KOLMOGOROV BEHAVIOR OF NEAR-WALL TURBULENCE AND ITS APPLICATION IN TURBULENCE MODELING

The near-wall behavior of turbulence is re-examined in a way different from that proposed by Hanjalic and Launder¹ and followers^2,3,4,5. It is shown that at a certain distance from the wall, all energetic large eddies will reduce to Kolmogorov eddies (the smallest eddies in turbulence). All the important wall parameters, such as friction velocity, viscous length scale, and mean strain rate at the wall, are characterised by Kolmogorov microscales. According t o this Kolmogorov behavior of near-wall turbulence, the turbulence quantities, such as turbulent kinetic energy, dissipation rate, etc. at the location where the large eddies become “Kolmogorov” eddies, can be estimated by using both direct numerical simulation (DNS) data and asymptotic analysis of near-wall turbulence. This information will provide useful boundary conditions for the turbulent transport equations. As a n example, the concept is incorporated in the standard κ - εmodel which is then applied t o channel and boundary layer flows. Using appropriate boundary conditions (based on Kolmogorov behaviour of near-wall turbulence), there is no need for any wall-modification to the κ - ε equations (including model constants). Results compare very well with the DNS and experimental data.     

Measurement of spectrum with particle image velocimetry

The purpose of this paper is to show that the measurement of turbulent spectrum using wholefield velocity techniques such as particle image velocimetry (PIV) is possible. Toward this end, data from the axial plane of a self-similar turbulent axisymmetric jet, at a Reynolds number, based on Taylor microscale of 30 has been analyzed. The two-dimensional velocity data are first high-pass filtered, which educes the vortices. An automated method is then used to identify the vortices and measure their properties. By directly measuring the energy of the vortices, it is possible to plot the turbulence spectrum. The spectrum presented here shows the presence of energy containing and inertial regimes. However, the smallest scales have not been resolved in the measurements. The slope of the spectrum in the inertial subrange is about −1.6. The number of vortices in the two regimes have also been measured. The number of vortices in the energy containing regime is substantially smaller than those in the inertial subrange. The technique has been verified by analyzing another dataset. These results show that the direct measurement of vortex properties with reasonable confidence is possible using PIV and an appropriate vortex eduction technique.     

AN ANALYTICAL DESIGN METHODOLOGY FOR AN ANNULAR COMBUSTOR

SUMMARY

This paper describes a computational procedure for the optimization of the performance parameters of a simulated annular combustor. This method has been applied to analyze the influence of the performance parameters and geometries on the annular combustor characteristics and provide a good understanding of combustor internal flow fields, and therefore it can be used for guiding the combustor design process. The approach is based on the solution of governing nonlinear, elliptic partial differential equations for 3-D axisymmetric recirculating turbulent reacting swirling flows and the modelling of turbulence, combustion, thermal radiation and pollutant formation. The turbulence effects are introduced via the modified two-equation κ-ε model. Turbulent combustion is modelled using the κ-ε-g model and a two-step turbulent combustion model is employed for the excess emission of carbon monoxide CO. For the evaluation of the NO pollutant formation rate, the NO pollutant formation model, which takes into account the influence of turbulence, presented here. The radiative heat transfer is handled by the heat flux model. The predictions of the combustor character-istics and performance parameters are made using the present approach.

Predictions of velocity, length of the recirculation zone, combustion efficiency and wall temperature are compared with measurements. Agreement between the predictions and experimental data is very satisfactory.     

Implicit unfactored implementation of two-equation turbulence models in compressible Navier–Stokes methods

An implicit unfactored method for the coupled solution of the compressible Navier–Stokes equations with two-equation turbulence models is presented. Both fluid-flow and turbulence transport equations are discretized by a characteristics-based scheme. The implicit unfactored method combines Newton subiterations and point-by-point Gauss–Seidel subrelaxation. Implicit-coupled and -decoupled strategies are compared for their efficiency in the solution of the Navier–Stokes equations in conjunction with low-Re two-equation turbulence models. Computations have been carried out for the flow over an axisymmetric bump using the k–ϵ and k–ω models. Comparisons have been obtained with experimental data and other numerical solutions. The present study reveals that the implicit unfactored implementation of the two-equation turbulence models reduces the computing time and improves the robustness of the CFD code in turbulent compressible flows. © 1998 John Wiley & Sons, Ltd.     

Numerical computation of an impinging jet with uniform wall suction

The paper presents numerical predictions of a turbulent axisymmetric jet impinging onto a porous plate, based on a finite volume method of solving the Navier-Stokes equations for an incompressible air jet with the K–ε turbulence model. The velocity and pressure terms of the momentum equations are solved by the SIMPLE (semi-implicit method for pressure-linked equation) method. In this study, non-uniform staggered grids are used. The parameters of interest include the nozzle-to-wall distance and the suction velocity. The results of the present calculations are compared with available data reported in the literature. It is found that suction effects reduce the boundary layer thickness and increase the velocity gradient near the wall.     

A two-scale second-moment turbulence closure based on weighted spectrum integration

A two-scale second-moment turbulence closure has been derived based on the weighted integration of the dynamic equation for the covariance spectrum. The goal is to close the Reynolds stress equations with two additional scalar equations that provide separately the scales of the spectral energy transfer and of the turbulence energy dissipation rate. Such a model should provide better prediction of nonequilibrium turbulent flows. The derivation consists of analytical integration of the wave-number-weighted covariance spectrum using a model of the spectral equations with an assumed simple representation of the shape of the energy spectrum. The resulting closure consists of a set of three tensorial equations, one for the Reynolds stress and two for length scale tensors, the latter representing the energy containing- and dissipative eddies respectively. The trace of the two tensor-scale equations leads to a set of two scalar scale parameters. In the equilibrium limit, the model reduces to the standard second-moment single-scale closure. The approach makes it also possible to derive the scale equations in a more systematic manner as compared with the common single-scale and other multi-scale models. The performance of the model in capturing the scale dynamics is illustrated by predictions of several generic homogeneous and inhomogeneous unsteady flows, demonstrating the expected response of the two scale equations. PACS 03.50.De, 04.20-q, 42.65-k     

空化、超空化流动的数值模拟方法研究

基于结构化网格,运用可压缩流N—S方程及k-ε湍流模型对流场进行求解,在低压区域引入一种基于混合密度函数的空化模型对轴对称体的空化、超空化流动进行了数值模拟．通过将半球圆柱的计算结果与实验数据和前人的计算结果进行对比,验证了所发展的数值方法的可靠性．最后,采用非定常的数值方法,研究了钝头体射弹的空化、超空化流动特性,并模拟了其超空泡的发展过程．     

Analysis of unsteady gas–liquid flows in a rectangular tank: Comparison of Euler–Eulerian and Euler–Lagrangian simulations

We study the dynamics of gas–liquid flows experimentally and computationally in a rectangular bubble column where the gas source is introduced at the corner. The flow in this reactor is complex and inherently unsteady in nature. The two-dimensional liquid phase velocity field is calculated by an Eulerian approach solving the unsteady Reynolds Averaged Navier Stokes equations. The conservation equations are closed using a two parameter turbulence model. The two-way coupling was accounted for by adding source terms in the conservation equations of the continuous phase to take into account the interaction with the dispersed phase. Bubble tracking is achieved through a Lagrangian approach. Here the equations of motion are solved taking into account the drag, pressure, buoyancy and gravity forces. The time-averaged flows along with the variables which characterize turbulence are analyzed for a wide range of gas flow-rates using Euler–Lagrangian simulations. These simulation predictions are validated with Euler–Eulerian simulations where the gas-phase distribution is captured as a void fraction and PIV experiments. The motion of bubbles induces turbulence in the flow. The applicability of two parameter models for turbulence like the standard k–ε model on time-averaged flow properties is addressed. From the results of the time averaged velocity field, turbulence intensity, turbulent viscosity and gas hold-up profiles, it is concluded that the Euler–Lagrangian model is applicable at lower gas flow-rates. The Euler–Eulerian approach was found to be valid at lower as well as higher gas flow-rates.     

轴对称再入舱模型气动热特性数值模拟研究

应用基于块结构网格的有限体积求解方法,对热化学非平衡环境下轴对称再入舱模型的气动热特性进行了数值模拟。控制方程为带化学反应的多组元轴对称N-S方程,空间离散采用VanLeer迎风格式,时间推进为隐式LU-SGS格式;采用7组元7化学反应模型及Park双温模型模拟再入流场的热化学非平衡效应。对Hollis MP-1模型的气动热特性进行了数值模拟,分别就网格效应、湍流模型、流场的热力学性质对流场的气动力、热环境的影响进行了深入研究。研究结果表明:SST模型与k-w1998模型能更准确地计算再入流场热流峰值的位置与大小;在再入舱模型的局部区域,采用热力学非平衡模型计算的物面压强与热流结果要明显低于热力学平衡模型的结果。     

Error quantification of 3D homogeneous and isotropic turbulence measurements using 2D PIV

Particle image velocimetry (PIV) has become a popular non-intrusive tool for measuring various types of flows. However, when measuring three dimensional flows with 2D PIV, there is inherent measurement error due to out-of-plane motion. Errors in the measured velocity field propagate to turbulence statistics. Since this can distort the overall flow characteristics, it is important to understand the effect of this out-of-plane error. In this study, the effect of out-of-plane motion on turbulence statistics is quantified. Using forced isotropic turbulence direct numerical simulation (DNS) flow field data provided by the Johns Hopkins turbulence database (JHTDB), synthetic image tests are performed. Turbulence statistics such as turbulence kinetic energy, dissipation rate, Taylor microscale, Kolmogorov scale, and velocity correlations are calculated. Various test cases were simulated while controlling three main parameters which affect the out-of-plane motion: PIV interrogation window size, camera inter-frame time, and laser sheet thickness. The amount of out-of-plane motion was first quantified, and then the error variation according to these parameters was examined. This information can be useful when examining fully three dimensional flows such as homogeneous and isotropic turbulence via 2D PIV.     

Compressibility effects due to turbulent fluctuations

This paper deals with intrinsic effects of compressibility, i.e. with dilatation fluctuations in response to pressure fluctuations. Three different types of turbulent flows are considered in more detail: homogeneous turbulent shear flow, wall-bounded turbulent shear flow and shock/turbulence interaction. A survey of the present knowledge in this field, mainly based on DNS data, is given. Using the linear inviscid perturbation equations a direct link between fluctuations of dilatation and of velocity in the direction of mean shear is presented for homogeneous shear flow. This relation might form the basis for a more universal pressure-dilatation model. It is conjectured that the insignificance of intrinsic compressibility effects in wall-bounded supersonic shear flow is mainly due to the impermeability constraint of the wall. To this end, a linear stability analysis of supersonic channel flow along cooled, but permeable walls has been performed based on Coleman et al.'s [5] mean flow data. It shows an increase in the moduli of eigenfunctions related to compressibility, like pressure, and in moduli of quantities derived from eigenfunctions such as ‘pressure dilatation’ and squared dilatation. Although these results do not prove our hypothesis they provide hints in this direction. Shock/turbulence interaction is viewed as a source of compressibility. Former DNS data of Hannappel and Friedrich [10] for shock/isotropic turbulence interaction showing the effect of compressibility on the amplification of fluctuations are interpreted based on linear perturbation equations.     

A scanning PIV method for fine-scale turbulence measurements

A hybrid technique is presented that combines scanning PIV with tomographic reconstruction to make spatially and temporally resolved measurements of the fine-scale motions in turbulent flows. The technique uses one or two high-speed cameras to record particle images as a laser sheet is rapidly traversed across a measurement volume. This is combined with a fast method for tomographic reconstruction of the particle field for use in conjunction with PIV cross-correlation. The method was tested numerically using DNS data and with experiments in a large mixing tank that produces axisymmetric homogeneous turbulence at \(R_\lambda \simeq 219\) . A parametric investigation identifies the important parameters for a scanning PIV set-up and provides guidance to the interested experimentalist in achieving the best accuracy. Optimal sheet spacings and thicknesses are reported, and it was found that accurate results could be obtained at quite low scanning speeds. The two-camera method is the most robust to noise, permitting accurate measurements of the velocity gradients and direct determination of the dissipation rate.     

Band-pass filtered velocity statistics in decaying turbulent box

For homogeneous isotropic turbulence study, the acquisition of band-pass filtered velocity increments (FVI) in a non-forced turbulent box is still a challenge both experimentally and numerically. Turbulence and associated physical processes, at a given instant, are permanently contaminated by a forcing process which can seldom be universal. The situation tends to be the origin of intermittency and the non-Gaussian probability density distribution for acceleration and velocity gradients. To reveal implied mechanism, grid turbulence is adapted to observe non-perturbed homogeneous isotropic turbulence. The velocity increments (VI) can be obtained following Comte-Bellot and Corrsin (GCBC) by means of two point-two time shifted velocity measurements. It is difficult to obtain decaying turbulence (DT) at large turbulent Reynolds number without pollution coming from walls. Nevertheless it is also significant to investigate DT in low Reynolds number regimes to determine non-polluted tendencies. The similarity of DT between particle image velocimetry (PIV) and hot wire anemometry measurements by GCBC are presented. Here we focus our tendency on VI and FVI probability density function (PDF) shapes in this letter. In conclusion, the tendency to Gaussian shape in inertial zone wavenumbers, demonstrates that there will be no intermittency if turbulent cascade is not perturbed.     

Short-and resonant-range interactions between scales in turbulence and their applications

Interactions between different scales in turbulence were studied starting from the incompressible Navier-Stokes equations. The integral and differential formulae of the shortrange viscous stresses, which express the short-range interactions between contiguous scales in turbulence, were given. A concept of the resonant-range interactions between extreme contiguous scales was introduced and the differential formula of the resonant-range viscous stresses was obtained. The short- and resonant-range viscous stresses were applied to deduce the large-eddy simulation ( LES ) equations as well as the multiscale equations, which are approximately closed and do not contain any empirical constants or relations. The properties and advantages of using the multiscale equations to compute turbulent flows were discussed. The short-range character of the interactions between the scales in turbulence means that the multiscale simulation is a very valuable technique for the calculation of turbulent flows. A few numerical examples were also given.     

An explicit algebraic Reynolds stress model in turbulence

A new algebraic Reynold stress model is constructed with recourse to the realizability constraints. Model coefficients are made a function of strain and vorticity invariants through calibration by reference to homogeneous shear flow data. The anisotropic production in near‐wall regions is accounted for substantially by modifying the model constants C_{ε(1, 2)} and adding a secondary source term in the ε equation. Hence, it reduces the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. To facilitate the evaluation of the turbulence model, some extensively used benchmark cases in the turbulence modelling are convoked. The comparisons demonstrate that the new model maintains qualitatively good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2006 John Wiley & Sons, Ltd.     

Application of compressible Reynolds-averaged governing equations to turbulent mixed convection in supercritical fluids in heated vertical tubes

Accurate estimation of thermal-hydraulic characteristics of supercritical flows has long been an attractive but elusive subject to many researchers in spite of tremendous effort devoted to the development of suitable turbulence models. One of the key reasons for the difficulty is a lack of measured turbulence data, which might have been used to formulate adequate turbulence models suitable for highly buoyant fluids. Turbulence models are typically based on the log-law, while the velocity profile in buoyant fluids substantially deviates from the log-law because of significant density variation in a turbulent boundary layer. In this paper, axisymmetric compressible Reynolds-Averaged governing equations were employed together with the property-dependent turbulent Prandtl number to reproduce experimental data representing heat transfer deterioration and consequential sudden temperature increase. The additional turbulence terms associated with turbulent mass flux appeared in the governing equations were modeled using the simple gradient diffusion hypothesis (SGDH). The proposed model successfully reproduced the experimental data. The various turbulence properties are presented and discussed.     

两方程湍流模型混合隐式迭代/解析算法研究

采用通量修正二阶TVD格式和混合隐式时间推进/解析算法求解k-ε湍流模型的输运方程以克服其计算刚性. 通过过膨胀状态下轴对称喷管内流场、带圆转方型面二元喷管内外流场两个算例, 验证了其配合改进的多重网格算法对计算效率的提高. 计算结果与试验数据进行了对比.      

Prediction of the particle-laden jet with a two-equation turbulence model

A two-equation turbulence model for two-phase flows has recently been proposed by Elghobashi & Abou-Arab (1983). They derived the exact equations of the kinetic energy of turbulence and the rate of dissipation of that energy, and modeled the turbulent correlations, resulting from time-averaging, up to third order. In order to validate the proposed model, a turbulent axisymmetric gaseous jet laden with spherical uniform-size solid particles is studied here. The predictions of the mean flow properties of the two-phases and the turbulence kinetic energy and shear stress of the carrier phase show good agreement with the experimental data.     

Applications of a higher‐order bounded numerical scheme to turbulent flows

This paper applies the higher‐order bounded numerical scheme Weighted Average Coefficients Ensuring Boundedness (WACEB) to simulate two‐ and three‐dimensional turbulent flows. In the scheme, a weighted average formulation is used for interpolating the variables at cell faces and the weighted average coefficients are determined from a normalized variable formulation and total variation diminishing (TVD) constraints to ensure the boundedness of the solution. The scheme is applied to two turbulent flow problems: (1) two‐dimensional turbulent flow around a blunt plate; and (2) three‐dimensional turbulent flow inside a mildly curved U‐bend. In the present study, turbulence is evaluated by using a low‐Reynolds number version of the k–ω model. For the flow simulation, the QUICK scheme is applied to the momentum equations while either the WACEB scheme (Method 1) or the UPWIND scheme (Method 2) is used for the turbulence equations. The present study shows that the WACEB scheme has at least second‐order accuracy while ensuring boundedness of the solutions. The present numerical study for a pure convection problem shows that the ‘TVD’ slope ranges from 2 to 4. For the turbulent recirculating flow, two different mixed procedures (Method 1 and Method 2) produce a substantial difference for the mean velocities as well as for the turbulence kinetic energy. Method 1 predicts better results than Method 2 does, comparing the analytical solution and the experimental data. For the turbulent flow inside the mildly curved U‐bend, although the predictions of velocity distributions with two procedures are very close, a noticeable difference of turbulence kinetic energy is exhibited. It is noticed that the discrepancy exists between numerical results and the experimental data. The reason is the limit of the two‐equation turbulence model to such complex turbulent flows with extra strain‐rates. Copyright © 2001 John Wiley & Sons, Ltd.     

Modeling the Dynamics of Ensemble-Averaged Linear Disturbances in Homogeneous Shear Flow

In order to expand the predictive capability of single-point turbulence closure models to account for the early-stage transition regime, a methodology for the formulation and calibration of model equations for the ensemble-averaged disturbance kinetic energy and energy dissipation rate is presented. The calibration is based on homogeneous shear flow where disturbances can be described by rapid distortion theory (RDT). The relationship between RDT and linear stability theory is exploited in order to obtain a closed set of modeled equations. The linear disturbance equations are solved directly so that the numerical simulation yields a database from which the closure coefficients in the ensemble-averaged disturbance equations can be determined. This revised version was published online in July 2006 with corrections to the Cover Date.