Vorticity Function Research Articles

Advancing neural network-based data assimilation for large-scale spatiotemporal systems with sparse observations

Data assimilation (DA) is a powerful technique for improving the forecast accuracy of dynamic systems by optimally integrating model forecasts with observations. Traditional DA approaches, however, encounter significant challenges when applied to complex, large-scale, highly nonlinear systems with sparse and noisy observations. To overcome these challenges, this study presents a new Neural Network-based Data Assimilation (DANet) model, specifically employing a Convolutional Long Short-Term Memory architecture. By leveraging the strengths of neural networks, DANet establishes the relationship among model forecasts, observations, and ground truth, facilitating efficient DA in large-scale spatiotemporal forecasting with sparse observations. The effectiveness of the DANet model is demonstrated through an initial case study of wind-driven oceanic flow forecasting, as described by a Quasi-Geostrophic (QG) model. Compared to the traditional Ensemble Kalman Filter (EnKF), DANet exhibits superior performance in cases involving both structured and unstructured sparse observations. This is evidenced by reduced Root Mean Square Errors (RMSEs) and improved correlation coefficients (R) and Structural Similarity Index. Moreover, DANet is seamlessly integrated with the QG model to operationally forecast vorticity and stream function in the long term, further confirming the accuracy and reliability of the DANet model. DANet achieves operational forecasting 60 times faster than EnKF, underscoring its efficiency and potential in DA advancement.

Investigating the Influence of Counterflow Regions on the Hydrodynamic Performance of Biomimetic Robotic Fish.

Biomimetic robotic fish are a novel approach to studying quiet, highly agile, and efficient underwater propulsion systems, attracting significant interest from experts in robotics and engineering. These versatile robots showcase their ability to operate effectively in various water conditions. Nevertheless, the comprehension of the swimming mechanics and the evolution of the flow field of flexible robots in counterflow regions is still unknown. This paper presents a framework for the self-propulsion of robotic fish that imitates biological characteristics. The method utilizes computational fluid dynamics to analyze the hydrodynamic efficiency of the organisms at different frequencies of tail movement, under both still and opposing flow circumstances. Moreover, this study clarifies the mechanisms that explain how changes in the aquatic environment affect the speed and efficiency of propulsion. It also examines the most effective swimming tactics for places with counterflow. The results suggest that the propulsion effectiveness of robotic fish in counterflow locations does not consistently correspond to various tail-beat frequencies. By utilizing vorticity maps, a comparative analysis can identify situations when counterflow zones improve the efficiency of propulsion.

The Spatial Variation of Large‐ and Meso‐Scale Plasma Flow Vorticity Statistics in the High‐Latitude Ionosphere and Implications for Ionospheric Plasma Flow Models

AbstractThe ability to understand and model ionospheric plasma flow on all spatial scales has important implications for operational space weather models. This study exploits a recently developed method to statistically separate large‐scale and meso‐scale contributions to probability density functions (PDFs) of ionospheric flow vorticity measured by the Super Dual Auroral Radar Network (SuperDARN). The SuperDARN vorticity data are first sub‐divided depending on the Interplanetary Magnetic Field (IMF) direction, and the separation method is applied to PDFs of vorticity compiled in spatial regions of size 1° of geomagnetic latitude by 1 hr of magnetic local time, covering much of the high‐latitude ionosphere in the northern hemisphere. The resulting PDFs are fit by model functions using maximum likelihood estimation (MLE) and the spatial variations of the MLE estimators for both the large‐scale and meso‐scale components are presented. The spatial variations of the large‐scale vorticity estimators are ordered by the average ionospheric convection flow, which is highly dependent on the IMF direction. The spatial variations of the meso‐scale vorticity estimators appear independent of the senses of vorticity and IMF direction, but have a different character in the polar cap, the cusp, the auroral region, and the sub‐auroral region. The paper concludes by discussing the sources of the vorticity components in the different regions, and the consequences for the fidelity of ionospheric plasma flow models.

An Implicit Factorized Transformer with Applications to Fast Prediction of Three-dimensional Turbulence

Transformer has achieved remarkable results in various fields, including its application in modeling dynamic systems governed by partial differential equations. However, transformer still face challenges in achieving long-term stable predictions for three-dimensional turbulence. In this paper, we propose an implicit factorized transformer (IFactFormer) model, which enables stable training at greater depths through implicit iteration over factorized attention. IFactFormer is applied to large eddy simulation of three-dimensional homogeneous isotropic turbulence (HIT), and is shown to be more accurate than the FactFormer, Fourier neural operator (FNO), and dynamic Smagorinsky model (DSM) in the prediction of the velocity spectra, probability density functions of velocity increments and vorticity, temporal evolutions of velocity and vorticity root-mean-square value and isosurface of the normalized vorticity. IFactFormer can achieve long-term stable predictions of a series of turbulence statistics in HIT. Furthermore, IFactFormer showcases superior computational efficiency compared to the conventional DSM in large eddy simulation.

Machine learning-based vorticity evolution and super-resolution of homogeneous isotropic turbulence using wavelet projection

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 Taylor microscale Reynolds number 92. The predicted results are assessed by using the flow visualization of vorticity and statistics, e.g., probability density functions of vorticity and enstrophy spectra. It is found that the predicted results are in good agreement with DNS results. The small-scale flow topology considering the second and the third invariants of the velocity gradient tensor likewise shows an approximate match. Furthermore, we apply the pre-trained neural networks to coarse-grained vorticity data using super-resolution. It is shown that the super-resolved flow field well agrees with the reference DNS field, and thus small-scale information and vortex tubes are well regenerated.

Blowup analysis for a quasi-exact 1D model of 3D Euler and Navier–Stokes

We study the singularity formation of a quasi-exact 1D model proposed by Hou and Li (2008 Commun. Pure Appl. Math. 61 661–97). This model is based on an approximation of the axisymmetric Navier–Stokes equations in the r direction. The solution of the 1D model can be used to construct an exact solution of the original 3D Euler and Navier–Stokes equations if the initial angular velocity, angular vorticity, and angular stream function are linear in r. This model shares many intrinsic properties similar to those of the 3D Euler and Navier–Stokes equations. It captures the competition between advection and vortex stretching as in the 1D De Gregorio (De Gregorio 1990 J. Stat. Phys. 59 1251–63; De Gregorio 1996 Math. Methods Appl. Sci. 19 1233–55) model. We show that the inviscid model with weakened advection and smooth initial data or the original 1D model with Hölder continuous data develops a self-similar blowup. We also show that the viscous model with weakened advection and smooth initial data develops a finite time blowup. To obtain sharp estimates for the nonlocal terms, we perform an exact computation for the low-frequency Fourier modes and extract damping in leading order estimates for the high-frequency modes using singularly weighted norms in the energy estimates. The analysis for the viscous case is more subtle since the viscous terms produce some instability if we just use singular weights. We establish the blowup analysis for the viscous model by carefully designing an energy norm that combines a singularly weighted energy norm and a sum of high-order Sobolev norms.

Зарождение вихреобразования на границе кругового цилиндра при его обтекании потоком вязкой жидкости

The paper considers the problem of the generation of vortex formation on the surface of a circular cylinder placed in a flow of incompressible viscous fluid. To study this process, a new time-iterative approach is formu-lated for solving the 2D problem of flow around the circle based on the BIE method with Green's functions for the outer part of the cylinder with the Dirichlet boundary condition on its surface. A new approach, based on the transient Navier-Stokes equation in the variables stream function - vortex, using classical methods of potential theory and the boundary element method, allows us to obtain a complete system of three integral equations that determine the fields of stream functions and vorticity at a new time step, if these fields are known from the previous step. The assumption about the initial unexcited state of the flow (the vorticity function in the entire flow region is zero) allows us to obtain an analytical solution at the first time iteration step. The analysis of the obtained solution depending on the time step allows us to formulate a hypothesis about the nature of the generation of vortex formation on the surface of the circular cylinder in a flow of incompressible viscous fluid. The results obtained describe the appearance on the surface of the cylinder of two generalized vortices, having infinitely high intensity and infinitely small localization near the surface of the cylinder, which “instantly” break away from the surface of the cylinder, with a significant decrease in intensity, but with a large coverage of the flow around the cylinder.

Rotational effect of a cylinder on hydro-thermal characteristics in a partially heated square enclosure using CNT-water nanofluid

Rotating cylinder movement in a cavity flow is an exciting field of study in heat transfer. Considerable research has been carried out on rotating cylinders under MHD mixed convection in various types of enclosures. However, considering partially heated square enclosure and magnetic field using CNT-water nanofluid is very limited. This study's goal is to assess the hydrothermal phenomena in a square enclosure with a rotating cylinder. Simulation has been conducted for different rotational speeds (Ω) and dimensionless times (τ) to observe the thermal and fluid flow behaviour. The Galerkin Residual based finite element method has been used to conduct numerical calculations. The results are shown as isotherms, streamlines, and average Nusselt number at the cylinder wall. Moreover, the drag force at the moving wall, and the fluid properties such as the root mean square (rms) of velocity, the temperature, the vorticity functions, and the average fluid temperature are also presented. The heat transfer rate, drag force, rms velocity, and temperature increase with the rise of rotational speed and dimensionless time rise. Maximum vorticity occurs at Ω = 8 and τ = 1. The maximum vorticity function increases 12 times with the increasing rotational speed. Higher rotational speed leads to increased average fluid temperature. The case of Ω = 8, τ = 1 shows the most temperature variance, while Ω = 1, τ = 0.1 has the least. Increasing rotational speed results in higher drag force on the cylinder's surface. At Ω = 4, the drag force is 2.8 times greater than at Ω = 2. Overall, the fluid flow and thermal performance boost up while the rotating speed of the cylinder is higher.

Point-vortex statistical mechanics applied to turbulence without vortex stretching

In turbulent systems with inverse cascades, energy will pile up at large scales if no large-scale sink is present. We observe that in forced-dissipative three-dimensional turbulence from which vortex-stretching is removed, such condensation is observed, associated with an inverse cascade of helicity. The large-scale structure of this condensate is characterized by a hyperbolic sine relation between vorticity and velocity, analogous to the sinh relation between vorticity and stream function observed in freely decaying 2D turbulence in periodic domains. We generalize a 2D point-vortex statistical mechanics approach to our 3D system. It is shown that the predictions of this approach are in agreement with observations of both the forced-dissipative system, after appropriate averaging, and of the freely decaying system.

Analysis of physiological pulsating flow of fractional Maxwell fluid in a locally narrow artery

The purpose of this paper is to provide a novel reference for the early diagnosis and treatment of atherosclerosis. Two-dimensional governing equations of fractional-order Maxwell fluid flow in a local stenotic artery are established, taking real physiological pulsating blood flow at inlet into consideration. Drawing support from the finite difference method as well as the L1 formula, vorticity and stream functions are introduced to acquire numerical solutions for velocity, stream function, and pressure. The distribution of blood flow in narrowed arteries within a real physiological pulse cycle is discussed. Furthermore, the influences of the degree of stenosis δ, the stenosis length parameter L0, fractional order parameter α, and relaxation time λ on crucial medical indicators, including the time average of the wall shear stress, oscillatory shear index, relative residence time, and pressure distribution are revealed. The results show that the deceleration and reversal phases of real physiological pulsatile flow critically affect the progression of arterial stenosis, and increasing the fractional order parameter α weakens the development of stenosis, while increasing λ has the opposite effect. This study is expected to serve as a reference for formulating standards of key medical indicators in the early diagnosis of vascular stenosis.

Separating Contributions to Plasma Vorticity in the High‐Latitude Ionosphere From Large‐Scale Convection and Meso‐Scale Turbulence

AbstractMeasurements of ionospheric flow vorticity can be used for studying ionospheric plasma transport processes, such as convection and turbulence, over a wide range of spatial scales. Here, we analyze probability density functions (PDFs) of ionospheric vorticity for selected regions of the northern hemisphere high‐latitude ionosphere as measured by the Super Dual Auroral Radar Network over a 6‐year interval (2000–2005 inclusive). Subdividing these PDFs for opposite polarities of the By component of the prevailing interplanetary magnetic field allows the separation into two distinct components: (a) A single‐sided Weibull distribution which relates to the large‐scale convection driven by magnetic reconnection; (b) A double‐sided and symmetric q‐exponential distribution which arises from meso‐scale plasma flow related to processes such as turbulence.

Maximum-principle-preserving high-order discontinuous Galerkin methods for incompressible Euler equations on overlapping meshes

In this paper, we construct a new local discontinuous Galerkin (LDG) algorithm to solve the incompressible Euler equation in two space dimensions on overlapping meshes. This method solves the vorticity, velocity field and potential function on different meshes. Different from the traditional LDG method, the overlapping meshes used in this paper make the velocity to be continuous along the interfaces of the primitive meshes. Therefore, the upwind fluxes can be applied. We derive two sufficient conditions to obtain the maximum principle of vorticity. The first one is the divergence-free numerical approximation of the velocity field. This condition further grants that the scheme of the vorticity equation keeps constant solutions. The second one is to preserve the positivity of the numerical vorticity. We select suitable time step sizes to construct positive numerical cell averages of the vorticity provided the vorticity in the previous time step is positive. Then a slope limiter can be applied to enforce the positivity of the numerical approximation of the vorticity. Thanks to the above two conditions, we can arbitrarily add constants to the vorticity function and construct high-order MPP LDG methods on overlapping meshes for the two-dimensional incompressible Euler equation in the vorticity stream function formulation. Numerical tests will be given to demonstrate the good performance of the proposed method.

A meshless generalized finite difference scheme for the stream function formulation of the Naiver-Stokes equations

In this study, a meshless numerical scheme, which is the combination of the generalized finite difference method, the fictitious-nodes technique, and the two-step Newton-Raphson method, was proposed to solve the stream function formulation of the Navier-Stokes equations. To reduce the numbers of unknowns and partial differential equations for the two-dimensional flow field of a viscous incompressible fluid, the stream function formulation, which is a fourth-order nonlinear partial differential equation with only one unknown variable, can be derived by introducing the definition of vorticity and stream function. By applying the generalized finite difference method, the derivatives in the stream function formulation can be simply expressed as a linear combination of functional data and weighting values at several nearest nodes. To deal with the two boundary conditions for the fourth-order partial differential equations, the fictitious-nodes technique was utilized to set up the fictitious nodes outside the computational domain, and cooperate with the generalized finite difference method to form a system of nonlinear algebraic equations. The aforementioned numerical procedure can yield a square Jacobian matrix, so the Newton-Raphson method can be adopted to efficiently acquire the numerical solutions. Several numerical examples were tested to verify the practicability of the proposed meshless numerical scheme.

An Analysis of Buoyancy-Driven Heat Behaviour and Flow Pattern of a Water-Copper Nanofluid in a Cylindrical Conduit

The flow of fluids and heat characteristics through free convection within an enclosed space has gained substantial study due to the various applications in manufacturing industries. This work examined the influence of buoyancy factors on normal convection in a heated tube filled with Copper (Cu) nanofluid. The method of finite difference was employed to characterize the regulating fluid formulae, and C++ programming language was employed to evaluate the Navier Stoke and continuity fields. This study examined Cu nanoparticles with particle sizes ranging from 1% to 10% and buoyancy values between 2.6 x 103 and 2.8 x 103 N. Cu nanofluid was used as the working fluid and the findings are presented as temperature gradient, Nusselt number, stream function, and vorticity curves. The findings revealed that an increase in the weight proportions of nanoparticles to 0.04 amplifies the buoyancy parameters to the highest value of 2.75 x 103 N; it yields a substantial enhancement in the heat transport rate by convection. Also, as the buoyancy factor increases, the temperature gradient, vorticity, and stream function of the nanofluid improve, while the local drag coefficient decreases. This study advances the understanding of buoyancy-driven convective flow and heat behavior in the technical design of floating vessels for safety and effectiveness.

Helical symmetry vortices for 3D incompressible Euler equations

In this paper, we study the desingularization of vortices for the 3D incompressible Euler equations in an infinite pipe. We construct a family of traveling-rotating helical vortices for the Euler equations with a general vorticity function, which tends asymptotically to singular helical vortex filament evolved by the binormal curvature flow. The results are obtained by using an improved vorticity method. Some asymptotic properties of this family of solutions have also been studied.

Drift-ordered fluid vorticity equation with energy consistency

Although drift-ordered fluid models are widely applied in tokamak edge turbulence simulations, the models used are acknowledged not to conserve energy or even electrical charge. The present paper aims to remove many of the existing pitfalls in drift-fluid models, however, with the objective of finding a solution simple enough to be implemented in numerical applications. Our main result is an improved version of the drift-Braginskii equations involving a generalized vorticity function. In the new drift-Braginskii system, the quasi-neutrality condition translates into a transport equation for a generalized vorticity, expressed in conservation form, and related to the total mass-weighted circulation. It is found that kinetic energy conservation can be achieved if the polarization flow is defined recursively. The resulting model conserves the kinetic energy associated with E×B and diamagnetic flows and retains the associated perpendicular kinetic energy flux.

Reduced‐order extrapolation technique for coefficient vectors of numerical solutions about fluid mechanics equation

AbstractFor convenience, this study mainly focuses on the reduced order of coefficient vectors for the classical finite spectral element Crank–Nicolson (CFSECN) solutions about the two‐dimensional (2D) nonstationary Stokes problem, resulting that the reduced order model can possess the same basis functions together with same precision as the CFSECN one. To this end, the CFSECN format of the 2D nonstationary Stokes problem about vorticity function and stream function and, the existence and stability as well as error estimations for the CFSECN solutions are first proposed. Afterwards, a reduced order extrapolation finite spectral element Crank–Nicolson (ROEFSECN) model is created by means of proper orthogonal decomposition method, and the existence together with stability as well as errors of the ROEFSECN solutions is analyzed by the matrix tool, resulting that the theory analysis turns into very simple. Finally, two numerical examples are provided to confirm the correctness for the theory consequences. Therefore, the study is absolutely distinguished from the existed ones.

A combination of large eddy simulation and physics-informed machine learning to predict pore-scale flow behaviours in fibrous porous media: A case study of transient flow passing through a surgical mask

A predictive method using physics-informed machine learning (PIML) and large eddy simulation (LES) is developed to capture the transient flow field through microscale porous media (PSPM). An image processing technique extracts the 3D geometry of the internal layers of the mask from 2D microscopy images, and then the fluid flow is first simulated numerically. The subsequently developed PIML method successfully predicts the transient flow patterns inside the porous medium. For the first time, 3D maps of time-dependent pressure, velocity, and vorticity are predicted across the fibrous porous medium. The results show that, compared to conventional computational fluid dynamics, the PIML method can reduce the computational cost by over 20 times. Further, the LES model can replicate the fine fluctuations caused by the flow passage through the porous medium. Therefore, the developed methodology allows for transient flow predictions in highly complex configurations at a substantially reduced cost. The results indicate that the PIML method can reduce the total computational time (including training and prediction) by 22.5 and 20.7 times over the standard numerical simulation, based on speeds of 0.1 and 0.5 m/s, respectively. Several factors including the inherent differences between CPUs and GPUs, algorithms and software, appear to influence this improvement.

Pair Correlation Function of Vorticity in a Coherent Vortex

Pair Correlation Function of Vorticity in a Coherent Vortex

Natural Convection of Heat-Generating Liquid of Variable Viscosity under Wall Cooling Impact

This research presents a computational investigation of the thermal convection of a heat-generating liquid having variable viscosity in a semi-cylindrical cavity. The analysis is carried out to obtain the time patterns of the average Nusselt number at the lower border of the chamber and understand the impact of the variable viscosity, the Prandtl number, and the Rayleigh number on this parameter. The natural convection in the cavity is defined by the set of non-dimensional equations based on the Boussinesq approach employing the non-primitive parameters such as vorticity and stream function. These governing equations are worked out numerically based on the finite difference technique. The time dependencies have been obtained at the Rayleigh number equal to 104, 105, and 106 and the Prandtl number taking values of 7.0, 70, and 700. The results obtained for variable and constant viscosity have been compared. Additionally, the paper represents maps of isotherms and streamlines for the mentioned values of the Rayleigh number. The influence of variable viscosity on the parameters of natural convection is poorly studied in closed systems; therefore, this research gives necessary data to understand the general time nature of the average Nusselt number at cooling surface of various parameters. Additionally in this research, the model for simulating the natural convection in non-primitive variables is presented in polar coordinates when the dynamic viscosity varies with temperature. The computational model designed could be used to simulate the free convection in systems with inner heat production such as chemical reactors, inductive metal melting facilities, or corium in-vessel retention to analyze the impact of various factors on the parameters of the natural convection in such systems.