Turbulent Kinetic Energy Equation Research Articles

Turbulent forced convection heat transfer in triangular cross sectioned helically coiled tube

Turbulent forced convection heat transfer in triangular cross sectioned helically coiled tube

Transition study of 3D aerodynamic configures using improved transport equations modeling

As boundary layer transition plays an important role in aerodynamic drag prediction, the proposal and study of transition prediction methods simulating the complex flow phenomena are prerequisite for aerodynamic design. In this paper, with the application of the linear stability theory based on amplification factor transport transition equations on the two-equation shear stress transport (SST) eddy-viscosity model, a new method, the SST-NTS-NCF model, is yielded. The new amplification factor transport equation for the crossflow instability induced transition is proposed to add to the NTS equation proposed by Coder, which simulates Tollmien–Schlichting wave transition. The turbulent kinetic energy equation is modified by introducing a new source term that simulates the transition process without the intermittency factor equation. Finally, coupled with these two amplification factor transport equations and SST turbulence model, a four-equation transition turbulence model is built. Comparisons between predictions using the new model and wind-tunnel experiments of NACA64(2)A015, NLF(2)-0415 and ONERA-D infinite swept wing and ONERA-M6 swept wing validate the predictive quality of the new SST-NTS-NCF model.

Fully Local Formulation of a Transition Closure Model for Transitional Flow Simulations

A new computational fluid dynamics compatible transition model for the natural transition and laminar separation bubble induced transition is established using two transport equations with consideration of analysis results through linear stability theory linear stability theory. One transport equation is introduced to describe the growth of the laminar kinetic energy in the boundary layer. In addition, another transport equation of intermittency factor is built to trigger the transition by controlling the source term of turbulent kinetic energy equation. Finally, applying the two transport equations to a shear stress transport turbulence model completes a four-equation transition turbulence model. Comparisons of predictions using the new model with wind-tunnel experiments of the Schubauer and Klebanoff flat plate, S809 airfoil, Aerospatiale-A airfoil, and DLR-F5 wing validate the predictive qualities of the new transition turbulence model.

Droplet–turbulence interaction in a confined polydispersed spray: effect of turbulence on droplet dispersion

The effect of entrained air turbulence on dispersion of droplets (with Stokes number based on the Kolmogorov time scale,$St_{{\it\eta}}$, of the order of 1) in a polydispersed spray is experimentally studied through simultaneous and planar measurements of droplet size, velocity and gas flow velocity (Hardalupaset al.,Exp. Fluids, vol. 49, 2010, pp. 417–434). The preferential accumulation of droplets at various measurement locations in the spray was examined by two independent methodsviz.counting droplets on images by dividing the image in to boxes of different sizes, and by estimating the radial distribution function (RDF). The dimension of droplet clusters (obtained by both approaches) was of the order of Kolmogorov’s length scale of the fluid flow, implying the significant influence of viscous scales of the fluid flow on cluster formation. The RDF of different size classes indicated an increase in cluster dimension for larger droplets (higher$St_{{\it\eta}}$). The length scales of droplet clusters increased towards the outer spray regions, where the gravitational influence on droplets is stronger compared to the central spray locations. The correlation between fluctuations of droplet concentration and droplet and gas velocities were estimated and found to be negative near the spray edge, while it was close to zero at other locations. The probability density function of slip between fluctuating droplet velocity and gas velocity ‘seen’ by the droplets signified presence of considerable instantaneous slip velocity, which is crucial for droplet–gas momentum exchange. In order to investigate different mechanisms of turbulence modulation of the carrier phase, the three correlation terms in the turbulent kinetic energy equation for particle-laden flows (Chen & Wood,Can. J. Chem. Engng, vol. 65, 1985, pp. 349–360) are evaluated conditional on droplet size classes. Based on the comparison of the correlation terms, it is recognized that although the interphase energy transfer due to fluctuations of droplet concentration is low compared to the energy exchange only due to droplet drag (the magnitude of which is controlled by average droplet mass loading), the former cannot be considered negligible, and should be accounted in two phase flow modelling.

A SIMPLE SCHEME TO IMPLEMENT A NONLOCAL TURBULENT CONVECTION MODEL FOR CONVECTIVE OVERSHOOT MIXING

ABSTRACT Classical “ballistic” overshoot models show some contradictions and are not consistent with numerical simulations and asteroseismic studies. Asteroseismic studies imply that overshoot is a weak mixing process. A diffusion model is suitable to deal with it. The form of diffusion coefficient in a diffusion model is crucial. Because overshoot mixing is related to convective heat transport (i.e., entropy mixing), there should be some similarity between them. A recent overshoot mixing model shows consistency between composition mixing and entropy mixing in the overshoot region. A prerequisite to apply the model is to know the dissipation rate of turbulent kinetic energy. The dissipation rate can be worked out by solving turbulent convection models (TCMs). But it is difficult to apply TCMs because of some numerical problems and the enormous time cost. In order to find a convenient way, we have used the asymptotic solution and simplified the TCM to a single linear equation for turbulent kinetic energy. This linear model is easy to implement in calculations of stellar evolution with negligible extra time cost. We have tested the linear model in stellar evolution, and have found that it can well reproduce the turbulent kinetic energy profile of the full TCM, as well as the diffusion coefficient, abundance profile, and stellar evolutionary tracks. We have also studied the effects of different values of the model parameters and have found that the effect due to the modification of temperature gradient in the overshoot region is slight.

LES データに基づく熱連成解析手法の予測性能改善に向けた検討

A numerical method based on turbulence statistics obtained by LES (Large Eddy Simulation) was proposed to predict time-mean temperature field of conjugate heat transfer problem in complex turbulent flows. For the evaluation of turbulent heat flux in the time-mean field, a state-of-art heat flux model was newly adopted to enhance the prediction capability of time-mean temperature fields. Turbulence kinetic energy was estimated by the approximate budget of transport equation for turbulence kinetic energy. It is confirmed that prediction of dissipation rate of turbulence kinetic energy was improved.

Flow property of fiber suspension in a turbulent channel flow under considering both Brownian and turbulent diffusions

The issue of fiber suspension flow has received great substantial attention in the last decades. In contrast with the abundant researches of normal size fiber suspensions flow, this paper is devoted to the small size fiber suspension composed of water and polyarmide fiber where Brownian motion plays an important role and thus cannot be neglected. The spatial distribution and orientation of fiber, streamwise mean velocity profile, turbulent kinetic energy, Reynolds stress and rheological property of fiber suspension in a turbulent channel flow are obtained and analyzed both numerically and experimentally. To simulate the small fiber suspension flow well, the well-known Reynolds averaged Navier-Stokes (N-S) equation governing the suspension flow is modified in consideration of the effect of fibers on base flow. The equation describing the probability density functions for fiber orientation is derived in view of the rotary Brownian diffusion. The general dynamic equation for fibers is reshaped in the effect of spatial Brownian diffusion. For the sake of the closure of the N-S equation, the turbulence kinetic energy and turbulence dissipation rate equations with fiber term are employed. The conditional finite difference method is adopted to discrete these partial differential equations. And the diffusion term and convective term are discretized by the central finite differences and the second-order upwind finite difference schemes, respectively. The second and fourth-order orientation tensors are integrated by the Simpson formula. Experiment is also performed to validate some numerical results. The results show that most fibers tend to align parallelly to the flow direction in the flow, especially in regions near the wall. Such a phenomenon is more obvious with the decreases of Reynolds number and fiber concentration, and with the increase of fiber aspect ratio. Fiber volume fraction distribution is non-uniform across the channel, and becomes more uniform with increasing Reynolds number, and with reducing fiber aspect ratio. The changes of fiber orientation distribution and spatial distribution are not sensitive to fiber aspect ratio for 5. Streamwise mean velocity profile in fiber suspension has a steeper slope than that in single phase flow, and the steepness increases as the fiber concentration and fiber aspect ratio increase, and as the Reynolds number decreases. The presence of fiber will reduce the turbulence kinetic energy and Reynolds stress. The effect of fiber on the turbulence suppression becomes more obvious with the increases of fiber concentration and aspect ratio, and with the decrease of Reynolds number. The first normal stress difference is less than 0.05 and much less than the shear stress. From the wall to the center of the channel, the shear stress increases while the first normal stress difference decreases. Both the shear stress and the first normal stress difference increase with increasing fiber concentration and aspect ratio. Shear stress increases while the first normal stress difference decreases with increasing Reynolds number. The effects of fiber concentration on the shear stress and the first normal stress difference are larger than the fiber aspect ratio.

Budget Of The Scalar Variance Equation In A Turbulent Patch Arising From Lee Wave Breaking

Based on averaged data of the direct numerical simulations, statistical moments are obtained in a turbulent patch arising after lee wave overturning in a flow with stable stratification and obstacle. Temporal evolution and spatial behavior of the scalar-variance transport equation budget have been studied. A priori estimations of algebraic approximations for scalar dissipation, scalar variance and turbulent-diffusion processes in the scalar-variance equation have been carried out. Such an analysis is helpful to explore the turbulent patch in terms of statistical moments, and to verify closure hypotheses in turbulence models. In the global balance of the scalar-variance equation, the compensation of production by dissipation and advection is shown, as for the turbulent kinetic energy equation. The ratio of turbulent time scales of the scalar and velocity fields varies from 0.2 to 2.2 within the wave breaking region, and the global value of this parameter is close to unity during the quasisteady period. The algebraic expression derived from the assumption of production and dissipation balance is incorrect leading to unphysical negative values, therefore the use of the full scalarvariance equation in the turbulent transport model is justified.

Shock wave-boundary layer interactions at the supersonic flow around three-dimensional configurations

The study suggests that the problems of interactions of turbulent flows at the supersonic flow around aircraft elements be solved by numerical analysis methods aided by a mathematical model that is based on the system of the Reynolds-averaged Navier-Stokes equations. The system includes the k-ω SST turbulence model for viscous compressible medium (with two scalar equations of turbulent kinetic energy and relative velocity of its dissipation with the modification that takes into account the transfer of shear stress). The paper presents applied testing and verification of the model along with examples of the problems on supersonic flows around a flat wall and a sphenoid superstructure as well as a perpendicular gas stream. We have identified physical characteristics of interactions between the condensation wave and the boundary turbulent layer, which manifest themselves in the formation of a complex structure of disconnecting and connecting zones of a boundary turbulent layer, which are characterized by respective lines of separation and joining on the wrap surface. The solutions adequately reflect the pattern of the supersonic flow of a compressible medium, condensation waves and vortex zones that are commonly observed during field studies. The comparative analysis of the results of numerical modeling and experimental data confirms the applicability of the mathematical model for complex tasks of the supersonic gas-dynamic state.

Streamwise Evolution of Statistical Events in a Model Wind-Turbine Array

Hot-wire anemometry data, obtained from a wind-tunnel experiment containing a $$3 \times 3$$ model wind-turbine array, are used to conditionally average the Reynolds stresses. Nine profiles at the centreline behind the array are analyzed to characterize the turbulent velocity statistics of the wake flow. Quadrant analysis yields statistical events occurring in the wake of the wind farm where quadrants 2 and 4 produce ejections and sweeps, respectively. The scaled difference between these two events is expressed via the $$\Delta R_{0}$$ parameter and is based on the $$\Delta S_{0}$$ quantity as introduced by M. R. Raupach (J Fluid Mech 108:363–382, 1981). $$\Delta R_{0}$$ attains a maximum value at hub height and changes sign near the top of the rotor. The ratio of quadrant events of upward momentum flux to those of the downward flux, known as the exuberance, is examined and reveals the effect of root vortices persisting to eight rotor diameters downstream. These events are then associated with the triple correlation term present in the turbulent kinetic energy equation of the fluctuations where it is found that ejections play the dual role of entraining mean kinetic energy while convecting turbulent kinetic energy out of the turbine canopy. The development of these various quantities possesses significance in closure models, and is assessed in light of wake remediation, energy transport and power fluctuations, where it is found that the maximum fluctuation is about $$30\%$$ of the mean power produced.

Non-equilibrium scaling laws in axisymmetric turbulent wakes

We present a combined direct numerical simulation and hot-wire anemometry study of an axisymmetric turbulent wake. The data lead to a revised theory of axisymmetric turbulent wakes which relies on the mean streamwise momentum and turbulent kinetic energy equations, self-similarity of the mean flow, turbulent kinetic energy, Reynolds shear stress and turbulent dissipation profiles, non-equilibrium dissipation scalings and an assumption of constant anisotropy. This theory is supported by the present data up to a distance of 100 times the wake generator’s size, which is as far as these data extend.

Four-way coupled simulations of small particles in turbulent channel flow: The effects of particle shape and Stokes number

This paper investigates the effects of particle shape and Stokes number on the behaviour of non-spherical particles in turbulent channel flow. Although there are a number of studies concerning spherical particles in turbulent flows, most important applications occurring in process, energy, and pharmaceutical industries deal with non-spherical particles. The computation employs a unique and novel four-way coupling with the Lagrangian point-particle approach. The fluid phase at low Reynolds number (Reτ = 150) is modelled by direct numerical simulation, while particles are tracked individually. Inter-particle and particle-wall collisions are also taken into account. To explore the effects of particles on the flow turbulence, the statistics of the fluid flow such as the fluid velocity, the terms in the turbulence kinetic energy equation, the slip velocity between the two phases and velocity correlations are analysed considering ellipsoidal particles with different inertia and aspect ratio. The results of the simulations show that the turbulence is considerably attenuated, even in the very dilute regime. The reduction of the turbulence intensity is predominant near the turbulence kinetic energy peak in the near wall region, where particles preferentially accumulate. Moreover, the elongated shape of ellipsoids strengthens the turbulence attenuation. In simulations with ellipsoidal particles, the fluid-particle interactions strongly depend on the orientation of the ellipsoids. In the near wall region, ellipsoids tend to align predominantly within the streamwise (x) and wall-normal (y) planes and perpendicular to the span-wise direction, whereas no preferential orientation in the central region of the channel is observed. Important conclusions from this work include the effective viscosity of the flow is not affected, the direct dissipation by the particles is negligible, and the primary mechanism by which the particles affect the flow is by altering the turbulence structure around the turbulence kinetic energy peak.

Hybrid LES/URANS simulation of turbulent natural convection by BEM

In this paper we have developed a hybrid LES/URANS turbulent model for a BEM based turbulent fluid flow solver. We employed the unified LES/URANS approach, where the interface between the LES and URANS regions is defined using a physical quantity, which dynamically changes during numerical simulation. The main characteristic of the unified hybrid model is that only one set of governing equations is used for fluid flow simulation in both the LES and URANS regions. Regions where turbulent kinetic energy is calculated by LES and URANS models are determined using a switching criterion. We used the Reynolds number based on turbulent kinetic energy and the Reynolds number based on total turbulent kinetic energy to establish the LES/URANS interface switching criterion. Depending on flow characteristics and with the use of switching criterion, we chose between sub-grid scale viscosity (SGS) and URANS effective viscosity. The SGS or URANS effective viscosity is used in the transport equation for turbulent kinetic energy and in governing equations for fluid flow. The developed numerical algorithm was tested by simulating turbulent natural convection within a square cavity. The hybrid turbulent model was implemented within a numerical algorithm based on the boundary element method, where single domain and sub-domain approaches are used. The governing equations are written in velocity–vorticity formulation. We used the false transient time scheme for the kinematics equation.

4D Flow MRI-based pressure loss estimation in stenotic flows: Evaluation using numerical simulations.

To assess how 4D flow MRI-based pressure and energy loss estimates correspond to net transstenotic pressure gradients (TPGnet) and their dependence on spatial resolution. Numerical velocity data of stenotic flow were obtained from computational fluid dynamics (CFD) simulations in geometries with varying stenosis degrees, poststenotic diameters and flow rates. MRI measurements were simulated at different spatial resolutions. The simplified and extended Bernoulli equations, Pressure-Poisson equation (PPE), and integration of turbulent kinetic energy (TKE) and viscous dissipation were compared against the true TPGnet . The simplified Bernoulli equation overestimated the true TPGnet (8.74 ± 0.67 versus 6.76 ± 0.54 mmHg). The extended Bernoulli equation performed better (6.57 ± 0.53 mmHg), although errors remained at low TPGnet . TPGnet estimations using the PPE were always close to zero. Total TKE and viscous dissipation correlated strongly with TPGnet for each geometry (r(2) > 0.93) and moderately considering all geometries (r(2) = 0.756 and r(2) = 0.776, respectively). TKE estimates were accurate and minorly impacted by resolution. Viscous dissipation was overall underestimated and resolution dependent. Several parameters overestimate or are not linearly related to TPGnet and/or depend on spatial resolution. Considering idealized axisymmetric geometries and in absence of noise, TPGnet was best estimated using the extended Bernoulli equation. Magnetic Resonance in Medicine published by Wiley Periodicals, Inc. on behalf of International Society for Magnetic Resonance.

One-Equation Turbulence Model Based on Extended Bradshaw Assumption

A one-equation turbulence model is presented. The modeling tries to adopt a more reliable source and less empirical coefficients. The governing equation is the turbulent kinetic energy equation, in which is closed phenomenologically. The stress intensity ratio is calibrated to be a function controlled by the local variables but not a constant 0.3. The extension of the Bradshaw assumption turns out to be of good accuracy, forming a new Reynolds stress constitutive relation. The two model coefficients are both extracted from one slice of the flat-plate boundary layer, yet extensive tests show that the model can significantly improve the precision of prediction for the boundary-layer flows, the adverse pressure gradient, shallow separation, transport effects, and massively separated flows.

Large-Eddy Simulation Study of Thermally Stratified Canopy Flow

A number of large-eddy simulations (LES) are performed for the calculation of the airflow over a horizontally homogeneous forest canopy for a wide range of thermal stability classes. For the first time, results from LES of a stably stratified canopy are also presented. Simulation results compare favourably to recent field measurements over a pine forest in south-eastern Sweden. The simple heat source model was found to perform adequately and to yield within-canopy heat-flux profiles typically observed for stable conditions in the field. Evidence was found for a layer of unstably stratified air in the canopy trunk space under stable stratification. The importance of a secondary wind-speed maximum is emphasized in stable conditions. Examination of the budget equation of turbulent kinetic energy (TKE) revealed that, during stable stratification, pressure transport plays an increasingly important role in supplying the canopy region with TKE.

Compressibility effects and turbulent kinetic energy exchange in temporal mixing layers

The kinetic energy exchange between the mean and fluctuating fields is analyzed using large-eddy simulation of a temporally developing compressible mixing layer. A solution database is generated by varying the convective Mach number over the interval of with an initial Reynolds number of Re = 100. Both mean and turbulent kinetic energy equations are considered, and the effect of compressibility on the production, pressure dilatation, and dissipation terms are considered. It is shown that the production term linearly decreases with the increase of within the interval of , which in turn leads to a linear reduction of the mixing layer growth rate. The pressure dilatation term in the mean kinetic energy equation is on average positive and transfers internal energy into mean kinetic energy by dilatation, whereas its counterpart in the turbulent kinetic energy equation is on average negative, and transfers turbulent kinetic energy into internal energy by compression. Although the turbulent viscous dissipation term is reduced by increasing the convective Mach number, it remains the primary kinetic energy dissipation mechanism at all convective Mach numbers. At the subgrid-scale level, it is found that the magnitude of the dynamic Smagorinsky model coefficient, Cs, monotonically decreases with the increase of and results in the reduction of all turbulent stress tensor components. Furthermore, increasing the convective Mach number results in a monotonic reduction of subgrid-scale dissipation of the turbulent kinetic energy, whereas the subgrid-scale dissipation of the mean kinetic energy stays almost unaffected. The sum of subgrid-scale pressure dilatation and pressure transport terms, which is modelled with a gradient transport model, is also suppressed by the increase of compressibility and the convective Mach number.

Statistical behavior of supersonic turbulent boundary layers with heat transfer at [formula omitted

Direct numerical simulations (DNS) of supersonic turbulent boundary layers (STBL) over adiabatic and isothermal walls are performed to investigate the effects of wall heat transfer on turbulent statistics and near wall behaviors. Four different cases of adiabatic, quasi-adiabatic, and uniform hot and cold wall temperatures are considered. Based on the analysis of the current database, it is observed that even though the turbulent Mach number is below 0.3, the wall heat transfer modifies the behavior of near-wall turbulence. These modifications are investigated and identified using both instantaneous fields (i.e. scatter plots) and mean quantities. Morkovin’s hypothesis for compressible turbulent flows is found to be valid for neither heated nor cooled case. It is further uncovered that although some near-wall asymptotic behaviors change upon using weak or strong adiabatic walls, respectively denote the isothermal and iso-flux walls, basic turbulent statistics are not affected by the thermal boundary condition itself. We also show that among different definition of Reynolds number used in STBL, the Reynolds number based on the friction velocity has some advantages data comparison regarding the first and second order statistical moments. More in depth analyses are also performed using the balance equation for turbulent kinetic energy (TKE) budget, as well as the dissipation rate. It is found that the dilatational to solenoidal dissipation ratio increases/decreases when heating/cooling the walls. The DNS of the current STBLs are available online for the community.

Representing Richardson Number Hysteresis in the NWP Boundary Layer

Abstract Turbulence in the planetary boundary layer (PBL) transports heat, momentum, and moisture in eddies that are not resolvable by current NWP systems. Numerical models typically parameterize this process using vertical diffusion operators whose coefficients depend on the intensity of the expected turbulence. The PBL scheme employed in this study uses a one-and-a-half-order closure based on a predictive equation for the turbulent kinetic energy (TKE). For a stably stratified fluid, the growth and decay of TKE is largely controlled by the dynamic stability of the flow as represented by the Richardson number. Although the existence of a critical Richardson number that uniquely separates turbulent and laminar regimes is predicted by linear theory and perturbation analysis, observational evidence and total energy arguments suggest that its value is highly uncertain. This can be explained in part by the apparent presence of turbulence regime-dependent critical values, a property known as Richardson number hysteresis. In this study, a parameterization of Richardson number hysteresis is proposed. The impact of including this effect is evaluated in systems of increasing complexity: a single-column model, a forecast case study, and a full assimilation cycle. It is shown that accounting for a hysteretic loop in the TKE equation improves guidance for a canonical freezing rain event by reducing the diffusive elimination of the warm nose aloft, thus improving the model’s representation of PBL profiles. Systematic enhancements in predictive skill further suggest that representing Richardson number hysteresis in PBL schemes using higher-order closures has the potential to yield important and physically relevant improvements in guidance quality.

Entrainment of air into an infrared suppression (IRS) device using circular and non-circular multiple nozzles

The present investigation predicts the mass entrainment into an IRS device which is used in naval or merchant ships. A high velocity jet emanating from the nozzle exit entrains ambient air to cool the hot combustion products of the gas turbines. Experiments have been carried out on a laboratory scale IRS device to measure the entrainment ratio. An optimum funnel overlap height (Hoverlap/Deq,nz=0) has been found out from the experiments which shows the highest mass suction rate at that overlap height. Moreover, the entrainment is observed to increase with the inlet mass flow rate of the nozzle. For a full scale IRS device, the conservation equations for mass, momentum and turbulent kinetic energy as well as its dissipation rate are solved using finite volume techniques in a three-dimensional computational domain by employing eddy viscosity based k–ɛ turbulence model with standard wall function. The parameters such as the nozzle aspect ratio (AR), geometrical shape of the nozzles, pitch circle diameter (PCD) and multiple nozzles affect the entrainment ratio significantly. With five triangular nozzles (AR=1), the entrainment rate is enhanced by 15.7% as compared to a single triangular nozzle (AR=1).