A finite element method for a three-dimensional second-order closure model of turbulent diffusion in a convective boundary layer

A three-dimensional second-order closure dispersion model is used to simulate the plume behaviour of a passive contaminant in a convective boundary layer. A time-splitting finite element method together with a non-linear filtering scheme is used to solve the three-dimensional second-order closure transport equations. The model results show good agreement with laboratory data for a ground level source.     

Parallel computation of atmospheric pollutant dispersion under unstably stratified atmosphere

Pollutant dispersion under unstably stratified atmosphere was investigated numerically using the finite element method. The effects of atmospheric stability on plume trajectory were studied using a three-dimensional second-order closure dispersion model. The numerical model was implemented using domain decomposition method and carried out using a parallel computer. The computation accelerates significantly and the size of computation can be largely increased as a result of the parallelism. A passive contaminant point source was placed at the middle of the convective boundary layer to simulate the atmospheric dispersion. The requirement of the input of dispersion coefficients in k-theory and Gaussian models was replaced with direct input of turbulence flow data. It was found that the present numerical model can predict several non-Gaussian plume behaviours and the computed results agreed well with findings from experimental observations. © 1998 John Wiley & Sons, Ltd.     

Sensitivity of plume dynamics to the parameterization of vertical mixing

A three‐dimensional primitive equation, baroclinic numerical model incorporating a range of turbulence closure schemes is used to investigate the effects of vertical diffusion of momentum and density upon the spread of a freshwater plume, with particular reference to the Ebro plume. Initial calculations show that there are some differences in the horizontal spread and vertical mixing of the plume when diffusion coefficients are computed from a two‐equation turbulence energy model compared with a one‐equation model. To understand results from the turbulence energy models, the sensitivity of the plume dynamics to variations in the coefficient of vertical eddy viscosity and diffusivity is also considered, with increases in these parameters having a significant effect upon the cross‐shore and along‐shore spread of the plume. Also, increasing these parameters changes the plume characteristics from supercritical to subcritical and reduces the occurrence of meandering and baroclinic instability along the plume's off‐shore edge. However, differences in the southerly spread (the direction of Kelvin wave propagation in the model) of the plume (although not its northerly spread) produced by changes in diffusion coefficients are small compared with the influence of changes in the bottom slope, upon the along‐shore southerly spread of the plume, which moves in the direction of Kelvin wave propagation in the near coastal region. Results from the series of calculations are used as a guide in experimental design, with reference to a planned experiment in the Ebro region involving a coastal HF Radar deployment, as well as off‐shore measurements. Calculations suggest that surface current measurements from a coastal HF Radar, together with a detailed survey of the density field associated with the plume, may be an appropriate, although indirect, means of determining suitable mixing coefficients to use in plume discharge problems. Detailed measurements of water depth variation will also be required. Copyright © 1999 John Wiley & Sons, Ltd.     

Buoyancy effects on turbulent swirling flows

A three-parameter model of turbulence applicable to free boundary layers has been developed and applied for the prediction of axisymmetric turbulent swirling flows in uniform and stagnant surroundings under the action of buoyancy forces. The turbulent momentum and heat fluxes appearing in the time-averaged equations for the mean motion have been determined from algebraic expressions, derived by neglecting the convection and diffusion terms in the differential transport equations for these quantities, which relate the turbulent fluxes to the kinetic energy of turbulence, k, the dissipation length scale of turbulence, L, and the temperature covariance, T ′². Differential transport equations have been used to determine these latter quantities. The governing equations have been solved using fully implicit finite difference schemes. The turbulence model is capable of reproducing the gross features of pure jet flows, buoyant flows and swirling flows for weak and moderate swirl. The behaviour of a turbulent buoyant swirling jet has been found to depend solely on exit swirl and Froude numbers. The predicted results indicate that the incorporation of buoyancy can cause significant changes in the behaviour of a swirling jet, particularly when the buoyancy strength is high. The jet exhibits similarity behaviour in the initial region for weak swirl and weak buoyancy strengths only, and the asymptotic case of a swirling jet under the action of buoyancy forces is a pure plume in the far field. The predicted results have been found to be in satisfactory agreement with the available experimental data and in good qualitative agreement with other predicted results.     

Large-eddy simulation of buoyancy-driven natural ventilation in an enclosure with a point heat source

Rising buoyant plumes from a point heat source in a naturally ventilated enclosure have been investigated using large-eddy simulation (LES). The aim of the work is to assess the performance and the accuracy of LES for modelling buoyancy-driven displacement ventilation of an enclosure and to shed more light on the transitional behaviour of the plume and the coherent structures involved. The Smagorinsky sub-grid scale model is used for the unresolved small-scale turbulence. The Rayleigh number, Ra is chosen to be in the range where spatial transition from laminar to turbulent flow takes place (Ra = 1.5 × 10⁹). The plume properties (source strength and rate of spread) as well as the ventilation properties (stratification height and temperature of stratified layer) estimated using the theory of Linden et al. are found to agree reasonably well with the LES results. The variation of the plume width with height indicates a linear variation of the entrainment coefficient rather than a constant value used by Linden et al. for a fully turbulent thermal plume. Flow visualisation revealed the nature of the large-scale coherent structures involved in the transition to turbulence in the plume. The most excited modes observed in the velocity, pressure and temperature fields spectra correspond to Strouhal number in the range 0.3 ≤ St ≤ 0.55 which is in agreement with those observed by Zhou et al. for a turbulent forced plume. Excited modes less than thisvalue (St = 0.2) were observed and may be due to low-frequency motions felt throughout the flow.     

A two-scale second-order moment two-phase turbulence model for simulating dense gas-particle flows

A two-scale second-order moment two-phase turbulence model accounting for inter-particle collision is developed, based on the concepts of particle large-scale fluctuation due to turbulence and particle small-scale fluctuation due to collision and through a unified treatment of these two kinds of fluctuations. The proposed model is used to simulate gas-particle flows in a channel and in a downer. Simulation results are in agreement with the experimental results reported in references and are near the results obtained using the single-scale second-order moment two-phase turbulence model superposed with a particle collision model (USM-θ model) in most regions. The project supported by the Special Funds for Major State Basic Research, China (G-1999-0222-08), and the Postdoctoral Science Foundation (2004036239) The English text was polished by Keren Wang     

From Two-Point Closures of Isotropic Turbulence to LES of Shear Flows

We first recall the EDQNM two-point closure approach of three-dimensional isotropic turbulence. It allows in particular prediction of the infrared kinetic-energy dynamics (with ak ₄ backscatter) and the associated time-decay law of kinetic-energy, useful in particular for one-point closure modelling. Afterwards, we show how the spectral eddy viscosity concept may be used for large-eddy simulations: we introduce the plateau-peak model and the spectral-dynamic models. They are applied to decaying isotropic turbulence, and allow recovery of the EDQNM infrared energy dynamics. Anew infrared k ₂ law for the pressure spectrum, predicted by the closure, is also well verified. Assuming that subgrid scales are not too far from isotropy, the spectral-dynamic model is applied to the channel flow at h ₊= 390, with statistics in very good agreement with DNS, while reducing considerably the computational time. We study with the aid of DNS and LES the case of the channel rotating about an axis of spanwise direction. The calculations allow to recover the universal linear behaviour of the mean velocity profile, with a local Rossby number equal to −1. We present also LES (using the Grenoble Filtered Structure-Function Model), of a turbulent boundary layer passing over a cavity. Finally, we make some remarks on the future of LES for industrial applications. This revised version was published online in July 2006 with corrections to the Cover Date.     

MULTIGRID CONVERGENCE ACCELERATION FOR TURBULENT SUPERSONIC FLOWS

A multigrid convergence acceleration technique has been developed for solving both the Navier–Stokes and turbulence transport equations. For turbulence closure a low-Reynolds-number q–ω turbulence model is employed. To enable convergence, the stiff non-linear turbulent source terms have to be treated in a special way. Further modifications to standard multigrid methods are necessary for the resolution of shock waves in supersonic flows. An implicit LU algorithm is used for numerical time integration. Several ramped duct test cases are presented to demonstrate the improvements in performance of the numerical scheme. Cases with strong shock waves and separation are included. It is shown to be very effective to treat fluid and turbulence equations with the multigrid method. A comparison with experimental data demonstrates the accuracy of the q–ω turbulence closure for the simulation of supersonic flows. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 1019–1035, 1997.     

An implicit mixed finite-volume–finite-element method for solving 3D turbulent compressible flows

The development of new aeronautic projects require accurate and efficient simulations of compressible flows in complex geometries. It is well known that most flows of interest are at least locally turbulent and that the modelling of this turbulence is critical for the reliability of the computations. A turbulence closure model which is both cheap and reasonably accurate is an essential part of a compressible code. An implicit algorithm to solve the 2D and 3D compressible Navier–Stokes equations on unstructured triangular/tetrahedral grids has been extended to turbulent flows. This numerical scheme is based on second-order finite element–finite volume discretization: the diffusive and source terms of the Navier–Stokes equations are computed using a finite element method, while the other terms are computed with a finite volume method. Finite volume cells are built around each node by means of the medians. The convective fluxes are evaluated with the approximate Riemann solver of Roe coupled with the van Albada limiter. The standard k–ϵ model has been introduced to take into account turbulence. Implicit integration schemes with efficient numerical methods (CGS, GMRES and various preconditioning techniques) have also been implemented. Our interest is to present the whole method and to demonstrate its limitations on some well-known test cases in three-dimensional geometries. © 1997 John Wiley & Sons, Ltd.     

On the entrainment assumption in Schatzmann’s integral plume model

The behaviour of stationary, non-passive plumes can be simulated in a reasonably simple and accurate way by integral models. One of the key requirements of these models, but also one of their less well-founded aspects, is the entrainment assumption, which parameterizes turbulent mixing between the plume and the environment. The entrainment assumption developed by Schatzmann and adjusted to a set of experimental results requires four constants and an ad hoc hypothesis to eliminate undesirable terms. With this assumption, Schatzmann’s model exhibits numerical instability for certain cases of plumes with small velocity excesses, due to very fast radius growth. The purpose of this paper is to present an alternative entrainment assumption based on a first-order turbulence closure, which only requires two adjustable constants and seems to solve this problem. The asymptotic behaviour of the new formulation is studied and compared to previous ones. The validation tests presented by Schatzmann are repeated and it is found that the new formulation not only eliminates numerical instability but also predicts more plausible growth rates for jets in co-flowing streams.     

One Equation Model for Turbulent Channel Flow with Second Order Viscoelastic Corrections

A modified second order viscoelastic constitutive equation is used to derive a k–l type turbulence closure to qualitatively assess the effects of elastic stresses on fully-developed channel flow. Specifically, the second order correction to the Newtonian constitutive equation gives rise to a new term in the momentum equation involving the time-averaged elastic shear stress and in the turbulent kinetic energy transport equation quantifying the interaction between the fluctuating elastic stress and rate of strain tensors, denoted by P _w , for which a closure is developed and tested. This closure is based on arguments of isotropic turbulence and equilibrium in boundary layer flows and a priori P _w could be either positive or negative. When P _w is positive, it acts to reduce the production of turbulent kinetic energy and the turbulence model predictions qualitatively agree with direct numerical simulation (DNS) results obtained for more realistic viscoelastic fluid models with memory which exhibit drag reduction. In contrast, P _w  < 0 leads to a drag increase and numerical breakdown of the model occurs at very low values of the Deborah number, which signifies the ratio of elastic to viscous stresses. Limitations of the turbulence model primarily stem from the inadequacy of the k–l formulation rather than from the closure for P _w . An alternative closure for P _w , mimicking the viscoelastic stress work predicted by DNS using the Finitely Extensible Nonlinear Elastic-Peterlin fluid model, which is mostly characterized by P _w  > 0 but has also a small region of negative P _w in the buffer layer, was also successfully tested. This second model for P _w leads to predictions of drag reduction, in spite of the enhancement of turbulence production very close to the wall, but the equilibrium conditions in the inertial sub-layer were not strictly maintained.     

用二阶矩模型研究Richtmyer-Meshkov不稳定性诱导湍流混合

发展了考虑密度脉动和各向异性湍流的二阶矩模型,强调了涉及湍流能量产生项的关联。采用该模型对Poggi等的激波管实验进行了模拟。通过与实验结果的比较分析,验证了采用的模型封闭、模型常数、数值算法和程序实现是合适的。在此基础上,进一步探讨了冲击马赫数和Atwood数对混合的影响。     

A USM-Θ two-phase turbulence model for simulating dense gas-particle flows

A second-order moment two-phase turbulence model for simulating dense gas-particle flows (USM- model), combining the unified second-order moment two-phase turbulence model for dilute gas-particle flows with the kinetic theory of particle collision, is proposed. The interaction between gas and particle turbulence is simulated using the transport equation of two-phase velocity correlation with a two-time-scale dissipation closure. The proposed model is applied to simulate dense gas-particle flows in a horizontal channel and a downer. Simulation results and their comparison with experimental results show that the model accounting for both anisotropic particle turbulence and particle-particle collision is obviously better than models accounting for only particle turbulence or only particle-particle collision. The USM- model is also better than the k--kp- model and the k--kp-p- model in that the first model can simulate the redistribution of anisotropic particle Reynolds stress components due to inter-particle collision, whereas the second and third models cannot.The project supported by the Special Funds for Major State Basic Research of China (G-1999-0222-08), the National Natural Science Foundation of China (50376004), and Ph.D. Program Foundation, Ministry of Education of China (20030007028)     

Second-Order Conditional Moment Closure Simulations of Autoignition of an n-heptane Plume in a Turbulent Coflow of Heated Air

Autoignition of an n-heptane plume in a turbulent coflow of heated air has been studied using the conditional moment closure (CMC) method with a second-order closure for the conditional chemical source term. Two different methodologies have been considered: (i) the Taylor expansion method, in which the second order correction was based on the solution of the full covariance matrix for the 31 reactive species in the chemical mechanism and hence was not limited to a few selected reactions, and (ii) the conditional PDF method, in which only the temperature conditional variance equation has been solved and its PDF assumed to be a β-function. The results compare favorably with experiment in terms of autoignition location. The structure of the reaction zone in mixture fraction space has been explored. The relative performance of the two methodologies is discussed.     

A calibration scheme for quantitative concentration measurements using simultaneous PIV and PLIF

A new approach for calibration of planar laser-induced fluorescence (PLIF) measurements is presented. The calibration scheme is based on the fact that there is a constant concentration flux through each cross-section of a fluorescent plume in a given flow field and makes use of simultaneous measurements of particle image velocimetry (PIV) and PLIF. The following are the advantages of the current technique: (1) it is experimentally less demanding and (2) it does not require in situ calibration for generating the calibration curves. The technique can be implemented in many experimental setups (both in water and gaseous flows) provided the geometry of the time-averaged scalar field is known. Using the calibration scheme, an analysis is carried out on the measurements of concentration fields in grid turbulence to validate the proposed technique. To demonstrate the feasibility of the scheme, the distributed second-order moments (μ ₂), and concentration and velocity correlations ( á u^￠c^￠ ñ \left\langle {u^{\prime}c^{\prime}} \right\rangle and á v^￠c^￠ ñ \left\langle {v^{\prime}c^{\prime}} \right\rangle ) are computed. Good agreement is found with previous studies. In addition, a quantitative appraisal of a simple closure approximation of the moment-based transport equation is also presented using simultaneous PIV and PLIF.     

SEMI-IMPLICIT FINITE VOLUME SHALLOW-WATER FLOW AND SOLUTE TRANSPORT SOLVER WITH k–ε TURBULENCE MODEL

A 3D semi-implicit finite volume scheme for shallow- water flow with the hydrostatic pressure assumption has been developed using the σ-co-ordinate system, incorporating a standard k–ε turbulence transport model and variable density solute transport with the Boussinesq approximation for the resulting horizontal pressure gradients. The mesh spacing in the vertical direction varies parabolically to give fine resolution near the bed and free surface to resolve high gradients of velocity, k and ε. In this study, wall functions are used at the bed (defined by the bed roughness) and wind stress at the surface is not considered. Surface elevation gradient terms and vertical diffusion terms are handled implicitly and horizontal diffusion and source terms explicitly, including the Boussinesq pressure gradient term due to the horizontal density gradient. The advection terms are handled in explicit (conservative) form using linear upwind interpolation giving second-order accuracy. A fully coupled solution for the flow field is obtained by substi- tuting for velocity in the depth-integrated continuity equation and solving for surface elevation using a conjugate gradient equation solver. Evaluation of horizontal gradients in the σ-co-ordinate system requires high-order derivatives which can cause spurious flows and this is avoided by obtaining these gradients in real space. In this paper the method is applied to parallel oscillatory (tidal) flow in deep and shallow water and compared with field measurements. It is then applied to current flow about a conical island of small side slope where vortex shedding occurs and velocities are compared with data from the laboratory. Computed concentration distributions are also compared with dye visualization and an example of the influence of temperature on plume dispersion is presented. © 1997 John Wiley & Sons, Ltd.     

A dynamic two-equation Sub Grid Scale model

In this paper we demonstrate that the transport equation of the generalised subgrid scale (SGS) turbulent stress tensor is form-invariant but not frame-indifferent under Euclidean transformations of the frame. A new closure equation between the generalized SGS turbulent stress tensor and the resolved kinematic quantities is proposed. The closure equation at the basis of the proposed model (Two-Equation Model, TEM): a) respects the principle of the turbulence frame indifference [1]; b) takes into account both the anisotropy of the turbulence velocity scales and turbulence length scales; c) removes any balance assumption between the production and dissipation of SGS turbulent kinetic energy; d) assumes scale similarity in the definition of the second-order tensor representing the turbulent velocity scales. In the proposed model: a) the closure coefficient C which appears in the constitutive equation is uniquely determined without using Germanos dynamic procedure [2]; b) the generalized SGS turbulent stress tensor is related exclusively to the generalized SGS turbulent kinetic energy (which is calculated by means of its balance equation) and the modified Leonard tensor; c) the viscous dissipation of the generalized SGS turbulent kinetic energy is calculated by solving the balance equation. The proposed model is tested for a turbulent channel flow at Reynolds numbers (based on friction velocity and channel half-width) ranging from 180 to 2340.Received: 11 February 2004, Accepted: 20 August 2004, Published online: 22 February 2005PACS: 02.60.Cb, 47.27.Eq, 47.11. + j Correspondence to: F. Gallerano     

Toward a Turbulence Constitutive Relation for Geophysical Flows

Rapidly rotating turbulent flows are frequently in approximate geostrophic balance. Single-point turbulence closures, in general, are not consistent with a geostrophic balance. This article addresses and resolves the possibility of a constitutive relation for single-point second-order closures for classes of rotating and stratified flows relevant to geophysics. Physical situations in which a geostrophic balance is attained are described. Closely related issues of frame-indifference, horizontal divergence, and the Taylor–Proudman theorem are discussed. It is shown that, in the absence of vortex stretching along the axis of rotation, turbulence is frame-indifferent. Unfortunately, no turbulence closures are consistent with this frame-indifference that is frequently an important feature of rotating or quasi-geostrophic flows. A derivation and discussion of the geostrophic constraint which ensures that the modeled second-moment equations are frame-invariant, in the appropriate limit, is given. It is shown that rotating, stratified, and shallow water flows are situations in which such a constitutive relation procedure is useful. A nonlinear nonconstant coefficient representation for the rapid-pressure strain covariance appearing in the Reynolds stress and heat flux equations, consistent with the geostrophic balance, is described. The rapid-pressure strain closure features coefficients that are not constants determined by numerical optimization but are functions of the state of turbulence as parametrized by the Reynolds stresses and the turbulent heat fluxes as is required by tensor representation theory. These issues are relevant to baroclinic and barotropic atmospheric and oceanic flows. The planetary boundary layers in which there is a transition, with height or depth, from a thermally or shear driven turbulence to a geostrophic turbulence is a classic geophysical example to which the considerations in this article are relevant. Received 14 October 1996 and accepted 9 June 1997     

A k–ℓ turbulence closure sensitized to non-simple shear flows

This paper introduces a two-equation turbulence model sensitized to deviations from simple shear flows. The closure is topography-parameter-free and is based on solving transport equations for the turbulence kinetic energy (k) and the turbulence length-scale (?). Brief model derivation details are given and test cases are presented to compare the model's performance to other closures and to experimental data. The flow examples demonstrate the advantage of the k–? model in non-simple shear flows.