Direct comparison between RANS turbulence model and fully-resolved LES

Transcription

1 International Journal of Gas Turbine, Propulsion and Power Systems July 2016, Volume 8, Number 2 Direct comparison between RANS turbulence model and fully-resolved LES Takuya Ouchi 1 Susumu Teramoto 2 and Koji Okamoto 3 1 IHI Corporation 2 Department of Aeronautics and Astronautics, The University of Tokyo 3 Department of Advanced Energy, The University of Tokyo Abstract Tip leakage flows are important in compressors because of their strong influence on cascade performance and stability. Since there are separations and three-dimensional shear layers in tip leakage flows, the validity of RANS is not self-evident. The objective of this paper is to evaluate the influence of a RANS turbulence model in tip leakage flows. To achieve this, both fully resolved LES and RANS are conducted under the same design conditions and Reynolds stresses and flow structures are directly compared using the resolved LES as the high-fidelity reference result. Elicited results indicate that RANS can reproduce the bulk exit flow angle and Reynolds stresses distribution in a qualitative manner. However, quantitative comparision showed that the RANS underestimate Reynolds stresses at strong shear layers, so there are differences in the exit total pressure loss and the velocity distribution of the tip leakage vortex that may affect its prediction of the vortex breakdown and rotor stall performance in the case of simulations near the stall point. Background Computational fluid dynamics (CFD) are widely used nowadays in the analysis of internal flows, including the case of the jet engine. In industry, RANS is mainly used because of its low computational cost. On the other hand, based on the recent development of computers, large eddy simulations (LES) that resolve large-scale fluctuations has also become applicable to flows with a high Reynolds number, such as refereces [1,2]. For example, LES of the compressor cascade using 800 million grid points [3] and detached eddy simulation of the multistage compressor using 18.7 million grid points [4] have been reported. Since RANS models all the fluctuations of typical turbulent flows, such as those in attached boundary layers, the handling of unsteady phenomena, such as separation flow and large eddy fluctuation, is known to be poor. However, considering the calculation cost, the use of resolved LES in the design is not realistic and even in the near future. Therefore, RANS will be mainly used to allow its improvement based on the comparison with experimental data or resolved LES. A typical flow for which the validity of RANS is questionable is a tip leakage flow. Tip leakage flow occurs in the gap between the blade tips and the casing wall due to the pressure difference between the pressure surface and the suction surface of the blade. A very large number of experimental studies have been reported over past decades, such as the visualization of the blade tip surface flow [5], the measurement of the wall pressure [6], and the velocity distributions of the clearance inlet or exit [6, 7] and the tip leakage vortex [7]. Fig.1 shows the typical flow structure near the blade tip of an axial compressor cascade. The tip leakage flow is said to account for 30% of the total loss of the compressor [8], and the relevance of the inception of the rotating Manuscript Received on December 4, 2015 Review Completed on June 15, 2016 stall was documented in prior work [9]. However, since it exhibits interactions with the mainstream, the boundary layer on the casing wall, the secondary flow, and blade tip separation vortex, flow phenomena are complex. In addition, the formation of a leakage vortex has been found to be dependent on various parameters, such as the blade tip speed, the inlet turbulence intensity, the boundary layer thickness, and others [10]. Fig.1 Schematic of near tip flow field Many numerical analyses on the tip leakage flow have also been carried out [11]. It has been indicated that the selection of the RANS turbulence model affects the result [12], and that some researchers have reported improvements using high-order RANS models [13, 14]. You et al. [8] have conducted LES of the tip leakage flow and reported that the turbulent energy that is generated by the strong shear at the clearance outlet contributes to the cascade loss. In this way, the tip leakage flow is a complex flow and includes a large-scale vortex and a shear layer, such as a tip leakage vortex and a tip separation vortex, which are three-dimensional and unsteady. In addition, its turbulent fluctuation is directly connected to cascade performance. It is interesting to quantify the extent to which the RANS model is able to simulate such a flow. However, the measurement of the tip leakage flow is difficult because the blade tip clearance is very narrow. For example, a 1.5 % span [15] and a 2-4% chord [16] are typical for aviation engines and for the low-pressure stage of the industrial gas turbine. Experimental studies of the linear cascade using a Laser Doppler velocimetry have also been performed [17], but only pointwise velocity data have Copyright 2016 Gas Turbine Society of Japan 1

2 Table 1 Specifications of GE rotor B Chord, C(mm) 254 mm Inlet flow angle (deg) 65.1 Stagger angle (deg) 56.9 Pitch 0.929C Span 0.3C Tip clearance height 0.016C been obtained. In this way, quantitative experimental data that can be used as a reference data is lacking, particularly at the tip clearance. It is therefore difficult to evaluate the validity of the Reynolds stresses of the RANS model by direct comparison with experimental data. Research objectives The objective of this paper is to strictly evaluate the validity of RANS turbulent model through direct comparison between a RANS result and corresponding fully-resolved LES result. Resolved LES and RANS of the compressor cascade clearance flow are conducted under the same conditions, and the Reynolds stresses elicited by both simulations are directly compared. The influences of the RANS model on the tip leakage flow, the flow structure as well as the overall cascade performance are discussed. The resolved LES results is regarded as high-fidelity reference numerical solution of the governing equation in this work. Therefore, no arguments are elaborated on whether LES or RANS closely resemble the elicited experimental results. Methods of Analysis Flow configuration The design flow condition of the tip cross-section of the final stage of the GE rotor B that was used in the experiment [18] is adopted for the simulation of this investigation. Table 1 lists the specifications for the GE rotor B. A preliminary RANS simulation showed that the extent of the tip leakage vortex is approximately 15% span for this flow condition. Therefore the blade aspect ratio is decreased from 1.0 to 0.3 as long as the tip leakage vortex does not deform largely. The Reynolds number, based on the chord length, is Computational method Calculation code is identical to that used in reference [19]. The governing equations in the LES and RANS equations are the threedimensional, compressible Navier Stokes equations. Spatial differences are calculated by SHUS[20] with a third-order MUSCL scheme in RANS, and a sixth-order compact scheme[21] in LES, except for the vicinity of the hub side boundary for numerical stability. The spatial filter in LES is a tenth-order accuracy implicit tridiagonal filter[22] with a filter coefficient of No SGS model is used in this LES. The ADI-SGS[23] method is used for time integration. The time accuracy in the LES is maintained by Newton sub-iterations with a three level backward-differencing formula. The RANS simulation is based on backward Euler formula and local time stepping. The RANS simulation is fully converged to steady state, and no unsteady fluctuation is observed in the RANS solution. The Spalart Allmaras (SA) model[24] is used as a RANS model. Computational grid and parallelization Figures 2 and 3 show the grids. LES and RANS use the same grid in order to eliminate the grid difference influences. The interaction between blade boundary layer and the leakage vortex takes place mainly between the endwall and and 15%span, therefore the streamwise and spanwise grid resolutions over the blade surface are x + < 50 and y + < 20 in this region. The spanwise spacing is Fig.2 Computational grid with a close view near the tip (every twelve points) gradually stretched upto y + < 50 at the hub. The grid resolution parallel to the endwall is ξ +, η + < 20 for both two direction since the grid lines do not align to the flow direction over the endwall. The minimum grid spacing is C from the solid wall corresponding to y + 1. The total number of grid points is approximately 800 million, and 26 million points are used at the tip clearance. Since it is assumed that the blade surface boundary layer in the actual compressor is fully turbulent, the boundary layer tripping is set both in LES and RANS to promote the transition on the inlet casing wall and 10% chord position of the blade surface. The grid is divided into 3440 zones for parallel computations on the FX10. In order to maintain the high-order accuracy in the zone boundary, six points are overlapped and data are exchanged at every time step. When the overlap region consists of noncoincident grid points, the data exchange is conducted using linear interpolation. It is confirmed in preliminary simulations[25], that the turbulent fluctuations are fully resolved, such as the turbulent boundary layers on both blade surfaces and on the casing wall, and at shear layers at the tip clearance and around the tip leakage vortex. An FX10 supercomputing system at the University of Tokyo is used for the work presented in this paper. Parallel computing is performed by MPI using 215 nodes that correspond to 3440 CPUs. A total of 100,000 time steps required 48 h to compute. Boundary conditions Total pressure, total density, and flow angle, are fixed at the inlet boundary, and the static pressure is fixed at the outlet boundary. The solid wall boundary conditions at the casing and blade surfaces are treated as nonslip adiabatic. The hub-side boundary condition is treated as the slip wall. The pitchwise boundaries are set as periodic boundaries. The inlet Mach number is 0.4. The dimensionless time increment, based on the inlet sound speed and the chord length is , and the maximum Courant number is approximately 12 at the clearance. The time averaged data are created by averaging 100,000 steps corresponding to a flow during a period of 0.7 s. 2

3 Fig.4 Inflow velocity and Reynolds stress distribution appears in the second figure. Three large-scale vortices, a tip separation vortex over the blade tip, a leakage vortex, and a counter vortex exist, and the leakage vortex is dominant. The comparison of the time-averaged LES and RANS indicates that they are almost identical, so RANS is able to reproduce the position and size of the three large-scale vortices. Fig.3 Computational grid of spanwise distribution with a close view near the tip (every four points) Results Inflow turbulent boundary layer Figure 4 shows the velocity and Reynolds stress distribution of the inflow turbulent boundary layer at 0.25C upstream of the leading edge. The solid line depicts velocity and the dotted line is the Reynolds stress. The Reynolds shear stress in RANS can be calculated using the following equation: u v = ν t du dy Based on the fact that both lines exhibit a good agreement with LES and RANS, it is deduced that inflow turbulent boundary layer is well resolved, and that both simulations are conducted under the same inflow conditions. In addition, in the comparison of the clearance flow described in the next section, the influence due to the difference of the incoming turbulent boundary layer is considered to be almost eliminated. In addition, the peak value of the Reynolds stress is located at y/chord=0.01 and it is just within the blade tip clearance (the black dotted line in the figure indicates the spanwise position of the tip surface). This means that the inflow turbulent boundary layer and the clearance jet interfere strongly under this inflow condition. Large scale vortex structure The visualization of the large-scale vortex is shown first in order to understand the dominant structure of the flow field. Fig.5 shows the vortex structure by means of the isosurface of the second invariant of the velocity gradient tensor. The instantaneous and time-averaged LES and RANS are shown from the top to the bottom. Since the instantaneous LES is covered with fine vortexes, it is difficult to understand the dominant vortex structure. However, temporal averaging results in the large-scale vortex structure that (1) Distribution of velocity and Reynolds stresses Before the comparison of the Reynolds stresses, Fig.6 shows the velocity distribution in order to understand the velocity field, especially the formation of the leakage vortex. First, there is a large positive w-component velocity area at the clearance, showing that the clearance jet is blowing strongly. After blowing, the w-component of the jet decreases and the v-component begins to increase instead. In this area, the jet is beginning to roll up from the casing. The jet then begins to swirl in a clockwise direction, so negative W- and V- components appear. In this way, the leakage vortex is formed that constitutes the dominant flow structure in this flow field. In addition, there are strong shear layers in the tip clearance and around the leakage vortex. These are important regions where RANS modeling is considered to be questionable. Fig.7 shows a comparison of the Reynolds stresses of LES and RANS. All of the components are noticeable within the strong clearance shear layer, the clearance outlet, and around the leakage vortex. Since these distributions are similar in LES and RANS, it is found that qualitative Reynolds stress distributions can be reproduced by RANS. However, a quantitative evaluation indicates that RANS tends to underestimate Reynolds stresses, and the difference exceeds 5% in instances where the jet is rolling up and at the tip separation vortex in the clearance. Comparing the velocity distributions in Fig.6, these are qualitatively similar, but the peak of the spanwise velocity where the jet rolls up becomes excessive and equal to approximately 10% in RANS. This is considered to be the result of the underestimation of the turbulent diffusion by Reynolds stresses. A detailed comparison at the tip clearance is presented next. Tip leakage vortex and clearance jet Figure 8 shows the Reynolds stress at the 50% chord crosssection in clearance and the surface streamlines of the tip surface. First, a large Reynolds stress is seen in the clearance that is caused by the shear layer of the tip separation vortex. It is underestimated in RANS by approximately 50% as compared to the peak. For example, the nondimensional value of v w is 0.03 in LES and in RANS. On the other hand, looking at the surface streamlines, it can be seen that the separation vortices are reattached at the position 3

4 Fig.6 velocity distribution Tip separation vortex Counter vortex Leakage vortex Fig.5 Vortex structure comparison (top: LES instantaneous, middle: LES time-averaged, bottom: SA, colormap indicative of static pressure) indicated by R in the LES and RANS figures. However, in RANS, the position of the reattachment becomes closer to the suction side edge, which means that the separation vortex becomes excessive in RANS. The difference of the R position is approximately 10% of the blade thickness. Based on the analyses listed above, RANS underestimates Reynolds stress, that is, turbulent diffusion. In turn, turbulent diffusion is considered to cause the delay of the reattachment of the separation vortex. The Reynolds stress term in the right hand side of pitchwise momentum equation appears as x u w + z w w. In Fig.7, difference in the Reynolds stress inside the clearance gap is the most noticeable for u w component, although its chordwise (x ) gradient is small compared with its pitchwise (z ) gradient. Considering that only the chordwise gradient of u w contributes to the pitchwise velocity, the difference of u w does not directly influence the w velocity. Fig.9 shows the w-component velocity distribution and the Reynolds stress distribution at the clearance outlet at a 50% chord position. Reynolds stresses differ significantly between the LES and RANS. However, it can be seen that the velocity distribution is almost the same. Fig.10 shows the chordwise distribution of the leakage mass flow rate as the percentage against the inlet mass flow rate. This also has the same distribution in the LES and RANS. Based on the above, although the size of the separation vortex is somewhat overestimated by the underestimation of the Reynolds stresses in RANS, the impact on the velocity distribution at the clearance outlet is small. As a result, the leakage flow rate distribution is well-reproduced. In particular, the leakage flow rate is the indicator of the momentum, and energy needs to be supplied to the leakage vortex. Therefore, since the reproducibility of the leak flow rate was observed, it can be said that the driving force of the leakage vortex is almost identical in both the LES and the RANS. 4

5 Fig.8 The difference of reattachment position at the clearance Fig.9 Distribution of Reynolds stress and velocity at the clearance outlet Fig.7 Reynolds stresses distribution Leakage vortex strength and velocity distribution Fig.11 shows the chordwise distribution of the pressure coefficient Cp at the leakage vortex center. Herein, the leakage vortex center is defined by the minimum static pressure at each of the chord cross-sections. Comparing the LES and RANS, the Cp of RANS is greater by 0.1. This suggests that the leakage vortex in RANS is weaker than that of LES. Moreover, it can be seen that a strong adverse pressure gradient occurs along the leakage vortex center. Fig.12 shows the u-component of the velocity distribution. The viewing is the same as that for Fig.6 and Fig.7. The low-velocity regions shown in blue can be seen around the leakage vortex, and correspond to areas where the clearance jet begins to roll up and swirl. The distribution of the LES and RANS are also similar, but in order to see more detail, the spanwise distribution at the vortex center cross-section is shown in Fig.13. The lines used are indicative of 50, 75, and 100%, of chord section. The effect of the blockage due to the leakage vortex appears at approximately 10% of the blade height from the casing. These low-velocity regions are ex- Fig.10 Tip leakage flow rate distribution panding downstream, reaching 20% of the span at the 100% chord position. In addition, the local minimum velocity in the leakage vortex is smaller downstream. It is considered to be caused by the adverse pressure gradient shown in Fig.11. For the comparison of LES and RANS, the spanwise position where the velocity recovers to the mainstream is matched. Based on this, it is found that RANS 5

6 Fig.14 Distribution of the total pressure loss coefficient Fig.11 Comparison of vortex strength U/U LE V x/c=0.25 x/c=0.5 x/c=0.75 x/c= C Fig.12 Velocity distribution of the u-component W near the design point, the vortex breakdown is not observed. However, considering that there is a velocity difference of 30% on this flow condition, in the cases of calculations near stall points, it is possible that the occurrence of the vortex breakdown will be slower in RANS. Total pressure loss coefficient Fig.14 shows the distribution of the total pressure loss coefficient C L. The definition of C L is the subtraction of the inlet total pressure at a -50% chord position and the local total pressure, divided by the inlet dynamic pressure, as indicated by the equation listed below: C L = P t inlet P t 1 2 ρ inletu inlet 2 Large losses are generated from the tip separation vortex, the leakage vortex center, and the counter vortex. When LES and RANS are compared, their distributions are qualitatively similar. Thus, for example, in the leakage vortex, the loss increases near the center of the vortex. From a quantitative perspective, RANS tends to underestimate the loss, and the maximum difference is approximately 0.1 at the leakage vortex center. (2) Fig.13 The difference of u-component velocity of the leakage vortex can reproduce the influence on the mainstream due to the blockage of the leakage vortex. On the other hand, the local minimum velocity in the leakage vortex shows quantitative differences, which can be for example, 30% in excess at a 100% chord section. It is considered that the leakage vortex will collapse when this velocity becomes zero, but since the flow condition setting in this paper is Cascade performance Figure 15 shows a mass averaged flow angle distribution at -50% and at 150% of the chord position. The influences due to the casing turbulent boundary layer and the leakage vortex appear from y/span=0-0.2 at 150% of the chord section. Since the leakage vortex is clockwise when viewed from downstream, the distribution exhibits a clockwise direction at approximately y/span= Comparing LES and RANS, RANS becomes excessive at approximately 3 degree near the casing wall, but the difference becomes less than 1 degree at the other midspan side. Therefore, RANS is able to evaluate the bulk flow angle well. Figure 16 shows the distribution of the mass averaged total pressure loss coefficient at a chord position of 150%. The definition is obtained by changing the local total pressure of equation (2) to the mass averaged total pressure. In the same way as in the case of the flow angle, large losses occur between y/span=0-0.2 generated by the leakage vortex and the casing turbulent boundary layer. In terms of the comparison of LES and RANS, RANS underestimates the loss, and the difference is 0.05 corresponding to 25%. It is considered that this is a result of the underestimation of the Reynolds stresses, that is, turbulent diffusion. Conclusion As a result of the evaluation of the validity of the RANS model by direct comparison with the fully resolved LES at the tip leakage flow, the following conclusions are obtained: 6

7 Fig.15 Mass averaged flow angle distribution Fig.16 Distribution of the mass-averaged loss coefficient The Reynolds stress distribution in the shear layer of the tip separation vortex and the leakage vortex is able to be reproduced qualitatively, but quantitatively it is underestimated up to 50% The reattachment position of the tip separation vortex is slightly different but the clearance exit velocity and the flow rate are almost the same Since RANS underestimates the strength of the leakage vortex, the local minimum of the u-component velocity is excessive by approximately 30% in design point. It is possible to delay the vortex breakdown and alter the performance at the stall point There is small difference in the bulk flow angle but RANS underestimates the total pressure loss by approximately 25% Acknowledgement The computations in this study were performed by the Fujitsu FX10 supercomputer in the Information Technology Center at the University of Tokyo, Japan. This research was mainly supported by Strategic Programs for Innovative Research (SPIRE) of the High Performance Computing Initiative (HPCI) (Project IDs: hp120296, hp130001, hp and hp150219). References [1] Tucker, P., Computation of unsteady turbomachinery flows: Part 2-LES and hybrids, Progress in Aerospace Sciences, Vol. 47, No. 7, oct 2011, pp [2] McMullan, W. and Page, G., Towards Large Eddy Simulation of gas turbine compressors, Progress in Aerospace Sciences, Vol. 52, 2012, pp [3] Gourdain, N., Validation of large-eddy simulation for the prediction of compressible flow in an axial compressor stage, Proceedings of ASME Turbo Expo 2013, [4] Yamada, K., Furukawa, M., Nakakido, S., Matsuoka, A., and Nakayama, K., LARGE-SCALE DES ANALYSIS OF UNSTEADY FLOW FIELD IN A MULTI-STAGE AXIAL FLOW COMPRESSOR AT OFF-DESIGN CONDITION US- ING K COMPUTER, Proceedings of ASME Turbo Expo 2015, Montreal, Canada, 2015, pp [5] Kang, S. and Hirsch, C., Experimental Study on the Three- Dimensional Flow Within a Compressor Cascade With Tip Clearance : Part I - Velocity and Pressure Fields, Journal of Turbomachinery, Vol. 115, 1993, pp [6] Kang, S. and Hirsch, C., Tip Leakage Flow in Linear Compressor Cascade, Journal of Turbomachinery, Vol. 116, 1994, pp [7] Kang, S. and Hirsch, C., Experimental Study on the Three- Dimensional Flow Within a Compressor Cascade With Tip Clearance : Part II - The Tip Leakage Vortex, Journal of turbomachinery, Vol. 115, No. 92, 1993, pp [8] You, D., Wang, M., Moin, P., and Mittal, R., Large-eddy simulation analysis of mechanisms for viscous losses in a turbomachinery tip-clearance flow, Journal of Fluid Mechanics, Vol. 586, aug 2007, pp [9] Yamada, K., Kikuta, H., Furukawa, M., Gunjishima, S., and Hara, Y., Effects of Tip Clearance on the Stall Inception Process in an Axial Compressor Rotor, Proceedings of ASME Turbo Expo 2013, Vol. 6 C, 2013, p. V06CT42A035. [10] Lakshminarayana, B., Zaccaria, M., and Marathe, B., The Structure of Tip Clearance Flow in Axial Flow Compressors, Journal of Turbomachinery, Vol. 117, No. July 1995, 1995, pp [11] Inoue, M. and Furukawa, M., Physics of Tip Clearance Flow in Turbomachinery, ASME 2002 Joint U.S.-European Fluids Engineering Division Conference, [12] Denton, J. D., SOME LIMITATIONS OF TURBOMA- CHINERY CFD, Proceedings of ASME Turbo Expo 2010, 2010, pp [13] Borello, D., Hanjalic, K., and Rsipoli, F., Computation of tip-leakage flow in a linear compressor cascade with a secondmoment turbulence closure, International Journal of Heat and Fluid Flow, Vol. 28, 2007, pp [14] Liu, Y., Yu, X., and Liu, B., Turbulence Models Assessment for Large-Scale Tip Vortices in an Axial Compressor Rotor, Journal of Propulsion and Power, Vol. 24, No. 1, 2008, pp [15] Layachi, M. Y. and Bölcs, A., EFFECT OF THE TIP CLEARANCE ON THE CHARACTERISTICS OF A 1. 5 COMPRESSOR STAGE WITH REGARD TO THE INDEX- ATION, The 9th of International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, 2002, pp [16] Storer, J. A. and Cumpsty, N. A., Tip Leakage Flow in Axial Compressors, Journal of Turbomachinery, Vol. 113, 1991, pp

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