• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    A PARTIALLY-AVERAGED NAVIER-STOKES MODEL FOR HILL AND CURVED DUCT FLOW*

    2011-06-27 05:54:02MAJiameiWANGFujunYUXinLIUZhuqing
    水動力學研究與進展 B輯 2011年4期

    MA Jia-mei, WANG Fu-jun, YU Xin, LIU Zhu-qing

    College of Water Conservancy and Civil Engineering, China Agricultural University, Beijing 100083, China, E-mail: jiameima@163.com

    A PARTIALLY-AVERAGED NAVIER-STOKES MODEL FOR HILL AND CURVED DUCT FLOW*

    MA Jia-mei, WANG Fu-jun, YU Xin, LIU Zhu-qing

    College of Water Conservancy and Civil Engineering, China Agricultural University, Beijing 100083, China, E-mail: jiameima@163.com

    Turbulent flows past hill and curved ducts exist in many engineering applications. Simulations of the turbulent flow are carried out based on a newly developed technique, the Partially-Averaged Navier-Stokes (PANS) model, including separation, recirculation, reattachment, turbulent vortex mechanism. The focus is on how to accurately predict typical separating, reattaching and secondary motion at a reasonable computational expense. The effect of the parameter, the unresolved-to-total ratio of kinetic energy (fk), is examined with a given unresolved-to-total ratio of dissipation (fε) for the hill flow with a much coarser grid system than required by the LES. An optimal value of fkcan be obtained to predict the separation and reattachment locations and for more accurate simulation of the resolved turbulence. In addition, the turbulent secondary motions are captured by a smaller fkas compared with the RANS method with the same grid.

    Partially-Averaged Navier-Stokes (PANS), flow separation, recirculation, reattachment, resolved turbulence

    Introduction

    The Reynolds-Averaged Navier-Stokes (RANS) equations are widely used for numerical predictions in engineering applications due to the low computational cost. However, it is found that the RANS is unsuitable for resolving fluctuating scales of motion[1]. The Large Eddy Simulation (LES) has the ability to accurately predict the resolved flow structures, but its application is limited because of too much computing cost[2,3]. Cheaper unsteady turbulent prediction methods are required for practical flow computations. Several hybrid methods that combine the best features of the RANS and the LES were developed, such as Detached Eddy Simulation (DES)[4], Very Large Eddy Simulation (VLES)[5], Partially Averaged Navier-Stokes (PANS) model[6-10].

    The PANS model was developed recently basedon the RANS k?ε closure equations with various modeled-to resolved scale ratios[6,7]. Its equations vary smoothly from the RANS to the Direct Numerical Simulation (DNS) by two controllable parameters: the unresolved-to-total ratio of kinetic energy (fk) and that of dissipation (fε)[6-8]. In this method, only the coefficients have to be changed depending on the choices of fkand fε[11]. The PANS model was studied in some typical turbulent flows. Girimaji[7,8]applied the PANS model to the flow over a surfacemounted cube, the flow past a square cylinder and a turbulent square jet, and demonstrated that the PANS can capture the resolved turbulence. Basu et al.[12]used the PANS model to a cavity. Frendi et al.[13]compared three hybrid approaches (DES, URANS and PANS), and showed that the PANS model gives better results than the other two turbulence models for a turbulent flow over a backward facing step. Song and Park[14]focused their investigation on how to determine the control parameter fkin the flow past a square cylinder.

    In many engineering applications, the presence of a separating, reattaching and vortex flow causes anincreased unsteadiness, possibly would lead to noise and vibrations, such as in draft tubes of hydraulic turbines, inlet passages of a pump station[15-17]. Thus, the prediction of unsteady separating, reattaching and vortex flow is important in practical applications with a reasonable computational cost.

    The hill flow is a typical case including separating, recirculating, reattaching and accelerating turbulence mechanisms. Frohlich et al.[18]and Breuer et al.[19]simulated the separated flow using the LES code with a very finer grid of more than 1.3×106nodes. The PANS method gives a very effective prediction in simulating the resolved turbulence with a coarse grid for many kinds of flows. However, there is no way to determine the appropriate value of the unresolved- tototal ratio of kinetic energy for predicting the separated or reattaching flow while the PANS model is used. The accuracy for predicting the secondary motion is another key problem.

    In this study, we investigate the turbulence flow past a hill and curved duct by using the PANS model and discuss these problems based on the study. Different values of fkare used in the compution of the hill flow, with a coarser grid system than what is required by the LES. The flow through a curved duct exists in several engineering applications[20], such as centrifugal pumps, hydraulic turbines, aircraft intakes, and river bends[21]. The comparisons at the small fkvalues between the PANS model and the RANS model are made.

    The article is organized as follows. The numerical equation is described in Section 1, and the computational methodology and cases are presented in Section 2. The results of the calculations of the hill and curved duct flow are discussed in Section 3. Finally, conclusions are drawn in Section 4.

    1. Numerical equations

    The governing equations of the time-averaged formulation in fluid flow represent mathematical statements of conservation laws of physics, including mass, momentum, and energy conservation equations. In this article, an incompressible turbulent flow is considered, the mass (continuity) equation and momentum RANS equations are written as

    The PANS models are based on the same equations as the RANS. In the original form of the PANS, the two-equation turbulence model is replaced by solving for the unresolved (partially-averaged) kinetic energy kuand the dissipation εu, so that νu= Cuk2w/εw. In the derivation of the transport equations for kuanduε, two parameters: the unsolved-to-total ratio of kinetic energy, fk, and that of dissipation, fε, are defined by[6-8]

    where subscript u indicates PANS statistics and kuanduε are the unresolved kinetic energy and its dissipation rate, respectively.

    In the PANS model, the kuequation is identical to the original k equation. In theuε equation, the following coefficients are used to turn the two-equation model into the RANS

    wherekσ andεσ represent the Prandtl numbers, Cε1and Cε2are the coefficients of dissipation, the same as in the RANS model.

    Therefore, the PANS closure equations for kuanduε are

    whereuPis the production term, and can be written as

    It is noted thatfkandfεare used as the resolution control parameters in the PANS model, which indicates that 0≤fk≤fε≤1. The smallerfk, the finer the filter is. RANS simulations are carried out atfk=1 and the DNS is calculated atfk=0. Smaller values offεwould be required for the resolution of dissipative scales of motion.

    In this article, the value offεis taken as 1.0 and the coefficientsCε1,Cε2,kσ,εσandCμtake the same values as in the parent RANS model as follows

    Fig.1 3-D schematic diagram of the hill flow

    Fig.2 The grid of the hill flow in a 2-D slice, with the grid plotted with every other line

    2. Computational methodology and cases

    An incompressible, finite volume code is used in computations[22], with the second-order central difference scheme used for space discretization for all terms except the convection terms in thekuanduεequations, where the hybrid central/upwind scheme is used. The Crank-Nicolson scheme is used for time discretization of all equations. The numerical procedure is based on an implicit, fractional step technique with a multigrid pressure Poisson solver[23]and a nonstaggered grid arrangement.

    Two different flow configurations are simulated with the PANS model, under different boundary conditions in the following two cases.

    Fig.3 Geometry of the curved duct flow

    Fig.4 The grid of the hill flow in a 2-D slice, the grid is plotted with every three nodes

    The first case is the hill flow, involving flow separation in a channel with periodic hills mounted on the bottom wall in the streamwise direction. The turbulence flow phenomena, separation, recirculation, reattachment and acceleration, are included in the periodic hill flow. The geometry of the hill flow is shown in Fig.1, and the grid is given in Fig.2. The computational domain extends from a crest to the next crest of two consecutive hills, separated by a distance ofL=9h. Plane and curved wall surfaces are bounded on the upper and lower sides. The computational grid isNx×Ny×Nz=160×80×32 in the streamwises, wall-normal and spanwise directions, respectively. It is noted that the present grid is much coarser than the LES grid, in which the grid isNx×Ny×Nz=196×128×186 for the same computationalregion[17]. The Reynolds number, based on the hill height,h, and the bulk velocity,Ub, above the hill crest isReb=Ubh/ν=10595. The time step is set to Δt=6.0×10?3. Statistical analysis is made over a time period of further 20 flow-through times after 20 flow-through times.

    Fig.5 Streamfunction contours for the hill flow (LES data is from Froehlich et al.[18])

    The second case is a square curved duct flow of a 90o-curved bend, in which the secondary motion (streamwise vorticity) is generated. Figure 3 shows the geometry of the curved duct considered in the present study. The bend is of square cross-section and the ratio of the bend radius to the duct heightrc/H=2.3. The duct height isH=40mm , the inner and outer radii areri=72mm andro=112mm . The configuration was proposed by Raisee et al.[24]. The upstream length of the computational domain is set to 3.75H, the downstream length is selected as 7.5H. As shown in Fig.4, due to the symmetry of the curved duct, only half of the cross-section is considered with a body fitted mesh, consisting of 162× 82×34 nodes in streamwise, wall-normal and spanwise directions, respectively. In the streamwise direction, 41 nodes are located along the upstream-length, 46 nodes cover the curved section and 75 nodes are set along the downstream-length of the duct. The Reynolds number, based on the centerline velocity (U0) and the duct’s height (H), is 40 000.

    For both cases, no-slip conditions are specified on the walls for the velocity components. In the hill flow, the values ofkuandεuon the wall surface are set, byku,w=0 andεu,w=2νku,1/y12, respectively, whereku,1is the value ofkuandy1is the wall distance for the first near-wall node. Periodic boundary conditions are imposed on the streamwise and spanwise boundaries. In the curved ducts, the inlet condition is imposed at the duct inlet by setting theU-velocity to the bulk velocity andVandWvelocities to zero. The initial value of the turbulent kinetic energy isk0=(0.03U0)2. The dissipationε0is assumed to beε0=k02/3/l, in whichl=0.1H.

    3. Results and discussions

    This section presents the simulation results with the PANS model for the hill and curved duct flows. The main concern is to show how to get accurate predictions for typical cases of separating, reattaching and secondary motion at a reasonable computational expense. An optimal value offkwill be obtained for predicting the separation and reattachment locations.

    The results are compared to available LES data (for the hill flow[18]).

    The values offkandfεare given in the computation, and it is obvious that these two parameters should vary with the grid resolution in accordance with the resolved kinetic energy and the dissipation rate. Various values offkare used in the following calculations in order to test the effect of this parameter on the modelling, andfεis set to 1.0 in all computations.

    3.1Periodic hill flow

    Figure 5 gives the streamfunction contours of the hill flow, including one of the LES solution. The LES solution shows that the flow is separated atx=0.2hshortly after the hill crest and is reattached atx= 4.7hafter the hill foot. The recirculation zone occupies almost 50% of the streamwise domain. Atfk= 0.4 and 0.5, accurate separation locations can be seen, and with other values offk, one sees a delayed separation. Nevertheless, the reattachment location is first delayed (fk=0.4, 0.5 and 0.6) and then moves towards the hill foot. Withfk=1.0, no separation and reattachment locations are shown. It should be recalled that the PANS model returns to the base model (the RANS model). The number of recirculation bubbles increases with the increase offkfirstly and then decreases (0.6<fk≤1.0). Anyway, the simulations give very good predictions for separation and reattachment flows with a coarse grid atfk=0.4 and 0.5.

    The pressure and the friction coefficients on the lower wall at differentfkare shown in Figs.6 and 7, respectively. The pressure coefficients on the lower wall assume almost a constant over most of the upstream half of the recirculation zone for the LES result, then a rapidly rise is seen after the crest, which leads to separation. Atfk=0.4, the pressure coefficient takes a smaller value in the upstream half of the recirculation region for pressure coefficient, and then approaches the LES data. The friction coefficient sees a rather irregular, geometry-induced variation of the near-wall velocity, as shown in Fig.7. The reverse flow in the recirculation zone approaches the downhill side, decelerates and almost reverses direction, indicating secondary vortices immediately after the separation. Atfk=0.4, the results agree well with the LES data in all computational region. However, the pressure and the friction coefficients are over-estimated atfk=1.0, which gives a steady simulation.

    The distributions of the mean flow and Reynolds stresses are compared in what follows in order to see the capability of the PANS modelling in resolving turbulence flows. Different locations are considered for the hill flow, withx=0.05h, 2.0h, 6.0hand 8.0h. The mean velocity components,uandv, in the streamwise and normal-wall directions, repectively, as well as the Reynolds stresses are compared with the LES data. Atx=0.05h, a short distance after the hill crest, the results may help to see the possible influence of the flow separation. Atx=2.0h, the second location, in the center of the recirculation zone, the results would be especially interesting, since a free shear layer above the region, a reverse flow below the free shear layer and a boundary layer on the top wall are concerned. The third location is after the reattachment point,x=6.0h, the profile here may recovery the boundary layer developed from the reattachment point and the motion of flow before reaching the next hill crest.x=8.0his a strong acceleration region of flow.

    Fig.6 Pressure coefficient along bottom wall

    Fig.7 Friction coefficient along bottom wall

    Fig.8 Comparison of mean velocities

    The mean velocities are shown in Fig.8, forx= 0.05h, 2.0h, 6.0hand 8.0h. It is shown that atfk= 0.4 and 0.5, more reasonable predictions are obtained than at other fkforu andv. Under-predictions of mean velocities can be found for fk=0.6 near the lower wall, but over-predictions are observed for fk=1.0 there.

    3.2 Curved duct flow

    The turbulence flow in the 90o-curved duct with square cross-section is simulated and the results are discussed in this section. Firstly, it is verified that the PANS moodel can be used for a large curvature change case (90-curved duct). The results of the RANS method using Fluent are compared with those of the PANS method at fk=1.0. A small value of fkis considered, namely fk=0.01. It should be noted that the simulation with so small value of fkin the PANS model is almost the same as by the DNS.

    Fig.9 Comparison of Reynolds stresses

    The velocity contours, obtained with the RANS model and with the PANS model at fk=1.0, at the vertical mid-symmetry plane of the duct, are compared in Fig.10. As can be observed, the total velocity contours are quite similar. The direction of velocity is changed due to the bend of duct. The flow accelerates along the suction (convex) surface within the bend. Obviously, the opposite is observed along the pressure (concave) surface. Due to the effect of the centrifugal force, the pressure decreases along the inner side at the bend inlet, and subsequently increases along the outer side. The secondary motion develops within the bend. After the bend of the duct, the velocity is gradually reduced along the concave side and increasedalong the convex side of the straight duct. Both computational approaches produce more or less similar streamwise (U) and normal-wall (V) velocity contours and secondary velocity cells. Therefore, the PANS model can be used to predict the turbulence flow with a large curvature change.

    Fig.10 Comparison of velocity contours

    Four cross-sections are selected along the flow direction of the curved duct, namely, S1: the upstream of the bend at L1=3.5H, S2: the bend part at θ= 30o, S3: the bend part at θ=60o, S4: the downstream of the bend at L2=0.25H, as shown in Fig.11.

    Figure 12 shows the profiles of velocity and spiral vortex at the four cross-sections obtained with the PANS model at fk=0.01 and fk=1.0. It clearly demonstrates that at small fk, more resolved turbulent scales can be achieved than with the RANS model. The influence of the bend is not very clear at the S1plane, but the PANS simulation with a small fkcan predict the profile of a spiral vortex near the wall (Figs.12(a) and 12(b)). At the S2plane (θ=30o), the turbulence is remarkably enhanced due to the action of the bend, the PANS simulation at a small fkgives a more resolved turbulent flow, whereas the RANS model can only predict the large-vortex motion (Figs.12(c) and 12(d)) . With the development of the flow, the center of the vortex moves towards the outer surface at the S3plane (θ=60o) as a result of the centrifugal force, and it is observed that the PANS simulation at a small fkenjoys higher accuracy in simulating the turbulence than the RANS model in a curved duct (Figs.12(e) and 12(f)). A similar result is also obtained at S4(Figs,12(g) and 12(h)). The trend of velocity contours is similar with the two approaches, while the PANS simulation at a small fksees strong fluctuations.

    Fig.11 The sketch of cross-sections locations

    For the hill flow computations, five constant values of fkare used, and the results are compared with the LES data. A much coarser grid system than what is required by the LES is considered. The results show that the unresolved- to-total ratio of kinetic energy at f k=0.4 and 0.5 c an be used to a ccurately predict the separation and reattachment locations.In

    4. Conclusions

    Fig.12 The profiles of velocity and spiral vortex at different cross-sections

    The PANS model and its key controlled parameters are investigated in the context of simulating the turbulent flow past the hill and curved duct, as can be found in many engineering applications including flow separation, recirculation, reattachment and secondary motion (vortex). addition, reasonable predictions of mean velocities,uandv, and Reynolds stresses are obtained at four locations, namely,x=0.05h, 2.0h, 6.0hand 8.0h. It demonstrates that atfk=0.4 a very good prediction is made for the resolved turbulence and atfk=1.0 the results of the RANS simulations are obtained.

    The turbulence flow in the 90o-curved duct with square cross-section is simulated. The results show that the turbulence flow with secondary motions are predicted by the PANS model, the three-dimensionality of the flowfield is captured, together with the fine scale structures and the unsteady vortex near the wall at a small value offk.

    The present investigation clearly indicates that the accurate prediction for separating and reattaching is achieved atfk=0.4 and 0.5 with a reasonable computational expense. In addition, atfk=0.4, very good simulations are obtained for resolved turbulence. The turbulent secondary motions are captured at smallfk, comparable to those obtained with the RANS method with the same grid. This observation combining with its easier implementation in an RANS code proves that the practical flow computations can be made by an optimal unresolved-to-total ratio of kinetic energy with a given unresolved-to-total ratio of dissipation.

    Acknowledgments

    The author would like to acknowledge the financial support given by Swedish National Infrastructure for Computing (SNIC) for computer time at C3SE (Chalmers Center for Computational Science and Engineering).

    [1] ABDOL-HAMID K. S., ELMILIGUI A. Calculations of high-temperature jet flow using hybrid Reynolds-Averaged Navier-Stokes formulations[J].Journal of Aircraft,2008, 45(1): 64-70.

    [2] DAVIDSON L., BILLSON M. Hybrid LES-RANS using synthesized turbulent fluctuations for forcing in the interface region[J].International Journal of Heat and Fluid Flow,2006, 27(6): 1028-1042.

    [3] ZENG Cheng, LI C. W. A hybrid RANS-LES model for combining flows in open-channel T-junctions[J].Journal of Hydrodynamics,2010, 22(5 Suppl.): 154-159.

    [4] XU Chang-yue, CHEN Li-wei and LU Xi-yun. Largeeddy and detached-eddy simulations of the separated flow around a circular cylinder[J].Journal of Hydrodynamics,Ser. B,2007, 19(5): 559-563.

    [5] RUPRECHT A., HELMRICH T. and BUNTIC I. Very large eddy simulation for the prediction of unsteady vortex motion[C].The 12th International Conference on Fluid Flow Technologies.Budapest, Hungary,2003.

    [6] GIRIMAJI S. S. Partially-Averaged Navier-Stokes model for turbulence: Implementation and validation[C].43rd AIAA Aerospace Science Meeting and Exhibit.Reno, Nevada, USA, 2005.

    [7] GIRIMAJI S. S. Partially-Averaged Navier-Stokes model for turbulence: A Reynolds-Averaged Navier-Stokes to direct numerical simulation bridging method[J].Journal of Fluids Engineering,2006, 73(2): 413-421.

    [8] GIRIMAJI S. S., JEONG E. and STRINIVASAN R. Partially-Averaged Navier-Stokes method for turbulence: Fixed point analysis and comparison with unsteady Partially Averaged Navier-Stokes[J].Journal of Applied Mechanics,2006, 73(2): 422- 429.

    [9] LAKSHMIPATHY S., GIRIMAJI S. S. Partially Averaged Navier-Stokes (PANS) method for turbulence simulations: Flow past a circular cylinder[J].Journal of Fluids Engineering,2010, 132(12): 121202.

    [10] JEONG J., GIRIMAJI S. S. Partially averaged Navier-Stokes (PANS) method for turbulence simulations-Flow past a square cylinder[J].Journal of Fluids Engineering,2010, 132(12): 121203.

    [11] FROHLICH J., TERZI D. Hybrid LES/RANS methods for the simulation of turbulent flows[J].Progress in Aerospace Science,2008, 44(5): 349-377.

    [12] BASU D., HAMED A. and DAS K. Assessment of Partially-Averaged Navier-Stokes (PANS) multiscale modeling transonic turbulent separated flows[C].5th Joint ASME/JSME Fluids Engineering Conference.San Diego, Califorina, USA, 2007.

    [13] FRENDI A,. TOSH A. and GIRIMAJI S. S. Flow past a backward-facing step: Comparison of PANS, DES and URANS results with experiments[J].International for Computational Methods in Engineering Science and Mechanics,2007, 8(1): 23-38.

    [14] SONG C. S., PARK S. O. Numerical simulation of flow past a square cyclinder using Partially-Averaged Navier-Stokes model[J].Journal of Wind Engineering and Industrial Aerodynamics,2009, 97(1): 37-47.

    [15] ILIESCU M. S. CIOCAN G. D. Analysis of the cavitating draft tube vortex in a Francis turbine using particle image velocimetry measurements in two-phase flow[J].Journal of Fluids Engineering,2008, 130(2): 0211051.

    [16] LACERY R. M., GRIFFIN F. M. Resolving vibration problems on a 25 MW Kaplan unit[J].International Journal on Hydropower and Dams,2003, 10(3): 84-88.

    [17] ANSAR M., NAKATO T. Experimental study of 3D pump-intake flows with and without cross flow[J].Journal of Hydraulic Engineering,2001, 127(10): 825-834.

    [18] FROHLICH J., MELLEN C. and RODI W. et al. Highly-resolved large eddy simulation of separated flow in a channel with streamwise periodic constrictions[J].Journal of Fluid Mechanics,2005, 526: 19-66.

    [19] BREUER M., KNIAZEV B. and ABEL M. Development of wall models for LES of separated flows using statistical evaluations[J].Computers and Fluids,2007, 36(5): 817-837.

    [20] HUANG Sui-liang, JIA Ya-fei and CHAN Hsun-Chuan et al. Three-dimensional numerical modeling of secondary flows in a wide curved channel[J].Journal of Hydrodynamics,2009, 21(6): 758-766.

    [21] ZHU Zuo-jin, YANG Hong-xing and CHEN Ting-yao. Direct numerical simulation of turbulent flow in a straight square duct at Reynolds number 600[J].Journal of Hydrodynamics,2009, 21(5): 600-607.

    [22] DAVIDSON L., PENG S. H. Hybrid LES-RANS: A one-equation SGS model combined with ak?ωmodel for predicting recirculating flows[J].International Journal for Numerical Methods in Fluids,2003, 43(9): 1003-1018.

    [23] EMVIN P. The full multigrid method applied to turbulent flow in ventilated enclosures using structured and unstructured grids[D]. Ph. D. Thesis, Gothenburg, Sweden: Chalmers University of Technology, 1997.

    [24] RAISEE M., ALEMI H. and IACOVIDES H. Prediction of developing turbulent flow in 90o-curved ducts using linear and non-linear low-Rek?εmodels[J].International Journal for Numerical Methods in Fluids,2006, 51(12): 1379-1405.

    January 22, 2011, Revised April 16, 2011)

    * Project supported by the National Natural Science Foundation of China (Grant Nos. 51079152, 51079151).

    Biography: MA Jia-mei (1982-), Female, Ph. D.

    Crresponding author: WANG Fu-jun,

    E-mail: wangfj@cau.edu.cn

    2011,23(4):466-475

    10.1016/S1001-6058(10)60137-0

    山东省| 沙田区| 建瓯市| 乌拉特中旗| 鞍山市| 平遥县| 外汇| 长丰县| 青龙| 孝昌县| 张家界市| 太仆寺旗| 福泉市| 谷城县| 南阳市| 社旗县| 辛集市| 双流县| 平潭县| 明水县| 汕尾市| 临潭县| 金山区| 灵宝市| 金坛市| 房产| 阳谷县| 罗城| 寻乌县| 枣庄市| 高青县| 遂川县| 嘉兴市| 瓦房店市| 桃园市| 定远县| 鹤山市| 石城县| 且末县| 即墨市| 额敏县|