Moment Closure Model Research Articles

ANALYSIS OF SYNTHETIC JET FLOW FIELD: APPLICATION OF URANS APPROACH

The present paper reports the time dependent simulation of a turbulent plane synthetic jet using an unsteady Reynolds averaged Navier-Stokes approach on the basis of the first and second moment closure turbulence models. All the applied turbulence models can capture a global feature of the long time averaged flow field quite well. However, the standard k – ε model yields a disappointing prediction of the turbulence field with inaccurately high levels of turbulence kinetic energy and normal Reynolds stress distributions. The second moment closure model with quadratic nonlinear pressure strain approximation shows the most reasonable prediction of the phase averaged flow and turbulence fields.

Numerical study of energy separation in a vortex tube with different RANS models

The aim of the present paper is to investigate numerically the energy separation mechanism and flow phenomena within a vortex tube. A 3D computational domain has been generated considering the quarter of the geometry and assuming periodicity in the Azimuthal direction which was found to exhibit correctly the general behaviour expected from a vortex tube. Air is selected as the working fluid. The flow predictions reported here are based upon four turbulence models, namely, the k– ɛ, k– ω and SST k– ω two-equations models and the second moment closure model ( RSM). The models results are compared to experimental data obtained from the literature. Four cases have been considered by changing the inlet pressure from 200 up to 380 kPa. It has been observed that all the above mentioned models are capable of predicting fairly well the general flow features but only the advanced RSM model is capable of matching correctly the measured cold and hot outlet temperatures. All the other models over predict the mean temperature difference by values up to twice the measured data.

An algorithm for LES of premixed compressible flows using the Conditional Moment Closure model

A novel numerical method has been developed to couple a recent high order accurate fully compressible upwind method with the Conditional Moment Closure combustion model. The governing equations, turbulence modelling and numerical methods are presented in full. The new numerical method is validated against direct numerical simulation (DNS) data for a lean premixed methane slot burner. Although the modelling approaches are based on non-premixed flames and hence not expected to be valid for a wide range of premixed flames, the predicted flame is just 10% longer than that in the DNS and excellent agreement of mean mass fractions, conditional mass fractions and temperature is demonstrated. This new numerical method provides a very useful framework for future application of CMC to premixed as well as non-premixed combustion.

Prediction of soot particle size distribution for turbulent reacting flow in a diesel engine

There is increasing interest in obtaining the size distribution as well as the number density and mass fraction of soot particles created by diesel engines. This study addresses this problem and particle size distributions are computed and validated in typical operating conditions of a diesel engine. The program KIVA is modified and integrated with CHEMKIN in order to be able to consider detailed gas phase kinetics and soot precursors using the conditional moment closure (CMC) model for turbulence/chemistry coupling. The soot simulator, SWEEP is then used to calculate the size distribution as a post-processing step at any given mixture fraction. The conditional results obtained at a mixture fraction of 0.12 are generally consistent with measurements over a range of engine operating conditions. However, the effect of differential diffusion in the probability density function weighted quantities affects the quality of the results obtained as a function of engine speed.

Deterministic epidemic models on contact networks: Correlations and unbiological terms

The relationship between system-level and subsystem-level master equations is investigated and then utilised for a systematic and potentially automated derivation of the hierarchy of moment equations in a susceptible-infectious-removed (SIR) epidemic model. In the context of epidemics on contact networks we use this to show that the approximate nature of some deterministic models such as mean-field and pair-approximation models can be partly understood by the identification of implicit anomalous terms. These terms describe unbiological processes which can be systematically removed up to and including the nth order by nth order moment closure approximations. These terms lead to a detailed understanding of the correlations in network-based epidemic models and contribute to understanding the connection between individual-level epidemic processes and population-level models. The connection with metapopulation models is also discussed. Our analysis is predominantly made at the individual level where the first and second order moment closure models correspond to what we term the individual-based and pair-based deterministic models, respectively. Matlab code is included as supplementary material for solving these models on transmission networks of arbitrary complexity.

New phenomena in variable-density Rayleigh–Taylor turbulence

This paper presents several issues related to mixing and turbulence structure in buoyancy-driven turbulence at low to moderate Atwood numbers, A, found from direct numerical simulations in two configurations: classical Rayleigh–Taylor instability and an idealized triply periodic Rayleigh–Taylor flow. Simulations at A up to 0.5 are used to examine the turbulence characteristics and contrast them with those obtained close to the Boussinesq approximation. The data sets used represent the largest simulations to date in each configuration. One of the more remarkable issues explored, first reported in (Livescu and Ristorcelli 2008 J. Fluid Mech. 605 145–80), is the marked difference in mixing between different density fluids as opposed to the mixing that occurs between fluids of commensurate densities, corresponding to the Boussinesq approximation. Thus, in the triply periodic configuration and the non-Boussinesq case, an initially symmetric density probability density function becomes skewed, showing that the mixing is asymmetric, with pure heavy fluid mixing more slowly than pure light fluid. A mechanism producing the mixing asymmetry is proposed and the consequences for the classical Rayleigh–Taylor configuration are discussed. In addition, it is shown that anomalous small-scale anisotropy found in the homogeneous configuration (Livescu and Ristorcelli 2008 J. Fluid Mech. 605 145–80) and Rayleigh–Taylor turbulence at A=0.5 (Livescu et al 2008 J. Turbul. 10 1–32) also occurs near the Boussinesq limit. Results pertaining to the moment closure modelling of Rayleigh–Taylor turbulence are also presented. Although the Rayleigh–Taylor mixing layer width reaches self-similar growth relatively fast, the lower-order terms in the self-similar expressions for turbulence moments have long-lasting effects and derived quantities, such as the turbulent Reynolds number, are slow to follow the self-similar predictions. Since eddy diffusivity in the popular gradient transport hypothesis is proportional to the turbulent Reynolds number, the dissipation rate and turbulent transport have different length scales long after the onset of the self-similar growth for the layer growth. To highlight the importance of turbulent transport, variable density energy budgets for the kinetic energy, mass flux and density-specific volume covariance equations, necessary for a moment closure of the flow, are provided.

A note on modeling double diffusive mixing in the global ocean

Though ubiquitous in the global oceans, double diffusive mixing has been largely ignored or poorly represented in the models of turbulent mixing in the ocean and in 3-D ocean models, until recently. Salt fingers occur in the interior of many marginal seas and ocean basins, the Tyrrhenian Sea and the subtropical Atlantic being two examples. Diffusive convection type of double diffusion occurs in the upper layers of many sub-polar seas and polar oceans due to cold melt water from sea ice. Consequently, it is important to be able to properly parameterize double diffusive mixing in basin scale and global ocean models, so that the water mass structure in the interior of the ocean can be properly simulated. This note describes a model for double diffusive mixing in the presence of background shear, based on Mellor–Yamada type second moment closure, more specifically Kantha (2003), Kantha and Clayson (2004) second moment closure models of resulting turbulence, following Canuto et al. (2008a) but employing a different strategy for modeling the pertinent terms in the second moment equations. The resulting model is suitable for inclusion in ocean general circulation models.

Erratum to: A note on stokes production of turbulence kinetic energy in the oceanic mixed layer: observations in the Baltic Sea

Shear- and convection-driven turbulence coexists with wind-generated surface gravity waves in the upper ocean. The turbulent Reynolds stresses in the oceanic mixed layer can therefore interact with the shear of the wave-generated Stokes drift velocity to extract energy from the surface waves and inject it into turbulence, thus augmenting the mean shear-driven turbulence. Stokes production of turbulence kinetic energy (TKE) is difficult to measure in the field, since it requires simultaneous measurement of the turbulent stress and the Stokes drift profiles in the water column. However, it is readily inferred using second moment closure models of the oceanic mixed layer provided: (1) wave properties are available, along with the usual water mass properties, and radiative and air–sea fluxes needed to drive the mixed layer model and (2) the model skill can be assessed by comparing the model results against the observed dissipation rates of TKE. Comprehensive measurements made during the Reynolds 2002 campaign in the Baltic Sea have made the estimation of Stokes production possible, and in this paper, we report on the effort and the conclusions reached. Measurements of air–sea exchange parameters and water mass properties during the campaign allowed a mixed layer model to be run and the turbulent stress in the water column to be inferred. Simultaneous wave spectrum measurements enabled Stokes drift profile to be deduced and wave breaking to be included in the model run, and the Stokes production of TKE in the water column estimated. Direct measurements of the TKE dissipation rate from an upward traversing microstructure profiler were used to assure that the model could reproduce the turbulent dissipation rate in the water column. The model results indicate that the Stokes production of TKE in the mixed layer is of the same order of magnitude as the shear production and must therefore be included in mixed layer models.

Implicit algebraic model for predicting turbulent heat flux in film cooling flow

AbstractThe present study addresses a new effort to improve the prediction of turbulent heat flux in the film cooling flow by applying the implicit algebraic flux (IAF) model of Rogers et al. A three‐dimensional symmetry case is investigated using a film hole length‐to‐diameter ratio of 1.75 and an injection angle of 35∘. The low Reynolds number second moment closure (SMC) model with a wall‐reflection term is employed for simulating the turbulent flow field right up to the wall. Results obtained from the IAF model are compared with two other algebraic turbulent heat flux models, namely, the simple eddy diffusivity (SED) with a constant turbulent Prandtl number and the generalized gradient diffusion hypothesis (GGDH). Comparisons of the turbulent heat flux components calculated by these models show that the major difference appears in the streamwise turbulent heat flux. These models demonstrate a significant effect on the prediction of film cooling effectiveness. The SED model with a constant prescribed value for the turbulent Prandtl number fails to predict the cooling air spreading in the lateral direction while by employing the GGDH and IAF models, the spreading of the cooling air and the decay of the effectiveness in the core region are reasonably predicted. A combination of the SMC and IAF models for simulating the turbulent flow and heat transfer is capable of predicting the streamwise and lateral film cooling effectiveness in very good agreement with the available experimental data. Copyright © 2009 John Wiley & Sons, Ltd.

Implementation Issues of the Conditional Moment Closure Model in Large Eddy Simulations

Different ways of transferring information regarding the mixture fraction, its sub-grid scale variance and the scalar dissipation rate are examined in terms of a Large Eddy Simulation (LES)/Conditional Moment Closure (CMC) calculation. In such a simulation, information must be transferred from a fine LES grid to a usually coarser CMC grid. Different options of calculating conditional and unconditional quantities in the CMC resolution are assessed by filtering experimental mixture fraction and scalar dissipation rate data at various resolutions. It was found that when a presumed shape for the Filtered Density Function at the CMC resolution is used, special care must be given to the mixture fraction variance. It was also found that the Amplitude Mapping Closure model can be used for the conditional scalar dissipation rate. LES/CMC with detailed chemistry of a bluff-body stabilised burner was performed using two different ways of calculating the turbulent diffusivity. The structure of the flame is realistic, with little difference noticed when using the two diffusivities.

Analysis of Calcium Alternans in a Cardiac Myocyte Model that Uses Moment Equations to Represent Heterogenous Junctional SR Calcium

The recently introduced “probability density approach” to modeling local control of CICR in cardiac myocytes [Williams et al. Biophys. J. 92(7):2311-28, 2007] and associated moment closure technique [95(4):1689-703, 2008] can reproduce whole cell voltage-clamp protocols high-gain Ca release that is graded with changes in membrane potential. This modeling formalism represents heterogeneous local Ca signals in a population of diadic subspaces and junctional sarcoplasmic reticulum (jSR) depletion domains using a system of differential-algebraic equations for the time-evolution of the zeroth, first, and second moments of probability density functions for jSR [Ca] jointly distributed with calcium release unit (CaRU) state. Here we show that a cardiac myocyte model that uses moment equations to represent heterogenous jSR Ca can exhibit Ca alternans when periodically stimulated by depolarizing voltage pulses, and makes predictions regarding the distribution of jSR [Ca] across a large population CaRUs as a function of stimulation frequency and cellular parameters such as the rate of diffusive transfer between network and junctional SR. Factors promoting alternating Ca responses in the moment closure model are analyzed and compared to analogous mechanism in a minimal “common pool” model with comparatively simple SR and PM fluxes. We derive load-release and release-reuptake curves for both models, and investigate how model parameters influence these relations and the existence and stability of steady-state periodic Ca responses during repetitive depolarizing voltage pulses. Specifically, we find that increasing SR Ca leak, RyR sensitivity, and maximum release flux decreases the steepness of load-release curves, shifts load-release curves to smaller SR loads, and increases the critical simulation frequency resulting in Ca alternans.

Conditional moment closure modelling of soot formation in turbulent, non-premixed methane and propane flames

Presented are results obtained from the incorporation of a semi-empirical soot model into a first-order conditional moment closure (CMC) approach to modelling turbulent, non-premixed methane–air and propane–air flames. Soot formation is determined via the solution of two transport equations for soot mass fraction and particle number density, with acetylene and benzene employed as the incipient species responsible for soot nucleation, and the concentrations of these calculated using a detailed gas-phase kinetic scheme involving 70 species. The study focuses on the influence of differential diffusion of soot particles on soot volume fraction predictions. The results of calculations are compared with experimental data for atmospheric and 3 atm methane flames, and propane flames with air preheated to 323 K and 773 K. Overall, the study demonstrates that the model, when used in conjunction with a representation of differential diffusion effects, is capable of accurately predicting soot formation in the turbulent non-premixed flames considered.

Autoignition of turbulent hydrogen jet in a coflow of heated air

Autoignition of hydrogen, leading to flame development under turbulent flow conditions is numerically investigated including a detailed chemical mechanism. The chosen configuration consists of a turbulent jet of hydrogen diluted with nitrogen which is issued into a coflow of heated air. Numerical simulations are performed with the Conditional Moment Closure model, to capture the transient evolution of the flow. Turbulence closure is achieved using the k– ɛ model. Simulations revealed that the injected hydrogen mixes with coflowing air, autoignites and a stable diffusion flame is established. Sometimes, flashback of the ignited mixture is observed, whereby the flame travels upstream and stabilizes. It is found that the constants assumed in various modeling terms can severely influence the degree of mixing. Hence, certain modifications to these constants are suggested, and improved predictions are obtained. The sensitivity of autoignition length to the coflow temperature is investigated. The predicted autoignition lengths show a reasonable agreement with the experimental data and LES results.

A-priori analysis of conditional moment closure modeling of a temporal ethylene jet flame with soot formation using direct numerical simulation

Modeling soot formation in turbulent nonpremixed combustion is a difficult problem. Unlike most gaseous combustion species, soot lacks a strong state relationship with the mixture fraction due to unsteady formation rates which overlap transport timescales, and strong differential diffusion between gaseous species and soot. The conditional moment closure model (CMC) has recently been applied to the problem of turbulent soot formation. A challenge in CMC modeling is the treatment of differential diffusion. Three-dimensional direct numerical simulation (DNS) of a nonpremixed ethylene jet flame with soot formation has been performed using a 19 species reduced ethylene mechanism and a four-step, three-moment, semi-empirical soot model. The DNS provides full resolution of the turbulent flow field and is used to perform a-priori analysis of a recent CMC model derived from the joint scalar PDF transport equation. Unlike other approaches, this CMC model does not require additional transport equations to treat differentially diffusing species. A budget of the terms of the CMC equation for both gaseous species and soot is presented. In particular, exact expressions for unclosed terms are compared to typical closure models for scalar dissipation, cross-dissipation, differential diffusion, and reactive source terms. The differential diffusion model for gaseous species is found to be quite accurate, while that for soot requires an additional model for the residual term.

Conditional Moment Closure Modeling for a Three-Dimensional Turbulent Non-premixed Syngas Flame with a Cooling Wall

The fully coupled conditional moment closure (CMC) modeling with an unstructured-grid finite volume method has been developed to realistically simulate the structure of complex, turbulent non-premixed syngas-air flames, where flame structures could be considerably influenced by turbulence, transport history, and heat transfer. In this study, conservative CMC formulation is used to improve robustness and accuracy in the boundary treatment of CMC-related fluxes. To correctly account for the transport effect, the CMC transport equations fully coupled with equations for the flow and mixing fields are numerically solved. Moreover, to reduce the excessive computational burden required for three-dimensional computations of the fully coupled CMC approach with detailed chemistry, parallel strategies are implemented into the general CMC formulation. The present fully coupled CMC formulation together with a parallel processing procedure successfully demonstrates the capability to realistically predict detailed struc...

Derivation of the conditional moment closure equations for spray combustion

In this work we derive the fundamental equations for conditional moment closure (CMC) modelling of individual phases set in a two-phase flow. The derivation is based on the instantaneous transport equations for the single phase that involve a level set/indicator function technique for accounting for interfaces. Special emphasis is put on spray combustion with the CMC equations formulated for the gas phase. The CMC equations are to be viewed as an adjunct to existing methods for the modelling of the dynamics of sprays: they provide a refinement of the modelling of chemical reactions in the gas phase. The resulting CMC equations differ significantly from those already in use in the literature. They contain, of course, unclosed terms that need to be modelled. Investigation of the unclosed terms associated with evaporation at the droplet surface is well beyond the capabilities of laboratory measurement or direct numerical simulation. It is proposed that modelling of these terms be based on the well-established ‘laws’ of similarity between heat and mass transfer: an example is detailed for one example of the general modelling of the spray dynamics. Other unclosed terms are important throughout the gas phase. Models used for these terms in single-phase flows are reviewed and it is proposed that any modifications needed for these models be investigated by DNS of suitable model problems having good resolution of the flow and mixing in the inter-droplet space. It is proposed that a spray analogue of the scalar mixing layer that has been widely studied in single-phase flows be used as the model problem for such DNS studies and also for LES and RANS modelling.

Spatially implicit plankton population models: Transient spatial variability

Ocean plankton models are useful tools for understanding and predicting the behaviour of planktonic ecosystems. However, when the regions represented by the model grid cells are not well mixed, the population dynamics of grid cell averages may differ from those of smaller scales (such as the laboratory scale). Here, the ‘mean field approximation’ fails due to ‘biological Reynolds fluxes’ arising from nonlinearity in the fine-scale biological interactions and unresolved spatial variability. We investigate the domain-scale behaviour of two-component, 2D reaction–diffusion plankton models producing transient dynamics, with spatial variability resulting only from the initial conditions. Failure of the mean field approximation can be quite significant for sub grid-scale mixing rates applicable to practical ocean models. To improve the approximation of domain-scale dynamics, we investigate implicit spatial resolution methods such as spatial moment closure. For weak and moderate strengths of biological nonlinearity, spatial moment closure models generally yield significant improvements on the mean field approximation, especially at low mixing rates. However, they are less accurate given weaker transience and stronger nonlinearity. In the latter case, an alternative ‘two-spike’ approximation is accurate at low mixing rates. We argue that, after suitable extension, these methods may be useful for understanding and skillfully predicting the large-scale behaviour of marine ecosystems.

Moment closure for FENE models of complex fluids

AbstractIn this talk, we present some joint work with current and former colleagues at Penn State (C. Liu, P. Yu and Y. Hyon) on moment closure approximations of multiscale complex fluid models. Moment closure not only provides a computationally feasible approach to simulate many multiscale phenomena but also offers insight to the understanding of the underlying physical processes. We will discuss some of our recent works on developing systematic moment closure models for micromacro FENE models of complex fluids and demonstrate their effectiveness for a wide range of parameter regimes. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Crucial properties of the moment closure model FENE-QE

We investigate four crucial properties for testing and evaluating a moment closure approximation of the FENE dumbbell model for dilute polymer solutions: non-negative configuration distribution function, energy dissipation, accuracy of approximation and computational expense. Through mathematical analysis, numerical experiments and comparisons with closure model FENE-P and FENE-YDL, we prove that the FENE-QE approximation has non-negative configuration distribution function, approximates the energy dissipation behavior of original kinetic theory and provides good accuracy. To improve the efficiency of this closure approximation, we introduce a piecewise linear approximation technique that greatly reduces the computational cost. This extension of FENE-QE, FENE-QE-PLA, is the closure model we recommend for simulating dilute polymer solutions.

Capturing Effects of Rotation in Sudden Expansion Channels Using Detached Eddy Simulation

T HE effects of system rotation are of interest in flows in hydro/ turbo machinery especially in the design of centrifugal compressor impellers and runner blades. When the direction of rotation is in tandem with the fluid rotation vector, that is, the vorticity, the flow is stabilized, whereas the flow is destabilized if the directions are opposing. As with boundary layers, shear layers are also stabilized or destabilized by rotation. Rotation (or streamline curvature) generates extra strain rates that significantly affect turbulent stress production. Bradshaw [1] formulated an analogy between meteorological parameters and parameters describing rotation about the axis normal to the plane of rotation. He defined an effective Richardson number (Ri) for flows undergoing rotation that is used to define amodifiedmixing length [l l0 1 Ri ].Most of the later studies also propose a similar definition, where the effects of rotation are modeled by formulating corrections using a rotation Richardson number. Nilsen and Andersson [2] used an algebraic second moment closure model to predict the rotational effects on backward facing step flows. It was observed that rotation induced variation in the mean flow pattern was a result of significant changes in turbulent fluctuations in the free shear layer. However quantitative predictions of the reattachment length showed only partial agreement with the experimental data (Rothe and Johnston [3]). A similar study was carried out on rotating flows using a v–f model by Iaccarino et al. [4]. Both the original and a modified version of the model were used to predict the reattachment length downstream of a backward facing step. The modified model predicted the reattachment length better than the originalmodel. However itwas observed that the predictions of the modified model, like the algebraic second moment (ASM) model of Nilsen and Andersson [2] showed only partial agreement with the experiments. Therefore in spite of the continual development of Reynolds-averagedNavier–Stokes (RANS)models, predicting the effects of rotation in turbulent flows is a major challenge. The eddies formed in themassively separated regions downstream of the step are geometry specific. The “detached” eddies are not as universal as the eddies in typical thin shear layers where RANS models are calibrated. This is the reasonwhyRANSmodels are often observed to fail in flows undergoing massive separation. Large eddy simulation (LES) resolves the large energy carrying eddies while modeling the smaller isotropic eddies using a subgrid scale stress model. LES is a viable and a reliable method for simulating flows undergoing massive separation. However the near-wall resolution necessary for LESmakes it prohibitively expensive at high Reynolds numbers. A solution to the computational challenges associated with the reliable prediction ofmassively separated turbulent flows is detached eddy simulations (DES). DES sensitizes a RANS model to grid length scales, thereby allowing it to function as a subgrid scalemodel in critical regions of interest. This allows the natural instabilities of the flow in this region to develop, the energy cascade to grow, and improves the quality of the solution in this region. Though DES was initially proposed for the Spalart–Allmaras model, it can be easily extended to other models (Strelets [5]), by appropriately defining a turbulent length scale. The credibility of the approach has been validated by numerous applications in the literature for external and internal flows. The objective of the current study is to investigate the capabilities of DES in capturing the effects of rotation induced Coriolis forces on turbulent separation and reattachment in a backward facing step geometry. To evaluate the accuracy of the scheme in predicting the effects, the reattachment lengths predicted by DES are compared with experimental measurements by Rothe and Johnston [1] and also compared with other RANS models from the literature.