Second-moment Turbulence Closure Research Articles

Decomposition of Estuarine Circulation and Residual Stratification under Landfast Sea Ice

Abstract For Arctic estuaries that are characterized by landfast sea ice cover during the winter season, processes generating estuarine circulation and residual stratification have not yet been investigated, although some of the largest estuaries in the world belong to this class. Landfast sea ice provides a no-slip surface boundary condition in addition to the bottom boundary, such that frictional effects are expected to be increased. For this study of estuarine circulation and residual stratification under landfast sea ice, first, a simple linear analytical model is used. To include tidally varying scenarios, a water-column model is applied with a second-moment turbulence closure to juxtapose free-surface and ice-covered estuaries. Well-mixed and strongly stratified tidally periodic scenarios are analyzed by means of a decomposition of estuarine circulation into contributions from gravitational circulation, eddy viscosity–shear covariance (ESCO), surface stress, and river runoff. A new method is developed to also decompose tidal residual salinity anomaly profiles. Estuarine circulation intensity and tidally residual potential energy anomaly are studied for a parameter space spanned by the Simpson number and the unsteadiness number. These are the major results of this study that will support future scenario studies in Arctic estuaries under conditions of accelerated warming: (i) residual surface drag under ice opposes estuarine circulation; (ii) residual differential advection under ice destabilizes the near-surface flow; (iii) reversal of ESCO during strong stratification does not occur under landfast sea ice; (iv) tidal pumping (s-ESCO) contributes dominantly to residual stratification also with sea ice cover. Significance Statement Our work gives a first qualitative and quantitative understanding of how landfast sea ice cover on tidal estuaries impacts on the generation of estuarine circulation and residual stratification. Along the Arctic coasts, where some of the world’s largest estuaries are located, these processes play a significant role for the economy and ecology by means of transports of sediments, nutrients and pollutants. Due to Arctic amplification, the conditions for ice-covered estuaries are strongly changing in a way that the ice-covered periods may be shorter in the future. Our results intend to motivate field observations and realistic model studies to allow for better predicting the consequences of these changes.

Influence of multiphase turbulence modelling on interfacial momentum transfer in two-fluid Eulerian-Eulerian CFD models of bubbly flows

Eulerian-Eulerian two-fluid computational fluid dynamic (CFD) models are increasingly used to predict bubbly flows at an industrial scale. In these approaches, interface transfer is modelled with closure models and correlations. Normally, the lateral void fraction distribution is considered to mainly result from a balance between the lift and wall lubrication forces. However, and despite the numerous models available that achieve, at least in pipe flows, a reasonable predictive accuracy, agreement on a broadly applicable and accurate modelling approach has not yet been reached. Additionally, the impact of turbulence modelling on the lateral void fraction distribution has not, in general, been examined in detail. In this work, an elliptic blending Reynolds stress model (EB-RSM), capable of resolving the turbulence field in the near-wall region and improved to account for the contribution of bubble-induced turbulence, is evaluated against best-practice k-ε and high-Reynolds second-moment turbulence closures. Lift and wall lubrication forces are initially deliberately neglected in the EB-RSM. Comparisons for flows in pipes and a square duct show that the EB-RSM reproduces the lateral void fraction distribution, including the peak in the void fraction in the near-wall region, and reaches an accuracy comparable to the other two models noted above. In rod bundles, even if none of the models considered performs with sufficient accuracy, the EB-RSM detects features of the flow that are not predicted by the other two approaches. Overall, the results demonstrate a much more prominent role of the turbulence structure and the induced cross-sectional pressure field on the lateral void fraction distribution than is normally considered. These effects need to be accounted for if more physically-consistent modelling of bubbly flows is to be achieved. The lift force is added to the EB-RSM in the final part of the paper, to provide a two-fluid formulation that can be used as the basis for additional developments aimed at improving the accuracy and general applicability of two-fluid CFD models.

Wind-Wave Effects on Estuarine Turbulence: A Comparison of Observations and Second-Moment Closure Predictions

AbstractObservations of turbulent kinetic energy, dissipation, and turbulent stress were collected in the middle reaches of Chesapeake Bay and were used to assess second-moment closure predictions of turbulence generated beneath breaking waves. Dissipation scaling indicates that the turbulent flow structure observed during a 10-day wind event was dominated by a three-layer response that consisted of 1) a wave transport layer, 2) a surface log layer, and 3) a tidal, bottom boundary layer limited by stable stratification. Below the wave transport layer, turbulent mixing was limited by stable stratification. Within the wave transport layer, where dissipation was balanced by a divergence in the vertical turbulent kinetic energy flux, the eddy viscosity was significantly underestimated by second-moment turbulence closure models, suggesting that breaking waves homogenized the mixed surface layer to a greater extent than the simple model of TKE diffusing away from a source at the surface. While the turbulent transport of TKE occurred largely downgradient, the intermittent downward sweeps of momentum generated by breaking waves occurred largely independent of the mean shear. The underprediction of stress in the wave transport layer by second-moment closures was likely due to the inability of the eddy viscosity model to capture the nonlocal turbulent transport of the momentum flux beneath breaking waves. Finally, the authors hypothesize that large-scale coherent turbulent eddies played a significant role in transporting momentum generated near the surface to depth.

Turbulence and mixing in a freshwater-influenced tidal bay: Observations and numerical modeling

In situ observations and numerical simulations of turbulence are essential to understanding vertical mixing processes and their dynamical controls on both physical and biogeochemical processes in coastal embayments. Using in situ data collected by bottom-mounted acoustic Doppler current profilers (ADCPs) and a free-falling microstructure profiler, as well as numerical simulations with a second-moment turbulence closure model, we studied turbulence and mixing in the Xiamen Bay, a freshwater-influenced tidal bay located at the west coast of the Taiwan Strait. Dynamically, the bay is driven predominantly by the M2 tide, and it is under a significant influence of the freshwater discharged from the Jiulong River. It is found that turbulence quantities such as the production and dissipation rates of the turbulent kinetic energy (TKE) were all subject to significant tidal variations, with a pronounced ebb-flood asymmetry. Turbulence was stronger during flood than ebb. During the flooding period, the whole water column was nearly well mixed with the depth-averaged TKE production rate and vertical eddy viscosity being up to 5×10−6 W kg−1 and 2×10−2 m2 s−1, respectively. In contrast, during the ebb strong turbulence was confined only to a 5–8 m thick bottom boundary layer, where turbulence intensity generally decreases with distance from the seafloor. Diagnosis of the potential energy anomaly showed that the ebb-flood asymmetry in turbulent dissipation and mixing was due mainly to tidal straining process as a result of the interaction between vertically shared tidal currents and horizontal density gradients. The role of vertical mixing in generating the asymmetry was secondary. A direct comparison of the modeled and observed turbulence quantities confirmed the applicability of the second-moment turbulence closure scheme in modeling turbulent processes in this weakly stratified tidally energetic environment, but also pointed out the necessity of further refinements of the model.

Residual Sediment Fluxes in Weakly-to-Periodically Stratified Estuaries and Tidal Inlets

Abstract In this idealized numerical modeling study, the composition of residual sediment fluxes in energetic (e.g., weakly or periodically stratified) tidal estuaries is investigated by means of one-dimensional water column models, with some focus on the sediment availability. Scaling of the underlying dynamic equations shows dependence of the results on the Simpson number (relative strength of horizontal density gradient) and the Rouse number (relative settling velocity) as well as impacts of the Unsteadiness number (relative tidal frequency). Here, the parameter space given by the Simpson and Rouse numbers is mainly investigated. A simple analytical model based on the assumption of stationarity shows that for small Simpson and Rouse numbers sediment flux is down estuary and vice versa for large Simpson and Rouse numbers. A fully dynamic water column model coupled to a second-moment turbulence closure model allows to decompose the sediment flux profiles into contributions from the transport flux (product of subtidal velocity and sediment concentration profiles) and the fluctuation flux profiles (tidal covariance between current velocity and sediment concentration). Three different types of bottom sediment pools are distinguished to vary the sediment availability, by defining a time scale for complete sediment erosion. For short erosion times scales, the transport sediment flux may dominate, but for larger erosion time scales the fluctuation sediment flux largely dominates the tidal sediment flux. When quarter-diurnal components are added to the tidal forcing, up-estuary sediment fluxes are strongly increased for stronger and shorter flood tides and vice versa. The theoretical results are compared to field observations in a tidally energetic inlet.

Large eddy simulations of wake-stabilised flares

This paper investigates the potential benefits, as well as limitations, of using the LES technique to predict the combustion efficiency, emissions and temperatures of wake-stabilised flares. LES calculations have been performed for two jet-to-cross-flow momentum flux ratios, with the results being compared with experimental data, where available, and with predictions of a closed RANS model using a second-moment turbulence closure. The flame fragmentation and the secondary flame zone attached to the release pipe at high jet-to-cross-flow momentum flux ratios, and their influence on the evolution of the counter-rotating vortex pair found in such flames, are discussed in detail, with realistic predictions being produced by the LES. The global mixing characteristics and the combustion efficiency of such flames are also compared with available data with reasonable agreement found. In all cases, LES predictions are found to be superior to equivalent RANS results, although the extra computational effort required in predicting species concentrations and flare efficiencies, in particular, may not be warranted.

Coupled RANS–LPT modelling of dilute, particle-laden flow in a duct with a 90° bend

A dilute, particle-laden flow in a square duct with a 90° bend is modelled using a RANS approach, coupled to a second-moment turbulence closure, together with a Lagrangian particle tracking technique, with particle dispersion modelled using a stochastic approach that ensures turbulence anisotropy. Detailed predictions of mean and fluctuating fluid and particle velocities are validated through comparisons of predictions with experimental measurements made for gas–solid flows in a vertical-to-horizontal flow configuration. Reasonable agreement between predicted first and second moments and data is found for both phases, with the consistent application of anisotropic and three-dimensional modelling approaches resulting in predictions that compare favourably with those of other authors, and which provide fluctuating particle velocities in acceptable agreement with data.

External intermittency prediction using AMR solutions of RANS turbulence and transported PDF models

External intermittency in turbulent round jets is predicted using a Reynolds-averaged Navier-Stokes modelling approach coupled to solutions of the transported probability density function (pdf) equation for scalar variables. Solutions to the descriptive equations are obtained using a finite-volume method, combined with an adaptive mesh refinement algorithm, applied in both physical and compositional space. This method contrasts with conventional approaches to solving the transported pdf equation which generally employ Monte Carlo techniques. Intermittency-modified eddy viscosity and second-moment turbulence closures are used to accommodate the effects of intermittency on the flow field, with the influence of intermittency also included, through modifications to the mixing model, in the transported pdf equation. Predictions of the overall model are compared with experimental data on the velocity and scalar fields in a round jet, as well as against measurements of intermittency profiles and scalar pdfs in a number of flows, with good agreement obtained. For the cases considered, predictions based on the second-moment turbulence closure are clearly superior, although both turbulence models give realistic predictions of the bimodal scalar pdfs observed experimentally.

Prediction of external intermittency using RANS-based turbulence modelling and a transported PDF approach

This paper investigates the modelling of external intermittency in turbulent round jets using a RANS approach coupled to solutions of the transported probability density function (PDF) equation for scalar variables. Solutions to the descriptive equations are obtained using a finite-volume method, combined with an adaptive mesh refinement algorithm, applied in both physical and compositional space. The effects of intermittency on the flow field are accommodated using intermittency-modified eddy viscosity and second-moment turbulence closures, as well as through modifications to the mixing model embodied within the transported PDF equation. Predictions of the model are validated against data on the velocity and scalar fields in jets, as well as against measurements of scalar PDFs and intermittency profiles, with reasonable agreement obtained. From the cases examined, predictions of the second-moment closure are superior, although both approaches provide realistic predictions of the bimodal features to the measured PDFs.

The Description of Mixing in Stratified Layers without Shear in Large-Scale Ocean Models

Abstract Large-scale geophysical flows often exhibit layers with negligible vertical shear and infinite gradient Richardson number Ri. It is well known that these layers may be regions of active mixing, even in the absence of local shear production of turbulence because, among other processes, turbulence may be supplied by vertical turbulent transport from neighboring regions. This observation is contrasted by the behavior of most turbulence parameterizations used in ocean climate modeling, predicting the collapse of mixing of mass and matter if the Richardson number exceeds a critical threshold. Here, the performance of a simple model without critical Richardson number is evaluated, taking into account the diffusion of turbulence into layers without shear production and therefore avoiding the suppression of mixing at large values of Ri. The model is based on the framework of second-moment turbulence closures, focusing on the consistent modeling of the turbulent length scale for strongly stratified turbulence. Results are compared to eddy-resolving simulations of stratified shear flows that have recently become available. The model is simple enough for inclusion in ocean climate models.

Elliptic-blending second-moment turbulence closure using an algebraic anisotropic dissipation rate tensor model**This paper is a modified version of the original article from the Proceedings of the 5th International Symposium on Turbulence and Shear Flow Phenomena held in Munich, Germany on 27–29 August 2007.

This study describes the amendment of an algebraic anisotropic dissipation rate model (ADRM) and its application to various turbulent flows to test the model's performance. Modeling anisotropies for the turbulence dissipation rate is considered by an analysis of the exact transport equation for the dissipation rate tensor. The second-moment closure, which is based on the explicit amended ADRM, is proposed and it is closely linked to the elliptic-blending model that is used for the prediction of Reynolds stresses.To develop and calibrate the present elliptic-blending second-moment closure that uses the amended ADRM, firstly, the distributions of both the mean velocity and Reynolds stress are solved for flows in a fully developed non-rotating channel and a straight square duct. And then, the fully developed turbulent flows in a rotating channel and a rotating straight square duct are predicted to test the ability of the explicit amended ADRM that is combined with the rotation effect.The prediction results are directly compared with the DNS and the large-eddy simulation (LES) to assess the performance of the new model predictions and to show their reasonable agreement with the DNS and LES data for all the flow fields that are analyzed for the present study.

Computation of tip-leakage flow in a linear compressor cascade with a second-moment turbulence closure

Strong interaction among tip-leakage vortex, mainstream, passage vortex and turbulence in and behind the blade tip and casing in compressors is one of the major causes of energy losses, affecting the overall gas turbine performance. A number of experiments and recent large-eddy simulations of simplified generic tip-clearance configurations have substantially advanced the understanding of the complex vortical and turbulence structures and energy conversion in and behind a very narrow clearance space. However, the prediction of phenomena in real turbomachinery still relies on the one-point (Reynolds-averaged Navier–Stokes, RANS) Computational Fluid Dynamics which is still burdened by uncertainties related primarily to inadequate turbulence modelling, and also to insufficient accuracy of the numerical algorithms. In this paper we address both these issues by testing an advanced turbulence model and a numerical method in the computations of a tip-clearance flow in a three-dimensional linear compressor cascade. For closing the mean momentum equations we adopted a Second Moment Turbulence Closure (SMC) incorporating low- Re-number and wall proximity effects, which has earlier proved to reproduce accurately a series of generic two- and three-dimensional anisotropic and non-equilibrium flows relevant to turbomachinery. In order to diminish numerical uncertainties, the computations are performed using Petrov-Galerkin stabilized finite element code featuring equal order Q1–Q1 pairs. The solution scheme is based on an in-house parallel Overlapping Domain Decomposition approach featuring a FGMRes linear solver. The results, compared with experiments and the conventional linear low- Re-number eddy viscosity model demonstrate that the SMC here adopted is capable to reproduce all the phenomenological features related to tip-leakage flows, especially the evolution of the strong three-dimensionality and turbulence anisotropy, and can be considered a prospective model for predicting real turbomachinery flows.

First- and second-order elliptic conditional moment closure calculations of piloted methane diffusion flames

Presented are results obtained from the application of a first- and higher-order conditional moment closure (CMC) approach to the modeling of three methane diffusion flames at differing levels of local extinction. In addition to the analysis of higher-order chemistry applications, the results obtained from an elliptic formulation of the CMC equation are considered next to parabolic results presented in previous work. All predictions are based upon second-moment turbulence and scalar-flux closures, and the chemistry applied to represent mean production rates of species is a 16-step reduced mechanism. A second-order chemistry is implemented on the basis of a two-step kinetic representation of methane combustion, used to correct first-order rates. In general, predictions obtained using the second-order model improve significantly upon first-order results for both major and minor species under fuel-rich conditions. The simplified chemistry employed does not however fully capture the effects of local extinction, and suggestions are made regarding the further developments required to permit the accurate prediction of highly strained flames using CMC methods.

Validation of an anisotropic model of turbulent flows containing dispersed solid particles applied to gas–solid jets

A mathematical model of turbulent flows containing dispersed solid particles is described together with its application to gas–solid jets. Flow fields are predicted by solution of the density-weighted transport equations expressing conservation of mass and momentum, with closure achieved through the k– ɛ turbulence model and a second-moment closure. The particle phase is calculated using a Lagrangian particle tracking technique which involves solving the particle momentum equation in a form that accounts for the spatial, temporal and directional correlations of the Reynolds stresses experienced by a particle. The two phases are coupled via modification of the fluid-phase momentum equations. Predictions of the complete model are validated against available experimental data on a number of single-phase and two-phase, gas–solid jet flows with various particle loadings, and both mono- and poly-dispersed particle size distributions. Overall, predictions of the models compare favourably with the data examined, with results obtained from the anisotropic second-moment turbulence closure being superior to eddy viscosity-based predictions.

Sensitivity of a coupled physical–biological model to turbulence: high-frequency simulations in a northern Adriatic station

This paper investigates the impacts of different turbulence models on the biological state at an ocean station in the northern Adriatic sea, named S3, comparing them with other uncertainties inherent to coupled physical–biological simulations. The numerical tool is a 1-D model resulting from the coupling of two advanced numerical models. The hydrodynamic part is modelled using the General Ocean Turbulence Model (www.gotm.net), in a version adopting state-of-the-art second-moment Turbulence Closure Models (TCMs). Marine biogeochemistry is parameterized with the Biogeochemical Flux Model (http://www.bo.ingv.it/bfm), which is a direct descendant of ERSEM (European Regional Sea Ecosystem Model). Results, obtained by forcing the model with hourly wind and solar radiation data and assimilating salinity casts, are compared against monthly observations made at the station during 2000–2001. Provided that modern second-moment TCMs are employed, the comparisons indicate that both the physical and the biological dynamics are relatively insensitive to the choice of the particular scheme adopted, suggesting that TCMs have finally ‘converged’ in recent years. As a further example, the choice of the nutrient boundary conditions has an impact on the system evolution that is more significant than the choice of the specific TCM, therefore representing a possible limitation of the 1-D model applied to stations located in a Region of Freshwater Influence. The 1-D model simulates the onset and intensity of the spring–summer bloom quite well, although the duration of the bloom is not as prolonged as in the data. Since local dynamics appears unable to sustain the bloom conditions well into summer, phytoplankton at the station was most likely influenced by river input or advection processes, an aspect that was not found when the S3 behaviour was adequately modelled using climatological forcings. When the focus is in predicting high-frequency dynamics, it is more likely that lateral advection cannot be neglected. While the physical state can be satisfactorily estimated at these short time scales, the accurate estimation of the biological state in coastal regions still appears as rather elusive.

Computation of Flow and Heat Transfer in Rotating Rectangular Channels (AR=4:1) With Pin-Fins by a Reynolds Stress Turbulence Model

Computations with multi-block chimera grids were performed to study the three-dimensional turbulent flow and heat transfer in a rotating rectangular channel with staggered arrays of pin-fins. The channel aspect ratio (AR) is 4:1, the pin length to diameter ratio (H∕D) is 2.0, and the pin spacing to diameter ratio is 2.0 in both the stream-wise (S1∕D) and span-wise (S2∕D) directions. A total of six calculations have been performed with various combinations of rotation number, Reynolds number, and coolant-to-wall density ratio. The rotation number and inlet coolant-to-wall density ratio varied from 0.0 to 0.28 and from 0.122 to 0.20, respectively, while the Reynolds number varied from 10,000 to 100,000. For the rotating cases, the rectangular channel was oriented at 150deg with respect to the plane of rotation to be consistent with the configuration of the gas turbine blade. A Reynolds-averaged Navier-Stokes (RANS) method was employed in conjunction with a near-wall second-moment turbulence closure for detailed predictions of mean velocity, mean temperature, and heat transfer coefficient distributions.

Analysis of kinetic mechanism performance in conditional moment closure modelling of turbulent, non-premixed methane flames

This paper presents results obtained from the application of a first-order conditional moment closure approach to the modelling of two methane flames of differing geometries. Predictions are based upon a second-moment turbulence and scalar-flux closure, and supplemented with full and reduced chemical kinetic mechanisms, ranging from a simple 12-step to a complex 1207-step mechanism. Alongside analysis of the full kinetic schemes' performance, is an appraisal of the behaviour of their derivatives obtained using mechanism-reduction techniques. The study was undertaken to analyse the practicality of incorporating kinetic models of varying complexity into calculations of turbulent non-premixed flames, and to make comparison of their performance. Despite extensive studies of the predictive ability of such schemes under laminar flame conditions, systematic evaluations have not been performed for turbulent reacting flows. This paper reflects upon the impact that selection of chemical kinetics has upon subsequent calculations and concludes that, although application of reduced schemes is more than adequate to reproduce experimental data, selection of the parent mechanism is of paramount importance to the prediction of minor species. Although widely used schemes are well documented and validated, their performances vary considerably. Thus, careful consideration must be made to their application and origins during the evaluation of combustion models.

Prediction of Cascade Flows With Innovative Second-Moment Closures

We report on the performances of two second-moment turbulence closures in predicting turbulence and laminar-to-turbulent transition in turbomachinery flows. The first model considered is the one by Hanjalic and Jakirlic (HJ) [Comput. Fluids, 27(2), pp. 137–156 (1998)], which follows the conventional approach with damping functions to account for the wall viscous and nonviscous effect. The second is an innovative topology-free elliptic blending model, EBM [R. Manceau and K. Hanjalic, Phys. Fluids, 14(3), pp. 1–11 (2002)], here presented in a revised formulation. An in-house finite element code based on a parallel technique is used for solving the equation set [Borello et al., Comput. Fluids, 32, pp. 1017–1047 (2003)]. The test cases under scrutiny are the transitional flow on a flat plate with circular leading edge (T3L ERCOFTAC-TSIG), and the flow around a double circular arc (DCA) compressor cascade in quasi-off-design condition (i=−1.5°) [Zierke and Deutsch, NASA Contract Report 185118 (1989)]. The comparison between computations and experiments shows a satisfactory performance of the HJ model in predicting complex turbomachinery flows. The EBM also exhibits a fair level of accuracy, though it is less satisfactory in transition prediction. Nevertheless, in view of the robustness of the numerical formulation, the relative insensitivity to grid refinement, and the absence of topology-dependent parameters, the EBM is identified as an attractive second-moment closure option for computation of complex 3D turbulent flows in realistic turbomachinery configurations.

Predictions of flow and heat transfer in multiple impinging jets with an elliptic-blending second-moment closure

We present numerical computations of flow and heat transfer in multiple jets impinging normally on a flat heated surface, obtained with a new second-moment turbulence closure combined with an elliptic blending model of non-viscous wall blocking effect. This model provides the mean velocity and turbulent stress fields in very good agreement with PIV measurements. The exploration of several simpler closures for the passive thermal field, conducted in parallel, confirmed that the major prerequisite for the accurate prediction of the temperature field and heat transfer is to compute accurately the velocity and stress fields. If this is achieved, the conventional anisotropic eddy-diffusivity model can suffice even in complex flows. We demonstrate this in multiple-impinging jets where such a model combination provided the distribution of Nusselt number over the solid plate in good agreement with experiments. Extension of the elliptic blending concept to full second-moment treatment of the heat flux and its truncation to a quasi-linear algebraic model is also briefly discussed.

Flow and Heat-Transfer Computations in Rotating Rectangular Channels with V-Shaped Ribs

Computations were performed to study three-dimensional turbulent flow and heat transfer in a rotating rectangular channel with V-shaped ribs. The channel aspect ratio is 4:1, the rib-height-to-hydraulic-diameter ratio e/Dh is 0.078, and the rib-pitch-to-height ratio P/e is 10. A total of eight calculations have been performed with various combinations of rotation number, Reynolds number, coolant-to-wall density ratio, and channel orientation. The rotation number and inlet coolant-to-wall density ratio were varied from 0.0 to 0.28 and from 0.122 to 0.40, respectively, whereas the Reynolds number was varied from 10 4 to 5 × × 10 5 . Three channel orientations (90, − −135, and 135 deg from the rotation direction) were also investigated. A multiblock Reynolds-averaged Navier‐Stokes method was employed in conjunction with a near-wall second-moment turbulence closure for detailed predictions of the mean velocities, mean temperature, turbulent Reynolds stresses, and heat fluxes.