numerical simulation of fluid flow and heat transfer processes

Advances in Mechanical Engineering

Numerical Simulation of Fluid Flow and Heat Transfer Processes

Guest Editors: Bo Yu, Tomoaki Kunugi, Toshio Tagawa, Shuyu Sun, Moran Wang, and Yi Wang

Numerical Simulation of Fluid Flow andHeat Transfer Processes

Advances in Mechanical Engineering

Numerical Simulation of Fluid Flow andHeat Transfer Processes

Guest Editors: Bo Yu, Tomoaki Kunugi, Toshio Tagawa,Shuyu Sun, Moran Wang, and Yi Wang

Copyright 2013 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in Advances in Mechanical Engineering. All articles are open access articles distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly cited.

Editorial Board

Koshi Adachi, JapanMehdi Ahmadian, USARehan Ahmed, UKM. Affan Badar, USAClaude Bathias, FranceAdib Becker, UKLeonardo Bertini, ItalyL. A. Blunt, UKMarco Ceccarelli, ItalyHyung H. Cho, Republic of KoreaSeung B. Choi, KoreaBogdan I. Epureanu, USAM. R. Eslami, IranA. Faghri, USAAli Fatemi, USASiegfried Fouvry, FranceIan Frigaard, CanadaJiin Y. Jang, Taiwan

Zhongmin Jin, UKEssam E. Khalil, EgyptXianwen Kong, UKCheng-Xian Lin, USAJaw-Ren Lin, TaiwanOronzio Manca, ItalyAristide F. Massardo, ItalyKim Choon Ng, SingaporeC. T. Nguyen, CanadaHirosi Noguchi, JapanHakan F. Oztop, TurkeyDuc Truong Pham, UKRobert L. Reuben, UKBidyut B. Saha, SingaporeDirk J. Schipper, The NetherlandsSteven R. Schmid, USAA. Seshadri Sekhar, IndiaC. S. Shin, Taiwan

Ray W. Snidle, UKMargaret M. Stack, UKNeil Stephen, UKKumar K. Tamma, USAYaya Tan, ChinaCho W. Solomon To, USAYoshihiro Tomita, JapanShandong Tu, ChinaMoran Wang, ChinaFengfeng Xi, CanadaGongnan Xie, ChinaHiroshi Yabuno, JapanWei Mon Yan, TaiwanJianqiao Ye, UKByeng D. Youn, USABo Yu, ChinaZhongrong Zhou, China

Contents

Numerical Simulation of Fluid Flow and Heat Transfer Processes, Bo Yu, Tomoaki Kunugi,Toshio Tagawa, Shuyu Sun, Moran Wang, and Yi WangVolume 2013, Article ID 497950, 3 pages

On Full-Tensor Permeabilities of Porous Media from Numerical Solutions of the Navier-StokesEquation, Yi Wang, Shuyu Sun, and Bo YuVolume 2013, Article ID 137086, 11 pages

Analysis on Shift of Nature Modes of Liquid Sloshing in a 3D Tank Subjected to Oblique HorizontalGround Motions with Damping Devices, Chih-Hua Wu, Odd Magnus Faltinsen, and Bang-Fuh ChenVolume 2013, Article ID 627124, 24 pages

Numerical Simulation of PAHs Formation and Effect of Operating Conditions in DI-Diesel EnginesBased on a Comprehensive Chemical Mechanism, Bei-Jing Zhong and Jun XiVolume 2013, Article ID 567159, 19 pages

Temperature Dependence of Ascending Bubble-Driven Flow Patterns Found in Champagne Glasses asDetermined through Numerical Modeling, Fabien Beaumont, Catalin Popa, Gerard Liger-Belair,and Guillaume PolidoriVolume 2013, Article ID 156430, 10 pages

DNS Study of the Turbulent Taylor-Vortex Flow on a Ribbed Inner Cylinder, Takahiro Tsukahara,Manabu Ishikawa, and Yasuo KawaguchiVolume 2013, Article ID 628490, 12 pages

Mathematical Modeling of the High Temperature Treatment of Birch in a Prototype Furnace,Duygu Kocaefe, Yasar Kocaefe, Ramdane Younsi, Noura Oumarou, and S. Thierry LekounougouVolume 2013, Article ID 194610, 8 pages

Numerical Study ofThermal Behavior in Alternating Current Light-Emitting Diodes, Farn-Shiun Hwuand Hung-Lin HsiehVolume 2013, Article ID 426767, 5 pages

Numerical Analysis of Flow around a Moving Object by an Immersed Boundary Method with the LevelSet Method, Atsuki Iijima, Tomomitsu Sato, and Toshio TagawaVolume 2013, Article ID 868240, 9 pages

Large Eddy Simulation of Inertial Particle Preferential Dispersion in a Turbulent Flow over aBackward-Facing Step, Bing Wang, Huiqiang Zhang, and Xilin WangVolume 2013, Article ID 493212, 8 pages

Numerical Simulation of Mixed Convection in a Rotating Cylindrical Cavity: Influence of PrandtlNumber, Gustavo Urquiza, Laura Castro, Juan Garca, Miguel Basurto, and Enoc BogarinVolume 2013, Article ID 950765, 8 pages

An Analytical Approximation for Continuous FlowMicrowave Heating of Liquids, G. Cuccurullo,L. Giordano, and G. ViccioneVolume 2013, Article ID 929236, 8 pages

Pressure Change in Tee Branch Pipe in Oscillatory Flow, Daisuke Sakamoto, Chongho Youn,and Toshiharu KagawaVolume 2013, Article ID 257283, 11 pages

Modeling and Numerical Simulation of the Grinding Temperature Field with Nanoparticle Jet of MQL,C. H. Li, J. Y. Li, S. Wang, and Q. ZhangVolume 2013, Article ID 986984, 9 pages

Numerical Simulation on the Food Package Temperature in Refrigerated Display Cabinet Influenced byIndoor Environment, Chang Zhijuan, Wu Xuehong, Lu Yanli, Ma Qiuyang, and Zhang WenhuiVolume 2013, Article ID 708785, 7 pages

Numerical Simulation of Gas-Liquid-SolidThree-Phase Flow in DeepWells, Jianyu Xie, Bo Yu,Xinyu Zhang, Qianqian Shao, and Xianzhi SongVolume 2013, Article ID 951298, 10 pages

Comparisons of LES and RANS Computations with PIV Experiments on a Cylindrical Cavity Flow,Wen-Tao Su, Xiao-Bin Li, Feng-Chen Li, Xian-Zhu Wei, Zhi-Ying Zheng, and Xin ZhangVolume 2013, Article ID 592940, 10 pages

Aerodynamic Performance Prediction of Straight-Bladed Vertical Axis Wind Turbine Based on CFD,L. X. Zhang, Y. B. Liang, X. H. Liu, Q. F. Jiao, and J. GuoVolume 2013, Article ID 905379, 11 pages

Numerical Simulation of the Transient Process of Power Failure in a Mixed Pump, Xudan Ma, Jintao Liu,and Leqin WangVolume 2013, Article ID 743201, 10 pages

Evaluation of Artificial Caudal Fin for Fish Robot with Two Joints by UsingThree-DimensionalFluid-Structure Simulation, Yogo Takada, Noboru Fukuzaki, Toshinori Ochiai, Tomoki Tajiri,and Tomoyuki WakisakaVolume 2013, Article ID 310432, 9 pages

Performance Analysis and Application ofThree Different Computational Methods for Solar HeatingSystem with Seasonal Water Tank Heat Storage, Dongliang Sun, Jinliang Xu, and Peng DingVolume 2013, Article ID 857941, 13 pages

Vortex-Induced Vibrations of a Square Cylinder with Damped Free-End Conditions, S. Manzoor,J. Khawar, and N. A. SheikhVolume 2013, Article ID 204974, 12 pages

Study on the Fluidic Component of the Complete Fluidic Sprinkler, Hong Li, Chao Wang, Chao Chen,and Zhenhua ShenVolume 2013, Article ID 658591, 8 pages

Optimal Model of Operation Parameters of Gathering Pipeline Network with Triple-Line Process,Yongtu Liang, Cen Lu, Kuijie Ren, Qiao Xiao, and Guoxi HeVolume 2013, Article ID 573542, 7 pages

Numerical Study on the Mixed Convection Heat Transfer between a Sphere Particle and High PressureWater in Pseudocritical Zone, Liping Wei, Youjun Lu, and Jinjia WeiVolume 2013, Article ID 527182, 10 pages

Effect of Step-Change Radiation Flux on Dynamic Characteristics in Tower Solar Cavity Receiver,Zhengwei Chen, Yueshe Wang, Yun Hao, and Qizhi WangVolume 2013, Article ID 402094, 10 pages

Contents

Comparison Study on Linear Interpolation and Cubic B-Spline Interpolation Proper OrthogonalDecomposition Methods, Xiaolong Wang, Yi Wang, Zhizhu Cao, Weizhong Zou, Liping Wang, Guojun Yu,Bo Yu, and Jinjun ZhangVolume 2013, Article ID 561875, 10 pages

Wavelet Analysis on Turbulent Structure in Drag-Reducing Channel Flow Based on Direct NumericalSimulation, Xuan Wu, Bo Yu, and Yi WangVolume 2013, Article ID 514325, 10 pages

Analyses on Heating Energy Saving of Two HotWaxy-Crude Oil Pipelines Laid Parallel in One Ditch,Changzheng Sun and Bo YuVolume 2013, Article ID 948980, 10 pages

Effective Resistance of Gas Flow in Microchannels, Xiao-Dong Shan and Moran WangVolume 2013, Article ID 950681, 7 pages

Turbulence Modulation by Small Bubbles in the Vertical Upward Channel Flow, Mingjun Pang,Jinjia Wei, and Bo YuVolume 2013, Article ID 379839, 7 pages

The Polymer Effect on Nonlinear Processes in Decaying Homogeneous Isotropic Turbulence,Wei-Hua Cai, Feng-Chen Li, Hong-Na Zhang, Yue Wang, and Lu WangVolume 2013, Article ID 921524, 8 pages

Understanding ofThermal Conductance ofThin Gas Layers, Xiaodong Shan and Moran WangVolume 2013, Article ID 692842, 7 pages

AModified -Model for Computation of Flows with Large Streamline Curvature, Jun-Lian Yin,De-Zhong Wang, Yu-Lin Wu, and D. Keith WaltersVolume 2013, Article ID 592420, 10 pages

Numerical Study on the Effect of Wax Deposition on the Restart Process of a Waxy Crude Oil Pipeline,Qing MiaoVolume 2012, Article ID 973652, 10 pages

Numerical Model on Frost Height of Round Plate Fin Used for Outdoor Heat Exchanger of MobileElectric Heat Pumps, Moo-Yeon LeeVolume 2012, Article ID 863731, 7 pages

Experimental Validation of Volume of Fluid Method for a Sluice Gate Flow, A. A. Oner,M. S. Akoz, M. S. Kirkgoz, and V. GumusVolume 2012, Article ID 461708, 10 pages

Hindawi Publishing CorporationAdvances in Mechanical EngineeringVolume 2013, Article ID 497950, 3 pageshttp://dx.doi.org/10.1155/2013/497950

EditorialNumerical Simulation of Fluid Flow and Heat Transfer Processes

Bo Yu,1 Tomoaki Kunugi,2 Toshio Tagawa,3 Shuyu Sun,4 Moran Wang,5 and Yi Wang1,4

1 National Engineering Laboratory for Pipeline Safety, Beijing Key Laboratory of Urban Oil and Gas Distribution Technology,China University of Petroleum, Beijing 102249, China

2Department of Nuclear Engineering, Kyoto University, C3-d2S06, Kyoto Daigaku-Katsura, Nishikyo-Ku, Kyoto 615-8540, Japan3Department of Aerospace Engineering, Tokyo Metropolitan University, 6-6 Asahigaoka, Hino, Tokyo 191-0065, Japan4Computational Transport Phenomena Laboratory, Division of Physical Science and Engineering,King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

5 Department of Engineering Mechanics and CNMM, Tsinghua University, Beijing 100084, China

Correspondence should be addressed to Bo Yu; [email protected]

Received 27 June 2013; Accepted 27 June 2013

Copyright 2013 Bo Yu et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Fluid flow and heat transfer processes are ubiquitous in natureand engineering. They exist in many aspects of industrialoperations and daily life. Numerical simulations of theseprocesses have been important methods for fundamentaland applicable researches. This special issue focuses on thelatest achievements in the two aspects. We received 63 activesubmissions from the United States of America, Canada,Mexico, France, Italy, Norway, Saudi Arabia, Turkey, China,Japan, Pakistan, Republic of Korea, and so forth and finallyaccepted 35 research articles to publish them in the specialissue after peer reviews. The topics cover the researcheshaving solid theoretical fundaments including turbulent fluidflow and heat/mass transfer and the researches having strongbackgrounds of applications.

In the field of turbulent fluid flow, 10 articles have beenpublished. The following articles make efforts on directnumerical simulation (DNS), the Reynolds averaged Navier-Stokes (RANS) model, and large eddy simulation (LES) ofturbulence. The article DNS study of the turbulent Taylor-vortex flow on a ribbed inner cylinder by T. Tsukahara etal. shows the investigation of turbulent Taylor-vortex flowsover regularly spaced square ribs mounted on a rotatinginner cylinder surface. The authors find that Taylor vorticesremaining over roughened cylinder surfaces can lead to lesspressure drag and an enhanced backflow in the recirculationzone. The article Turbulence modulation by small bubbles inthe vertical upward channel flow by M. Pang et al. presents themechanisms of the liquid turbulence modulation induced by

the addition of small bubbles. Intensified turbulence near thewall and slightly weakened turbulence in the channel regionare discovered. In the article entitled A modified - modelfor computation of flows with large streamline curvature by J.-L. Yin et al., the authors propose an improved RANS modelfor system rotation and streamline curvature effects and pro-vide an effective way for turbulence modeling. In the articleentitled Large eddy simulation of inertial particle preferentialdispersion in a turbulent flow over a backward-facing step byB. Wang et al., LES of a turbulent flow with inertial particledispersion over a backward-facing step is performed. Theresearch conclusions are useful for further understandingthe two-phase turbulence physics and establishing accurateengineering prediction models of particle dispersion. In thearticle Comparisons of LES and RANS computations withPIV experiments on a cylindrical cavity flow by W.-T. Su etal., RANS and LES methods are compared. The results showthat LES is more suitable for predicting the complex flowcharacteristics inside complicated three-dimensional (3D)geometries. In the article Experimental validation of volumeof fluid method for a sluice gate flow by A. A. Oner et al.,two-dimensional (2D) open channel flow under a verticalsluice gate can be successfully analyzed by the volume offluid (VOF) method-based modeling after the experimentalvalidation. The following four articles focus on aerodynamicsor drag reduction. Aerodynamic performance prediction ofstraight-bladed vertical axis wind turbine based on CFD by L.X. Zhang et al. demonstrates that the leading edge separation

2 Advances in Mechanical Engineering

vortex and its movement on the airfoil surface have a signifi-cant impact on the aerodynamic performance because bladesexperience mild and deep stalls at low tip speed ratio. Thearticle Vortex-induced vibrations of a square cylinder withdamped free-end conditions by S. Manzoor et al. summarizesthe vortex-induced vibrations of a square cylinder in a windtunnel and suggests proper revision of the wake modelused for analytical lift force predictions. In the article Thepolymer effect on nonlinear processes in decaying homogeneousisotropic turbulence by W. H. Cai et al., the authors studythe behaviors of nonlinearities affected by polymer additivesin decaying homogenous isotropic turbulence. They findthat polymer has a negative effect on enstrophy and strainproduction, that is, depression of nonlinearity. In the articleWavelet analysis on turbulent structure in drag-reducingchannel flow based on direct numerical simulation by X.Wu et al., wavelet transformation is applied to decomposevelocity fluctuation time series into ten different frequencycomponents including approximate components and detailedcomponents. Features of turbulent multiscale structures areshown intuitively by continuous wavelet transform, verifyingthat turbulent structures become much more regular in drag-reducing flow.

In the field of heat/mass transfer, 9 articles have beenpublished. The articles Numerical analysis of flow arounda moving object by an immersed boundary method withthe level set method by A. Iijima et al. and Comparisonstudy on linear interpolation and cubic B-spline interpolationproper orthogonal decomposition methods by X. Wang etal. are investigations of numerical methods for fluid flowand heat transfer. The former develops a new immersedboundary method with advantages of accuracy, flexibility,and rapidness combined with the level set method. Thelatter concludes that the proper orthogonal decompositionmethod with cubic B-spline interpolation is more accuratethan that with linear interpolation. In the article entitledNumerical model on frost height of round plate fin used foroutdoor heat exchanger of mobile electric heat pumps byM.-Y. Lee, the numerical model for prediction of the frostgrowth of the round plate fin is established. The predictionon the frost height with time is improved by using thefrost thermal conductivity reflecting the void fraction anddensity of ice crystal with frost growth. In the article entitledUnderstanding of thermal conductance of thin gas layers byX. Shan and M. Wang, the authors study heat conductionsin a thin gas layer at micro- and nanoscales between twostraight walls by atomistic modeling. They indicate that twodominating factors to the thermal conductivity reduction ofthin gas layers are the temperature jump on wall surfacesand the properties changing significantly by the confinedspace. In the article Modeling and numerical simulation ofthe grinding temperature field with nanoparticle jet of MQLby C. H. Li et al., the heat transfer model of surface grindingtemperature field with nanoparticle jet flow of MQL and theproportionality coefficient model of energy input workpieceis established. It is found that MQL grinding conditionswith additive nanoparticles demonstrate great impact onthe weakening of temperature effect on the grinding zone.In the article An analytical approximation for continuous

flow microwave heating of liquids by G. Cuccurullo et al.,a numerical and analytical model is developed to simulatetemperature profiles in continuous laminar pipe flow duringmicrowave heating. The simplified analytical model can leadto an easy way to predict the heat transfer through the pipe. Inthe article On full-tensor permeabilities of porous media fromnumerical solutions of the Navier-Stokes equation by Y. Wanget al., a new method combining Navier-Stokes equation andDarcys law is proposed to compute full-tensor permeabilityof porous media instead of simplified tensor in tradition.It is found that anisotropy becomes pronounced especiallywhen convection is dominant. The followed two articlescontribute on numerical studies of mixed convection heattransfer. The article Numerical study on the mixed convectionheat transfer between a sphere particle and high pressure waterin pseudocritical zone by L. Wei et al. makes efforts onmixed convection heat transfer between supercritical waterand particles in supercritical water fluidized bed reactor,which is a new but rare focused topic recently. The resultsshow that buoyancy force has a remarkable effect on flowand heat transfer processes, and variation of specific heatand conductivity plays a main role in determination of heattransfer coefficient. Article Numerical simulation of mixedconvection in a rotating cylindrical cavity: influence of Prandtlnumber by G. Urquiza et al. describes the influence of thePrandtl number on flow in critical state on a cavity containinga cooling fluid, and the heat transfer in the inferior wallincreases as the Prandtl number increases and aspect ratiodecreases.

Additional 16 articles with strong backgrounds of applica-tions in industry or daily life are also published. Five articlesdevote efforts to petroleum industry. In the article Numericalstudy on the effect of wax deposition on the restart process ofa waxy crude oil pipeline by Q. Miao, the restart processof a wax crude oil pipeline is investigated numerically by anew model for wax deposition. The temperature drop duringthe shutdown process and the transient inlet pressure arepresented, and the effect of the wax deposition on the safetyof the restart process is clarified. In the article Analyseson heating energy saving of two hot waxy-crude oil pipelineslaid parallel in one ditch by C. Sun and B. Yu, a newtechnology laying two oil pipelines in one ditch employed bythe petroleum companies of China is numerically presented.It is found that two hot crude oil pipelines laid parallel in oneditch can dramatically save heating energy when comparedwith two pipelines laid, respectively, in two separate ditches.In the article Numerical simulation of gas-liquid-solid three-phase flow in deep wells by J. Xie et al., a new model forgas-liquid-solid annulus flow in the deep wells is establishedconsidering the effect of the cuttings on the pressure drop. Itis found that temperature and pressure are greatly affected bywell depth, drilling mud density, and gas kick while the effectof the cuttings on the total pressure drop is small. In the articlePressure change in tee branch pipe in oscillatory flow by D.Sakamoto et al., the authors propose a simulation methodto predict the pressure changes in a pneumatic branchpipe under oscillatory flow. The results contribute to theunderstanding of unsteady flow of branch pipes in pneumaticsystems. In the article Optimalmodel of operation parameters

Advances in Mechanical Engineering 3

of gathering pipeline network with triple-line process by Y.Liang et al., a mathematic model for the optimal operation ofthe gathering pipeline network is proposed and applied to theoptimal operation analysis in North China Oilfield. Opera-tion cost can be reduced by 2076 RMB/d, which demonstratesthat this method contributes to the production cost reductionof old oilfields in their high water-cut stage. The articleEffective resistance of gas flow in microchannels by X.-D.Shan and M.R. Wang studies fluid flow in microscale insteadof macroscale for the previous five articles. They turn acomplicated micromechanical problem into simple availableformulae for designs and optimization of microengineering.Four articles discuss fluid flow in affiliations such as pump,tank, engine, and sprinkler. In the article Numerical simula-tion of the transient process of power failure in a mixed pumpby X. Ma et al., the authors use a hydraulic-force couplingmethod to simulate the transient process of power failurecondition. They conclude that the rotational speed decreasesmuch faster than the flow rate in power failure accidents. Inthe article Analysis on shift of nature modes of liquid sloshingin a 3D tank subjected to oblique horizontal ground motionswith damping devices by C.-H. Wu et al., the study of sloshingfluid in tanks with internal structures is extended from 2D to3D. In the article Numerical simulation of PAHs formationand effect of operating conditions in di-diesel engines based on acomprehensive chemical mechanism by B.-J. Zhong and J. Xi,numerical simulations of polycyclic aromatic hydrocarbon(PAH) formation in a Chaochai 6102 bzl direct injectiondiesel engine are performed. PAHs first increase and thendecrease with the increase in diesel crank angle. The dieselengine operating conditions have a significant effect on PAHformation. In the article Study on the fluidic component ofthe complete fluidic sprinkler by H. Li et al., the offset jet withcontrol stream is analyzed in the simplified model. The yawangle and the attachment angle of the offset jet flow increasewith the pressure increase and vary little when the pressureis more than 0.5 MPa. Six articles are applicable researchesrelating artificial fish, solar energy, wood preservation, foodpackage, champagne glasses and light. The article Evaluationof artificial caudal fin for fish robot with two joints by usingthree-dimensional fluid-structure simulation by Y. Takadaet al. confirms that a good caudal fin for fish robot withtwo active joints is a rigid fin with a flexible materialon the root by using the 3D fluid-structure interactionanalysis. The article Performance analysis and applicationof three different computational methods for solar heatingsystem with seasonal water tank heat storage by D. Sun etal. compares three different computational methods for asolar heating system with seasonal water tank heat storage.In the article Mathematical modeling of the high temperaturetreatment of birch in a prototype furnace by D. Kocaefe etal., a reliable and predictive model is developed to simulatenumerically the high-temperature heat treatment process ofwood preservation. In the article Numerical simulation onthe food package temperature in refrigerated display cabinetinfluenced by indoor environment by Z. Chang et al., the foodpackage temperature is investigated by numerical simulationunder different conditions to study the relation betweenthe food package temperature and ambient environment.

A numerical modeling of bubble-driven flow patterns in aglass of champagne has been carried out by F. Beaumontet al. in the article Temperature dependence of ascendingbubble-driven flow patterns found in champagne glasses asdetermined through numerical modeling. The velocities ofthe liquid phase significantly vary with the champagne tem-perature. In the article Numerical study of thermal behaviorin alternating current light-emitting diodes by F.S. Hwu andH.-L. Hsieh, thermal characteristics of an alternating currentlight-emitting diode chip based on a 3D unsteady numericalsimulation are discussed. Results show that the AC LED hasa better performance under a higher frequency than under alower frequency.

Bo YuTomoaki KunugiToshio Tagawa

Shuyu SunMoran Wang

Yi Wang


Research ArticleOn Full-Tensor Permeabilities of Porous Media fromNumerical Solutions of the Navier-Stokes Equation

Yi Wang,1,2 Shuyu Sun,2 and Bo Yu1

1 National Engineering Laboratory for Pipeline Safety, Beijing Key Laboratory of Urban Oil and Gas Distribution Technology,China University of Petroleum, Beijing 102249, China

2 Computational Transport Phenomena Laboratory, Division of Physical Science and Engineering,King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

Correspondence should be addressed to Shuyu Sun; [email protected]

Received 25 January 2013; Revised 28 May 2013; Accepted 3 June 2013

Academic Editor: Toshio Tagawa

Copyright 2013 Yi Wang et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A numerical method is proposed to compute full-tensor permeability of porous media without artificial simplification. Navier-Stokes (N-S) equation and Darcys law are combined to design these numerical experiments. This method can successfully detect thepermeability values in principle directions of the porous media and the anisotropic degrees. It is found that the same configuration ofporous media may possess isotropic features at lower Reynolds numbers while manifesting anisotropic features at higher Reynoldsnumbers due to the nonlinearity from convection. Anisotropy becomes pronounced especially when convection is dominant.

1. Introduction

Permeability is a key parameter of porous media because itrelates with many parameters, such as the infiltration, dielec-tric strength and thermophysics. It may strongly affect thedesign and production process of fibrous reinforcements,cement paste and so forth and related scientific researches.The permeability may vary over several orders of magnitude,and it plays an important role in petroleum migration andreservoir performance [1]. Therefore, the accurate measure-ment of permeability is always a hot topic in both industrialand academic fields. Many of these researches adopt experi-mental approaches, such as free-space methods, open-endedcoaxial probe techniques, cavity resonators, transmission-line techniques, and gas-dynamic methods [2, 3].

Jensen and Heriot-Watt [4] proposed a statistical modelfor small-scale permeability using minipermeameter andcore plug measurements. They suggested the minipermeame-ter measurement as a better choice. Wang et al. [5] discussedtwo experimental methods to determine the absolute valuesof in-plane permeability. They concluded that both theradial flow measurement method and the unidirectional flow

measurement method were recommended to obtain reliablepermeability data. Ferland et al. [6] proposed a concurrentmethod to estimate permeability at low cost. However, theexperimental procedures can introduce uncertainty signifi-cantly on the estimated permeability. They suggested somespecial treatments to increase the reliability of experimen-tal data. Some other researchers [710] also realized thatpermeability is difficult to measure although its definitionis simple. Many factors, such as flow rate, pressure, fluidproperties, handling process by human factors, edge effect,wall effect, single-equipment reproducibility, and between-equipment repeatability, could strongly influence the resultsof measurements. Therefore, the ability to obtain consis-tent permeability data depends on skilled and experiencedexperimental design, reproducible preparation of specimens,operation of equipment, and evaluation of measured rawdata. This is the reason many published data observedby different persons often have significant differences forthe same material. Weitzenbock et al. [11] pointed outthat numerous practical problems caused three-dimensional(3D) permeability measurements to be very difficult. Then,they proposed an approach to measure the permeability


by a two-dimensional (2D) radial flow method, which allowsthe experimental axes not to align with the principal directionof permeability [12, 13].

Some researchers made efforts on measuring transportproperties (including permeability) by numerical simulation.Keehm et al. obtained good results in estimating permeabilityand electric conductivity of complicated pore geometriesusing Lattice-Boltzmann method (LBM) [14]. Kameda et al.applied LBM to estimate 3D permeability through 2Dimages of small fragments of rocks. They obtained a validpermeability-porosity trend by using a significant number ofsuch small fragments in statistical sense [15]. Saenger et al.estimated permeability through LBM flow simulation andcompared mechanical and transport properties for the samedigital rocks [16].

Since all physical parameters can be easily and exactlycontrolled in numerical simulations, the shortcomings of lab-oratory experiments mentioned previously can be avoided.In particular, numerical methods have the advantage ofseparately studying different physical processes coexistingin nature, which are uneasily separated in laboratory. It iseasier to form standardization of measurement methods andobtain unchangeable results. It will be a very efficient andeconomical way for measuring properties of porous mediacombining with the digital rock physics [17, 18]. In the presentpaper, we studied how to establish this digital laboratorymethod for measuring permeability of porous media throughdirectly solving the N-S equation, other than reconstructingit by LBM method. To the best knowledge of the authors,all studies of permeability prediction have been concernedonly with diagonal permeability tensor actually. Full tensorof permeability has not been studied extensively since ithas more general meanings for practical applications. Thus,we expand our research object from the simple diagonalpermeability tensor to more general full permeability tensor.We select gravity in periodic domain as the driven force,instead of pressure widely used in previous studies [118].This can avoid the edge effect encountered in experimentalmeasurements. As a first trial, we assume incompressiblesingle-phase flow to pass through the porous media inrectangular geometry, but the methodology can be extendedto more complex applications, for example two-phase flowusing diffuse interface models, which will be pursued in ourfuture researches. The basic principles and proposed methodsin this study will be presented next.

2. Principles and Methods

2.1. General Principles. The basic law that is used by experi-mental and numerical studies for measuring permeability isthe Darcys law:

u = k ( g) , (1)where u is Darcy velocity (m/s), k is the permeability tensor(m2), g is the gravitational acceleration (m/s2), is thepressure gradient (Pa/m), is the density of fluid (kg/m3),and is the dynamic viscosity of fluid (Pa s).

The N-S equation (2) and the continuity equation (3) thatdescribe fluid flow in the continuum sense are as follows (is the Laplace operator):

u + (u ) u = 1 + u + g, (2) u = 0. (3)Conventional methods for measuring permeability neglectgravity and drive the fluid flow by pressure because it is easierto control, especially for laboratory experiments. Here, weselect to use the full expressions of (1)(3) to determine thepermeability. Periodic boundary condition is adopted so thatboundary pressure needs not to be considered. It is prettyeasier for numerical implementation and more sensible innature because subsurface formation is in fact a periodicsystem or statistically periodic one. The general procedure isstated below.

Step 1. Solve (2) and (3) for incompressible single-phase flowin porous media and obtain several velocity fields as samples.

Step 2. Obtain volumetric velocity of the whole domainaccording to the samples obtained in Step 1.

Step 3. Solve permeability for the whole domain by directlysetting the volumetric velocity as the Darcy velocity in (1).

The detailed numerical methods used in these three stepswill be introduced in the next section.

2.2. Numerical Methods. As stated previously, we only con-sider simple 2D rectangular cases in the present study. Theincompressible single-phase flow in porous media is modeledas the flow passing through a square barrier in a squaredomain with periodic boundary conditions. The numericalmodel is shown in Figure 1(a). is the domain side lengthwhile is the side length of the barrier. means periodicboundary.

On this kind of domain, the vectors and tensors in (1)(3)have the following forms:

u = [] , g = [] ,k = [ ] , = [[[[[

]]]]]. (4)

Here = cos , = sin , = 9.807m/s2, , and are the diagonal components of the permeability tensorwhile and are the off-diagonal components of thepermeability tensor. These four variables are all independentof each other. Three typical directions of gravity are chosen tocalculate the full-tensor permeability. They are represented by = 3/2,, and/4, respectively, and named Sample a, andSample b, and Sample c as shown in Figure 1(b).


L

LO

x

y

P

P

P

P

a

a

(a)

Sample c:gx = gcos(/4)gy = gsin(/4)

Sample b:gx = g,gy = 0

y

x

Sample a:gx = 0, gy = g

O

(b)

y

x

O

(c)

Figure 1: Calculation methodology: (a) numerical model; (b) sampling; and (c) coordinate rotation.

The numerical algorithm for solving (2)(3) in Step 1is projection method with fully explicit spatial discretization(but solve pressure implicitly) using second-order finitedifference central type scheme. Cyclic tridiagonal matrixalgorithm (CTDMA) is applied to accelerate the convergenceof pressure iteration. The mesh size is set as 80 80. Flowis considered to reach the steady state as the temporaltruncation |u/| averaged over the whole domain is lowerthan 106m/s2.

The volumetric velocity in Step 2 is defined as follows:

U = 1 u, (5)where U = [ ] is the volumetric velocity with the compo-nents and in the and directions. is the volumeof the whole domain. Equation (5) has a discrete expressionin this study:

= () , = () , (6)where and are the total number of and on gridcells, respectively. The values of and can be determinedby solving the N-S equation so that and can bedetermined.

The volumetric velocity in (5) is actually the Darcyvelocity in (1); that is, u = U. In the Darcy scale averaged inthe whole domain, pressure gradient should be zero becauseof the periodic boundary condition; that is, = 0. Thus, (1)can be rewritten as

U = kg. (7)For Sample a (() = ), (7) becomes

() = (), (8a)() = (). (8b)

For Sample b (() = ), (7) becomes() = (), (9a)() = (). (9b)

For Sample c (() = cos(/4), () = sin(/4)), (7)becomes

() = () + (), (10a)() = () + (). (10b)

Theoretically, (8a), (8b), (9a), and (9b) are adequate to deter-mine the four components of the full-tensor permeability.However, () and () are always very close to zeros sincethey are orthogonal to the bulk flow directions and largenumerical errors may exist. Thus, and determined by(8a) and (9b) are always very close to zeros. This apparentlyviolates the fact that and may be far from zeros forsome cases. The application of (8a) and (9b) actually makesthe full-tensor permeability decay to a diagonal permeabilityso that they should be canceled. To predict the off-diagonalcomponents accurately, (10a) and (10b) are utilized sincethe bulk flow in Sample c has equivalent projections inthe and directions so that () and () are bothaccurate. Therefore, the final equations to determine the fourcomponents of the permeability tensor should be the fourequations (8b), (9a), (10a), and (10b), which are used inStep 3. The procedure is calculating by (9a) and by(8b) and substituting them to (10a) and (10b) to obtain and , respectively.

In practice, it is of interest to detect the maximumpermeability, the minimum permeability, principle direc-tion, and anisotropy so that essential features of reservoir,which are independent of coordinate systems and samplings,can be described. Therefore, the original permeability tensorcan be transformed by rotating the original coordinate


Table 1: Characteristic parameters.

Case (m/s) (m) Re Da ReDa1/2Case 1 1.16 106 2.50 108 2.90 108 6.25 104 7.25 1010Case 2 6.62 107 5.00 108 3.31 108 2.50 103 1.66 109Case 3 3.96 107 7.50 108 2.97 108 5.63 103 2.23 109Case 4 2.32 107 1.00 107 2.32 108 1.00 102 2.32 109Case 5 1.28 107 1.25 107 1.60 108 1.56 102 2.00 109Case 6 6.24 108 1.50 107 9.36 109 2.25 102 1.40 109Case 7 2.51 108 1.75 107 4.39 109 3.06 102 7.69 1010Case 8 7.12 109 2.00 107 1.42 109 4.00 102 2.85 1010Case 9 8.58 1010 2.25 107 1.93 1010 5.06 102 4.34 1011

to a new coordinate according to tensors characteristics[19]. The angle between the coordinate system andthe coordinate system is represented by the angle (Figure 1(c)). Therefore, a series of transformed permeabilitytensors ktrans = [ ] can be obtained by continuouslyrotating the coordinate system, that is, continuouslychanging the value of angle . Once 0, 0 isachieved at a certain angle ; it represents the case that fluidflows no longer in any tangential directions but only in thetwo normal directions and so that maximum and min-imum components of permeability among all transformedpermeability tensors can be examined (max = max(, ),min = min(, )). In this case, represents the prin-ciple direction. To examine the principle direction precisely, = 0360 with interval 1 is adopted. Another importantparameter is the anisotropy of the permeability in the porousmedia. Here, we define the ratio to determine theanisotropic degree of permeability: = max/min. Thetensor transformation formula [19] is shown as follows: = cos cos + cos cos+ cos cos + cos cos, (11) = cos cos + cos cos+ cos cos + cos cos , (12) = cos cos + cos cos+ cos cos + cos cos , (13) = cos cos + cos cos+ cos cos + cos cos, (14)where = /2 , = /2 + .3. Results and Discussion

Two important factors are discussed here: porosity andReynolds number. To study them separately, two groups ofnumerical cases are designed. One is changing porosity bychanging the size of barrier but keeping constant length ofdomain. The other one is changing the length of domain but

keeping constant porosity. These two factors are discussed inthe following two sections. Fluid passing through the porousmedia is water with density = 1000 kg/m3 and dynamicviscosity = 0.001Pa s.3.1. Discussion on Porosity. The domain size is set as =106m. The size of barrier is set as = (1/10) (9/10)with interval (1/10) so that 9 cases named Case 1Case 9 aregenerated. According to the size of the computational domainand spatial element sizes on the fixed mesh, time step is set asits maximum value available: = 1011 s.

The flow simulation results by directly solving the Navier-Stokes equation are shown in Figure 2. The contours rep-resent the magnitudes of velocity (i.e., the characteristicspeed to be defined later), while the vectors represent thedirections of velocity. It can be seen that the flow fields arewell represented in the 9 cases. The square barriers are clearlyidentified by the vectors. We define characteristic speed as = 2 + 2, characteristic length as the hydraulic radius = 2/(4) = /4, Reynolds number as Re = /,and Darcy number as Da = (/)2. The characteristicparameters are listed in Table 1 (results for Samples a, b, andc are the same so that the data in Table 1 are any of them).The very low Reynolds number (order of magnitude 1010108) indicates that the flows are very slow laminar flows. Itcan be seen that the dimensionless number ReDa1/2 has theorder of magnitude 1011109, which satisfies the suggestedrestriction of Darcys law ReDa1/2 1 by Bear and Cheng[20].

Porosity is defined as the ratio of void space volume to thetotal volume as follows:

= 1 22 . (15)Thus, the 9 cases relate to 9 different values of porosity.Full permeability tensors obtained from flow simulationresults of the 9 cases are listed in Table 2. It can be seenthat the diagonal components of the permeability tensor alldecrease with decreasing porosity. Their values show thatthe studied porous media fall within the ranges of oil rocks(1011m21014m2), sandstone (1014m21016m2), andgood limestone (1016m21018m2) [20]. The diagonal


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(a)Figure 2: Continued.


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Figure 2: Flow fields for different porosities from Navier-Stokes simulation: columns a, b, and c represent Sample a, Sample b, and Sample c;numbers 19 represent Case 1Case 9; the contour shows total speed while the vector shows directions of total velocity.components of transformed permeability tensor are shown inFigure 3 for the 9 porosities; there always exist = and = , = . These mean that the two diagonal com-ponents of the full permeability tensor are the same and arealways equal to each other along with the rotating coordinate.This indicates that the permeable properties are all the samefor any direction so that principle direction does not exist.Therefore, the full-tensor permeabilities are always diagonaltensors with equivalent components for the 9 porosities.The off-diagonal components, which are 36 orders ofmagnitudes lower than the diagonal components (Table 2),

should be considered as numerical errors. In this case, theoff-diagonal components should be considered as zeros.Revealed from the mathematical meaning, physical meaningis that the porous media in the 9 cases are all isotropic.Corresponding characteristic parameters are listed inTable 3.

3.2. Discussion on Reynolds Number. Besides the porosity,Reynolds number is also a key factor which may stronglyaffect velocity fields so that the results of permeabilitymay be affected. Thus, it is important to check whether


Table 2: Full-tensor permeabilities at different porosities.

Case (m2) (m2) (m2) (m2)Case 1 0.99 1.1808 1013 1.1808 1013 5.6848 1016 5.6363 1016Case 2 0.96 6.7481 1014 6.7481 1014 3.1129 1017 2.8981 1017Case 3 0.91 4.0362 1014 4.0362 1014 6.2013 1019 2.6772 1019Case 4 0.84 2.3702 1014 2.3702 1014 2.7576 1019 3.5447 1020Case 5 0.75 1.3026 1014 1.3026 1014 8.3818 1020 1.6370 1020Case 6 0.64 6.3596 1015 6.3596 1015 7.1193 1021 2.1080 1020Case 7 0.51 2.5614 1015 2.5614 1015 5.8343 1021 3.6183 1020Case 8 0.36 7.2643 1016 7.2643 1016 1.0640 1021 1.0052 1020Case 9 0.19 8.7493 1017 8.7493 1017 6.4756 1023 1.9648 1022

Table 3: Characteristic permeabilities and anisotropies at different porosities.

Case max (m2) min (m2) (= max/min)Case 1 0.99 1.1808 1013 1.1808 1013 1.0000Case 2 0.96 6.7481 1014 6.7481 1014 1.0000Case 3 0.91 4.0362 1014 4.0362 1014 1.0000Case 4 0.84 2.3702 1014 2.3702 1014 1.0000Case 5 0.75 1.3026 1014 1.3026 1014 1.0000Case 6 0.64 6.3596 1015 6.3596 1015 1.0000Case 7 0.51 2.5614 1015 2.5614 1015 1.0000Case 8 0.36 7.2643 1016 7.2643 1016 1.0000Case 9 0.19 8.7493 1017 8.7493 1017 1.0000

Case 1: k = k = 1.1808 1013 m2

Case 2: k = k = 6.7481 1014 m2

Case 3: k = k = 4.0362 1014 m2

Case 4: k = k = 2.3702 1014 m2

Case 5: k = k = 1.3026 1014 m2

Case 6: k = k = 6.3596 1015 m2

Case 7: k = k = 2.5614 1015 m2

Case 8: k = k = 7.2643 1016 m2

Case 9: k = k = 8.7493 1017 m2

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ponents o

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Figure 3: Diagonal components of transformed permeability tensorat different porosities.

the Reynolds number, actually controlled by the domainsize, may change the conclusions in Section 3.1. Therefore,the domain size is gradually scaled up from = 106m by

keeping porosity at 0.64 (Case 6 in Tables 2 and 3). The resultsare shown in Table 4.

From 106m to 104m, the diagonal components of thefull-tensor permeability just increase two orders of magni-tudes once the domain size increases one order of magnitude.The off-diagonal components should still be considered asnumerical errors and set to be zeros as stated previously.These three cases are quite similar, and their difference is onlythe magnitude. It can be verified by the flow fields in Figures4(a1), 4(b1), 4(c1)Figures 4(a3), 4(b3), 4(c3), Figures 5(a)5(c), and Table 5 that the domain size scaling up from 106mto 104m just leads the velocity to increase proportionally.They are all isotropic. However, the situation is different whenthe domain size increases up to 103m (see line 5 in Table 4).The off-diagonal components are no longer several orders ofmagnitudes smaller than the diagonal components so that thefull-tensor permeability is no longer diagonal. Figures 4(a4)4(c4) show that the fluid tends to depart the solid walls andthe vortex expands. Figure 5(d) shows that the components ofthe transformed permeability tensor are no longer constant.Maximum and minimum values of the diagonal components(max, min) can always be obtained at the same time oncethe off-diagonal components become zeros. Their values(Table 5) show that the porous media at the domain lengthof 103m enter the range of clean stand (109m21012m2)which is a better aquifer than the previous ones [20].Correspondingly, four angles (I, II, III, and IV) can beobtained, which are shown in Table 6. Surprisingly fromTable 5, the porous media start to show anisotropic property( = 1.9241) when the domain length reaches 103m.


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Figure 4: Flow fields for different Reynolds numbers from Navier-Stokes simulation: columns a, b, and c represent Sample a, Sample b, andSample c; numbers 14 represent domain size = 106m, 105m, 104m, and 103m, respectively; the contours show total speed while thevectors show directions of total velocity.

It can be summarized from Table 6 that is always equalto angle II or IV when max equals . These two angles areon the same line across the second and the fourth quadrantsin the original coordinate (see Figure 1(c)). This line isthe direction where permeability achieves maximum value,that is, the easiest direction where fluid can pass through, sothat this is the principle direction. Accordingly, the directionfluid receives the most resistance (i.e., the direction for min)is the line represented by anglesI andIII, so that this line isthe direction orthogonal to the principle direction.

When the length of domain is greater than or equal to102m, the flow simulation is divergent. This is probablybecause the mesh size 80 80 is not dense enough for largerdomain. Larger domains with finer grid, which are very timeconsuming, will be studied in the future.

To discuss the previous phenomenon that differentReynolds numbers may generate essentially different perme-ability tensors, or in other words predict different types ofporous media for the same configuration, four characteristicparameters are shown in Table 7. and V are defined as


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kk

kk

I II III IV

()

(d)

Figure 5: Components of transformed permeability tensor at different Reynolds numbers: (a) = 106m, (b) = 105m, (c) = 104m,and (d) = 103m.

Table 4: Full-tensor permeabilities at different Reynolds numbers.

(m) (m2) (m2) (m2) (m2)106 6.3596 1015 6.3596 1015 7.1193 1021 2.1080 1020105 6.3596 1013 6.3596 1013 7.5167 1019 2.1064 1018104 6.3596 1011 6.3596 1011 7.4280 1017 3.6338 1016103 6.0366 109 6.0366 109 1.9077 109 1.9077 109102 Divergence Divergence Divergence Divergence

Table 5: Characteristic permeabilities and anisotropies at different Reynolds numbers.

(m) max (m2) min (m2) (= max/min)106 0.64 6.3596 1015 6.3596 1015 1.0000105 0.64 6.3596 1013 6.3596 1013 1.0000104 0.64 6.3596 1011 6.3596 1011 1.0000103 0.64 7.9443 109 4.1289 109 1.9241


Table 6: Angles at zero off-diagonal components of transformed permeability for = 103m. (m) I II III IV 103 45 135 225 315 135, 315

Summary max = min = max = min = max = min = max = min = = II or IVTable 7: Characteristic parameters at different Reynolds numbers.

(m) Re ReDa1/2 V106 9.36 109 1.40 109 2.79 109 2.79 109105 9.36 106 1.40 106 2.79 106 2.79 106104 9.36 103 1.40 103 2.79 103 2.79 103103 8.88 1.33 1.35 1.35

the ratios of the mean convection effect to the mean diffusioneffect in the momentum equation (2):

= (/) + V (/)(/) (2/2 + 2/2) ,V = (V/) + V (V/)(/) (2V/2 + 2V/2) ,

(16)

where the superscript represents the average value overthe whole domain. It is clear in Table 7 that the fluid flowsare actually diffusion dominated for the domain lengthssmaller than 103m since and V are all much smallerthan 1. However, and V are greater than 1 at the domainlength of 103m. This means that the flow is dominated bythe convection effect of fluid. It is well known from theirintentions that the diffusion effect tends to transport variablesto all directions uniformly while the convection effect tendsto transport variables along specific directions. Therefore,the existence of convection dominant flow may be the mainreason for the anisotropy. It is worth to point out that Table 7also verifies the conclusion made by Bear and Cheng [20] thatDarcys law is usually valid as long as Reynolds number islower than 1 (occurred at the domain lengths 106m104mwhere Re = 9.36 1099.36 103) but sometimes as highas 10 (occurred at the domain length 103m where Re = 8.88and the suggested restriction ReDa1/2 1 is violated).4. Conclusion

A numerical method is proposed to compute the permeabilityin the form of full tensor. The flow simulation results showthat flow fields can be well represented by solving the Navier-Stokes equation directly. Original and transformed perme-ability tensors are obtained so that maximum and minimumcomponents of permeability with principle directions andanisotropies are detected successfully. With this information,the directions of largest and smallest resistances for fluid flowin porous media can be inferred easily.

Through the analyses on the porosity effect and theReynolds number effect, it is found that porous media withthe same porosity and the same configuration can manifestdifferent levels of anisotropy at different Reynolds numbers.At the Reynolds numbers that diffusion dominates flow,isotropy is a good description. At the Reynolds numberthat convection dominates flow, anisotropy occurs. Thus,it is important to pay attention to Reynolds numbers forporous media applications. This is especially important forapplications with large Darcy velocities, such as flow andtransport in packed columns and in fractured geologicalmedia.

5. Future Issues to Be Addressed

The previous discussions show that the proposed methodin the present study is an easy way to determine full-tensor permeability numerically and related characteristics.However, many issues still need to be addressed in futureworks. Only two of them are briefly listed below as anexample.

(1) Method for solving the Navier-Stokes equation inporous media should be improved to accelerate thecomputation. Current speed is not acceptable forengineering applications. Multigrid method needs tobe developed for the flow around solids so that theiteration of pressure can be largely accelerated. High-resolution time integration scheme is expected tolargely reduce the number of time steps needed. Sinceonly steady-state results are needed for obtaining thefull-tensor permeability, a method directly solving thesteady-state Navier-Stokes equation is also expectedso that a large amount of time integrations can beavoided.

(2) Shape, number, position, and so forth of inner barri-ers should be intensively studied to know the appli-cation and limitation of this method and to improveprecision. Denser meshes are also expected to studywhat will happen for larger Reynolds numbers.


Acknowledgments

The work presented in this paper has been supported in partby the project entitled Simulation of Subsurface Geochem-ical Transport and Carbon Sequestration, funded by theGRP-AEA Program at KAUST. The work has also been sup-ported in part by National Science Foundation of China (no.51206186, no. 51174206) and Science Foundation of China,University of Petroleum, Beijing (no. 2462012KYJJ0403, no.2462012KYJJ0404).

References

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Research ArticleAnalysis on Shift of Nature Modes of Liquid Sloshing in a 3DTank Subjected to Oblique Horizontal Ground Motions withDamping Devices

Chih-Hua Wu,1 Odd Magnus Faltinsen,2 and Bang-Fuh Chen3

1 Institute of High Performance Computing, ASTAR, 1 Fusionopolis Way, No. 16-16 Connexis, Singapore 1386322 Centre for Ships and Ocean Structures & Department of Marine Technology, NTNU, 7491 Trondheim, Norway3 Asia-Pacific Ocean Research Center (APORC), National Sun Yat-sen University, Kaohsiung 802, Taiwan

Correspondence should be addressed to Bang-Fuh Chen; [email protected]

Received 25 January 2013; Accepted 27 March 2013

Academic Editor: Yi Wang

Copyright 2013 Chih-Hua Wu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The extended study of Wu et al. (2012) of sloshing fluid in tanks with internal structures from 2D to 3D is presented in the paper.The phenomenon of liquid sloshing in a 3D tank with various damping devices is solved by the time-independent finite differencemethod combined with the ghost (fictitious) cell approach. Two types of damping devices, a tank bottom-mounted baffle anda vertically surface-piercing plate, are considered in the study. In this work, the experimental measurement of liquid sloshingin a 3D tank with the baffle is carried out to further validate the present simulation. The comparison of the results between theexperimental measurement and the present computation shows good accuracy. The effect of the vertically tank bottom-mountedbaffle or the vertically surface-piercing plate on various sloshing waves for the tank under horizontal oblique excitation is discussedand investigated. The phenomena of the shift of the nature frequency of the tank with damping devices due to various obliquehorizontal excitations under different sloshing waves are presented in detail. The sloshing wave type is varied due to the influenceof the baffle or the plate, and the coexistence of two types of sloshing waves is found for the tank under larger excitation frequencies.

1. Introduction

Liquid sloshing is the most prominent phenomenon of liquidmotion in either stationary or moving tanks subjected toforced external perturbations. The study of sloshing phe-nomenon in tanks is related to a wide range of applica-tions such as in ships, rockets, satellites, trucks, and evenstationary-petroleum containers. Resonant free-surface flowsin tanks in aircrafts, missiles, and rockets have been thefocus of extensive research. The amplitude of the sloshing,in general, depends on amplitude and frequency of thetank motion, liquid-fill depth, liquid properties, and tankgeometry. These parameters have significant effects on thedynamic stability and performance of moving vehicles car-rying containers. One of the passive devices to reduce theinfluence of sloshing impact on structures or suppress thestrength of liquid sloshing is inserted in internal obstaclesin containers, such as baffles, plates, rings, and wire screens.

The tanks mounted with sloshing-damping devices are calledTuned Liquid Damper (TLD).

Tuned Liquid Dampers (TLDs) are economical andeffective dynamic vibration absorbers. The main function ofa passive damping device is to absorb portion of the inputenergy associated with external dynamic excitation actingon the structure. Examples for external excitations are windand earthquakes. By doing so, the passive damping deviceminimizes or eliminates the possibility of structural damages.TLDs were used to stabilize marine vessels against rockingand rolling motions [1, 2] in offshore platforms [3, 4] andin tall structures [59]. Often the TLD is used as a waterstorage tank preventing the use of a higher-viscosity liquid.Several approaches have been implemented to increase theenergy dissipated by the sloshing fluid, including roughnesselements [10], surface contaminants [11], wave breaking inshallow water TLDs [12], and nets or screens [7, 1315].Akyildiz and Unal [16, 17] investigated the pressure variations


in both baffled and unbaffled rectangular tank numericallyand experimentally. They observed that the effects of thevertical baffle are most pronounced in shallow water, andconsequently the pressure response is reduced by using thebaffles. Liquid viscosity cannot be neglected when flow-damping devices are mounted with a tank with fluid andenergy is dissipated by viscous action. Celebi and Akyildiz[18] revealed that flow over a vertical baffle produces a shearlayer and energy is dissipated by viscous effect of the fluid.They concluded that, in an increased fill depth, the rollingamplitude and frequency of the tank with or without baffleconfigurations directly affect the degrees of nonlinearity ofthe sloshing phenomena. As a result, a phase shift in forcesand moments occurred. Armenio and La Rocca [19] adoptedthe finite difference method to solve the 2-D RANS equationsand validated the numerical results with their experimentalmeasurement.

The control of the sloshing behavior with baffles is alsoa subject of interest in the recent years, because of thecomplexity and highly nonlinear nature of the problem. Choand Lee [20] reported a parametric investigation on thetwo-dimensional nonlinear liquid sloshing in baffled tanksunder horizontal forced excitation by using fully nonlinearpotential flow theory. In their study, the liquid motion anddynamic pressure variation in the vicinity of the baffle tipare more significant than those below the baffle tip. Cho etal. [21] did a further study on the resonance characteristicsof liquid sloshing in a 2D baffled tank under surge motionby the linearized potential flow theory. The various positions,baffle heights, and number of baffles were considered intheir work and the fundamental resonant frequency, and thepeak elevation of sloshing decreases with these parameters.However, the viscous effect on liquid sloshing could notbe resolved based on potential theory. Younes et al. [22]experimentally explored the hydrodynamic damping due tovertical baffle arrangements in a rectangular tank with slosh-ing fluid. The arrangement of upper-mounted and lower-mounted vertical baffles of different heights and numberswere considered in their experiment. They found that thetwin-sided upper mounted baffles and center-holed lowermounted baffle arrangement yield a maximum damping ratioon sloshing. More recently, Akyildiz [23] investigated theeffect of the vertical baffle height on liquid sloshing in arolling 2D rectangular tank, and the nonlinear liquid sloshingwas solved by the volume of fluid (VOF) technique. He solvedthe complete Navier-Stokes equations in primitive variablesby using of finite difference approximations with the movingcoordinate system. He concluded that the blockage effect ofthe baffle on the liquid convection is predominant to the tipvortex when the baffle height increases. Wu et al. [24] carriedout that the fictitious cell approach associated with a coor-dinate transformation technique was successfully adopted tosolve for the sloshing liquid in 2D tanks with baffles. Thenumerical scheme was validated by their experiment work.The effects of the number of the baffles on the sloshingamplitude were studied. In the study of two baffles, the largestwave damping might occur when the distance between twobaffles is 0.2 L. In addition, the influence of baffle heighton the shift of the first natural mode of the baffled tank

(1) under different water depths is carried out by spectralanalyses of sloshing elevation. Several empirical formulas arederived by curve fitting, and they can be used to predict theshift of the fundamental mode of the liquid sloshing in tankswith baffles.

However, the 3D numerical simulation of viscous liquidsloshing in a tank with internal structures is still very limitedin the literature. Liu and Lin [25] investigated liquid sloshingin a baffled tank with large-eddy simulation (LES). In theirstudy, the vertical baffle is a more effective tool in reducingthe sloshing amplitude. Jung et al. [26] utilized commercialsoftware, ANSYS Fluent, to solve liquid sloshing in a 3D tankwith baffles under only lateral excitation. The behavior of tipvortex, free surface elevation depending on the baffle height,and the pressure exerted on the tank wall were discussed indetail.

In the present study, we straightforwardly extend thenumerical model of Wu et al. [24] from 2D to 3D to exploresloshing dynamics in tanks with two damping devices, atank bottom-mounted baffle, and a surface-piercing flat plate.The excitation of a three-dimensional tank (Figure 1) withdifferent dimensionless excitation amplitudes; with multi-ple degrees of freedom for the excitation direction; withexcitation frequencies near and far from the first naturalfrequency; with arbitrary water depths are considered in thiswork. In the three-dimensional model, the time-independentfinite difference method [24] is utilized to incorporate the3D Navier-Stokes equations and the fully nonlinear kine-matic and dynamic free surface boundary conditions forincompressible fluid in a rectangular tank with a squarebase. The time varying moving boundary is mapped ontoa time-independent domain through proper transformationfunctions, and a special finite difference approximation ismade in order to overcome the difficulty of maintaining theaccuracy of the finite difference expression for the secondderivative when the difference mesh is stretched near theboundary. The treatment of flow field around flow dampingdevices is carried out by a fictitious cell approach which issimilar to the ghost cell approach [27]. The second orderupwind scheme is also used to deal with the convective terms.The main focus of this paper is to discuss the effect of avertically tank bottom-mounted baffle and a surface-piercingflat plate on the nature modes of various sloshing wavesfor a tank subjected to oblique horizontal excitation. Notonly is the numerical simulation studied in this work, theexperiment setup for a tank with a baffle is also investigatedto further validate the accuracy of the developed numericalscheme.

Section 2 introduces the equations of motion which arewritten in a moving frame of reference attached to theaccelerating tank. The fully nonlinear free surface boundaryconditions are listed in this section. Besides, the fictitiouscell approach is implemented to deal with the interfaces offluid and structure (baffle, tank bottom, and tank walls).The comprehensive benchmark tests of the present numericalscheme are demonstrated in Section 3. The detailed influ-ences of the flow damping devices on various sloshing wavesare also dissected in this section. Section 4 summarizes thekey conclusions.


Z

X

Y

X

Z

y

z

x

II

N

E

EE

AA

BB

LL

D G

BB

FF

CC

G

d0d0

PL

db

(Xb, Yb, Zb)

Free surface

(a) (b)

x1 x2

z1z2

y1

y2

Figure 1: Definition sketches of the tank and the coordinate system. (a) A vertically tank bottom-mounted baffle; (b) a vertically surface-piercing flat plate.

2. Mathematical Formulation

In this work, the sloshing phenomenon in a rigid 3D tankwith partially filled liquid is analyzed, and two flow dampingdevices, a vertically tank bottom-mounted baffle, and asurface-piercing flat plate are considered. As illustrated inFigure 1, the breadth of the tank is , the tanks width is ,and 0 is the still liquid depth. is the baffle height, and is the width of the plate. The gas flow including the possibilityof gas pockets is neglected. The horizontal oblique excitationangle is measured between the excitation direction and -coordinate. The laminar flow is assumed, and the Navier-Stokes equations in a tank-fixed coordinate system can beexpressed as

+ + V + = 1 + (22 + 22 + 22) ,V + V + VV + V= 1 + ( 2V2 + 2V2 + 2V2) , + + V + = 1 + (22 + 22 + 22 ) ,

(1)

where , V, and are the relative velocity components in ,, and directions, and are the relative accelerationcomponents of the tank in and directions, is thepressure, is the liquid density, is kinematic viscosity of

the liquid, and is the components of the acceleration dueto gravity.

The continuity equation for incompressible flow is

+ V + = 0. (2)Taking partial derivatives of (1) with respect to , , and ,respectively, and summing the results, one can obtain thefollowing equation to solve for the pressure:

22 + 22 + 22 = ( + V + ) ( V + VV + V) ( + V + ) .

(3)

2.1. BoundaryConditions. We assume that the surface tensioneffect is neglected. The kinematic condition states that theliquid particles at free surface remain on the free surface andcan be expressed as

+ + = V, (4)where = (, , ) 0 is the elevation of free sur-face measured from the initial liquid depth. The dynamiccondition requires that the normal stress is equal to theatmospheric pressure, and the two tangential stresses are zero


along the free surface boundary. The dimensionless dynamicconditions can, then, be derived and expressed as follows:

= Fr2+ 2 [2 + 2 + V + ( + ) ( + V) (V + ) ]

(Re (2 + 2 + 1))1,(5)

= V + 2 ( V) + ( + ) + (V + ) 1 2 ,(6)

= V + 2 ( V) + ( + ) + (V + ) 1 2 ,(7)

where Fr is the Froude number and Re is the Reynoldsnumber that are defined as

Fr = 0 ,Re = 0 ,

(8)

where = 0 ( is the angular velocity, and 0 is theexcitation displacement of the tank) is the maximum velocityof the tank, denotes a partial derivative of with respect to, and the others have same meanings. In the present study,(5) is used to determine the hydrodynamic pressure at thefree surface, while (6) and (7) are used to extrapolate thehorizontal velocity (, ) at the free surface from the flowdomain.

2.2. The Coordinate Transformation and Computational Algo-rithm. As well known, the way of accurately predicting freesurface elevation in 3D tanks with external forcing is stilla big challenge due to time dependence of free surface,especially when flow damping devices are involved andcoupled with the sloshing flow. In the present study, weextend the numerical scheme of [24] from 2D to 3D by usingsimple mapping functions to remove the time dependenceof the free surface of the liquid domain. The time-varyingliquid surface can be mapped onto a cube by the propercoordinate transformations. The convenience of coordinatetransformation is to map a wavy and time-dependent liquiddomain onto a time-independent unit cubic domain. As listedin Figure 1, the distance from the tank west wall to the baffle(plate) center is and from the south wall to the baffle(plate) center is , and the baffle height is . We dividethe liquid domain into eight parts based on the location and

the height of the baffle or plate. The mapping functions ofcoordinate transformation of eight parts can be expressed as

1 = 1 , 2 = 2 ,1 = 1 1 + 0 (, , ) , 2 = 2 ,

1 = 1 , 2 = 2 .(9)

Through the mapping functions in (9), one can transformthe west wall to 1 = 0, the baffle (plate) center to 1 =1 and 2 = 0, the east wall to 2 = 1, the free surfaceto 1 = 0, the baffle tip to 1 = 1 and 2 = 0,the tank bottom to 2 = 1, the south wall 1 = 0 tothe baffle center to 1 = 1 and 2 = 0, and the northwall to 2 = 1. In this way, the computational domain isinvariant (eight unit squares), and the more advantage ofthe transformations is to deal with the tank with internalstructures of various positions and scales and to avoidthe internal structure surrounded by the irregular meshes.Furthermore, combining with the stretching technique [28],the stretching grids can be arranged around the structureboundaries with the sharp corners. The thickness of the baffleor plate is set only at 1% of the tanks length and is, therefore,negligible compared with the length of the tank.

In this three-dimensional analysis, the liquid flow issolved in a unit cubic mesh in the transformed flow domain.All computations use the dimensionless equations in the -- coordinate system. All the numerical results presentedin this work are in the dimensionless form [28], and thedimensionless equations can be referred to [24, 29] thatare omitted in the text. Central difference approximationsare used for the space derivatives, except at the boundarywhere the fictitious cell approach [24, 27] is employed. Astaggered grid system is used in the analysis. That is, thepressure is defined at the centre of a finite difference gridcell (of dimensions (, , and )), whereas the velocitycomponents , , and are calculated 0.5, 0.5,and 0.5 behind, above, or backward of the cell centre.The Crank-Nicholson second order finite difference schemeand the Gauss-Seidel point successive overrelaxation iterativeprocedure are used to calculate the velocity and pressure,respectively. The detailed numerical scheme is similar to thatreported [24, 28, 29] and is omitted here.

3. Results and Discussion

3.1. Experiment Investigation. It is difficult to be solved bytheoretical and numerical studies that the complex andintricate phenomenon of the nonlinearity behavior of res-onant sloshing waves occur. The experimental investigationof sloshing with damping devices is very limited and mostlyfocuses on lateral excitation (only surge motion). In reality,as the tank is excited by accelerations due to an earthquakeor waves, the excitation directions of the tank are variedwith time. In view of this, an experiment was conceived andattempted to carry out the preliminary investigation on the


Figure 2: Photograph of the experiment setup of a baffled tank.

effect of damping devices on sloshing in tanks subjected tovarious horizontal excitation angles and to further validatethe accuracy of the present numerical work.

The photograph of the experiment setup is shown inFigure 2. The excitation direction of shaking table is designedto be altered by an aluminum alloy rotational table. Thematerial of the baffled tank is acryl with 20 mm thickness toprevent the tank deformation from the hydrostatic pressureand hydrodynamic impact of the liquid, and that of thebaffle is fibreglass that can avoid the occurrence of baffledeformation due to hydrodynamic forces. The maximummoving distance () of the shaking table is 30 mm, and thehighest revolution of the motor is 2000 r.p.m. The frequencylevel depends on the limitation of the maximum velocityimplemented by the AC motor and the motor reducer.In this experiment work, the maximum velocity ( =) of the shaking table is about 30 mm/s which indicatesthat if the excitation displacement () becomes large, thecorresponding excitation frequency has to be reduced. Themeasurement of wave elevation is carried out by wave probes,and the locations of wave probes, 1 and 2, are depictedin Figure 3. The comparison of wave history at 1 betweenthe experimental measurement and the present numericalscheme for a baffled tank subjected to an oblique excitationof 15 is illustrated in Figure 4, and a good agreement isdemonstrated.

3.2. Effect of a Vertically Tank Bottom-Mounted Baffle on Shiftof Nature Modes of Sloshing Waves. In this section, the effectof a vertically tank bottom-mounted baffle parallel to the tankbreadth () on shift of the nature modes of a tank with liquidis discussed. The natural modes (,) of 3-D tank can beexpressed as

, = 2 + 22,2, = , tanh (,0) , (10)

where , are the natural modes components of - and -axes, respectively. Wu et al. [24] analyzed the influence ofbaffle height on the shift of natural modes in 2D tanks withsloshing fluid and concluded that the shift of the lowestnatural mode 1,0 of the tank apparently increases with thegrowth of baffle height. For the present design of 3D baffledtank, the effect of baffle height

numerical simulation of fluid flow and heat transfer processes

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