Turbulent Kinetic Energy Balance Research Articles

Assessment of Turbulent Kinetic Energy Balance within the Boulder Array in Gravel Bed Stream

Assessment of Turbulent Kinetic Energy Balance within the Boulder Array in Gravel Bed Stream

Reynolds number dependence of turbulent kinetic energy and energy balance of 3-component turbulence intensity in a pipe flow

Measurement data sets are presented for the turbulence intensity profile of three velocity components ( $u$ , $v$ and $w$ ) and turbulent kinetic energy (TKE, $k$ ) over a wide range of Reynolds numbers from $Re_\tau =990$ to 20 750 in a pipe flow. The turbulence intensity profiles of the $u$ - and $w$ -component show logarithmic behaviour, and that of the $v$ -component shows a constant region at high Reynolds numbers, $Re_\tau >10\,000$ . Furthermore, a logarithmic region is also observed in the TKE profile at $y/R=0.055\unicode{x2013}0.25$ . The Reynolds number dependences of peak values of $u$ -, $w$ -component and TKE fit to both a logarithmic law (Marusic et al., Phys. Rev. Fluids, vol. 2, 2017, 100502) and an asymptotic law (Chen and Sreenivasan, J. Fluid Mech., vol. 908, 2020, R3), within the uncertainty of measurement. The Reynolds number dependence of the bulk TKE $k^+_{bulk}$ , which is the total amount of TKE in the cross-sectional area of the pipe also fits to both laws. When the asymptotic law is applied to the $k^+_{bulk}$ , it asymptotically increases to the finite value $k^+_{bulk}=11$ as the Reynolds number increases. The contribution ratio $\langle u'^2\rangle /k$ reaches a plateau, and the value tends to be constant within $100< y^+<1000$ at $Re_\tau >10\ 000$ . Therefore, the local balance of each velocity component also indicates asymptotic behaviour. The contribution ratios are balanced in this region at high Reynolds numbers as $\langle u'^2\rangle /k\simeq 1.25$ , $\langle w'^2\rangle /k\simeq 0.5$ and $\langle v'^2\rangle /k \simeq 0.25$ .

Effect of the Density Ratio on Emulsions and Their Segregation: A Direct Numerical Simulation Study

Using direct numerical simulation (DNS) in combination with the volume of fluid method (VoF), we investigate the influence of the density ratio between the carrier and dispersed phase on emulsions, where the baseline simulation approximately corresponds to the ratio of water-in-gasoline emulsions. For this purpose, homogeneous isotropic turbulence (HIT) is generated using a linear forcing method, enhanced by a proportional–integral–derivative (PID) controller, ensuring a constant turbulent kinetic energy (TKE) for two-phase flows, where the TKE balance equation contains an additional term due to surface tension. Then, the forcing is stopped, and gravitational acceleration is activated. The proposed computational setup represents a unique and well-controlled configuration to study emulsification and segregation. We consider four different density ratios, which are applied in industrial processes, to investigate the influence of the density ratio on the statistically steady state of the emulsions, and their segregation under decaying turbulence and constant gravitational acceleration. At the statistically steady state, we hold the turbulence constant and study the effects of the density ratio ρd/ρc, on the interface area, the Sauter mean diameter (SMD), and the statistical droplet size distribution. We find that all are affected by the density ratio, and we observe a relation between the SMD and ρd/ρc. Furthermore, we assume a dependence of the critical Weber number on the density ratio. In the second part of our work, we study the segregation process. To this end, we consider the change in the center of mass of the disperse phase and the energy release, to analyze the dependence of segregation on the density difference Δρ/ρd. We show that segregation scales with the density difference and the droplet size, and a segregation time scale has been suggested that collapses the height of the center of mass for different density ratios.

Turbulent Kinetic Energy Budgets over Gentle Topography Covered by Forests

Abstract Large-eddy simulations of flow over a “horizontally” uniform model forest are used to investigate the effects of gentle topography on the turbulent kinetic energy (TKE) budget within the canopy roughness sublayer. Despite significant differences between simulations using idealized sinusoidal topography and real topography of the Amazon forest, results indicate that the effects of topography are located predominantly in the upper canopy and above, and are mostly caused by mean advection of TKE. The “horizontally” averaged TKE budget from idealized and real gentle topographies are almost identical to that for flat terrain, including a clear inertial layer above the roughness sublayer in which shear production is balanced by local dissipation. At topography crests, where observational towers are usually located, mean vertical advection of TKE can be as important as horizontal advection. We propose the use on an approximate TKE balance equation to estimate mean advection from single tower measurements, and introduce a new advection index that can be used as a proxy to quantify the importance of the topography on the TKE budget.

Mean and Turbulent Characteristics of a Bottom Mixing‐Layer Forced by a Strong Surface Tide and Large Amplitude Internal Waves

AbstractWe present 15 days of both mean and turbulent field observations bottom mixing‐layer at a gently sloping 250 m deep continental shelf site, energized by tides and nonlinear internal waves (NLIWs). The tidal frequency forcing was due to the combined effects of the barotropic tide and a mode‐1 internal tide (IT), while the NLIWs were predominantly mode‐1 waves of depression. The bottom mixing‐layer thickness varied at both semidiurnal and sub‐tidal ∼O(10)d frequencies, with an average thickness of around 10 m. Compression and expansion of the mixing‐layer by both the IT and NLIWs affected the mean velocity profiles in the mixing‐layer, while the phasing between the barotropic and baroclinic flows led to an asymmetry in mean velocity profiles between periods of rising and falling isotherms. With the exception of periods of flow reversal, the turbulent kinetic energy balance and turbulent stress observations were consistent with the existence of an inertial‐sublayer with thickness of approximately 10%–15% of the mixing‐layer thickness ( ∼1 m), even beneath NLIWs. In the outer portion of the mixing‐layer—that is, above the inertial‐sublayer—NLIWs modulated the local turbulence spectra. We discuss the observations in the context of a predictive model for mixing‐layer thickness. The analysis suggests that the high‐frequency variability in mixing‐layer thickness was dominated by internal wave pumping, though strength of the ambient stratification and the frequency of the forcing were important controls on the time‐averaged (sub‐tidal) variation.

Analysis of the particles-induced turbulence in confined gas-solid fluidized beds by PR-DNS

Particles-induced turbulence in fluidized beds is vital in dense particulate flow and still lacks detailed investigation. In this study, a multi-direct forcing immersed boundary method coupled with a soft-sphere model is employed in simulations of shallow and deep bubbling beds of movable spheres. Five different particle-resolved direct numerical simulations are performed by varying the inlet gas velocity, the bed extent, and the number of particles. The particles-induced turbulence under the wall effect in the bubbling beds is analysed, including the mean and second-order statistics of velocity, momentum balance, turbulent kinetic energy balance, and kinetic energy transfer between particles and fluid. It is found that the turbulence intensity is greater at the location where the bubble bursts. An analysis of the Reynolds-averaged (RA) momentum equation shows that the mean streamwise feedback force term and the mean pressure gradient term are the most critical terms in particles' presence. An analysis of the turbulent kinetic energy (TKE) equation shows that the pressure dilatation term is the source term generating the TKE, while the interphase exchange term is the dissipation term reducing the TKE in particles' presence. A detailed analysis of the kinetic energy transfer between particles and fluid is finally conducted.

Transient 3D simulations of turbulent premixed flames of gas-to-liquid (GTL) fuel in a fan-stirred combustion vessel

This work aims to conduct turbulent flame studies of GTL fuel and 50/50 diesel-GTLblend and compare them to diesel under a wide range of equivalence ratio (0.7 < Ф < 1.3) and turbulence intensities (0.5 m/s < u′ < 3.0 m/s). For this purpose, Zimont Turbulent Flame Speed Closure (Zimont TFC) model is adapted and implemented into ANSYS Fluent to study the influence of turbulence on premixed combustion. The model is verified against the experimental results of the turbulent flame speeds for methane-air mixture using a cylindrical fan-stirred combustion vessel. Results show that (i) rich diesel and lean GTL fuels are characterized by a faster flame development and higher turbulent flame speeds; (ii) at the same elapsed time, ReT and Da are found to be greater for stoichiometric GTL fuel compared to diesel and 50/50 diesel GTL blend, which indicates that the flame propagates towards the vessel’s wall at a faster rate and the chemistry has dominated turbulence in a shorter time, and (iii) at low turbulence intensity level (u′=0.5 m/s), the flame morphology is defined by a wrinkled flamelet regime in Borghi diagram. However, at moderate and higher turbulence levels (u′=1.5 m/s and u′=3.0 m/s, respectively), the flame structure is defined by the corrugated flamelets regime. Furthermore, the turbulence homogeneity and isotropy have been investigated throughout the combustion vessel by conducting a study for the turbulent kinetic energy balance.

A Generalized Mixing Length Closure for Eddy‐Diffusivity Mass‐Flux Schemes of Turbulence and Convection

AbstractBecause of their limited spatial resolution, numerical weather prediction and climate models have to rely on parameterizations to represent atmospheric turbulence and convection. Historically, largely independent approaches have been used to represent boundary layer turbulence and convection, neglecting important interactions at the subgrid scale. Here we build on an eddy‐diffusivity mass‐flux (EDMF) scheme that represents all subgrid‐scale mixing in a unified manner, partitioning subgrid‐scale fluctuations into contributions from local diffusive mixing and coherent advective structures and allowing them to interact within a single framework. The EDMF scheme requires closures for the interaction between the turbulent environment and the plumes and for local mixing. A second‐order equation for turbulence kinetic energy (TKE) provides one ingredient for the diffusive local mixing closure, leaving a mixing length to be parameterized. Here, we propose a new mixing length formulation, based on constraints derived from the TKE balance. It expresses local mixing in terms of the same physical processes in all regimes of boundary layer flow. The formulation is tested at a range of resolutions and across a wide range of boundary layer regimes, including a stably stratified boundary layer, a stratocumulus‐topped marine boundary layer, and dry convection. Comparison with large eddy simulations (LES) shows that the EDMF scheme with this diffusive mixing parameterization accurately captures the structure of the boundary layer and clouds in all cases considered.

Analysis of the effect of intermittency in a high-pressure turbine blade

High-pressure turbine blades are subject to large thermomechanical loads that may threaten their mechanical integrity. The prediction of the heat transfer on the blade surface, crucial to ensure its durability, thus requires an accurate description of the flow physics around the blade to be reliable. In an effort to better qualify the use of computational fluid dynamics in this design context as well as the need for an improved understanding of the flow physics, this paper investigates a transonic highly loaded linear turbine blade cascade that has been found difficult to predict in the literature using large-eddy simulations. Indeed, the configuration results in shocks and acoustic waves on the suction side of the blade, features that are commonly encountered in high-pressure turbines. Turbulent spots are observed on the suction-side boundary layer with an inlet turbulence intensity of 6%. The turbulent spots are shown to have a complex and highly unsteady effect on the shock/boundary-layer interaction, disrupting flow detachment and creating laminar spots downstream of the shock. To address these transient flow phenomena, conditional averages based on the intermittency level are introduced to show that accurate heat transfer predictions require an accurate prediction of the rate of turbulent-spot production. The analysis then focuses on the effect of intermittency on the turbulent kinetic energy exchanges in the near-wall region as the turbulent kinetic energy balance must be addressed in Reynolds-averaged Navier–Stokes models.

Kinetic energy balance in turbulent particle-laden channel flow

The present study investigates the influence of particle additives on the transfer, conversion, and dissipation of kinetic energy (KE) of a turbulent gas–solid channel flow. We derived the equations of KE, mean-flow KE, and turbulent KE (TKE) of the particle-laden flow and further performed two-way coupled direct numerical simulations of channel flow laden with four-million particles with Stokes number St = 30 (corresponding to a mass loading ratio of around one) with an Eulerian–Lagrangian approach. We found that, in the unladen flow, more than half of the input energy is directly dissipated in the mean flow, whereas the rest is converted to maintain the turbulence. By contrast, in the laden flow, both mean dissipation and energy supply are comparable with the unladen flow. However, the turbulence production is greatly reduced in the particle-laden flow. Another sink term due to the presence of the particle–fluid interactions corresponds to the rest loss of the total energy supply. The results reveal the particle-induced redistribution of mean KE, which is transferred from the mean flow to particles in the channel core, whereas the flow gains energy from particles in the near-wall region. In total, there is a loss of the mean-flow energy due to the presence of the inertial particles. Regarding TKE balance, the particles, gaining energy from the mean flow, transfer the energy to the fluid across the channel, which contributes around one third of the TKE source. The present results provide a general picture of KE balance of a particle-laden channel flow.

A relationship for the diffusion coefficient in eddy diffusion based indoor dispersion modelling

Turbulent or eddy diffusion models are used to predict spatially resolved exposures to toxic airborne materials in indoor environments. The single parameter that governs mixing in these models is the eddy diffusion coefficient. Some relationships that enable this coefficient to be predicted have been proposed in the literature, but wider applicability of these has not previously been tested. In this paper an automated computational fluid dynamics tool was used to calculate the eddy diffusion coefficient in a range of isothermal, mechanically ventilated rooms. Available models for the diffusion coefficient were then tested and the most applicable was found to be one based on a turbulent kinetic energy balance. This relationship was only appropriate when the characteristic length was set to a dimension of the air supply inlet, instead of the length usually applied, i.e. the room height. The validity of this relationship was further demonstrated using experimental test cases and by applying standard error metrics. The eddy diffusion approach can now be used with improved confidence in a wider range of scenarios than was possible before.

Bulk models of sheared boundary layer convection

The paper discusses approaches to the construction of integral models of the convective boundary layer (CBL), based on the concept of complete mixing. To test analytic bulk models and the basic hypotheses of similarity, we use the results of eddy modeling (LES – Large Eddy Simulation). The empirical constants of the CBL integral models obtained according to the LES data for the case of free convection, are in good agreement with the previously published data of laboratory experiments. It is also shown that the flow of kinetic energy from the upper boundary of the CPS, carried out by gravitational waves, is small compared with other components of the balance of turbulent kinetic energy (TKE) in the convective layer. Parametrization of TKE generation for the case of sheared convective boundary layer in terms of the friction velocity and the average wind velocity in the CBL derived; resulting dimensionless constants are obtained from LES data. The results of the work allow us to formulate an integral model of the shear KPS suitable for practical use.

Bulk Models of Sheared Boundary Layer Convection

The paper discusses approaches to the construction of integral models of the convective boundary layer (CBL), based on the concept of complete mixing. To test analytic bulk models and the basic hypotheses of similarity, we use the results of eddy modeling (LES – Large Eddy Simulation). The empirical constants of the CBL integral models obtained according to the LES data for the case of free convection, are in good agreement with the previously published data of laboratory experiments. It is also shown that the flow of kinetic energy from the upper boundary of the CPS, carried out by gravitational waves, is small compared with other components of the balance of turbulent kinetic energy (TKE) in the convective layer. Parametrization of TKE generation for the case of sheared convective boundary layer in terms of the friction velocity and the average wind velocity in the CBL derived; resulting dimensionless constants are obtained from LES data. The results of the work allow us to formulate an integral model of the shear KPS suitable for practical use.

POD Analysis of Entropy Generation in a Laminar Separation Boundary Layer

Separation of laminar boundary layer is a great source of loss in energy and power machinery. This paper investigates the entropy generation of the boundary layer on the flat plate with pressure gradient. The velocity of the flow field is measured by a high resolution and time related particle image velocimetry (PIV) system. A method to estimate the entropy generation of each mode extracted by proper orthogonal decomposition (POD) is introduced. The entropy generation of each POD mode caused by mean viscous, Reynolds normal stress, Reynolds sheer stress, and energy flux is analyzed. The first order mode of the mean viscous term contributes almost 100% of the total entropy generation. The first three order modes of the Reynolds sheer stress term contribute less than 10% of the total entropy generation in the fore part of the separation bubble, while it reaches to more than 95% in the rear part of the separation bubble. It indicates that the more unsteady that the flow is, the higher contribution rate of the Reynolds sheer stress term makes. The energy flux term plays an important role in the turbulent kinetic energy balance in the transition region.

An Analytical Model With a Critical Richardson Number Closure for Sediment‐Stratified Open Channel Flow

AbstractAn analytical model is developed for sediment‐stratified open channel flow, with an objective to explore the parameter regime where the self‐induced stratification controls the vertical structure. The model exploits a critical Richardson number closure motivated by a suite of numerical experiments. It is shown that, with Rouse number (relative settling speed) below one, a buoyancy length scale LB, defined as the Monin‐Obukhov length scaled by a factor of 0.1, separates the water column into two regions. Below LB, the buoyancy destruction by sediment stratification is of secondary importance (as indicated by the factor 0.1), and the concentration and velocity closely resemble the unstratified Rousean and logarithmic distributions; Above LB, the sediment feedback in the turbulence kinetic energy balance helps maintain a critically stratified state, and the vertical profiles deviate markedly from the unstratified scaling. Incorporating this closure then leads to analytical expressions for concentration and velocity that compare well with the numerical model and prior studies. The influence of sediment stratification on vertical structure is governed by a dimensionless height LB/h (h is water depth). A smaller LB/h reflects a greater portion of critically stratified water column, indicative of a stronger stratification control. Large near‐bottom buoyancy destruction, as indicated by a large bottom Richardson number, and slow vertical decay, as indicated by a small Rouse number, lead to an expansion of critically stratified region and thus favor sediment stratification. It is shown further that, in sediment‐stratified regime, different concavities of concentration profiles are possible.

Air turbulence characteristics at multiple sites in and above the Amazon rainforest canopy

Atmospheric turbulence characteristics within and above rain forest canopies are investigated at several sites located in the Amazon region of Brazil. Turbulence data provided by bi- and three-dimensional sonic anemometers, which were deployed at heights ranging from near the forest floor to about 80 m, are analyzed to describe the principal features of atmospheric turbulence, sensible heat flux (H), and components of the turbulent kinetic energy (TKE) budget equation. The analyses focused on weak (WW) and strong (SW) wind conditions to achieve the research objectives of evaluating the turbulence structure above and below the rain forest canopy and estimating the degree of coupling between air layers above the forest and deep in the canopy. Turbulence statistical moments show that atmospheric eddies, generated above the canopy, hardly penetrate the region below 0.5h (h is the canopy height). Forest-atmosphere exchanges of heat differ depending on the observed wind regimes. Sensible heat fluxes decrease with canopy depth for SW conditions and are approximately constant with the height for WW above the canopy. Sensible heat flux profiles reveal a transition layer (around 0.6h) which sometimes exchanges heat with the upper and sometimes with the lower forest canopy, depending on time of day and weather conditions. TKE balance results show that during the daytime period in SW conditions the shear production is at least an order of magnitude greater than the buoyancy above the forest canopy. This turbulence, however, is practically all dissipated in the region above 0.5h. Thus, the air layer from the soil surface to 0.5h is largely decoupled from the upper part of the forest canopy. This feature of having the bottom of the canopy mostly decoupled from the air aloft in the dense and tall rain forest can exert control on the residence times and turbulent transport of plant-emitted gases out of the forest canopy.

Turbulence in the presence of internal waves in the bottom boundary layer of the California inner shelf

Turbulence measurements were collected in the bottom boundary layer of the California inner shelf near Point Sal, CA, for 2 months during summer 2015. The water column at Point Sal is stratified by temperature, and internal bores propagate through the region regularly. We collected velocity, temperature, and turbulence data on the inner shelf at a 30-m deep site. We estimated the turbulent shear production (P), turbulent dissipation rate (e), and vertical diffusive transport (T), to investigate the near-bed local turbulent kinetic energy (TKE) budget. We observed that the local TKE budget showed an approximate balance (P ≈ e) during the observational period, and that buoyancy generally did not affect the TKE balance. On a finer resolution timescale, we explored the balance between dissipation and models for production and observed that internal waves did not affect the balance in TKE at this depth.

Sediment Suspension by Straining‐Induced Convection at the Head of Salinity Intrusion

AbstractThe tidal straining can generate convective motions and exert a periodic modification of turbulence and sediment transport in estuarine and coastal bottom boundary layers. However, the evidence and physics of convection and sediment suspension induced by tidal straining have not been straightforward. To examine these questions, mooring and transect surveys have been conducted in September 2015 in the region of the Yangtze River plume influence. Field observations and scaling analyses indicate an occurrence of convective motions at the head of saline wedge. Theoretical analyses of stratification evolution in the saline wedge show that unstable stratification and resultant convection are induced by tidal straining. Vertical turbulent velocity and eddy viscosity at the head of saline wedge are both larger than their neutral counterparts in the main body, largely enhancing sediment suspension at the head of saline wedge. Moreover, sediment suspension in both neutral and convection‐affected flows is supported by the variance of vertical turbulent velocity, rather than the shearing stress. Finally, the stability correction functions in the Monin‐Obukhov similarity theory can be simply derived from the local turbulent kinetic energy balance to successfully describe the effects of tidal straining on turbulent length scale, eddy viscosity, and sediment diffusivity in the convection‐affected flow. These recognitions may provide novel understanding of estuarine turbidity maxima, and the dynamical structure and processes for coastal hypoxia.

Baroclinic Frontal Instabilities and Turbulent Mixing in the Surface Boundary Layer. Part II: Forced Simulations

AbstractGeneration of ocean surface boundary layer turbulence and coherent roll structures is examined in the context of wind-driven and geostrophic shear associated with horizontal density gradients using a large-eddy simulation model. Numerical experiments over a range of surface wind forcing and horizontal density gradient strengths, combined with linear stability analysis, indicate that the dominant instability mechanism supporting coherent roll development in these simulations is a mixed instability combining shear instability of the ageostrophic, wind-driven flow with symmetric instability of the frontal geostrophic shear. Disruption of geostrophic balance by vertical mixing induces an inertially rotating ageostrophic current, not forced directly by the wind, that initially strengthens the stratification, damps the instabilities, and reduces vertical mixing, but instability and mixing return when the inertial buoyancy advection reverses. The resulting rolls and instabilities are not aligned with the frontal zone, with an oblique orientation controlled by the Ekman-like instability. Mean turbulence is enhanced when the winds are destabilizing relative to the frontal orientation, but mean Ekman buoyancy advection is found to be relatively unimportant in these simulations. Instead, the mean turbulent kinetic energy balance is dominated by mechanical shear production that is enhanced when the wind-driven shear augments the geostrophic shear, while the resulting vertical mixing nearly eliminates any effective surface buoyancy flux from near-surface, cold-to-warm, Ekman buoyancy advection.

Direct numerical simulation of stratified flow past a sphere at a subcritical Reynolds number of 3700 and moderate Froude number

Direct numerical simulation of flow past a sphere in a stratified fluid is carried out at a subcritical Reynolds number of 3700 and $Fr=U_{\infty }/ND=1,2$ and 3 to understand the dynamics of moderately stratified flows with $Fr=O(1)$. Here, $U_{\infty }$ is the free stream velocity, $N$ is the background buoyancy frequency and $D$ is the sphere diameter. The unstratified flow past the sphere consists of a separated shear layer that transitions to turbulence, a recirculation zone and a wake with a mean centreline deficit velocity, $U_{0}$, that decreases with downstream distance as a power law. With increasing stratification, the separated shear layer plunges inward vertically and its roll up is inhibited, the recirculation zone is shortened and the mean wake decays at a slower rate of $U_{0}\propto (x_{1}/D)^{-0.25}$ in the non-equilibrium (NEQ) region. The transition from the near wake where $U_{0}$ has a decay rate similar to the unstratified case to the NEQ regime occurs as an oscillatory modulation by a steady lee wave pattern with a period of $t=2\unicode[STIX]{x03C0}/N$ that leads to accelerated $U_{0}$ between $Nt=\unicode[STIX]{x03C0}$ and approximately $Nt=2\unicode[STIX]{x03C0}$. Far downstream, the wake is dominated by coherent horizontal motions. The acceleration of $U_{0}$ by the lee wave and the lower turbulence production in the NEQ regime, thereby less loss to turbulence, prolongs the lifetime of the wake relative to its unstratified counterpart. The intensity, temporal spectra and structure of turbulent fluctuations in the wake are assessed. Buoyancy induces significant anisotropy among the velocity components and between their vertical and horizontal profiles. Consequently, the near wake ($x_{1}/D&lt;10$) exhibits significant differences in turbulence profiles relative to its unstratified counterpart. Spectra of vertical velocity show a discrete peak in the near wake that is maintained further downstream. The turbulent kinetic energy (TKE) balance is computed and contributions from pressure transport and buoyancy are found to become increasingly important as stratification increases. The findings of this investigation will be helpful in designing accurate initial conditions for the temporally evolving model of stratified wakes.