Input Boundary Research Articles

A learning based numerical method for Helmholtz equations with high frequency

High-frequency issues have been remarkable challenges in numerical methods for partial differential equations. In this paper, a learning based numerical method (LbNM) is proposed for Helmholtz equation with high frequency. The main novelty is using Tikhonov regularization to stably learn the solution operator by utilizing relevant information especially the fundamental solutions. Then applying the solution operator to a new boundary input could quickly update the solution. Based on the method of fundamental solutions and the quantitative Runge approximation, we give the error estimate. This indicates interpretability and generalizability of the present method. Numerical results validate the error analysis and demonstrate the high-precision and high-efficiency features.

Improving heat transfer prediction across catalytic SiC interface through a multiscale framework leveraging machine learning and data fusion techniques

Accurate aerothermal prediction under hypersonic flow conditions is crucial for any thermal protection materials design and engineering. The surface catalytic effect, where dissociated atoms recombine into their molecular forms at the air-solid interface, is an exothermic reaction and plays an important role in high-enthalpy aerodynamic environment prediction. Taking SiC, a widely recognized high-temperature thermal protection material as an example, a multiscale fusion approach for aerothermal prediction is proposed. The approach utilizes reactive molecular dynamics method to construct an interface catalysis model, and the Bayesian maximum entropy to integrate experimental and simulational data into an optimized database. Subsequently, using the radial basis function neural network algorithm, a machine learning-based reaction kinetics model with precise analysis of surface catalysis is trained. The obtained catalytic recombination efficiency is used as a boundary input for computational fluid dynamics simulation, enabling rapid and accurate prediction of hypersonic thermal environment. The result shows that the predicted surface heat flux by this novel approach is consistent with the benchmark studies, but comes with significantly reduced computation time. The stagantion heat flux predictions based on the assumptions of full catalytic wall, finite rate catalytic wall (from the proposed multiscale fusion method), and non-catalytic wall are determined to be 8.65×106 W/m2, 7.51×106 W/m2, and 4.02×106 W/m2, respectively. Comparing these with the wind tunnel benchmark value of 7.48×106 W/m2, the error from the proposed multiscale fusion method is 0.30 %, indicating significant enhancement in prediction accuracy through the proposed upscaling method. The new approach could not only reveal complex exothermic reaction processes at the interface, but also enhance the prediction accuracy at much higher computational efficiency, providing an alternative for multiscale modeling of complex flow and heat transfer at the interface.

Monitoring river discharge from space: An optimization approach with uncertainty quantification for small ungauged rivers

The number of in-situ stations measuring river discharge, one of the Essential Climate Variables (ECV), is declining steadily, and numerous basins have never been gauged. With the aim of improving data availability worldwide, we propose an easily applicable and transferable approach to estimate reach-scale discharge solely using remote sensing data that is suitable for filling gaps in the in-situ network. We combine 20 years of satellite altimetry observations with high-resolution satellite imagery via a hypsometric function to observe large portions of the reach-scale bathymetry. The high-resolution satellite images, which are classified using deep learning image segmentation, allow for detecting small rivers (narrower than 100m) and can capture small width variations. The unobserved part of the bathymetry is estimated using an empirical width-to-depth function. Combined with precise satellite-derived slope measurements, river discharge is calculated at multiple consecutive cross-sections within the reach. The unknown roughness coefficient is optimized by minimizing the discharge differences between the cross-sections. The approach requires minimal input and approximate boundary conditions based on expert knowledge but is not dependent on calibration. We provide realistic uncertainties, which are crucial for data assimilation, by accounting for errors and uncertainties in the different input quantities. The approach is applied globally to 27 river sections with a median normalized root mean square error of 12% and a Nash–Sutcliffe model efficiency of 0.560. On average, the 90% uncertainty range includes 91% of the in-situ measurements.

Stability and Preservation of Fuzzy Membership Functions in Adaptive Gaussian Derivative Filters

This paper examines the stability and preservation of fuzzy membership functions in Gaussian filters, particularly focusing on the Adaptive Gaussian Derivative (AGD) filter. Gaussian filters are essential for smoothing and noise reduction in image processing. The AGD filter, which adapts based on local image statistics, outperforms traditional methods. Key concepts like fuzzy sets, Boundary Input Boundary Output (BIBO) stability, and convolution are defined, and the mathematical formulation of the Gaussian and its derivative is presented. The AGD filter's BIBO stability ensures bounded outputs for bounded inputs, guaranteeing consistent behavior. It also preserves fuzzy membership function properties, maintaining convexity and boundedness through linearity and continuity. Frequency response analysis using Fourier Transform confirms the AGD filter retains the Gaussian shape in the frequency domain, preserving image smoothness. Theorems and proofs validate the AGD filter's stability and its capability to preserve fuzzy membership functions, ensuring reliable processing in applications such as medical image analysis. These properties make the AGD filter a robust tool for advanced image processing tasks.

PDE-Based Boundary Adaptive Consensus Control of Multiagent Systems With Input Constraints.

The leader-follower adaptive consensus control problem is addressed for partial differential equations (PDEs) multiagent systems (MASs), and these agents are composed of flexible manipulator systems with input nonlinearity, boundary uncertainties, and time-varying disturbances. Because of the spatial variables in the model, the design of adaptive protocols is more difficult than that of ordinary differential equation (ODE) MASs. By designing the Lyapunov function, a novel distributed boundary control (BC) protocol is constructed, which not only ensures the consensus of angular positions but also suppresses the boundary vibration of each agent. The hybrid effects of dead zones and input saturation on flexible manipulator systems are addressed using the approximation properties of neural networks (NNs). In addition, the disturbance adaptive laws are proposed to provide a control solution for bounded and time-varying disturbances. Furthermore, by applying the Lyapunov stability theory, the uniformly bounded stability of the multiflexible manipulator can be ensured. Finally, the feasibility of the presented control approach is verified using numerical examples.

Real-time integrated water availability – Salt intrusion modelling and management during droughts

Climate change and population increase are major challenges when guaranteeing a sustainable water supply in densely populated river basins worldwide, where different water users such as industry, drinking water production and agriculture typically come into conflict. During dry periods, real time monitoring and short term forecasting of the water availability are crucial to prepare and take appropriate action. Integrated water quantity and water quality models of complex surface water systems are indispensable to achieve this. However, existing model techniques often focus on a single aspect of the water system, are computationally too demanding and are hard to integrate with other sub models. This paper proposes a framework for developing efficient, integrated water quantity/quality models in support of real-time management and forecasting of complex surface water systems. Its three main components are: a low-flow prediction for the supplying river(s) during dry periods, a conceptual water quantity-quality model with semi-automated retrieval of necessary boundary conditions and input data, and classification models to predict the water quality state of the considered system during forecasts or scenario analyses. The novelty of the methodology lies in the use of parsimonious, computationally efficient sub-models that still allow to consider numerous operational rules, competing water users and water supply and quality constraints. The framework was tested in the context of water availability and salt intrusion management during droughts in the Campine Canals in the Scheldt-Meuse-Rhine delta, considering boundary conditions of shipping traffic, pumping at locks, hydropower stations, level regulation, and water abstractions for drinking water supply, industry, agriculture and nature reserves. Despite the complexity of the considered system, the model is able to describe the state of the water system in terms of water levels, discharges and salt intrusion impacts on water production with an acceptable accuracy. The research shows that the Campine Canal network is susceptible to water scarcity due to the increased risk of low flows in summer as a consequence of climate change. Local decision makers should adopt both effective adaptation strategies as well as real-time forecasts in acute drought situations to increase the region’s defence and resiliency against water shortages.

Approximate Bayesian Computation for structural identification of ancient tie-rods using noisy modal data

Masonry arches and vaults are common historic structural elements that frequently experience asymmetric loading due to seismic action or abutment settlements. Over the past few decades, numerous studies have sought to enhance our understanding of the structural behavior of these elements for the purpose of preventive conservation. The assessment of the structural performance of existing constructions typically relies on effective numerical models guided by a set of unknown input parameters, including geometry, mechanical characteristics, physical properties, and boundary conditions. These parameters can be estimated through deterministic optimization functions aimed at minimizing the discrepancy between the output of a numerical model and the measured dynamic and/or static structural response. However, deterministic approaches overlook uncertainties associated with both input parameters and measurements. In this context, the Bayesian approach proves valuable for estimating unknown numerical model parameters and their associated uncertainties (posterior distributions). This involves updating prior knowledge of model parameters (prior distributions) based on current measurements and explicitly considering all sources of uncertainties affecting observed quantities through likelihood functions. However, two significant challenges arise: the likelihood function may be unknown or too complex to evaluate, and the computational costs for approximating the posterior distribution can be prohibitive. This study addresses these challenges by employing Approximate Bayesian Computation (ABC) to handle the intractable likelihood function. Additionally, the computational burden is mitigated through the use of accurate surrogate models such as Polynomial Chaos Expansions (PCE) and Artificial Neural Networks (ANN). The research focuses on setting up numerical models for simple structural systems (tie-rods) and inferring unknown input parameters, such as mechanical properties and boundary conditions, through Bayesian updating based on observed structural responses (modal data, strains, displacements). The main novelties of this research regard, on the one hand, the proposal of a methodology for obtaining a reliable estimate of the axial force in ancient tie-rods accounting for different sources of uncertainty and, on the other hand, the application of ABC to obtain the structural identification inverse problem solution.

Coronal Models and Detection of the Open Magnetic Field

A plethora of coronal models, from empirical to more complex magnetohydrodynamic (MHD) ones, are being used for reconstructing the coronal magnetic field topology and estimating the open magnetic flux. However, no individual solution fully agrees with coronal hole observations and in situ measurements of open flux at 1 au, as there is a strong deficit between the model and observations contributing to the known problem of the missing open flux. In this paper, we investigate the possible origin of the discrepancy between modeled and observed magnetic field topology by assessing the effect on the simulation output by the choice of the input boundary conditions and the simulation setup, including the choice of numerical schemes and the parameter initialization. In the frame of this work, we considered four potential field source surface-based models and one fully MHD model, different types of global magnetic field maps, and model initiation parameters. After assessing the model outputs using a variety of metrics, we conclude that they are highly comparable regardless of the differences set at initiation. When comparing all models to coronal hole boundaries extracted by extreme-ultraviolet filtergrams, we find that they do not compare well. This mismatch between observed and modeled regions of the open field is a candidate contributing to the open flux problem.

Influence of the Presence of Different Signatures on the Heat Transfer Profile of Laminar Flow Inside a Microchannel

Abstract The process of transferring heat and mass involves a high-pressure decline. Hence microchannels are utilized in extremely efficient heat and mass transfer processes, such as in the systems of the lungs and kidneys. Due to their high surface-to-volume ratio and compact volume, microchannels have demonstrated superior thermal performance. In this work, micro-fins of different topologies and arrangements were used inside microchannels for modulating heat transfer from the surface of the fins, in the presence of flowing media. The influx of media was simulated in sinusoidal form, reproducing simplified form of physiological circulatory blood flow. The study was conducted in numerical domain, while heat transfer and mass transfer equations were solved using pardiso solver. Temperature modulation at the fin surfaces was conducted to examine the transient profile of heat transfer, as function of both variable fluid velocity as well as variable feeding heat sources. In case of different input boundary conditions, the effect of heat distributions on fluid flow with respect to spatial distribution of micro-fins within a microchannel was evaluated. Results revealed the difference in heat transfer profiles in microchannels in presence of different sets of fin configurations. It was found that rectangular fins have the highest heat transfer in fluid at all its spatial configurations; while semi-ellipsoidal-shaped fins had shown the least heat transfer profile at same surface area. At the same time, it was also observed that the rate of heat dissipation was faster and limited in semi-ellipsoidal-shaped fins configuration; while the heat transfer rate was found symmetrical and low in presence of rectangular fins. Hence, this article emphasizes the modulation of temperature and velocity variations within the working fluid by emphasizing the thermo-fluid coupling effects in microchannels.

Estimation of blocked forces for NVH source characterization of a beam axle using analytical methods

Component-Based Transfer Path Analysis (TPA) has been of growing interest recently because, unlike classical TPA, it enables source characterization independent of the receiver structure. The source energy in component TPA is typically characterized by blocked forces. As part of virtual prototyping, OEMs have interest in building a library of component/subsystem models that may be "plugged" into larger system models. For an OEM supplier, it is imperative to have the capability to provide subsystem models along with source energy characterized in component blocked force prior to availability of physical parts to support virtual validation and optimization efforts for seamless system integration. In this paper, three methods of blocked force calculation are considered: Direct, Free Velocity and In-situ. A Finite Element Analysis (FEA) model is used to Directly calculate the blocked forces exerted by a rear beam axle at the vehicle attachment points, then to simulate how they might be estimated in a dynamometer test stand using the Free Velocity or In-situ methods, with the aim of offering practical guidance for both scenarios. Assumptions regarding input and output boundary conditions, along with three versus six degree-of-freedom (DOF) attachment paths, affecting both the analytical and test-based estimation of blocked forces, are also scrutinized.

Unsteady flow of nanofluid over a sheet of variable thickness with nonlinear kinematics

The transport of heat in an unsteady flow of nanofluid has been discussed in this paper. The flow is maintained over a non-flat, variably porous, and moving sheet in the present simulation. Moreover, the sheet has both constant temperature and nanoparticle concentration at its surface. Unsteadiness in all field variables (i.e. the flow, temperature, and nanoparticles concentration) has been produced by the time-dependent quantities, specified at the surface of the sheet (i.e. kinematics of sheet). The well-known Buongiorno model of nanofluid has been undertaken in the present studies. Furthermore, the simulation includes the particular cases of uniform and linear (polynomial & algebraic functions) stretching/shrinking and injection/suction velocities. Besides that observations of the same nature have been recorded for the non-flat sheet. The boundary layer form of the governing PDEs has been utilized to evaluate the exact status of field variables in the flow domain. Appropriate conditions are imposed by considering the geometry of the flow problem. The problem at hand is simplified by introducing a new set of functions for the field variables because of boundary inputs. After this treatment, the system of boundary value ODEs, which contains several dimensionless numbers (parameters), has appeared, and the problem is solved numerically for various values of parameters, involved in it. More precisely, effects of unsteadiness, wall roughness, surface deformation parameters, Prandtl, Schmidt, thermophoretic, and Brownian motion numbers have been seen on profiles of four field variables. Besides that, the numerical results are also obtained for the skin friction coefficient, Nusselt, and Sherwood numbers. The present modelled problem and its solution are compared with classical models of that particular nature and with their solutions.

Adaptive observer design for coupled ODE–hyperbolic PDE systems with application to traffic flow estimation

This paper studies the state estimation problem for a class of coupled linear ODE–hyperbolic PDE systems with unknown in-domain parameters. Based on the swapping transformation and the least squares parameter estimation method, a Luenberger-type adaptive boundary observer is designed to estimate the ODE–PDE states under the unknown parameters. The exponential convergence conditions w.r.t. the observer feedback gains are given by employing the Lyapunov technique. Finally, an application to traffic state estimation of the Aw–Rascle–Zhang (ARZ) model involving the unknown relaxation time and the boundary input speed deviation is provided to validate the theoretical result.

Site effect on seismic response of buried nuclear power plant structure

The development of multifunctional, multipurpose small modular next generation nuclear reactors has become a trend in designing small-scale nuclear power stations. In this study, taking a new next-generation buried nuclear power structure as the target structure, a 3D finite element model of the site-nuclear power plant structure system was established, and its response at different sites was studied. Using the internal substructure method, this study initially explored the impact of earthquake input and artificial boundary locations on the structural response. The results indicate that the internal substructure method enhances computational efficiency without compromising accuracy. For this model, it is recommended to input earthquake motion close to the structure, set the lateral boundary location to three times the width of the structure, and set the bottom boundary location to two times the depth of the structure. Subsequently, an investigation into the response of the structure at varying site wave velocities is conducted. The results show that as the wave velocity increases, the acceleration response intensifies, while the displacement response diminishes. However, the relationship is nonlinear; the response exhibits less variation with increasing site wave velocity. The seismic response of a structure is profoundly influenced by site conditions, emphasizing the imperative need for careful consideration of specific site characteristics in seismic analyses. In addition, the response pattern of the overall structure is significantly different from that of conventional large surface-sited nuclear power plants, and further research should be conducted on the seismic response characteristics of buried nuclear power plants.

A systematic methodology for port-Hamiltonian modeling of multidimensional flexible linear mechanical systems

This article introduces a novel systematic methodology for modeling a class of multidimensional linear mechanical systems that directly allows to obtain their infinite-dimensional port-Hamiltonian representation. While the approach is tailored to systems governed by specific kinematic assumptions, it encompasses a wide range of models found in current literature, including ℓ-dimensional elasticity models (where ℓ = 1, 2, 3), vibrating strings, torsion in circular bars, classical beam and plate models, among others. The methodology involves formulating the displacement field using primary generalized coordinates via a linear algebraic relation. The non-zero components of the strain tensor are then calculated and expressed using secondary generalized coordinates, enabling the characterization of the skew-adjoint differential operator associated with the port-Hamiltonian representation. By applying Hamilton's principle and employing a specially developed integration by parts formula for the considered class of differential operators, the port-Hamiltonian model is directly obtained, along with the definition of boundary inputs and outputs. To illustrate the methodology, the plate modeling process based on Reddy's third-order shear deformation theory is presented as an example. To the best of our knowledge, this is the first time that a port-Hamiltonian representation of this system is presented in the literature.

Uncertainty quantification of the lattice Boltzmann method focussing on studies of human-scale vascular blood flow

Uncertainty quantification is becoming a key tool to ensure that numerical models can be sufficiently trusted to be used in domains such as medical device design. Demonstration of how input parameters impact the quantities of interest generated by any numerical model is essential to understanding the limits of its reliability. With the lattice Boltzmann method now a widely used approach for computational fluid dynamics, building greater understanding of its numerical uncertainty characteristics will support its further use in science and industry. In this study we apply an in-depth uncertainty quantification study of the lattice Boltzmann method in a canonical bifurcating geometry that is representative of the vascular junctions present in arterial and venous domains. These campaigns examine how quantities of interest—pressure and velocity along the central axes of the bifurcation—are influenced by the algorithmic parameters of the lattice Boltzmann method and the parameters controlling the values imposed at inlet velocity and outlet pressure boundary conditions. We also conduct a similar campaign on a set of personalised vessels to further illustrate the application of these techniques. Our work provides insights into how input parameters and boundary conditions impact the velocity and pressure distributions calculated in a simulation and can guide the choices of such values when applied to vascular studies of patient specific geometries. We observe that, from an algorithmic perspective, the number of time steps and the size of the grid spacing are the most influential parameters. When considering the influence of boundary conditions, we note that the magnitude of the inlet velocity and the mean pressure applied within sinusoidal pressure outlets have the greatest impact on output quantities of interest. We also observe that, when comparing the magnitude of variation imposed in the input parameters with that observed in the output quantities, this variability is particularly magnified when the input velocity is altered. This study also demonstrates how open-source toolkits for validation, verification and uncertainty quantification can be applied to numerical models deployed on high-performance computers without the need for modifying the simulation code itself. Such an ability is key to the more widespread adoption of the analysis of uncertainty in numerical models by significantly reducing the complexity of their execution and analysis.

Open boundary conditions for atmospheric large-eddy simulations and their implementation in DALES4.4

Abstract. Open boundary conditions were developed for atmospheric large-eddy simulation (LES) models and implemented into the Dutch Atmospheric Large-Eddy Simulation model. The implementation was tested in a “Big Brother”-like setup, in which the simulation with open boundary conditions was forced by an identical control simulation with periodic boundary conditions. The results show that the open boundary implementation has minimal influence on the solution. Both the mean state and the turbulent structures are close to the control simulation, and disturbances at the in- and outflow boundaries are negligible. To emulate a setup in which the LES is coupled to a coarser model, the influence of coarse boundary input was tested by smoothing the output of the periodic control simulation both temporally and spatially before feeding it as input to the simulation with open boundary conditions. When smoothing is applied over larger spatial and longer temporal scales, disturbances start to form at the inflow boundary and an area exists where turbulence needs to develop. Adding synthetic turbulence to the smoothed input reduces the size of this area and the magnitude of the disturbances.

Periodic Optimal Control of a Plug Flow Reactor Model with an Isoperimetric Constraint

We study a class of nonlinear hyperbolic partial differential equations with boundary control. This class describes chemical reactions of the type “A→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A \\rightarrow $$\\end{document} product” carried out in a plug flow reactor (PFR) in the presence of an inert component. An isoperimetric optimal control problem with periodic boundary conditions and input constraints is formulated for the considered mathematical model in order to maximize the mean amount of product over the period. For the single-input system, the optimality of a bang-bang control strategy is proved in the class of bounded measurable inputs. The case of controlled flow rate input is also analyzed by exploiting the method of characteristics. A case study is performed to illustrate the performance of the reaction model under different control strategies.

Multidirectional cascade of square MIMO LTI subsystems: Transfer function representation

This paper considers continuous transfer function representation for a cascade composed of N mutually interconnected square multiple-input multiple-output (MIMO) linear time-invariant (LTI) subsystems with general multidirectional input–output configurations. Such cascaded structure can arise, e.g., from the spatial discretization of one-dimensional distributed parameter systems (DPSs) described by partial differential equations (PDEs) with boundary conditions representing different configurations of boundary input signals. As shown in the paper, the transfer function matrix of the resulting cascade can be calculated using the Redheffer star product of the subsystems’ transfer function matrices, which simplifies into the usual matrix product for the unidirectional cascade. For the particular case of 2 × 2 cascade composed of rational transfer function subsystems, the recursive formulas for its boundary and distributed transfer functions have been derived for both uni- and multidirectional configurations. The well-posedness and the stability criteria are also analyzed, proving to be more restrictive for the multidirectional cascade than for the unidirectional one due to the internal positive feedback in the first case. As shown in the attached example, the presented results can be used to approximate one-dimensional distributed parameter systems using rational transfer function subsystems representing the spatial sections of the original DPS.

Numerical and Experimental Investigation of Performance and Flooding Phenomena of a PEM Fuel Cell with and without Micro-Porous Layers

This work presents the results of manufacturing a single Proton Exchange Membrane Fuel Cell (PEMFC) with Micro-Porous Layers (MPLs) and an active area of 25 cm2, and the experimental study required to build its polarization curve. Based on the physical model data, a numerical model of this PEMFC is created in the ANSYS PEM Fuel Cell module. Numerical simulations were performed with boundary conditions consistent with the experimental conditions on the test station. The calculation and experimental result comparison of the polarization curves for voltages ranging from 0.29 V to 0.94 V proved that the utilized numerical model is highly reliable. The simulation of PEMFC without MPLs was conducted according to other stable input parameters and boundary conditions. The results show that the PEMFC performance decreases significantly due to the flooding phenomenon inside PEMFC without MPLs compared to PEMFC with MPLs. Such phenomena are challenging to observe experimentally. Numerical modeling can be further used to optimize the fuel cell components.

Стационарное протекание несжимаемой вязкой жидкости сквозь двухмерный разветвленный канал

The results of a numerical experiment for the problem of an incompressible viscous fluid stationary flow through a branched planar (two-dimensional) channel are presented. The region in which the flow occurs simulates either blood vessels or a river delta. The finite element method and a modification of the penalty method, as well as the splitting method, are used for calculations. The implementation of the calculation algorithm is using with the help the package FreeFem++. The main goal, in addition, of course, to study the properties and structure of the stationary flow, is to demonstrate the effectiveness of the proposed modification of the penalty method. It is assumed that the region has one input boundary section through which the liquid flows into the region, and several (five) boundary sections through which the liquid flows out of the region. The remaining sections of the region boundary are considered impermeable to liquid. The boundary condition corresponding to the Poiseuille flow in a plane rectilinear channel is set at the input section. Three types of boundary conditions are considered on the output boundary secctions. 1. The boundary conditions correspond to the conditions of conservation of motion of luid particles i. e. the material derivative of the velocity is zero. 2. The boundary conditions correspond to the setting of the flow velocity. 3. The boundary conditions correspond to the setting of the same pressure. The stationary solution is constructed by the relaxation method. In fact, a non-stationary problem is solved over a sufficiently large time interval. As an initial condition for a non-stationary problem, a flow is chosen that is a Poiseuille flow in some region near the input boundary. The dependence of the convergence rate of the relaxation method on the initial data is investigated. It is found that the Poiseuille flow, given at the input section of the boundary of the region, induces similar Poiseuille flows at the output boundaries sections of the region in all these cases of boundary conditions. For a region with five sections of the output boundaries for some configuration of the region, the presence of stationary vortex flows (‘maelstorm’) is found.