Journal of Thermal Science Vol.16, No.3

193―202

DOI: 10.1007/s11630-007-0193-0

Article ID: 1003-2169(2007)03-0193-10

Impeller-diffuser interaction: analysis of the unsteady flow structures based on their direction of propagation N. Bulot and I. Trébinjac Ecole Centrale de Lyon, Laboratoire de Mécanique des Fluides et d'Acoustique ECL / UCBL / INSA / UMR CNRS 5509 – 69134 Ecully cedex, France

The present study is focused on the analysis of the deterministic fluctuations arising from the rotor-stator interaction within a transonic centrifugal compressor stage. A spectral analysis applied to the unsteady flow field leads to the values of the rotation speed of most energetic modes. From these values, the various structures are classified according to their direction of propagation which leads to a comprehensive description of the underlying mechanisms involved in the interaction.

Keywords: centrifugal compressor, transonic flow, rotor-stator interaction, unsteady flow

Introduction The unsteady behaviour of the flow field in any turbomachine arises from both random and deterministic fluctuations with various length scales. Averaging procedures can be used to segregate the different sources of unsteadiness. Firstly random phenomena in the time- dependent field (including turbulence) can be filtered out using an ensemble-averaging operator. Then the deterministic fluctuations can be separated into those that are correlated with the rotation of the machine and those that are not. In stable operating conditions of the machine (which excludes rotating stall for instance), blade row interaction generates deterministic fluctuations which are correlated with the rotation speed. The present study is focused on the analysis of that source of unsteadiness resulting from the interaction between the distorted flow at the exit of a centrifugal impeller and the potential perturbation induced by the vaned diffuser of a transonic compressor stage. Blade row interaction is a result of individual blades encountering the flow non-uniformities produced by other blade rows. Actually, blade row interaction includes potential interaction, vortical interaction

Received: May 2007 I. Trebinjac: Associate Professor

(associated with wakes, tip leakage and passage vortices) and shock-wave interaction. A good description of the various blade row interactions in low-speed low Mach number compressors is given by Tan [1]. In contrast, there have been far fewer publications which give results of blade row interaction within high Mach number high loaded turbomachines, especially in centrifugal configurations. Beside the contribution of the blade row interaction to the overall performance (contribution which may be positive or negative [2]), the interaction may lead to instabilities when the operating point approaches the surge line. This last point motivated the present analysis in order to reach a comprehensive description of the underlying mechanisms involved in the interaction at a stable operating point. The identification of the mechanisms and their quantification in an energetic point of view, should lead to evaluate their role in the instability inception. In the first part of the paper, the test case is briefly related. Then, the unsteady computation of the complete stage using a 3D Navier-Stokes code with a phase-lagged technique is described. The second part presents the data reduction process applied to the numerical results in the interaction flow region. Finally, the mechanisms involved

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Nomenclature Latin letters A,B convected structures

Greek letters

-1

α,β

propagated structures

c

speed of sound (m.s )

σ

strain tensor (Pa)

f

arbitrary flow variable

τ

shear stress tensor (Pa)

k,n

integers

ρ

density (kg.m-3)

m

spatial mode

Ω

rotation speed (rad.s-1)

M

Mach number

MB / SB

impeller main / splitter blades

co

convected structure variable

DV

diffuser vanes

m

spatial mode variable

MCC

metal coincidence criterion

t

turbulent field

NR

number of impeller blades

pole

metal coincidence pole variable

NS

number of diffuser vanes

pro

propagative structures variable

p

static pressure (Pa)

s

curvilinear variable

SL

sonic line

R,S

impeller / diffuser variable

t

time (s)

ref

variable of reference

T

time period (s)

Subscripts

Superscripts -1

U

circumferential speed of impeller (m.s )

+

direction of progressive propagation

Vr

-1

radial velocity (m.s )

–

direction of regressive propagation

Vt

tangential velocity (m.s-1)

*

purely unsteady interaction field

-1

W

relative velocity (m.s )

axi

axisymmetric field

(r , θ , z)

cylindrical coordinates (m , rad , m)

R

averaged field in impeller frame

S

averaged field in diffuser frame

e

-1

internal energy (J.kg )

in the interaction are described.

Test case The test case is a single-stage centrifugal compressor designed and built by Turbomeca. It is composed of a backswept splittered unshrouded impeller (composed of 2NR blades) and a vaned diffuser (composed of NS vanes). The results presented in this paper were obtained at an off-design operating point (ΩR=0.927ΩR-nominal) for which the overall pressure ratio is 6.75 and the specific values of the mass flow and rotation speed are respectively 0.38 and 0.30. A 3D sketch of the stage is given in Fig.1. The geometry has been deformed for reasons of confidentiality.

Numerical simulation Unsteady 3D simulations were performed with the elsA software (Cambier and Gazaix [3]), developed by

Fig. 1

3D sketch of the centrifugal compressor.

ONERA and focused on compressible fluid dynamics simulation. The simulation of 3D turbulent viscous flows solves the Reynolds-Averaged Navier-Stokes equations, associated with one of the turbulence models available. The results shown in this paper come from simulations performed using the Smith k-l model (Smith [4]) which leads to the closest results compared with the experi-

N. Bulot et al. Impeller-diffuser interaction: analysis of the unsteady flow structures based on their direction of propagation

mental results.

195

The turbulent viscosity is evaluated using the following relation μt = μχf μ , where:

Governing equations For turbomachinery applications, the RANS equations are written in the rotating frame of reference of each row: ∂ρ + ∇ ⋅ ρW = 0 (1) ∂t

( ) ∂ρW + ∇ ⋅ (ρW ⊗ W + σ − τ ) = ρ (Ω r − 2Ω ∧ W ) ∂t ∂ρe + ∇ ⋅ (ρeW + (σ − τ )⋅ W + q + q ) = ρΩ r ⋅ W ∂t t

t

2 R

R

(2)

t

2 R

(3)

where σ = p I − τ . The closure of the previous system requires the specification of the Reynolds tensor τ t and the turbulent heat flux qt . By analogy with the

Newtonian description of viscous stresses in a laminar flow, the Reynolds stress tensor is assumed to be proportional to the rate-of-strain tensor: 2 τ t = 2μt D − ⎛⎜ μ t ∇ ⋅ W + ρk ⎞⎟ I 3⎝

(4)

⎠

The first term constitutes the Boussinesq hypothesis, where μt is the turbulent eddy viscosity, which has to be modeled. The last term provides the proper trace of the Reynolds stress tensor. k is the turbulent kinetic energy. Similarly, the Boussinesq model for the turbulent heat flux is qt = − K t ∇T , in analogy with the laminar heat flux, where Kt is the turbulent thermal conductivity defined by K t = c p μt / Prt and Prt is the turbulent Prandtl number.

χ=

ρ 2kl

(7)

μB11/3 1/ 4

⎛ c 4 f + c 2 χ2 + χ4 ⎞ (8) fμ = ⎜ 1 4 1 42 2 ⎜ c + c χ + χ 4 ⎟⎟ 2 ⎝ 1 ⎠ The damping function f1 and constants are defined as: f1 =

2 ⎛ ⎛ l ⎞ ⎞ ⎜ −50⎜ ⎟ ⎟ ⎜ ⎝ kd ⎠ ⎟⎠ e⎝

(9) {c1 = 25.5 ,c2 = 2 ,σ l = σ k = 1.43 , B1 = 18 , E2 = 1.2} (10) This model requires the specification of each mesh node distance to the wall, d. Stage mesh A multi-domain approach on structured meshes is used, H, C and O topologies are employed (Fig. 2). The near-wall region around the blades is described by an O-block, to allow a precise description of the viscous effects and to capture the turbulent gradients. C-blocks are used to connect O-blocks of the main blade and the splitter. Upstream, downstream and interface regions are defined with classical H-blocks. The tip clearance is meshed with an O-H topology to provide a good connexion with the O-block around the blades.

With the assumption of a constant Prandtl number, the closure of the system is reduced to the determination of μt and k. This is the goal of any turbulence model based on the Boussinesq hypothesis. Smith k-l turbulence model This two-equations model was proposed by Smith [4], in which the transported quantities are the turbulent kinetic energy k and a turbulence characteristic length scale, l: uur ur ⎛ ⎛ μ ⎞ ur ⎞ ∂ρ k ur + ∇ ⋅ ρ kW − ∇ ⎜⎜ ⎜ μ + t ⎟ ∇k ⎟⎟ = ∂t σk ⎠ ⎠ ⎝⎝ uur ρ (2k )3/ 2 ur =τ t : ∇W − − 2μ ∇ k B1l

(

Fig. 2 H-C-O Topology of the mesh.

)

(5) 2

uur ur ⎛ ⎛ μ ⎞ ur ⎞ ∂ρ l ur + ∇ ⋅ ρ lW − ∇ ⎜ ⎜ μ + t ⎟ ∇l ⎟ = ⎜ ∂t σ l ⎠ ⎟⎠ ⎝⎝

(

)

⎞ ur uur ⎟ + ρ l∇ ⋅W ⎟ B1 ⎠ 2 ur ur ur μ μ ⎛ l ⎞ + 2 t ∇l ⋅∇k − t ⎜ ⎟ ∇l σlk σ l l ⎝ kd ⎠ = ( 2 − E2 )

ρ 2k ⎛

⎛ l ⎞ ⎜1 − ⎜ ⎟ ⎜ ⎝ kd ⎠ ⎝

2

(6)

The global stage mesh is composed of 15 blocks with more than 1 700 000 nodes. Numerical scheme The equations are discretized in the relative rotating frame using the Jameson’s centered space scheme in a “cell-centered” approach. In order to stabilize the scheme, 2nd order and 4th order dissipative terms are added at the 4 steps of the Runge-Kutta time-integration scheme. An implicit residual smoothing technique is added to this explicit scheme. For the impeller-diffuser interaction computation, the phase lagged approach is used. In this approach, the

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computation domain is limited to a single blade passage for each row. At a stable operating point and assuming uniform inlet conditions, the unsteady effects are only due to the impeller-diffuser interaction. Then, the flow is time-periodic in the frame of reference of the rows TS = 2π / ΩR NR being the period in the diffuser frame and TR = 2π / ΩR NS being the period in the impeller frame (NS and NR are respectively the number of stator and rotor blades). As a consequence of the time-periodicity in each frame, a phase-lag exists between two adjacent blade passages. For each row, this phase-lag is the time taken by a blade of the next row to cover the pitch of the row, modulo the time-period of the row. Basically, the phase-lagged technique consists in storing the flow values on the periodic boundaries and on the impeller-diffuser interface boundaries in order to deal with the phase-lag existing between adjacent blade passages. In order to manage the stored data, the constant time-step used by the solver, Δt is conveniently defined such as TS = NS Nq Δt and TR = NR Nq Δt, with Nq an integer intended to satisfy the CFL stability criterion. When NS and NR are prime numbers, the number of iterations to describe a thorough revolution of the impeller is then equal to NS NR Nq. As the direct storage may lead to very large requirements in terms of memory for 3D applications, the data storage can be lowered to a reasonable amount by a Fourier harmonic decomposition. Periodic unsteady signal is defined by its first highest harmonics. On each row periodic boundaries, the boundary condition based on the characteristic relations is similar to a classical continuity treatment. In practice, the number of stored harmonics is greater or equal to the blade number. The stored values of the harmonics are actualized at each time-step in the boundary condition process and these harmonics should remain constant at computation convergence. The unsteady computation of the stage flow field required at least 5 rotating wheel revolutions to get acceptable convergence. It led to more than 250 hours CPU on NEC-SX6 ONERA computer.

Data post-processing The unsteady RANS simulations using the phase lagged approach lead to the knowledge of the time-dependent data over a stator pitch. It is possible to reconstruct the field over a complete turn of the machine by using the spatial-temporal periodicity property of the flow field: ⎛ 2π 2π ,t + f (θ , t ) = f ⎜⎜θ + Ω N S R NS ⎝

⎞ ⎟ ⎟ ⎠

(11)

The generated flow field which is unsteady but deter-

ministic may be decomposed as proposed by Adamczyk [5] [6]. At given (r,z) coordinates, this decomposition is written in the absolute frame of reference as: f ( θ, t ) = { f axi + f S ( θ ) + f R ( θ − Ω R t ) + f * ( θ, t ) (12) 123 14 4244 3 1 424 3 (1)

(2)

(3)

(4)

The first term is the axisymmetric field. Terms (2) and (3) represent the spatial fluctuations of the time-averaged field in the frame of reference of the vaned diffuser and impeller respectively. Finally the term (4) represents the purely unsteady part of the field, which is time-dependent whatever the frame of reference. An equivalent decomposition, based on Fourier transform, was proposed by Tyler and Sofrin [7]. The main result of this decomposition is the identification of the spatial harmonics m, which may be classified into three sets. The first set is composed of the m = kNS harmonics which represent the time-averaged flow in the absolute frame. The second set includes the m = nNR harmonics involved in the time-averaged flow field in the relative frame of reference. The third set includes the m = nNR+kNS harmonics (with n, k ≠ 0) involved in the impeller diffuser interaction. The reconstructed signal from each set corresponds to the terms (2), (3) and (4) of Eq. 2 respectively. The angular rotation speed of the mth spatial mode is given by: nN R Ωm = ΩR (13) nN R + kN S In this paper, we show the results of the post- processing applied to the numerical data extracted at mid gap between the impeller trailing edge and the diffuser leading edge. Results are given in Fig. 3 regarding the reduced static pressure 1 at 50% section height. Fig. 3-a gives a blade to blade pressure map at a given time t, in which the sonic line is superimposed with a white line. The data extracted at mid gap (black line) at the given time t over the stator pitch are located on the white dotted line in Fig. 3-b. By repeating the extraction at the various time steps included in the rotor time period, the time-dependent pressure field may be plotted in a time / space plot (Fig. 3-b) over a stator pitch (abscissa) and a rotor time period (ordinate). In this figure, the inclined lines represent the wake blade locations, whereas the two vertical lines represent the trace of the time-averaged sonic line. Figs. 3-c, 3-d, 3-e, represent the Adamczyk fluctuation terms, i.e. the terms (2), (3) and (4) of Eq. 2 respectively. Due to the fact that the three fluctuation fields are orthogonal to each other, their cross products are null. Consequently, the sum of the three fluctuation variances 1

2 The reduced static pressure is defined as : p /(ρref U ref / 2)

N. Bulot et al. Impeller-diffuser interaction: analysis of the unsteady flow structures based on their direction of propagation

197

[

Fig. 3 Extraction and post-processing procedure applied to the fluctuating pressure field at 50% blade height.

is equal to the total variance.

(f

S

+ fR + f*

Spectral analysis

) = ( f ) +( f ) +( f ) 2

S 2

(

S

R 2

R

S

* 2

*

R

*

)

+2 f f + f f + f f 14444244443

(14)

=0

The pressure varies slightly in the impeller frame of reference and contributes only to 1.4% of the total pressure variance. The diffuser pressure fluctuations contribute to 87.7% of the total variance, which is explained by the presence of the shock wave. Finally, 10.9% of the pressure variance is contained in the purely unsteady field. This value is high enough not to be neglected, as it is often done (at least partially) when solving the “Averaged-Passage Equation System” developed by Adamczyk [5] [6]. In order to extract the dominant mth harmonics involved in the impeller-diffuser interaction, a discrete Fourier transform is thus applied to the purely unsteady field.

Fig. 4 shows the spectrum of the spatial Fourier transform applied to the purely unsteady pressure field at a given time t. The powerless harmonics (lower than 5% of the most powerful harmonic) are concealed in the gray domain for reasons of confidentiality. The abscissa gives the harmonic order and the ordinate is the amplitude. Amongst all the harmonics, the six highest modes are extracted and listed in Table 1 which also gives their dimensionless rotation speed defined as : Ωm/ΩR = nNR/m (with m = kNS+nNR). In order to understand the values of the mode rotation speed, let us first quantify the averaged rotation speed values of convection structures and progressive propagation waves. In a first approach, they may be estimated as Ωpro+/ΩR=(Vtaxi+caxi)/Uref≈1.45 and Ωco/ΩR=Vtaxi/Uref ≈ 0.74 respectively.

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Journal of Thermal Science, Vol.16, No.3, 2007

⎞ ⎛ kπ − 2kπb MCC k = ⎜⎜ t k = ,θ k = [2π ] ⎟⎟ NR NSΩR NS ⎠ ⎝ (15) Here, b is an integer obtained with the Bezout’s identity 1 . In the present case, as ∆θk,k+1 ≈2 π/3, three rotating coincidence poles 2 may be identified (Fig. 5) which rotation speed is defined as: Ω pole ΩR

= =

Fig. 4

Spatial spectrum of the purely unsteady pressure field at given time step.

Table 1 Dominant harmonics involved in the purely unsteady pressure field. mode

m

Ωm/ΩR

(1)

4NR-NS

1.76

(2)

6NR-NS

1.40

(3)

2(4NR-NS)

1.76

(4)

4NR+2NS

0.61

(5)

3(4NR-NS)

1.76

(6)

2NR-NS

7.33

Fig. 5 Sketch of the three rotating metal coincidence poles.

Thus, it may first be concluded that the mode (4NR+2NS) is convected whereas the modes (4NR–NS) and its multiples and (6NR–NS) are propagation structures. As the mode (2NR–NS) has a rotation speed equal to 7.33ΩR, it cannot be transported by any flow phenomenon. The other effect which may lead to rotating waves is due to the geometrical configuration between impeller and diffuser blades. To qualify this geometrical configuration, let us define a simple metal coincidence criterion noted MCCk. If (tk,θk) are the time and space coordinates of the kth metal coincidence, such as(tk,θk)|k=0 = (0,0), MCCk is calculated as:

1 Δθ k ,k +3 [2π ]

ΩR

Δt k ,k +3

⎞⎛ N N Ω 1 ⎛ − 6 πb ⎜ − 2π ⎟⎟⎜⎜ R S R ⎜ 3π ΩR ⎝ NS ⎠⎝

=−

⎞ ⎟⎟ ⎠

(16)

2 N R (3b + N S ) ≈ 7.33 3

Thus, the rotation speed of mode (2NR-NS) is linked with the clocking between the impeller and diffuser blades.

Physical analysis Characteristic directions Once the speed of the rotating modes has been determined using spectral analysis, the aerodynamic structures can be classified according to the type of transport involved (convection, acoustic, ...). In the physical plane, it leads us to analyze the phenomena regarding the associated characteristic flow directions which are: (1) θ = cst : characteristic direction of steady structures, thus attached to the diffuser (2) θ – ΩRt = cst : direction of structures rotating at the impeller speed (3) θ – VtaxiΩRt/U = cst : direction of convected structures at the local flow velocity (in the inter row gap, in a first approximation the only tangential velocity is taken into account) (4) θ – (Vtaxi ± caxi)ΩRt/U = cst : these two directions correspond to progressive or regressive acoustic waves. Whereas the direction of the acoustic wave propagation is clearly visible in a pressure field, the direction of the structures transported by convection is less visible. Thus, in order to complete the description, maps of velocity components are added. Figures 6, 7 and 8 show the purely unsteady part (term (4) of Eq. (2)) of the pressure, radial and tangential velocity components 3 , at 50% section height, at mid inter row gap. Each variable is given in a reduced form 4 . In order to make easier the identification of the various directions, all the maps are plotted 1 ''if 2NR and NS have no common denominators, there are two intergers a and b such that NSa + 2NRb =1'' 2 Local space-time domain near a metal coincidence 3 In the inter-row gap, the axial velocity component is negligible. 4 Reduced radial and tangential velocity are defined as : Vr/Uref and Vt/Uref respectively.

N. Bulot et al. Impeller-diffuser interaction: analysis of the unsteady flow structures based on their direction of propagation

over one and half time period (ordinate axis) and two and half diffuser blade pitch (abscissa axis). The trace of the rotor blade wakes are shown with oblique dotted black lines. The location of the diffuser vane leading edge is marked out with vertical black lines. The areas delimited with white contours correspond to the subsonic domain around the diffuser vane leading edge. Acoustic wave propagation is highlighted in the pressure fluctuation map (Fig. 6). The wave named α+ is particularly observable (and much more observable than the rotor blade wake trace). In a slightly way, one can identify waves named α− and β−. Superscripts + and – respectively indicate the progressive and regressive nature of the pressure wave. The α+ wave is generated in the vicinity of a metal coincidence, at the same time as the α− wave. After having travelled over one diffuser pitch at a mean rotational speed of 1.7ΩR , the α+ wave interacts the adjacent diffuser blade leading edge, which results in a β− wave. It may be noted that the α+ wave creates temporarily a supersonic passage through the subsonic domain. The α+ wave exists up to two diffuser blade pitches after its generation. According to their rotational speed, α− and β− waves are regressive waves. They are located in the subsonic area (Vt
199

Fig. 6 Map of the purely unsteady reduced pressure, at 50% of blade height and at mid inter-row gap.

Fig. 7 Map of the purely unsteady reduced radial velocity, at 50% blade height and at mid inter-row gap.

Interaction mechanisms The previous section led to identify the most important unsteady structures (α+, α-, β-, A and B) involved in the impeller-diffuser interaction. An analysis to understand the origins at these structures is now proposed. Figure 9 shows six time steps of the pressure gradient in the interaction zone, at 50% blade height. The pressure . The sign of

Fig. 8 Map of the purely unsteady reduced tangential velocity, at 50% blade height and at mid inter-row gap.

that variable indicates if the pressure gradient has the same direction as the velocity vector, which leads us to clearly identify the downward/upward step of the pressure waves. The white curves show the leading edge de-

tached shock wave. The black curves, doted white curves and fined doted white curves mark out the α+, α- and βwaves respectively.

gradient is calculated as:

⎛⎜ ∇ p ⎞⎟ = ∇ p ⋅ V / V ⎝ ⎠s

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Journal of Thermal Science, Vol.16, No.3, 2007

Fig. 9 Temporal evolution of the pressure gradient (in the flow direction) at 50% blade height

The first time step map shows the shock wave at a time just before its chopping by the impeller blade. At the next time step (Fig. 9-b), the trailing edge intersects the strong part of the shock wave which is thus reflected on the blade pressure surface, leading to an α+ wave (noted α+2), then to an α- wave (visible in the third time step map, Fig.9-c) emanating from the vane leading edge. A wave noted α+1 may be observed in Fig. 9-b. This wave propagates without any obstacle up to the fourth time step. At the fifth time step (Fig. 9-e), the wave passes the shock wave and hits the vane leading edge, leading to the β-1 wave, which rotates backwards. In the last map (Fig. 9-f), the α+1 wave has been cut into two branches α+1-a and α+1-b The α+1-b branch moves quicker than the α+1-a branch because it is in a supersonic flow while α+1-a is in a subsonic flow. The origin of structures A and B which are transported by convection may be analyzed from Fig. 10 which gives the reduced radial velocity at three time steps. In each map, the sonic line is marked out with a black line. Fig. 10-a corresponds to a time before the blade trailing edge has crossed the vane leading edge shock wave. Fig. 10-b gives the pattern at a metal coincidence. Finally, Fig. 10-c corresponds to a time after the passage of the impeller blade. From these figures, it may be observed that:

·The sonic line shape is dramatically deformed at the passage of the impeller blade.

·The blade trailing edge tears a subsonic bubble apart. This subsonic bubble is then transported at the impeller rotation speed and leads to the structure named ‘B’.

·The wake material of the jet and wake structure emanating from the impeller causes a steep gradient which is the origin of the structure named ‘A’ which is then transported at the mean flow velocity. It may also be noted that the moving wake leads to a periodic increase in the absolute Mach number ahead of the shock which leads to its periodic deformation. Finally, due to the discrepancy between the flow field which develops into two adjacent impeller blade passages (i.e. between the passage from the suction side of a main blade to the pressure side of the adjacent splitter blade and the passage from the suction side of the splitter blade to the pressure side of the main blade), the period of the pattern is based on the main blade number (also see Figs. 7 & 8). To the onset of dynamic instabilities on the diffuser vanes As seen above, the acoustic wave α+ results from the interaction between the vane leading edge shock wave and the impeller pressure side. After its generation, it moves downstream within the diffuser with a velocity

N. Bulot et al. Impeller-diffuser interaction: analysis of the unsteady flow structures based on their direction of propagation

201

Fig. 10 Temporal evolution of the radial velocity at 50% blade height

Fig. 11 Temporal evolution of the pressure gradient (in the flow direction) at 50% blade height in the vaned diffuser

equal to the local flow velocity plus the speed of sound. Fig. 11 shows three time steps of the pressure gradient in the flow direction 1 within a given passage between a vane named DV2 and the adjacent vane named DV3. A series of α+ waves is observed: waves α+a come from the vane DV2 whereas waves α+b-a come from the vane DV1. The interaction of these waves with the vane pressure side boundary layer leads to unsteady bubbles of reversed flow as highlighted in Fig. 11. Actually, each wave is composed of an upward step and a downward step. The upward step corresponds to a favorable pressure gradient whereas the downward step corresponds to an adverse pressure gradient. Thus, the boundary layer which develops on the pressure side separates at the downward step passage and reattaches at the upward step

passage, leading to a pulsating behavior. Moving towards the vane trailing edge, the size of the unsteady separation bubbles increases due to the decrease in mean velocity. The phenomenon will probably become more evident on the hub and shroud walls because of the hub and shroud recirculations brought about by the interaction between the vane shock wave and the impeller blade. Moreover, going to the surge condition, the vane shock wave moves upstream and then, the shock wave-impeller blade interaction strengthened. The consequence will be a reinforcement of waves α+ leading to a more severe boundary layer unsteady separation. These last two points are part of current research.

1

Defined as : ⎛⎜ ∇ p ⎞⎟ = ∇ p ⋅ V / V ⎝ ⎠s

Conclusion Unsteady flow structures involved in the impeller-vaned diffuser interaction in a transonic centrifugal compressor stage has been analyzed in detail. Applying

202

various data processing techniques, they have been classified into three sets: - the structures convected by the flow, - the structures of acoustic nature, - the structures due to the clocking effects. The physical analysis of these rotating structures emphasizes the mechanisms which are responsible for the flow field unsteadiness. The interaction of the acoustic waves - brought about by the interaction between the vane shock wave and the impeller blade - with the vane pressure side boundary layer leads to a pulsating behavior of separated bubbles within the diffuser. Due to the fact that the onset of strength instabilities such as surge may come from the flow unsteadiness inside the diffuser, the analysis presented in this paper is thought to be useful to describe the surge inception.

Acknowledgement We would like to thank TURBOMECA which supported this research, together with ONERA which collaborated on the numerical simulation. Dr Pascale Kulisa (LMFA) is acknowledged for numerical support.

References

Journal of Thermal Science, Vol.16, No.3, 2007

Turbomachines. Lecture Series 2005-3, Von Karman Institute for Fluid Dynamics, Brussels (2005). [2] Trebinjac, I., Charbonnier, D. and Leboeuf, F.: Unsteady Rotor-Stator Interaction in High Speed Compressor and Turbine. Journal of Thermal Science, vol. 14, N°4, pp. 289—297, (2005). [3] Cambier, L. and Garaix, M.: elsA: An Efficient Object-Oriented Solution to CFD Complexity. 40th AIAA, Aerospace Science Meeting & Exhibit, Reno, USA, (2002). [4] Smith, B.R.: Prediction of Hypersonic Shock Wave Turbulent Boundary Layer Interactions with the k-l Two-Equations Turbulence Model. 33rd AIAA, Aerospace Sciences Meeting and Exhibition, 1995-02232, Reno. [5] Adamczyk, J. J.: Model Equation for Closing Flows in Multistage Turbomachinery. Technical Memorandum, NASA, (1984). [6] Adamczyk, J. J., Mulac, R. A. and Celestina, M. L.: A model for Closing the Inviscid Form of the Average-Passage Equation System. ASME paper 86-GT-227, (1986). [7] Tyler, J. M. and Sofrin, T. G.: Axial Flow Compressor Noise Studies. SAE Transaction 70, pp. 309 — 332,

[1] Tan, C. S.: Effects of Aerodynamic Unsteadiness in Axial

(1962).