Journal of Thermal Science Vol.24, No.5 (2015) 442‒451

DOI: 10.1007/s11630-015-0807-x

Article ID: 1003-2169(2015)05-0442-10

Full-Annulus Simulation of the Surge Inception in a Transonic Centrifugal Compressor I. Trébinjac1, E. Benichou1, and N. Buffaz2 1. Laboratoire de Mécanique des Fluides et d’Acoustique, UMR CNRS 5509 Ecole Centrale de Lyon, UCBLyon I, INSA, 36 av. Guy de Collongue, 69134 Ecully Cedex, France 2. Turbomeca, groupe Safran, 64511 Bordes, France © Science Press and Institute of Engineering Thermophysics, CAS and Springer-Verlag Berlin Heidelberg 2015

Full annulus simulations of the flow which develops in a transonic centrifugal compressor are performed at two stable operating points (peak efficiency and near surge) and during the path to surge. At stable conditions, the flow field properties are analyzed by comparisons with experimental data and numerical simulations using a phase lagged approach previously carried out. Regarding the stage overall performance, an excellent agreement is obtained between the numerical results (both with time lagged approach and full-annulus calculation) and the experiments.

From the full-annulus simulations, the change in flow pattern from peak efficiency to surge is found

to be perfectly similar to that obtained from the simulations using the time lagged approach. In particular, provided that the operating point is stable, the flow proves to be chorochronic. The full-annulus simulations were continued after a unique small change in the throttle law applied at the exit of the numerical domain. The mass flow, pressure ratio and efficiency then significantly drop all the more the time progresses. The simulation becomes unstable and the surge inception well underway. The path to surge is found to be due to the enlargement of the boundary layer separation on the suction side of the diffuser vanes in accordance with the conclusions drawn from the chorochronic simulations and experiments. But as the time progresses, the flow loses its chorochronic character. Stall cells rotating at around 7% of the rotor speed develop and lead to surge in around 5 revolutions.

Keywords: centrifugal compressor, surge, transonic, rotating stall, full-annulus simulation, URANS

Introduction In compressors, as the mass flow is reduced, the pressure rise increases up to a given point at which further reduction of mass flow leads more or less rapidly to more or less severe instabilities. The instabilities are typically classified into two main classes: rotating stall and surge which actually are very different phenomena. Rotating stall corresponds to a flow field which is no longer axi-

symmetric but has a circumferentially non-uniform pattern rotating along the annulus which then contains regions of stalled flow. Depending on the number and size of the stalled cells, the compressor is affected by either part-span rotating stall or full-span rotating stall. On the contrary, when fully developed the surge process is axisymmetric and is characterized by a variation with time of the overall annulus mass flow. The surge process may be violent (process known as ‘deep surge’ with reversed

Received: June 2015 I. Trébinjac: Professor www.springerlink.com

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Full-Annulus Simulation of the Surge Inception in a Transonic Centrifugal Compressor

Nomenclature Latin letters D Diameter P Static pressure m Mass flow dt Temporal timestep of the simulations Mabs Absolute Mach number N Blade number Nqo Integer involved in the temporal timestep PE / NS Peak Efficiency / Near Surge r Perfect gas constant T Static temperature averaged mass flow) or mild (process known as ‘mild surge’ with the operating points orbiting around the surge point). Actually the classification into two classes is not strictly tight because most of the time rotating stall evolves into surge. Because instabilities can be catastrophic for performance and even damaging for the machine, predicting the onset of instability is an essential part of the description of the compressor performance map. Many works have been dedicated to the description and analysis of the instabilities and their inception but most of them concern axial compressors [1]. In centrifugal compressors, things are a little bit different: centrifugal compressors may satisfactorily operate with stall in the rotor because the largest part of the pressure rise comes from centrifugal effects by change of radii. Consequently in centrifugal compressors (i) surge is the most likely instability which may occur at low mass flow rate even if it is not exclusive [2, 3, 4] and (ii) the vaned diffuser is the component which most likely triggers the instability at high rotation speed. Predicting the surge onset remains a difficult task because surge depends on the compressor (geometry and operating conditions) and the system in which it is included [5, 6, 7, 8]. Experimental and numerical investigations are complementary, each of them comprising constraints and restrictions. Experiments give only partial information, may be dangerous when moving toward surge, but remain essential for at least validating the simulations. CFD (Computational Fluid Dynamics) offers three main categories of simulations: RANS (Reynolds-Averaged Navier-Stokes), LES (Large Eddy Simulation) and DNS (Direct Numerical Simulation), the former being the only one which is affordable in case of multi-row turbomachinery simulations. It is based on the Reynolds decomposition and turbulence models are added to close the set of equations. The modeling obviously leads to bias which effect on the surge onset is yet unknown. Moreover, due to the rotor-stator interactions, time-de-

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Greek letters Heat capacities ratio  η Stage isentropic efficiency π Pressure ratio Ω Rotation speed ω Relaxation coefficient used in the CFD valve law Subscripts 1 Relative to stage inlet (impeller inlet) 3 Relative to stage outlet (diffuser exit) R/S Relative to Rotor / Stator t Stagnation value pendent calculations are required. Assuming that the unsteady effects are only due to the rotation, a phase lagged approach may be used [9, 10]. But near the surge line unsteady phenomena uncorrelated with the blade passing frequency may occur. Therefore, to capture all of the spatial and temporal contents, the calculation domain has to be extended to all the blade passages (full-annulus simulations). The present paper gives the results of full-annulus simulations of the flow in a transonic centrifugal compressor stage. In a first part, the test case is introduced and the numerical procedure is described. Then the results are given at stable operating points and discussed comparing with experiments and numerical results obtained with a phase-lagged approach. Finally the transient behavior during the path to surge is described.

Test case The test case is a centrifugal compressor stage, used in a helicopter engine, designed and built by Turbomeca. It is composed of an impeller with axial inflow, a radial vaned diffuser and an axial diffuser as in the real geometry which is therefore axisymmetric. The backswept, unshrouded impeller is composed of NR=11 main blades and NR=11 splitter blades. The radial vaned diffuser is composed of NS=19 vanes. A meridional view of the stage is given in figure 1.

Fig. 1

Meridional view of the compressor.

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The compressor stage was mounted on a 1MW test rig equipped with 67 steady sensors (temperature, pressure, and vibration measurements) dedicated to monitoring and overall performance measuring. The rotation speed, the flow rate, the pressure and the temperature were measured at ±0.01%, ±0.5%, ±0.05% and ±1K respectively. Unsteady pressure measurements up to 150 KHz were carried out in the inducer of the impeller (between the leading edge of the main blade and the leading edge of the splitter blade), in the vaneless diffuser and in the vaned radial diffuser. The internal flow was also investigated by Laser Doppler Anemometry. Experimental results can be found in [21, 22, 23]. Numerical procedure Computations were performed with the elsA software developed at ONERA [12]. The code is based on a cellcentered finite volume method and solves the compressible RANS equations on multiblock structured meshes. The set of equations are resolved in the relative frame of each row. Figure 2 gives a meridional sketch of the computation domain which is composed of the inlet domain, the impeller, the radial vaned diffuser and the exit domain.

Fig. 3 Initial mesh blocks (a) and zoom in at the impeller LE (b)

Fig. 2 Meridional sketch of the computation domain

Mesh configuration The initial mesh is based on an H-C-O topology in the impeller and an H-O topology in the diffuser (figure 3). The tip gap is modeled with an H-O topology. This initial mesh was then split and centered on the channels instead of on the blades, in order to save memory (harmonics storage at the periodic boundaries in chorochronic simulations) and to make flow visualization easier (figure 4). The impeller and diffuser meshes include respectively 3.7M and 1.1M points per passage, which leads to 70M cells in the full-annulus simulation. The impeller gap region is meshed with 29 points in the spanwise direction. The gap size is smaller than 0.5% span height at the impeller leading edge and around 10% span height at the impeller trailing edge. Turbulence modeling The k-l model of Smith [13] (chosen according to previous work [14]) is used. The turbulent inlet values are determined from a free-stream turbulence rate of 5% resulting from hot wire measurements in the test rig.

Fig. 4 Impeller - diffuser interface in the final mesh

Spatial scheme The Jameson second-order centered spatial scheme is used. A numerical viscosity corrected by Martinelli's method [15] is added to stabilize the computation and capture the possible discontinuities such as shock waves. Temporal integration In order to reconstruct more easily a whole blade passing period in both frames, the time step is chosen as follows, with Nqo a natural integer: 2 (1) dt  N R N S Nqo In chorochronic simulations, a scalar lower-upper symmetric successive over-relaxation method (LUSSOR) proposed by Yoon and Jameson [16] is associated to the first-order backward – Euler scheme. A time step convergence study led to choose the parameter Nqo equal to 200. However, previous works [17, 18] proved that a second-order time integration scheme enables to obtain a better convergence with a parameter Nqo up to ten times

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Full-Annulus Simulation of the Surge Inception in a Transonic Centrifugal Compressor

smaller. Thus, chorochronic simulations were also performed with the second-order Gear scheme using a value of Nqo equal to 20 (determined also from time step convergence tests). For each physical time step, eight subiterations are performed in the inner loop, which results in a decrease in residuals of two orders of magnitude. These simulations led exactly to the same results than using the backward – Euler scheme, which validated the change in time integration. Consequently, in order to make the full-annulus simulations more CPU-efficient, the second-order scheme was kept. Boundary conditions The inlet condition is composed of the velocity angles, the standard stagnation pressure and temperature. The outlet condition is composed of a uniform value of static pressure, calculated with a linear throttle law. Indeed, when the operating point moves toward surge, the slope of the stage pressure ratio characteristic can be close to zero or even positive. Thus, a fixed static pressure exit condition is not adapted anymore. Instead, the following is prescribed at the outlet:   m (n)  P ( n)  P(n  1)  P(n)    Pref (2)   m ref   where ω is a relaxation coefficient, Pref and m ref respectively static pressure and mass flow reference values, and n, n+1 two consecutive time steps. This expression can be written as:  m (n) / m ref  P(n  1) (3)  1    1   P ( n)  P(n) / Pref  so that in the end the throttle law is controlled by the parameter m /  ts stage . The simulated operating point can move from choked point to surge by simply decreasing the value of this ratio. The walls are described with non-slip and adiabatic conditions. In chorochronic simulations, the phase-lagged approach is used. Assuming that the flow is periodic in time and space, the flow in a blade passage can be deduced from the neighboring passage shifted in time. This enables to compute only a single blade passage per row. At the circumferential periodic boundaries, Fourier series are used to store the temporal signals, keeping a number of harmonics equal to four times the opposite row blade number. In the full-annulus simulations, the impeller – diffuser interface is completely described over 360° with a sliding mesh technique and the impeller is rotating at each time step. Parallelization of the full-annulus simulation The mesh of the full-annulus simulation is the same as in chorochronic simulations but spatially replicated ac-

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cording the stator and rotor rows periodicities. Since the whole domain is taken into account, no flow periodicity hypothesis is necessary. But interpolations are realized through the sliding interface in order to transmit information from one frame to the other, using a distribution of fluxes and thus ensuring conservativity [19, 20]. This interpolation step is done at the end of each time step computation and concerns only the interface blocks. For this reason and because of the size of the mesh, a special effort has been made on the parallelization quality. After a CPU speed-up study involving different configurations, 512 scalar processors were used. The computational domain includes 4096 mesh blocks, 504 of them being part of the sliding mesh interface and running on 504 distinct processors. Table 1

Numerical parameters used in the simulations Chorochronic

Full-annulus

Mesh size

5 million cells

70 million cells

Turbulence model

k – l Smith

Spatial scheme

2nd-order Jameson

k – l Smith 2nd-order Jameson

Temporal integration Nqo Impeller – Diffuser Interface CPU time consumption (PE / NS)

1st-order Backward Euler 200 Chorochronic interface 900 / 1400 h (vectorial)

2nd-order Gear 20 Sliding mesh 400.000 / 1.100.000 h (scalar)

Table 1 summarizes the parameters for both types of simulation. Since chorochronic simulations have been run on vectorial processors, no proper comparison can be done concerning the CPU time consumption.

Overall performances Comparisons between experiments, steady simulations and phase-lagged simulations In order to properly compare experimental and numerical results, the raw data have to be post-treated in a strictly similar way. Due to the fact that some experimental data are missing, assumptions have to be done. As an example, the mean value of the static pressure at a given section is calculated assuming a linear evolution of the pressures measured on the shroud and hub wall surfaces. Hence, using a similar data post-processing, the ts , as a function of the reduced mass pressure ratio,  stage flow coming from the experiments (squares) the steady (triangles) and the phase-lagged (circles) simulations is given in figure 5. The reduced mass flow is defined as:

m red 

m  rTt1 D12S pt1

(4)

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Fig. 5 Pressure ratio of the compressor stage (square: experiments, circle: chorochronic simulations, triangles: steady simulations).

The agreement between the experiments and the chorochronic results is very good contrary to the steady simulations which results greatly overestimate the operating range. In the steady approach the flow communication between the two blade rows uses a mixing plane method consisting in applying a circumferential average to the conservative flow quantities but preserving radial variations. This method prevents the diffuser bow shock wave to extend upstream the inter-row plane and proved to be the origin of the shift in overall performance [21]. The good agreement between the unsteady numerical results using a phase lagged approach and the experiments is also obtained concerning the internal flow pattern which was investigated by Laser Doppler Anemometry and fast pressure sensors [21, 22, 23]. Hence, the chorochronic simulations may serve to evaluate the full-annulus simulations instead of using the experiments: it thus avoids doing assumptions during the data postprocessing. Comparisons between full-annulus simulations and phase-lagged simulations As stated above, in order to avoid doing assumptions during the data post-processing due to the lack of some experimental data, the full-annulus simulation is compared with only the chorochronic simulation. The mean numerical values hence come from time and area-averaged values. Two full annulus simulations were performed, at peak efficiency (PE) and near surge (NS). The stage pressure ratio and efficiency calculated at different extracted time steps lead to the same values as shown by the superimposition of the green squares in figure 6. For both operating points, the overall performances are very similar to those obtained with the phase-lagged approach. Figure 7 gives the time-dependent pressure signals and enables to check the chorochronic behavior of the flow at

Fig. 6 Pressure ratio (top) and isentropic total-to-total stage efficiency (bottom) from full annulus (green squares) and chorochronic (black dots) simulations

stable operating conditions. The full-annulus result is given in green at PE and NS, at 95% section height at the rotor leading edge (a), at 50% section height at mid-vaneless diffuser (b) and at 50% section height at the vaned diffuser exit (c). NR*NS instants have been extracted over 360°, then placed over an angular sector (four rotor blade passages for (a) and (b), 60% of the annulus for (c)), thanks to the adequate spatial and temporal shift. The signal given by the phase-lagged calculation is superimposed in black. Other drawings have been done for different extractions at different revolutions; they always lead to the same results to wit: -at the rotor leading edge and at mid-vaneless diffuser the full-annulus calculation gives the same results as the chorochronic calculation. The differences never exceed the differences due to a change in temporal scheme for a given simulation. -at the diffuser exit, the disparity between both calculations increases. The differences come from the interactions of the waves resulting from the reflection of the vane leading edge shock wave on the rotor blade pressure surface. These waves go through the diffuser channels (cf. figure 8) and interact at the exit. This interaction is badly predicted by the chorochronic calculation. Due to the strengthening of the shock as the mass flow is reduced, the intensity of the waves increases, which explains that the differences between the full annulus calculation and the phase lagged approach are higher at NS than at PE.

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Full-Annulus Simulation of the Surge Inception in a Transonic Centrifugal Compressor

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Fig. 7 Pressure signals from full annulus (green) and chorochronic (black) calculations at the rotor leading edge (a),mid-vaneless diffuser (b) and diffuser exit (c).

Figure 8 shows the absolute Mach number at 90% section height in the diffuser at an arbitrary time step, at PE and NS, from the full-annulus simulations. The above mentioned waves are visible by the blue spots clearly marked on the vane suction side at NS. The pink curves show the zones of reverse flow. As in chorochronic calculations [22, 23], it may be seen that from PE to NS the flow separation moves from the pressure side to the suction side of the vanes. From this stable operating condition, let us now look at the path to surge predicted by the full-annulus simulation. The path to surge From the last stable operating point, a unique throttle

parameter change is imposed at the exit boundary and the full-annulus simulation is let free to progress during 5.5 rotor revolutions. The overall performances during that transient operating condition are given in figure 9. The instantaneous mass flow is taken upstream the impeller, and the stage pressure ratio and efficiency are calculated at each extracted time step, as previous. From the last stable operating point (identified by t=t0) almost two rotations are necessary for the simulation to be compatible with the prescribed throttle condition. Then the mass flow, pressure ratio and efficiency dramatically decrease and all the more rapidly as the time progresses. Figure 10 shows the absolute Mach number at 50%

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paration spots clearly enlarge. Between timestep (b) (t0+2.65 rotations) and timestep (f) (t0+5.52 rotations) six rotating stall cells develop over the entire span. They rotate in the same direction as the rotor. As an example, at timestep (e), the cells completely obstruct the vane passages 1, 4, 7, 10, 14, 17. At timestep (f), the passages 2, 5, 8, 11, 14-15, 18 are blocked. The cells therefore rotate at around 7% of the impeller rotation speed. The origin of the cells (effect of over-incidence) and their rotation speed (ΩR) is compatible with the observations given in [24, 25]. However in the present case, the cells enlarge very rapidly and trigger the surge in only a few rotations. The stall cells lead to a deviation of the flow in the adjacent vane channel on the right hand side. As an example at time step (f) the passage 2 which was observed to be blocked is associated with a negative radial velocity whereas the passage 3 is characterized by a high positive value (figure 11 – left showing the radial velocity at

Fig. 8 Full-annulus simulations. Absolute Mach number in the diffuser at 90%section height.

Fig. 9 Full-annulus simulations. Overall performances during surge inception.

section height at six time steps from t0. The white contours indicate the reverse flow zones. At timestep (a) (t0+1.93 rotations) the flow field is very similar to the stable field (figure 8-NS), then the vane suction side se-

Fig. 10

Instantaneous Mach number during surge inception.

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Full-Annulus Simulation of the Surge Inception in a Transonic Centrifugal Compressor

timestep (f)). The fluid flow is highlighted in figure 12 which shows the absolute velocity vectors (black arrows) and the sketch of the flow topology (white arrows). The vane channel 2 is blocked yielding the fluid to flow towards the vane channel 3. The vane channel 1 keeps operating normally and exhibits a pressure increase (figure 11– right showing the static pressure at time step (f)) as normally required in a diffuser. The flow structure is therefore organized in a set of three successive vane channels which leads to the development of 6 cells as observed in figures 10 and 11.

Fig. 11

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 2 2  (5) ,t  f  , t   f     NS R NS   Hence the deterministic field may be expanded into Fourier series as originally proposed by Tyler and Sofrin [27]. The flow field is thus described as a superposition of circumferential spinning lobed structures which number of lobes m is a linear combination of the number of blades of the rotor NR and the stator NS: (6) m = n.NR+k. NS The angular rotation speed of any spatial mode m is given by: n NR m   (7) n  NR  k  NS Two main characteristic directions are clearly visible in figures 13 and 14: m=0, thus m =0 associated to the diffuser vanes m = 2NR + 2Ns=60 thus m = 0.37, associated to the interaction mode between the rotor blades (main blades and splitter blades) and twice the diffuser vane number.

Radial velocity (left) and static pressure (right) at timestep (f)

Fig. 12

Fig. 13

Space-time diagram of radial velocity during surge inception

Fig. 14

Space-time diagram of static pressure during surge inception

Absolute velocity vectors at time step (f).

Another representation of the formation of these 6 stall cells is given with the radial velocity (figure 13) and the static pressure (figure 14) plotted in a space-time diagram at the impeller-diffuser interface at mid-span. The abscissa represents 2 and the ordinate gives the time from t0 to the time step (f). The time steps (a) to (f) are marked with the horizontal dashed black lines. The black and white contours in figure 13 respectively stand for the absolute Mach number Mabs = 1 and the radial velocity Vr = 0. From t0 to the time step (b) the flow field clearly exhibits a spatial-temporal periodicity which may be written as:

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From the time step (d), the alternations previously described (figures 10 and 11) between blocked channels (low/negative radial velocity) and high massflow channels set in. Consequently, the apparition of new spatial modes makes sense with the new structure of the flow field. Figure 15 shows the temporal evolution of the spatial modes of the pressure at the impeller – diffuser interface. At t0, the main spatial harmonics are directly linked to the diffuser vane number and its multiples (19, 38, 57, 76, 95). As the time progresses from time step (a) to time step (f), the amplitude of the mode 6 dramatically increases. It is thought that this number of stall cells comes from the blade numbers of the compressor. Indeed 6 is approximately the greater common divisor of (11, 19). As moving towards surge, the modes 13 and 16 coming from the interactions between stall cells and blades (13 = 19 - 6, 16 = 22 - 6) also increase. This change in spatial modes repartition is highlighted in figure 16

which shows the temporal evolution of the dominant harmonics associated to the rotor blade number (22), the stator vane number (19) and the interactions (6, 13, 16).

Conclusions The analysis of full-annulus simulations performed at stable operating points (peak efficiency and near surge) and during the path to surge of a high-pressure centrifugal compressor stage led to the following conclusions: - Provided that the operating point is stable the flow field proves to verify the spatial-temporal periodicity properly. - The major change in the stable flow pattern when moving from peak efficiency towards near surge is the transfer of the diffuser vane boundary layer separation from pressure side to suction side. - During the path to surge the flow field loses its chorochronic character. - In the transient regime the diffuser flow gets organized into blocked passages (low/negative radial velocity) and high momentum flow channels leading to the formation of a pattern with six modes rotating at around 7% of the impeller rotation speed. - The spectrum of the flow field at the ultimate time step before the numerical surge clearly shows dominant spatial modes associated to the stall cells and their interactions with the blade harmonics. This work proves that full-annulus simulations can bring valuable information especially close to the surge line where instabilities occur. In the present case, they confirm that the diffuser is involved in the surge inception and open some perspectives concerning flow control.

Acknowledgements Fig. 15 Temporal evolution of the spatial modes of the pressure at the impeller – diffuser interface

We would like to thank TURBOMECA which supported this study. This work was granted access to the HPC resources of CINES under the allocation 20122a6356 and 2013-2a6356.

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Fig. 16

Temporal evolution of dominant spatial modes during surge inception

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