Parallel Computations of Unsteady Three-Dimensional Flows in a High Pressure Turbine Dongil Chang and Stavros Tavoularis Department of Mechanical Engineering, University of Ottawa, Ottawa, ON Canada Stavros.Tavoularis@uottawa.ca Abstract. Steady (RANS) and unsteady (URANS) numerical simulations of three-dimensional transonic flow in a high-pressure turbine have been conducted. The results are compared by evaluating differences in contours of total pressure loss, contours of static pressure and wakes. Significant differences have been observed in these parameters, both qualitative and quantitative, which demonstrate that URANS simulations are necessary when the effect of stator-rotor interaction on flows in turbines is strong. Keywords: Turbine, coherent vortices, blade, vane, RANS, URANS. 1 Introduction Turbomachines are used extensively in transportation (e.g., aircraft engines and propellers), energy generation (e.g., steam turbines and wind turbines), and fluid moving (e.g., pumps, fans and blowers). They span, in size, the range from micrometers for a micro-turbine to 150 m for a wind turbine, and they are available in a great variety of shapes. Industrial design of turbomachines relies heavily on semi-empirical design tools, notably on correlations among various parameters, which are based on experimental results, sometimes collected under idealized conditions. The accuracy of such approaches faces serious challenges and their applicability is limited within their range, which reduces their usefulness for new designs. With advances in computational resources and three-dimensional simulation tools, large-scale steady CFD (Computational Fluid Dynamics) simulations are becoming more common in the design process of turbomachines. As a result of the continuing efforts to produce lighter and smaller gas turbines for the aviation industry, the turbine stage loading is increased, while the space between blade rows is reduced. This leads to increased effects of unsteadiness and threedimensionality. Unsteady three-dimensional effects are particularly strong in flow fields through transonic high-pressure turbines, in which they have been associated with the presence of coherent vortices, wakes, and trailing-edge shock waves. The vibratory stresses on a turbine blade, which are intimately connected with the lifetime of the engine and the scheduling of maintenance inspections, are highly dependent on the flow unsteadiness, due to the stator-rotor interactions. This interaction is also known as forced response and is more important in a high-pressure turbine than in a D.J.K. Mewhort et al. (Eds.): HPCS 2009, LNCS 5976, pp. 20 29, 2010. Springer-Verlag Berlin Heidelberg 2010
Parallel Computations of Unsteady Three-Dimensional Flows 21 low-pressure turbine. To determine the vibratory stress level on turbine blade surfaces, one requires the time-dependent surface pressure distribution on the blades. Experimental determination of this distribution is extremely difficult, which makes unsteady CFD simulations a valuable tool for high-pressure turbine performance analysis and design. In recent years, advanced unsteady CFD methods, including Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES), have been introduced in the study of turbomachinery flows in low-pressure turbines [1-3] and in high-pressure turbine cascades [4]. Nevertheless, their application to the simulation of flows in high-pressure turbines has not yet been accomplished, because they require excessive computer resources. A less powerful but more practical approach, which is feasible with the use of currently available computer systems, is the numerical solution of the Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations, with the inclusion of a suitable turbulence model. The computational resources and CPU time required for URANS simulations are much larger than those required for steady (RANS) ones, and it is necessary to investigate whether the benefits from the use of the former are worth the additional efforts and cost. In the present article, which follows a previous one [5], we will compare timeaveraged results of URANS simulations with corresponding results of RANS simulations in order to assess the significance of unsteadiness on turbine analyses. 2 Computational Procedures The computational geometry was the same as that in our previous simulations [5]. Simulations were conducted using the commercial code FLUENT 6.3. Considering the compressible nature of the flow, the implicit coupled solver was used by solving the momentum and energy equations at the same time. For high accuracy, a secondorder upwind scheme was chosen for space discretization and the second-order implicit Euler scheme was employed for temporal discretization. The simulations are the solutions of the URANS equations with the SST turbulence model, which is a combination of the k-ω model in the near-wall region and the k-ε model in the core flow region [6]. Rotational periodic boundary conditions were employed, based on the assumption that, like the geometry, the velocity distribution was also azimuthally periodic. No-slip conditions were applied at all walls. The adiabatic boundary condition was applied on all walls, following common practice in turbomachinery analyses [7]. The total pressure and the total temperature were specified at the inlet, while the static pressure was specified at the outlet assuming radial equilibrium. The inlet Mach number was approximately 0.1. Figure 1 shows the computational domain, which consisted of an inlet section containing a stator with one vane passage, a rotating rotor section having three blade passages, and a non-rotating outlet duct section. The inlet plane was located in the middle of the 180 degree duct, which connected the combustor with the high-pressure transonic turbine stage. The outlet duct section had a length of two blade axial chords and was included to eliminate any possible interactions of the flow in the rotor with reflected pressure waves generated on the blades at the turbine outlet.
22 D. Chang and S. Tavoularis Fig. 1. Computational geometry: (1) inlet; (2) sliding interface between the stator and the rotor; (3) sliding interface between the rotor and the extended duct; (4) outlet; rotational periodic boundaries have been coloured by light green RANS simulations were performed using the mixing plane implementation and steady solutions served as the initial input for URANS simulations. In the URANS simulations, the rotor domain was given a constant angular velocity and sliding meshes were used to model the interface regions. Instantaneous solutions were advanced in time using a dual time stepping technique, which accounts for the relative motion of rotor and stator and other sources of unsteadiness. The initial time step was set sufficiently small for the solution to converge within 20 iterations for every time step. It was increased subsequently to reduce the overall computing time, but the increase was gradual to ensure convergence within each time step. 200,000 time steps per rotor blade passing were used initially, subsequently reduced to 500, which were small enough to resolve time-varying flow features. The convergence of solution was checked by monitoring the temporal fluctuations of the mass flow rate at the outlet, shown in Fig. 2(a). Inspection of this figure demonstrated that a time-periodic mass flow rate was achieved at 6 t / τ v, where τ v is the vane passing period (i.e., the time it takes for one rotor blade to pass past two adjacent stator vanes). Convergence levels of the solution s periodicity were also determined by following the suggestion of Clark and Grover [8]. Figure 2(b) shows plots of convergence levels for the mean mass flow rate, cross-correlation factor (CCF) at zero lag, and fractional power of the power spectrum, as well as the overall convergence level (OCE; i.e., the minimum value of the previous convergence levels). All examined convergence levels at 6 t / τ v reached values which were much closer to 1 (perfect periodicity) than the convergence threshold (CT) of 0.950, which Clark and Grover [8] suggested to be sufficient for overall convergence.
Parallel Computations of Unsteady Three-Dimensional Flows 23 Fig. 2. Time evolution (a) and convergence levels (b) of the mass flow rate at the outlet An unstructured mesh was composed of tetrahedral elements in the core region and prismatic hexahedral elements in the boundary regions. 10 prism mesh elements were allocated across the gap regions between the blade tip and the rotor casing. Three different meshes (1.2, 1.7 and 2.5 Million cells) were used to examine the sensitivity of mesh size on the numerical accuracy and the medium mesh with 1.7 Million cells was selected as combining a high accuracy and a relatively low computing time [5]. The parallel-computing performance of FLUENT 6.3 with MPI was examined in a Sunfire cluster at HPCVL (High Performance Computing Virtual Laboratory), configured with four Sun Fire 25000 servers with 72 dual UltraSPARC-IV+ processors running at 1.5 GHz. Because the maximum allowable thread number for each user of HPCVL is limited to 20, the selected test cases had numbers of threads N within the range between 5 and 20. Figure 3 shows the parallel performance of FLUENT in a Sunfire cluster in terms of speedup and efficiency. The speedup is defined as the ratio of the wall clock time for N = 5 and the wall clock time for N > 5 (it is noted that the usual definition of speedup is with respect to one thread, but single-thread simulations were not performed in this work), whereas the efficiency is defined as the ratio of the speedup and N. When N=5, it took 30.4 minutes of wall clock time for one time step. Figure 3(a) indicates a very good speedup performance with increasing N within this range. The parallelization efficiency in Fig. 3(b) indicates that the efficiency did not change substantially and that it reached a maximum for N=15. The same job that was run on the Sunfire cluster was also run on a cluster at the University of Ottawa, configured with six Dell HP M8120N servers with 12 quad core Intel 6600 processors running at 2.4 GHz. Simulations on the Dell cluster with 20 threads were about 1.7 times faster than those on the Sunfire cluster. Subsequently, all main computations were run on the Dell cluster. It took about 9 days to collect data for one vane passing period. The steady RANS simulations were run on a personal computer with 2 quad core Intel 6600 processors at 2.4 GHz. It took 23 hours for about 4000 iterations, which were required for the convergence of the solutions.
24 D. Chang and S. Tavoularis Fig. 3. Dependence of speedup (a) and efficiency (b) on the number of threads used for the Sunfire cluster 3 Results The modified Q criterion, introduced by Chang and Tavoularis [9], allowed us to identify all well-known types of quasi-stationary vortices in axial gas turbines, including the stator casing horseshoe and passage vortices, the rotor casing and hub horseshoe and passage vortices, the blade tip leakage vortices and vortices shed by the trailing edges of vanes and blades. The modified Q is defined as m ( Ω Ω S S ) 1 Q = c 2 ij ij 1 where U U i j 1 Ωij = and U U i j S 2 ij = + are the rotation tensor and the x j x i 2 x j xi rate of strain tensor, respectively, and c q is an empirical factor. In the original Q criterion, c q = 1 and positive values of Q identify vortices. However, the identification of vortices by the original Q criterion is highly dependent on the choice of the Q value, called the threshold. Using values of c q 1 increases or decreases the relative effect of the rate of strain and may identify realistic coherent vortices, which are less sensitive to the threshold choice [9]. In the present work, the value c q = 0. 63 and the threshold value of 1 s -2 were chosen as appropriate. All these vortices in the rotor section have been marked in Fig. 4. In addition to these quasi-stationary vortices, Chang and Tavoularis [5], using URANS, documented two types of unsteady vortices, the axial gap vortices and the crown vortices. The formation of both types is attributed to interactions between vane wakes and passing blades. The axial gap vortices are roughly aligned with the leading edges of the rotor blades, whereas the crown vortices are located in the mid-section of the blade passage near the blade crown. These vortices have also been marked in Fig. 4. q ij ij (1)
Parallel Computations of Unsteady Three-Dimensional Flows 25 Fig. 4. Vortices in the rotor (coloured by axial vorticity): hub horseshoe (1), hub passage (2), casing horseshoe (3), casing passage (4), tip leakage (5), axial gap (6), crown (7) Coherent vortices identified in the rotor using RANS and URANS are presented in Fig. 5. The sizes and the locations of the rotor casing passage vortices and the rotor hub passage vortices are comparable in both simulations. However, the horseshoe vortices and the tip leakage vortices in RANS are smaller than those in URANS; the tip leakage vortices in URANS start rolling up near the mid-chord section, while the tip leakage vortices in RANS first appear half way between the mid-chords and the trailing edges of blades. Finally, the axial gap vortex observed in URANS is not at all identified in RANS. Fig. 5. Vortices in the rotor (vortices coloured by axial vorticity): (a) steady contours from RANS; (b) instantaneous contours from URANS
26 D. Chang and S. Tavoularis Fig. 6. Iso-contours of static pressure, normalized by the total inlet pressure, at 50% span: (a) steady contours from RANS; (b) time-averaged contours from URANS Contours of the time-averaged static pressure at 50% span are depicted in Fig. 6. The static pressure in the aft-midsection on the suction side of the vanes from RANS is lower than the corresponding time-averaged pressure from URANS. There is also a sudden increase of static pressure near the trailing edges of the vanes in RANS but not in URANS. All these difference in static pressure are mainly due to the unsteadiness, which is not accounted for in RANS. Fig. 7. Iso-contours of entropy, normalized by the gas constant, at 50% span: (a) steady contours from RANS; (b) time-averaged contours from URANS Figure 7 shows entropy distributions at 50% span for the steady and unsteady simulations. Relatively high entropy indicates high viscous losses and marks boundary layers, wakes and other viscous regions in the flow. While there are almost no differences in the vane wakes within the stator for URANS and RANS, the wakes of the blades are appreciably stronger for the URANS and the losses within the blade passages are also higher in the URANS.
Parallel Computations of Unsteady Three-Dimensional Flows 27 Figure 8 shows isocontours of the total pressure loss coefficient in the rotor reference frame. The relative total pressure loss coefficient at the rotor outlet is defined as ~ P02 P Y r 03r r = ~ ~ (2) P P 03r ~ where the mass-weighted pressure is P = P ρ V da / ρv da and the relative total ~ pressure is P0 r = P0 r ρ V da / ρv da. The total pressure loss from URANS is highest near the midspan of the rotor outlet and decreases towards the endwalls. This high loss may be related to the rotor blade wakes shown in Fig. 8. In contrast, the total pressure losses from RANS show two peaks. The differences in mass-weighted total pressure coefficients are presented in Table 1, which indicates that the total pressure losses from URANS are measurably higher than the ones from RANS. The percent difference of the relative total pressure loss coefficients at the rotor outlet was 10.9%, which is comparable to the value 10% reported by Pullan [10] for rotor-only simulations. 3 Fig. 8. Iso-contours of relative total pressure loss coefficient on the rotor outlet plane(a) steady contours from RANS; (b) time-averaged contours from URANS; view is towards upstream Table 1. Comparison of mass-weighted total pressure loss coefficients Total pressure loss (Y) at stator outlet Relative total pressure loss (Y r ) at rotor outlet RANS 0.6053 0.4477 URANS 0.6257 0.5025 Difference (%) 3.3 10.9
28 D. Chang and S. Tavoularis Finally, Fig. 9 shows the static pressure distributions on the blade suction side, determined from RANS and URANS. The URANS results indicate that a local lowpressure region is formed at mid-span toward the leading edge of blades and is isolated from the low-pressure region formed by the two passage vortices; in the RANS results, the three low-pressure regions are connected. It was reported by Chang and Tavoularis [5] that the local low-pressure regions are highly correlated to the locations of blade passage vortices, tip leakage vortices, and crown vortices. Another visible difference is that the pressure in the forward part of the blade tips is stronger for URANS than one for RANS. Fig. 9. Iso-contours of static pressure normalized by total inlet pressure: (a) steady contours from RANS; (b) time-averaged contours from URANS 4 Conclusions Numerical simulations in a high-pressure turbine have been completed. The present article focuses on comparisons of steady-state results and time-averaged results. The main conclusions are as follows. - The time-averaged static pressure distribution on the suction side of the vanes predicted by URANS was different from the corresponding static pressure distribution in RANS. - Rotor wakes and total pressure loss at the midspan were stronger in the unsteady simulations. - The time-averaged static pressure distribution on the blades predicted by URANS was different from the corresponding static pressure distribution in RANS. In the front parts of the blade tips, the static pressure was higher in the unsteady simulations. By comparison to steady simulations, unsteady simulations have the obvious advantage of resolving the temporal fluctuations of the pressure, velocity and other properties, which can then be used in vibratory stress analyses, an essential aspect of high-pressure turbine design.
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