Applications of URANS on predicting unsteady turbulent separated flows

Transcription

1 Acta Mech Sin (2009) 25: DOI /s RESEARCH PAPER Applications of URANS on predicting unsteady turbulent separated flows Jinglei Xu Huiyang Ma Received: 10 January 2008 / Revised: 26 August 2008 / Accepted: 13 October 2008 / Published online: 4 March 2009 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2009 Abstract Accurate prediction of unsteady separated turbulent flows remains one of the toughest tasks and a practical challenge for turbulence modeling. In this paper, a 2D flow past a circular cylinder at Reynolds number 3,900 is numerically investigated by using the technique of unsteady RANS (URANS). Some typical linear and nonlinear eddy viscosity turbulence models (LEVM and NLEVM) and a quadratic explicit algebraic stress model (EASM) are evaluated. Numerical results have shown that a high-performance cubic NLEVM, such as CLS, are superior to the others in simulating turbulent separated flows with unsteady vortex shedding. Keywords URANS Nonlinear eddy viscosity turbulence model Separated flows Circular cylinder 1 Introduction Separated turbulent vortical flows widely exist in various civil and military applications of fluid engineering, such as in chemical engineering, coastal engineering, city planning, and aeronautics. Research on the unsteady separated vortical flows is of great importance in academic study and engineering application. As a classical benchmark of the unsteady vortex shedding flow with massive separation, flow past a circular cylinder has been extensively studied both experimentally and numerically, because of its geometric simplicity and abundance of interesting flow features. Numerical simulation plays an important role in the investigation of unsteady turbulent flows with massive separation. The direct J. Xu H. Ma (B) Department of Physics, Graduate University of the Chinese Academy of Sciences, Beijing, China hyma@gucas.ac.cn numerical simulation (DNS) of turbulence is restricted to simple flow geometry and a low Reynolds number due to its huge consume of computer resource; Large eddy simulation (LES) is much less CPU-consuming than DNS, however, large amount of grid points and too small the time step are required to ensure appropriate resolution in both time and space, limiting the application of LES to many industrial applications. Turbulence models (RANS) have remained for over three decades the mainstay of the computational fluid dynamics to solve engineering turbulent problems due to their considerable robustness and reasonable accuracy, and have supported the commercial software industry of computational fluid dynamics. So far different levels of turbulent models have been developed and achieved great success in the engineering applications. However the simulation of the unsteady vortical flows with massive separation put forward a great challenge for turbulence models theory. Reynolds-averaged methodology has the well-recognized deficiencies, the idea of averaging has simplified the resolution of turbulence and eliminates many significant information of the actual flow structures, because only averaged flow variables rather than the instantaneous fluctuations are obtained. As a result, the traditional RANS turbulence model, such as the k ε two-equation model, has come short when dealing with complex turbulent flows, such as flows with separation (there is strong adverse pressure gradient), strong streamline curvature and rotation effects, large-scale vortex shedding and time dependant characteristics. Another fundamental difficulty is that the model coefficients are calibrated by DNS or experiments of some typical fully developed and steady turbulent flow, which makes the turbulence models insensitive to transition onset location and further weakens the accuracy of RANS in predicting time dependent flows. Despite all of the inherent deficiencies of RANS described above, it is the only practical choice for the numerical

2 320 J.Xu,H.Ma simulation of separated vortical flows in engineering. At present there are some encouraging progresses for URANS approach [1 3]. It is well known that a significant development in the turbulence modeling theory since the 1990s is the nonlinear eddy viscosity turbulence model (NLEVM), which is essentially equivalent to the explicit algebraic stress model (EASM). NLEVM describes the Reynolds stress explicitly in an algebraic expression in terms of the mean strain and spin tensors, in which the second-order terms describe the anisotropy of the turbulences structure and the third-order terms depict the effects of the streamline curvature and the rotation. The advantage of NLEVM lies in that, with a well-developed robust CFD code for the conventional two-equation turbulence models, it can be readily implemented to simulate complex turbulent flows by merely adding the nonlinear terms into the expression for the Reynolds stress, without significantly consuming the CPU and losing the computational robustness. In the present paper, numerical simulation of twodimensional flow past a circular cylinder is conducted at Re = 3,900, for which detailed DNS and experimental data are available. Though Reynolds number of 3,900 is low for most technical applications, the flow is already very complex, which is appropriate to assess the performance of turbulence models in predicting unsteady separated turbulent flow. Zdravkovich [4] has pointed out that when Reynolds number varies from 350 to , the flow past a circular cylinder is in the transition-in-shear-layers region, in which the separated boundary layers remains laminar, while a transition takes place along the free shear layers with shedding vortexes leaving the body as large-scale turbulent vortices. The turbulence models assessed here are zero-equation model (BL), one-equation model (SA), linear k ω model (SST), quadratic NLEVM (SZL) [5], cubic NLEVM (CLS) [6] and the quadratic EASM (GS) [7]. It should be noted that throughout the numerical simulation no modification to any model has been made by referring to DNS results or experimental data. 2 Numerical results and discussion The numerical method to solve RANS equations and the transportation equations of k and ε is the finite volume method with structured grids. It is advanced in time with an implicit three-factor approximate factorization method. In order to implement time-accurate computation, dual-time stepping with subiterations and multigrid are employed, and the second order temporal accuracy is achieved. The thirdorder upwind-biased spatial differencing is used for the convective and pressure terms, and second-order differencing for the viscous terms. The flux difference-splitting method of Roe is chosen to obtain fluxes at the cell faces. Fig. 1 Pressure coefficient on the cylinder surface It is known that three-dimensional effects are present in various experiments of fluid flows past a circular cylinder, and the longer the cylinder length is, the more the 2D behavior appears. For the purpose of the turbulence models assessment, only two-dimensional flow is considered, so a 2D O-grid is adopted here. After the grid independence test, grids of are found to be fine enough. The grids extend 20 diameters upstream and downstream with the cut-line set in front of the cylinder. Fine grid spacing is needed near the viscous wall to utilize the damping function, and the distance of the first grid to the wall is 10 5 which is corresponding to a y + less than 1.0. The turbulence intensity level is chosen as 0.02%, and the numerical results are found non-sensitive to it. The constant time step used in the calculation is 0.01, which corresponds to about 450 steps per cycle of the shedding frequency, and sub-iterations are set to 10 during the dual-time stepping computation. Now the numerical results are discussed. The pressure distribution on the cylinder surface is shown in Fig. 1, where the symbol of circle denotes DNS [8], and it is evident that only the CLS model predicts the pressure distribution in good agreement with DNS. Failure of other models arises from the over-predicted separation angle, which changed significantly the flowfield structures. The recirculation region averaged within one vortex shedding period is shown in Fig. 2,in which CLS model predicts the flow pattern well matched by the LES [9] results, but SST model exhibits a rather different flow pattern. In the numerical simulation of flow around a circular cylinder, people used to adjust the prediction of separation angle either by employing various curvature corrections to the models or by altering the free-stream turbulence intensity. By taking the free-stream turbulence intensity as 0.5% and the µ t /µ as 10, Marongiu et al. [10] obtained a reasonable numerical results even with the isotropic standard k ε model, although the results were not as good as CLS model. The fact shows that an accurate prediction of the separation angle is the key to the accurate simulation of flows around a cylinder, and a cubic turbulence model, which is capable of capturing the anisotropy and the strong streamline curvature

3 Applications of URANS on predicting unsteady turbulent separated flows 321 Fig. 2 Streamlines of the time-mean flow: a CLS model and b SST model Table 1 Mean flow quantities from DNS, LES, RANS and experiments (taken from Ref. [14]) Data from C D C Pb Lr/D St Exp ± ± ± ± ± DNS [8], Case I DNS [18] LES [13] LES[10] CLS B-L S-A SST SZL EASM effects in the wake, shows its great potential in the simulation of turbulent flows with massive separation. The numerical results of the mean drag coefficient C D, base pressure coefficient C Pb, separation angle, length of mean recirculation region Lr and Strouhal number St are presented in Table 1. It is surprising that the overall predictions of the cubic CLS model are in fairly good agreement with the experiments and DNS, while LES of Germano et al. [11] predicts a smaller length of the mean recirculation region Lr and a little larger drag coefficient. However, all of the other models, including the second-order anisotropic models, predict incorrect separation angles and subsequently too small recirculation regions. Let us focus on the mean velocity and the Reynolds normal stress profiles. The streamwise data are more reliable than that of the cross-flow direction. For example, the experimental uncertainty in the measurements of the streamwise velocity is about 5% while the uncertainty of the cross-flow velocity is more than 50% according to Beaudan and Moin [12]. Therefore only the streamwise mean velocity profiles are discussed here. The streamwise velocity profile from CLS model is presented in Fig. 3. In this figure, the DNS data come from case I of Ma et al. [8], the experimental data in the very near wake from Lourenco and Shih [13] at locations x/d 3.0 denoted by squares and the others from Ong and Wallace [14] denoted by circles. It is observed that the CLS results agree well with the DNS and the experiments, especially in the very near wake. One of the most important indicators to evaluate the model s performance in the circular cylinder s numerical simulation is the mean streamwise velocity distribution along the centerline behind the cylinder, which is shown in Fig. 4. The symbols of circle, triangle, and gradient denote experiments of Ong and Wallance [14], Govardhan and Williamson (taken from [8]) and Lourenco and Shih [13], respectively. LES of Kravchenko and Moin [15] (case 2) denoted by solid line is also presented in this figure. It can be seen that the experimental data are quite scattered. It was pointed out that the velocity measurements in this flow field are extremely difficult due to the large angle of attack of the velocity and relative low flow velocity magnitudes, especially at locations very near the cylinder and along the centerline of its wake [14]. Besides, the cylinder s aspect ratio in different experiments is also different: in the Lourenco and Shih experiment it was 20.5 while in the Govardhan and Williamson experiments it was 10. These may explain the large difference in the measurement of the length of the recirculation region. In addition, the length of the recirculation region depends not only on the cylinder aspect ratio but also on the cylinder diameter as well, referring to a systematic study of Noca et al. [16]. At present, it is not clear what other sources contribute to the discrepancy in these three sets of experimental data,

4 322 J.Xu,H.Ma Fig. 3 Mean streamwise velocity profile: a X/D = 1.06, b X/D = 1.54, c X/D = 2.02, d X/D = 4.00, e X/D = 7.00, f X/D = 10.0 Fig. 4 Mean streamwise velocity on the centerline but no matter how much uncertainty is involved, the velocity on the centerline is strongly dependent on the recirculation region, and it is evident SST model performed poorly. The velocity profile obtained from CLS model is in very good agreement with Ong and Wallance which contains minor 3D effects in contrast with Govardhan and Williamson, and we believe that CLS model predicts the velocity a little bigger than the true value due to the under-predicted velocity fluctuations, taking into account the reliability of the LES and experiments. In order to give a reasonable explanation for this discrepancy we plot the velocity fluctuations (i.e. Reynolds stress) in Figs. 5, 6, and 7. Symbols in these figures denote the same date origin as in Fig. 4. It is seen that at X/D = 1.06 and 2.02, the Reynolds normal stress is smaller than LES and the experimental data near Y/D = 0.0. The under-predicted fluctuations lead to insufficient local flow mixing, which justifies the corresponding under-predicted velocity magnitude near Y/D = 0.0. Though the CLS model has predicted the variation trend of the Reynolds stress along Y axis, referring to Figs. 5, 6, and 7, the under-predicted fluctuations peak is evident. The farther downstream, the smaller the fluctuations peak will be. This is also the reason why the mean velocity profile since X/D 4.00 is not as good as in the very near wake. This phenomenon also appeared in our

5 Applications of URANS on predicting unsteady turbulent separated flows 323 Fig. 5 Reynolds shear stress profile Fig. 6 Reynolds normal stress profile Fig. 7 Reynolds normal stress profile previous study of a turbulent U-duct flow [17] due to the curvature effects, which indicates that the coefficients of the third-order terms in the Reynolds stress constitutive relation may be further tuned to better capture the curvature effects although the cubic CLS model has shown its excellent performance. It is believed that even though the second-order terms of the NLEVM are necessary to predict the anisotropy features in the separate region, the capture of streamline curvature effects is responsible not only for the separation position but also for the prediction of the vortices abound with curved streamlines in the wake. This fact makes the third-order terms essential, and a cubic of NLEVM is required in the accurate calculation of the vortical separation flows. At last we want to explore the inconsistent behavior of the different models from the view point of turbulent kinetic energy (TKE). Since the standard k and ε transport equations are adopted for most of the models evaluated here, the essential difference of them lies in the Reynolds stress constitutive relation, in which the second-order terms describe the anisotropy of the turbulence structure, and the third-order terms depict the streamline curvature and the rotation effects as mentioned above. Obviously, the high performance of CLS model is due to the contribution of its third-order terms. As observed in experiments, the appearance of streamline curvature can drastically change the structure of turbulence. A concave curvature destabilizes the flow field and enhances the turbulence intensity and its characteristic scales, whereas a convex curvature stabilizes the flow field, restrains the turbulent mixing and damps the Reynolds shear stress and the turbulent kinetic energy. In our previous study of the turbulent U-duct flow [17], the non-cubic eddy viscosity models over-predicted the TKE near the convex wall, therefore put off the separation point and significantly changed the flow structure, which is similar to the present cylinder case. As shown in Fig. 8, the horizontal coordinate denotes TKE, the y-coordinate R means the distance to the centre of the cylinder. Figure 8a, b, and c present the results for three different angle, starting from the leading edge. Up to 90 where separation has not occurred or has just begun, the models of SST, SA and EASM have significantly over-predicted the TKE near the convex wall of the cylinder. As a result, the separation positions have been put off by these models. In addition to the delayed separation position, the over-predicted turbulence scales have destabilized the shear layers and thus have shortened the recirculation region. When reaches 100, where the predicted separation from the models of SST and SA has just begun, the turbulence predicted by the CLS model has well developed. The magnitude of the predicted TKE from the CLS model is larger than that from the SST and SA model. 3 Concluding remarks Flow past a circular cylinder, as a classical benchmark of the unsteady separated flow, is numerically investigated at Re = 3,900 by employing the URANS technique, and the performances of different levels of turbulence models are assessed. The numerical results given by CLS model, which is a well-known third-order nonlinear eddy-viscosity turbulence model, agree well with DNS and experiments for the region in the vicinity of the cylinder. It is quite inspiring that the well-developed cubic eddy viscosity model, such as the CLS model, may be applied to the numerical simulation of

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