Transport Equation For Kinetic Energy Research Articles

Flow Field Characteristics at the Spanwise Edge of a Vegetative Canopy Model

This study investigates the complex flow field across a spanwise vegetative model canopy edge focusing on turbulent transport processes. Utilizing stereoscopic particle image velocimetry, the three velocity components were measured in wall-parallel planes at various elevations within canopy across the spanwise canopy edge. Conventional ensemble averaged results were contrasted with those obtained by conditionally averaged flow properties across instantaneous internal interfaces in the flow to understand their contribution to the ensemble average. The conditional average captured the strong gradients in mean velocities, Reynolds stresses, vorticity, swirling strength, and turbulent kinetic energy production across the dynamically changing instantaneous interface. In contrast, the conventional ensemble average smeared out the strong gradients. Small magnitudes of advective terms in the turbulent kinetic energy transport equation suggested weak secondary transverse flows in the present model canopy. The turbulent flow structure across the spanwise canopy edge was further investigated using Quadrant-Hole analysis for both averaging approaches. Conventional ensemble averaged results indicated a shift from sweep to ejection dominance when moving from canopy into the open patch, while the conditional average showed only sweep dominated transport. In contrast to a homogeneous canopy layout, below canopy height at the canopy edge, sweeps and ejections lose their dominance in vertical turbulent transport. The present results show that the dynamics of internal interfaces govern the ensemble averaged results and a possible implementation into existing models is proposed. The present results are expected to increase understanding of spanwise turbulent transport and aid in developing strategies to mitigate desertification.

A modification of production term for the shear stress transport turbulence model based on a new wall-law for adverse pressure gradient

This paper presents an attempt to improve the predictive accuracy of the Menter shear stress transport (SST) turbulence model for adverse pressure gradient (APG) flows. The foundation of this work lies in the newly modified law of the wall and turbulent kinetic energy (TKE) transport equation within the APG turbulent boundary layer. The modification of the law of the wall is achieved through a newly proposed variable λ, which is derived theoretically and calibrated using direct numerical simulation data. Based on the new law of the wall, we analyze the TKE transport equation of the SST model at APG and demonstrate that the analytical solution satisfying the law of the wall cannot achieve equilibrium in the current SST model. Further theoretical analysis reveals that this imbalance is caused by the TKE production term Pk, and the decomposition of the wall friction coefficient Cf indicates that Pk directly affects the prediction of flow separation. Based on the analysis and the new law of the wall, an additional term for Pk is suggested to correct Pk and improve the model's predictive accuracy for APG flows. Numerical validation results show that the correction leads to an increase in Pk within the APG turbulent boundary layer, resulting in more accurate predictions of Cf and the mean velocity profile in the inner layer of the turbulent boundary layer.

Data-driven approach for modeling Reynolds stress tensor with invariance preservation

The present study represents a data-driven turbulent model with Galilean invariance preservation based on machine learning algorithm. The fully connected neural network (FCNN) and tensor basis neural network (TBNN) (Ling et al., 2016) are involved and applied. The models are trained based on five kinds of flow cases with Reynolds Averaged Navier–Stokes (RANS) and high-fidelity data. The mappings between two invariant sets, mean strain rate tensor and mean rotation rate tensor as well as additional consideration of invariants of turbulent kinetic energy gradients, and the Reynolds stress anisotropy tensor are trained. The prediction of the Reynolds stress anisotropy tensor is treated as user’s defined RANS turbulent model with a modified turbulent kinetic energy transport equation. The results show that both FCNN and TBNN models can provide more accurate predictions of the anisotropy tensor and turbulent state in square duct flow and periodic flow cases compared to the RANS model. The machine learning based turbulent model with turbulent kinetic energy gradient related invariants can improve the prediction precision compared with only mean strain rate tensor and mean rotation rate tensor based models. The TBNN model is able to predict a better flow velocity profile compared with FCNN model due to a prior physical knowledge.

Comparing local energy cascade rates in isotropic turbulence using structure-function and filtering formulations

Two common definitions of the spatially local rate of kinetic energy cascade at some scale $\ell$ in turbulent flows are (i) the cubic velocity difference term appearing in the ‘scale-integrated local Kolmogorov–Hill’ equation (structure-function approach), and (ii) the subfilter-scale energy flux term in the transport equation for subgrid-scale kinetic energy (filtering approach). We perform a comparative study of both quantities based on direct numerical simulation data of isotropic turbulence at Taylor-scale Reynolds number 1250. While in the past observations of negative subfilter-scale energy flux (backscatter) have led to debates regarding interpretation and relevance of such observations, we argue that the interpretation of the local structure-function-based cascade rate definition is unambiguous since it arises from a divergence term in scale space. Conditional averaging is used to explore the relationship between the local cascade rate and the local filtered viscous dissipation rate as well as filtered velocity gradient tensor properties such as its invariants. We find statistically robust evidence of inverse cascade when both the large-scale rotation rate is strong and the large-scale strain rate is weak. Even stronger net inverse cascading is observed in the ‘vortex compression’ $R>0$ , $Q>0$ quadrant, where $R$ and $Q$ are velocity gradient invariants. Qualitatively similar but quantitatively much weaker trends are observed for the conditionally averaged subfilter-scale energy flux. Flow visualizations show consistent trends, namely that spatially, the inverse cascade events appear to be located within large-scale vortices, specifically in subregions when $R$ is large.

Fluctuation characteristics induced by energetic coherent structures in air-core vortex: The most complex vortex in the tidal power station intake system

The air-core vortex is the most complex vortex in the intake system of tidal power stations. Nearly all power plants prohibit the operation of units with air-core vortices, as these vortices not only contain air but also induce significant inflow fluctuations that can potentially lead to significant damage to the units. The current understanding of the fluctuation characteristics and mechanisms induced by energetic coherent structures in the air-core vortex flow is very limited. To address this, the coupled level-set and volume-of-fluid large-eddy simulations are conducted to investigate the energetic coherent structures that cause power output fluctuations. Two critical frequencies have been identified in the energy spectra of the velocity field: the air-core vortex meandering frequency and the shear vortex shedding frequency. The level and trend of the spectrum density at low frequencies are determined by the large-scale energy structure represented by the air-core vortex. The shear vortices are initially formed from a shear separation layer that eventually evolves into streamwise vortices downstream. When the air-core vortex enters the pipe, it breaks up and merges with the near-wall coherent structures. The enhanced velocity and pressure fluctuations, induced by air-core vortices and coherence structures, are analyzed using the turbulent kinetic energy transport equation, revealing that the production term dominates the development of turbulent kinetic energy. Moreover, the diffusion term is responsible for transferring energy through the evolution of turbulent structures. The energy loss induced by fluctuations is investigated using the entropy production method. We find that the direct dissipation of entropy production is mainly induced by air-core vortices, while the side-wall coherent structures in the pipe contribute to the turbulent dissipation.

Turbulent boundary layer flow over a three-dimensional sinusoidal surface

The sinusoidal roughness effect is investigated using a direct numerical simulation (DNS) of a spatially developing turbulent boundary layer (TBL) over three-dimensional sinusoidal roughness. The validity of Townsend's outer-layer similarity hypothesis is assessed based on comparisons of mean and second-order flow statistics, with a DNS of smooth-wall TBL data set at a similar Reynolds number. The total, Reynolds and dispersive stress tensors are calculated using the double-averaging procedure. The mean and second-order statistical similarities in the outer layer between rough-wall and smooth-wall TBLs are generally observed. The transport between total, turbulent and dispersive kinetic energy is investigated utilising triple-decomposed kinetic energy transports equations. The transport behaviour of turbulent kinetic energy (TKE) is significantly affected by the local mean shear induced by the surface roughness. However, the TKE transport shows good collapse with the smooth-wall case in the outer region of the flow. On the other hand, the transport of dispersive kinetic energy, including local production, redistribution and dissipation, are confined within the roughness sublayer. The intercomponent transfer between TKE and dispersive kinetic energy is quantified from the triple-decomposed kinetic energy transport equations. The intercomponent energy transfer is associated with the local spatial gradients of the turbulent momentum fluxes generated near the roughness canopy.

Thermohydraulic performance assessment of a solar air heater with equilateral‐triangular, trapezoidal, and square sectional ribs on the absorber plate: A comparative study

AbstractHeat transfer systems have been employed in numerous industrial and residential settings to convert and recover heat. One effective approach for enhancing heat transfer rates involves incorporating artificial roughness in the form of regularly spaced ribs on the heated surface. In this particular study, the researchers conducted numerical simulations to analyze the turbulent heat and flow properties within a rectangular duct of a solar air heater (SAH). The duct featured three different types of transverse rib shapes: equilateral‐triangular, trapezoidal, and square sectional ribs. The governing equations have been solved numerically using a finite‐volume technique via ANSYS FLUENT. For solving the transport equations of turbulent kinetic energy and its dissipation rate, the RNG k– turbulence model was utilized. At a broad range of Reynolds numbers (6000 ≤ Re ≤ 18,000), relative roughness height (), and a fixed relative roughness pitch (), the Nusselt number (), friction factor (), and thermal performance factor (TPF) have been examined. The maximal Nu was obtained for equilateral‐triangular ribs, while the minimum friction factor was seen for square sectional ribs. The highest TPF of 1.591 was achieved with equilateral‐triangular ribs at and e/Dh = 0.032. Contours of kinetic energy, turbulent intensity, velocity, and pressure produced significant findings for comprehending the SAH's physical phenomena.

Compressible Rayleigh–Taylor instability subject to isochoric initial background state

The effect of compressibility on the single-mode Rayleigh–Taylor instability is examined using two (2D) and three-dimensional (3D) direct numerical simulations. To isolate compressibility from background stratification effects, this work employs a constant density profile on each side of the interface. The numerical simulations are performed at various Reynolds numbers using the gas kinetic method for static Mach numbers up to M = 0.4. The most important finding is that compressibility acting in isolation enhances the instability and perturbations grows faster with increasing Mach number, unlike previous results with background isothermal state, which show suppression of the instability at higher static Mach numbers. In addition, compressibility is also shown to increase the bubble-spike asymmetry. While the instability grows faster for the 3D case, the findings are qualitatively similar in 2D and 3D. The dynamical reasons underlying the effect of compressibility are elucidated by examining the evolution of vorticity and turbulent kinetic energy transport equations.

A new composite modelling strategy for engineering computation of turbulence modulation in dilute solid-liquid two-phase flows

Turbulence modulation is an important physical phenomenon in solid-liquid two-phase flows, including two co-effects of wake enhancement and resistance dissipation. This physical phenomenon is usually modelled in an integrated way by adding two corresponding source terms to the turbulent kinetic energy (TKE) transport equation. However, this modelling strategy may easily cause the problem of explicit interference between the two co-effects, while the influencing factors of turbulence modulation are also not fully reflected, affecting the prediction accuracy of turbulence intensity. A new composite modelling strategy for turbulence modulation in dilute solid-liquid two-phase flows is proposed in this article. In the Euler-Euler framework, the two co-effects of turbulence modulation are modelled in a separate way to avoid the explicit interference. The resistance dissipation term is independently added to the TKE transport equation as a TKE change rate, while the wake enhancement term is independently added to the eddy viscosity as a TKE increment. On this basis, the appropriate mathematical expressions are determined, the influencing factors of the two effects are considered individually to reflect the dynamic characteristics of particles as fully as possible, and the empirical coefficients are calibrated for the condition of dilute solid-liquid two-phase flows to improve the applicability to the flow medium. Four typical flow cases are tested, and it is demonstrated that the new composite modelling strategy can better reflect the modulation of particles on the TKE of liquids, thereby providing higher-precision simulation results without significant increase in computational cost for dilute solid-liquid two-phase flows.

Physics-informed neural networks for the Reynolds-Averaged Navier–Stokes modeling of Rayleigh–Taylor turbulent mixing

Recently, Physics-informed neural networks (PINNs) have proven to be an efficient machine-learning method for solving partial differential equations. However, this method can be quite challenging when solving complex problems with shock/material discontinuities or multi-scale features, such as turbulence. In this paper, we propose an improved PINNs framework for solving the Reynolds-averaged Navier–Stokes (RANS) equations for turbulent mixing induced by the Rayleigh–Taylor (RT) instability. The RANS model is based on the closure form of the K–L model. However, the transport equations of the turbulent kinetic energy K and turbulent length scale L are not included and are instead predicted directly by neural networks, thus resulting in an inverse problem. Several modifications are made to the original PINNs to improve its applicability to RT turbulent mixing and accelerate the training optimization process. We first examine the applicability of the PINNs for solving multi-material Euler equations without considering turbulence. Then, PINNs is applied to the RT turbulent mixing problem using training data from the traditional K–L model. The results confirm the ability of the PINNs to predict the entire spatiotemporal field using limited training data. Next, we further train the PINNs using data from the implicit large eddy simulation (ILES), which yields a PINN-based turbulence model that performs better than the traditional K–L model. These results shed light on further applications of PINNs for complex problems, particularly those with limited measurement data and unknown physical models.

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.

Spectral analysis of turbulence kinetic and internal energy budgets in hypersonic turbulent boundary layers

The spectral transport equations of turbulence kinetic and internal energy (TKE and TIE) are rigorously derived in the compressible turbulent flows, exhibiting a full structural similarity in terms of two-point correlations of velocity and a sound-speed-like variable. The scale dependence of TKE and TIE flow across space and scales, and among components is revealed comprehensively in a similar manner, in hypersonic turbulent boundary layers. Special attention is paid to the effect of wall cooling on the energy evolution, which is observed to exist throughout the boundary layer primarily by means of modifying the small-scale motions.

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.

Significance of recirculating micro-bubbles in oxygenation from poly-dispersed plumes

Poly-disperse bubble plumes are simulated to explore the effect of interactions between multiple bubbles of varying sizes on the liquid flow, plume dynamics, and oxygen transfer in aeration processes. An Eulerian-Lagrangian model based on the multiphase particle-in-cell method (MPPIC) implemented in OpenFOAMTM has been extended to account for bubbly flow. Bubbles have been approximated as spherical particles whose diameters change in response to pressure and nitrogen and oxygen mass dissolution and transfer. The mass transfer model relies on a Sherwood number correlation which accounts for the individual bubble size and relative velocity and is parameterized against the liquid phase temperature. A Large Eddy Simulation (LES) approach based on a transport equation for the turbulent kinetic energy has been adopted to model the effect of the sub-grid scales (SGS) of turbulence on the liquid flow and bubble motion. The bubble injection is defined in terms of the air mass transfer and the bubble size distribution. The model has been compared against experimental measurements of the evolution in oxygen concentration in a 1m deep water tank in response to bubble plumes generated from a porous ceramic plates at two different flow rates. The interaction between milli- and micro-bubbles over a wide range of sizes has been shown to create significant differences in the aeration process. A method to correct for bubble recirculation has been proposed and allows for good agreement between measured and simulated volumetric transfer rates. The simulation has shown the important role that the interplay between bubble sizes takes in determining the oxygen transfer efficiency. Larger standard deviations from Gaussian bubble size distributions have been shown in particular to reduce recirculation from smaller bubbles thereby reducing the aeration efficiency at an equivalent flow rate. An analysis of the age distribution by bubble size plumes suggest that the largest bubble have an important role in increasing the entrainment of smaller bubbles limiting their recirculation and reducing their retention time.

Analysis of the kinetic energy recovery behind a tidal stream turbine for various submergence levels

Tidal turbines are commonly deployed at sea sites with water depths of up to 50 m to ease their deployment and quick maintenance operations. In these relatively shallow water depth conditions, the vertical expansion of tidal stream turbine wakes is restricted by the proximity of the rotor blades to the bottom bed and free-surface layer. These physical constrains can lead to changes in the flow mechanisms that drive momentum recovery behind the turbines, e.g. limiting the vertical fluxes of velocity. Understanding how the wake recovers depending on the submergence ratio is of utmost importance to designing the future multi-row tidal turbine arrays. Here, we adopt high-fidelity Large-Eddy Simulations (LES) with an Actuator Line Method (ALM) to represent the turbine's rotor to analyse the mean flow and transport equation for mean kinetic energy (MKE) behind a single bottom-fixed tidal turbine for four water depth values. Our results show that the close proximity of the turbine blade tip to the free-surface can notably constrain the wake expansion, with very shallow conditions leading to a limited contribution to the MKE replenishment of the turbulent momentum exchange over the vertical direction. Conversely, under such shallow conditions, the horizontal flux of MKE is enhanced over the lateral boundaries of the downstream wake. Our study evidences that the ratio of water depth to turbine diameter plays a relevant role in future tidal arrays and needs to be correctly accounted for in numerical models to provide reliable results.

Numerical investigation of cavitation-vortex structures around a sphere with boundary data immersion method

The boundary data immersion method (BDIM) coupled with the artificial compressibility method is applied to investigate the interaction of the cavitation-vortex dynamics around a sphere. The convolution of governing equation with a nascent delta kernel is employed to smooth the transition region between the solid and the fluid subdomain. Therefore, the BDIM addresses a key problem with smooth immersion boundary method in representing the discontinuity of the velocity gradient ∂u/∂y at the wall when considering flows with intermediate to high Reynolds numbers. In addition, a pseudo-time derivative of the pressure multiplied by an artificial-compressibility factor is added to the continuity equation. The developed coupling algorithm can accurately capture the unsteady cavitation characteristics, which is consistent with the previous experimental data. Subsequently, the typical vortex identification methods and the turbulent kinetic energy (TKE) transport equation are applied to analyze the interaction between the cavitation dynamics and turbulence vortices structures. A series of turbulent vortices are captured with the cloud cavity shedding. Moreover, it is found that the unsteady cavity topology will promote the generation of TKE.

A new dynamic subgrid-scale model using artificial neural network for compressible flow

The subgrid-scale (SGS) kinetic energy has been used to predict the SGS stress in compressible flow and it was resolved through the SGS kinetic energy transport equation in past studies. In this paper, a new SGS eddy-viscosity model is proposed using artificial neural network to obtain the SGS kinetic energy precisely, instead of using the SGS kinetic energy equation. Using the infinite series expansion and reserving the first term of the expanded term, we obtain an approximated SGS kinetic energy, which has a high correlation with the real SGS kinetic energy. Then, the coefficient of the modelled SGS kinetic energy is resolved by the artificial neural network and the modelled SGS kinetic energy is more accurate through this method compared to the SGS kinetic energy obtained from the SGS kinetic energy equation. The coefficients of the SGS stress and SGS heat flux terms are determined by the dynamic procedure. The new model is tested in the compressible turbulent channel flow. From the a posterior tests, we know that the new model can precisely predict the mean velocity, the Reynolds stress, the mean temperature and turbulence intensities, etc.

Polymer-turbulence interactions in a complex flow and implications for the drag reduction phenomenon

We present direct numerical simulation data for turbulent duct flow of a finite-extensibility non-linear elastic dumbbell model with the Peterlin approximation (FENE-P) fluid in the high drag reduction regime. While the secondary flow pattern is qualitatively similar to that in a Newtonian fluid, its magnitude is significantly reduced, resulting in a less uniformly distributed velocity profile and hence smaller gradients at the wall. The Reynolds stress tensor in the polymer-laden flow was found to be increasingly anisotropic with most of the turbulent kinetic energy retained in the streamwise component, u′u′¯. We introduce a novel approach for investigating polymer stretching using the anisotropy invariant map of the polymer stress tensor and observe the persistence of both uniaxial and biaxial extension. Analysis of the transport equation for the mean kinetic energy indicates that polymer stretching and relaxation is a highly dissipative process; hence, the introduction of an additional channel for dissipation in a flow is key to drag reduction.

Effects of Body Forces on Turbulent Kinetic Energy Transport in Premixed Flames

The effects of buoyancy on turbulent premixed flames are expected to be strong due to the large changes in density between the unburned and fully burned gases. The present work utilises three-dimensional direct numerical simulations of statistically planar turbulent premixed flames under decaying turbulence to study the influence of buoyancy on the evolution of turbulent kinetic energy within the flame brush. Four sets of turbulence parameters have been studied, with four different values of Froude number for each set. It is found that for a given set of turbulence parameters, flame wrinkling increases with an increase in body force magnitude in the case of unstable stratification, which is also reflected in the increased values of turbulent burning velocity and flame surface area. An increase in body force magnitude in the case of stable stratification acts to reduce the extent of flame wrinkling. Turbulent kinetic energy and its dissipation rate are found to be affected by both the magnitude and direction of the body force. For low turbulence intensities considered here, turbulent kinetic energy increases from the leading edge of the flame brush before decaying eventually towards the product side of the flame brush. For high turbulence intensities, the turbulent kinetic energy is found to decay across the flame brush, and it is also found that the effect of body force on the evolution of turbulent kinetic energy is marginal in the case of high turbulence intensities. The effects of body force magnitude and direction on the statistical behaviours and closures of the various terms of the turbulent kinetic energy transport equation are analysed, and existing models have been modified to account for Froude number effects, where necessary.

A Comparative Study of the Buoyancy-Opposed Wall Jet using Different Turbulent Models

A comparative study of the buoyancy-opposed wall jet has been carried out using RANS methods (including RNG model, Realizable k-ε model, and two low Reynolds number k-ε models) and LES methods (including the subgrid scale model developed by Smagorinsky et al.(1963), Germano et al. (1991) and Kim et al. (1997)). The capability of each turbulence model to predict the flow field and temperature field in mixing stage was investigated. The results show that the k-ε series model can accurately predict the velocity distribution of flow field under isothermal case. However, in the case of buoyancy, due to the assumption of turbulent normal stress isotropy, the trend of temperature change in the mixing region and transition position existed an obvious deviation with experimental data. The LES methods, solved directly the large scale vortices, take into account the influence of turbulence stress anisotropy in the mixing region on the temperature change and capture the temperature change trend over the whole domain accurately. Due to the application of the subgrid kinetic energy transport equation, KET model has certain advantages in numerical simulation of similar engineering flow phenomenon.