Development and Analysis of Wall Models for Internal Combustion Engine Simulations Using High-speed Micro-PIV Measurements

The performance, efficiency and emissions of internal combustion (IC) engines are affected by the thermo-viscous boundary layer region and heat transfer. Computational models for the prediction of engine performance typically rely on equilibrium wall-function models to overcome the need for resolving the viscous boundary layer structure. The wall shear stress and heat flux are obtained as boundary conditions for the outer flow calculation. However, these equilibrium wall-function models are typically derived by considering canonical flow configurations, introducing substantial modeling assumptions that are not necessarily justified for in-cylinder flows. The objective of this work is to assess the validity of several model approximations that are commonly introduced in the development of wall-function models for IC-engine applications. This examination is performed by considering crank-angle resolved high-resolution micro-particle image velocimetry (µ-PIV) measurements in a spark-ignition direct-injection single cylinder engine. Using these measurements, the performance of an algebraic equilibrium wall-function model commonly used in RANS and LES IC-engine simulations is evaluated. By identifying shortcomings of this model, a non-equilibrium differential wall model is developed and two different closures are considered for the determination of the turbulent viscosity. It is shown that both wall models provide adequate predictions if applied inside the viscous sublayer. However, the equilibrium wall-function model consistently underpredicts the shear stress if applied in the log-layer. In contrast, the non-equilibrium wall model provides improved predictions of the near-wall region and shear stress irrespective of the wall distance and the piston location. By utilizing the experimental data, significant adverse pressure gradients due to the large vortical motion inside the cylinder (induced by tumble, swirl and turbulence) are observed and included in the non-equilibrium wall model to further improve the model performance. These investigations are complemented by developing a consistent wall heat transfer model, and simulation results are compared against the equilibrium wall-function model and Woschni’s empirical correlation.     

A numerical study of the turbulent flow around the stern of ship models

The present work is concerned with the numerical calculation of the turbulent flow field around the stern of ship models. The finite volume approximation is employed to solve the Reynolds equations in the physical domain using a body-fitted, locally orthogonal curvilinear co-ordinate system. The Reynolds stresses are modelled according to the standard k-ε turbulence model. Various numerical schemes (i.e. hybrid, skew upwind and central differencing) are examined and grid dependence tests have been performed to compare calculated with experimental results. Moreover, a direct solution of the momentum equations within the near-wall region is tried to avoid the disadvantages of the wall function approach. Comparisons between calculations and measurements are made for two ship models, i.e. the SSPA and HSVA model.     

Advanced near-wall modeling for engine heat transfer

Recent developments in the engine heat transfer modeling tend to improve existing wall heat transfer models (temperature wall functions) which mostly rely on the standard or low-Re variants of k-ε turbulence model. Presently applied mesh resolutions already allow for first near-wall computational cells reaching the buffer or locally even viscous/conductive sub-layer, thus increasing the importance of more sophisticated modeling approach. As temperature gradient-induced density and fluid property variations become significant, wall heat transfer is strongly influenced by property variations (viscous/conductive sub-layer) and predictive capability of the turbulence model (buffer region), standard wall laws being inadequate anymore, even for attached boundary layers. The present approach relies on the k-ζ-f turbulence model and formulates a compressible wall function of Han and Reitz in the framework of hybrid wall treatment. The model is validated against spark ignition (SI) engine heat transfer measurements. Predicted wall heat flux evolutions on the cylinder head exhibit very good agreement with the experimental data, being superior to similar numerical predictions available in the published literature.     

The structure of a three-dimensional boundary layer subjected to streamwise-varying spanwise-homogeneous pressure gradient

A spatially-evolving three-dimensional boundary layer, subjected to a streamwise-varying spanwise-homogeneous pressure gradient, equivalent to a body force, is investigated by way of direct numerical simulation. The pressure gradient, prescribed to change its sign half-way along the boundary layer, provokes strong skewing of the velocity vector, with a layer of nearly collateral flow forming close to the wall up to the position of maximum spanwise velocity. A wide range of flow-physical properties have been studied, with particular emphasis on the near-wall layer, including second-moments, major budget contributions and wall-normal two-point correlations of velocity fluctuations and their angles, relative to wall-shear fluctuations. The results illustrate the complexity caused by skewing, including a damping in turbulent mixing and a significant lag between strains and stresses. The study has been undertaken in the context of efforts to develop and test novel hybrid LES–RANS schemes for non-equilibrium near-wall flows, with an emphasis on three-dimensional near-wall straining. Fundamental flow-physical issues aside, the data derived should be of particular relevance to a priori studies of second-moment RANS closure and the development and validation of RANS-type near-wall approximations implemented in LES schemes for high-Reynolds-number complex flows.     

Analytical and numerical study of generalized Cauchy problems occurring in gas dynamics

The existence and uniqueness theorems for solutions in the class of analytic functions were proved for two generalized Cauchy problems with conditions imposed on two surfaces. Using the theorems proved, implicit difference schemes are constructed for the numerical solution. A program was designed, and the corresponding numerical calculations were performed.     

Combined compact difference method for solving the incompressible Navier–Stokes equations

This paper presents a numerical method for solving the two‐dimensional unsteady incompressible Navier–Stokes equations in a vorticity–velocity formulation. The method is applicable for simulating the nonlinear wave interaction in a two‐dimensional boundary layer flow. It is based on combined compact difference schemes of up to 12th order for discretization of the spatial derivatives on equidistant grids and a fourth‐order five‐ to six‐alternating‐stage Runge–Kutta method for temporal integration. The spatial and temporal schemes are optimized together for the first derivative in a downstream direction to achieve a better spectral resolution. In this method, the dispersion and dissipation errors have been minimized to simulate physical waves accurately. At the same time, the schemes can efficiently suppress numerical grid‐mesh oscillations. The results of test calculations on coarse grids are in good agreement with the linear stability theory and comparable with other works. The accuracy and the efficiency of the current code indicate its potential to be extended to three‐dimensional cases in which full boundary layer transition happens. Copyright © 2011 John Wiley & Sons, Ltd.     

三阶WENO格式精度分析与改进

高分辨率激波捕捉格式对含激波流场的数值模拟具有重要意义。通过理论推导分析了WENO-JS3格式和WENO-Z3格式的精度,发现两种格式在光滑流场区域（包含极值点处）具有相同的理论精度且均低于三阶设计精度,WENO-Z3格式由于增大了非光滑模板的非线性权重使其计算精度有所提高。在理论推导的基础上,提出了WENO-Z+3格式及其改进格式（WENO-Z+3P1和WENO-Z+3P32）,且改进格式在光滑流场区域能满足所设计的三阶精度要求。选用一维平面黎曼问题及双马赫反射等经典算例,验证了本文提出的WENO-Z+3格式及其改进格式相较其他格式具有耗散低和对流场结构分辨率高的特性。     

On the mechanism of turbulence suppression by spanwise surface oscillations

The attenuation of turbulent pulsations in near-wall flows by means of spanwise periodic surface oscillation is examined. A direct numerical simulation of the flow in a circular pipe with imposed rotational oscillations has shown that for Re=4000 and the optimal oscillation frequency, the degree of turbulence attenuation increases with increase in the oscillation amplitude until the flow relaminarizes. The estimated optimal frequency ω⁺=0.06. The results of applying the theory of the development of near-wall coherent structures agree qualitatively with those of numerical simulation. It is concluded that the intensity of the pulsations is reduced because the spanwise movements weaken the longitudinal vortices which cause turbulent bursts in near-wall flows. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 37–44, March–April, 2000. The research was carried out with financial support from the Russian Foundation for Basic Research (project No. 99-01-01095).     

Structural Wall-modeled LES Using a High-order Spectral Difference Scheme for Unstructured Meshes

The combination of the high-order unstructured Spectral Difference (SD) spatial discretization scheme with Sub-Grid Scale (SGS) modeling for Wall-Modeled Large-Eddy Simulation (WMLES) is investigated. Particular focus is given to the use of wall-function approaches and to the relevant optimal coupling with the numerical scheme and the SGS model, a similarity mixed type model featuring newly designed discrete filters with specified cutoff length scale. To take full advantage of the discontinuous Finite Element (FE) structure which characterizes the SD scheme, wall-modeling is accomplished within the first wall element by using the information from the farthest solution points from the wall. Compared to the customary used first off-wall node, this point provides more accurate information to the wall-function, thus improving the quality of the solution. Two different law-of-the-wall are tested, a classical three-layers wall-function based on the equilibrium assumption and a more general formulation to account for the pressure gradient in more complex configurations. Moreover, the mixed scale-similarity SGS model is used in the entire computational domain without any particular adjustment inside the wall-modeled region. Numerical tests on the classical test case of the turbulent channel flow at different Reynolds numbers and on the channel with periodic constrictions at Re _h = 10,595 give evidence that the results are extremely sensitive to the choice of the solution points used to provide the informations to the law-of-the-wall. In particular, it is shown that significant improvements in the results can be attained by solving the wall-function away from the wall, rather than at the first off-wall solution point as customary done. The combination of the selected wall-modeling strategies and the similarity mixed formulation proves to be remarkably accurate, even in the presence of boundary layer separation, thus opening the path to further exploit the high-order SD platform, as well as a broad range of other similar methodologies, for WMLES. Extensions of the methodology are envisaged to include more sophisticated wall-modeling approaches incorporating turbulent sensors to switch to no-slip conditions in laminar regions.     

New Answers on the Interaction Between Polymers and Vortices in Turbulent Flows

Numerical data of polymer drag reduced flows is interpreted in terms of modification of near-wall coherent structures. The originality of the method is based on numerical experiments in which boundary conditions or the governing equations are modified in a controlled manner to isolate certain features of the interaction between polymers and turbulence. As a result, polymers are shown to reduce drag by damping near-wall vortices and sustain turbulence by injecting energy onto the streamwise velocity component in the very near-wall region.     

Some developments in computational modeling of turbulent flows

In this paper, some recent developments of two turbulence closure schemes at ICOMP, NASA Lewis will be discussed. One is the Reynolds-stress algebraic equation model and the other is the Reynolds-stress transport equation model. Various model constraints required by the rapid distortion theory, the invariant theory and the realizability principle, etc. will be described in the model development. The models discussed are for high-turbulent Reynolds number flows, so that the near-wall turbulence and the low-Reynolds-number turbulence are not discussed here.     

Computation of backward-facing step flows by a second-order Reynolds stress closure model

This paper scrutinizes the predictive ability of the differential stress equation model in complex shear flows. Two backward-facing step flows with different expansion ratios are solved by the LRR turbulence model with an anisotropic dissipation model and the near-wall regions of the separated side resolved by a near-wall model. The computer code developed for solving the transport equations is based on the finite-volume-finite-difference method. In the numerical solution of the time-averaged momenum equations the Reynolds stresses are treated partially as a diffusion term and partially as a source term to avoid numerical instability. Computational results are compared with experimental data. It is found that the near-wall region of the separated side resolved by the near-wall model, the LRR model with a simple modification of an anisotropic dissipation model can predict backward step flows well.     

On the Relation between Fronts and High-Shear Layers in Wall Turbulence

Direct numerical simulation results of turbulent channel flow are analyzed in order to examine the relation between two kinds of near-wall flow structures, namely the instantaneous shear layers and the fronts which are derived from two-point statistics of the streamwise velocity component. The near-wall shear layers are analyzed by flow visualizations and conditional sampling, while the fronts are examined by means of space-time correlations and spatial two-point correlation functions. The present study focuses on the analysis of the propagation speed and the spatial shape of the structures. Concerning the propagation speed it is shown that the results obtained from flow visualizations are in close agreement with the propagation velocities derived from space-time correlation functions. The comparison of VISA results for the instantaneous shear with spatial structures obtained from two-point correlations of the streamwise velocity and the shear gives evidence that the fronts are intimately related to the pronounced near-wall shear layers.     

Comparison of three solvers for viscoelastic fluid flow problems

The computational efficiency of three numerical schemes has been examined for the solution of a linearized system of equations resulting from the finite element discretization of a viscoelastic fluid flow problem. The first scheme is a modified frontal solver, which solves the linear system of equations directly. The other two, one based on a biconjugate gradient stabilized (BiCGStab) method and another based on a generalized minimal residual (GMRES) method, are iterative schemes. The stick-slip problem and the four-to-one contraction problem were analyzed and the viscoelastic fluid was assumed to obey the Oldroyd-B model. The two iterative schemes are superior to the direct scheme in terms of CPU time consumed and the BiCGStab scheme is even faster than the GMRES scheme. The range of convergence for both iterative schemes is compatible with that of the direct scheme.     

MICRO-SCALE NUMERICAL STUDY OF THE EFFECT OF ERYTHROCYTE MECHANICAL PROPERTIES ON THE NEAR-WALL MOTION OF PLATELET

采用浸入式边界法,模拟了多个红细胞和血小板在毛细血管内流动过程中的相互影响。通过改变红细胞体积比和红细胞的力学特性,分析了红细胞力学特性对血小板在与内皮细胞发生粘附前的动力学行为的影响机理,包括：红细胞对血小板趋壁效应的影响,血小板在流动过程中的变形情况,并从血小板所受垂直壁向合力的角度深入研究了红细胞和血小板之间的相互作用。数值模拟的结果表明,增加血流中的红细胞体积比,减小了血小板和血管壁之间的距离,增大了血小板的变形,血小板所受垂直壁向合力呈现剧烈波动,两者之间的挤压显著加强;而增大红细胞硬度,使得血小板的离壁距离增大。     

Some Results Based on Near-wall Turbulence

In this paper some results based on the near-wall mean characteristics of a bounded turbulent flow are presented. In the study empirical polynomials and experimental data for an attached wall-bounded flow are used with the objective of studying the time-scales similarities in the very-near-wall region. As a result of this analysis a new parameter to characterize the high to low-Reynolds turbulence transition in the context of turbulence models is proposed. A relation for the Reynolds stress in the buffer region is also proposed, which allows a mean velocity profile through the buffer region to be obtained. This mean velocity profile joins the logarithmic ones at the beginning of the inertial sub-layer and fits appropriately to experimental data. Another result derived from the previous analysis is an expression for the eddy viscosity through the very-near-wall region. Comparison of this expression with those relations used by four known low-Reynolds models reveals that it has a very good performance.     

Friction Drag Variation via Spanwise Transversal Surface Waves

The introduction of spanwise velocity is a promising technique to effect the near-wall turbulent flow field to influence friction drag. However, the essential physical mechanism which significantly reduces friction drag has not been understood, yet. It is the objective of this numerical study to improve the fundamental knowledge on the drag reduction mechanism. The investigation is based on spanwise traveling transversal surface waves which are applied to modify the near-wall flow field and to influence friction drag. Two actuation configurations are analyzed in detail. Compared with an unactuated flat plate boundary layer simulation the first wave setup, which represents a low frequency wave at an amplitude larger than the viscous sublayer, leads to a reduced wall-shear stress resulting in friction drag reduction of up to 9%. The second wave setup, which possesses a higher frequency and an amplitude in the range of the viscous sublayer, yields an increase of friction drag of about 8%. Unlike previous investigations which focus on excitation setups to lower friction drag, the comparison of the two wave setups in this study allows to identify the effects which on the one hand, lead to drag reduction and on the other hand, result in drag increase. That is, due to the pronounced differences the major effects determining the friction distribution are more evident. The two key features for drag reduction are the damping of the wall-normal vorticity fluctuations above the entire surface and the decrease of turbulence production. Furthermore, the effect of rearranging streamwise vorticity, which has been stated to be responsible for drag reduction, is found to occur at increasing and decreasing drag, i.e., it is not the effect that lowers the friction drag.     

Assessment of ghost-cell based cut-cell method for large-eddy simulations of compressible flows at high Reynolds number

Large-eddy simulations (LES) of high Reynolds number flows are performed using a non-body conformal method in conjunction with a wall model. We use a simple wall function to model the wall-shear stress and the truncation error of the numerical discretization to model the sub-grid scale turbulence (implicit LES), although these can be easily replaced if necessary. The validation cases are: turbulent flow through an inclined channel, turbulent flow over a wavy surface, and supersonic flow over a circular cylinder. Since the near-wall grids are naturally coarse, the key is to use a method that is capable of capturing the flow dynamics accurately in the vicinity of the interface. Towards the purpose, we develop a Cartesian cut-cell method, referred to as the ghost-cell based cut-cell method (GC-CCM), in the context of fully compressible solutions of Navier–Stokes equations. This method employs ghost-cells inside the solid interface such that the local spatial reconstruction remains consistent everywhere including in the vicinity of the boundary. In order to capture the near-wall flow behavior more accurately with coarse grids, this method decomposes cell faces of merged cells and computes fluxes through each decomposed segment separately. The objective of this work is to qualify whether the proposed method can accurately represent the high Reynolds number flows in the vicinity of immersed interfaces. To analyze the performance of the proposed method, we compare the results to the corresponding numerical results from the two other non-body conformal methods, namely the ghost-cell based immersed boundary method (GCIBM) and standard cut-cell method (S-CCM), that are implemented in the same numerical solver. The comparison demonstrates that the proposed method is capable of capturing near-wall flows relatively accurately with coarse grids.     

Development of a class of multiple time‐stepping schemes for convection–diffusion equations in two dimensions

In this paper we present a class of semi‐discretization finite difference schemes for solving the transient convection–diffusion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection–diffusion (CD) equation to the inhomogeneous steady convection–diffusion‐reaction (CDR) equation after using different time‐stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one‐dimensional framework. For the sake of increasing accuracy, the exact solution for the one‐dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one‐dimensional problem. Development of the proposed time‐stepping schemes is rooted in the Taylor series expansion. All higher‐order time derivatives are replaced with spatial derivatives through use of the model differential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection–diffusion‐reaction differential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions. Copyright © 2006 John Wiley & Sons, Ltd.