Equations Of Fluid Dynamics Research Articles

Conservation laws for a perturbed resonant nonlinear Schrödinger equation in quantum fluid dynamics and quantum optics

The current paper retrieves the conservation laws for the extended version of the resonant nonlinear Schrödinger's equation for description of physical processes in quantum fluid dynamics and quantum optics. The method of multipliers recovers three fundamental conservation laws. Analytical solutions of equation are found taking into account traveling wave reduction. The conserved quantities are subsequently computed from the soliton solution of the model equation that is derived in this work too.

Course control in a self-consistent model of cuttlefish movement

We developed a simulation model to mimic cuttlefish movement. We developed a simulation model to mimic cuttlefish movement, representing an elongated body with two undulatory fins that generate propulsive forces for underwater movement. Our mathematical model concurrently solved equations for both body mechanics and fluid dynamics, using the Navier–Stokes equations to describe the latter. To implement this self-consistent model, we utilized deformable mesh techniques. This enabled us to compute both the apparatus’s movement performance characteristics and hydrodynamic flow parameters, such as vorticity and pressure fields. Our study focused on examining how oscillations of the left and right fins, each with different parameters, impact the apparatus’s maneuverability. We found that differences in frequencies between the left and right fins resulted in a peak turning angle velocity. We also explored how the interplay between hydrodynamic forces influences the apparatus’s course control.

Fokker-Planck Central Moment Lattice Boltzmann Method for Effective Simulations of Fluid Dynamics

We present a new formulation of the central moment lattice Boltzmann (LB) method based on a minimal continuous Fokker-Planck (FP) kinetic model, originally proposed for stochastic diffusive-drift processes (e.g., Brownian dynamics), by adapting it as a collision model for the continuous Boltzmann equation (CBE) for fluid dynamics. The FP collision model has several desirable properties, including its ability to preserve the quadratic nonlinearity of the CBE, unlike that based on the common Bhatnagar-Gross-Krook model. Rather than using an equivalent Langevin equation as a proxy, we construct our approach by directly matching the changes in different discrete central moments independently supported by the lattice under collision to those given by the CBE under the FP-guided collision model. This can be interpreted as a new path for the collision process in terms of the relaxation of the various central moments to “equilibria”, which we term as the Markovian central moment attractors that depend on the products of the adjacent lower order moments and a diffusion coefficient tensor, thereby involving of a chain of attractors; effectively, the latter are nonlinear functions of not only the hydrodynamic variables, but also the non-conserved moments; the relaxation rates are based on scaling the drift coefficient by the order of the moment involved. The construction of the method in terms of the relevant central moments rather than via the drift and diffusion of the distribution functions directly in the velocity space facilitates its numerical implementation and analysis. We show its consistency to the Navier-Stokes equations via a Chapman-Enskog analysis and elucidate the choice of the diffusion coefficient based on the second order moments in accurately representing flows at relatively low viscosities or high Reynolds numbers. We will demonstrate the accuracy and robustness of our new central moment FP-LB formulation, termed as the FPC-LBM, using the D3Q27 lattice for simulations of a variety of flows, including wall-bounded turbulent flows. We show that the FPC-LBM is more stable than other existing LB schemes based on central moments, while avoiding numerical hyperviscosity effects in flow simulations at relatively very low physical fluid viscosities through a refinement to a model founded on kinetic theory.

Deciphering postoperative hemodynamics in pediatric intensive care: insights from patient-specific computational fluid dynamics models for congenital heart disease

Abstract Purpose Precise hemodynamic control is imperative for optimizing outcomes in children undergoing cardiac surgery, particularly in the pediatric intensive care unit (PICU). Computational fluid dynamic (CFD) models tailored to individual pediatric patients following definitive repairs for congenital heart disease hold the potential to discern and analyze the intricacies of their hemodynamic systems. This study aims to explore the applicability of patient-specific CFD models in determining optimal treatment strategies to enhance patient hemodynamics. Methods Ten postoperative pediatric patients (6 females and 4 males, median age 4.5±3.6 months, median height 62.5±5.3 cm, and median weight 6.2±1.3 kg) diagnosed with ventricular septal defect (VSD, n=5), atrial septal defect (ASD, n=2), and atrioventricular septal defect (AVSD, n=3) were enrolled. Continuous radial arterial waveforms and sporadic arterial blood draws, including lactate levels, were recorded in the PICU postoperatively. Fluid dynamics equations were employed to model major and peripheral blood vessels, utilizing a 0–1-dimensional hemodynamic model validated in previous studies and refined for this investigation. Sensitivity analysis based on correlation coefficients identified significant parameters, adjusted for replicating pressure waveforms. Comparison and re-approximation was made with actual radial artery pressure waveforms from 78 arterial waveforms and blood draws. Updated parameters were then correlated with postoperative time and lactate levels in the ICU. Results Replicated waveforms demonstrated high agreement with original arterial waveforms, with mean, maximal, and minimal blood pressure percentages of 96.7±1.8%, 96.7±2.4%, and 95.8±2.4%, respectively. Notably, systemic vascular resistance and compliance exhibited a substantial relationship with time elapsed after admission to the PICU, with correlation coefficients (R) of 0.26 and 0.43, respectively. Lactate levels demonstrated a decrease with reduced peripheral vascular resistance and an increase in vascular compliance, with R of 0.23 and 0.32, respectively, aligning with expected clinical improvement and stability. Conclusion This study successfully attained precise replication of radial artery blood pressures and intricate pressure waveforms using patient-specific CFD models in pediatric patients following congenital heart disease repairs. The insights gained shed light on the pivotal hemodynamic parameters that impact both stability and instability. This research sets the stage for theoretically informed treatments by amalgamating real-time biological information with patient-specific CFD models.Re-approximation of arterial waveformCorrelation of hemodynamic parameters

Optimal decay of the Boltzmann equation

The Boltzmann equation is a typical example of partially dissipative equations, where the linearized collision operator is positive definite with respect to the microscopic part and the dissipation of the hydrodynamic part is discovered from the coupling structure between the transport operator and the linearized collision operator. Guo and Wang (Comm. Partial Differential Equations, 37, 2012) developed a general energy method for proving the optimal time decay rates of the solution to such type of equations in the whole space; however, the decay rate of the highest order spatial derivatives of the solution is not optimal. In this paper, by incorporating the high‐low frequency decomposition in the energy estimates, both linearly and nonlinearly, we prove the optimal decay rates of any high order spatial derivatives of the low frequency part of the solution to the Boltzmann equation and the almost exponential decay rate of the high frequency part, which imply in particular the optimal decay rate of the highest order spatial derivatives of the solution. Moreover, the velocity‐weighted assumption of the initial data required in Guo and Wang (Comm. Partial Differential Equations, 37, 2012) is removed by capturing the time‐weighted dissipation estimates via the time‐weighted energy method. The method can be applied to the compressible Navier–Stokes equations and many partially dissipative equations in kinetic theory and fluid dynamics.

Closed-Boundary Reflections of Shallow Water Waves as an Open Challenge for Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) have recently emerged as a promising alternative to traditional numerical methods for solving partial differential equations (PDEs) in fluid dynamics. By using PDE-derived loss functions and auto-differentiation, PINNs can recover solutions without requiring costly simulation data, spatial gridding, or time discretization. However, PINNs often exhibit slow or incomplete convergence, depending on the architecture, optimization algorithms, and complexity of the PDEs. To address these difficulties, a variety of novel and repurposed techniques have been introduced to improve convergence. Despite these efforts, their effectiveness is difficult to assess due to the wide range of problems and network architectures. As a novel test case for PINNs, we propose one-dimensional shallow water equations with closed boundaries, where the solutions exhibit repeated boundary wave reflections. After carefully constructing a reference solution, we evaluate the performance of PINNs across different architectures, optimizers, and special training techniques. Despite the simplicity of the problem for classical methods, PINNs only achieve accurate results after prohibitively long training times. While some techniques provide modest improvements in stability and accuracy, this problem remains an open challenge for PINNs, suggesting that it could serve as a valuable testbed for future research on PINN training techniques and optimization strategies.

Algebraic calming for the 2D Kuramoto-Sivashinsky equations

Abstract We propose an approximate model for the 2D Kuramoto–Sivashinsky equations (KSE) of flame fronts and crystal growth. We prove that this new ‘calmed’ version of the KSE is globally well-posed, and moreover, its solutions converge to solutions of the KSE on the time interval of existence and uniqueness of the KSE at an algebraic rate. In addition, we provide simulations of the calmed KSE, illuminating its dynamics. These simulations also indicate that our analytical predictions of the convergence rates are sharp. We also discuss analogies with the 3D Navier–Stokes equations of fluid dynamics.

Learning fluid physics from highly turbulent data using sparse physics-informed discovery of empirical relations (SPIDER)

We show how a complete mathematical model of a physical process can be learned directly from data via a hybrid approach combining three simple and general ingredients: physical assumptions of smoothness, locality and symmetry, a weak formulation of differential equations and sparse regression. To illustrate this, we extract a complete system of governing equations of fluid dynamics – the Navier–Stokes equation, the continuity equation and the boundary conditions – as well as the pressure-Poisson and energy equations, from numerical data describing a highly turbulent channel flow in three dimensions. Whether they represent known or unknown physics, relations discovered using this approach take the familiar form of partial differential equations, which are easily interpretable and readily provide information about the relative importance of different physical effects. The proposed approach offers insight into the quality of the data, serving as a useful diagnostic tool. It is also remarkably robust, yielding accurate results for very high noise levels, and should thus be well suited for analysis of experimental data.

Phase trajectories, chaotic behavior, and solitary wave solutions for (3+1)-dimensional integrable Kadomtsev–Petviashvili equation in fluid dynamics

This paper focuses on finding exact soliton solutions and further examining the qualitative characteristics of these solutions for a nonlinear partial differential equation known as the (3+1)-dimensional integrable Kadomtsev–Petviashvili equation that portrays a unique dispersion effect. The (3+1)-dimensional integrable Kadomtsev–Petviashvili equation models the evolution of weakly nonlinear dispersive waves in three spatial dimensions “x,y,z” and time “t”. It balances nonlinearity and dispersion, leading to solitary wave solutions that are stable and localized. This equation is crucial in understanding complex wave phenomena in fields like fluid dynamics, plasma physics, and optics. Initially, the traveling wave transformation is employed to convert the nonlinear partial differential equations into the non-linear ordinary differential equations. Then, we use the Riccati equation approach to find solitary wave solutions for the resulting ordinary differential equation. To better comprehend the physical significance of these derived solutions, we illustrate them using various visual representations. Diverse novel collections of solitary wave solutions are obtained such as bright solitons, dark solitons, and dark singular solitons. Second, we convert the nonlinear ordinary differential equation into a linear dynamical system using the planar dynamical transformation. Then, we undertake a qualitative analysis of the dynamical system to study its chaotic characteristics and bifurcation phenomena. Phase portraits of bifurcation are observed at the equilibrium points of the planner dynamical system. We discuss various tools to identify the chaos (random and unpredictable behavior) in dynamical systems. Graphical visualization of qualitative analysis of the system is also depicted in 3-D phase portraits, 2-D phase portraits, Poincaré mapping, and time series. Additionally, the Runge–Kutta technique is used to do a thorough sensitivity analysis of the dynamical system. Using this analytical procedure, it is verified that the stability of the solution has negligible impact while there is little change in the beginning circumstances. This paper aims to provide vital views to scientists and researchers seeking to advance their experimental operations. Furthermore, this research might be expanded by incorporating solitons and situations and investigating wave dynamics. The methods employed provide a direct and uncomplicated approximation of all solutions when compared to the previously established techniques. Various combinations and values of the physical parameters are utilized to explore the optical soliton solutions of the resultant system.

Numerical simulation of vehicle hood underwater active propulsion separation

Abstract In order to study the separation process of the underwater active propulsion hood, this paper analyzes the main stages of the underwater separation system. By using unstructured dynamic mesh technology, the control equations of computational fluid dynamics are coupled with the six-degree-of-freedom motion equations to calculate the influence of different installation angles of the propulsion device and the distance of thrust deviation from the center of gravity on the separation of the hood. The results show that hydrodynamics will exert a moment on the hood, causing the hood to rotate. Excessive rotation of the hood can lead to collisions with the vehicle. Raising the installation angle of the propulsion device and applying thrust to resist the rotation of the hood can reduce the angular velocity of the hood, improve the effectiveness of the thrust, and optimize the separation performance of the system.

Gyrotactic micro-organism dusty fluid flow over an extended surface

The conventional magnetohydrodynamics (MHD) equations are modeled for the dusty fluid flow past a stretching sheet embedded in a porous medium. These equations are coupled with motile microorganisms, temperature and concentration equations for both fluid and dust particles dynamical equations. The two-phase dusty fluid flow is subjected to magnetic effect, porosity factor, 2nd order chemical reaction, Brownian diffusion, thermophoretic effect and Arrhenius activation energy. An appropriate transformation is used to reset the system of partial differential equations (PDEs) into non-linear system of ODEs (ordinary differential equations). The flow equations are numerically solved by using the parametric continuation method (PCM). For validity of the applied methodology, the results are compared using numerical and graphical results of Matlab built-in package “BVP4c”. Both results show accuracy up to four decimal places. Furthermore, from graphical results, it can be noticed that the fluid particles interaction parameter causes an attractive force among fluid particles which prevents the dust particles phase flow. This technique can be used for biological separations, filters, fluid dust separators and crude oil purification.

Density Waves of Vortex Fluids on a Sphere

Abstract We aim to find one highly nontrivial example of the solutions to the vortex fluid dynamical equation on the unit sphere (S^2) and compare it with the numerical simulation. Since the rigid rotating steady solution for vortex fluids on S^2 is
already known to us, we consider the perturbations above it. After decomposing the perturbation of the vortex number density and vortex charge density into spherical harmonics, we find that the perturbations are propagating waves. To
be precise, the velocities for different single-mode vortex number density waves are all the same, while the velocities for single-mode vortex charge density waves depend on the degree of the spherical harmonics l, which is a signal of the existence of dispersion. Meanwhile, we find that there is a beat phenomenon for the positive (or negative) vortex density wave. Numerical simulation based on the canonical equations for point vortex model agrees perfectly with our theoretical calculations.

Analysis of pedestrian second crossing behavior based on physics-informed neural networks

Pedestrian two-stage crossings are common at large, busy signalized intersections with long crosswalks and high traffic volumes. This design aims to address pedestrian operation and safety by allowing navigation in two stages, negotiating each traffic direction separately. Understanding crosswalk behavior, especially during bidirectional interactions, is essential. This paper presents a two-stage pedestrian crossing model based on Physics-Informed Neural Networks (PINNs), incorporating fluid dynamics equations to determine characteristics such as speed, density, acceleration, and Reynolds number during crossings. The study shows that PINNs outperform traditional deep learning methods in calculating and predicting pedestrian fluid properties, achieving a mean squared error as low as 10–8. The model effectively captures dynamic pedestrian flow characteristics and provides insights into pedestrian behavior impacts. The results are significant for designing pedestrian facilities to ensure comfort and optimizing signal timing to enhance mobility and safety. Additionally, these findings can aid autonomous vehicles in better understanding pedestrian intentions in intelligent transportation systems.

GPU-enabled extreme-scale turbulence simulations: Fourier pseudo-spectral algorithms at the exascale using OpenMP offloading

Fourier pseudo-spectral methods for nonlinear partial differential equations are of wide interest in many areas of advanced computational science, including direct numerical simulation of three-dimensional (3-D) turbulence governed by the Navier-Stokes equations in fluid dynamics. This paper presents a new capability for simulating turbulence at a new record resolution up to 35 trillion grid points, on the world's first exascale computer, Frontier, comprising AMD MI250x GPUs with HPE's Slingshot interconnect and operated by the US Department of Energy's Oak Ridge Leadership Computing Facility (OLCF). Key programming strategies designed to take maximum advantage of the machine architecture involve performing almost all computations on the GPU which has the same memory capacity as the CPU, performing all-to-all communication among sets of parallel processes directly on the GPU, and targeting GPUs efficiently using OpenMP offloading for intensive number-crunching including 1-D Fast Fourier Transforms (FFT) performed using AMD ROCm library calls. With 99% of computing power on Frontier being on the GPU, leaving the CPU idle leads to a net performance gain via avoiding the overhead of data movement between host and device except when needed for some I/O purposes. Memory footprint including the size of communication buffers for MPI_ALLTOALL is managed carefully to maximize the largest problem size possible for a given node count.Detailed performance data including separate contributions from different categories of operations to the elapsed wall time per step are reported for five grid resolutions, from 20483 on a single node to 327683 on 4096 or 8192 nodes out of 9408 on the system. Both 1D and 2D domain decompositions which divide a 3D periodic domain into slabs and pencils respectively are implemented. The present code suite (labeled by the acronym GESTS, GPUs for Extreme Scale Turbulence Simulations) achieves a figure of merit (in grid points per second) exceeding goals set in the Center for Accelerated Application Readiness (CAAR) program for Frontier. The performance attained is highly favorable in both weak scaling and strong scaling, with notable departures only for 20483 where communication is entirely intra-node, and for 327683, where a challenge due to small message sizes does arise. Communication performance is addressed further using a lightweight test code that performs all-to-all communication in a manner matching the full turbulence simulation code. Performance at large problem sizes is affected by both small message size due to high node counts as well as dragonfly network topology features on the machine, but is consistent with official expectations of sustained performance on Frontier. Overall, although not perfect, the scalability achieved at the extreme problem size of 327683 (and up to 8192 nodes — which corresponds to hardware rated at just under 1 exaflop/sec of theoretical peak computational performance) is arguably better than the scalability observed using prior state-of-the-art algorithms on Frontier's predecessor machine (Summit) at OLCF. New science results for the study of intermittency in turbulence enabled by this code and its extensions are to be reported separately in the near future.

Remarks on Regularization by Noise, Convex Integration and Spontaneous Stochasticity

AbstractThis note is devoted to a discussion of the potential links and differences between three topics: regularization by noise, convex integration, spontaneous stochasticity. All of them deal with the effect on large scales of a small-scale perturbation of fluid dynamic equations. The effects sometimes have something in common, like convex integration and spontaneous stochasticity, sometimes they look the opposite, as in regularization by noise. We are not aware of rigorous links or precise explanations of the differences, and hope to drive new research with this comparative examination.

Variable coefficient-informed neural network for PDE inverse problem in fluid dynamics

Variable-coefficient equations are crucial in the field of fluid dynamics as they accurately capture the spatial and temporal variations of fluid properties. In many cases, there exist some constraints among multiple coefficients and embedding these constraints into neural networks poses a challenge. In this paper, we design a variable coefficient-informed neural network (VCINN) to address the inverse problem of variable-coefficients partial differential equation in fluid dynamics. The VCINN framework integrates the physics-informed neural network (PINN) with the constraints among multiple coefficients, encoding both constraints and physics information into the neural networks. Compared to classical PINN, VCINN enjoys such several advantages as parallelization capacity, embedding constraint information and efficient hyperparameter adjustment. Through series of examples, the capability of the approach to recover coefficients from observations has been demonstrated. Numerical results indicate that the present method achieves higher accuracy and lower training error compared to classical PINN.

A formulation for fluid–structure interaction problems with immersed flexible solids: Application to splitters subjected to flow past cylinders with different cross-sections

In the finite element method framework, a fluid–structure formulation is developed by coupling an Eulerian fixed-mesh fluid approach with a Lagrangian deforming-mesh description for a flexible solid. The coupled formulation is solved using a staggered scheme during time. For the fluid solution stage, the solid walls are considered as a time-variable internal boundary. The velocity and pressure fields are obtained by solving the weak form of the fluid dynamic equations in which the solid velocity is imposed on the internal boundary via a penalization term. For the solid solution stage, the displacement field is obtained by solving the discrete solid dynamic equations which consider traction forces computed by integrating pressures and viscous stresses on the nodes belonging to the solid walls. This novel technique is firstly applied to analyze a flexible splitter under the shedding of a flow past square cylinder due to this problem is considered as a benchmark in the literature. The present solutions agree with those computed using body-fitted techniques, thus validating the proposal. Secondly, flexible splitter motions under the shedding of flow past cylinders with different cross-sections and splitter lengths are comprehensively studied. Overall, the computed results confirmed that the hydrodynamic coefficients on the cylinders were reduced because of the presence of the splitter.

Computational framework for the generation of one-dimensional vascular models accounting for uncertainty in networks extracted from medical images.

One-dimensional (1D) cardiovascular models offer a non-invasive method to answer medical questions, including predictions of wave-reflection, shear stress, functional flow reserve, vascular resistance and compliance. This model type can predict patient-specific outcomes by solving 1D fluid dynamics equations in geometric networks extracted from medical images. However, the inherent uncertainty in in vivo imaging introduces variability in network size and vessel dimensions, affecting haemodynamic predictions. Understanding the influence of variation in image-derived properties is essential to assess the fidelity of model predictions. Numerous programs exist to render three-dimensional surfaces and construct vessel centrelines. Still, there is no exact way to generate vascular trees from the centrelines while accounting for uncertainty in data. This study introduces an innovative framework employing statistical change point analysis to generate labelled trees that encode vessel dimensions and their associated uncertainty from medical images. To test this framework, we explore the impact of uncertainty in 1D haemodynamic predictions in a systemic and pulmonary arterial network. Simulations explore haemodynamic variations resulting from changes in vessel dimensions and segmentation; the latter is achieved by analysing multiple segmentations of the same images. Results demonstrate the importance of accurately defining vessel radii and lengths when generating high-fidelity patient-specific haemodynamics models. KEY POINTS: This study introduces novel algorithms for generating labelled directed trees from medical images, focusing on accurate junction node placement and radius extraction using change points to provide haemodynamic predictions with uncertainty within expected measurement error. Geometric features, such as vessel dimension (length and radius) and network size, significantly impact pressure and flow predictions in both pulmonary and aortic arterial networks. Standardizing networks to a consistent number of vessels is crucial for meaningful comparisons and decreases haemodynamic uncertainty. Change points are valuable to understanding structural transitions in vascular data, providing an automated and efficient way to detect shifts in vessel characteristics and ensure reliable extraction of representative vessel radii.

A novel study of analytical solutions of some important nonlinear fractional differential equations in fluid dynamics

The space-time fractional Burger-like equation and the space-time coupled Boussinesq equation with conformable derivative, both of which are significant in fluid dynamics, are investigated in this work using the improved [Formula: see text] method. The process works effectively and produces soliton solutions. The method was successfully and consistently implemented with Maple, a symbolic computing tool. The solutions also contain a few of graphics. Numerous novel exact solutions to these equations, distinct from those found earlier with the proposed approach, have been provided. The study’s findings add to the body of knowledge by offering insightful justifications for various types of nonlinear systems. The results demonstrated the value of the proposed method as a mathematical tool and the ease, dependability, and speed increases that result from carrying out these tasks using a symbolic computing program. Notably, it applies to many nonlinear evolution problems in mathematical physics.

About regular expansion of a two-phase ball

The work is devoted to the research of an expansion into a vacuum of a ball, which is filled with a two-phase fluid. In assumptions that the dynamics passes in a regular mode, the velocities of the phases are linear functions by space coordinate and the first phase spreads into the void faster than the second phase there is obtained a solution of equations of two-phase fluid dynamics, which describes an expansion of a ball into a vacuum. This solution generalizes the known Sedov solution, which determines an expansion of a gas cloud into a vacuum, to the case of a ball filled with a gas suspension. In the last part of the work there are derived asymptotic formulas determining a relationship between the velocities of the phases and the radii of two balls.