Gust Response Analysis of a Turbine Cascade

NASA/TM—2001-210562

January 2001

R.S.R. Gorla

Cleveland State University, Cleveland, Ohio

T.S.R. Reddy

The University of Toledo, Toledo, Ohio

D.R. Reddy and A.P. Kurkov

Glenn Research Center, Cleveland, Ohio
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NASA/TM—2001-210562

January 2001

National Aeronautics and

Space Administration

Glenn Research Center

Prepared for the

46th International Gas Turbine and Aeroengine Technical Congress,

Exposition, and Users Symposium

sponsored by the American Society of Mechanical Engineers

New Orleans, Louisiana, June 4–6, 2001

R.S.R. Gorla

Cleveland State University, Cleveland, Ohio

T.S.R. Reddy

The University of Toledo, Toledo, Ohio

D.R. Reddy and A.P. Kurkov

Glenn Research Center, Cleveland, Ohio

Gust Response Analysis of a Turbine Cascade
Acknowledgments

One of the authors (RSRG) is grateful to Dr. Shantaram S. Pai, Mr. Jeff Rusick, and Mr. Oral Mehmed of

NASA Glenn Research Center for their interest and financial support.

Available from

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This report is a preprint of a paper intended for presentation at a conference. Because

of changes that may be made before formal publication, this preprint is made

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This report is a formal draft or working

paper, intended to solicit comments and

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This report contains preliminary

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analysis proceeds.

Available electronically at http://gltrs.grc.nasa.gov/GLTRS
NASA/TM—2001–210562 1

GUST RESPONSE ANALYSIS OF A TURBINE CASCADE

R.S.R. Gorla

Department of Mechanical Engineering

Cleveland State University

Cleveland, Ohio 44115

T.S.R. Reddy

Department of Mechanical Engineering

The University of Toledo

Toledo, Ohio 43606

D.R. Reddy and A.P. Kurkov

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

SUMMARY

A study was made of the gust response of an annular turbine cascade using a two-dimensional Navier

Stokes code. The time-marching CFD code, NPARC, was used to calculate the unsteady forces due to the fluid flow.

The computational results were compared with a previously published experimental data for the annular cascade

reported in the literature. Reduced frequency, Mach number and angle of incidence were varied independently and

the gust velocity was sinusoidal. For the high inlet velocity case, the cascade was nearly choked.

NOMENCLATURE

A,B amplitudes

Ch blade chord length

M Mach number

N number of cylinders

p static pressure

S blade spacing

T temperature

U1 axial inlet velocity

t time

u,v axial and tangential velocity components

x,y axial and tangential coordinate directions

Greek Symbols:

α phase angle

β incidence

φ flow coefficient

ρ density

σ solidity (Ch/S)

ωh angular velocity

Subscripts:

1 inlet conditions

2 conditions downstream of rotor
NASA/TM—2001–210562 2

w rotor conditions

r radial conditions

h blade properties

INTRODUCTION

Periodic wake-passing from upstream blade rows in the gas turbine flow field strongly influences the

unsteady forces on the surfaces of the downstream blades. Manwaring and Wisler (1993) conducted a series of

experiments involving a gust inlet velocity in a large speed compressor and turbine rig and provided aerodynamic

forced response measurements on downstream stators. Kurkov and Lucci (1997) provided experimental data on a

gust response of an annular turbine cascade in a transonic flow regime. They varied the reduced frequency, Mach

number and incidence angle independently. The mean flow was documented by measuring blade surface pressures

and cascade exit flow.

The unsteady aerodynamics generated by a strong blade row coupling and the interactions due to the highly

unsteady flow induce vibrations, which can lead to high cycle fatigue. In order to predict blade life or design blades

for longer high cycle fatigue life, accurate predictions of the multiblade row unsteady aerodynamics are crucial.

Turbomachinery aeroelastic analyses have been carried out at NASA Glenn Research Center for the past decade.

Kielb and Chiang (1992) reviewed research related to the forced response analyses of turbomachinery. Hall and

Verdon (1990) provided an analysis for gust response for cascades. Giles (1988) computed the interaction of an

unsteady wake with a rotor. Reddy and Srivastava (1994) presented an analysis of a rotor-stator stage interaction

by solving Euler equations. Reddy and Srivastava (1996) extended their previous work to include the effects of

incoming gust. Barter et al. (2000) studied the interaction effects in a transonic turbine stage by means of a threedimensional, viscous, time-accurate computational code. Their results showed that interaction effects must be taken

into account in order to accurately predict the unsteady loading on the upstream blade row. Haldeman et al. (2000)

reported a combined experimental and computational effort quantifying unsteady airfoil interactions.

The present work was undertaken in order to study the fluid/structure interaction for the turbomachinery

blade configuration using the state of the art CFD techniques. The time-marching CFD code, NPARC was used to

calculate the unsteady forces due to the fluid flow. The computational results are compared with the experimental

data of Kurkov and Lucci (1997).

GOVERNING EQUATIONS

The differential equations used are the Reynolds mass-averaged Navier-Stokes equations inside a rotating

blade passage and may be written as:

ρ ρ

ρ ρ ρε μ δ ρ

ρ ρ μ

, ,

,, , ,

, , Pr ,

, , ,

t ii

it i j j ijk j k i i j ji k k ij i j j i

t ij

eff

j

U

U UU U p U U U uu F

e p Ue T

+( ) =

( ) +( ) + =− + + − 





 − 



 



 +

( ) − +( ) = 

















0

2 2

3

Ω







−( ) + + +− 











 





=

,, , j j j i i i i j ji pU U F U U U U , , k k, ij j

p RT

μ δ

ρ

2

3

where Ui = mean velocity, ui = fluctuating velocity, e = total energy, Ωi = angular velocity. For stationary blades,

we set Ωi = 0. It is well known that neither the conventional mixing length type turbulence model nor any standard

two-equation type turbulence model describes turbulence stresses properly in the region behind a shock wave or in

the separated flow regions. Several recent studies have shown that significant improvement can be achieved when

the standard two-equation model is modified to include the low Reynolds number effects. The following are the

transport equations for the turbulent shear stress:
NASA/TM—2001–210562 3

ρ ρ

μ

σ

ρ ρε μ

ρε ρ ε

μ

σ

ε ρε ρε ε

ν ν

ε

k U k k uu U k

l

U C

k

uuU

k C f

k Cu l

l

t jj

eff

k

j j i j ji

t ij

eff

j j i j ji

( ) +( ) = 



 



 + [ ] − −

( ) +( ) = 



 



 + [ ] − + ( ) −

, , ,,

, , ,, exp * / ,

2

2

2

1 2

4

2













Standard expressions were used for μeff, f and C’s. These details are omitted in order to conserve space.

The mean flow and turbulence equations were integrated in time using a fully coupled approximately factored

implicit backward Euler method.

CODE DESCRIPTION

Two-dimensional time-marching flow simulations were performed with NPARC version 3.1 code. NPARC

solves the Euler or Navier-Stokes equations in conservation law form on a multi block body fitted grid system. The

fluid is assumed to be ideal and Newtonian and the Fourier heat conduction law is assumed for heat transfer. The

flow can be assumed to be laminar, turbulent or inviscid. A variety of turbulence models, including the k-ε model

can be selected. The flow simulations were performed in two steps. First, a converged steady state flow solution

was achieved. In the second step, a convective gust perturbation of known strength was introduced at the upstream

boundary and allowed to propagate down the cascade. The steady state simulation to initialize the flow field was

done with the approximate factorization algorithm (ISOLVE = 1) using local time stepping (IVARDT = 2) with

DTCAP = 1. Typically, about 2500 time steps were required to reduce the L2 norm to about 10–7. For the time

accurate computations, a five step Jameson algorithm second order accurate in time was selected (ISOLVE = 5). The

time step throughout the grid block was set to a constant equal to the CFL limit specified by DTCAP (IVARDT = 4

and DTCAP = 0.3) at the location of maximum change in the flow variables.

GRID GEOMETRY DESCRIPTION AND BOUNDARY CONDITIONS

Figure 1 shows a single grid block used for computations. The airfoil upper and lower surfaces are located

along lines BC and FG, respectively where solid wall boundary conditions are prescribed. Lines AE and DH represent inflow and outflow boundaries, respectively. The flow incidence angle is specified and the backpressure is

adjusted until the average Mach number along the inflow boundary AE matches a specified value. Periodicity is

imposed between lines AB and EF and lines CD and GH.

Computational grids were generated using the GRIDGEN2D grid generation program. A uniform grid

spacing was used in each coordinate direction. A grid refinement study was completed to ensure that the computed

results were independent of the grid density. A grid of 41×118 was selected for the cascade simulation. There were

60 points on the airfoils, 29 points between the inlet and leading edge and the rest between the trailing edge and

exit plane.

For treating the velocity perturbations, namely, the gust for the present problem, the wake profile was fitted

to experimental velocity profile at the inlet in the following form:

u

U

N y

R

t

v

U

N y

R

t

w

w

1

1

1

2

1

1

=+ + 





 + 



 

















=+ + 





 + 



 

















ASin

BSin

ω α

ω α

In the above equations, A and B are the amplitudes; U1 the axial inlet velocity and α1 and α2 the phase angles.
NASA/TM—2001–210562 4

DESCRIPTION OF THE EXPERIMENTAL SETUP

The air entered the bellmouth and passed through the annulus to the rotor consisting of radial pins, either

0.317 or 0.476 cm (1/8 or 3/16 in.) in diameter. The wakes from the pins were convected by the main flow into a

turbine cascade. The rotor pins and the turbine cascade are shown in figure 2. The rotor induced a swirl of 3 to 5°

into the mean flow. The cascade turning angle was 61°, the blade chord angle relative to the axial direction was 38°

and the distance between the rotor and the cascade leading edge was 3.9 axial chords. The chord length was 4.75 cm

and annulus outside and inside diameters were 40.64 and 27.13 cm, respectively. The cascade consisted of 23 blades

that extended in the radial direction spanning the annulus. The power was delivered to the rotor through a long shaft

that extended vertically through the inner pipe of the annulus. The shaft exits the outer pipe at the 90°-bend close to

the floor level and is terminated by a belt-driven pulley, which transmits the power from the electric motor.

The number of pins on the rotor was varied among 24, 12 and 6. The pin diameter for the 24-pin configuration was 0.317 cm and for 12 and 6-pin configurations, it was 0.476 cm. The pin diameter was increased for the last

two configurations in order to reduce the number of harmonics in the wake. The reduced frequency

ω

ω

φ = = N c

U

w r h w N

1

0 375 . was varied by changing the number of pins on the rotor, Nw. Here, ωr

 is the shaft angular

velocity, ch the blade chord, U1 the axial inlet velocity and φ the flow coefficient. Subscript 1 denotes the station

downstream of the bellmouth and upstream of the rotor. The reduced frequency values were 10, 5 and 2.5 for the

available values of Nw. The inlet Mach number was either 0.2 or 0.27. At the higher inlet Mach number the flow in

the blade passage throat was near choking at the midspan radius. The flow coefficient was kept constant at 0.9 and

the shaft speeds for the two Mach numbers were 4290 and 5790 rpm. The Reynolds numbers corresponding to low

and high inlet velocities were 2.7×105 and 3.6×105, respectively based on the blade chord. Positive and negative

values of incidence correspond to two directions of rotation.

The inlet condition was defined by the barometric pressure, inlet temperature at the bellmouth inlet and

several static pressure measurements on the inside and outside wall of the annulus downstream of the bellmouth.

The cascade exit flow was surveyed at the blade midspan radius over a circumference in excess of one blade pitch

with a five-hole probe. This station was one axial chord length downstream of the cascade trailing edge. The locations of the steady state and dynamic ports on the instrumented blades are shown in figure 3. Kurkov and Lucci

(1997) have documented in more detail the measurements taken.

RESULTS AND DISCUSSION

Figure 4 displays the steady state results for the pressure coefficient distribution for inlet Mach number

M1 = 0.2 and inlet angle of attack of 4.59°. Figure 5 shows the steady state results for the pressure coefficient distribution for inlet Mach number M1 = 0.27 and inlet angle of attack of 5.17°. The computed results agree with the

experimental data quite well in both cases. The spikes in the pressure coefficient distribution in the vicinity of the

trailing edge may be due to the definition of the airfoil surface coordinates in generating the grid.

Figures 6 to 9 show results for the distribution of the amplitude of the unsteady pressure coefficient (percentage of the inlet dynamic head) for inlet Mach number M1 = 0.2 and reduced frequency of 10 based on chord

length. Figures 10 to 13 show results for the distribution of the amplitude of the unsteady pressure coefficient for

inlet Mach number M1 = 0.27 at the same reduced frequency. The computed results agree with the experimental

results within 20 percent error over most of the blade length, both on the suction and the pressure surfaces. However, in the vicinity of the leading edge of the blade, the discrepancy seems to be higher.

Figures 14 to 17 show results for the distribution of the phase angle for inlet Mach number M1 = 0.2 and

reduced frequency of 10. Figures 18 to 21 show the same results for inlet Mach number M1 = 0.27. The quantitative

agreement between the computed and measured values of the phase angle is poor although qualitatively they are

showing similar trends.
NASA/TM—2001–210562 5

CONCLUDING REMARKS

In this paper, we have considered an annular cascade configuration subjected to unsteady gust inflow

conditions. The gust response calculation has been implemented into the time marching CFD code, NPARC. The

computed steady state results for the pressure distribution demonstrated good agreement with experimental data.

The computed results for the amplitudes of the unsteady pressure over the blade surfaces agreed with experimental

data within 20 percent error margin, except in the vicinity of leading edge. There exists considerable quantitative

discrepancy between the computed and experiments results for phase angle, although the trends are similar.

REFERENCES

Barter, J.W., Vitt, P.H. and Chen, J.P., 2000: “Interaction Effects in a Transonic Turbine Stage,” 2000–GT–0376,

ASME Turbo Expo, Munich, Germany.

Giles, M.B., 1988: “Calculation of Unsteady Wake/Rotor Interactions,” Journal of Propulsion and Power, Vol. 4,

pp. 356–362.

Hall, K.C. and Verdon, J.M., 1990: “Gust Response Analysis for Cascades Operating in Nonuniform Mean Flows,”

Unsteady Aerodynamic Phenomena in Turbomachines, AGARD-CP–468, Paper No. 10.

Haldeman, C.W., Dunn, M.G., Abhari, R.S., Johnson, P.D. and Montesdeoca, X.A., 2000: “Experimental and Computational Investigation of the Time-Averaged and Time-Resolved Pressure Loading on a Vaneless Counterrotating Turbine,” 2000–GT–0445, ASME Turbo Expo, Munich, Germany.

Kielb, R.E. and Chiang, H.D., 1992: “Recent Advances in Turbomachinery Forced Response Analyses,” AIAA

92–0012, 30th Aerospace Sciences Meeting & Exhibit, Reno, NV.

Kurkov, A.P. and Lucci, B.L., 1997: “Measurement of Gust Response on a Turbine Cascade,” ASME Journal of

Turbomachinery, Vol. 119, pp. 238–246.

Manwaring, S.R. and Wisler, D.C., 1993: “Unsteady Aerodynamics and Gust Response in Compressors and Turbines,” ASME Journal of Turbomachinery, Vol. 115, pp. 724–740.

Reddy, T.S.R. and Srivastava, R., 1994: “Analysis of a Rotor-Stator Stage Using a Two Dimensional Euler

Aeroelatic Solver,” AIAA–94–1638, 35th Structures, Structural Dynamics and Materials Conference, Hilton

Head, S.C.

Reddy, T.S.R. and Srivastava, R., 1996: “Gust and Structural Response Analysis of a 2D Cascade Using an Euler

Aeroelatic Solver,” AIAA–96–1492, 37th Structures, Structural Dynamics and Materials Conference, Salt Lake

City, UT.
NASA/TM—2001–210562 6

Figure 1.—(a) Cascade geometry for a single blade passage. (b) Grid.

(b)

(a)

s

c

B C

A

Inlet Exit

i

E



F G H

D
NASA/TM—2001–210562 7

Figure 2.—Experimental setup.

x = 1.325 in., phase reference

3.9 axial

chords

3.9 axial

chords

Rotating pins

Cascade midspan section

x

y

Figure 3.—Instrumentation ports.

Dynamic

Steady 1 2

3

4

5

9

8 7 6

10 11
NASA/TM—2001–210562 8

–2

–1

0

1

2

3

4

5

6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

X

–Cp

Pressure

NPARC

Exp-pressure

Exp-suction

Figure 4.—Steady state pressure prediction (M1 = 0.2, i = 4.59°).

–2

–1

0

1

2

3

4

5

6

7

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

–Cp

NPARC

Exp-pressure

Exp-suction

Figure 5.—Steady state pressure prediction (M1 = 0.27, i = 5.17°).

Fraction of chord
NASA/TM—2001–210562 9

Figure 6.—Amplitude of the unsteady pressure coefficient distribution 

 on suction surface (M1 = 0.2, i = 3.24°).

0

1

2

3

4

5

6

7

8

0.0 0.5 1.0 1.5 2.0 2.5

Distance along chord

Amplitude

NPARC

Experiment

Figure 7.—Amplitude of the unsteady pressure coefficient distribution 

 on pressure surface (M1 = 0.2, i = 3.24°).

0

1

2

3

4

5

6

7

8

0.0 0.5 1.0 1.5 2.0 2.5

Distance along chord

Amplitude

NPARC

Experiment
NASA/TM—2001–210562 10

Figure 8.—Amplitude of the unsteady pressure coefficient distribution 

 on suction surface (M1 = 0.2, i = 3.24°).

0.0 0.5 1.0 1.5 2.0 2.5

Distance along chord

Amplitude

NPARC

Experiment

0

2

4

6

8

10

12

14

Figure 9.—Amplitude of the unsteady pressure coefficient distribution 

 on pressure surface (M1 = 0.2, i = 3.24°).

0.0 0.5 1.0 1.5 2.0 2.5

Distance along chord

Amplitude

NPARC

Experiment

0

1

2

3

4

5

6

7

8

9

10
NASA/TM—2001–210562 11

Figure 10.—Amplitude of the unsteady pressure coefficient distribution 

 on suction surface (M1 = 0.27, i = 2.98°).

0.0 0.5 1.0 1.5 2.0 2.5

Distance along chord

Amplitude

NPARC

Experiment

0

2

4

6

8

10

12

Figure 11.—Amplitude of the unsteady pressure coefficient distribution 

 on pressure surface (M1 = 0.27, i = 2.98°).

0.0 0.5 1.0 1.5 2.0 2.5

Distance along chord

Amplitude

NPARC

Experiment

0

2

4

6

8

10

12
NASA/TM—2001–210562 12

Figure 12.—Amplitude of the unsteady pressure coefficient distribution 

 on suction surface (M1 = 0.27, i = 2.98°).

0.0 0.5 1.0 1.5 2.0 2.5

Distance along chord

Amplitude

NPARC

Experiment

0

2

4

6

8

10

12

14

16

Figure 13.—Amplitude of the unsteady pressure coefficient distribution 

 on pressure surface (M1 = 0.27, i = 2.98°).

0.0 0.5 1.0 1.5 2.0 2.5

Distance along chord

Amplitude

NPARC

Experiment

0

2

4

6

8

10

12

14

16
NASA/TM—2001–210562 13

Figure 14.—Phase angle distribution on suction surface (M1 = 0.2, i = 3.24°).

0.0 0.5 1.0 1.5 2.0 2.5

Distance along chord

Phase angle

NPARC

Experiment

–0.4

–0.2

0.0

0.2

0.4

0.6

Figure 15.—Phase angle distribution on pressure surface (M1 = 0.2, i = 3.24°).

0.0 0.5 1.0 1.5 2.0 2.5

Distance along chord

Phase angle

NPARC

Experiment

–0.2

–0.1

0.0

0.1

0.2

0.3

0.4

0.5
NASA/TM—2001–210562 14

Figure 16.—Phase angle distribution on suction surface (M1 = 0.2, i = 3.24°).

0.0 0.5 1.0 1.5 2.0 2.5

Distance along chord

Phase angle

NPARC

Experiment

–0.6

–0.4

–0.2

0.0

0.2

0.4

0.6

0.8

Figure 17.—Phase angle distribution on pressure surface (M1 = 0.2, i = 3.24°).

0.0 0.5 1.0 1.5 2.0 2.5

Distance along chord

Phase angle

NPARC

Experiment

–0.6

–0.4

–0.2

0.0

0.2

0.4

0.6

0.8
NASA/TM—2001–210562 15

Figure 19.—Phase angle distribution on pressure surface (M1 = 0.27, i = 2.98°).

0.0 0.5 1.0 1.5 2.0 2.5

Distance along chord

Phase angle

NPARC

Experiment

–0.2

–0.1

0.0

0.1

0.2

0.3

0.4

0.5

Figure 18.—Phase angle distribution on suction surface (M1 = 0.27, i = 2.98°).

0.0 0.5 1.0 1.5 2.0 2.5

Distance along chord

Phase angle

NPARC

Experiment

–0.6

–0.4

–0.2

0.0

0.2

0.4

0.6

0.8

1.0
NASA/TM—2001–210562 16

Figure 20.—Phase angle distribution on suction surface (M1 = 0.27, i = 2.98°).

0.0 0.5 1.0 1.5 2.0 2.5

Distance along chord

Phase angle

NPARC

Experiment

–0.5

–0.4

–0.3

–0.2

–0.1

0.1

0.0

0.2

0.3

0.4

Figure 21.—Phase angle distribution on pressure surface (M1 = 0.27, i = 2.98°).

0.0 0.5 1.0 1.5 2.0 2.5

Distance along chord

Phase angle

NPARC

Experiment

–1.6

–1.2

–0.8

–0.4

0.0

0.4
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 OF REPORT

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Unclassified Unclassified

Technical Memorandum

Unclassified

National Aeronautics and Space Administration

John H. Glenn Research Center at Lewis Field

Cleveland, Ohio 44135–3191

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 AGENCY REPORT NUMBER

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National Aeronautics and Space Administration

Washington, DC 20546–0001

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

January 2001

NASA TM—2001-210562

E–12520

WU–323–71–00–00

22

A03

Unsteady aerodynamics; Gust response; Turbine cascade

Unclassified -Unlimited

Subject Categories: 02 and 07 Distribution: Nonstandard

Prepared for the 46th International Gas Turbine and Aeroengine Technical Congress, Exposition, and Users Symposium sponsored by

the American Society of Mechanical Engineers, New Orleans, Louisiana, June 4–6, 2001. R.S.R. Gorla, Department of Mechanical

Engineering, Cleveland State University, Cleveland, Ohio 44115; T.S.R. Reddy, Department of Mechanical Engineering, The University

of Toledo, Toledo, Ohio 43606; D.R. Reddy and A.P. Kurkov, NASA Glenn Research Center. Responsible person, D.R. Reddy, organization code 5430, 216–433–8133.

R.S.R. Gorla, T.S.R. Reddy, D.R. Reddy, and A.P. Kurkov

Gust Response Analysis of a Turbine Cascade

A study was made of the gust response of an annular turbine cascade using a two-dimensional Navier Stokes code. The

time-marching CFD code, NPARC, was used to calculate the unsteady forces due to the fluid flow. The computational

results were compared with a previously published experimental data for the annular cascade reported in the literature.

Reduced frequency, Mach number and angle of incidence were varied independently and the gust velocity was sinusoidal.

For the high inlet velocity case, the cascade was nearly choked.