Turbulent Equation Research Articles

Anisotropic boundary layer mesh generation for reliable 3D unsteady RANS simulations

This paper proposes a Computational Fluid Dynamics (CFD) framework with the aim of combining consistency and efficiency for the numerical simulation of high Reynolds number flows encountered in engineering applications for aerodynamics. The novelty of the framework is the combination of a Reynolds-Averaged Navier–Stokes (RANS) model with an anisotropic mesh adaptation strategy handling arbitrary immersed geometries by building the corresponding boundary layer meshes. The numerical algorithm consists of robust and accurate solution of the unsteady incompressible Navier–Stokes equations supplemented with a Spalart–Allmaras turbulence model and boundary layer remeshing relying on a specifically designed metric. The flow solver is formulated as a Variational Multiscale (VMS) finite element method for the momentum balance and the incompressibility constraint, and as an upwind Petrov–Galerkin method for the nonlinear turbulent equation. The boundary layer remeshing strategy is flexible as it allows the adaptation of arbitrary coarse meshes by modifying the size and the orientation of elements along the immersed boundary to ensure a smooth gradation along the curvature of the body's geometry. The solver is capable of handling highly stretched anisotropic elements and is shown to successfully predict both mean and fluctuating drag/lift coefficients. Laminar and turbulent test cases in 2D and 3D are presented to assess the performance of this framework against experimental results relevant to external aerodynamics, including an airship and a flying drone.

Research on Turbulent Lubrication of Misaligned Journal Bearing Considering the Axial Motion of Journal

Abstract Based on the coordinate transformation method, the generalized turbulent lubrication equation considering the axial motion of the journal is derived. The finite-difference method is used to solve the generalized turbulent lubrication equation. The variations of turbulent lubrication performance with axial displacement for different axial movement velocity of the journal, journal misalignment angle, and initial mid-plane eccentricity ratio are obtained. The results show that when the axial movement velocity of the journal increases, the mid-plane eccentricity ratio of the bearing and the minimum film thickness remain unchanged, the average Reynolds number, maximum film pressure, load capacity, frictional power, and side leakage increases. As the axial displacement of the journal increases, the greater the misalignment angle of the journal, the greater the degree of misalignment, maximum film pressure, load capacity, and misalignment moment of the bearing. The greater the initial mid-plane eccentricity ratio, the greater the degree of journal misalignment, maximum film pressure, load capacity, frictional power, and misalignment moment.

A buoyancy–shear–drag-based turbulence model for Rayleigh–Taylor, reshocked Richtmyer–Meshkov, and Kelvin–Helmholtz mixing

A phenomenological turbulence model for Rayleigh–Taylor, reshocked Richtmyer–Meshkov, and Kelvin–Helmholtz instability-induced mixing is developed using a general buoyancy–shear–drag model. Analytic solutions to the simplified model equations corresponding to each instability are derived separately, which are then used to calibrate the model coefficients to predict specific values of the mixing layer growth parameters and exponents. The buoyancy–shear–drag equations for the bubble and spike mixing layer widths are then solved numerically, and a turbulent diffusivity (or viscosity) is constructed dimensionally as the product of the mixing layer width, h, and its time-derivative, dh/dt. Surrogate turbulent fields are then constructed by multiplying a presumed, approximate self-similar spatial profile by appropriate functions of h and dh/dt. Using several simplifying approximations, the turbulent diffusion equations satisfied by the mean mass fraction and mean shear velocity are solved analytically in a reference frame moving with the mean advection velocity, and it is shown that the mean fields evolve in space and time as expected. The explicit modeling and solution of turbulent transport equations (e.g., for the turbulent kinetic energy and its dissipation rate or a lengthscale) are not required. By using separate equations for the bubble and spike evolution, the model naturally captures the asymmetry induced by an increasing density contrast between the heavy and light fluids. The model also includes molecular dissipation and diffusion, and therefore, can describe transition to fully-developed turbulence if the initial turbulent diffusivity/viscosity is much smaller than the molecular diffusivity/viscosity. This model is applied to constant-acceleration Rayleigh–Taylor, impulsively reshocked Richtmyer–Meshkov, and variable-density Kelvin–Helmholtz mixing layers to demonstrate its utility. It is shown that the numerical solutions of the model calibrated using specific values of the instability growth parameters and exponents: (1) produces mixing layer widths in agreement with the expected self-similar growth power laws; (2) gives spatiotemporal profiles of turbulent fields that are expected and consistent with previous results; (3) predicts the expected power-law growths and decays of the spatially-integrated turbulent fields, and; (4) gives spatiotemporal profiles of the mean fields that are expected and consistent with previous results. This relatively simple model can be implemented numerically, and is expected to be useful for fundamental as well as astrophysical and inertial confinement fusion applications.

Improved Eddy-Viscosity Modelling of Turbulent Flow around Porous–Fluid Interface Regions

The RANS modelling of turbulence across fluid-porous interface regions within ribbed channels has been investigated by applying double (both volume and Reynolds) averaging to the Navier–Stokes equations. In this study, turbulence is represented by using the Launder and Sharma (Lett Heat Mass Transf 1:131–137, 1974) low-Reynolds-number $$k-\varepsilon $$ turbulence model, modified via proposals by either Nakayama and Kuwahara (J Fluids Eng 130:101205, 2008) or Pedras and de Lemos (Int Commun Heat Mass Transf 27:211–220, 2000), for extra source terms in turbulent transport equations to account for the porous structure. One important region of the flow, for modelling purposes, is the interface region between the porous medium and clear fluid regions. Here, corrections have been proposed to the above porous drag/source terms in the k and $$\varepsilon $$ transport equations that are designed to account for the effective increase in porosity across a thin near-interface region of the porous medium, and which bring about significant improvements in predictive accuracy. These terms are based on proposals put forward by Kuwata and Suga (Int J Heat Fluid Flow 43:35–51, 2013), for second-moment closures. Two types of porous channel flows have been considered. The first case is a fully developed turbulent porous channel flow, where the results are compared with DNS predictions obtained by Breugem et al. (J Fluid Mech 562:35–72, 2006) and experimental data produced by Suga et al. (Int J Heat Fluid Flow 31:974–984, 2010). The second case is a turbulent solid/porous rib channel flow to examine the behaviour of flow through and around the solid/porous rib, which is validated against experimental work carried out by Suga et al. (Flow Turbul Combust 91:19–40, 2013). Cases are simulated covering a range of porous properties, such as permeability and porosity. Through the comparisons with the available data, it is demonstrated that the extended model proposed here shows generally satisfactory accuracy, except for some predictive weaknesses in regions of either impingement or adverse pressure gradients, associated with the underlying eddy-viscosity turbulence model formulation.

Compressibility effects on the transition to turbulence in a spatially developing plane free shear layer

The compressibility effects on the transition to turbulence in a spatially developing, compressible plane free shear layer are investigated via direct numerical simulation using a high-order discontinuous spectral element method for three different convective Mach numbers of 0.3, 0.5, and 0.7. The location of the laminar–turbulent transition zone is predicted by the analyses of vorticities, Reynolds stresses, and the turbulent dissipation rate. In the turbulence transition and self-similar turbulence regions, the effects of compressibility on the flow properties, such as the velocity autocorrelation function, integral time scale, momentum thickness, Reynolds stress, and turbulent kinetic energy budget, are investigated. The compressibility effects on the onset and length of the turbulence transition zone are studied based on the analyses of such flow properties. The mean velocity, momentum thickness, and Reynolds stress profiles compare well with published experimental data. Vorticity contours and iso-surface of the second invariant of velocity gradient tensor identify the characteristic of flow structures. The two-point correlation functions of velocity components, the one-dimensional (1D) spanwise energy spectrum, and the balance of the turbulent kinetic energy transport equation validate the domain size and resolution of the adopted grid for turbulence simulation. An increase in the convective Mach number leads to a reduction in the sizes of the largest-scale structures, resulting in a significant decrease in Reynolds stresses and turbulence production. The onset of turbulence transition and the location where the transition completes shift downstream, while the length of the transition zone increases with increasing convective Mach number.

Analysis of impeller blade parameters and tip clearance of turboexpander in organic Rankine cycle system

AbstractThe optimization of a turboexpander can significantly improve the performance of the organic Rankine cycle (ORC) system, which can use in low‐temperature geothermal energy for high‐efficiency power generation. Therefore, this paper studies and optimizes the expander in the ORC of low‐temperature geothermal energy around 90°C. Firstly, the influence of impeller parameters on the performance of the expander is analyzed. Then, the grid is divided by the turbine mesh, and numerical simulation is performed by CFX. The gas state equation selects the Peng–Robinson equation, and the turbulence equation selects the SST model. Finally, the effect of tip clearance on the performance of the expander was studied. Research on impeller blade parameters shows that with the increase in the outer diameter of the impeller blade outlet, the isentropic efficiency of the expander decreases, and the output power increases. As the increase in the angle between the direction of the impeller blade and the meridian plane, the outlet velocity of the blade increases, the temperature decreases, and the efficiency and control of the turboexpander will decrease. The meridional section width has few effects on the performance of the expander. The study of the tip clearance by numerical simulation shows that the existence of tip clearance will cause clearance flow between pressure surface and suction surface that interferes with the mainstream direction. When the tip clearance increases from 0 mm to 1.2 mm, the efficiency of the expander is reduced by 8.9%.

A Numerical Investigation of Synthetic Jet Effect on Dynamic Stall Control of Oscillating Airfoil

At high angles of attack, the dynamic stall phenomenon could be appearing owing to the vortex shedding particularly in an oscillating airfoil. The consequences of this event are a considerable decrease in the lift and an increase in the drag as well as the pitching moment coefficients. The flow was assumed to be unsteady and turbulent at a Mach number 0.2 and for Reynolds number 1million. This research was done for a range of angle of attack 15o±10o. In order to carry out the numerical analysis of the problem, the 2-D compressible turbulent Navier-Stokes equations based on “Roe” scheme with second-order accuracy were solved. Turbulence modeling was carried out using the three-equation k-kL-ω model. Regarding the obtained results, it was observed that this flow control method had a significant ability in eliminating the dynamic stall. It was also revealed that the phase difference between the jet and airfoil oscillations is more affected by the dynamic stall decrement. In these changes, use of the SJ with 0.1 momentum coefficient, led to the highest amplitude of lift at φ=-30°, and the multiplication of drag amplitude and amplitude of moment coefficient at φ=-10° offered the best performance in addition to the considerable decrease.

Employing the Method of Characteristics to Obtain the Solution of Spectral Evolution of Turbulent Kinetic Energy Density Equation in an Isotropic Flow

This study aims to review the physical theory and parametrizations associated to Turbulent Kinetic Energy Density Function (STKE). The bibliographic references bring a broad view of the physical problem, mathematical techniques and modeling of turbulent kinetic energy dynamics in the convective boundary layer. A simplified model based on the dynamical equation for the STKE, in an isotropic and homogeneous turbulent flow regime, is done by formulating and considering the isotropic inertial energy transfer and viscous dissipation terms. This model is described by the Cauchy Problem and solved employing the Method of Characteristics. Therefore, a discussion on Linear First Order Partial Differential Equation, its existence, and uniqueness of solution has been presented. The spectral function solution obtained from its associated characteristic curves and initial condition (Method of Characteristics) reproduces the main features of a modeled physical system. In addition, this modeling allows us to obtain the scaling parameters, which are frequently employed in parameterizations for turbulent dispersion.

Numerical prediction of interaction between turbulence structures and vortex cavitation

ABSTRACTA novel numerical method to perform direct numerical simulations (DNS) of turbulent flow was established to predict the characteristics of compressible gas–liquid two-phase flow. A cavitation model was introduced to reproduce the phase change in the cavitation phenomena, and a DNS of wall turbulence of water with cavitation was performed. Vortex cavitation was observed in the wall turbulence. The growing cavity in the core of the vortex changed the vortex intensity. After vortex cavitation finished, the friction coefficient for the cavitating flow became lower than that of non-cavitating flow. The variation tendency of turbulent statistics corresponded to that of experimental measurements. For a small friction coefficient of the cavitating flow, the redistribution term in the streamwise component of the turbulent kinetic energy equation was reduced owing to the vortex cavitation. Consequently, it was proved that the occurrence of vortex cavitation modified the redistribution term, resulting in turbulence modification.

Terrain numerical simulation based on RANS/LES hybrid turbulence model

Based on the computational fluid dynamics (CFD) method, the terrain of 2 km near the proposed humanoid landscape bridge in Xiamen is taken as the research object. Firstly, the three-dimensional topographic map is established. Because the selected terrain is relatively flat, the bridge area is adopted in the form of spline curve. The boundary is extended, and the exponential rate of wind speed boundary conditions is applied to the boundary to avoid non-0m/s wind speed from the inflow near wall boundary. The RANS/LES hybrid mode is used to numerically analyze the external flow field of the terrain. The K - ε - v2 turbulence model of the three equations is used as the RANS part of the RANS/LES hybrid turbulence model. According to the Boussinesq hypothesis and dimension-analysis, the differential form equations of V2 are rationally simplified, and the algebraic form of the normal stress equation of V2 and the new turbulent viscosity equation are obtained and written into FLUENT in UDF form. By comparing and analyzing the standard K -ε turbulence model and the SSTκ-ε turbulence model, the wind speed amplification factor and the variation law of wind parameters along the height are studied, and the canyon effect and the reduction effect of the wind field are discussed. The results show that under the same calculation conditions, the mixed turbulence model can accurately reflect the change of airflow in the near-wall wall, and more accurately capture the wind field away from the bridge near the wall.

NUMERICAL MODELING OF AIR POLLUTION FROM DUMPS

Purpose. Rock dumps are long-term sources of air pollution. A significant amount of harmful substances enters the atmosphere. Particularly intense is the pollution of atmospheric air due to dusting dumps. An important task is the development of methods for predicting the dynamics of atmospheric air pollution during the emission of harmful impurities from dumps. When developing methods for predicting the dynamics of air pollution from dumps. It is very important to create a universal model that would allow on a single computing platform to calculate the process of atmospheric air pollution for various impurities that are released from dumps. Another important problem is the choice of the place for optimal placement of future dumps in order to minimize their negative impact on the environment. Methodology. Numerical modeling of the formation of atmospheric pollution zones near rock dumps is based on the application of the equation of turbulent dispersion of an impurity in the atmosphere. To solve the problem of choosing the rational location of the dumps, the conjugate equation is used. Numerical integration is carried out using an implicit difference scheme. Findings. Numerical models are developed. The first numerical model allows one to evaluate the effect of rock dumps on air pollution. The second numerical model allows, on the basis of a one-time calculation, to determine the rational location of the future dump from the point of view of its minimal impact on air pollution. Originality. Two numerical models are proposed that are based on the application of the turbulent diffusion equation and the conjugate equation. These models make it possible to quickly determine the effect of rock dumps on air pollution. The models take into account the convective transport of the pollutant, atmospheric turbulent diffusion, the intensity of the emission of impurities from the dump. Practical value. The developed numerical models are implemented in the form of program codes. These program codes can be used when performing serial calculations on computers of low and medium power, i.e. computer technology, which is available to organizations involved in environmental issues in technologically saturated regions. When carrying out calculations based on the constructed numerical models, typical initial data are used regarding the intensity of the emission of impurities from dumps, weather conditions characteristic of a specific region of the country.

Numerical Simulation of the Effect of Process Parameters on Cooling Rate and Secondary Dendrite Arm Spacing in High-Speed Twin Roll Strip Casting of Al–15 wt % Cu Alloy

Abstract Twin roll casting is a process used to produce thin strips of metals by continuously pouring melt on to rotating rolls. In order to make the process more productive and economical, high roll speed is recommended. The numerical simulation of high-speed twin roll casting is performed by analyzing fluid flow, heat transfer, and solidification behavior of Al–Cu hypo-eutectic alloy. The flow field, temperature, liquid fraction distribution, and cooling rate are analyzed by solving governing transport equations of continuity, momentum, energy, and turbulence. The low-Re Turbulence model is used to capture turbulence effects in the process and enthalpy-porosity technique used to account for the rise in viscosity due to phase change. The effect of melt pool height and roll velocity on average cooling rate along the strip surface is investigated. It is found that the increase in melt pool height and roll speed increases the average cooling rate along strip surface due to rise in heat transfer up to certain roll velocity but beyond that process fails due to breakout. The average cooling rate of process affects the microstructure and properties of strips. It is found that higher cooling rates result in a decrement of secondary dendrite arm spacing (SDAS) of 1 mm thin strip along strip surface results in the fine and homogeneous microstructure.

A Functional Approach to Vertical Turbulent Transport of Scalars in the Atmospheric Surface Layer

Eddy covariance has been the de facto method of analyzing scalar turbulent transport data. To refine the information available from these data, we derive a simplified version of the turbulent scalar-transport equation for the surface layer, which employs a more explicit form of signal decomposition and dispenses with Reynolds averaging in favour of an averaging operator based on the relevant scalar-flux driving variables. The resulting method, termed functional covariance, provides five areas of improvement in flux estimation: (i) Better representation of surface fluxes through closer correspondence of turbulent exchange with variations in the driving variables. (ii) An approximate 25% reduction in flux uncertainty resulting from improved independence of turbulent-flux samples. (iii) Improved data retention through less onerous quality control (stationarity) testing. (iv) Improved estimation of low-frequency flux contributions through reduced uncertainty and avoidance of driving-variable nonstationarity. (v) Potential elimination of flux-storage estimation when state driving-variables are used to define the functional-covariance flux averaging. We describe the important considerations required for application of functional covariance, apply both functional- and eddy-covariance methods to an example dataset, compare the resulting eddy- and functional-covariance estimates, and demonstrate the aforementioned benefits of functional covariance.

Large-time behavior of solutions to the three-dimensional turbulent flow equations

Large-time behavior of solutions to the three-dimensional turbulent flow equations

On the mean field equation with variable intensities on pierced domains

We consider the two-dimensional mean field equation of the equilibrium turbulence with variable intensities and Dirichlet boundary condition on a pierced domain −Δu=λ1V1eu∫ΩϵV1eudx−λ2τV2e−τu∫ΩϵV2e−τudxin Ωϵ=Ω∖⋃i=1mB(ξi,ϵi)¯u=0on ∂Ωϵ, where B(ξi,ϵi) is a ball centered at ξi∈Ω with radius ϵi, τ is a positive parameter and V1,V2>0 are smooth potentials. When λ1>8πm1 and λ2τ2>8π(m−m1) with m1∈{0,1,…,m}, there exist radii ϵ1,…,ϵm small enough such that the problem has a solution which blows-up positively and negatively at the points ξ1,…,ξm1 and ξm1+1,…,ξm, respectively, as the radii approach zero.

VarRhoTurbVOF: A new set of volume of fluid solvers for turbulent isothermal multiphase flows in OpenFOAM

The volume of fluid (VOF) method is a popular approach for multiphase flow modeling. The open-source computational fluid dynamics (CFD) software, OpenFOAM, implements a variety of VOF-based solvers and provides users a wide range of turbulence models. Since isothermal multiphase flows under the VOF framework belong to the variable-density incompressible flow category, the isothermal VOF-based solvers in OpenFOAM fail to use the correct turbulence models. varRhoTurbVOF is designed to solve this issue and with the hope to replace all the corresponding existing solvers in the future. With the object-oriented paradigm, varRhoTurbVOF guarantees the usability, reusability and maintainability of the codes. Aside from turbulence modeling, all other features in the original solvers are preserved in varRhoTurbVOF. Program summaryProgram Title: varRhoTurbVOFProgram Files doi:http://dx.doi.org/10.17632/4t8z8vzyvs.1Licensing provisions: GPLv3Programming language: C++Supplementary material:http://dx.doi.org/10.17632/7mp25kyb4p.4Nature of problem: Under the VOF framework, the flow of the isothermal mixture belongs to the variable-density incompressible flow category. For such flows, VOF-based solvers of OpenFOAM fail to construct the correct governing equations for turbulence modeling. varRhoTurbVOF contains a set of newly designed VOF-based solvers which could use the desired governing equations for turbulence quantities.Solution method: varRhoTurbVOF creates a new class for variable-density incompressible turbulence models, which allows reusing the existing turbulence model template classes. A set of VOF-based solvers are then created to be able to construct variable-density incompressible turbulence models.

Engineering equations for the filamentation collapse distance in lossy, turbulent, nonlinear media.

The propagation of high peak-power laser beams in real atmospheres has been an active research area for a couple of decades. Atmospheric turbulence and loss will induce decreases in the filamentation self-focusing collapse distance as the refractive index structure parameter and volume extinction coefficient, respectively, increase. This paper provides a validated analytical method for predicting the filamentation onset distance in lossy, turbulent, nonlinear media. It is based on a modification of Petrishchev's and Marburger theories. It postulates that the ratio of the peak power to critical power at range in turbulence is modified by a multiplicative, rather than additive, gain factor. Specifically, this new approach modifies the Petrishchev's turbulence equation to create the required multiplicative factor. This is necessary to create the shortened filamentation onset distance that occurs when a laser beam propagates through the cited nonlinear medium. This equation then is used with the Marburger distance and the Karr et al loss equations to yield the filamentation onset distance estimate in lossy, turbulent, nonlinear environment. Theory validation is done against two independent sets of computer simulation results. One comes from the NRL's HELCAP software and the other from MZA's Wave Train modeling software package. This paper also shows that once the zero-turbulence onset distance is set based on link loss, the addition of turbulence creates essentially the same PDFs at similar median distances for each loss case. This result had not been previously reported. This is the first quantitative comparison between closed form equations and computer simulation results characterizing filament generation in a lossy, turbulent, nonlinear medium.

Large eddy simulation of the tip-leakage cavitating flow with an insight on how cavitation influences vorticity and turbulence

• The tip-leakage cavitating flow is simulated by LES with Cartesian cut-cell mesh. • Three types of cavitating vortical flows, including TLV, TSV and IV, are analyzed from Euler/Lagrangian viewpoints. • Three stages for the spatial–temporal evolution of tip-leakage cavitating flow are discussed. • The physical mechanism for cavitation-turbulence interaction is clarified. Cavitation within a tip leakage flow remains a challenging issue in a variety of axial hydraulic machines. It is still not possible nowadays to predict cavitation occurrence in such a flow with acceptable accuracy. In the present study, we have carried out numerical simulations of a tip leakage cavitating flow, generated by a straight NACA0009 hydrofoil. We have used the LES method combined with the Schnerr–Sauer cavitation model. The numerical results agree well with experimental data. The evolution of the tip leakage cavitating flow, involving tip-leakage vortex (TLV), tip-separation vortex (TSV) and induced vortex (IV), is analyzed from Eulerian and Lagrangian viewpoints. The results show that the spatial evolution of the tip leakage cavitating flow can be divided into three stages: Stage Ⅰ, Independent development of the TLV and TSV; Stage Ⅱ, Fusion of the TLV and TSV; and Stage Ⅲ, Development of the IV and dissipation of the TLV. The Lagrangian coherent structures (LCSs) obtained from the numerical results indicate that the TLV cavitation significantly influence the local flow patterns. The vorticity transport equation was then used to further analyze the influence of the cavitation on the vortices. The results demonstrate that the stretching term dominates the TLV evolution and the dilatation term is responsible for the vorticity reduction inside the TLV cavity. The results also show how the cavitation influences the local turbulence and that the transport term in the turbulent kinetic energy equation influences the turbulence distribution near the TLV cavity.

Kinetic Theory of Parallel Momentum Transport due to Collisionless Electromagnetic Turbulence in Slab Geometry

A quasi-linear theory of parallel momentum transport due to electromagnetic turbulence is developed. The ion turbulent parallel momentum transport equation is derived from Vlasov kinetic equation. ...

Influence of the Shapes of Arcing Contacts on Prestrike Phenomenon in SF6 Circuit Breakers

Prestrike phenomenon in circuit breakers (CBs) can cause electromagnetic transients in power systems. Therefore, the investigation of this phenomenon has considerable importance. The scope of this paper is to investigate the effects and importance of arcing contact shape on the prestrike in SF6 gas CBs. In order to simulate the prestrike with high accuracy, a multi-physics model is developed precisely. In this model, equations of motion of the SF6 gas, the electric charge conservation equation, turbulent equations, heat transfer equation in fluids, and the governing equations of an electric circuit in which the CB works are considered simultaneously. The numerical study is performed on a 72.5-kV single pressure puffer type SF6 CB. Three different types of arcing contacts are investigated in this paper. Numerical results show that the arcing contact shapes have an important effect on the prestrike in gas type CBs.