void regions in particle-laden incompressible isotropic turbulence. Three-dimensional direct numerical simulations of homogeneous isotropic turbulence at high Reynolds number (Reλ≳200) are performed. Lagrangian...

Fully adaptive computations of the resistive magnetohydrodynamic (MHD) equations are presented in two and three space dimensions using a finite volume discretization on locally refined dyadic grids. Divergence cleaning is used to control the incompressibility constraint of the magnetic field. For automatic grid adaptation a cell-averaged multiresol...

Fully adaptive computations of the resistive magnetohydrodynamic (MHD) equations are presented in two and three space dimensions using a finite volume discretization on locally refined dyadic grids. Divergence cleaning is used to control the incompressibility constraint of the magnetic field. For automatic grid adaptation a cell-averaged multiresol...

Fluid-structure interactionKolomenskiy, Dmitry of the flapping wings of a hovering bumblebee is considered.Ravi, Sridhar Kinematic reconstruction of the wing motion using synchronized high-speed video recordings is described,Xu, Ru that provides the necessary input data for numerical modelling.Ueyama, Kohei Computational fluid dynamicsJakobi, Timot...

This volume collects the most important contributions from four minisymposia from ICIAM 2019. The papers highlight cutting-edge applications of Cartesian CFD methods and describe the employed algorithms and numerical schemes. An emphasis is laid on complex multi-physics applications like magnetohydrodynamics, combustion, aerodynamics with fluid-str...

The main goal of the proposed research is to investigate and develop error estimators and mesh adaptation strategies within a discontinuous Galerkin (DG) formulation in order to reduce the computational cost, for a prescribed level of accuracy. We are interested in how multiwavelets (MWs) and their properties may shed new light on the adaptation pr...

Wing flexibility plays an essential role in the aerodynamic performance of insects due to the considerable deformation of their wings during flight under the impact of inertial and aerodynamic forces. These forces come from the complex wing kinematics of insects. In this study, both wing structural dynamics and flapping wing motion are taken into...

Inertial particle data from three-dimensional direct numerical simulations of particle-laden homogeneous isotropic turbulence at high Reynolds number are analysed using Voronoi tessellation of the particle positions and considering different Stokes numbers. A finite-time measure to quantify the divergence of the particle velocity by determining the...

The proper orthogonal decomposition (POD) is a powerful classical tool in fluid mechanics used, for instance, for model reduction and extraction of coherent flow features. However, its applicability to high-resolution data, as produced by three-dimensional direct numerical simulations, is limited owing to its computational complexity. Here, we prop...

EDITOR'S RECOMMENDATION The flight of insects has enlightened the flying dream of human beings for centuries. Wing flexibility is often used by insects to increase their flight efficiencies. However, the mechanism of the increased efficiencies still remains mysterious. Prof. Kai Schneider's group studies the aerodynamics of a tethered flapping bumb...

We propose an efficient semi-Lagrangian method for solving the two-dimensional incompressible Euler equations with high precision on a coarse grid. This new approach evolves the flow map using a combination of the Characteristic Mapping (CM) method [1] the gradient-augmented level-set (GALS) method [2]. The flow map possesses a semigroup structure...

Multiscale statistical analyses of inertial particle distributions are presented to investigate the statistical signature of clustering and void regions in particle-laden incompressible isotropic turbulence. Three-dimensional direct numerical simulations of homogeneous isotropic turbulence at high Reynolds number ($Re_\lambda \gtrsim 200$) with up...

Wing flexibility plays an essential role in the aerodynamic performance of insects due to the considerable deformation of their wings during flight under the impact of inertial and aerodynamic forces. These forces come from the complex wing kinematics of insects. In this study, both wing structural dynamics and flapping wing motion are taken into a...

Inertial particle data from three-dimensional direct numerical simulations of particle-laden homogeneous isotropic turbulence at high Reynolds number are analyzed using Voronoi tessellation of the particle positions, considering different Stokes numbers. A finite-time measure to quantify the divergence of the particle velocity by determining the vo...

Dynamic mesh adaptation methods require suitable refinement indicators. In the absence of a comprehensive error estimation theory, adaptive mesh refinement (AMR) for nonlinear hyperbolic conservation laws, e.g. compressible Euler equations, in practice utilizes mainly heuristic smoothness indicators like combinations of scaled gradient criteria. As...

A turbulent flow mixes in general more rapidly a passive scalar than a laminar flow does. From an energetic point of view, for statistically homogeneous or periodic flows, the laminar regime is more efficient. However, the presence of walls may change this picture. We consider in this investigation mixing in two-dimensional laminar and turbulent wa...

The secret to the spectacular flight capabilities of flapping insects lies in their wings, which are often approximated as flat, rigid plates. Real wings are however delicate structures, composed of veins and membranes, and can undergo significant deformation. In the present work, we present detailed numerical simulations of such deformable wings....

The secret to the spectacular flight capabilities of flapping insects lies in their wings, which are often approximated as flat, rigid plates. Real wings are however delicate structures, composed of veins and membranes, and can undergo significant deformation. In the present work, we present detailed numerical simulations of such deformable wings....

We present a wavelet-based adaptive method for computing 3D flows in complex, time-dependent geometries, implemented on massively parallel computers. The incompressible fluid is modeled with an artificial compressibility approach in order to avoid the solution of elliptical problems. No-slip and in/outflow boundary conditions are imposed using volu...

We propose an efficient semi-Lagrangian method for solving the two-dimensional incompressible Euler equations with high precision on a coarse grid. The new approach evolves the flow map using the gradient-augmented level set method (GALSM). Since the flow map can be decomposed into submaps (each over a finite time interval), the error can be contro...

The question of the relative importance of coherent structures and waves has for a long time attracted a great deal of interest in astrophysical plasma turbulence research, with a more recent focus on kinetic scale dynamics. Here we utilize high-resolution observational and simulation data to investigate the nature of waves and structures emerging...

Computational magnetohydrodynamics (MHD) for space physics has become an essential area in understanding the multiscale dynamics of geophysical and astrophysical plasma processes, partially motivated by the lack of space data. Full MHD simulations are typically very demanding and may require substantial computational efforts. In particular, computa...

It is shown that the wings of bumblebees during flapping undergo pitching (feathering angle) rotation that can be characterized as a fluid-structure interaction problem. Measurements of shape, size and inertial properties of the wings of bumblebees Bombus ignitus are described that provide the necessary input data for numerical modelling. A computa...

Computational magneto-hydrodynamics (MHD) for space physics has become an essential area in understanding the multi-scale dynamics of geophysical and astrophysical plasma processes, partially motivated by the lack of space data. Full MHD simulations are typically very demanding and may require substantial computational efforts. In particular, comp...

A space-time fully adaptive multiresolution method for evolutionary non-linear partial differential equations is presented introducing an improved local time-stepping method. The space discretisation is based on classical finite volumes, endowed with cell average multiresolution analysis for triggering the dynamical grid adaptation. The explicit ti...

We develop a volume penalization method for inhomogeneous Neumann boundary conditions, generalizing the flux-based volume penalization method for homogeneous Neumann boundary condition proposed by Kadoch et al. [J. Comput. Phys. 231 (2012) 4365]. The generalized method allows us to model scalar flux through walls in geometries of complex shape usin...

The fluid–structure interaction problem of the flapping wings of bumblebees is considered, with focus on the action of elastic joints between wings and body. Morphological measurements and kinematic reconstruction of the wing motion using synchronized high-speed video recordings are described. They provide the necessary input data for numerical mod...

A space–time fully adaptive multiresolution method for evolutionary non-linear partial differential equations is presented introducing an improved local time-stepping method. The space discretisation is based on classical finite volumes, endowed with cell average multiresolution analysis for triggering the dynamical grid adaptation. The explicit ti...

We develop a volume penalization method for inhomogeneous Neumann boundary conditions, generalizing the flux-based volume penalization method for homogeneous Neumann boundary condition proposed by Kadoch et al. (2012) [4]. The generalized method allows us to model scalar flux through walls in geometries of complex shape using simple, e.g. Cartesian...

Magnetohydrodynamics is an important tool to study the dynamics of plasma Space Physics. In this context, we introduce a three-dimensional magnetohydrodynamic solver with divergence-cleaning in the adaptive multiresolution CARMEN code. The numerical scheme is based on a finite volume discretization that ensures the conservation of physical quantiti...

A numerical approach for solving evolutionary partial differential equations in two and three space dimensions on block-based adaptive grids is presented. The numerical discretization is based on high-order, central finite-differences and explicit time integration. Grid refinement and coarsening are triggered by multiresolution analysis, i.e. thres...

Flapping insects are remarkably agile fliers, adapted to a highly turbulent environment. We present a series of high resolution numerical simulations of a bumblebee interacting with turbulent inflow. We consider both tethered and free flight, the latter with all six degrees of freedom coupled to the Navier--Stokes equations. To this end we vary the...

A numerical approach for solving evolutionary partial differential equations in two and three space dimensions on block-based adaptive grids is presented. The numerical discretization is based on high-order, central finite-differences and explicit time integration. Grid refinement and coarsening are triggered by multiresolution analysis, i.e. thres...

Flapping insects are remarkably agile fliers, adapted to a highly turbulent environment. We present a series of high-resolution numerical simulations of a bumblebee interacting with turbulent inflow. We consider both tethered and free flight, the latter with all six degrees of freedom coupled to the Navier-Stokes equations. To this end, we vary the...

Energy dissipation caused by boundary layer instability at vanishing viscosity – ERRATUM - Volume 857 - Natacha Nguyen van yen, Matthias Waidmann, Rupert Klein, Marie Farge, Kai Schneider

Wavelet basis functions are a natural tool for analyzing turbulent flows containing localized coherent structures of different spatial scales. Here, wavelets are used to study the onset and subsequent transition to fully developed turbulence from a laminar state. Originally applied to neutral fluid turbulence, an iterative wavelet technique decompo...

Recent simulations have demonstrated that coherent current sheets dominate the kinetic-scale energy dissipation in strong turbulence of magnetized plasma. Wavelet basis functions are a natural tool for analyzing turbulent flows containing localized coherent structures of different spatial scales. Here, wavelets are used to study the onset and subse...

For adaptive multiresolution schemes we propose a local timestepping scheme based on natural extensions of Runge-Kutta methods. We consider reaction-diffusion equations in two space dimensions and assess the precision and efficiency of the new method. The obtained results are compared with those using classical finite volume schemes on a uniform gr...

Magnetohydrodynamics is an important tool to study the dynamics of Space Physics. In this context, we introduce a three-dimensional magnetohydrodynamic solver with divergence-cleaning in the adaptive multiresolution CARMEN code. The numerical scheme is based on a finite volume discretization that ensures the conservation of physical quantities. The...

Direct numerical simulation of pore-scale turbulence is performed in a unit cell of a face-centered cubic lattice at three different pore Reynolds numbers (300, 500, and 1000). The pore-geometry gives rise to very low porosity resulting in rapid acceleration and deceleration of the flow in different regions. Eulerian statistics of mean velocity and...

The question of the relative importance of coherent structures and waves has for a long time attracted a great deal of interest in astrophysical plasma turbulence research, with a more recent focus on kinetic scale dynamics. Here we utilize high-resolution observational and simulation data to investigate the nature of waves and structures emerging...

High resolution direct numerical simulations of rotating and flapping bumblebee wings are presented and their aerodynamics is studied focusing on the role of leading edge vortices and the associated helicity production. We first study the flow generated by only one rotating bumblebee wing in circular motion with $45^{\circ}$ angle of attack. We the...

We present numerical simulations of simplified models for swimming organisms or robots, using chordwise flexible elastic plates. We focus on the tip vortices originating from three-dimensional effects due to the finite span of the plate. These effects play an important role when predicting the swimmer's cruising velocity, since they contribute sign...

The volume penalization method, which allows to impose no-slip boundary conditions, is assessed for wall-bounded flows. For the numerical solution of the penalized equations a spectral method is used. Considering a two-dimensional Poiseuille flow, the solution of the Navier-Stokes penalized equation is computed analytically and the convergence of t...

We thank the organizers of Cemracs and the Centre International de Rencontres Mathématiques for kind hospitality. We acknowledge Dr. Thomas Engels for giving insightful comments on this work.

70th Annual Meeting of the APS Division of Fluid Dynamics (November 19, 2017 — November 21, 2017) V0079: Bumblebee flight in turbulence: high resolution numerical simulations Authors • Thomas Engels, ISTA, Technische Universität Berlin, Berlin, Müller-Breslau-Strasse 12, 10623 Berlin, Germany & LMD-CNRS, Ecole Normale Supérieure, 24 rue Lhomond, 7...

We present numerical simulations of the three-dimensional Galerkin truncated incompressible Euler equations that we integrate in time while regularizing the solution by applying a wavelet-based denoising. For this, at each time step, the vorticity filed is decomposed into wavelet coefficients, that are split into strong and weak coefficients, befor...

We present numerical simulations of the three-dimensional Galerkin truncated incompressible Euler equations that we integrate in time while regularizing the solution by applying a wavelet-based denoising. For this, at each time step, the vorticity filed is decomposed into wavelet coefficients, that are split into strong and weak coefficients, befor...