'. 

NASA/CR-2003-212101 

Modeling of Unsteady Three-Dimensional 

Flows in Multistage Machines 

Kenneth C. Hall and Edmund T. Pratt 

Duke University, Durham, North Carolina 

January 2003 

https://ntrs.nasa.gov/search.jsp?R=20030014838 2018-05-15T15:36:58+00:00Z
NASA/CR-2003-212101 

Modeling of Unsteady Three-Dimensional 

Flows in Multistage Machines 

Kenneth C. Hall and Edmund T. Pratt 

Duke University, Durham, North Carolina 

Prepared under Grant NAG3-2627 

National Aeronautics and 

Space Administration 

Glenn Research Center 

January 2003 
The Propulsion and Power Program at 

NASA Glenn Research Center sponsored this work. 

Available from 

NASA Center for Aerospace Information 

7121 Standard Drive 

National Technical Information Service 

5285 Port Royal Road 

Hanover, MD 21076 Springfield, VA 22100 

Available electronically at htq.-,: I Igltrs.grc.nasa.gov 
Abstract 

Modeling of Unsteady Three-Dimensional Flows 

in Multistage Machines 

Kenneth C. Hall, Principal Investigator 

Professor and Chair 

Department of Mechanical Engineering and Materials Science 

Edmund T. Pratt, Jr. School of Engineering 

Duke University 

Durham, NC 27708-0300 

This report describes progress made on NASA Glenn Research Center Grant 

NAG3-2627 for the period April 2001 through September 2002. This grant was funded 

by NASA as part of the GUIde Consortium III program. This report details work to date, 

as well as plans for the upcoming reporting period. 

Introduction 

Despite many years of development, the accurate and reliable prediction of 

unsteady aerodynamic forces acting on turbomachinery blades remains less than 

satisfactory, especially when viewed next to the great success investigators have had in 

predicting steady flows. Hall and Silkowski (1997) have proposed that one of the main 

reasons for the discrepancy between theory and experiment and/or industrial experience 

is that many of the current unsteady aerodynamic theories model a single blade row in an 

infinitely long duct, ignoring potentially important multistage effects. However, unsteady 

flows are made up of acoustic, vortical, and entropic waves. These waves provide a 

mechanism for the rotors and stators of multistage machines to communicate with one 

another. In other words, wave behavior makes unsteady flows fundamentally a multistage 

(and three-dimensional) phenomenon. 

In this research program, we have has as goals (1) the development of 

computationally efficient computer models of the unsteady aerodynamic response of 

blade rows embedded in a multistage machine (these models will ultimately be capable of 

analyzing three-dimensional viscous transonic flows), and (2) the use of these computer 

codes to study a number of important multistage phenomena. 

Recently, Hall and Smelova (1999) have developed a three-dimensional unsteady 

multistage Euler analysis under an AFOSR grant. For a given multistage fan or 

compressor, they first generate a computational mesh for each blade row. The steady and 

unsteady multistage flows are computed using the Euler and linearized Euler equations 

and conventional computational fluid dynamic (CFD) techniques, with so-called mixing 

planes (the inter-row computational boundaries of the computational grid) used to couple 

together the solutions computed in the individual blade rows. For the unsteady flow 

solution, several linearized unsteady flow calculations are performed simultaneously, one 

NASAlCR-2003-212101 
corresponding to each spinning mode in the model. The only coupling among the various 

spinning modes is at the inter-row boundaries. The method is very efficient. A typical 

unsteady multistage flow calculation might take on the order of ten times the 

computational time required for a single row steady flow computation. 

For flows in which nonlinear effects are thought to be important, most researchers 

have relied on time marching techniques. However, these methods, as previously 

discussed, are computationally expensive. Recently, Hall, Thomas and Clark (2000) have 

developed a novel technique for computing unsteady nonlinear flows in turbomachinery 

cascades. The unsteady flow is represented by a Fourier series in time with frequencies 

that are harmonics of the excitation frequency -- blade passing frequency in the case of 

wake/rotor interaction, or the blade vibratory frequency in the case of flutter. Borrowing 

from the structural dynamics community, they use a harmonic balance technique to write 

a set of coupled partial differential equations for the unknown flow solution at each 

harmonic. A pseudo-time term is introduced into the harmonic balance equations so that 

the equations may be solved by using conventional time marching computational fluid 

dynamic techniques. 

Both the linearized multistage analysis and the single row harmonic balance 

analyses described above have a number of distinct advantages over more conventional 

time-domain solutions. First, because the solutions are computed in the frequency 

domain, the time-marching algorithm is only used to converge the solution to steady 

state. Thus, acceleration techniques, including local-time time marching with multi-grid 

acceleration, can be used. Second, complex periodicity conditions may be applied for 

each harmonic, so that the computational domain for each blade row may be reduced to a 

single blade passage. Finally, for many applications, just a few spinning modes in the 

multistage analysis, or higher harmonics in the harmonic balance analysis, need to be 

incorporated where engineering accuracy is sufficient. The result is that these methods 

are potentially two orders-of-magnitude faster than conventional time-marching methods, 

making them ideal for both routine design work and parametric studies to gain insight 

into complex flow physics. 

Research Progress and Challenges During this Reporting Period 

In this research program, we will further develop both the multistage and 

harmonic balance analyses, and to use them to study problems of importance to the gas 

turbine industry. At the GUIde Consortium Meeting held in August 2001, we gave an 

oral presentation of our plans for this grant, including a timeline for completion of the 

major tasks (see Table 1). The steering committee recommended that the tasks be 

completed in a different order, with an emphasis on the early completion of Task 5, the 

development of a linearized multistage Navier-Stokes analysis of multistage flows. This 

Task is dependent on Task 3, the development of a block structured version of the 

multistage code. We have, therefore, revised our timeline for the completion of the major 

tasks (see Table 2). 

During the present reporting period, our efforts have been focused on Tasks 3 and 

Task 5. With regard to Task 3, in a previous AFOSR grant, investigators at Duke 

University developed a three-dimensional linearized multistage analysis of inviscid 

(Euler) unsteady flow in turbomachinery. However, a number of modifications to the 

previous computer code are required if the code is to model viscous flows. First, the 

original computational code used an H-grid structure. H-grids, while relatively simple to 

use in a CFD code, do not have adequate resolution to resolve boundary layers, especially 

near the leading and trailing edges of an airfoil. Therefore, in the present research, we are 

implementing using an H-O-H grid structure, with an O-grid around the airfoils, with HNASNCR-2003-212101 2 
grid extensions upstream and downstream of the O-grids. During this reporting period, 

we have made good progress on this task. We now have a single blade row Navier-Stokes 

code that uses and H-O-H grid structure. We have included the additional viscous terms 

in the Navier-Stokes, including Reynolds stress terms modeled using the SpalartAllmaras turbulence model. We are currently in the process of debugging this analysis, 

and hope to have both steady and linearized unsteady three-dimensional viscous analysis 

capability shortly. 

Simultaneously, we have continued development of an inviscid multistage 

analysis. (This analysis will be the combined with the single row block structured NavierStokes analysis to form a multistage Navier-Stokes analysis.) Some preliminary results 

have been obtained. Figure 1 shows a typical example. Shown is the computed unsteady 

pressure difference across the surface of a vibrating rotor airfoil that has a helicoidal 

shape. For this example, the rotor has 38 airfoils, and the downstream stator has 50 

airfoils. In the non-rotating frame of reference, the flow is entirely axial. Both the rotor 

and the stator do no turning of the flow field, and the steady axial Mach number is 0.35. 

The hub to tip ratio is 0.5. The rotor airfoils vibrate in torsion. Also shown in Figure 1 

is the semi-analytical solution due to Namba (private communication). Note the 

generally good agreement between the two theories. In this example, nine spinning 

modes are used to couple the unsteady flows in the rotor and stator blade rows. 

Shown again in Figure 2 is the same solution shown in Figure 1, along with the 

solution computed for the rotor in isolation, i.e., without multistage effects. Note the 

substantial differences in the two solutions. Clearly multistage effects have a dramatic 

influence on the unsteady aerodynamic response of the rotor, and should not be 

neglected. 

Shown in Figure 3 is the convergence history for the shown in Figure 1. Note that 

the flow is computed in the frequency domain; frequencies appear as parameters so that 

the flow may be solved as a steady state problem. Thus, the goal is to drive the residual 

of the solution to zero. The solution is seen to converge more than six orders of 

magnitude in only 2000 iterations. Furthermore, only a single blade passage is required 

for each blade row, making the calculation extremely efficient. 

In August 2002, the principal investigator gave a brief presentation of the 

progress of this project at the GUIde Consortium annual meeting. These vu-graphs are 

attached for reference. The project has suffered a minor setback, which was reported at 

the annual GUIde Consortium meeting. The doctoral student working on the project, 

Dmytro Voyotovich, a native of the Ukraine, returned to the Ukraine for a short visit in 

June of 2002. To return to the United States, he was required to apply for a new student 

visa. However, because of heightened security measures in the wake of September 11 th, 

the time required to obtain such visas has been greatly increased He has been recently 

notified that he will receive his visa shortly. However, as of October 20,2002, he but still 

has not received his visa, and cannot return to complete his studies. In his absence, a 

post-doctoral student, Kivanc Ekici, has largely taken over Dmytro's portion of the 

project, and is now making good progress. 

The main technical issue that we have considered in recent months was to revisit 

the issue of the coupling of unsteady solutions in the individual blade rows via spinning 

modes at the interface boundary between the individual blade rows. The results for model 

problems, when compared to known solutions, were reasonably close, but did show some 

small but systematic errors, and more troubling, did not converge to the correct solution 

as the number of spinning modes was increased. Instead, small but persistent errors 

remained in the solution. We have systematically looked for this error, and concluded 

NASAlCR-2003-212101 3 
that the error occurs only when we use multiple spinning modes in our solution, and 

further, that the error is most likely being introduced at the interface boundaries that 

connect the individual blade rows. We have just recently found and corrected this 

problem. 

Figure 4 shows a typical example. Shown is the computed unsteady twodimensional aerodynamic force for "Configuration B," a model multistage test case 

(stator/rotor/stator) presented in an earlier paper by Silkowski and Hall (1997). In this 

example, a single mode is used to couple solutions in the three blade rows. Also shown 

for comparison are the computed results using previous two-dimensional theories using 

semi-analytical flow solvers. Figure 5 shows the same example, but now with nine 

coupling modes. Note the excellent agreement we now achieve, at least with previous 

two-dimensional theories. We are in the process of performing similar validation tests on 

a semi-analytical three-dimensional solution due to Namba for unsteady multistage flow. 

Having eliminated a systematic error in our technique, we will now concentrate 

our efforts on extending the method to three-dimensional Navier-Stokes Analyses. We 

have already completed large portions of this analysis. We have developed an unsteady 

linearized Navier-Stokes analysis, although the method does not yet model tip flows, nor 

does it account for no-slip conditions on the hub and tip casings. These features will be 

added and combined with the current Euler analysis to produce the desired threedimensional Navier-Stokes analysis of unsteady flows in multistage machines (Tasks 3 

and 5). 

References 

Hall, K. c., and Silkowski, P. D., "The Influence of Neighboring Blade Rows on the 

Unsteady Aerodynamic Response of Cascades," Journal of Turbomachinery, Vol. 119, 

No. I, January 1997, pp. 85-93. 

Hall, K. C., and Smelova, N., "Time-Linearized Analysis of Unsteady Multistage Flows 

in Turbomachinery," presented at the Fourth National Turbine Engine High Cycle 

Fatigue (HCF) Conference, Monterey, California, 9-11 February 1999. 

Hall, K.C., Thomas, J. P., and Clark, W. S., "Computation of Unsteady Nonlinear Flows 

in Cascades Using a Harmonic Balance Technique," to appear AIAA Journal, 2002. 

Whitehead, D. S., 1987 "Classical Two-Dimensional Methods," AGARD Manual on 

Aeroelasticity in Axial Flow Turbomachines, Volume I, Unsteady Turbomachinery 

Aerodynamics (AGARD-AG-298), M. F. Platzer and F. O. Carta, ed., Neuilly sur Seine, 

France, ch. 3. 

NASAJCR-2003-212101 4 
Table 1. Original timeline for research. 

Project Timeline 

TASK # DESCRIPTION 1 / 2 / 3 / 4 5/6/7/8/9/10/11/12/13/14/15/16 

/ I 

1 Investigate the influence of vane 

clocking and spacing on flutter and 

forced response of representative 

turbomachines. 

2 Investigate the propagation of low 

engine order excitations through 

multiple stages. 

3 Develop block-structured version of 

multistage linearized Euler 

analysis. 

4 Develop harmonic balance version 

of multistage analysis. 

5 Develop Navier-Stokes version of 

linearized and harmonic 

balance multistage analyses. 

Prepare final report. 

~ 

Table 2. Revised timeline for research. 

Project Timeline 

TASK # DESCRIPTION 1/2/3/4/5/6/7/8 9/10/11/12 13/14/15/16 

1 Investigate the influence of vane 

clocking and spacing on flutter and 

forced response of representative 

turbomachines. 

2 Investigate the propagation of low 

engine order excitations through 

multiple stages. 

3 Develop block-structured version of 

multistage linearized Euler 

analysis. 

4 Develop harmonic balance version 

of multistage analysis. 

5 Develop Navier-Stokes version of 

linearized and harmonic 

balance multistage analyses. 

Prepare final report. 

~ 

NASAlCR-2003-212101 5 
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0. 

0. 

o 

-- 3D Euler (9 Modes) 

0. Tip (Namba, 9 Modes) 

o Midspan (Namba, 9 Modes)1 

0. Hub (Namba, 9 Modes) 

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Axial chord 

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-3 f- J I o Midspan (Namba. 9 Modes) 

0. Hub (Namba, 9 Modes) 

0.2 0.4 0.6 0.8 

Axial chord 

Figure 1. Computed unsteady pressure jump across airfoil surface. Shown are real and 

imaginary parts of unsteady pressure difference across airfoil surface. Also shown is 

semi-analytical solution due to Namba (personal communication). 

NASAlCR-2003-212101 6 
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Figure 2. Computed unsteady pressure jump across airfoil surface. Shown are real and 

imaginary parts of unsteady pressure difference across airfoil surface. Also shown is 

uncoupled solution, i.e., solution computed without multistage effects. 

NASAlCR-2003-2121 0 1 7 
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Number of iterations 

Figure 3. Convergence history of unsteady multistage calculation. 

NASA/CR--2003-212101 8 
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Interblade Phase Angle, C1 (deg) 

Figure 4. Real and imaginary part of unsteady aerodynamic lift for plunging rotor blade 

embedded in a multistage machine (stator/rotor/stator). Current three-dimensional Euler 

analysis is compared to an isolated blade row analysis (LINSUB [Whitehead, 1987]), and 

the coupled mode method of Hall & Silkowski (1997). Note almost exact agreement with 

previously derived multistage analysis. These solutions were computed using one 

coupling mode. 

NASAlCR--2003-212101 9 
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Interblade Phase Angle, 0" (deg) 

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Figure 5. Real and imaginary part of unsteady aerodynamic lift for plunging rotor blade 

embedded in a multistage machine (stator/rotor/stator). Current three-dimensional Euler 

analysis is compared to an isolated blade row analysis (LINSUB [Whitehead, 1987]), and 

the coupled mode method of Hall & Silkowski (1997). Note almost exact agreement with 

previously derived multistage analysis. These solutions were computed using nine 

coupling modes. 

NASAlCR--2003-212101 10 
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Modelling of Unsteady Three-Dimensional Flows in 

Multistage Machines 

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Kenneth C. Hall 

Department of Mechanical Engineering and Materials Science 

Edmund T. Pratt, Jr. School of Engineering 

Duke University 

GUide Consortium III, Annual Review 

July 31 - August 1, 2002 
I Goal of Research I 

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8 

~ • Multistage effects. N 

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• Viscous effects. 

• Nonlinear effects. 

Will bring together and further develop two technologies previously developed at Duke University: 

• Frequency domain (time-linearized) analysis of multistage flows. 

• Nonlinear harmonic balance analysis of unsteady viscous flows in 

single blade row. 

Final code will be able to model nonlinear, viscous, three-dimensional, 

multistage flows much more efficiently than conventional time-marching 

algorithms. 
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I Description of Multistage Coupling Mechanism I 

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waves (spinning modes). 

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o Generate single passage computational grid for each blade row and 

compute steady flow through multistage geometry. 

o For each blade row, solve several unsteady linearized Navier-Stokes 

problems in parallel on each grid, one for each spinning mode. 

o We use Ni's Lax-Wendroff "pseudo-time" time marching with local 

ti me steppi ng and multi pie grid acceleration. 

o At each iteration, pass inter-row wave information at far-field between various spinning mode solutions. 

o Iterate until converged. 

Method is computationally efficient, several orders of magnitude more 

efficient than time marching simulations. 

I Note spinning modes are only coupled at inter-row boundaries. I 
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interblade phase angle, deg. 

Work per cycle (aerodynamic damping) for blade rotor blade of 

IGV /rotor /stator configuration vibrating in first bending mode. 
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In harmonic balance approach, assume unsteady periodic flow may be 

represented by Fourier series in time, i.e. 

p(x, y, t) 2: Rn(x, y)ejwnt 

n 

Harmonic balance equations then take the form 

8U 8F(U) 8G(U) + S(U) = 0 -+ + a aT ax y 

If n harmonics are kept in solution, then 2n + 1 coefficients are stored 

for each flow variable (1 for mean flow, 2n for real and imaginary parts 

of unsteady harmonics). 

Note harmonics are coupled via nonlinearities in governing equations. 
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IIiSimultaneous Dual Time-Step" Form of Harmonic Balance I 

Computation of harmonic fluxes difficult and computationally expensive, 

especially for viscous flows. Alternatively, could store solution at 2n + 1 

equally spaced points in time over one temporal period. 

-

U == EU* 

U* == E-1iJ 

Where the matrices E and E-1 are discrete Fourier transform and inverse Fourier transform operators. Thus, pseudo-time harmonic balance 

eq uations become 

oEU* + oEF* + oEG* + jwNEU* = 0 

8T 8x 8y 

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8U* 8F* 8G* 

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8U* 8F* 8G* 

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where 

S* = jw[Erl[N][EJ U* ~ au* 

y ,I 

a/at 

Here we use spectral operator to compute time derivative. Using finite 

difference does not work well. Use of spectral difference operator allows 

for very coarse temporal discretization. 

• Note that si nce on Iy IIsteady-state" sol ution is desi red, ca n use local time stepping, multiple-grid acceleration techniques, and residual smoothing to speed convergence. 

• For 2D and 3D cascades, only a single blade passage is required, 

with complex periodicity conditions along periodic boundaries. 

• Because we work in the frequency domain, essentially exact nonreflecting boundary conditions are available. 
I Example: Transport of Stator Wakes in Low-Speed Turbine Rotor I 

z 

~ Hodson's low-speed turbine. M == 0.146, w G/V == 8, (J == 270°, Gaussian f velocity and density defect in stator wake. Three modes used in harmonic 

§ balance analysis. Single passage 129x33 computational grid. I 

N 

..... 

N 

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~.. ~J . ....•...... ~ ~_I ··· ......... r ....... 

Linearized Analysis 

40.00 

39.27 

38.55 

37.82 

37.09 

36.36 

35.64 

34.91 

34.18 

33.45 

32.73 

32.00 

Harmonic Balance Analysis 

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I Nonlinear Unsteady Flow Using Harmonic Balance I 

0.8 

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:::::-

c.. 

~ 0.6 :J 

(/) 

(/) 

Q) 

... 

Cl.. 0.5' 

s: 

0 

iI o ci = 0.01 (deg) 

c: o ci = 0.1 (deg) 

m ') ci = 0.5 (deg) Q) Ll ci = 1.0 (deg) ~ 

0.3 

0.0 0.2 0.4 0.6 0.8 1.0 

Airfoil Surface Location, xlc 

~ 10 ~ 8 

0 0]1 

:U ]1 8 o ci = 0.01 (de g) o ci = 0.01 (deg) .Ii o ci = 0.1 (deg) .Ii oci=0.1 (deg) .3: .3: ~ 6 < ci = 0.5 (deg) ~ ,j ci = 0.5 (deg) 

c: Ll ci = 1.0 (deg) S Ll ci = 1.0 (deg) 

Q) 4 E a: 

a> 2 ~ 2 ... 

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a.. -2 Cl.. 

>- >- -2 

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(ij -8 Cl -6 

0.0 0.2 0.4 0.6 0.8 1.0 m Q) 0.0 0.2 0.4 0.6 0.8 1.0 E a: 

Airfoil Surface Location, xlc Airfoil Surface Location, xlc 

Unsteady pressure distribution for front stage compressor rotor airfoils vibrating 

in pitch with w == 1.0 and (J' == 30°. Top: zeroth harmonic. Bottom: first 

harmonic. 
z 

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~ VJ 

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N 

I Nonlinear Unsteady Flow Using Harmonic Balance I 

0.8 0.4 

..-.. 0.4 0.2 ..-.. l(j l(j 

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c 

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11 a=1.0 (deg) 5 Harmonics 

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~ <> a=O.5 (deg) 5 Harmonics 

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ca 

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Interblade Phase Angle, (j (deg) Interblade Phase Angle, (j (deg) 

First harmonic of unsteady pitching moment for front stage compressor rotor 

airfoils vibrating in pitch with w == 1.0. 

180 
z 

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I Nonlinear Unsteady Flow Using Harmonic Balance I 

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Pitch Amplitude, a (deg) Pitch Amplitude, a (deg) 

First harmonic of unsteady pitching moment for front stage compressor rotor 

airfoils vibrating in pitch with a = 30°. 

1.0 
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r 

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I Nonlinear Unsteady Flow Using Harmonic Balance I 

:::::: -4 II 

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Iteration Number 

Convergence history for steady flow solver (left) and harmonic balance flow solver 

(right) with a == 30°, w == 1.0, and ~a == 1.0°. 
z 

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en 

~ 

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8 

w 

I 

tv 

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~ 

I Statement of Work I 

The following tasks will be performed over a four-year period: 

1. We will use the existing and newly developed three-dimensional 

linearized Euler and Navier-Stokes multistage analysis codes developed under the AFOSR grant and this program to investigate 

and characterize the influence of vane clocking on the forced response and flutter of representative multistage machines. 

2. We will use the existing and newly developed three-dimensional 

linearized Euler and Navier-Stokes multistage analysis codes to 

investigate the propagation of low-engi ne order excitations th rough 

multiple stages. 

3. We will further develop the existing multistage linearized multistage 

analysis. Specifically, we will develop a block-structured version of 

the analysis (the existing method uses only H-grids). The blockstructured version will improve the accuracy of the analysis, and 

eventually enable tip clearances and other flow path features to be 

modelled. 
z 

;J> 

en 

~ 

VJ 

~ I 

N 

-N 

-

o 

..... 

N 

0\ 

I Statement of Work, Continued I 

The following tasks will be performed over a four-year period: 

4. We will further develop the unsteady linearized block-structured 

version of the multistage analysis, converting the code into a nonlinear harmonic balance analysis, which will be capable of analyzing nonlinear unsteady flows for the special case of vane/rotor or 

vane/rotor/vane problems with equal vane counts. 

5. We will further develop both the linearized and nonlinear multistage 

analyses, converting the Euler codes into Navier-Stokes codes. 
z 

> [/J 

~ 

I Timeline I 

Project Timeline 

~ I TASK # I DESCRlPllON w 

I 

N 

-

N 

-o 

-

N 

--l 

1 Investigate the influence of vane 

clocking and spacing on flutter and 

forced response of representative 

turbomachines. 

2 I Investigate the propagation of low 

engine order excitations through 

multiple stages. 

3 I Develop block-structured version of 

multistage linearized Euler 

analysis. 

4 I Develop harmonic balance version 

of multistage analysis. 

5 I Develop Navier-Stokes version of 

linearized and harmonic 

balance multistage analyses. 

Prepare final report. 

Steering committee recommended changes to this timeline, pushing linearized Navier-Stokes work up. 
I Timeline I 

z 

>- [/) 

~ I TASK # I DESCRIPTION I 1 

:::0 

I 

N 1 Investigate the influence of vane 8 

'JJ clocking and spacing on flutter and , 

N 

-t-.J 

- forced response of representative 0 

- turbomachines. 

2 I Investigate the propagation of low 

engine order excitations through 

multiple stages. 

3 I Develop block-structured version of 

~I multistage linearized Euler 

analysis. 

4 I Develop harmonic balance version 

of multistage analysis. 

5 I Develop Navier-Stokes version of 

linearized multistage analyses. 

Prepare final report. 
z 

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e 

1 I 

n 

r 

8 

\.N 

I 

tv 

..... 

tv 

..... 

0 

..... 0 

/' 

- ~ c.. 

<I 

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a: 

-1 ~ f 

tv 

\0 III 

-2 III 

0 0.2 

I Namba Test Case: (J" == 94.7 deg·1 

~ /)1 

0 0 0 -c.. 

<I 

-- 0 E 

1-----==-30 Eul~~ (-H G~id)l 

o Tip (Namba) I 

<> Midspan (Namba) 

o Hub (Namba) 

0.4 0.6 0.8 1 

Axial Chord 

1 If 

~ o ' 

-1 

-2 

-3 

-4 

0 

~----------,---- --~-------,------------ -~, - ------------'---1 

o 0 0 0 

<>-

o 

0.2 

~ 

Q 

Q 

1=--30 Eu~r (H-Grid) 

, 0 Tip (Namba) 

0.4 

<> Midspan (Namba) 

o Hub (Namba) 

0.6 0.8 

Axial Chord 

1 

1 
I Namba Test Case: (J == 180 deg·1 

z 

~ 

V1 

~ 1 1 

f 

0 0 N 

8 0 0 0 

~I 0 w 0 I 

N 

...... 

N 

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..... 0 

/ v.----- ;-, ~I 

-1 

-0. -0. <I <I '-' '-' 

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a: 

-2 

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-1 

( -~--------I 

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0 

() Midspan (Namba) () Midspan (Namba)r o Hub (Namba) o Hub (Namba) I 

-2 If -4 

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 

Axial Chord Axial Chord 
I Namba Test Case: (j == 94.7 deg·1 

z 

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Cf.l 

n 

~ 1 I 1 

:;0 

I 

N 

8 

w 

I 

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-

N 

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0 

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<l <l 

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-2 0 

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<> Midspan (Namba) I <> Midspan (Namba) I 

o Hub (Namba) , o Hub (Namba) i 

~----------~-----~ 

-2 -4 

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 

Axial Chord Axial Chord 
J8QwnN SUO!lBJ8lI10 

OOS OOO~ OOS~ OOOG OOSG I I I I 

-

-

-

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O~- -

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8- -

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CD 

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0-

c: 

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-

N 

M 

o -

N 

-

I N 

-

~, 

~ ~ rJl 

<t: 

Z 
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N 

-

I N 

-

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~ VJ 

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Z 
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N 

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8 

J 

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<r: 

z 

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I Progress to Date I 

r During first year, have most of pieces in place for multistage linearized 

8 Navier-Stokes analysis. w 

I 

N 

-

§ • We have lIc1eaned up" original H-grid Euler code developed in 

W 

0'1 

earlier project. We use this code for IIboot strap" verification of 

H-O-H codes. 

• We have developed H-O-H time-linearized Euler multistage code. 

Still debugging. Final debugging in progress. 

• We have developed H-O-H Navier-Stokes solver. Development well 

underway (single blade row). Have steady code well in hand. Currently working on time-linearized version. 

• Will in the next quarter work to couple multiple stages in H-O-H 

Navier-Stokes code. 

Have had minor setback. Ph.D. student working on this project is stuck 

in Ukraine with visa problems. 
z 

;l> 

en 

~ 

1 Plans for Year 21 

f In Year 2, we will begin development of nonlinear multistage analysis, and 

8 will use linearized analysis to investigate problems of interest to industry. VJ 

I 

tv 

-

§ • We will complete development of linearized multistage NavierVJ 

-...l 

Stokes analysis. 

• Will develop harmonic balance form of multistage analysis. 

• We will begin to use linearized analysis to examine low-engine order 

excitations in engines (long wavelength disturbances can travel long 

distances through machine). 

• We will begin to investigate the influence of blade row spacing on 

flutter and forced response. 
REPORT DOCUMENTATION PAGE I Form Approved 

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gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this 

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1. AGENCY USE ONLY (Leave blank) 12. REPORT DATE 13. REPORT TYPE AND DATES COVERED 

January 2003 Interim Contractor Report-April 200 I-September 2002 

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS 

Modeling of Unsteady Three-Dimensional Flows in Multistage Machines 

VVBS-22-708-87-23 

6. AUTHOR(S) NAG3-2627 

Kenneth C. Hall and Edmund T. Pratt 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION 

REPORT NUMBER 

Duke University 

Durham, North Carolina 27708 E-13754 

9. SPONSORINGIMONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORINGIMONITORING 

AGENCY REPORT NUMBER 

National Aeronautics and Space Administration 

Washington, DC 20546-0001 NASA CR-2003-212101 

11. SUPPLEMENTARY NOTES 

Project Manager, Anatole Kurkov, Structures and Acoustics Division, NASA Glenn Research Center, organization code 

5930,216-433-5695. 

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE 

Unclassified - Unlimited 

Subject Category: 34 Distribution: Nonstandard 

Available electronically at http://gItrs.grc.nasa.gov 

This publication is available from the NASA Center for AeroSpace Information. 301-621-0390. 

13. ABSTRACT (Maximum 200 words) 

This report describes progress made on NASA Glenn Research Center Grant NAG3-2627 for the period April 2001 

through September 2002. This grant was funded by NASA as part of the GUIde Consortium III program. This report 

details work to date, as well as plans for the upcoming reporting period. 

14. SUBJECT TERMS 

Fluid mechanics; Heat transfer 

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 

OF REPORT OF THIS PAGE OF ABSTRACT 

Unclassified Unclassified Unclassified 

NSN 7540-01-280-5500 

15. NUMBER OF PAGES 

43 

16. PRICE CODE 

20. LIMITATION OF ABSTRACT 

Standard Form 298 (Rev. 2-89) 

Prescribed by ANSI Std. Z39-18 

298-102