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Printed in U.S.A.

93-GT-92

THREE DIMENSIONAL TIME-MARCHING INVISCID ANDVISCOUS SOLUTIONS FOR UNSTEADY FLOWS

AROUND VIBRATING BLADES

L. He and J. D. DentonWhittle Laboratory

Cambridge UniversityCambridge, United Kingdom

AbstractA 3-dimensional non-linear time-marching method of solving

the thin-layer Navier-Stokes equations in a simplified form has been

developed for blade flutter calculations. The discretization of the

equations is made using the cell-vertex finite volume scheme in space

and the 4-stage Runge-Kutta scheme in time. Calculations are carried

out in a single-blade-passage domain and the phase-shifted periodic

condition is implemented by using the shape correction method. The

3-D unsteady Euler solution is obtained at conditions of zero

viscosity, and is validated against a well-established 3-D semi-

analytical method. For viscous solutions, the time-step limitation on

the explicit temporal discretization scheme is effectively relaxed by

using a time-consistent two-grid time-marching technique. A

transonic rotor blade passage flow (with tip-leakage) is calculated

using the present 3-D unsteady viscous solution method. Calculated

steady flow results agree well with the corresponding experiment and

with other calculations. Calculated unsteady loadings due to

oscillations of the rotor blades reveal some notable 3-D viscous flow

features. The feasibility of solving the simplified thin-layer Navier-

Stokes solver for oscillating blade flows at practical conditions is

demonstrated.

Nomenclature

C Chord length

Cpi unsteady pressure coefficient

dA differential area element with unit outward normal vector

dV differential volume element

E internal energy

f frequency

n„ the unit vectors in x direction

static pressure

radial coordinate

S span

S, --- viscous terms in a body force form

U --- primitive flow variable

velocity in x direction

--- absolute velocity at inlet

velocity in 8 direction

w -- velocity in radial direction

axial coordinate

Cm --- turbulence eddy viscosity

laminar viscosity coefficient

0 circumferential coordinate

p density

inter blade phase angle

shear stress

angular frequency

Superscript

n --- index of time step

Subscript

time averaged

mg --- moving grid

order of the Fourier component

on walls

IntroductionPredictions of unsteady flows around vibrating blades are

essential for turbomachinery blade flutter predictions. Numerical

methods of calculating unsteady flows through vibrating blades are

currently under active development. Most of the methods available

Presented at the International Gas Turbine and Aeroengine Congress and ExpositionCincinnati, Ohio May 24-27, 1993

This paper has been accepted for publication in the Transactions of the ASMEDiscussion of it will be accepted at ASME Headquarters until September 30,1993

Copyright © 1993 by ASME

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are for a 2-dimensional model. Notable examples are the work by

Verdon and Caspar (1982) and Whitehead (1982). Under practical

turbomachinery working conditions, three dimensional effects are

important and need to be modelled. Calculations of 3-dimensional

unsteady flows are usually simplified by assuming that a flow field

would behave in a 2-dimensional fashion at each spanwise section,so the strip method (i.e. performing 2-D calculations at each section)

can be used. It has been demonstrated that the strip theory can lead to

serious errors for blade flutter calculations (e.g. Namba, 1987).

For simple 3-D blade geometries (e.g. flat plate cascade) at

purely subsonic or supersonic flow conditions, fully 3-D inviscid

solutions for oscillating blade flows can be efficiently and accurately

obtained by using semi-analytical theories. Examples of this kind are

those by Namba (1976, 1987) and Chi (1991). These semi-analytical

methods can provide some basic insight into 3-D effects on blade

aeroelastic behaviours, but their capability in predicting blade flutter

under practical flow conditions is rather limited.

Recently three dimensional numerical methods for blade flutter

calculations have begun to emerge. 3-D inviscid Euler solutions have

been developed by Giles (1991) and Hall and Lorence (1992) using

a time-linearized model, and by Gerolymos (1992) using a time-

marching method. So far no work on 3-D unsteady viscous flow

solutions for blade flutter calculations has been reported.

The objective of the present work is to develop a nonlinear time-

marching viscous flow solution method for blade flutter calculations.

The major effort is focused on enhancing solution efficiency which is

believed to be a very important factor affecting the feasibility of

unsteady time-marching solvers for industrial use. For the phase-

shifted periodic boundary condition, the "Shape Correction" method

developed for the previous 2-D Euler solver (He, 1989) is extended

to the present 3-D solution to save computer storage. Another feature

of the present work is that the time-step limitation on the explicit

time-marching scheme, is effectively relaxed by using a time-

consistent two-grid technique.

Basic Governing Equations and DiscretizationFor convenience of simulating 3-D flows in turbomachinery

blade passages, a cylindrical coordinate (x, 0, r) in an absolute

system is adopted. An integral form of the 3-D Navier-Stokes

equations over a moving finite volume AV is:

(p pu pvpu puu+p puv

U = pvr F puvr G = (pvv+p)r

Pw puw pvw

E ) (pE+p)u (pE+p)v 1and

r pw

puw(

0H pwvr

pww+p

Si = 0pv2/r

(pE+p)w

\ 0 )

It should be mentioned that the circumferential moving-grid velocity

v mg includes both the blade vibration velocity and the rotation

velocity.

The viscous term S v in a body force form can be modelled at

different levels of approximation. In the present work, a simplified

form of the thin-layer Navier-Stokes equations as used in Denton's

viscous flow solver for steady turbomachinery flows (Denton, 1990)

is adopted. Firstly, the viscous terms in the energy equation are

neglected. This simplification is introduced in the light that the work

done by the viscous shear forces would be roughly cancelled by the

heat conduction at a Prandtl number near unity. Furthermore, the

viscous terms in the three momentum equations are modelled under

the thin-layer assumption. So only the viscous shear stress terms in

the direction tangential to solid surfaces are included. For laminar

flows, if we have the shear stress T corresponding to a solid surface:

1 = 11 anau,

(2)

where n is the normal distance from the surface, u s is the absolute

velocity. Then the corresponding viscous terms in the body force

form are approximated by :

fff S v dV = §A

T, dA6,v

where

(3)

rAy

UdV+§A

,(F-Uu mg )n„±(G-Uv mg )n e+(H-Uw mg) n r 1.d A

(Si +S v )dVAv

where:

(1) a, I3, T1 are angles between the velocity and the x, 0 and r directions

respectively.

For turbulent flows the shear stress is modelled in an eddy

viscosity form:

tcosaTv = trcosr3

tcosti0

2


= P fl)

aus (4)

The eddy viscosity is at the present stage obtained by using a simple

mixing-length model:

Ern = (0.4111)2 I (5)

This mixing-length model is simpler than most of other

turbulence models currently available, but it is felt that the number of

mesh grid points likely to be available for a 3-D unsteady viscous

solution would probably not justify a more sophisticated model. The

eddy viscosity e n, is "cut off" when the mesh point is far from the

wall.

The spatial discretization for the 3-D equations is made by using

the cell-vertex finite volume scheme. The temporal integration is

performed using the 4-stage Runge-Kutta scheme. The discretized

form of Eq.1 is:

u n+1/4 =u n AVn 1 At

AVn+I/4 4 AN/a -opt - Rv n - Dn )

(6a)

u n+113 =u n AVn 1 A At „„.n+113 K„ Dn) (6b)v n

AVn+1 /3 3 AV n+1 /3

un±1/2 =un AVn 1 At ( R .ri+1/2 R n Dn ) (6cAvn.4-1/2 2 AVn+I/2 I

v

un+ 1 _un AVnAt , R ,

1n+1 Rvn Dn )

4Vn+1 6,Vn+1 k

(6d)

where

R, = { IF - U u mg IAA, + [G - U vnig [AA0 +11-1 - U wmg]AA r )

+ S, AV

Rv = E Tv AA

The summation is taken along the finite volume boundary. The

numerical damping term Dn is treated in a similar way to that in the

previous 2-D Euler solution (He, 1989). The second and fourth order

adaptive smoothing (Jameson 1983) is used in the streamwise

direction and a simple second order smoothing is used in the

pitchwise and the radial directions. Similar to the numerical damping

term, the viscous terms are only updated at the first stage.

On solid surfaces (blade surfaces, hub and casing), a slip

condition is used, i.e. only the normal velocity are set to be zero. The

wall shear stress is determined by using an approximate form of the

log-law (Denton, 1990):

Tw— - 0.001767 + 01 n. 1:1( R3 el7w)

0n) R7

+ ( .(25e6w121)4 0.5 p w u w 2

where u, and p, are velocity and density at mesh points one grid

spacing (normal distance An ) away from the wall and

Rew—pwuwAn

. For laminar flows the wall shear stress is simply

evaluated by 't = !..tuw

. It has been found from steady flowAn

solutions (e.g. Denton 1990) that viscous flow behaviour can be

adequately modelled by this slip condition, which needs fewer mesh

points in the near wall region than does the standard non-slip wall

condition.

At the inlet and outlet boundaries, the non-reflecting boundary

conditions are preferred to prevent artificial reflections of out-going

waves. The 1-D and 2-D non-reflecting boundary condition methods

are currently available (e.g. Giles, 1990). While the 3-D non-

reflecting boundary conditions are still a subject of research. In the

present calculations, the 1-D non-reflecting condition (Giles, 1990) is

adopted for its simplicity of implementation.

Phase-Shifted Periodic ConditionA single-blade-passage computational domain is adopted. It is

assumed that the unsteadiness to be dealt with satisfies a temporal

and circumferentially spatial periodicity, characterized by a frequency

f and an inter-blade-phase angle a. Thus the phase-shifted periodic

condition can be applied at the periodic boundaries. It is recognized

that there is a limitation of the phase-shifted periodicity. That is, self-

excited periodic vortex shedding phenomena can only be

accommodated if they are locked in with the imposed fundamental

frequency and its higher order harmonics. The phase-shifted periodic

condition can be applied by using the conventional "Direct Store"

method, proposed by Erdos et al (1977). However the large

computer storage required by the "Direct Store" method presents a

severe limitation for 3-D solutions. To avoid using large computer

storage, a new method called 'Shape Correction' has been developed

for the previous 2-D Euler solver (He,1989). Here we extend the

Shape Correction method to 3-D solutions.

Let us denote that the single-passage computational domain is

bounded by the lower (indicated by "L") and the upper (indicated by

"U") periodic boundaries upstream and downstream of the blade

row. At these periodic boundaries, the flow primitive variables are

approximated by a Nth order Fourier series in time:

NUU(x.r,t)=Uo (x,r)+E[A n (x,r) sin (nux+a) +B n (x,r) cos (ncot+cs) J

n=1

(8a)

N

UL(x,r,t) = Up (x,r) + E [A n (x,r) sin (ncot) +B n (x,r) cos (ncot) 1n=1

(8 b)

(7)

3


Instead of directly storing the variables for one period of time as

in the "Direct Store" method, only the temporal Fourier components

are stored. During the time-marching solution process, the flow

variables at the periodic boundaries are corrected by the stored

temporal variation (i.e. the Fourier components). And the new

Fourier coefficients are obtained by a straightforward temporal

integration of the flow variables over one period, (e.g. for the lower

boundary):

Np

(1) LAn = — U sin (not) At (9a)

A,

C '1G

- r - — r

rINN g

d I h

Bc

1-r -"1

D H

_xNp

B n = uL cos (nom) At

(9b) Fig.1. Two-Grid Finite Volume

where N p is the number of time-steps in one period. These new

Fourier coefficients will then be used to correct the old ones. In this

way, the stored temporal shape and the current solution correct each

other until a periodic state is reached. The step by step

implementation procedure is given by He (1989). It should be

mentioned that another advantage of the "Shape Correction" method,

apart from saving computer storage, is that it can be extended to

general situations with multiple-perturbations at different frequencies

and inter-blade phase angles (He, 1992a).

Two-Grid Time Integration

For a typical finite volume, the mesh size in the three directions is

comparable in the mainly inviscid part of the flow field. The

streamwise mesh grids can usually be more or less uniformly

distributed. However in the radial and circumferential directions

highly refined mesh points have to be placed in the near wall regions

in order to resolve thin viscous layers. For time accurate explicit

time-integration schemes like the one adopted in the present method,

the time step is limited by the smallest mesh size due to the numerical

stability requirement (CFL condition). Thus the corresponding time-

step for unsteady flow solutions will be very small. It is recognized

that a time-step length dictated by the mesh size in the mainly inviscid

part of flow field is sufficiently small to give an adequate temporal

resolution. Hence the efficiency of the explicit time-marching method

would be enhanced if the usable time-step length in the near-wall fine

mesh regions can be increased. For this purpose, we adopt a two-

grid time integration method.

The basic fine mesh is the one on which the flow variables are

stored and the fluxes are evaluated. The coarse mesh is taken in such

a way that each mesh cell contains several pitchwise and radial cells

of the fine mesh. As shown in Fig.1, a small finite volume

`abcdefgh' of the fine mesh is contained in a big finite volume

ABCDEFGH of the coarse mesh. For simplicity we now consider

only one stage temporal integration over a fixed computational cell. If

evaluated on a cell of the fine mesh as in a direct time-marching

solution, the temporal change of flow variables is:

(un+1 un )fine AAvtfrinc R fine (10)

where Rfi„ is the net flux for the finite volume on the fine mesh.

While if the temporal change is evaluated on a big cell of the coarse

mesh which contains the small cell on the fine mesh, we have:

n-" nAtcoars(U - ti )coarseAvcoAtcoars e

Rcoarse

where Rwars, is the net flux for the big cell on the coarse mesh.

The time step Atfi ne is the time-step allowed on the fine mesh

limited by the smallest mesh spacing. While At em.„ is dictated by the

mesh size of the coarse mesh and therefore can be much larger than

Atfi ne . Suppose we want to run an unsteady solution with a time step

At. Based on the smallest mesh size of the fine mesh (either in the

radial direction or in the circumferential direction), At gives a Courant

number CFL, much larger than CFL° , the one dictated by the

numerical stability. The idea is that the temporal integration should be

formulated in such a way as if the solution were time marched firstly

on the fine mesh up to its stability limit At f,„ and then on the coarse

mesh using At c„,„, to make up the desired time step At. The time

consistence condition is thus:

At = Atcoarse + Atfine

CFL°Take Atf,„ - At CFL

4


APIACP ] –

u o„, 2 A m

The final formulation for evaluating the temporal change on the fine

mesh is:

(un+1 u n Rfinc CFL° Reparse 0 CFL°) At (12)'fine – AV ime CFL — AVcoarse CFL

The implementation of the above formula is very easy, if we

recall that the conservation relations give:

N, N,

AV,ouse =

Rcoarse = /Rime (13)

1=1

i = 1

where N, is the number of the small cells contained in the big cell.

This two-grid formulation is applied at each stage of the Runge-Kutta

integration. In the mainly inviscid flow part where the mesh size in

the three directions is comparable, the basic one-grid time-marching

formulation is recovered.

It should be pointed out that for a steady flow solution, this two-

grid scheme is equivalent to a direct solution on the fine mesh

because the residual, which drives the solution, is formed based on

the net fluxes on the fine mesh. For an unsteady flow solution, the

timewise accuracy on the fine mesh is no longer guaranteed. The loss

in the temporal accuracy depends on the local ratio between fine and

coarse mesh sizes. The length scale on which the temporal resolution

is lost would be the mesh spacing length on the coarse mesh. As long

as the coarse mesh spacing is taken to be much smaller than the

physical wave length of interest, this loss in time accuracy should be

acceptable. More details and validations of this two-grid method for

the 2-dimensional full Navier-Stokes calculations have been recently

given by He (1992b, 1993).

Validation of 3 -D Euler Solution

For 3-D flows, validations of unsteady flow calculation methods

become more difficult, because 3-D unsteady experimental data are

currently hardly available in the published literature. Therefore

comparisons between numerical methods and analytical or semi-

analytical linear theories for simple cascade geometries at inviscid

flow conditions must play an essential part in validations of 3-D

unsteady solution methods. The present authors have proposed the

following test case.

The geometry is of a simple linear flat plate cascade placed

between two parallel solid walls as shown in Fig.2. The geometric

parameters are

Chord length:

C= 0.1m

Stagger angle: y = 45°

Pitch / chord ratio:

P/C=1

Span/ chord ratio:

S/C =3

The inlet flow Mach number is 0.7 and the incidence is zero.

The blades are oscillated in a 3-D mode. Each 2-D section is subject

P

A

Meridional View (A-A)

Blade-Blade View (B-B)

Fig.2. 3-D Flat Plate Cascade Test Case Geometry

to torsion mode around its leading-edge. The torsion amplitude is

linearly varied along the span. At one end ("hub section"), the

amplitude is 0. at the other end ("tip section"), the amplitude is 1°.

coCThe reduced frequency ( K= ) is 1. Two inter blade phase

angles, a = 0° ; 180°, are chosen.

For this kind of flat plate cascade geometries at zero incidence

flow condition, time-linearized semi-analytical theories should be

very accurate and can be practically regarded as "exact" solutions. A

well-known 3-D semi-analytical lifting-surface method has been

developed by Namba (e.g. Namba, 1977, 1987), who provided his

results for this 3-D case (Namba, 1991).

The present calculations were carried out by using the inviscid

Euler equations (setting viscous terms S,, to be zero). The mesh

density is 81x25x21 in the streamwise, pitchwise and radial

directions respectively. Calculated unsteady pressure differences

(jump) at each 2-D section are presented in the form of

where AP I is the first harmonic pressure jump across blade; A m is

the torsion amplitude at the tip (i.e. 1°) in radian. The spanwise

position of each 2-D section is given by :

AR = s

where R is the distance measured from the hub section.

Fig.3 shows the chordwise distributions of the real and

imaginary parts of the unsteady pressure jump at six spanwise

sections (AR = 0.0; 0.2; 0.4; 0.6; 0.8; 1.0) for the inter blade phase

angle a=0°. The corresponding results for a =180° are given in

Fig.4. Also presented in these figures are Namba's semi-analytical

results. For both inter blade phase angles, the present Euler

calculations agree very well with Namba's theory.

5


10.0 10.0

7.5 7.5

5.0 5.0

2.5 2.5

0.0 0.0

-2.5 .2.5

-5.0 -5.0

.7 5 .7.5

.10.0 .10.0

00 0.5

we

(a) AR-0.0

10.0 10.0

7.5 7.5

5.0 5.0

2.5

0.08

2.5

0.0

-2.5 .2.5

.5.0 -5.0

-7.5 .7 .5

-10 0 .10.0

0 0

0.5

we(b) AR=0.2

1 0

o 0

0.5

VG 0 0

0 0 0.5

VC

0.5

we

o

20 0

15.0

10.0

;75.0

0.0

.5.0

10.0

-15.0

.20 0

(a) AR=0.0

(b) AR-0 2

10.0

7.5

5.0

2.5

0.0

-2.5

-5.0

-7 5

-10.0

0 0 0 5

xc1 D

20 0

15 0

10.0

5.0

• 0 0

L77

•

-5 0

10 0

.75 0

-20 0

0 0 0.5 1 0

we

7_

10.0

7 5

5.0

2 5

0 0

2 5

5.0

.7 5

-10 0

0 0 0.5

X/C

20.0 1

15.0

10 0

;5 5.0

• 0.0

-5.0

-10.0

.15.0

-20.0

00 0.5

we

1 0

(c) AR=0.4

10.0

7 5

5.0

2.5

0.0

.2.5

-5.0

-7.5

-10.0

0.0 0.5

1 0

we(c) AR=0.4

1 01 005

'Sc

05

we

10.0

7,5

5.0

2.5

0.0

-2.5

-5.0

-7.5

.10.0

00

10.0

7.5

5 0

2.5

0.0

-2.5

.5.0

-7.5

-10.0

00

8

1 0 1 0 0.50.0 1 00.5

'Sc

0.5

we1 0

VC0.5

x/C

10.0

7.5

5.0

0.0

-2.5

.5

-5.0

.7.5

-10.0

0 0

20.0

15.0

- 10.0

3 5.0

0.0

-5.0

.10.0

-15 0

-20.0

10.0

7.5

5.0

2.5

0.0

-2.5

-5.0

.7.5

-10.0

00

10.0

7.5

5.0

2.5

0.0

-2.5

.5.0

-7.5

-10.0

0 0

10.0

7.5

5.0

2.5

0.0

-2.5

-5.0

-7.5

.10.0

0 0 0.5

VC

1 0

(e) AR=0.8

10 0

7.5

5.0

2.5

0.0

-2.5

-5.0

-7 5

-10.0

00 0.5

VC

1 0

we

7.5

5.0

2.5

Cf 0.0

-5.0

-7.5

-10.0

0.5 1 0 0 0

(e) AR-0.8

10.0

0.5

X/C

1 0

20.0

15.0

10.0

U 5.0

• 0.0

T, -5.0

-10.0

-15.0

-20.0

0.5 1.0 0 0

we (f)

10.0

75

5.0

25

0.0

-2.5

.5.0

-7.5

-10.0

0.5

we

1 0

(f) AR-1 0

0.5

we

0.5

X/C

10.0 1

].5

5.0 12.5 10.0

-2.5

-5.0

-7.5

-10.0

0.0

20.0

15.0

10.0 -5.0

0.0

-5 0

.10.0

.15 0

.20.0 4-

10.0

75

5.0

2.5

00

-2 5

-5 0

.7.5

-10.0

0 0

_ Present; 00 Namba - Present; 00 Namba

(d) AR-0.6 (d) AR-0.6

Fig.4. Real and Imaginary Parts of Unsteady PressureJump Coefficient at Different Spanwise Sections (a =0') Jump Coefficient at Different Spanwise Sections (a =180')Fig.3. Real and Imaginary Parts of Unsteady Pressure

6


(a). Meridional View at Midpassage

(b). Blade-Blade View at Midspan

Fig.5. Computational Mesh for a Transonic Rotor

Casing// /// ////

Steady and Unsteady Viscous Results for a

Transonic FanA check on the present viscous flow solution has been made by

calculating the steady flow through a transonic fan rotor, known as

"NASA Rotor-67", of which the steady flow field had been

extensively measured at NASA Lewis (Strazisar et al, 1989). A

computational domain with mesh points 73x25x25 was used in the

calculation. Fig.5a shows the meridional view of the mesh. The

blade-to-blade view of the mesh at the midspan section is given in

Fig.5b.

For this experimental case, the rotor blades have about 0.8 % tip

clearance. To take this into account, a very simple tip-clearance

model is adopted in the present calculation. As shown in Fig.6, the

radial mesh spacing (a-c or b-d) of the computational cells adjacent to

the tip is taken to be the tip gap (i.e. 0.8 % of the span). For steady

flows, the tip-leakage effect is approximately included by equating

flow variables at a and b (c and d). For unsteady flows, this simple

tip-clearance model is implemented by applying the phase-shifted

periodic condition for two mesh lines near the tip.

The flow condition chosen for the present calculation is that at the

rotor's peak efficiency. The calculation is performed assuming that

the flow is fully turbulent. At the inlet the measured flow angle,

stagnation pressure and stagnation temperature in the absolute system

are specified. And the measured static pressure at the outlet is

specified. Fig.7 shows the calculated steady Mach number contours

at the blade-to-blade sections of 10%, 30% and 70% span measured

from the tip. Also shown in Fig.7 is the corresponding Laser

Anemometry measurement results (Strazisar et al, 1989). The results

near the hub agree reasonably well with the experiment. At the

section near the tip the present calculated shock position is more

rearward than that experimentally observed, indicating that the tip-

leakage effect might be under predicted. This is not unexpected

considering the simple tip-clearance treatment adopted in the

calculation. It may also be argued that the simple mixing-length

turbulence model adopted may be another major reason for the

discrepancy. It is however noticed that the flow patterns for this case

predicted recently by Jennions and Turner (1992) who adopted the

full Navier-Stokes equations with the k e turbulence model, are

similar to the present ones.

In order to demonstrate the feasibility of the present 3-D unsteady

viscous solver for blade flutter calculations under practical

conditions, an unsteady calculation was carried out for this transonic

fan. After the above steady solution is obtained, the blades are

oscillated in a torsion mode at each radial section. The torsion axis

for all the sections is a radial line which goes through the mid point

of the chord on the hub section. The amplitude is linearly varied from

0' at the hub to 0.5' at the tip. The oscillation frequency is taken to be

1000Hz, which based on parameters at the rotor-tip section gives a

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Fig.6. A Tip-Leakage Model

7


-- Viscous- Inviscid

Laser Measurement

Present Calculation

(a). 10% Span

FLOW

(b). 30% Span

(c). 70% Span

Fig.7. Mach Number Contours (interval =0.05) at ThreeSpanwise Sections (position measured from the tip)

reduced frequency of 1.5. The inter-blade-phase-angle is 180°. It

should be mentioned that for this unsteady viscous solution, the time-

step by using the two-grid method is increased by a factor of 10

compared to the direct explicit time-marching solution. A periodic

solution with 10 periods of oscillation requires a total of 48 CPU

hours on a single processor of the Alliant FX-80 computer which is

40% faster than a personal computer PC-486 for the present code.

Fig.8 shows calculated spanwise distributions of real and

imaginary parts of the unsteady moment coefficient. The moment

coefficient at each spanwise section is defined in the same way as that

for a 2-D cascade case (Fransson, 1984) except that the blade

vibration amplitude for normalization is taken from the tip section.

Also presented in Fig.8 are results from the 3-D inviscid Euler

1.00

0.50

0.00

-0.50

-1.00

0.00 0.25 0.50 0.75

1.00

(R - Rhub)/( 13 19- R hub)

(a) Real Part

0.50

0.00

2 -0.50

-1.00

-1.50

0.00 0.25 0.50 0.75

1.00

(R - Rhub)/(Rnp - Rhub)

(b) Imaginary Part

Fig.8. Calculated Spanwise Distribution

of Unsteady Moment Coefficient

solution obtained at the same steady and unsteady conditions. For the

inviscid Euler solution, a mesh with same number of points and a

uniform distribution in the circumferential and radial directions is

used. It has been well established that for transonic fans, inviscid and

viscous steady flow methods predict very different shock structures.

So the marked difference in the unsteady results between the inviscid

and viscous solutions for the present case is not unexpected. The

imaginary part of the moment coefficient is a direct measure of the

aerodynamic damping. The negative values (i.e. positive damping) of

the imaginary part are predicted along the whole span by the viscous

solution and virtually along the whole span by the inviscid solution.

Thus both the solutions suggest that the rotor blades under these

unsteady conditions would be aeroelastically stable. It should be

noted, however, that the maximum differences in terms of magnitude

and phase of the unsteady loading between the inviscid and the

viscous solution are in the regions near the tip and hub where the

8


flow is dominated by 3-D viscous effects. The tip-leakage effect can

be clearly seen from the results of the viscous solution. The unsteady

loading is diminished at the tip section. In addition, there is a local

region of high loading near the tip, probably induced mainly by the

variation of the tip leakage flow due to the blade vibration. The 3-D

inviscid Euler solution, on the other hand, shows a much smoother

spanwise loading distribution.

Concluding Remarks

This paper describes the first-known three-dimensional

unsteady viscous flow solution method for blade flutter calculations.

The simplified thin-layer Navier-Stokes equations are discretized in

the cell-vertex finite volume scheme in a single-blade-passage

computational domain, and temporally integrated in the 4-stage

Runge-Kutta scheme.

The phase-shifted periodic condition is implemented by using

the Shape Correction method, which greatly reduces the computer

storage requirement compared to the conventional "Direct Store"

method.

The time-step limitation on the explicit time-marching scheme

due to thin-viscous-layer resolution is considerably relaxed by using

a time-consistent two-grid time-marching technique.

Two test cases have been used to validate the present method.

The first one is a proposed 3-D oscillating flat plate cascade.

Calculated unsteady pressure distributions using the Euler equations

(setting zero viscosity) are in very good agreement with a well-

established semi-analytical theory. The second test case is for a

transonic fan rotor with tip-leakage. Calculated steady viscous flow

results agree well with the corresponding experiment and with other

calculations. Calculated unsteady loadings due to oscillations of the

rotor blades reveal some significant 3-D viscous effects. The

feasibility of the present simplified thin-layer Navier-Stokes solver

for oscillating blade flows at practical conditions has been

demonstrated.

AcknowledgementThe authors wish to thank Professor M. Namba for providing his

results for the 3-D flat plate cascade test case. The present work has

been sponsored by Rolls-Royce plc. L. He has been in receipt of a

Rolls-Royce Senior Research Fellowship at Girton College,

Cambridge University. The technical support provided by Drs. Peter

Stow, Alex Cargill and Arj Suddhoo of Rolls-Royce is gratefully

acknowledged.

References

Chi, R.M. (1991), "An Unsteady Lifting Surface Theory for DuctedFan Blades" ASME Paper 91-GT-131.

Denton, J. D. (1990), "The Calculation of Three DimensionalViscous Flow Through Multistage Turbomachines", ASME Paper90-GT-19.

Erdos, J. 1., Alzner, E. and McNally, W. (1977), "NumericalSolution of Periodic Transonic Flow Through a Fan Stage," AIAAJournal, Vol.15, No.11.

Fransson, T. H. (1984), "Two-Dimensional and Quasi Three-Dimensional Experiments of Standard Configuration for AeroelasticInvestigations in Turbomachine-Cascades," Proc. of the 3rd Symp.on Unsteady Aerodynamics and Aeroelasticity of Turbomachines andPropellers, Cambridge, U.K., 1984.

Giles, M. B. (1990), "Nonreflecting Boundary Conditions for Euler

Equation Calculations", AIAA Journal, Vol.28, No.12, Dec. 1990.

pp.2050-2058.

Giles, M. B. (1991), Private Communication, MassachusettsInstitute of Technology.

Gerolymos, G. A. (1992), "Advances in the Numerical Integrationof the 3-D Euler Equations in Vibrating Cascades", ASME Paper 92-GT-170.

Hall, K.C. and Lorence C. B. (1991), "Calculation of Three-Dimensional Unsteady Flows in Turbomachinery Using theLinearized Harmonic Euler Equations", ASME Paper 92-GT-136.

He, L. (1989), "An Euler Solution for Unsteady Flows aroundOscillating Blades", ASME Paper 89-GT-279, ASME Journal ofTurbomachinery, Vol.112, No.4, pp 714-722.

He, L. (1992a), "A Method of Simulating Unsteady TurbomachineryFlows with Multiple Perturbations", AIAA Journal, Vol.30, No.12.

He, L. (1992b), "Numerical Solutions for Unsteady TurbomachineryFlows", Lecture Notes on " Turbomachinery C.F.D.", AdvancedCourse for Industry, Cambridge, U.K. June, 1992.

He, L. (1993), "A New Two-Grid Acceleration Method forUnsteady Navier-Stokes Calculations", AIAA Journal of Propulsionand Power, Vol.9, No.2.

Jameson, A. (1983), "Numerical Solution of the Euler Solution forCompressible Inviscid Fluids", Report MAE 1643, PrincetonUniversity.

Jennions, I.K. and Turner, M.G. (1992),"Three-DimensionalNavier-Stokes Computations of Transonic Fan Flow Using an

Explicit Flow Solver and an Implicit k c Solver", ASME Paper 92-GT-309.

Namba, M., (1976) "Lifting Surface Theory for Unsteady Flows inRotating Annular Cascade", in Proceedings of the 1st IUTAMSymposium on Aeroelasticity in Turbomachines, Paris, 1976.

Namba, M., (1987), "Three-Dimensional Methods," in"Aeroelasticity in Axial-Flow Turbomachines," AGARD-AG-298,Vol.] (Platzer, M. F. and Carta, F. 0. ed.).

Namba, M. (1991), Private Communication, Kyushu University.

Strazisar, A. J., Wood, J.R., Hathaway, M. D. and Suder, K.L.(1989), " Laser Anemometer Measurements in a Transonic Axial-Flow Fan Rotor", NASA Technique Report 2879.

Verdon, J. M. and Caspar, J. R. (1982), "Development of a LinearUnsteady Aerodynamics for Finite-Deflection Cascades," AIAAJournal, Vol.20, No.9.

Whitehead, D. S. (1982), "The Calculations of Steady and UnsteadyTransonic Flow in Cascades," Report CUED/A-Turbo/TR 1 18,Cambridge University Engineering Department.

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