Navier-Stokes Research Articles

Assessment of Cavitation Erosion Using Combined Numerical and Experimental Approach

This paper aims to numerically assess the cavitation damage of hydrodynamic machines and hydraulic components and its development in time, based on cavitation erosion tests with samples of used materials. The theoretical part of this paper is devoted to the numerical simulation of unsteady multiphase flow by means of computational fluid dynamics (CFD) and to the prediction of the erosive effects of the collapses of cavitation bubbles in the vicinity of solid surfaces. Compressible unsteady Reynolds-averaged Navier–Stokes equations (URANS) are solved together with the Zwart cavitation model. To describe the destructive collapses of vapor bubbles, the modeling of cavitation bubble dynamics along selected streamlines or trajectories is applied. The hybrid Euler–Lagrange approach with one-way coupling and the full Rayleigh–Plesset equation (R–P) are therefore utilized. This paper also describes the experimental apparatus with a rotating disc used to reach genuine hydrodynamic cavitation and conditions similar to those of hydrodynamic machines. The simulations are compared with the obtained experimental data, with good agreement. The proposed methodology enables the application of the results of erosion tests to the real geometry of hydraulic machines and to reliably predict the locations and magnitude of cavitation erosion, so as to select appropriate materials or material treatments for endangered parts.

Energy Conservation for 3D Euler and Navier–Stokes Equations in a Bounded Domain: Applications to Beltrami Flows

Energy Conservation for 3D Euler and Navier–Stokes Equations in a Bounded Domain: Applications to Beltrami Flows

Performance evaluation of a scramjet engine utilizing varied cavity aft wall divergence with parallel injection in a reacting flow field

Abstract The effect of implications of the dual cavity with aft wall divergence in a parallel injection reacting flow has been numerically investigated. This research intends to emphasize the behaviour of the supersonic flow under varying divergence angles of the cavity aft wall. A two-dimensional Reynolds-Averaged Navier-Stokes (RANS) equation and an SST k-ω turbulence model with a single-step chemical reaction for hydrogen-air are utilized for the simulation. Followed by the strut injector, the cavities are positioned symmetrically inside the combustor. The bottom cavity aft wall divergence varies, whereas the top wall is mounted with a rectangular cavity. The performance of cavity locations is compared with the baseline DLR model. From the evaluation of numerical outcomes along with different cavity configurations, it has been noted that the 15-degree cavity divergence angle enhances the recirculation zone which leads to improved mixing performance. Also, the cavity improves the combustion stability by increasing the flow residence time. From this numerical analysis, it is associated that an almost 20 % reduction in combustor length is achieved, however, an 18 % rise in the pressure loss is noted because of emanating shock waves from the cavity edges.

Pullback Measure Attractors and Periodic Measures of Stochastic Non-autonomous Tamed 3D Navier–Stokes Equation

Pullback Measure Attractors and Periodic Measures of Stochastic Non-autonomous Tamed 3D Navier–Stokes Equation

Influence of speed and line spacing on aerodynamic forces in high speed train passing in tunnels

As train speeds increase, the aerodynamic effects generated during the passage of high-speed trains in tunnels become more pronounced. This article investigates the influence of line spacing on the aerodynamic forces during the passage of high-speed trains in tunnels at different speeds. The research methodology is based on the three-dimensional transient compressible Reynolds-averaged Navier-Stokes (RANS) equations and the k-ω Shear Stress Transport (SST) turbulence model, using overset grid technology to simulate the train passing process in the tunnel. The results show that the aerodynamic drag, lateral force, and lift experienced by the train increase as the line spacing decreases and as the speed increases. line spacing has the greatest impact on aerodynamic lift, followed by lateral force, with drag having the smallest impact. Compared to line spacing, speed has a much larger effect on aerodynamic forces. The aerodynamic forces experienced by the train during tunnel passage are proportional to the square of the train’s speed and have an exponential relationship with the line spacing during train passage.

A numerical investigation on rheological turbulent flow through a 90° mixing elbow

Mixing elbows are widely used in various industrial processes to mix two different Newtonian or non-Newtonian fluids under turbulent flow conditions. This study numerically investigates the fluid flow and mixing characteristics in 90-degree elbows with different bend curvatures (curvature ratios of 3, 6, and 9) and Reynolds numbers ranging from 1 × 104 to 5 × 105 for both fluid types. The Reynolds-Averaged Navier–Stokes equations are solved using the turbulent k-epsilon model in ANSYS Fluent. Results show that the bend curvature creates an adverse pressure gradient that drives secondary flows, which become more pronounced with higher Reynolds numbers and lower curvature ratios. Higher Reynolds numbers improve mixing quality, as indicated by velocity and temperature variations quantified using standard deviations. For Reynolds numbers in the range of 104, velocity fluctuations initially increase, but at higher values (~ 105), they decrease, even by up to 64% as the Reynolds number rises from 1 × 105 to 5 × 105. Similarly, increasing the curvature ratio enhances mixing efficiency, with a 32% reduction in velocity fluctuations when the curvature ratio increases from 6 to 9. Furthermore, shear-thinning fluids (n=0.6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 0.6$$\\end{document}) exhibit superior mixing performance, with a standard deviation of velocity fluctuations at the outlet of 0.32, compared to 0.54 for shear-thickening fluids (n=1.4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 1.4$$\\end{document}). In terms of temperature fluctuations also, a similar trend is observed. These findings are valuable for optimizing mixing and fluid transport in industries such as chemical processing and food production, where effective mixing of complex fluids is critical. This study provides insights for improving the design of mixing systems that handle fluids with various rheological characteristics.

Exact planetary waves and jet streams

We investigate exact nonlinear waves on surfaces locally approximating the rotating sphere for two-dimensional inviscid incompressible flow. Our first system corresponds to a $\beta$ -plane approximation at the equator, and the second to a $\gamma$ approximation, with the latter describing flow near the poles. We find exact wave solutions in the Lagrangian reference frame that cannot be written down in closed form in the Eulerian reference frame. The wave particle trajectories, contours of potential vorticity and Lagrangian mean velocity take relatively simple forms. The waves possess a non-trivial Lagrangian mean flow that depends on the amplitude of the waves and on a particle label that characterizes values of constant potential vorticity. The mean flow arises due to potential vorticity conservation on fluid particles. Solutions over the entire space are generated by assuming that the flow far from the origin is zonal and there is a region of uniform potential vorticity between this zonal flow and the waves. In the $\gamma$ approximation, a class of waves is found that, based on analogous solutions on the plane, we call Ptolemaic vortex waves. The mean flow of some of these waves, which we can describe in highly nonlinear scenarios due to the exact nature of the solutions, resembles polar jet streams. Several illustrative solutions are used as initial conditions in the fully spherical rotating Navier–Stokes equations, where integration is performed via the numerical scheme presented in Salmon & Pizzo (Atmosphere, vol. 14, issue 4, 2023, 747). The potential vorticity contours found from these numerical experiments vary between stable permanent progressive form and fully turbulent flows generated by wave breaking.

Numerical and Experimental Simulation of Supersonic Gas Outflow into a Low-Density Medium

This study is aimed at developing methods for the experimental and numerical simulation of the outflow of underexpanded gas jets into a rarefied medium. The numerical method is based on using Navier–Stokes equations in the continuum flow regime and the direct simulation Monte Carlo method in the transitional flow regime. The experimental method includes the modeling of jet flows in the LEMPUS-2 gas-dynamic setup with electron beam diagnostics for the jet density measurements. The results of the experimental modeling for the nozzles of various diameters confirm that a key parameter determining the jet structure is the Reynolds number based on the characteristic length ReL. The results of the numerical simulations agree well with the experimental data both for the maximum values of the ReL considered (approximately 30) when a barrel jet structure with Mach disks is formed and for the minimum values (approximately 4) when no Mach disks are formed. In the entire range of parameters, significant thermal nonequilibrium is observed at all jet segments where the measurements are performed.

Mathematical Models for Removal of Pharmaceutical Pollutants in Rehabilitated Treatment Plants

This paper aims to investigate appropriate mathematical models devoted to the optimization of some cleaning processes related to pharmaceutical contaminant removal. In our recent works, we found the rehabilitation of the existing cleaning plants as a viable solution for the removal of this type of micropollutants from waters by introducing efficient techniques such as adsorption on granulated active carbon filters and micro-, nano-, or ultrafiltration. To have these processes under better control and to assure the transfer from small- to large-scale treatment stations, specific mathematical models are necessary. Starting from Navier–Stokes equations and imposing proper boundary conditions, some mathematical physics problems are obtained for which adequate solving methods via variational methods and surjectivity results are proposed. The importance of these solution characterizations consists in their continuation in adequate numerical methods and the possibility to visualize the result by using a CFD program.

Comparison of Contact Angle Models in Two‐Phase Flow Simulations Using a Conservative Phase Field Equation

ABSTRACTIn phase field methods based on a second‐order Allen‐Cahn (AC) equation, contact angles are prescribed mostly via a geometric formulation. However, it is of great interest to utilize the surface‐energy formulation, which is often employed in the Cahn‐Hilliard (CH) phase field method, in the AC phase field method. This article thus put forward a surface‐energy formulation of contact angles. The model was compared with the geometric one in a number of impact problems, including both normal and oblique impacts. The governing equations were discretized using a finite difference method on a half‐staggered grid. The Navier–Stokes equation was tackled using an explicit projection method. The major findings are as follows. First, the geometric model can maintain a fixed contact angle throughout contact line motion, while the surface‐energy one predicts a changeable contact angle, with a fluctuation of about 5°. In the oblique drop impact, contact angle hysteresis was captured even if a static contact angle was applied in the surface‐energy formulation.

A variational principle of minimum for Navier–Stokes equation and Bingham fluids based on the symplectic formalism

A variational principle of minimum for Navier–Stokes equation and Bingham fluids based on the symplectic formalism

New Estimates of the Potential Schrödinger Equation

The Schrödinger equation, a fundamental construct in quantum mechanics, plays a pivotal role in understanding the properties and behaviors of quantum systems across various scientific fields, including but not limited to physics, economics, and geophysics. This research paper delves into the exploration of novel invariants associated with the Schrödinger equation, uncovering previously unrecognized symmetries and properties that open new pathways for theoretical and applied scientific exploration. The investigation reveals that the energy of bound states remains invariant under transformations of the coordinate system, highlighting a universal symmetry embedded within the equation. This discovery, coupled with an analysis of the variation in scattering amplitudes’ phase with different coordinate systems, not only enriches the theoretical framework of quantum mechanics but also has significant practical implications across various domains such as fluid dynamics, material science, and medical imaging. Furthermore, by applying the principles of the Poincaré-Riemann-Hilbert boundary-value problem, the study offers a novel approach to estimate potentials within the Schrödinger equation, advancing the three-dimensional inverse problem of quantum scattering theory. This methodological innovation allows for a more profound understanding of the unitary scattering operator, facilitating enhanced modeling of complex systems and phenomena. The implications of these findings extend beyond theoretical physics, offering transformative insights and applications in areas ranging from fluid dynamics, where it aids in refining models for the Navier-Stokes equations, to seismic exploration, tomography, and ultrasound imaging, where it enhances phase selection and interpretation of measurement results. In essence, this research not only contributes to a deeper understanding of the Schrödinger equation’s fundamental properties but also paves the way for interdisciplinary advancements, demonstrating the profound impact of theoretical discoveries on practical applications in diverse scientific and technological domains.

Rheological properties of phosphorus-containing oligoester(meth)acrylate for processing by vacuum infusion

Objectives. To obtain polymer composite materials (PCM) with enhanced physicomechanical properties using the vacuum assisted resin transfer molding (VaRTM) method, binders must have a viscosity of up to 500 mPa∙s. In some cases, this leads to restrictions on the use of certain materials or requires the use of temporary diluents. This is closely related to the deterioration of other required composite characteristics, such as increased flammability. Three phosphorus-containing oligoester(meth)acrylates PhOEM-1, PhOEM-2, and PhOEM-3 were synthesized with significant differences in viscosity characteristics in the series PhOEM-1 << PhOEM-2 << PhOEM-3. The polymer based on PhOEM-1 exhibits inferior physicomechanical properties despite having lower viscosity. Hence, the aim of the study was to investigate the viscosity characteristics of mixtures of methacrylate binders of the same nature but different structures and functionalities. This was done by studying the rheological properties of the original oligoester(meth)acrylates and their mixtures taken in various ratios. The method used was to optimize compositions via a simplex lattice (Scheffe’s plan), in order to obtain PCM using the VaRTM technology.Methods. The study of rheological properties of phosphorus-containing oligoester(meth)acrylates and their mixtures was conducted using the method of rotational viscometry on a Brookfield LVDV-II + Pro viscometer with a spindle 27 at different shear rates ranging from 0 to 70 s−1 and temperatures from 30 to 70°C. Rheological studies were also conducted on a Lamy Rheology GT300 PLUS (GEL TIMER) viscometer in the same range of shear rates and temperatures.Results. It was established that the objects under investigation can be characterized by viscosity values ranging from 96 to 2137 mPa∙s depending on the temperature. The nature of the viscous flow of phosphorus-containing oligoester(meth)acrylates and their mixtures is similar to that of Newtonian liquids only at certain shear rates. The effective activation energies of the viscous flow of binders and their mixtures were calculated, and the influence of temperature on the viscosity of binders was determined.Conclusions. The study identified the features and nature of the flow curves of phosphorus-containing oligoester(meth)acrylate binders of the same nature but different structures and functionalities, as well as of their mixtures. The optimal composition ranges of threecomponent mixtures of phosphorus-containing oligoester(meth)acrylates for use in the VaRTM technological process in producing polymer composite materials within the temperature range of 30 to 70°C were defined. The optimal compositions and temperature conditions for obtaining polymer composite materials using the VaRTM technology were also identified. This enables the production of polymer products with complex geometric shapes and varying sizes.

Shape optimized acoustic metagratings for anomalous refraction under strong thermoviscous effects

The recent development of microacoustic metagratings opens up promising possibilities for manipulating acoustic wavefronts passively, particularly in applications such as flat acoustic lenses and ultra-high frequency ultrasound imaging. The emergence of two-photon polymerization has made it feasible to precisely manufacture microscopic structures, as required when metagratings are scaled to MHz frequencies in airborne ultrasound. Nevertheless, the downsizing process presents another hurdle as the increased thermoviscous effects result in substantial losses that must be considered during the design phase. In this study, we propose two designs for microacoustic metagratings that refract a normally incident wave towards –35 ° at 2 MHz, consisting of single-body and two-body meta-atoms. The designs are created by employing shape optimization techniques that incorporate the linearized Navier–Stokes equations in every iteration starting from a neutral geometry. This ensures that the evolution of geometric key features responsible for anomalous refraction fully accounts for thermoviscous effects, as would be the case during evolution in nature where the full set of physics is always active. Subsequently, we experimentally evaluate the effectiveness of these metagratings by employing a capacitive micromachined ultrasonic transducer as the sound source and an optical microphone as the detector, covering a frequency range from 1.8 to 2.2 MHz. Our findings confirm the single-body geometry reported in the literature and show an alternative geometry for two-body design, showcasing the successful utilization of two-photon polymerization for manufacturing microscopic acoustic metamaterials.

Dynamic interaction patterns of oblique detonation waves with boundary layers in hypersonic reactive flows

Due to their high thermal cycle efficiency and compact combustor, oblique detonation engines hold great promise for hypersonic propulsion. Previous numerical simulations of oblique detonation waves have predominantly solved the Euler equations, disregarding the influence of viscosity and boundary layers. This work aims to study how the interaction between the oblique detonation wave and the boundary layer influences the detonation wave structures in confined spaces. Two-dimensional numerical simulations considering detailed chemistry are performed in a stoichiometric H2/air mixture. The results indicate that the wedge-induced oblique detonation wave generates a strong adverse pressure gradient upon impacting the upper wall, leading to boundary layer separation. The separation zone subsequently induces an oblique shock wave near the upper wall, and an increase in separation angle will cause the transition from an oblique shock wave to an oblique detonation wave. The formation of the separation zone reduces the actual flow area and may even lead to flow choking; its obstructive effect is similar to that of the Mach stem in inviscid flow. To establish a connection between the viscous recirculation zone and the inviscid Mach stem, we introduce a dimensionless parameter, η, based on the inviscid assumption. It is defined as the ratio of the inviscid Mach stem height to the channel entrance height. This parameter can be used to identify three wave systems in a viscous flow field: separation shock-dominated wave systems, separation detonation-dominated wave systems, and unstable Mach stem-dominated wave systems. Among these, the appearance of detonation Mach stems leads to flow choking, and the shock-detonation wave system continually moves upstream, ultimately causing the failure of the oblique detonation combustion. The findings of this study provide new insights into the investigation of the influence of viscosity on the flow structure of oblique detonation waves.

A novel Lax–Wendroff type procedure of two-derivative time-stepping schemes for Euler and Navier–Stokes equations

A novel Lax–Wendroff type procedure of two-derivative time-stepping schemes for Euler and Navier–Stokes equations

Physics-informed neural networks for weakly compressible flows using Galerkin–Boltzmann formulation

In this work, we study the Galerkin–Boltzmann formulation within a physics-informed neural network (PINN) framework to solve flow problems in weakly compressible regimes. The Galerkin–Boltzmann equations are discretized with second-order Hermite polynomials in microscopic velocity space, which leads to a first-order conservation law with six equations. Reducing the output dimension makes this equation system particularly well suited for PINNs compared with the widely used D2Q9 lattice Boltzmann velocity space discretizations. We created two distinct neural networks to overcome the scale disparity between the equilibrium and non-equilibrium states in collision terms of the equations. We test the accuracy and performance of the formulation with benchmark problems and solutions for forward and inverse problems with limited data. We compared our approach with the incompressible Navier–Stokes equation and the D2Q9 formulation. We show that the Galerkin–Boltzmann formulation results in similar L2 errors in velocity predictions in a comparable training time with the Navier–Stokes equation and lower training time than the D2Q9 formulation. We also solve forward and inverse problems for a flow over a square, try to capture an accurate boundary layer, and infer the relaxation time parameter using available data from a high-fidelity solver. Our findings show the potential of utilizing the Galerkin–Boltzmann formulation in PINN for weakly compressible flow problems.

Experimental Study of Airfoil Aerodynamic Behavior under Oscillating Motion in Ground Effect

When a flying vehicle approaches a water or land surface, it induces changes in the fluid flow field pattern known as the "ground effect." This research analyzes the ground effect phenomenon, exploring its impact on aerodynamic coefficients and flow patterns around the NACA0012 airfoil in an incompressible subsonic regime under static and dynamic conditions with pitch movements. Numerical simulations and experimental testing in an incompressible subsonic wind tunnel were deployed. The flow field solution is derived from the Navier-Stokes equations, incorporating the Transition SST turbulence model. Initially, the impact of the ground effect phenomenon was investigated at varying distances from the surface in the static state. Subsequently, the airfoil underwent a sinusoidal pitching oscillation at each distance with a specified frequency and amplitude. This allowed for examining its aerodynamic characteristics over time. The static analysis results reveal alterations in the curve's behavior and pressure distribution on the airfoil surface at close distances to the surface. This is attributed to the ground effect phenomenon, which reduces lift force to a certain height and then increases. Dynamic analysis further demonstrates changes in lift coefficient oscillation amplitude. It also exhibits a minimum and maximum lift point phase difference as the airfoil approaches the surface.

Multiple solutions of nonlinear coupled constitutive relation model and its rectification in non-equilibrium flow computation

In this study, the multiple solutions of Nonlinear Coupled Constitutive Relation (NCCR) model are observed and a way for identifying the physical solution is proposed. The NCCR model proposed by Myong is constructed from the generalized hydrodynamic equations of Eu, and aims to describe rarefied flows. In the non-equilibrium regime, the NCCR equations are more reliable than the Navier-Stokes equations. And the NCCR equations have an advantage of efficiency over the discrete velocity methods and stochastic particle methods. However, the NCCR model is a complicated nonlinear system. Many assumptions have been used in the schemes for solving the NCCR equations. The corresponding numerical methods may be associated with unphysical solution and instability. At the same time, it is hard to analyze the physical accuracy and stability of NCCR model due to the uncertainties in the numerical discretization. In this study, a new numerical method for solving NCCR equations is proposed and used to analyze the properties of NCCR equations. More specifically, the nonlinear equations are converted into the solutions of an objective function of a single variable. Under this formulation, the multiple solutions of the NCCR system are identified and the criteria for picking up the physical solution are proposed. Therefore, a numerical scheme for solving NCCR equations is constructed. A series of flow problems in the near continuum and low transition regimes with a large variation of Mach numbers are conducted to validate the numerical performance of proposed method and the physical accuracy of NCCR model.

Dynamics of fluid–structure interaction in paintbrush

Fluid–structure interactions are fundamental in both natural phenomena and industrial applications, particularly in dip-coating processes where withdrawal velocity and drainage dynamics are crucial. Understanding these interactions is essential for optimizing various processes, from enhancing the efficiency of industrial coatings to developing advanced materials with tailored properties. Here, we examine the capillary-induced dynamics of fiber bundles using paintbrush-like structures. We submerge fiber bundles in water and withdraw them at various velocities, observing that water trapped within the bundles leads to capillary-driven fiber assembly. Our experiments show that the bundle diameter after emergence increases with higher withdrawal speed due to viscous effects. We develop a theoretical model that accurately predicts the dynamics of fiber assembly driven by capillary and viscous flows. The mathematical model agrees well with our experimental results, demonstrating the complex interplay between capillary forces and fiber packing. We anticipate that our findings improve the understanding of fluid-structure interactions in fibrous media, providing physical insights that can be applied to more complex systems such as nanopattern collapse and nano/micro-manufacturing.