Turbulent Diffusion Term Research Articles

Variational calculus in hybrid turbulence transport models with passive scalar

Non-zonal methods of hybrid Reynolds Averaged Navier–Stokes (RANS) equations-Large Eddy Simulation (LES) with subfilter transport equations enable to simulate turbulent flows out of spectral equilibrium on relatively coarse grid resolution. They can operate from RANS to LES depending on a control parameter linked to the grid step size. Variational analysis gives a useful framework to specify the functional dependence of this parameter to the grid size. This is the case for the partially integrated transport modelling method originally established in spectral space for homogeneous turbulence. The partitioning control function then monitors the ratio of the subfilter part to the total turbulent energy and the same process is developed in the present work for the thermal or transported scalar variance as well. So, we demonstrate here that this control function mechanism can be derived by variational calculus from a mathematical physics formalism developed both for first-order and second-order moment closures. We show that the result is entirely consistent with the spectral model derivation made in previous papers and that this approach can be transposed to almost any subfilter transport model developed in hybrid RANS-LES methodology in full generality. As a result, it will be evidenced also that resolved turbulent diffusion terms play a significant role in the acting mechanisms of turbulence and cannot be therefore neglected, even if these terms do not explicitly appear in LES subfilter closure because they are computed by the simulation itself and not modelled by the subfilter model. In addition, numerical simulations have been performed for illustrating the theoretical results of the variational analysis.

A well-balanced and positivity-preserving numerical model for overland flow under vegetation effects

In this study, we used the depth-averaged shallow water equations for modeling flows through vegetation field. The vegetation effects on flow are modeled using Morison’s equation taking into account drag and inertia forces which depend on both vegetation and flow properties. We compute and compare different formulations for the stem drag coefficient based on the Froude number or the vegetation volume fraction. Vegetation-induced turbulence is taken into account by adding diffusion terms in the momentum equations. The resulting system of equations is solved using a well-balanced and positivity preserving finite volume method to guarantee the balance between the flux and bed topography source terms, and the positivity of the computed water depth. In our approach, the drag force and bed friction source terms are combined into a unified form. We propose to discretize the obtained term using an implicit temporal method where an analytical technique is used. Special discretization techniques are used for the inertia force and turbulent diffusion terms. Numerical simulations are performed to validate the accuracy of the proposed numerical model. We investigate and compare different formulations for the stem drag coefficient in the vegetation model. Our results confirm the capability of the proposed numerical model for simulating overland flows under vegetation effects.

Modeling of turbulent diffusion terms for one-dimensional interfacial area transport equation in vertical round channels

The two-fluid model is comprised of the mass, momentum, and energy equations for liquid and gas phases separately. The terms of interfacial transfer govern mass, momentum, and energy transfer across the gas-liquid interface. The interfacial area concentration is critical to formulating the interfacial transfer terms. The one-dimensional Interfacial Area Transport Equation (IATE) has excellent potential in dynamically predicting the interfacial area concentration. The sink and source terms modeling for the interfacial area concentration is vital in the successful IATE development. The current IATE in bubbly flows considers the sink terms caused by bubble coalescence owing to random bubble collision and wake entrainment. The source term is caused by bubble breakup owing to turbulent impact. The turbulent diffusion term has not been considered in the current practice of one-dimensional IATE. This study developed the equation to predict the turbulent diffusion terms in the one-dimensional IATE for bubbly flow in vertical round channels. The predictive capability of the developed equation was validated with the collected database. The non-dimensionalized turbulent diffusion terms predicted by the developed equation agreed with those directly calculated by the experimental data. The developed turbulent diffusion term was also implemented in the one-dimensional IATE. The interfacial area concentrations predicted by the one-dimensional IATE with the turbulent diffusion term had a satisfactory agreement with the experimental results by the statistical parameters of the mean relative deviation of 3.76% and the mean absolute relative deviation of 10.1%.

CFD modeling of traffic tidal flow: Assessment of pollutant dispersion

Pollutant distribution remains poorly understood when traffic tidal flows (TTFs) happen. By conducting computational fluid dynamic (CFD) simulations, the research efforts first focus on how different-side traffic-produced flow and turbulence (TPFT) affect in-canyon airflow and corresponding pollutant dispersion. Second, the composite effects of non-uniform traffic emission and TPFT are investigated. Finally, the influences of TTFs while varying different street canyon geometry and approaching wind condition is explored.The results demonstrate that the turbulent diffusion terms enhanced by the traffic movement contribute to the pollutant dispersion around traffic lanes. Besides, both-side TPFTs push leeward pollutants towards traffic flow “downstream” due to the unidirectional advection terms along the traffic direction. Simultaneously considering the non-uniform traffic emission and TPFT, either leeward- or windward-congested TTFs has a higher concentration at the pedestrian level close to the congested traffic lane. Besides, the TTF with windward congestion has a higher volume-average concentration of the whole street canyon. With varying building separation, street canyon aspect ratio, and incoming wind direction, the TTFs still result in a larger pollutant accumulation above the pedestrian level, which is nearby the congested traffic flow.

Exact Eddy-Viscosity Equation for Turbulent Wall Flows—Implications for Computational Fluid Dynamics Models

For turbulent channel flow, pipe flow, and zero-pressure gradient boundary layer, Heinz yielded recently analytical formulas for the eddy viscosity as a product of a function of (the wall-normal distance scaled in inner units) and a function of (the same scaled in outer units). By calculating the eddy-viscosity turbulent diffusion term, an exact high-Reynolds-number equation with one production and two dissipation terms is constructed for those flows. One dissipation term is universal, peaks near the wall, and scales mainly with . The second, smaller one, is flow dependent, peaks in the wake, and scales mainly with . The production term is flow dependent, peaks in between, and scales similarly. The universal dissipation term implies a length scale analogous to the von Karman length scale used in the scale-adaptive simulation models of Menter. This length scale also appears in the production term. This confirms the relevance of these length scales. An asymptotic analysis of all terms in the budget in the limit of infinite Reynolds numbers is provided. This yields a test bench of existing Reynolds-averaged Navier–Stokes models with a similar equation. It is shown that some models, e.g., the one of Spalart and Allmaras, do not respect the flow physics: they display a production peak in the near-wall region. The most promising model, a scale-adaptive simulation model, is modified. As a step forward toward a solution to the wall damping problem, the equation of our model behaves much more correctly in the near-wall region.

Dynamics of a single bubble rising in a quiescent medium

In the present work, an experimental analysis was performed to characterise the flow field around a single bubble of different diameters ∼ 2.77–3.53 mm) rising in a quiescent medium aiming to determine the effect of bubble size on kinetic energy distribution. The velocity field was measured using a non-intrusive particle image velocimetry (PIV) technique and kinetic energy spectrum was determined in both transverse and longitudinal directions applying a Fast Fourier Transformation (FFT). Both small- and large-scale motions of the flow field were identified and separated using a discreate wavelet transformation (DWT) method. It was found that the energy spectrum of the large-scale motions depended on the bubble size while the small-scale energy spectrum was nearly independent of it. The slopes of the energy spectrum were found to be close to -5/3 and -3 for the large- and small-scale regimes, respectively and the transition of slope was observed to occur at the wavenumber corresponding to the bubble diameter. Using the measured velocity field data, a turbulence kinetic energy (TKE) budget analysis was performed involving five components namely kinetic energy production, turbulent transport, pressure diffusion, viscous diffusion, and energy dissipation. Overall, it was observed that in the vicinity of bubble surface, turbulence production term was not entirely balanced by the dissipation term; and turbulent transport and pressure diffusion term also had significant contributions.

Application of Turbulent Diffusion Term of Aerosols in Mesoscale Model

AbstractThe presence of unfavorable meteorological conditions triggers pollution, and then further weakens turbulence, especially in the stable boundary layer (SBL), which is a frequent situation in heavy pollution episodes in China. The inapplicability of the classical Monin‐Obukhov similarity theory and the uncertainty of the planetary boundary layer height can lead to large deviation of turbulent diffusion in the SBL in numerical simulations. However, in current mesoscale models, no term has been used to accurately describe the turbulent diffusion of aerosols. Therefore, we use the Mixing‐Length theory to obtain the turbulent diffusion term of aerosols based on high‐resolution observational data, and, for the first time, embed this term into a mesoscale model, which makes the turbulent diffusion process of aerosols more truly depicted. Results from a two‐way coupled atmospheric‐chemistry mesoscale model demonstrate that the turbulent diffusion term of aerosols can improve the problem of overestimated PM2.5 concentration in Eastern China.

Low complexity model to study scale dependence of phytoplankton dynamics in the tropical Pacific.

We demonstrate that a simple model based on reaction-diffusion-advection (RDA) equation forced by realistic surface velocities and nutrients is skilled in reproducing the distributions of the surface phytoplankton chlorophyll in the tropical Pacific. We use the low-complexity RDA model to investigate the scale relationships in the impact of different drivers (turbulent diffusion, mean and eddy advection, primary productivity) on the phytoplankton chlorophyll concentrations. We find that in the 1/4^{∘} (∼25km) model, advection has a substantial impact on the rate of primary productivity, while the turbulent diffusion term has a fairly negligible impact. Turbulent diffusion has an impact on the phytoplankton variability, with the impact being scale propagated and amplified by the larger scale surface currents. We investigate the impact of a surface nutrient decline and some changes to mesoscale eddy kinetic energy (climate change projections) on the surface phytoplankton concentrations. The RDA model suggests that unless mesoscale eddies radically change, phytoplankton chlorophyll scales sublinearly with the nutrients, and it is relatively stable with respect to the nutrient concentrations. Furthermore, we explore how a white multiplicative Gaussian noise introduced into the RDA model on its resolution scale propagates across spatial scales through the nonlinear model dynamics under different sets of phytoplankton drivers. The unifying message of this work is that the low-complexity (e.g., RDA) models can be successfully used to realistically model some specific aspects of marine ecosystem dynamics and by using those models one can explore many questions that would be beyond computational affordability of the higher-complexity ecosystem models.

Direct numerical simulation of turbulent flow separation induced by a forward-facing step

Direct numerical simulation is used to investigate turbulent flow separations generated by a forward-facing step exposed to an incoming fully-developed turbulent plane channel flow at Reτ=180. The step height is 25% of the inlet channel height. The results are analyzed in terms of the topology of the Reynolds stresses, transport of turbulence kinetic energy (TKE) and unsteadiness of separation bubbles using a broad range of post-processing techniques, including the Lumley triangle, quadrant analysis, auto-correlation, joint probability density function, proper orthogonal decomposition and linear stochastic estimation. The incoming turbulence intensity is elevated as the step is approached and peaks along the mean separating streamline over the step. The topology of the Reynolds stresses switches from two-component to axisymmetric along the mean separating streamline over the step. The production term of TKE peaks in the separated shear layer, while the turbulent diffusion term tends to transport TKE away from the shear layer. An expanded separation bubble upstream of the step is typically associated with a contracted separation bubble over the step. This pattern is captured by the first proper orthogonal decomposition (POD) mode. In the first POD mode, the streamwise and vertical fluctuating velocities switch signs abruptly near the mean separating streamline over the step. This is characterized by two pairs of counter-rotating vortices: one pair of counter-rotating vortices leaning over the step and another pair of opposite-signed counter-rotating vortices near the top surface of the step.

Energy transport characteristics of converging Richtmyer–Meshkov instability

In this paper, the Richtmyer–Meshkov (RM) instability in spherical and cylindrical converging geometries with a Mach number of about 1.5 is investigated by using the direct numerical simulation method. The heavy fluid is sulfur hexafluoride, and the light fluid is nitrogen. The shock wave converges from the heavy fluid into the light fluid. The main focus is on the energy transport characteristics in the mixing layer during the entire development process from early instability to late-time turbulent mixing. First, the turbulence kinetic energy transport equation is analyzed, and it is found that the production and dissipation mechanisms of the turbulence induced by the spherical and cylindrical converging RM instabilities in the mixing layer are the same. The turbulent diffusion terms are crucial in the whole development processes of the mixing layers. Before the reflected shock waves transit the interfaces, the dissipation terms can be ignored relative to other terms, and after that, the dissipation terms are close to the production terms and play an important role. The compressibility terms are approximate to the production terms and promote the production of turbulence kinetic energy in the later stage. The viscous diffusion terms can be ignored throughout the process. Then, the enstrophy transport equation is researched, and it is found that, in the mixing layers, the baroclinicity terms play a leading role in the early stage, while the vortex stretching terms play a leading role in the later stage, and the vortex stretching term of the spherical converging geometry develops faster than that of the cylindrical converging geometry. The compressibility terms are positive in the early stage, which promote the production of enstrophy. After the reflected shock waves transit the interfaces, the compressibility terms become negative, which inhibit the production of enstrophy. In addition, the results of the present direct numerical simulation also show that the density fluctuation spectra in the centers of the mixing layers of the spherical and cylindrical converging RM instabilities present the obvious −5/3 scaling law.

Modeling of turbulent mixing with an improved K–L model

Turbulent mixing, induced by Rayleigh–Taylor (RT), Richtmyer–Meshkov (RM), and Kelvin–Helmholtz (KH) instabilities, broadly occurs in both natural phenomena, such as supernova explosions, and engineering applications, such as inertial confinement fusion (ICF). These three instabilities usually simultaneously exist and are highly coupled to drive and affect turbulent mixing, which raises a great challenge for turbulence modeling. In this study, an improved version of the K–L model is proposed. The modifications include that: (i) the deviatoric shear stress is considered to describe the KH instability; (ii) the concept of characteristic acceleration is introduced to better distinguish RT and RM instabilities; and (iii) an enthalpy diffusion is directly derived from the internal energy equation to model the turbulent diffusion term. Then, a unified set of model coefficients is systematically derived based on the self-similar analysis and physical observations. This model is validated by canonical RT, RM, and KH mixings and further investigated for more complex cases, including the RM mixing with multiple reshocks, the two-dimensional RT mixing called “tilted-rig,” and the simple spherical implosion, a much simplified version of an ICF implosion. Good agreement with the corresponding experimental and numerical data is achieved, revealing the ability of the present model to describe combined buoyancy, shock, and shear effects, which will contribute to a further application in real problems.

Impact of frontal round jet on vortex synchronization control around a cylinder: a turbulence perspective

The present study demonstrates that a round turbulent jet issued from a cylinder surface towards the upstream against the current can be a controlling parameter for vortex synchronization (VSC) in the wake region (WR) of the vertical cylinder. The results depict that the VSC state can be reached when the jet excitation frequency and the natural shedding frequency (NSF) are almost equal, or the jet excitation frequency is double that of the NSF. Precisely for VSC case, the turbulent production and dissipation terms of the turbulent kinetic energy budget equation contribute majorly to the total energy at the reattachment region with shifting of energy from the pressure and the turbulent diffusion terms. The enhancement of intermittency in turbulence for the VSC cases is also pertinent.

Turbulent mixing and dispersion mechanisms over flexible and dense vegetation

The present study investigates flow turbulence and dispersion processes in the presence of flexible and dense vegetation on the bed. The turbulent dispersion coefficients and the terms of the turbulent kinetic energy equation are determined by using data collected in a straight laboratory channel with living vegetation on the bed. Results show that the turbulent integral lengths assume an order of magnitude comparable to the stems’ characteristic dimension independently by the direction and the turbulence assumes an isotropic behavior. The coefficients of dispersion have a trend similar to that of the turbulent lengths and assume low values in the longitudinal, transversal and vertical directions. Results also show that, in the mixing layer, the shear and wake turbulence production terms balance the dissipation; the turbulent diffusion term also assumes low values and its sign varies along the vertical indicating a transport of turbulent energy both from the vegetation to the free surface and from the free surface to vegetation.

Prediction of four concentration moments of an airborne material released from a point source in an urban environment

Ιn case of the dispersion of an airborne material from a point source in an urban environment the reliable prediction of the concentration statistical distribution by a numerical dispersion model presupposes the capability of the model to predict at least four statistical moments (mean, variance, skewness and kurtosis). In the present study, the beta distribution, the selection of which is justified based on a previous study, is incorporated in the Reynolds Averaged Navier Stokes (RANS) methodology. The shape parameters of the beta distribution are calculated using the numerical results of the mean, variance and maximum concentration. The latest is calculated through a deterministic model which uses also the numerical results of the mean and variance concentration as well as a hydrodynamic time scale. The validation of the new hybrid model “RANS-beta” is performed using the experimental dataset of the MUST wind tunnel experiment. The performance of the “RANS-beta” model for the skewness is very good (FAC2 = 0.811) while for the kurtosis it is acceptable (FAC2 = 0.557). The discrepancies are observed mainly at the edges of the plume. Future research will be focused on the optimization of the mean flow field, turbulent quantities (Reynolds stresses, turbulent kinetic energy) and on new parameterizations for the turbulent diffusion term.

Numerical study on the propagation of solitary waves in the near-shore

The propagation and transformation of water waves in the near-shore region lead complicated wave-current field, which plays a significant role in the study of the mechanism of sediment transport and coastal evolution. Solitary wave is supposed as an initially incoming wave because of its widespread use in the near-shore. The Reynolds-averaged Navier-Stokes (RANS) equations were incorporated into a k-ε turbulence model and applied to simulate the run-up and hydrodynamic characteristics of solitary waves under the influences of different incident wave height and beach slope. The simulation results were validated with theoretical and experimental results from the literature. The effects of the non-linear terms associated with the curl of the fluid, Reynolds stresses and viscous stresses in the governing equations of fluid as well as the turbulent production term, convection term and diffusion term in the k-ε model were analyzed. The analysis shows that wave breaking is of great importance to the hydrodynamic characteristics of the wave-current field in the near-shore. The run-up heights of breaking solitary waves increase with beach slope, and the velocity and vorticity reach the maximum after wave breaking. The non-linear (curl and convection) terms and turbulent production term contribute most to the nonlinearities of the near-shore hydrodynamics, and increase with beach slope under these conditions. Nevertheless, under the conditions of wave non-breaking, the wave run-up height, the maximum of velocity and vorticity decrease with increasing beach slope. Meanwhile, the effects of fluid diffusion and Reynolds stresses increase gradually with increasing beach slope.

Modeling and simulation of particle dispersion in dense particle‐laden flow

AbstractTraditional drag force coefficient with regard to modeling gas–particle flows is generally established on the semiempirical or full empirical measurement data with isotropic hypothesis resulting in neglection of true particle dispersions, especially for anisotropic characteristics. A developed drag force coefficient to fully consider the anisotropics was applied to model and simulate dense particle‐laden flows. Coupled with a unified second‐order moment, 2‐fluid turbulent model consisting of a set of fluctuation velocity Reynolds transport equations was employed for predicting dense particle‐laden flows in pseudo‐2D horizontal chamber. Simulated results showed that they are in good agreement with measurement data. Particle Reynolds stresses have been redistributed by the momentum transfer interaction term between gas and particle phases, the turbulent diffusion term, and the particle collision term. Compared with traditional Wen's model, horizontal and vertical root‐mean‐square of particle velocities exhibited much more flatter and wider distribution behaviors. Due to limitation of two‐dimensional model and wall conditions, errors near wall regions should be improved in future work.

Experimental investigation of streamwise velocity fluctuation based on the Reynolds-number dependency in turbulent viscoelastic-fluid flow

This paper describes an experimental verification of energy supply mechanisms for the streamwise component of the turbulent kinetic energy (TKE) at different Reynolds numbers in viscoelastic-fluid flow. We investigated the characteristics of the streamwise turbulent velocity fluctuation by analyzing the production and turbulent diffusion terms in the TKE transport equation. In addition, we reported on the Reynolds-number dependency in a high Reynolds-number regime where direct numerical simulation cannot demonstrate changes in fluid properties. Based on the experimental verification, we proposed a conceptual model of the energy-exchange term between the TKE and the elastic energy, with focusing on the dependency of the fluid properties on the shear stress. This model is indirectly reflected in the streamwise TKE, the instantaneous velocity field, and the wave number relevant to energy-containing eddies. The main gain term of the TKE switches between the energy-exchange term and the production term dependently on the Reynolds number: as the Reynolds number exceeds the value which provides the maximum drag reduction rate, the production term becomes dominant and the magnitude of streamwise TKE becomes high compared to the water flow case.

Gasoline2: a modern smoothed particle hydrodynamics code

The methods in the Gasoline2 Smoothed Particle Hydrodynamics (SPH) code are described and tested. Gasoline2 is the most recent version of the Gasoline code for parallel hydrodynamics and gravity with identical hydrodynamics to the Changa code. As with other Modern SPH codes, we prevent sharp jumps in time steps, use upgraded kernels and larger neighbour numbers and employ local viscosity limiters. Unique features in Gasoline2 include its Geometric-Density-Average Force expression, explicit Turbulent Diffusion terms and Gradient-Based shock detection to limit artificial viscosity. This last feature allows Gasoline2 to completely avoid artificial viscosity in non-shocking compressive flows. We present a suite of tests demonstrating the value of these features with the same code configuration and parameter choices used for production simulations.

シェルピンスキー四面体を通して生成される乱流の乱流拡散項の調査

シェルピンスキー四面体を通して生成される乱流の乱流拡散項の調査

A Priori Direct Numerical Simulation Modeling of Scalar Dissipation Rate Transport in Head-On Quenching of Turbulent Premixed Flames

ABSTRACTThe statistical behavior and modeling of scalar dissipation rate (SDR) transport for head-on quenching of turbulent premixed flames by an inert isothermal wall have been analyzed in the context of Reynolds averaged Navier–Stokes simulations based on three-dimensional simple chemistry direct numerical simulation (DNS) data. It has been found that the density variation, scalar-turbulence interaction, reaction rate gradient, molecular diffusivity gradient, and molecular dissipation terms, i.e., , and , respectively, act as leading order contributors to the SDR transport away from the wall and the turbulent transport and molecular diffusion terms remain negligible in comparison to the other terms. The leading order contributors to the SDR transport have been found to be in a rough equilibrium away from the wall before the quenching is initiated but this equilibrium is not maintained during flame quenching. The predictions of the existing models for the unclosed terms of the SDR transport equation have been assessed with respect to the corresponding quantities extracted from DNS data. No existing models have been found to predict the near-wall behavior of the unclosed terms of the SDR transport equation. The models, which exhibit the most satisfactory performance away from the wall, have been modified to account for near-wall behavior in such a manner that the modified models asymptotically approach the existing model expressions away from the wall.