Hybrid RANS/LES of flow and heat transfer in round impinging jets

Fluid flow and convective heat transfer predictions are presented of round impinging jets for several combinations of nozzle-plate distances H/D = 2, 6 and 13.5 (where D is the nozzle diameter) and Reynolds numbers Re = 5000, 23,000 and 70,000 with the newest version of the k-ω model of Wilcox (2008) and three hybrid RANS/LES models. In the RANS mode of the hybrid RANS/LES models, the k-ω model is recovered. Three formulations are considered to activate the LES mode. The first model is similar to the hybrid models of Davidson and Peng (2003) and Kok et al. (2004). The turbulent length scale is replaced by the grid size in the destruction term of the k-equation and in the definition of the RANS eddy viscosity. As grid size, a maximum measure of the hexahedral grid cell is used. The second model has the same k-equation, but the eddy viscosity is the minimum of the k-ω eddy viscosity and the Smagorinsky eddy viscosity, following a proposal by Batten et al. (2004). The Smagorinsky eddy viscosity is formed with the cube root of the cell volume. The third model has, again, the same k-equation, but has an eddy viscosity which is an intermediate between the eddy viscosities of the first and second models. This is reached by using the cube root of the cell volume in the eddy viscosity formula of the first model.The simulation results are compared with experimental data for the high Reynolds number cases Re = 23,000 and Re = 70,000 and LES data for the low-Reynolds number case Re = 5000. The Reynolds numbers are defined with the nozzle diameter and the bulk velocity at nozzle outlet. At low nozzle-plate distance (the impingement plate is in the core of the jet), turbulent kinetic energy is overpredicted by RANS in the stagnation flow region. This leads to overprediction of the heat transfer rate along the impingement plate in the impact zone. At high nozzle-plate distance (the impingement plate is in the mixed-out region of the jet), the turbulence mixing is underpredicted by RANS in the shear layer of the jet which gives a too high length of the jet core. This also results in overprediction of the heat transfer rate in the impingement zone caused by too big temperature gradients at impingement.All hybrid RANS/LES models are able to correct the heat transfer overprediction of the RANS model. For good predictions at low nozzle-plate distance, it is necessary to sufficiently resolve the formation and development of the near-wall vortices in the jet impingement region. At high nozzle-plate distance, the essence is to capture the evolution and breakup of the flow unsteadiness in the shear layer of the jet, so that accurate mean and fluctuating velocity profiles are obtained in the impingement region. Although the models have a quite different theoretical justification and generate a quite different eddy viscosity in some flow regions, their overall results are very comparable. The reason is that in zones that are crucial for the results, the models behave similarly.     

Numerical computations of internal flows for axisymmetric and two-dimensional nozzles

MacCormack's explicit time-marching scheme is used to solve the full Navier–Stokes unsteady, compressible equations for internal flows. The requirement of a very fine grid to capture shock as well as separated flows is circumvented by employing grid clustering. The numerical scheme is applied for axisymmetric as well as two-dimensional flows. Numerical predictions are compared with experimental data and the qualitative as well as the quantitative agreement is found to be quite satisfactory. © 1997 John Wiley & Sons, Ltd.     

Computation of axisymmetric confined jets in a diffuser

The paper reports on a numerical study of turbulent confined jets in a conical duct with a 5° divergence. The flow has a large ratio of jet to ambient velocities at the entrance so that it gives rise to strong recirculation. The calculations are carried out with a general finite volume method designed for calculating incompressible elliptic flows with complex boundaries. Turbulence is simulated by the standard κ–? model. The sensitivity of the solution to numerical discretization errors is examined using three convection schemes, i.e. hybrid central/upwind differencing, QUICK and SOUCUP, on two grids consisting of 68 × 50 and 102 × 82 points respectively. An examination is also made of the influence of inlet boundary conditions on the predicted flow field. The computed results are compared with experimental data for mean axial velocity, turbulent shear stress and turbulent kinetic energy profiles. It is shown that the calculations reproduce the essential features of the flow observed in the experiments.     

Numerical simulation of turbulent impinging jet on a rotating disk

The calculations of quasi‐three‐dimensional momentum equations were carried out to study the influence of wall rotation on the characteristics of an impinging jet. The pressure coefficient, the mean velocity distributions and the components of Reynolds stress are calculated. The flow is assumed to be steady, incompressible and turbulent. The finite volume scheme is used to solve the continuity equation, momentum equations and k–ε model equations. The flow characteristics were studied by varying rotation speed ω for 0?ω?167.6 rad/s, the distance from nozzle to disk (H/d) was (3, 5, 8 and 10) and the Reynolds number Re base on V_J and d was 1.45 × 10⁴. The results showed that, the radial velocity and turbulence intensity increase by increasing the rotation speed and decrease in the impingement zone as nozzle to disk spacing increases. When the centrifugal force increases, the radial normal stresses and shear stresses increase. The location of maximum radial velocity decreases as the local velocity ratio (α) increases. The pressure coefficient depends on the centrifugal force and it decreases as the distance from nozzle to plate increases. In impingement zone and radial wall jet, the spread of flow increases as the angular velocity decreases The numerical results give good agreement with the experiment data of Minagawa and Obi (Int. J. of Heat and Fluid Flow 2004; 25 :759–766). Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical investigation of fully developed turbulent fluid flow and heat transfer in a square duct

Three-dimensional fully developed turbulent fluid flow and heat transfer in a square duct are numerically investigated with the author's anisotropic low-Reynolds-number k-ε turbulence model. Special attenton has been given to the regions close to the wall and the corner, which are known to influence the characteristics of secondary flow a great deal. Hence, instead of the common wall function approach, the no-slip boundary condition at the wall is directly used. Velocity and temperature profiles are predicted for fully developed turbulent flows with constant wall temperature. The predicted variations of both local wall shear stress and local wall heat flux are shown to be in close agreement with available experimental data. The present paper also presents the budget of turbulent kinetic energy equation and the systematic evaluation for existing wall function forms. The commonly adopted wall function forms that are valid for two-dimensional flows are found to be inadequate for three-dimensional turbulent flows in a square duct.     

COMPUTATION OF SUPERSONIC TURBULENT FLOWFIELD WITH TRANSVERSE INJECTION

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Computation of normal impinging jets in cross-flow and comparison with experiment

The PHOENICS code has been used to model the flow field surrounding subsonic and underexpanded jets impinging on a ground plane in the presence of a cross-flow, for cases with both a fixed ground plane and a ‘rolling road’. The standard k-ε turbulence model is used, without correction factors. It is confirmed that this overpredicts the free jet entrainment rate; the wall jet spreading rate is slightly underpredicted but the initial thickness is too high. Agreement with experiment is, nevertheless, much better than for previous calculations, showing the importance of the extent of the grid used. The ground vortex formed in cross-flow is shown to move with varying effective velocity ratio and with rolling road operation in the same manner as experimentally observed. Ground vortex self-similarity is also accurately predicted with the numerical modelling.     

An algorithm for rotating axisymmetric flows: model,validation and industrial applications

The study of axisymmetric flows is of interest not only from an academic point of view, due to the existence of exact solutions of Navier–Stokes equations, but also from an industrial point of view, since these kind of flows are frequently found in several applications. In the present work the development and implementation of a finite element algorithm to solve Navier–Stokes equations with axisymmetric geometry and boundary conditions is presented. Such algorithm allows the simulation of flows with tangential velocity, including free surface flows, for both laminar and turbulent conditions. Pseudo‐concentration technique is used to model the free surface (or the interface between two fluids) and the k–ε model is employed to take into account turbulent effects. The finite element model is validated by comparisons with analytical solutions of Navier–Stokes equations and experimental measurements. Two different industrial applications are presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical simulation of 3-D turbulent flows over dredged trenches

A 3-D free surface flow in open channels based on the Reynolds equations with thek-ε turbulence closure model is presented in this paper. Insted of the “rigid lid” approximation, the solution of the free surface equation is implemented in the velocity—pressure iterative procedure on the basis of the conventional SIMPLE method. This model was used to compute the flow in rectangular channels with trenches dredged across the bottom. The velocity, eddy viscosity coefficient, turbulent shear stress, turbulent kinetic energy and elevation of the free surface can be obtained. The computed results are in good agreement with previous experimental data.     

Numerical prediction of turbulent flow in a conical diffuser usingk-ε model

Fully developed incompressible turbulent flow in a conical diffuser having a total divergence angle of 8° and an area ratio of 4∶1 has been simulated by ak-ε turbulence model with high Reynolds number and adverse pressure gradient. The research has been done for pipe entry Reynolds numbers of 1.16×10⁵ and 2.93×10⁵. The mean flow velocity and turbulence energy are predicted successfully and the advantage of Boundary Fit Coordinates approach is discussed. Furthermore, thek-ε turbulence model is applied to a flow in a conical diffuser having a total divergence angle of 30° with a perforated screen. A simplified mathematical model, where only the pressure drop is considered, has been used for describing the effect of the perforated screen. The optimum combination of the resistance coefficient and the location of the perforated screen is predicted for high diffuser efficiency or the uniform velocity distribution.     

Numerical simulation of axisymmetric turbulent flow in combustors and diffusers

Numerical studies of turbulent flow in an axisymmetric 45° expansion combustor and bifurcated diffuser are presented. The Navier-Stokes equations incorporating a k–? model were solved in a non-orthogonal curvillinear co-ordinate system. A zonal grid method, wherein the flow field was divided into several subsections, was developed. This approach permitted different computational schemes to be used in the various zones. In addition, grid generation was made a more simple task. However, treatment of the zonal boundaries required special handling. Boundary overlap and interpolating techniques were used and an adjustment of the flow variables was required to assure conservation of mass flux. Three finite differencing methods—hybrid, quadratic upwind and skew upwind—were used to represent the convection terms. Results were compared with existing experimental data. In general, good agreement between predicted and measured values was obtained.     

Simulating two-dimensional turbulent flow by using the κ–ϵ model and the vorticity–streamfunction formulation

A numerical procedure to solve turbulent flow which makes use of the κ–? model has been developed. The method is based on a control volume finite element method and an unstructured triangular domain discretization. The velocity-pressure coupling is addressed via the vorticity-streamfunction and special attention is given to the boundary conditions for the vorticity. Wall effects are taken into account via wail functions or a low-Reynolds-number model. The latter was found to perform better in recirculation regions. Source terms of the κ and ε transport equations have been linearized in a particular way to avoid non-realistic solutions. The vorticity and streamfunction discretized equations are solved in a coupled way to produce a faster and more stable computational procedure. Comparison between the numerical predictions and experimental data shows that the physics of the flow is correctly simulated.     

Large Eddy Simulation of a plane impinging jet

Large Eddy Simulations of a plane turbulent impinging jet have been carried out using the dynamic Smagorinsky model. The statistical results are first validated with the measurements from the literature: mean and turbulent quantities along the jet axis and at different vertical locations are presented. This study is completed by the analysis of the wall shear stress at the impingement wall. The effect of the jet Reynolds number (3000Re13500) on the kinematic development of the jet is also discussed. To cite this article: F. Beaubert, S. Viazzo, C. R. Mecanique 330 (2002) 803–810.     

Direct measurement and computational analysis of turbulence length scales of a motored engine

This paper presents a direct measurement technique and the computational fluid dynamics (CFD) analysis of in-cylinder turbulence length scales of the flow inside a motored engine. A two-point simultaneous measurement technique was devised using a two-probe laser Doppler velocimetry (LDV) system. The engine was made transparent by replacing the liner with a quartz tube. The operating condition was set at the motoring speed of 500 rpm. This paper demonstrates the measurement of radially separated lateral and longitudinal integral length scales at the location of 13 mm beneath the center of the cylinder head. The measured integral length scales were then compared with the computational turbulence dissipation length scale resulted from a k– model in an engine simulation code, KIVA-3. The comparison shows a reasonable level of agreement in both tendency and magnitude in such a complex flow field.     

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.     

Predictive Capabilities of an Improved Cubic k–ε Model for Inert Steady Flows

Through an improved ε transport equation, a major quality enhancement of the cubic k–ε model, earlier developed in[13], is obtained. The ε-equation of [13],yielding good results for wall-bounded and rotating flows, is combined with the one derived by Shih et al. [20], which produces good results for free shear flows (e.g. the plane jet–round jet anomaly is resolved).Results are presented for the following flows: fully developed stationary and rotating channel and pipe, backward-facing step, sudden pipe expansion, smooth channel expansion and contraction, plane and round jet. Heat transfer predictions in turbulent impinging jets are also discussed. Accurate results are obtained for the mean flow quantities for all test cases, without case dependent model tuning. This revised version was published online in July 2006 with corrections to the Cover Date.     

Parameter estimation of engineering turbulence model

A parameter estimation algorithm is introduced and used to determine the parameters in the standardk-∈ two equation turbulence model (SKE). It can be found from the estimation results that although the parameter estimation method is an effective method to determine model parameters, it is difficult to obtain a set of parameters for SKE to suit all kinds of separated flow and a modification of the turbulence model structure should be considered. So, a new nonlineark-∈ two-equation model (NNKE) is put forward in this paper and the corresponding parameter estimation technique is applied to determine the model parameters. By implementing the NNKE to solve some engineering turbulent flows, it is shown that NNKE is more accurate and versatile than SKE. Thus, the success of NNKE implies that the parameter estimation technique may have a bright prospect in engineering turbulence model research.     

Strong convergence theorem for a generalized equilibrium problem and a <Emphasis Type="Italic">k</Emphasis>-strict pseudocontraction in Hilbert spaces

The main purpose of this paper is to study a new iterative algorithm for finding a common element of the set of solutions for a generalized equilibrium problem and the set of fixed points for a k-strict pseudocontractive mapping in the Hilbert space. The presented results extend and improve the corresponding results reported in the lit-erature.     

基于k-fold交叉验证的代理模型序列采样方法

在代理模型序列采样框架下,针对现有研究中的不足之处,通过引入k-fold交叉验证计算样本的预测误差,并结合泰森多边形法和最大距离最小化准则,发展了一种适用于任意代理模型的k-fold CV-Voronoi自适应序列采样方法。相较于传统序列采样方法,本文方法具有计算简单和自适应性强等显著优势。通过数值算例和工程算例对比分析发现所提序列采样方法具有较高的近似精度和计算效率,此外,进一步讨论了k-fold交叉验证中k的不同取值对于代理模型精度的影响,总结出k的最优取值范围以供参考。     

Assessment of Turbulence Models for Predicting the Interaction Region in a Wall Jet by Reference to LES Solution and Budgets

Highly-resolved LES and experimental data for a plane wall jet are used to study the characteristics of turbulence-closure proposals, mainly within the framework of second-moment-transport modelling. The study is motivated by the observed importance of diffusive Reynolds-stress transport in the interaction region between the outer shear layer and the near-wall layer of the wall jet, which gives the near-wall flow characteristics that are very different from those of a conventional boundary layer. Comparisons are presented for mean-flow quantities, second moments and budgets. Also included are a priori studies of approximations for the pressure-velocity interaction, pressure-fluctuation-driven transport and turbulent transport of the Reynolds stresses by triple correlations, the last observed to contribute significantly to the stress budgets. The study reveals major defects in the closure approximations for the pressure-velocity interaction terms, especially in the near-wall region. These defects result in a poor representation by the particular second-moment closures investigated of even the integral and mean-flow characteristics of the wall jet.