Kenneth C. Hall and Edmund T. Pratt

Duke University, Durham, North Carolina

Modeling of Unsteady Three-Dimensional

Flows in Multistage Machines

NASA/CR—2003-212101

January 2003
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Kenneth C. Hall and Edmund T. Pratt

Duke University, Durham, North Carolina

Modeling of Unsteady Three-Dimensional

Flows in Multistage Machines

NASA/CR—2003-212101

January 2003

National Aeronautics and

Space Administration

Glenn Research Center

Prepared under Grant NAG3–2627
Available from

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

The Propulsion and Power Program at

NASA Glenn Research Center sponsored this work.

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

Abstract

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

NASA/CR—2003-212101 1
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 HNASA/CR—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 11th,

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

NASA/CR—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.!1, 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 1, Unsteady Turbomachinery

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

France, ch. 3.

NASA/CR—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

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.

NASA/CR—2003-212101 5
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).

NASA/CR—2003-212101 6
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.

NASA/CR—2003-212101 7
Figure 3. Convergence history of unsteady multistage calculation.

NASA/CR—2003-212101 8
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.

NASA/CR—2003-212101 9
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.

NASA/CR—2003-212101 10
Modelling of Unsteady Three-Dimensional Flows in

Multistage Machines

–180 –90 0 90 180

interblade phase angle, deg. 

–10

–2–4–6–8

0

aerodynamic work per cycle

uncoupled1 mode7 modes

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

NASA/CR—2003-212101 11
Goal of Research

Goal of research is to develop analysis tool for predicting unsteady aerodynamic forces acting on airfoils of multistage machine.

• Multistage effects.• 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-marchingalgorithms.

NASA/CR—2003-212101 12
Description of Multistage Coupling Mechanism

◦ Neighboring blade rows are coupled via pressure, vorticity, and entropy

waves (spinning modes).

1

34

2

N = k0 + kB2 N = k0 + kB2

N = k0 + nB1 + kB2 N = k0 + nB1 + kB2

ω' = ω0 +(k0 + kB2 )Ωω' = ω0 +(k0 + kB2 )Ω ω = ω0

ω = ω0

N = k0

STATOR ROTOR

ω = ω0 − nB Ω1

5

ΩR

x

θ

′ θ

′

x

◦ Original disturbance with a single frequency and interblade phase angle

gives rise to response with multiple interblade phase angles (scattering)

and frequencies (shifting).

NASA/CR—2003-212101 13
C1 Compressor

Computational grid near mid-span of front stage (IGV/rotor/stator) of

modern compressor.

NASA/CR—2003-212101 14
Time-Linearized Multistage Modelling

◦ Generate single passage computational grid for each blade row and

compute steady flow through multistage geometry.

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

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

◦ We use Ni’s Lax-Wendroff “pseudo-time” time marching with local

time stepping and multiple grid acceleration.

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

◦ Iterate until converged.

Method is computationally efficient, several orders of magnitude more

efficient than time marching simulations.

Note spinning modes are only coupled at inter-row boundaries.

NASA/CR—2003-212101 15
C1 Compressor

–180 –90 0 90 180

interblade phase angle, deg. 

–

–10

8

–6–4–

0

2

aerodynamic work per cycle

uncoupled1 mode7 modes

Work per cycle (aerodynamic damping) for blade rotor blade ofIGV/rotor/stator configuration vibrating in first bending mode.

NASA/CR—2003-212101 16
Solution of Harmonic Balance Euler Equations

In harmonic balance approach, assume unsteady periodic flow may be

represented by Fourier series in time, i.e.

ρ(x, y,t) = 

n

Rn(x, y)ejωnt

Harmonic balance equations then take the form

∂

 ̃

U

∂τ +

∂

 ̃

F(

 ̃

U)

∂x +

∂

 ̃

G(

 ̃

U)

∂y +

 ̃

S(

 ̃

U)=0

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

NASA/CR—2003-212101 17
“Simultaneous Dual Time-Step” Form of Harmonic Balance

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−1  ̃U

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

equations become

∂EU∗

∂τ +

∂EF∗

∂x +

∂EG∗

∂y + jωNEU∗ = 0

Pre-multiplying by E−1 gives

∂U∗

∂τ +

∂F∗

∂x +

∂G∗

∂y + jωE−1NE   

∂/∂t

U∗ = 0

NASA/CR—2003-212101 18
“Simultaneous Dual Time-Step” Form of Harmonic Balance

∂U∗

∂τ +

∂F∗

∂x +

∂G∗

∂y + S∗ = 0

where

S∗ = jω[E −] 1[N][E]

  

∂/∂t

U∗ ≈ ∂U∗ ∂t

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 since only “steady-state” solution is desired, can 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.

NASA/CR—2003-212101 19
Example: Transport of Stator Wakes in Low-Speed Turbine Rotor

Hodson’s low-speed turbine. M = 0.146, ω G/V = 8, σ = 270◦, Gaussian

velocity and density defect in stator wake. Three modes used in harmonic

balance analysis. Single passage 129×33 computational grid.

 40.00 39.27 38.55 37.82 37.09 36.36 35.64 34.91 34.18 33.45 32.73 32.00

 40.00 39.27 38.55 37.82 37.09 36.36 35.64 34.91 34.18 33.45 32.73 32.00

Linearized Analysis Harmonic Balance Analysis

NASA/CR—2003-212101 20
Nonlinea

r Unsteady Flo

w Using Harmonic Balance

0.0 0.2 0.4 0.6 0.8 1.0

Airfoil Surface Location, x/c

0.80.70.60.50.40.3

Mean Flow Pressure, p/(qinletc) −

α = 0.01 (deg)

 −

α = 0.1 (deg)

 −

α = 0.5 (deg)

 −

α = 1.0 (deg)

0.0 0.2 0.4 0.6 0.8 1.0

Airfoil Surface Location, x/c

−8−6−4−

864202

10

Real Unsteady Pressure, Re(p1)/(qinlet αc −)

−

α = 0.01 (deg)

−

α = 0.1 (deg)

−

α = 0.5 (deg)

−

α = 1.0 (deg)

0.0 0.2 0.4 0.6 0.8 1.0

Airfoil Surface Location, x/c

−6−4−

864202

Imag Unsteady Pressure, Im(p1)/(qinlet αc −)

−

α = 0.01 (deg)

−

α = 0.1 (deg)

−

α = 0.5 (deg)

−

α = 1.0 (deg)

Unsteady

pressure distribution for front stage compressor rotor airfoils vibrating

in pitch with

ω = 1.0 and

σ = 3

0

◦. Top: zeroth harmonic. Bottom: first

harmonic.

NASA/CR—2003-212101 21
Nonlinea

r Unsteady Flo

w Using Harmonic Balance

−180

−120

−60

0

6

0 120 180

Interblade Phase Angle, 

σ (deg)

−2.4−2.0−1.6−1.2−0.8−

0.80.40.00.4

1 Real Moment, Re(M e.a.

)/(qinlet αc −)

−

α=0.01 (deg) 1 Harmonic

−

α=0.1 (deg) 5 Harmonics

−

α=0.5 (deg) 5 Harmonics

−

α=1.0 (deg) 5 Harmonics

−180

−120

−60

0

6

0 120 180

Interblade Phase Angle, 

σ (deg)

−1.4−1.2−1.0−0.8−0.6−0.4−

0.40.20.00.2

1 Imag Moment, Im(M e.a.

)/(qinlet αc −)

−

α=0.01 (deg) 1 Harmonic

−

α=0.1 (deg) 5 Harmonics

−

α=0.5 (deg) 5 Harmonics

−

α=1.0 (deg) 5 Harmonics

First harmonic of unsteady pitching moment for front stage compressor rotor

airfoils vibrating in pitch with

ω = 1.0.

NASA/CR—2003-212101 22
Nonlinea

r Unsteady Flo

w Using Harmonic Balance

0.0 0.2 0.4 0.6 0.8 1.0

Pitch Amplitude, −

α (deg)

−1.4−1.2−1.0−0.8−0.6−0.4

1 Real Moment, Re(M e.a.

)/(qinlet αc −)

1 Harmonic3 Harmonics5 Harmonics7 Harmonics

0.0 0.2 0.4 0.6 0.8 1.0

Pitch Amplitude, −

α (deg)

−

0.80.60.40.20.00.2

1 Imag Moment, Im(M e.a.)/(qinlet αc −

)

1 Harmonic3 Harmonics5 Harmonics7 Harmonics

First harmonic of unsteady pitching moment for front stage compressor rotor

airfoils vibrating in pitch with

σ = 3

0

◦.

NASA/CR—2003-212101 23
Nonlinea

r Unsteady Flo

w Using Harmonic Balance

0 2000 4000 6000 8000 10000

Iteration Number

−

−

10

9

−

8

−

7

−

6

−

5

−

4

Solution Residual, Log10(L2(Residual))

0 2000 4000 6000 8000 10000

Iteration Number

−

−

10

9

−

8

−

7

−

6

−

5

−

4

Solution Residual, Log10(L2(Residual)) 1 Harmonic 3 Harmonics 5 Harmonics 7 Harmonics

Convergence history for steady flo

w solver (left) and harmonic balance flo

w solver

(right) with

σ = 3

0

◦, ω = 1.0, and

∆

α = 1.0

◦.

NASA/CR—2003-212101 24
Statement of Work

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-engine order excitations through

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.

NASA/CR—2003-212101 25
Statement of

W

ork, Continued

The following tasks will

b

e

performed over

a four-yea

r

period:

4.

W

e will further develop the unsteady linearized block-structured

version of the multistage analysis, converting the code into

a nonlinea

r

h

armonic balance analysis, which will

b

e capable of analyzing nonlinea

r unsteady flows fo

r the special case of vane/roto

r

o

r

vane/rotor/vane

problems with equal vane counts.

5.

W

e will further develop

both the linearized and nonlinea

r multistage

analyses, converting the Euler codes into Navier–Stokes codes.

NASA/CR—2003-212101 26
Timeline

Project Timeline

TASK # DESCRIPTION 123456789 1 0 1 1 1 2 1 3 1 4 1 5 1 6

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.

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

NASA/CR—2003-212101 27
Timeline

TASK # DESCRIPTION 123456789 1 0 1 1 1 2 1 3 1 4 1 5 1 6

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 multistage analyses.Prepare final report.

NASA/CR—2003-212101 28
Namba

Test Case:

σ

= 94.7 deg.

0 0.2 0.4 0.6 0.8 1

Axial Chord

−

2

−

101

p) ∆ Re(

3D Euler (H Grid)Tip (Namba)Midspan (Namba)Hub (Namba)

0 0.2 0.4 0.6 0.8 1

Axial Chord

−

4

−

3

−

2

−

101

p) ∆ Im(

3D Euler (H Grid)Tip (Namba)Midspan (Namba)Hub (Namba)

NASA/CR—2003-212101 29
Namba

Test Case:

σ

= 180 deg.

0 0.2 0.4 0.6 0.8 1

Axial Chord

−

2

−

101

p) ∆ Re(

3D Euler (H Grid)Tip (Namba)Midspan (Namba)Hub (Namba)

0 0.2 0.4 0.6 0.8 1

Axial Chord

−

4

−

3

−

2

−

101

p) ∆ Im(

3D Euler (H Grid)Tip (Namba)Midspan (Namba)Hub (Namba)

NASA/CR—2003-212101 30
Namba

Test Case:

σ

= 94.7 deg.

0 0.2 0.4 0.6 0.8 1

Axial Chord

−

2

−

101

p) ∆ Re(

3D Euler (HOH Grid)Tip (Namba)Midspan (Namba)Hub (Namba)

0 0.2 0.4 0.6 0.8 1

Axial Chord

−

4

−

3

−

2

−

101

p) ∆ Im(

3D Euler (HOH Grid)Tip (Namba)Midspan (Namba)Hub (Namba)

NASA/CR—2003-212101 31
Namba

Test Case

0 500 1000 1500 2000 2500

Number of Iterations

−12−11−

−

10

9

−

8

−

7

−

6

−

5

−

4

Residual

NASA/CR—2003-212101 32
C1 Compressor. H-O-H computational grid

NASA/CR—2003-212101 33
C1 Compressor. Mach Number Distribution.

NASA/CR—2003-212101 34
C1 Compressor.

Velocit

y

Vectors.

NASA/CR—2003-212101 35
Progress to Date

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

Navier-Stokes analysis.

• We have “cleaned up” original H-grid Euler code developed in

earlier project. We use this code for “boot 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.

NASA/CR—2003-212101 36
Plans for Year 2

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

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

• We will complete development of linearized multistage NavierStokes 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.

NASA/CR—2003-212101 37
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Interim Contractor Report—April 2001–September 2002

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National Aeronautics and Space Administration

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

January 2003

NASA CR—2003-212101

E–13754

WBS–22–708–87–23

NAG3–2627

43

Modeling of Unsteady Three-Dimensional Flows in Multistage Machines

Kenneth C. Hall and Edmund T. Pratt

Fluid mechanics; Heat transfer

Unclassified -Unlimited

Subject Category: 34 Distribution: Nonstandard

Duke University

Durham, North Carolina 27708

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

5930, 216–433–5695.

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.