The mixing-plane method for calculating the three-dimensional flow through multistageturbomachinery is used to perform flutter analysis on a single stage transonic compressor.The turbomachine considered is composed of an inlet guide vane (IGV) and a compres-sor blade (NASA Rotor 67). The mixing-plane boundary condition enables steady andunsteady computation of the flow through multiple blade rows. This allows incorporationof multi-stage effects without having to perform computationally intensive fully unsteadymulti-stage flow computations. Forced motion flutter computations are performed for boththe isolated compressor and with the IGV. Comparison between the damping ratio showsa decrease in stability when the IGV is included in the computation.
I. Introduction
The modern desire to make turbomachinery blade rows both lighter and better performing pushes thedesigns to the extreme. As blade rows are designed to be lighter they also become less rigid. Also, asthe blade rows are made to operate under more sever conditions the blades are put under more stress.One negative consequence of these design trends is that blade rows may become more susceptible to flowinduced vibrations. These vibrations may fatigue the blades and/or lead to catastrophic failure. If this fateis to be avoided, the designer must be aware of the possible sources of the vibrations, and the mechanismsinvolved. Because turbomachines are highly dynamic, there are many ways in which harmful vibrations maybe internally generated. One important source of these vibrations is the unsteady flow field that results fromthe interaction of consecutive blade rows in multi-stage turbomachines.
The practice of using CFD solvers to obtain “steady” flow solutions for three-dimensional turbomachineryblade rows has been maturing for about twenty years. While these methods may not be in standard usefor industrial design yet, they have certainly been relied upon more and more in recent years. However, ashortfall of these solvers is that they are unable to account for the effects arising from the interaction ofmultiple blade rows. Since advanced turbomachines are typically comprised of many closely spaced stages,these effects should be considered as they play an important role in flow field, and thus may be critical tounderstanding the response of blade rows to flow induced vibrations.
In response to this need for understanding, work has been done in the area of fully unsteady flow solutionsto consider blade row interactions.1 These types of methods are able to resolve the complete unsteady flowfield and may give the designer a high fidelity view into the flow domain. Unfortunately, these methodsare highly expensive computationally and as such are not practical for a designer who needs performancepredictions in a matter of hours, not days.
In attempts to find somewhat more practical methods for considering the flow through multiple bladerows, several researchers have worked on approximate methods. One such method approximates the unsteadyflow field that occurs in multi-stage turbomachines with a steady flow by including a mixing layer betweenadjacent blade rows. The idea is that by adding this mixing layer between adjacent blade rows, the flowcan essentially mix out the circumferential variations, which are the cause of some of the unsteadiness ina real multi-stage machine. While this method smooths out the circumferential variations, it still retainsvariations in the axial and radial directions, enabling the researcher to examine how these variations effectthe multistage flow field.
∗Graduate Student Researcher, Mechanical & Aerospace Engineering Department, University of California Irvine†Professor, Mechanical & Aerospace Engineering Department, University of California Irvine, Associate AIAA-Fellow
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In 1992 work by Denton2 demonstrated an approach to this method he called an “Interrow MixingModel”. In that work he coupled an Euler Solver with various methods of approximate solution for viscousstresses. Variations in the circumferential direction were also accounted for at the mixing layer boundaryin order to eliminate improper loading on the leading and trailing edges of the blades. This early workdemonstrated that results for the “steady” flow through multi-stage turbomachines could be obtained bythis approach.
Later work by Chima3 used this approach to compute the flow through the space shuttle main enginefuel turbine. His work examined several different averaging methods as well as interface boundary conditionsin order to examine their effects on solution accuracy. Significant was his inclusion of a 1-D characteristicboundary condition4 and the mixing-plane interface which help eliminate distortions in the pressure fieldwhich result from a more commonly used constant pressure exit condition.
Recent work by Davis5 considered the flow through the NASA Stage 35 single stage compressor using amixing-plane approach for both steady and unsteady flows. Here the focus was on predicting the flow overa wide range of operating conditions (choke to stall) for a fixed speed. By modeling the flow physics moreclosely than isolated blade row studies, these simulations were able to more clearly identify the unsteadymechanisms responsible for the onset of stall in the machine.
While there has been considerable work done to resolve the flow field in multi-stage turbomachines, therehas been relatively little work done to examine the interaction of this flow with the structural dynamics ofturbomachines.
One example of this is the recent work by Vahdati, Sayma and Imregun.6, 7 Here, free vibration computa-tions were performed for several three dimensional multi-stage test cases. A viscous flow solver was coupledwith a linear, modal superposition, structural solver to simulate the aeroelastic system. As a testamentto the fidelity of this type of model, a good comparison was made to the available experimental data forthe maximum tip displacement observed under self excited vibration. One point which is emphasized bythe authors is that the low frequency unsteadiness arising from the flow is much less understood that thedeterministic unsteadiness linked to the shaft rotation rate.
The current work is motivated both by the desire to improve the understanding of multi-stage turbo-machine flows and the desire to analyze the fluid-structure interaction present in such machines. To thatend, a three-dimensional flow solver is to obtain the “steady” flow solution for a multi-stage turbomachineusing the mixing-plane approach described above. Proceeding on from this steady result, flutter analysis isconducted. Here only forced motion cases are performed. Isolated blade row forced motion computations arealso performed and the results are compared to determine what effects, if any, the incorporation of multipleblade rows has on the computed flutter sensitivity of a transonic compressor fan blade row.
II. Computational Method
II.A. Flow Solver
A density-based finite-volume method is used to solve the conservative laws for unsteady compressible flow.The Favre-averaged Navier-Stokes equations may be written as
∂
∂t
∫∫∫
V
WdV +
∫∫
S
F · ndS = 0
where V is an arbitrary control volume with boundary surface S, and n is the unit normal to the surface(directed outward). The state vector W is
W =
ρ
ρu
ρE
where ρ is density, u is the three dimensional velocity vector, E is the energy.The Flux vector F = Fi − Fv may be split into is inviscid and viscous components
Fi =
ρuT
ρuu + p
ρEu + puT
, Fv =
0
τ
(τ · u − q + (µ + σ∗)∇T k)T
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where τ is the shear stress tensor and q is the heat flux vector.The flow solver used to solve these equations is based on the code ParCAE which was developed and
used for the study of three-dimensional turbomachinery aeroelasticity.8 ParCAE is an explicit, multistage,multiblock, parallel flow solver which uses dual time stepping to resolve unsteady flows. An implementationof the Spalart-Almaras turbulence model has been used for all viscous simulations shown in this work.9
II.B. Structural Solver
The structural model Incorporated into ParCAE is a modal superposition method. Starting from the struc-tural equations of motion for a system with a finite number of degrees of freedom
M q̈ + Cq̇ + Kq = F
where q is the displacement vector. Using a modal approach, the solution has the form
q =
N∑
i=1
ηiΦi
where Φi are the eigenmodes and ηi are the generalized displacements. If we diagonalize this system usingthe matrix of eigenvectors Φ, we obtain an expression for the generalized displacement of the structure
η̈i + 2ζiωiη̇i + ω2i ηi = Qi
Where the aerodynamic forces F are projected onto the mode shapes to obtain the generalized forces Qi, foreach mode. Here, ωi are the natural frequencies, and ζi are the damping coefficients for each mode.
When solving the structural equations with ParCAE, the modes shapes for the structure are generatedfor each geometry as a preprocessing step and may be obtained from either a finite element solver or fromexperimental measurement. An explicit method is used to integrate the structural equations of motions intime. An efficient transfinite interpolation method is used to interpolate structural deformations across theentire grid after each time step, and thus avoid completely re-meshing the domain.
II.C. Boundary Conditions
II.C.1. Inlet/Outlet Boundary Conditions
For the flows considered here, a radial distribution of total pressure, total temperature and flow angles isspecified while one characteristic is extrapolated for the inlet boundary condition. At the outlet, the staticpressure is specified while the remaining four characteristics are extrapolated.
II.C.2. Interface Boundary Condition
At the mixing-plane interfaces, radial profiles of flow variables are specified as the initial condition. Whilethe code is running, updated profiles are obtained using “mixed-out” averaged flow variables3 in the firstinterior cell each block adjacent to the mixing-plane. These interior profiles are then passed to the ghost cellson the opposite side of the interface. As a result, when a converged solution is obtained, the circumferentiallyaveraged flow variables at the mixing plane will be equivalent on either side of the interface and thus mass,momentum and energy will be conserved. However, as pointed out by Denton,2 entropy will be generated bythis mixing process. A diagram of the use of ghost cells for this mixing-plane boundary condition is includedas Figure II.C.2.
In order to compute the “mixed-out” flow variables, the expressions given in Equation 1 are used. Here,a local coordinate frame which is aligned with the mixing-plane has been used to write the inviscid fluxesthrough the interface. The subscript n refers to direction normal to the surface. The subscript t refers to
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Downstream BlockUpstream Block
w(r)
w(r)
InterfaceMixingplane
Figure 1. Diagram of the update procedure at the mixing-plane interface
the radial direction, and θ refers to the circumferential direction, which is also tangent to the mixing-plane.
I1 = 1A
∫
ρ U · dA = ρ Un
I2 = 1A
∫
[ρUnU + p ex] · dA = ρ Un Un + p
I3 = 1A
∫
[ρUθU + p eθ] · dA = ρ Uθ Un
I4 = 1A
∫
[ρUtU + p et] · dA = ρ Ur Un
I5 = 1A
∫
[ρ E + p]U · dA = ρ E Un + p Un
(1)
where the expression for specific total energy E is
E =1
γ − 1
p
ρ+
1
2
(
U2n + U2
θ + U2t
)
These expressions taken together represent a quadratic equation for pressure, who’s solution is
p =1
γ + 1
[
I2 ±
√
I22 + (γ2 − 1) (c − 2I1I5)
]
wherec = I2
2 + I23 + I2
4
and the remaining flow variables are determined as
Un =I2 − p
I1
, Uθ =I3
I1
, Ut =I4
I1
, ρ =I21
I2 − p
II.D. Flutter Analysis
The flutter analysis performed here consists of so called forced motion tests. Forced motion tests are con-ducted by applying a prescribed motion to the structure in question and observing the resulting flow field.
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(a) Top View (b) Side View
X
Y Z
(c) Isometric View
Figure 2. Rotor 67 with IGV configuration
In this case, no structural dynamics model is needed as the motion is prescribed. The primary result for thistype of test is the damping ratio Ξ, which is defined as
Ξ = −1
πh2
∫ T
0
Cηdη (2)
where Cη is the force coefficient aligned with the generalized displacement η. and h is the normalized mag-nitude of oscillation. The damping ratio provides a measure of the aerodynamic damping on the structure.A negative damping ratio indicates that work is being done on the structure, and thus the condition may beclassified as unstable.
III. Results
ParCAE has been previously demonstrated for computation of 3D turbomachinery flutter8, 10 in isolatedblade rows. To demonstrate ParCAE’s ability to perform multi-stage computations, NASA’s Rotor 67transonic compressor blade row has been used.11, 12 Although the NASA Rotor 67 case was designed tooperate in an isolated configuration, for the purposes of this study, an artificial inlet guide vane (IGV) wascreated using airfoils from the NACA four digit airfoil series which varied in thickness from 6 percent atthe hub to 12 percent at the shroud. Although the purpose of an IGV is typically to align the inlet flowwhen a compressor is operating in an off-design condition, the IGV is kept parallel to the flow for this case.Top, side and isometric views of the geometry are given in Figure 2. For the forced-motion computations, afictitious torsional mode shape was used as torsional modes tend to be the most sensitive to vibrations forhighly twisted transonic compressor blades.
Steady flow computations were performed for both the isolated Rotor 67 case as well as the case with theIGV. The performance line for the isolated computations is compared with experimental data in Figure 3.Notice that near the choke and the maximum efficiency conditions, the agreement with experiment is quitegood. This agreement breaks down in the near stall condition. This may be due in part to the over prediction
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η
0.82
0.84
0.86
0.88
0.9
0.92
(a) Total Pressure Ratio
mdot/mdot,choke
P02
/P01
0.94 0.96 0.98 11.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
1.7
experimentcurrent
(b) Adiabatic Efficiency
Figure 3. Performance From Steady Computations for the Rotor 67 Compressor
of tip leakage flow, which may be exaggerated at higher pressure ratios.To give a more detailed description of the flow field, circumferentially averaged radial profiles of flow
variables were computed at the inlet and outlet of the Rotor domain for a near-stall operating condition.These are shown in Figure 4. The agreement for both the flow angle and Total temperature ratio are excellentfor this condition. The largest discrepancy is seen in the downstream position from mid-span to the tip forthe Total pressure ratio. This coincides with the under predicted total pressure ratio seen in Figure 3.
Even with the under prediction of the pressure ratio for the near stall conditions, the trends in theperformance are as expected and the agreement of the radial profiles is good. That being the case, theresults of forced-motion computations may still be instructive as to the effect of multiple blade rows onflutter behavior. These computations were performed for both the case with and without the IGV at themaximum efficiency point (about 99% of the choked mass flow rate). The results are shown below in Figure 5and Table 1.
Figure 5 shoes the time history of the modal component of the resultant aerodynamic force on thedeforming Rotor 67 blade for the last several periods of the unsteady computation. Here the green line is forthe isolated rotor case and the blue line is for the case with the IGV. Also plotted is the prescribed sinusoidalmagnitude of the modal deformation. Note that the magnitude of the forces are significantly larger for thecase without the IGV.
Using Equation 2 we are able to compute damping coefficients for both cases. Table 1 shows the results ofthis computation. The case with the IGV shows a significant decrease in the damping coefficient, representinga decrease in stability of approximately 70%.
Table 1. Damping Coefficient for Forced-Motion Case
Case ζ
Isolated 0.53579
With IGV 0.16046
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Figure 4. Radial profiles of flow variables in the near stall flow condition
IV. Conclusions and Future Work
In this work, the mixing-plane method has been used to perform steady and unsteady computations ofthe flow through a single stage transonic compressor comprised of a fictitious IGV and the NASA Rotor 67blade row. In the steady computations, an under prediction of performance was observed. This has been atleast partially attributed to the over prediction of the tip leakage flow in the high pressure flow conditions.The circumferentially averaged radial profile of total pressure also show this discrepancy at the near stallcondition from mid-span to the tip. However, the radial profiles of total temperature and flow variables werein very good agreement with the experiment.
The results of the forced-motion computations showed a significant decrease in stability when the fictitiousIGV was included in the computational domain. Even though these computations are very limited in scope,their implications are significant. The results indicate that multi-stage effects may play a large role in theaeroelastic stability of transonic compressors such as NASA’s Rotor 67. It also significant that this wasfound even when ignoring the unsteady circumferential disturbances which are ignored by the mixing-planemethod. If nothing else, this results gives motivation for further studies.
The authors intend to extend this work in several ways. First, by performing these forced-motion comp-tutation over a wide range of inter blade phase angles and operational conditions, a more complete picture
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-0.04
-0.02
0
0.02
0.04
0 1 2 3 4 5-1.5
-1
-0.5
0
0.5
1
1.5
F-F
0
Mod
al D
ispl
acem
ent (
eta)
Time
etaforce isolatedforce with igv
Figure 5.
of the stability domain for this configuration will be obtained. Secondly, by performing free-response compu-tations for the cases with and without the IGV, we will examine what indications the free response methodmay give as to multi-stage effects on the flutter sensitivity of transonic turbomachines. Also, a study ofthe effect of IGV spacing on the stability of these blade rows will be performed. Finally, fully unsteadycomputations are intended for the configurations described in this paper. Although, these computations arenot currently practical for design purposes, they may provide useful insight into fully unsteady flow field,which is not captured by the mixing-plane computations.
References
1Rai, M. M., “Unsteady Three-Dimensional Navier-Stokes Simulations of Turbine Rotor-Stator Interaction,” Journal of
Propulsion and Power , Vol. 5, No. 3, May-June 1989, pp. 307–319.2Denton, J. D., “The calculation of three dimensional viscous flow through multistage turbomachines,” Transactions of
the ASME , Vol. 114, January 1992, pp. 18–26.3Chima, R. V., “Calculation of Multistage Turbomachinery Using Steady Characteristic Boundary Conditions,” Tech.
Rep. NASA-TM-1998-206613, NASA, 1998.4Giles, M. B., “Non-Reflecting Boundary Conditions for Euler Equation Calculations,” AIAA Journal , Vol. 28, No. 12,
1990, pp. 2050–2058.5Davis, R. L. and Yao, J., “Prediction of Compressor Stage Performance from Choke Through Stall,” Journal of Propulsion
and Power , 2006.6Sayma, A. I., Vahdati, M., and Imregun, M., “An Integrated Nonlinear Approach For Turbomachinery Forced Response
Prediction. Part I: Formulation,” Journal of Fluids and Structures, Vol. 14, 2000, pp. 87–101.7Vahdati, M., Sayma, A. I., and Imregun, M., “An Integrated Nonlinear Approach For Turbomachinery Forced Response
Prediction. Part II: Case Studies,” Journal of Fluids and Structures, Vol. 14, 2000, pp. 103–125.8Sadeghi, M., Parallel Computation of Three-Dimensional Aeroelastic Fluid-Structure Interaction, Ph.D. thesis, Univer-
sity of California, Irvine, 2004.9Wilcox, D. C., Turbulence Modeling for CFD , DCW Industries, 1998.
10Sadeghi, M. and Liu, F., “Coupled Fluid-Structure Simulation for Turbomachinery Blade Rows,” No. AIAA Paper2005-0018, Reno, NV, Jan 2005.
11Reid, L. and Moore, R. D., “Performance of a Single-Stage Axial-Flow Transonic Compressor with Rotor and StatorAspect Ratios of 1.19 and 1.26, Respectively, and with Design Pressure Ratio of 1.82,” Tech. Rep. NASA-TP-1978-1338,NASA, 1978.
12Reid, L. and Moore, R. D., “Deisng and Overall Performance of Four Highly Loaded, High-Speed Inlet Stages for anAdvanced High-Pressure-Ratio Core Compressor,” Tech. Rep. NASA-TP-1978-1337, NASA, 1978.
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