Turbulent Flow Equations Research Articles

Similarity solutions of a class of unsteady laminar boundary layer

The study of laminar unsteady boundary layer flows is essential for understanding the transition from laminar to turbulent flow, as well as the origins of turbulence. However, finding solutions to this phenomenon poses significant challenges. In this study, we introduce a novel method that employs a similarity transformation to convert the two-dimensional unsteady laminar boundary layer equations into a single partial differential equation with constant coefficients. By applying this transformation, we successfully derive similarity solutions for flat plate boundary layer flow, expressed in terms of Kummer functions. For convergent boundary layer flow, we derive an approximate analytical solution that includes both shock wave and soliton wave solutions. The superposition of these solutions provides evidence for the existence of solitons or soliton-like coherent structures (SCS) within boundary layers. Additionally, this paper explores two- and three-dimensional laminar flows, as well as three-dimensional turbulent flow equations, revealing that they all incorporate third-order derivatives with respect to spatial coordinates. This finding suggests that all viscous fluid motions have the potential to exhibit solitons/like coherent structures (SCS).

The rheological kolmogorov scales of viscoelastic Oldroyd-B fluids

This article introduces a method to determine the Kolmogorov rheological scales for turbulent flow in Viscoelastic Oldroyd-B fluids. The findings reveal a noteworthy characteristic wherein the Kolmogorov rheological length is consistently smaller than that observed in Newtonian cases. Moreover, this length diminishes with an increase in the prominence of elastic effects. Leveraging these rheological scales, a detailed friction equation for turbulent flow in Oldroyd-B fluids is derived. The resultant friction relationship exhibits a high degree of agreement with existing theories. Notably, it delineates the Maximum Drag Reduction (MDR) scenario for the studied case (β=0.9). Additionally, the investigation delves into the onset of drag reduction effects, shedding light on the transitional phases in viscoelastic fluid flows.

Urban Air Pollution by Laser Photoacoustic Spectroscopy and Simplified Numerical Modeling of Gas Pollution in Urban Canyon

With rapid urbanization and industrialization, atmospheric pollution has emerged as a significant environmental challenge in Romania. Employing a laser photoacoustic spectroscopy detector, researchers analyzed ethylene, benzene, and toluene simultaneously across three distinct environmental settings in the country's southern region. This investigation spanned from March to August 2021, covering both spring and summer seasons. Measurements were taken at a breathing height of 1.5 meters above ground level. The highest concentrations of ethylene (116.82 ± 82.37 ppb), benzene (1.13 ± 0.32 ppb), and toluene (5.48 ± 3.27 ppb) were recorded at measurement point P1, situated within the city amidst residential buildings during the summer season. Additionally, the highest ozone levels (154.75 ± 68.02 ppb) were observed at point P3, located in an industrial area, during the summer. The behavior of gas concentrations is influenced by meteorological factors such as temperature, wind speed, and direction. The high toluene/benzene ratio suggests that traffic and industrial emissions are the primary sources of these pollutants. Notably, benzene and ozone concentrations exceeded prescribed limit values based on the measurements. Concurrently, a numerical model was employed to assess the impact of greenery on mitigating pollution in urban canyons. Specifically, the study focused on how wind velocity affects the dispersion of benzene pollutants in a street canyon. This study's governing equations utilized for air pollutant flow were the Reynolds-averaged Navier–Stokes (RANS) equations for compressible turbulent flow and moisture transport in air, implemented through Comsol software.

Rock Breaking Mechanism and Process Optimisation of Jet Cutting Basic Roof Rock under Submerged Jet Condition

The destruction of rock under the condition of a close submerged jet has become a hot topic of scientific research and engineering application in the past decade. With the unremitting efforts of a large number of experts and scholars around the world, gratifying progress has been made in the research of computational fluid dynamics (CFD) on the internal and external flow fields of the jet nozzle, the theoretical derivation of rock mechanics on the fracture initiation and propagation criteria of hydraulic fracturing, and the numerical simulation of jet erosion mechanism under the coupling of fluid and solid fields, however, for the rock mechanics hydraulic fracturing cutting engineering scale of non-oil drilling fracturing technology, the research on the fluid-solid coupling boundary conditions of fracturing fluid and hard dense rock under the flow state conditions of the submerged field inside and outside the borehole is not sufficient. In the calculation of the fluid-solid coupling boundary flow field under the non-submerged jet state, the control equation with Reynolds number between 2300-4000 shall be selected, while it belongs to the laminar flow state in the stage of hole sealing and pressurised fracturing. Therefore, Von-Mises equivalent plastic stress is selected in the mechanical model to calibrate the failure state of the rock-solid boundary, and the control equations of laminar flow and turbulent flow are selected to calibrate the fluid boundary. The mechanism of different stages of rock breaking by hydraulic fracturing jet can be further analysed in detail, and Comsol 6.0 multi-physical field simulation software is selected for verification. The research results will help deepen the understanding of rock breaking mechanism by jet and optimise the selection of parameters for field construction.

Numerical Modeling of Melt Flow and Mixing in an Inductively Stirred Ladle and Comparison with Gas and Mechanical Stirring

Fluid flow and mixing in an inductively stirred ladle are modeled mathematically. The effect of two modes of electromagnetic stirring on motion is simulated; one produces a predominantly swirling motion (as in mechanical, paddle‐stirred ladle), while the other generates an upwelling, toroidal motion (as in gas‐stirred ladle). Two different ANSYS software programs, namely, Maxwell‐3D and Fluent, are applied to carry out numerical simulations. In the former, 3D Maxwell's equations are solved to estimate, a priori, the distribution of Lorentz force within the melt phase due to the imposed three‐phase, alternating current excitation of copper coils, while in the latter, time‐averaged magnetohydrodynamic (MHD) turbulent flow equations are solved to yield the electromagnetically driven flow field. Model predictions are validated against published results and it is shown that the electromagnetic as well as turbulent‐MHD models developed in this work are robust and perform sufficiently accurately. Finally, numerically predicted results together with relevant published work indicated that at equivalent specific input stirring power and identical vessel shape and volume, mixing is fastest in electromagnetic stirring with a traveling magnetic field (producing an upwelling toroidal motion), followed by mechanical agitation, gas stirring, and electromagnetic stirring with a rotating magnetic field (inducing swirling flow).

Investigation of the performance of a horizontal-axis dual rotor wind turbine

Recent years have seen a rise in interest in wind energy as a useful alternative to harmful energies like fossil fuels. The dual rotor wind turbine (DRWT) offers more rapid rates of wind energy extraction. The current study intends to compare the performance of the turbine with and without the addition of a second rotor. Additionally, it examines how tip speed ratio and phase shift angle will affect DRWT performance. Realizable k-shear stress transport turbulence models are used to solve the three-dimensional, turbulent, stable, and incompressible flow equations for the performance of dual-rotor wind turbines. Domain-independence tests and an impartial mesh test are run to assess the results and ensure their accuracy. The researcher relies on previous studies while constructing the single rotor wind turbine model. This model uses an S826 airfoil. The front and rear rotors are given streamlined representations using ANSYS, according to the researcher. The independent mesh test indicates that the mesh density has 11.5 million elements. The experiment's results show that the DRWT has a significant effect on the efficiency of wind energy.

Data-driven discovery of turbulent flow equations using physics-informed neural networks

In the field of fluid mechanics, traditional turbulence models such as those based on Reynolds-averaged Navier–Stokes (RANS) equations play a crucial role in solving numerous problems. However, their accuracy in complex scenarios is often limited due to inherent assumptions and approximations, as well as imprecise coefficients in the turbulence model equations. Addressing these challenges, our research introduces an innovative approach employing physics-informed neural networks (PINNs) to optimize the parameters of the standard k−ω turbulence model. PINNs integrate physical loss functions into the model, enabling the adaptation of all coefficients in the standard k−ω model as trainable parameters. This novel methodology significantly enhances the accuracy and efficiency of turbulent flow simulations, as demonstrated by our application to the flow over periodic hills. The two coefficients that have been modified considerably are σω and α, which correspond to the diffusion and production terms in the specific dissipation rate equation. The results indicate that the RANS simulation with PINNs coefficients (k−ω−PINNs simulation) improves the prediction of separation in the near-wall region and mitigates the overestimation of turbulent kinetic energy compared to the base RANS simulation. This research marks a significant advancement in turbulence modeling, showcasing the potential of PINNs in parameter identification and optimization in fluid mechanics.

A fast three-dimensional flow field prediction around bluff bodies using deep learning

This study presents a deep learning approach for predicting the flow field in the incompressible turbulent three-dimensional (3D) external flow around right-rhombic prism-shaped bluff bodies. The approach involves treating the nodes of the unstructured grid in the computational fluid dynamics domain as a point cloud, which is used as an input for a neural network. The neural network is trained to map the spatial coordinates of the nodes to the corresponding velocity and pressure values in the domain. The PointNet, a reliable solution in 3D vision tasks, is selected as the neural network architecture. Implementing this architecture makes it feasible to use irregular positions of the nodes of an unstructured grid as an input without needing interpolation. A dataset, comprising 3511 cases, is generated for training and testing the network. This is achieved by changing the geometric parameters of a right rhombic prism and varying its angle to the flow stream. Then, the continuity and momentum equations for turbulent flow are solved using a solver. Given the need for a larger number of points to accurately represent a 3D flow, the architecture of PointNet is modified. This modification involves adding extra layers and adjusting the number of neurons inside the layers to overcome this challenge. Once the training is completed, given the unseen samples from the test dataset to the model, our model can predict the velocity and pressure of the flow field at a speed that exceeds our conventional solver by several orders of magnitude with a maximum relative error of 4.58%.

Stochastic Navier–Stokes Equations for Turbulent Flows in Critical Spaces

In this paper we study the stochastic Navier–Stokes equations on the d-dimensional torus with transport noise, which arise in the study of turbulent flows. Under very weak smoothness assumptions on the data we prove local well-posedness in the critical case Bq,pd/q-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {B}^{d/q-1}_{q,p}$$\\end{document} for q∈[2,2d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q\\in [2,2d)$$\\end{document} and p large enough. Moreover, we obtain new regularization results for solutions, and new blow-up criteria which can be seen as a stochastic version of the Serrin blow-up criteria. The latter is used to prove global well-posedness with high probability for small initial data in critical spaces in any dimensions d⩾2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\geqslant 2$$\\end{document}. Moreover, for d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}, we obtain new global well-posedness results and regularization phenomena which unify and extend several earlier results.

Improvement of the two-fluid momentum equation for turbulent bubbly flows

A two-fluid model is widely employed to predict gas–liquid two-phase flows. Conventional two-fluid equations, derived by assuming that all phases are interpenetrating continua, occasionally yield impractical results. For example, the bubble velocity is predicted to be higher than the water velocity for a horizontal fully developed bubbly flow. One solution for this irrational result is two-fluid momentum equations based on the motion of fluid particles. This study revisits the particle-motion-based two-fluid equations. Previously, it was validated only with respect to the difference between the bubble and liquid velocities. Recently, we observed an erroneous prediction of the void fraction distribution by the existing particle-motion-based momentum equations. This study, through rigorous analysis, discovered that this error could be resolved by neglecting the turbulent kinetic energy term in the particle-motion-based equation for the gas phase. Subsequently, the revised momentum equations were applied to the vertical and horizontal air–water bubbly flows. The conventional and revised momentum equations produced similar results for upward bubbly flows because the relative velocity between the bubble and water phases was mainly governed by buoyancy. For horizontal bubbly flows, the revised momentum equations led to a downward shift in the liquid velocity distributions, which was attributed to the modified momentum diffusion term. At a higher modification factor of turbulent dispersion, the velocity distributions exhibited an upward shift. We conclude that the optimization of physical models should be based on the revised two-fluid momentum equations.

Approximate Analytical Solution for Turbulent Flow in Channel

In this work, the approximate analytical solutions of the Generalized Hydrodynamic Equations (GHE) for turbulent flow in channel is presented. The general solution is a superposition of the laminar (parabolic) and turbulent (superexponential) solutions. The analytical solution compares well with the experimental data by Van Doorne (2007) and Nikuradse (1933).It is proposed to explain the nature of turbulence as oscillations between the laminar (parabolic) and turbulent (superexponential) solutions. The suggestion is confirmed by comparing the results for turbulent velocity fluctuations of the analytical formula with the experiment by Van Doorne (2007). The Navier-Stokes equations do not have the superexponential solution.The obtained analytical solution provides a complete structure of the turbulent boundary layer that compares well with the experiments by Wei and Willmarth (1989).

Time-mean equation and multi-field coupling numerical method for low-Reynolds-number turbulent flow in ferrofluid

Significant breakthroughs in the theory and applications of ferrofluid have broadened their usage in areas such as sealing and actuation. However, the development of numerical ferrofluid dynamics has been slow owing to inadequate multi-field coupling techniques and incomplete equations for turbulence in ferrofluid flow. The understanding of low-Reynolds-number turbulent flow mechanisms in ferrofluid at small scales, particularly in sealing and actuation applications, remains limited, therefore hindering further advancements. This article delves into the turbulent flow equations for general fluid and discusses the influence of different-scale vortices on the average fluid motion. An anisotropic turbulence model was introduced and verified using fluid flow around a cylinder. Magnetic and flow fields data were coupled through node ranking and interpolation methods. By introducing the interaction force of magnetic dipoles, the turbulent equations were refined within Euler grids, thereby establishing a numerical model for the turbulent motion of ferrofluids influenced by multiple fields. This model was applied to study the deformation and migration processes of ferrofluid under an external magnetic field. The variations in ferrofluid motion under magnetic forces were encapsulated, and macroscopic flow comparisons were made through experiments, which demonstrated good consistency. This research provides new methods and ideas for use in ferrofluid numerical studies. Additionally, it offers valuable technical support that can aid in developing industrial products such as sealing and driving devices based on ferrofluids.

Numerical study of external turbulent flow

Flow around a cylinder on a fixed-bed open- channel is numerically studied. A CFD code is executed to analyze the flow fields and the kinetic energy field around the cylinder. A system of equations of two-dimensional and horizontal flow is obtained by the integration of the continuity and momentum equation for turbulent flow using (k -) standard model. The turbulent stresses are then expressed with the eddy-viscosity concept. The numerical simulation was done for four flows cases with different velocities. From the numerical results, the expected alteration of the flow field due to the presence of the cylinder is evident, notably the wake behind the cylinder. The measured velocities in the longitudinal and transversal directions and the kinetic energy are compared with those of the numerical simulation.

Proposing novel approaches for solar still performance enhancement by basin water heating, glass cooling, and vacuum creation

Solar stills are widely recognized for their cost-effective approach to purifying brackish water. However, their ability to produce drinkable water has been limited. As an innovative solution, this paper offers the integration of basin water heating, glass cooling, simultaneous heating and cooling, and operating under negative gauge operational pressures, aiming to create a vacuum within the range of 60–120 kPa. The performance of domestic-sized single-slope solar stills is analyzed using two-dimensional numerical simulations. The analysis encompasses the governing equations for mass, momentum, and energy conservation, as well as additional equations for species transport, radiation, and turbulent flow. Additionally, this investigation combines numerical simulations with a machine learning prediction algorithm, spanning pressures of 10–60 kPa. The results demonstrate that the simultaneous use of glass cooling and water heating mechanisms leads to the most significant improvement in performance, achieving an impressive enhancement of 152.69 %. Moreover, basin water heating, operational pressure reduction, and glass cover cooling are found to enhance performance by 112.72 %, 31.82 %, and 27.16 %, respectively. Overall, this study underscores the potential to optimize solar still efficiency through a range of operational strategies, ultimately contributing to enhanced potable water production.

A novel model for air velocities in airblast events in block caving mining

An airblast is a sudden collapse of a large volume of rock that causes air to rapidly flow through an underground opening as a result of air being compressed in a small space. Through the solution of the Ergun equation for turbulent flows in porous media, we have modeled air gusts in this paper. To resolve the issue, we use the finite element method (FEM). At a drawbell’s associated drawpoints, air flows and velocities are calculated. A series of numerical simulations are run while taking into account different arrangements of the average size of the rocks in the fractured zone and the air gap length, Lag, which is implicitly represented as the boundary condition for incoming air velocity at the muckpile. We propose a novel mathematical function that, using scaling arguments, expresses the maximum air gust velocity at the drawpoints as a function of the air gap height and average particle diameter. Designing geomechanical risk mitigation strategies for airblast events during block caving mining can take advantage of the findings in this paper.

Analysis of melting of phase change material inserted a block via impinging turbulent slot jet

PurposeThe purpose of this study is to do a numerical analysis of the jet to a body filled with phase change material (PCM). The melting of the PCM filled body was investigated by the hot jet flow. Four different values of the Reynolds number were taken, ranging from 5 × 103 = Re = 12.5 103. Water, Al2O3 1%, Al2O3 2% and hybrid nanofluid (HNF; Al2O3–Ag mixture) were used as fluid types and the effects of fluid type on melting were investigated. At 60 °C, the jet stream was impinged on the PCM filled body at different Reynolds numbers.Design/methodology/approachTwo-dimensional analysis of melting of PCM inserted A block via impinging turbulent slot jet is numerically studied. Governing equations for turbulent flow are solved by using the finite element method via analysis and system fluent R2020.FindingsThe obtained results showed that the best melting occurred when the Reynolds number increased and the HNF was used. However, the impacts of using alumina-water nanofluid were slight. At Re = 12,500, phase completion time was reduced by about 13.77% when HNF was used while this was only 3.93% with water + alumina nanofluid as compared to using only water at Re = 5,000. In future studies, HNF concentrations will change the type of nanoenhanced PCMs. In addition, the geometry and jet parameters of the PCM-filled cube can be changed.Originality/valueEffects of impinging jet onto PCM filled block and control of melting via impinging hot jet of PCM. Thus, novelty of the work is to control of melting in a block by impinging hot jet and nanoparticles.

Theoretical analysis of unsteady buoyant turbulent heat and mass transport from a vertical plate using LRN k-ϵ model

The contemporary analysis has been developed to investigate the transient natural convective combined heat and mass transport of a turbulent flow adjacent to the plate by utilizing the low Reynolds number (LRN) k-ϵ model. The research problem on turbulent flow is assumed to be two dimensional and viscous incompressible. The governing turbulent equations such as continuity, momentum, energy, concentration, turbulent kinetic energy (TKE) and dissipation rate of TKE are considered as per the flow geometry. Due to the non-availability of analytical or direct numerical techniques, the corresponding highly nonlinear PDE’s are solved by adopting the standard implicit type of finite difference method namely Crank–Nicolson scheme. Also, this scheme is unconditionally stable and helps to keep the governing turbulent flow equations in discretized form and are solved via tridiagonal matrix algorithm. The simulated results for several control parameters including turbulent Reynolds number are graphically displayed and also analyzed the flow profiles. To understand the physical phenomenon of turbulent flow, the authors shown the turbulent behavior of average momentum, energy and mass transfer rates. It has been noted that, the average temperature and concentration fields boosted with rising values of . Also, TKE and dissipation rate of TKE profiles increase with enhancing turbulent values. In addition, the simulated turbulent flow results from the LRN k-ϵ model are compared with usual laminar flows as a special case and found to be in respectable agreement.

Numerical study of transient convective turbulent boundary layer flow along a vertical plate: analysis of kinetic energy and its dissipation rate

ABSTRACT The present article numerically investigates the turbulent buoyancy-driven (natural convection) flow along a vertical plate with a low Reynolds turbulence two-equation k-ϵ model. The deployed turbulence model is appropriate for low Reynolds number (LRN) turbulent flow adjacent to a solid boundary, and the turbulence kinetic energy (TKE) and dissipation rate of TKE are estimated using the momentum equations and are solved simultaneously with the mean flow conservation equations. Two-dimensional time-dependent viscous incompressible turbulent flow is simulated. This flow domain is governed by a highly non-linear group of partial differential equations, namely the time-averaged continuity, momentum, and energy, and also the flow property is determined through TKE, and dissipation rate of TKE equations. Since these equations are not solvable using analytical methods, an implicit second-order finite difference method is employed to solve the governing turbulent flow equations numerically. The simulated time-averaged velocity, temperature, TKE, and dissipation rate of TKE profiles along with friction factor and heat transfer rate are computed for different values of turbulent Reynolds and Prandtl numbers. Average velocity, temperature, turbulence energy, and dissipation rate under both transient and steady-state conditions are decreased with increment in . There is a decrement in average transient velocity and turbulence energy rates with increasing values, whereas the average temperature profile increases. Average steady temperature and turbulence energy profiles are also enhanced with rising values of . The majority of researchers focus on laminar flow and thus far turbulent flow has mainly been addressed through experiments only. To better elucidate the flow in this condition, the novelty of the present study is to utilize a numerical FDM procedure to probe more deeply into the turbulence characteristics in buoyancy-driven (natural convection) flow along a vertical plate with a low Reynolds turbulence two-equation k-ϵ model, as this topic is fundamental to many applications in science and engineering. Furthermore, the results of the turbulent flow simulations obtained from the current model are corroborated with existing results for the special case of laminar flow and a good correlation is achieved.

A comparison study of the behaviors of single-phase turbulent flow at low to moderate Reynolds numbers through a vertical pipe: 3D counters analysis

The study presented three-dimensional (3D) analysis of water's upward flowing through the vertical pipe under turbulent characteristic considerations. Both numerical constructed and improved the model of 3D for cylindrical coordinates of governing equations for incompressible turbulent flow with the Reynolds Average Navier-Stokes (RANS) model using the improved constants of the (k–ε) type. The present model is then compared with a previous study to give the feasibility of the present single-phase turbulent flow parameters. The pipe length is tested to measure how much it affected the turbulent parameters though one of the expected factors is the turbulent time scale. On the other hand, the model is numerically examined to determine the velocity profile, shear rate, and surface deformation of the water domain. While the pressure distribution, turbulent kinetic energy, and turbulent dissipation rate, these parameters are classified as the mechanic's system factors. The simulation is done with wide software used to simulate industrial is COMSOL 5.4 Multiphysics software. The results obtained increased the velocity of three inlet water velocities used ranging from (0.087, 0.105, and 0.123 m/sec) of upward flow. High fluctuation in the water flow moves along the entire pipe length and it can notice the sensitivity to any change in water properties or mechanical properties. The liquid upward flow in turbulent conditions is suffered from many characteristics such them related to liquid properties and others related to the mechanics of the application through the systems. The interaction between the fluid film (fluid boarded the pipe inner diameter) has been observed by the shear rate and liquid surface deformation

Numerical investigation on the performance of a Solar Chimney Power Plant System (SCPP)

AbstractIn this work, a parametric two‐dimensional computational fluid dynamics (CFD) analysis of solar chimney power plant was performed to describe the physical mechanism in order to establish realistic predictions for the operation of such plants in different location in Algeria. The developed numerical model includes a solar radiation model within the collector; an energy storage model; an air flow and heat transfer model, and a turbine model. It is based on the solution of the Navier‐Stokes equation for turbulent flow using the standard k − ϵ model. The Boussinesq model was adopted in this simulation for the density of the fluid. The boundary conditions were adapted according to the selected sites and the effect of this on the SCPP performance was studied. The numerical simulation was performed using the commercial computational fluid dynamics (CFD) code “Ansys Fluent 19.2”.