Turbulent Kinetic Energy Equation Research Articles

On a new one‐dimensional k$$ \boldsymbol{k} $$–ε$$ \boldsymbol{\varepsilon} $$ turbulence closure for building‐induced drag

AbstractVarious urban canopy parameterizations (UCPs) have been developed in the last two decades to take into account the modifications induced by buildings on mean flow and turbulent fields in mesoscale meteorological models, the typical spatial resolution of which cannot resolve urban structures explicitly. In particular, several multilayer UCPs have been proposed, successfully reproducing wind‐flow characteristics in the urban environment. However, they often rely on length scales for the calculation of the eddy viscosity and the dissipation rate, which need to be tuned for various urban configurations. The main objective of this work is to address this shortcoming, by developing a new one‐dimensional turbulence closure that takes building‐induced turbulence into account independently of turbulence length scales. This model directly solves not only the equation for turbulent kinetic energy but also the equation for its dissipation rate (– model). Similar closure schemes have been successfully adopted for vegetated canopies, but their applicability to urban canopies is still unknown. The performance of the new – model, with additional sources and sinks for wind speed, turbulent kinetic energy, and dissipation rate, is tested by means of single‐column simulations in idealized urban areas, using different building packing densities. Results are in good agreement with spatially averaged building‐resolving CFD simulations. In particular, vertical profiles of mean and turbulent variables show better results with respect to simulations using turbulence closures adopting a parameterization of turbulence length scales. The best improvements are obtained for wind speed, reducing errors to half the typical values for standard UCP for high packing densities, and for the dissipation rate. Furthermore, besides the enhancement in the reproduction of the mean flow for urban areas, the proposed turbulence closure does not need additional tuning of coefficients depending on the packing density, resulting in a more general and efficient scheme.

CFD of Fully Developed Channel Flow

Turbulent flow, characterized by random and chaotic fluid motion, is a common occurrence in many engineering applications, such as aerospace, wind turbines, hydroelectric power plants, and chemical mixing processes. In these applications and others, the turbulent flows often involve complex geometries, multiple scales of motion, and nonlinear interactions, which make it challenging to simulate and predict turbulent flow behaviour accurately. As a result, turbulence modelling plays a crucial role in these applications by providing an efficient and cost-effective way to model the complex physics of turbulent flows. Several turbulence models have been developed and are commonly used in computational fluid dynamics (CFD) simulations to predict the behaviour of turbulent flows. In this work, the Navier-Stokes equations and transport equations of turbulent kinetic energy and turbulent dissipation are solved using finite difference approximation. In-house MATLAB code is developed in this work to solve the two-dimensional, incompressible, fully developed turbulent channel flow for Reynolds number ranged from 6000 to 12000.

Mathematical modelling and simulation of leak detection system in crude oil pipeline

A first-order differential leak detection model that accurately detects leaks in a crude oil pipeline is presented. This model incorporates a leak factor KL in the axial direction, which is simulated by applying the finite element method of numerical solution using COMSOL multi-physics software. Additionally, the model includes the transport equation for turbulent kinetic energy and the rate of kinetic energy model. Eigenvalues for velocities and pressures were determined and plotted against time for various pipe segments. The system is stable when the Eigenvalue is zero, but a leak is declared when the Eigenvalue for pressure or velocity is less than one. The study shows that pressure measurements are more sensitive parameters for detecting leaks than velocity measurements, and the sinusoidal waveform characterizes leak behaviours for velocity.

Production and transport of turbulent kinetic energy from premixed flames subjected to mean pressure gradients

The influence of mean favorable pressure gradients on the production and transport of turbulent kinetic energy is experimentally investigated in a premixed bluff-body combustor. The combustor wall geometry is modified into converging, straight, or diverging configurations to experimentally alter the mean pressure gradient in the reacting flow field. For each configuration, turbulent kinetic energy production and transport dynamics are characterized with high-speed particle image velocimetry and simultaneous CH* chemiluminescence imaging. For all cases, the premixed reaction increases the turbulent kinetic energy and integral length scales in the flow. However, increasing the magnitude of the mean favorable pressure gradient increases the turbulent kinetic energy produced by the flame. The source of increased turbulent fluctuations from reactants to products is first investigated with a steady flow energy analysis, and is accompanied with a Reynolds decomposed investigation of the kinetic energy in the flow. As the mean pressure gradient increases, the turbulent kinetic energy in the flow increases due to stronger magnitudes of flame-generated turbulence in the form of baroclinic torque. The evolution of the flame-generated turbulence is examined by decomposing the transport equation of turbulent kinetic energy. Increasing the mean favorable pressure gradient augments the largest magnitude transport terms, and the budget of turbulent kinetic energy is primarily driven by pressure gradient work and viscous dissipation. The results here differ from previous direct numerical simulation (DNS) studies, and should ultimately motivate additional research to aid the development of computational models which can better predict turbulent flame-flow dynamics in confined environments with mean pressure gradients.

Role of turbulent kinetic energy modulation by particle–fluid interaction in sediment pick-up

Reliable prediction of the erosion rate of sediment beds is important for many applications in coastal and river engineering. Theoretical understanding of empirically derived scaling relations is still lacking. This applies in particular for the scaling anomaly between low and high Shields number conditions. In this work, the erosion process is studied from the perspective of the phase-averaged turbulent kinetic energy (TKE) equations. The multi-phase TKE equations are written in a form that allows for a direct comparison with the TKE equation that appears for a stratified single-phase flow under the Boussinesq approximation. This reveals that next to buoyancy destruction, several other TKE modulation mechanisms become important at high Shields numbers and concentrations. Two scaling laws are derived for both moderate and high Shields numbers, and are tested against a wide range of experimental data.

Field inversion for transitional flows using continuous adjoint methods

Transition modeling represents one of the key challenges in computational fluid dynamics. While numerical efforts were traditionally devoted to either improving Reynolds-averaged Navier–Stokes-based turbulence modeling or developing scale-resolving simulations, cautious attention has been recently given to field inversion and machine learning techniques. This paper discusses an updated development of field inversion model for transitional flows based on k–ω shear stress transport model using the continuous adjoint approach, instead of the typical discrete adjoint method. The original model is modified by multiplying the production term of the turbulent kinetic energy equation by a spatially varying discrepancy function η(x). The adjoint equations and the relevant boundary conditions are specifically derived and integrated in OpenFOAM. The present model is validated using two zero pressure-gradient and four non-zero pressure-gradient from flow-over-flat-plate T3-series test cases. The gradient descent method is employed in the optimization process to minimize the discrepancy in the calculated shear stress. The inferred solution indicates a smooth transition to turbulence at the reported critical Reynolds numbers. The optimized model significantly improves the predictions of skin-friction coefficients, originally incorporated in the objective function. To demonstrate the usefulness of the present approach, the investigation is further extended to determine both velocity and shear Reynolds-stress profiles, which to our knowledge has not been reported before. Furthermore, a reduction in the percentage error from 50.2% to 7.3% is well observed in the predicted boundary layer thickness considering the laminar regime in the T3C5 test case.

Effects of kinematic and magnetic boundary conditions on the dynamics of convection-driven plane layer dynamos

Rapidly rotating convection-driven dynamos are investigated under different kinematic and magnetic boundary conditions using direct numerical simulations. At a fixed rotation rate, represented by the Ekman number$Ek=5\times 10^{-7}$, the thermal forcing is varied from$2$to$20$times its value at the onset of convection ($\mathcal {R}=Ra/Ra_c=2\unicode{x2013}20$, where$Ra$is the Rayleigh number), keeping the fluid properties constant ($Pr=Pr_m=1$, where$Pr$and$Pr_m$are the thermal and magnetic Prandtl numbers). The statistical behaviour of the dynamos, including the force balance, energetics, and heat transport, depends on the boundary conditions that dictate both the boundary layer and the interior dynamics. At a fixed thermal forcing ($\mathcal {R}=3$), the structure and strength of the magnetic field produced by the dynamos, especially near the walls, depend on both velocity and magnetic boundary conditions. Though the leading-order force balance in the bulk remains geostrophic, the Lorentz force becomes comparable to the Coriolis force inside the thermal boundary layer with no-slip, perfectly conducting conditions. In this case, a term signifying the work done by the Lorentz force in the turbulent kinetic energy (TKE) equation is found to have some components that extract energy from the velocity field to produce the magnetic field, while some other components extract energy from the magnetic field to produce TKE. However, with no-slip, pseudo-vacuum conditions, all the components of the work done by the Lorentz force perform unidirectional energy transfer to produce magnetic energy from the kinetic energy of the fluid to sustain dynamo action. We find heat transfer enhancement in the rotating dynamo convection, as compared with non-magnetic rotating convection, with the peak enhancement lying in the range$\mathcal {R}=3\unicode{x2013}4$. For free-slip conditions, in the absence of an Ekman layer, the dynamo action may alter the heat transport significantly by suppressing the formation of large-scale vortices. However, the highest heat transfer enhancement is found at$\mathcal {R}=3$with no-slip, perfectly conducting walls, which can be attributed to a local magnetorelaxation of the rotational constraint due to enhanced Lorentz force inside the thermal boundary layer.

Predicting transition with wall-distance-free SST k-[formula omitted] model

Unlike the correlation-based γ–Reθ transition model, an exceptionally simplified approach is applied to extend the wall-distance-free (WDF) model toward transition predictions. An anisotropic stress-intensity parameter as a function of eddy-to-laminar viscosity ratio is introduced, preserving the “flow-structure-adaptive” characteristic. The prospective parameter is included with the constitutive relation for the eddy-viscosity. The proposed formulation is intrinsically plausible, having a dramatic influence on the prediction of bypass, separation-induced and natural transitions. The newly devised transitional WDF (TWDF) model is further modified to compute separation-induced transition over a low-Reynolds number (LRN) airfoil. The curiosity toward airfoil performance at an LRN has been executed with growing attention to marine autonomous systems. An extra viscous-production term Pklim is added to the turbulent kinetic energy (k) equation to ascertain proper development of k at the viscous sublayer region. Reliance on this Pklim is likely to be suitable for forecasting separation-induced transition over an LRN airfoil. Results illustrate that the TWDF model sustains decent agreement with the four-equation γ–Reθ transition model.

An efficient discrete unified gas-kinetic scheme for compressible turbulence

In this paper, we develop an efficient Boltzmann-equation-based mesoscopic approach to simulate three-dimensional (3D) compressible turbulence, using reduced Gauss–Hermite quadrature (GHQ) orders by redefining the second distribution in terms of the total energy in the double distribution function approach. This allows the use of two sets of 3D off-lattice discrete particle velocity models, namely, a 27 discrete velocity model of the seventh-order GHQ accuracy (D3V27A7) combined with a 13 discrete velocity model of the fifth-order GHQ accuracy (D3V13A5), to achieve full consistency with the Navier–Stokes–Fourier system. The source terms in the Boltzmann–Bhatnagar–Gross–Krook system are designed to adjust both the Prandtl number and bulk-to-shear viscosity ratio. Compressible decaying homogeneous isotropic turbulence (DHIT) is simulated at low and moderate turbulent Mach numbers to validate our code. It is observed that the simulation results are in good agreement with those in the existing literatures. Furthermore, the terms in the transport equation of turbulent kinetic energy are analyzed in detail, to illustrate four different transient stages from the initial random flow field to the developed DHIT. It is shown that the transient pressure-dilatation transfer happens rapidly, while the small-scale vortical structures take a longer time to establish physically. Compared to the existing literatures, our approach represents the most efficient mesoscopic scheme for compressible turbulence under the double distribution function formulation.

Model of interfacial term in turbulent kinetic energy equation and computation of dissipation rate for particle-laden flows

In this paper, we demonstrate that the original Troshko–Hassan model with the correction coefficient being unity should be chosen for an interfacial term in the turbulent kinetic energy equation at the statistically steady state based on the theoretical analysis and interface-resolved direct numerical simulations of the particle sedimentation in a periodic domain and an upward turbulent channel flow. In addition, the computational schemes for the viscous dissipation near the particle surface with the non-boundary-fitted mesh are examined, and the results show that the second-order single-sided differentiation for the velocity gradient at the fluid grids in the immediate vicinity of the particle surface can reduce the computational error significantly, compared to the two-sided differentiation.

A New K–ε Turbulence Parameterization for Mesoscale Meteorological Models

Abstract A new one-dimensional 1.5-order planetary boundary layer (PBL) scheme, based on the K–ε turbulence closure applied to the Reynolds-averaged Navier–Stokes (RANS) equations, is developed and implemented within the Weather Research and Forecasting (WRF) Model. The new scheme includes an analytic solution of the coupled equations for turbulent kinetic energy and dissipation rate. Different versions of the PBL scheme are proposed, with increasing levels of complexity, including a model for the calculation of the Prandtl number, a correction to the dissipation rate equation, and a prognostic equation for the temperature variance. Five different idealized cases are tested: four of them explore convective conditions, and they differ in initial thermal stratification and terrain complexity, while one simulates the very stable boundary layer case known as GABLS. For each case study, an ensemble of different large-eddy simulations (LES) is taken as reference for the comparison with the novel PBL schemes and other state-of-the-art 1- and 1.5-order turbulence closures. Results show that the new PBL K–ε scheme brings improvements in all the cases tested in this study. Specifically, the more significant are obtained with the turbulence closure including a prognostic equation for the temperature variance. Moreover, the largest benefits are obtained for the idealized cases simulating a typical thermal circulation within a two-dimensional valley. This suggests that the use of prognostic equations for dissipation rate and temperature variance, which take into account their transport and history, is particularly important with the increasing complexity of PBL dynamics.

The flow patterns and sediment deposition in the forebay of a forward-intake pumping station

In Northwest China, the concentration of sediment in the Yellow River is high. The flow patterns in the forebay and inlet tank of a pumping station on the river are disordered, and sediment deposition endangers the normal operation and safety of the pumping station. To address this problem, the three-dimensional two-phase water–sediment flow in the forebay of the pumping station is modeled using fluid simulation software, and diagrams of the sediment content and velocity streamline in the flow layers of different sections are obtained. Combined with the multiphase flow theory of mixtures and the realizable k–ε turbulent kinetic energy equation, the location and formation mechanism of each vortex, as well as the area and degree of sediment deposition in the forebay, are analyzed. The actual engineering and numerical simulation results are compared to verify the accuracy of the simulation, and the operating conditions of each pump port under different unit operating conditions are proposed. The results show that the deposition of sediment has different effects on the outlets of the pumps, but its impact on pump Nos. 4, 5, and 6 is small. After desilting, the velocity uniformity and deflection angle of pumps 4, 5, and 6 are improved to a certain extent. This study provides specific guidance for the construction and reconstruction of a pumping station forebay to avoid the impacts of backflow areas and sediment deposition to a certain extent.

The relationship among summer atmospheric boundary layer height over the Taklimakan Desert, its land surface parameters and Eurasian circulation

AbstractTo investigate the relationship among summer atmospheric boundary layer (ABL) over the Taklimakan Desert (TD), its land surface parameters and large scale meteorological field, the Urumqi Institute of Desert Meteorology of China Meteorological Administration conducted a comprehensive observation of boundary layer in Tazhong, the hinterland of TD from July 1 to 31, 2016. The correlations between the ABL height and surface sensible heat, latent heat, air temperature, turbulent kinetic energy (TKE) and buoyancy term are analyzed by using the observed GPS sounding data, the surface layer micro meteorological data in the hinterland of the TD and ERA5 reanalysis data so as to study the correlation of the ABL height and Eurasian circulation field. The research results indicate: (1) The height of summer ABL over the TD does not fully correspond with the variation tendency of surface sensible heat flux. In addition to surface sensible heat, large scale advection and residual mixed layer (RML) are also crucial factor influencing ABL. (2) The surface TKE over the TD is consistent with ABL height variation, with the correlation coefficient of 0.62. Hence the equation of ABL height and TKE is expressed as: Y = 1609.2X4 – 12,828X3 + 35,617X2 –39,514X + 15,858. (3) When the ABL height over the TD is quite high, 500 hPa altitude in Xinjiang is mostly controlled by anticyclone circulation, and the easterly low level jet (LLJ) tends to be formed at 850 hPa the next night. (4) The high correlation areas of summer ABL height over the TD and Eurasian temperature field are Xinjiang and Balkash Lake region. Significant positive correlation is identified between the ABL height and the temperature field at 850, 700, and 500 hPa of the above regions. The maximum correlation coefficient reaches 0.7. The study deepens the understanding of the relationship among the ABL process over the TD, land surface parameters and Eurasian circulation.

Adaptability evaluation of boundary layer schemes for simulation of sea and land breeze circulation in the west coast of the Yellow Sea

In the study, a three dimensional Doppler Wind Lidar placed at a coastal city Rizhao in west coast of Yellow sea was used to evaluate the wind field simulated by ten different planetary boundary layer parameterization schemes (PBLS) in the Weather Research and Forecasting model. It is found that the characteristics of the simulated wind field are very sensitive to the boundary layer parameterization schemes used in the model, especially for the sea-land breeze (SLB). Analysis shows that Grenier-Bretherton-McCaa (GBM), Mellor-Yamada Nakanishi and Niino Level 2.5 (MYNN2), and Yonsei University (YSU) produced more realistic results of the three dimensional wind field, as well as the occurrence time, duration and heights of SLB. This is because the entrainment effect is adopted in the GBM scheme, which improves the radiation process between the ground and the atmosphere and ensures the accuracy of turbulent diffusion calculation caused by the differences of temperature and humidity in coastal area. The YSU scheme introduces a local gradient correction term, so that the non-local thermals brings the surface hot air to the middle and upper part of the boundary layer. This process increases the turbulent mixing induced by thermal forcing in the simulation of SLB. The MYNN2 scheme specifies the relationship between the length scale and buoyancy, turbulence scale and surface layer in turbulent kinetic energy equation, and introduces a set of closure parameters representing buoyancy and shear. However, the disadvantages of this scheme are that the influence of the underlying surface is not considered in detail, and the entrainment heat flux at the top of the convective boundary layer is underestimated.

Effects of Lewis and Karlovitz numbers on transport equations for turbulent kinetic energy and enstrophy

Effects of Lewis and Karlovitz numbers on transport equations for turbulent kinetic energy and enstrophy

Impact of Wave–Current Interactions on the Detachment of Low-Salinity Water From Changjiang River Plume and Its Subsequent Evolution

The evolution of low-salinity water (LSW) detached from river plumes is critical to coastal systems and inner-shelf biogeochemical processes. In this study, the impact of wave–current interactions on the LSW detachment from the Changjiang River plume and its subsequent evolution is explored. Waves can advance the detachment of LSW through enhancing the vertical mixing, the northward intrusion of the inshore branch of the Taiwan Warm Current, and the upwelling. The wave-enhanced vertical mixing increases the core salinity of the initially isolated LSW and the growth rate of the core salinity during its subsequent evolution. Waves can lengthen the journey of the isolated LSW and accelerate its movement. Five wave–current interaction processes are compared and it is found that the wave-induced form drag contributes the most to the LSW detachment and its subsequent evolution. It enhances the vertical mixing and alters the wave-driven flow through the vertical transfer of wave-generated pressure to the momentum equation and the turbulent kinetic energy (TKE) equations. The wave dissipation working in the TKE equations weakens the northeastward flow off the Changjiang River estuary, which restrains the diluted water expansion. The current advection and refraction of wave energy contribute most to the enhancement of the double-core upwelling system.

Unstable turbulent channel flow response to spanwise-heterogeneous heat fluxes: Prandtl's secondary flow of the third kind

Turbulent secondary flows are defined as Prandtl's secondary flow of the first or second kind, the former produced by stretching and/or tilting of vorticity, the latter produced via spatial heterogeneity of Reynolds stresses. Both mechanisms are instantaneously active within inertia-dominated wall turbulence; Reynolds stress spatial heterogeneity is required for Reynolds-averaged secondary flows. Spanwise-variable surface roughness can induce turbulent stress spatial heterogeneity in the spanwise–wall-normal plane and provide sustenance for streamwise-aligned mean secondary flows. Herein, we demonstrate that turbulent secondary flows can also be sustained by spanwise variability in the surface heat flux in unstably stratified turbulent channels, defined hereafter as Prandtl's secondary flow of the third kind. Support for this mechanism is established with scaling arguments, while large-eddy simulation is used to model inertia-dominated channel turbulence responding to a lower boundary with uniform aerodynamic/hydrodynamic roughness but spanwise-variable surface heat flux. Transport equations for streamwise vorticity and turbulent kinetic energy, $k$ , outline the conditions needed for third-kind production: shear and buoyancy production over the elevated heat flux regions necessitates lateral entrainment of low- $k$ fluid, inducing mean counter-rotating secondary cells aligned such that upwelling and downwelling occur over the high and low heat flux regions, respectively. Buoyancy-driven production of $k$ alters aggregate flow response and thus is a distinctly different mechanism responsible for sustenance of secondary flows.

Parsing the Kinetic Energy Budget of the Ocean Surface Mixed Layer.

The total rate of work done on the ocean by the wind is of considerable interest for understanding global energy balances, as the energy from the wind drives ocean currents, grows surface waves, and forces vertical mixing. A large but unknown fraction of this atmospheric energy is dissipated by turbulence in the upper ocean. The focus of this work is twofold. First, we describe a framework for evaluating the vertically integrated turbulent kinetic energy (TKE) equation using measurable quantities from a surface mooring, showing the connection to the atmospheric, mean oceanic, and wave energy. Second, we use this framework to evaluate turbulent energetics in the mixed layer using 10 months of mooring data. This evaluation is made possible by recent advances in estimating TKE dissipation rates from long‐enduring moorings. We find that surface fluxes are balanced by TKE dissipation rates in the mixed layer to within a factor of two.

Assessment of Regularized Ensemble Kalman Method for Inversion of Turbulence Quantity Fields

This paper introduces an ensemble-based field inversion framework to augment the turbulence models by incorporating prior physical knowledge. Different types of prior knowledge such as smoothness, prior values, and sparsity are enforced to improve the inference of the eddy viscosity and laminar–turbulent intermittency. This work first assesses the method on the problems of inferring eddy viscosity in the Reynolds-averaged Navier–Stokes equation from the velocity observation data for separated flows over periodic hills. Further, the method is used to infer the intermittency field in the transport equation of turbulent kinetic energy from measurements of the friction coefficient for transitional flows over a plate. The results demonstrate the performance of the regularized ensemble method by enforcing prior knowledge into the inference. The method serves as a useful inverse modeling tool to augment the turbulence model from observation data.

Study of unsteady cavitating flow around Clark-Y hydrofoil using nonlinear PANS model with near-wall correction

In this paper, we describe the use of a new nonlinear partially-averaged Navier–Stokes (PANS) model with near-wall correction for simulating the cavitating flow around a Clark-Y hydrofoil. For comparison, the standard [Formula: see text]–[Formula: see text] PANS model is also used. The results demonstrate that compared to [Formula: see text]–[Formula: see text] PANS and experiment, the new PANS model shows better performance for cavitation flow, including time-averaged velocity, root mean square (rms) velocity and cavity shedding processing. Through the calculation of the lift and drag coefficient at [Formula: see text] and [Formula: see text], it can be concluded that the cavitation will decrease the lift and increase the drag of the hydrofoil, resulting in a decrease of the lift-to-drag ratio. From the analysis of different terms in both the turbulent kinetic energy (TKE) and dissipation rate transport equations of the cloud cavitation, it is found that the production term and the dissipation term are dominant in the turbulent transport, and they are mainly distributed in the vapor–liquid interface and the trailing edge of the hydrofoil.