Yun Long, Lin-feng Deng, Jun-qiang Zhang, Bin Ji, Xin-ping Long

1. State Key Lab of Water Resources and Hydropower Engineering Science, School of Power and Mechanical Engineering, Wuhan University, Wuhan 430072, China

2. Wuhan Second Ship Design and Research Institute, Wuhan 430064, China

Abstract: The large eddy simulation (LES) is used to resolve the flow structure in the cavitating turbulent flow around the Clark-Y hydrofoil coupled with a homogeneous cavitation model. A new method is proposed in this paper to calculate the LES error of the time-averaged streamwise velocity for the LES verification and validation (V&V). From the instantaneous cavity patterns, it is demonstrated that the predicted results agree fairly well with the experimental data. With this new proposed method, the LES errors can be easily and effectively calculated with a limited mesh number, and the method might be used in the other applications of the LES V&V. Results of the LES errors obtained by the new method show that the relatively steady flow can be simulated with small errors, while the complex flow structures at the cavity shedding region might lead to an increase of errors in the LES modeling. In addition, the distributions of the resolved Reynolds stresses are used to estimate the influences of the cavitation on the turbulent fluctuations. Results indicate that the turbulent fluctuations for the cavitating flow are much larger in magnitude as compared to the cases without cavitation.

Key words: Cavitating flow, large eddy simulation (LES), verification and validation (V&V), Reynolds stress and turbulent fluctuations

Introduction

The cavitation concerns dynamic and turbulent two-phase flows when the local pressure in the liquid is below the saturated vapor pressure[1-3]. The re-entrant jet mechanism[4-5]was shown to be an important factor for the cavity shedding, accompanied with the evolution of numerous large and fine vortex structures. The shock wave mechanism[6-7]was observed experimentally and numerically in recent years. In general, the cavitation can significantly increase the turbulent fluctuations[8]. Dittakavi et al.[8]found that the collapse of the cloud cavitation may lead to the production of the turbulence within the cavitating flow. However, the negative effect, with the cavitation reducing the production of the turbulence,is also observed by Iyer and Ceccio[9].

The LES was regularly applied in these complex cavitating flows along with the rapid development of the high performance computers, and many new modified and proposed numerical schemes and sub-grid models were reported to obtain satisfactory results[10]. However, lack of a powerful method to quantitatively assess the accuracy of the calculated results might produce under-resolved results,inaccurate due to the simple treatment of the sub-grid stress. The verification and validation (V&V) is a basic procedure to assess the reliability of the numerical simulation as we have demonstrated in our previous work[11]. The practical use of the LES V&V is a great challenge for the complicated unsteady cavitating flow, although much progress has been made in this respect recently[12-14]. Xing[13]proposed a general framework for the LES V&V. Various methods in that framework were extensively evaluated for fully developed turbulent channel flows[14]. It was shown that the five-equation and a robust threeequation method may be applied for the LES V&V.However the available methods to obtain the LES errors and carry out the LES V&V are still very few,and more useful and simpler methods need to be proposed for the application of the LES V&V.

This paper is our on-going effort to resolve the transient cavitating flow around the Clark-Y hydrofoil by the large eddy simulation, and a new method is proposed in this paper to calculate the LES errors for cavitating flows. Further, the influence of the cavitation on the resolved Reynolds stresses is also studied.

The LES method and the Smagorinsky-Lilly model in the ANSYS Fluent solver are adopted in this paper. The detailed introductions of the method and the model can be found in our previous work[11]. The Zwart-Gerber-Belamri cavitation model is adopted to model the mass transfer between the vapor and liquid phases, as is widely used and validated for its accuracy of modeling the complex cavitating flow.

The bounded central difference scheme for the momentum equations and the second order implicit scheme for the transient formulation are adopted in the ANSYS Fluent solver. Convergent results under the steady condition without cavitation are used as the initial conditions for the transient cavitation calculation. The time steps are listed in Table 1. The number of the maximum internal iteration steps in each time step is 40 with the 10-4RMS residual as the convergence criterion. The computational domain, the boundary setup and the mesh generation are similar to our previous work[11]. The angle of attack of the Clark-Y hydrofoil is 8° and the chord lengthCis 70 mm. The hydrofoil is placed in a channel, of 10.0Cin length and 2.7Cin height. The inlet velocity is set asU∞=10 m/s . The outlet is fixed at the static pressurepoutaccording to the cavitation number, and the cavitation number is set as 0.8 for the cavitating flow.A no-slip boundary is imposed on the foil surface and the free slip wall is set on the upper and lower walls.The detailed mesh number, the model constant and the calculation setup are given in Table 1. For comparison with the cavitating flow, the flow without cavitation withσ=2 is also computed for mesh 5.

Some innovative attempts were made to quantify the LES errors by Dutta and Xing[13-14], but the estimation of the quantitative LES errors is still not satisfactory for the explosive applications of the LES in various fields. The difficulty of applying the V&V for the LES lies in the fact that the numerical and modeling errors due to the numerical discretization and the sub-grid scale model contribution are both influenced by the mesh size. The interaction between these two sources of errors is a challenging problem.For the LES V&V and the calculation of the LES errors in an effective way, a new model for the LES errors is proposed by systematically changing the mesh size and the Smagorinsky constant in this paper.The LES errors, including the modeling error and the numerical error, can be calculated as follows:

whereSCdenotes the numerical benchmark,S1-S5correspond to the LES solutions obtained by using meshs 1-5, as shown in Table 1,cNandcMare the undetermined coefficients for the numerical and modeling errors,pNandpMare the orders of accuracy of the numerical and modeling errors,respectively. The time step Δtand the mesh sizehshare the constant refinement ratior(, as shown in Table 1).h*is calculated as.The change ratioαis also set asfor the Smagorinsky constant. The numerical error and the modeling error for mesh 5 correspond to the first and second terms on the right-hand side of Eq. (5),respectively. Then, the orders of accuracy of the numerical error and the modeling error based on Eqs.(1)-(5) for mesh 5 can be calculated as:

Table 1 Mesh and model constant information and calculation setup

It should be noted thatα=r.The numerical error and the modeling error based on Eqs. (1), (2) and (5) for mesh 5 can be calculated as:

Then we can use the above-mentioned formulas (Eqs.(6)-(9)) to calculate the LES errors. Theoretically, any variables (such as the velocity and the pressure) can be used to obtain the values ofpNandpM, the numerical error and the modeling error by Eqs. (6)-(9).However, as Long et al.[11]pointed out, it is very difficult and impractical to determine reliable values ofpNandpMfor the local variables (such as the transient velocity in each grid nodes) in practical applications, especially, for complex transient unsteady flows. A preferable choice to calculate the LES errors is to determine appropriate values ofpNandpMby using some macroscopic quantities. Here,we use the time-averaged lift coefficient, whereSis the spanwise length)to solve Eqs. (6), (7) for determining proper values ofpNandpM. ThenpNandpMare set as constants in Eqs. (8), (9), and we can use them to calculate the numerical and modeling errors for other local quantities (the streamwise velocity in the following discussions). This simplified procedure was adopted and applied in some other researches[12, 15].

The time-averaged lift coefficients for mesh 1-5 are presented in Table 2. By solving Eqs. (6), (7), the values ofpNandpMare 1.40 and 0.75. They will be used to calculate the numerical and modeling errors for the normalized streamwise velocity (u/U∞) .

Table 2 Time-averaged lift coefficients for meshs 1-5

To assess the accuracy of the calculated results,the periodic cavity evolutions obtained by the experimental photographs[16]and the numerical predictions are compared, as shown in Fig. 1. The predicted cavity patterns are visualized by the contours of the vapor volume fraction obtained by using mesh 5.Tis the cavity shedding frequency for the CFD and experimental data, and all results from Figs. 1(a) to 1(d) are presented in dimensionless forms for convenience.Figure 1 clearly shows the initiation, the growth and the shedding of the attached cavity above the hydrofoil suction surface. The initial attached cavity as shown in Fig. 1(a) gradually grows to the vicinity of the trailing edge (T.E.) from the leading edge (L.E.)as shown in Fig. 1(b). Then, due to the development of the re-entrant jet towards the L.E., the attached cavity sheds from the hydrofoil surface as shown in Fig. 1(d) and collapses downstream. Then, at the next cavity shedding cycle, the unsteady cavity repeats its process of growth and collapse. From Fig. 1, it can be seen that the predicted unsteady cavitation process agrees well with the experimental observations.

Fig. 1 (Color online) Comparison between the experimental and modeled cavity patterns within one typical cycle.σ = 0.8, α=8°, U ∞=10 m/s

A new LES error estimation method is proposed as is demonstrated before. In this part, the modeling error and the numerical error are calculated by this new method for the dimensionless streamwise velocity,and the results are shown in Fig. 2, wherex/cis the normalized distance to the foil leading edge, the abscissa is the error magnitude of the modeling error and the numerical error, and the vertical axis denotes the normalized distance to the foil surface. Fromx/c= 0 to 0.4, the magnitude of the modeling error gradually grows with the largest error reaching 0.2.Significantly, with the cavity thickness increasing from x/ c=0.4 to 1.0, the magnitude along the vertical direction dramatically changes. Even at the upper region away from the foil surface, the modeling and numerical errors have the magnitude of 0.2 or 0.1.These large errors are nearly of the same order of magnitude with the streamwise velocity. The similar tendency was observed in our previous work[11].

Fig. 2 Modeling error and numerical error for time-averaged u/ U∞ for mesh 5. σ=0.8, α=8°, U ∞=10 m/s

The separated flow and the cavitation change little over the flow time from x/ c=0 to 0.2. In other words, the relatively steady flow structure within these regions can be resolved with a small range of error by using the current meshes and models.Unfortunately, the inherent turbulent characteristics over time and space make the numerical calculation very difficult, especially for the quasi-periodic shedding cavitation flow. From x/ c=0.6 to 1.0, the flow structures see a dramatic change in time and space, with the development of vortices and the shedding and the collapse of the cloud cavitation. The complex flow structures are the causes for the large errors of the calculated results.

The turbulent fluctuations characterized by the Reynolds stresses are used in this section to study the influence of the cavitation on the calculated results.Figure 3 shows the distributions of the normalized resolved Reynolds stresses(abbreviated asin the following discussions) for the flows with and without cavitation.From the results in Fig. 3 for the cavitating flow, the streamwise turbulent fluctuation mainly occurs above the trailing edge, while the cross-stream turbulent fluctuation mainly occurs at the rear region of the streamwise fluctuation. It can be observed that the magnitude of the cross-stream fluctuation is larger than that of the streamwise fluctuation at the hydrofoil wake. The cavity shedding and collapse occurs periodically above the trailing edge, which increases the streamwise turbulence. However, the complex large scale vortex structure resulted by the cavitation shedding has its largest strength at the close rear region of the trailing edge, which increases the crossstream fluctuation. For the case without cavitation, the magnitude of the streamwise fluctuation remains larger at all locations as compared to the cross-stream fluctuation. Obviously, the fluctuations of flows without cavitation are significantly smaller than those with cavitation. In conclusion, the unsteady periodic shedding cavitation leads to the abundant production of turbulence within the flow.

Fig. 3 (Color online) Distribution of the resolved Reynolds stresses obtained by using mesh 5 for flows with and without cavitation

The LES coupled with the Zwart-Gerber-Belamri cavitation model is used to resolve the flow structure in the cavitating turbulent flow around the Clark-Y hydrofoil. The time evolu- tion of the transient cavitation is discussed. A new proposed method is applied to study the LES error for the streamwise velocity. Finally, the influences of cavitation on the turbulent fluctuations are analyzed. The main conclusions can be summarized as follows:

(1) The instantaneous cavity patterns show that our calculation results agree fairly well with the experimental data. The new proposed method is much easier and more effective to calculate the LES errors and to carry out the LES V&V, and can help the practical applications of the LES V&V.

(2) The LES errors grow from the leading edge to the trailing edge for the cavitating flow. The relatively steady flow structure near the leading edge region can be resolved with a small range of the LES errors using the current meshes and models. The complex flow structures, including the cavity shedding and the large scale vortex, are the causes of large errors of the calculated results.

(3) The periodic cavity shedding and collapse increase the streamwise turbulence above the trailing edge, and the complex large scale vortex is the major cause of the increase of the cross-stream fluctuation at the rear region of the trailing edge and the further wake. In addition, the cavitation contributes to a much larger magnitude of the turbulent fluctuations as compared to flows without cavitation.

Acknowledgments

The authors would like to thank Prof. Lian-di Zhou from China Ship Scientific Research Center for providing us many helpful discussions and comments.The numerical calculations in this paper have been done on the supercomputing system in the Supercomputing Center of Wuhan University.