TIME-ACCURATE CFD ANALYSIS OF A CENTRIFUGAL COMPRESSOR

Harri Pitkanen, Hannu Esa, Petri Sallinen, Jaakko LarjolaDepartment of Energy Technology, Lappeenranta University of Technology, Finland

Harri Heiska and Timo SiikonenLaboratory of Applied Thermodynamics, Helsinki University of Technology, Finland

A time-dependent flow in a radial compressor with a vaneless diffuser and a volute is simulated. Aquasi-steady approach, where the impeller is rotating and the rest is stationary, and a time-accuratesimulation are performed. In the latter case a sliding-mesh technique is utilized and a second-orderfully implicit time integration is applied. Dual time-stepping is performed inside a physical time step.In both cases the Baldwin-Lomax turbulence model is used. Results obtained from the simulationsare compared with the available experimental data.

Key-words: turbulent flow, time-accurate simulation, sliding mesh, radial compressor

INTRODUCTION

There are different ways of solving Navier–Stokes equations in a case of a centrifugal com-pressor. The simplest level of approximation is to utilize a rotating coordinate system andto model only a single flow passage. This can be treated as a steady-state problem. Con-sequently, moderate computational resources are required, but the variation in boundaryconditions that takes places during the rotational cycle cannot be taken into account. Thesecond level of approximation is to model also the diffuser and the volute, and to performa quasi-steady calculation for the impeller. In this approach the impeller is calculated in arotating coordinate system, but the rest is stationary. However, the whole impeller must bemodeled which makes this approach computationally demanding. In this case different im-peller passages have different boundary conditions depending on their angular position, butan approximation is made because the time history is not taken into account. Thus the coup-ling between the rotating impeller and the stator is unphysical. The most complete method ofsimulation is to rotate the computational mesh describing the rotating part of the machineryand to connect it with the stationary part by using a sliding mesh technique. The whole geo-metry must be modeled and, in addition, a time-accurate simulation requires considerablymore computing time than the quasi-steady approach.

Recently we presented result of the CFD analysis concerning a centrifugal compressorand a vaneless diffuser [1]. In the analysis the volute was taken into account with a con-stant static pressure recovery factor. The comparison between the predicted and measureddata showed a fairly good agreement. However, as far as the actual volute performance isconcerned, it varies according to the operating point of the compressor. There is a needfor more detailed analysis, which is the topic of the present paper. If the velocity after theimpeller is subsonic, the volute may have a significant influence on the impeller flow.

In this paper we present results about the quasi-steady and time-accurate analyses. Thesimulation method is based on Roe’s flux-difference splitting. Turbulence is modeled with theBaldwin-Lomax model. In the quasi-steady calculations the flow equations are integrated intime until the steady-state has been obtained. Convergence is accelerated by utilizing localtime-stepping and a multigrid. The basic solution methods are described in [2]. The time-accurate computation is based on dual time-stepping [3]. In the following we describe the

basic solution methods and present results for a centrifugal compressor with a volute.

SOLUTION METHODS

Governing Equations

In the present study either a steady-state solution in a rotating coordinate system or a time-accurate integration in a stationary Cartesian reference frame is utilized. In both cases theimpeller rotates around the x-axis with an angular velocity . The flow simulation is basedon the solution of the Reynolds averaged Navier–Stokes equations with Cartesian velocitycomponents

@U

@t+@(F�Fv)

@x+@(G�Gv)

@y+@(H�Hv)

@z= Q (1)

where U = (�; �u; �v; �w; E)T , and F , G, H are the flux vectors in each coordinate direc-tion, respectively. Here � is the density, ~V = u~i + v~j + w~k is the absolute velocity, and E isthe total internal energy. The rotational speed of the domain is ~ � ~r = [0;�z;y]T andthe convective, i.e. the relative speeds are ~V �

~ � ~r. In the case of the rotating coordin-ate system a source term Q = [0; 0; �w;��v; 0]T is introduced, but in the time-accuratecalculation Q = 0. The quasi-steady calculations are performed in the rotating coordinatesystem with Cartesian velocity components and modified convective speeds. In the case ofthe time-accurate solution the convective speeds are identical to those in a rotating coordin-ate system. Hence, the only difference in these two approaches concerns the source termQ.

The pressure is calculated from an equation of state p = p(�; e), where e is the specificinternal energy. For a perfect gas

p = ( � 1)�e (2)

where is the ratio of specific heats cp=cv. The molecular viscosity is calculated from Suth-erland’s formula.

The viscous fluxes Fv, Gv and Hv contain a laminar and a turbulent part. For the Reynoldsstresses we use Boussinesq’s approximation. The algebraic turbulence model developed byBaldwin and Lomax [4] is applied in order to calculate turbulent viscosity.

Turbulence Modeling

In spite of the evident simplicity, the algebraic turbulence models often give as goodresults as the more complex models, like the k � �- model, in flows past streamlined bodiesat high Reynolds numbers. The recent studies by the present authors [5] also indicatethat the effect of the turbulence modeling, at least at the design point, is not of paramountimportance.

Unlike in many other algebraic models, in the Baldwin-Lomax model there is no need toknow the thickness of the boundary layer, which makes the implementation easier. However,in the case of a complex geometry the implementation of an algebraic model is not straight-forward. In the present calculations the directional turbulent viscosities are replaced by thevector sum

�T = (�2TI

+ �2TJ

+ �2TK

)1=2 (3)

The viscosities in each grid direction �TI , �TJ , �TK are evaluated by starting separately fromthe lower and upper walls, and the evaluation is continued to the centreline of the flowpassage in that direction.

Numerical Methods

The quasi-steady approach and the time-accurate solution utilize the same basic steady-state algorithm. Since the differences between the two approaches are small, in the followingthe main features of the steady-state solution are firstly described. In the present solution, afinite-volume technique with a structured grid is applied. The flow equations have a discreteform

VidUi

dt=Xfaces

�S(F � Fv) + ViQi (4)

where the sum is taken over the faces of the computational cell. The inviscid part of the fluxis

F =

0BBBBBB@

�U�uU + nxp�vU + nyp

�wU + nzpeU + p �U

1CCCCCCA

(5)

Here nx~i + ny~j + nz

~k is the unit normal vector of the cell face, �U = nxu + nyv + nzw isthe velocity component normal to a stationary cell surface, and U is the convective velocityrelative to the moving cell surface. The functional form of the flux formula is the same in thecase of the time-accurate solution.

In the evaluation of the inviscid fluxes in a moving grid, Roe’s method [6] is applied. AMUSCL-type approach has been adopted for the evaluation of the primary flow variables(�; u; v; w; p) on the cell surfaces. The viscous fluxes are evaluated using a thin-layer ap-proximation. The thin-layer model is activated in all coordinate directions.

Eq. (4) is integrated in time implicitly by applying the DDADI-factorization [7]. This isbased on the approximate factorization and on the splitting of the Jacobians of the flux terms.The resulting implicit stage consists of a backward and forward sweep in every coordinatedirection. The boundary conditions are treated explicitly, and a spatially varying time stepis utilized. Furthermore, a multigrid acceleration of convergence is utilized. Hence, theintegration is not accurate in time, but provides an iterative way to approach the steady-state. More details of the solution methods are given in [2] and [8].

The time-accurate integration shares the same basic features described above, but nowthe pseudo time-integration is performed inside a physical time step and a true time derivat-ive is added on the right-hand side of Eq. (4). A three-level fully implicit scheme is appliedfor the time-integration [3]. Then the residual Ri defined by the right-hand side of Eq. (4) isas discretized in time as

Ri = �Vi3Un+1

� 4Un + Un�1

2�t+Xfaces

�S(F n+1� F n+1

v ) (6)

Here n indicates the time level and �t is a physical time step. ¿From Eq. (6) it is seen thatthe time-derivative term is treated as the source term in Eq. (4). In the time-accurate casethere are no additional source terms caused by the rotation.

In a time-accurate case inside a true time step a kind of steady-state integration is per-formed using the solution methods descibed above. The left-hand side of Eq. (4) is utilizedin the iteration exactly in the same way as in the steady-state calculation. The iteration isconverged as the residual of Eq. (6) is equal to zero. During the solution the grid blockdescribing the impeller rotates and a new position as well as geometrical properties are cal-culated as the calculation proceeds to the next time level. The impeller is connected to the

R (impeller)

R inlet

ω

VoluteStator

Outlet as modelled(not in scale)

R (Stator)

Splitter LE

modelledinlet

Volute

Blade LE

Inlet height

J

K1

2

34

5

Fig. 1: Geometry of the compressor in the meridional view (left), Surface grid (right), which showsthe inlet (1), the rotor outlet (2), the stator outlet (3), the volute (3 to 4), and the outlet cone (4 to 5).

Table 1: Performance data for the compressor

N ! T01 _m p01 p05 �15[Hz] [rad/s] [K] [kg/s] [Pa] [Pa]300 1884.96 288 2.78 97650 161150 1.65

rest of the domain using a sliding mesh technique. With the sliding mesh the impeller isrotated to a new position at every time step. As a result the grid lines between the impellerblocks and the stator are discontinuous. A mass conserving interpolation is made betweenthe connecting blocks at every time step [9].

DEFINITION OF THE TEST CASE

Geometry

As an application, a 2.0 pressure ratio centrifugal compressor with a scroll-type volute hasbeen simulated at the mass flow rate of 2:78 kg/s and the rotation speed of 18000 rpm.Boundary conditions for this case are given in Table 1. The outflow boundary condition is astatic pressure p5 at the end of the exit cone. The geometry and the position of the meas-urement points can be seen in Fig 1. The impeller is followed by a vaneless diffusor (stator)and a scroll type diffusor with an exit cone. The impeller has 7 full blades and 7 splitterblades with a backsweep of 30Æ from the radial direction. The hub and the tip diameters ofthe impeller leading edge are 24% and 54% of the impeller exit diameter, respectively.

Computational Grid

The whole radial compressor is modeled. The grid size is 457;472 computational cellsand it is divided into 25 different blocks. The block dimensions can be seen in Table 2 anda sketch of the volume grid in Fig. . All blocks are of the H-type, block 1 is between the inletand the impeller, blocks 2, 5, 8, 11, 14, 17 and 20 are between the blades, blocks 3, 4, 6, 7,

Table 2: Dimensions of the grid

Block i j k � Location1 320 16 12 53;760 Inlet2 16 16 12 6;912 Before splitter blade3 16 16 20 5;120 Suction side4 16 16 20 5;120 Pressure side: :: :: :: ::::23 320 16 12 53;760 Vaneless diffusor24 320 16 12 53;760 Vaneless diffusor25 344 16 32 176;128 Volute

Total 457;472

1

234

5

67

89

1011

1213

14

15 16

1718

19

20 21

22

2324

25 Sliding mesh boundary

Inlet,

Outlet

} Impeller

} Stator

Ro

tati

ng

mes

h

2

Block 1

3 4

(23)

(24)

Volute, Block 25

20

Sta

tio

nar

y

A B

5

6 7

A B

1

1’

2

5

3

21 22

Fig. 2: Position of the grid blocks (left), topology of the grid (right).

9, 10, 12, 13, 15, 16, 18, 19, 21 and 22 are between the blade and the splitter blade, blocks23, 24 describe the vaneless diffusor, and block 25 is the scroll type volute. The surface gridfor the whole compressor can be seen in Fig. 1.

The topology of the grid is such that there is boundary layer clustering on every surface,and between the block connectivities. Tip clearance is not modeled. The height of the firstrow of cells is 8 � 10�5m. As nondimensional distances this is on the average y+ = 20. Forthe accurate presentation of the boundary layers this is too much, but the grid is modeled inthis way in order to get sufficiently good qualitative result with the computational resourcesavailable.

COMPUTATIONAL RESULTS

Arrangement of the Computation

The computations were performed on the SGI Origin2000 machine (R10k) in a single pro-cessor mode. In a case of the quasi-steady analysis the Courant number, which governsthe size of the local time step, was 1.5. A converged result was obtained in 3500 itera-tion cycles. The computation time was 28 hours and the computational speed 63 �s/gridcell/iteration cycle.

In the time-accurate simulations the physical time step was set to 2�s, which corresponds

Fig. 3: Static pressure distributions on the volute shroud surface (left), and average, minimum andmaximum static pressure distributions in a time-accurate case (right).

to a rotation angle of 0:216Æ/time step. The length of the simulation period was as short as0.0128 seconds, which corresponds to a 3.83 rotations around the rotating axis or to 6394time steps. Starting from the quasi-steady result, the computation reached a sufficientlyperiodic behaviour already in three impeller rounds. Inside a time step 40 internal iterationswere made. This was necessary because, as the number of iterations was decreased to 20,the computation finally blew up. With 40 internal iterations the computation took 1393 s/timestep and the total computation four calender months.

Boundary Conditions

The present calculations are performed at design conditions. As the inflow boundaryconditions, the total entahlpy, mass flow and the angle between the velocity components(=zero in this case) are given. The pressure is extrapolated from the flow field and thedensity is iterated from the total enthalpy and pressure. The outlet plane is at the end ofthe exit cone where the outlet pressure is set to 160 kPa, and the rest of the flow variablesare assumed to have a constant gradient. At the solid walls the velocity is set to zero and asecond-order extrapolation from the flow field is used for the pressure. In the time-accuratesimulations the same boundary condition as in the quasi-steady case are applied.

Results

The CFD results have been compared with the measured total pressure ratio, and withthe static pressure along the volute surface. The calculated total pressure ratio �15 = 1:653agrees very well with the measured value of 1.65. The static pressure was measured at fivelocations in the circumferential direction on the volute surface shown in Fig. 1. Since the inletpressure in the computations is not exactly the same as in the measurements, the resultsare scaled with the corresponding inlet total pressure p01. A comparison between the resultsis given in Fig. 3. It is seen that there is a difference between the quasi-steady pressure dis-tribution and the time-accurate result. The time-accurate result is perhaps somewhat closerto the experiments. A closer look at the time-accurate result shows that the pressure on thevolute shroud surface does not vary considerably. In Fig. 3 (right) the pressure distributionsare plotted using the minimum and the maximum as well as the average values of pressure.These distributions are almost indistinguishable.

0.18

0.2

0.22

0 200 400 600 800 1000

A+B

0.38

0.4

0.42

0

A+B

B

A

Rotation angle (degrees)

mas

s flo

w r

ate

(kg/

s)

Fig. 4: Mass flow rate at the impeller channel exits (left), and instantaneous pressure rises in differentimpeller channels (right). A and B are channels on both sides of the splitter blades, A+B is the totalmass flow rate.

The mass flow rate in the impeller channels varies, since the outlet pressure is differentat different circumferential positions. The mass flow rates in all the channels as a functionof the rotation angle are given in Fig. 4. There are some differences in the mass flow ratesof different channels, which can only be of a numerical origin. However, these differencesare so small that the computation can be considered to be succesfull. The flow history inFig. 4 corresponds to two and a half rotations. It is seen that the mass flow rate exhibitstwo maximums during one rotation. In Fig. 4 (right) the instantaneous pressure distributionsin the impeller channels are given. The pressure rise in the impeller channel varies byabout 5% depending on the circumferential position. Static pressure histories at different

Circumferential position 90 degrees

180 degrees

270 degrees

360 degrees

Fig. 5: Non-dimensional static pressure histories at circumferential positions 90, 180, 270 and 360degrees.

circumferential positions of the impeller exit are given in Fig. 5. It is seen that the oscillationscaused by the by-pass of the impeller blade are small in comparison with the differencesin pressure at different circumferential locations. At the location of 90Æ the effect of the

Cr

C

Cu

Fig. 6: Stator exit velocity components (left), and flow angle (right) at a circumferential location 180degree between the hub and the shroud surfaces.

Fig. 7: Static pressure contours in the middle of the rotor, stator and on the corresponging voluteplane. Quasi-steady (left) and time-accurate result at t = 0:01278 s (right).

splitter blade is visible, but at the other locations the frequency of the pressure oscillationscorresponds only to the effect of seven impeller blades. In the pressure histories there aresome long-term changes, which are not visible in the mass flow histories. The reason forthis behaviour is not clear. It is possible that 3.83 impeller rotations is not enough to producecompletely cyclic varying flow conditions in the stator and volute.

The stator exit velocity components and flow angles at different circumferential positionsare given in Fig. 6. These are also compared with the experimental data. It should be notedthat the measurements using a cobra-probe were made in vacuum operation and the resultscan be compared mainly qualitatively. It can be seen that the flow is ill-predicted at thislocation. The experimental velocity profile is almost flat as the predicted velocity shows amaximum near the hub surface. The deviation is largest in the radial velocity component cr,which according to the calculation is almost zero near the shroud durface. The instantaneoustime-accurate distributions differ only slightly from the quasi-steady results.

The calculated static pressure contours are compared in Fig. 7. It is seen that there areonly minor differences between the time-accurate and quasi-steady results.

CONCLUSIONS

Quasi-steady and time-accurate simulations were made for a centrifugal compressor with avaneless diffusor and volute. In the time-accurate case a second-order fully implicit methodwith dual time-stepping was applied successfully. The time-accurate results show periodicoscillations in impeller mass flows and pressure distributions. The motion of the impellerblades and even the splitter blades is visible in the stator pressure histories, but the amplitudeof these oscillations is small as compared to the pressure level and the changes in the outletpressure of the impeller channels. It can be conluded that the quasi-steady simulation ofthe whole compressor is almost as accurate as the time-accurate calculation. It should benoted that the differences between the results would be much larger in a case of a statorwith vanes. The pressure rise and the mass flow vary in both simulations depending on thecircumferential position of the channel. This variation is difficult to take into account in atraditional calculation, where only a single impeller passage is modeled.

The results were also compared with the experimental data. The pressure distribution atthe shroud surface of the volute is fairly well predicted, but there is considerable deviationfrom the experiments in the stator velocity distribution. This is probably mainly caused by thesimple turbulence modeling. The grid (457,472 cells) is too coarse for the accurate repres-entation of the boundary layers, but that does not explain the qualitatively wrong behaviourof the calculated velocities.

The present simulations were performed in a single-processor mode with a workstation.In that case the computational time is several months. With a parallel computation this timecan be reduced to a couple of weeks or even to some days. In that case also the grid canbe much denser. Utilizing parallel computation a time-accurate simulation is already nowa feasible way to investigate various time-dependent phenomena inside turbomachinery. Aquasi-steady approch where the whole compressor is modeled offers a cost-effective altern-ative and obviously in many cases the difference from a time-accurate result is small.

REFERENCES

[1] Pitkanen, H., “The CFD Analysis of the Impeller and Vaneless Diffuser of an IndustrialWater-Treatment Compressor,” in ASME Paper No. 97-WA/PID-3, 1997.

[2] Siikonen, T. and Pan, H., “Application of Roe’s Method for the Simulation of Viscous Flowin Turbomachinery,” in Proceedings of the First European Computational Fluid DynamicsConference, (Brussels), pp. 635–641, Elsevier Science Publishers B.V., Sept. 1992.

[3] Hoffren, J., Siikonen, T., and Laine, S., “Conservative Multiblock Navier–Stokes Solverfor Arbitrarily Deforming Geometries,” Journal of Aircraft, Vol. 32, No. 6, 1995, pp. 1342–1350.

[4] Baldwin, B. and Lomax, H., “Thin Layer Approximation and Algebraic Model for Separ-ated Turbulent Flows,” Jan 1978. AIAA Paper 78-257.

[5] Rautaheimo, P., Salminen, E., and Siikonen, T., “Numerical Simulation of the Flow in theNASA Low-Speed Centrifugal Compressor,” Helsinki University of Technology, Laborat-ory of Applied Thermodynamics, 1998. ISBN 951–22–4360–1.

[6] Roe, P., “Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes,”Journal of Computational Physics, Vol. 43, No. 2, 1981, pp. 357–372.

[7] Lombard, C., Bardina, J., Venkatapathy, E., and Oliger, J., “Multi-Dimensional Formula-tion of CSCM — An Upwind Flux Difference Eigenvector Split Method for the Compress-ible Navier–Stokes Equations,” in 6th AIAA Computational Fluid Dynamics Conference,(Danvers, Massachusetts), pp. 649–664, Jul 1983. AIAA Paper 83-1895-CP.

[8] Siikonen, T., “An Application of Roe’s Flux-Difference Splitting for the k � � TurbulenceModel,” International Journal for Numerical Methods in Fluids, Vol. 21, No. 11, 1995,pp. 1017–1039.

[9] Rai, M., “A Relaxation Approach to Patched-Grid Calculations with the Euler Equations,”Journal of Computational Physics, Vol. 66, Dec 1986, pp. 99–131.