Category UNSTEADY AERODYNAMICS, AEROACOUSTICS AND AEROELASTICITY OF TURBOMACHINES

Steady state operating fbw conditions were set up by adjusting the inlet to­tal pressure, inlet total temperature, and outlet static pressure. Both stagnation pressure and temperature were measured in the settling chamber using a total pressure probe and a T-type thermocouple which gave an accuracy of ±0.7K on the temperature reading. The outlet static pressure was measured using a pressure tap located on the upper and side walls at x=290mm. Unsteady op­erating flow conditions were thereafter estimated by measuring the change in back pressure between the extreme positions (vertical and horizontal) of the downstream rod and then setting the averaged value order to match the steady state operating point. The experimental operating conditions are summarized in table 1.

Once the operating conditions were set up, the acquisition procedure for unsteady pressure measurements basically consisted in sliding the 2D bump throughout the width of the channel and record the transducer output voltage together with TTL reference signal for each of the operating conditions sum­marized in table 1. Schlieren visualizations were performed at the very same operating conditions. As the resolution of the CCD camera decreases with the frame rate, a translation device was used in order to focus the image onto the region of interest in the test section. As a result, shock motion were recorded throughout the whole channel’s height.

The data reduction for unsteady pressure measurements consisted of, first, converting the output voltages from the transducers into pressure signals using the coefficients obtained during the static calibration. An ensemble average (EA) of the time-serie data was then performed for each channel using the ref­erence TTL signal from the motor. The obtained single unsteady cycle hence represents an average of all unsteady cycles. Thereafter, a Discrete Fourier Serie Decomposition (DFSD) was performed on the EA signal computed pre­viously and the amplitude and phase angle of the few first harmonics were evaluated. Additionally, a Fast Fourier Transform (FFT) was performed on the entire time fhctuating signal to evaluate all frequency components. At this point, the transfer function (TF) throughout each capillarity tube was evaluated depending on the respective amplitude of the fundamental and finally, both the DFSD components (amplitude and phase of all harmonics) as well as the FFT signal (amplitude only) were corrected using the damping and phase lag values evaluated at the corresponding frequency.

The data reduction procedure for high speed Schlieren visualizations con­sisted in extracting the instantaneous shock position at different location of the channel’s height, perform an EA to obtain a single unsteady cycle and conduct an harmonic analysis on the resulting time-serie signal.

Finally, all data was made dimensionless by dividing the amplitude of each harmonic of the DFSD on pressure by the amplitude of the fundamental at the outlet, and subtracting the phase angle of the outlet pressure signal for each harmonic respectively. As a result, the data issued from harmonic analysis presented in this paper actually corresponds to the pressure amplification and phase lag relative to a reference at the outlet.

With the aim at simulating potential interaction in turbomachines, the "quasi steady” shock wave was put into oscillations using a rotating elliptical cam placed at x=625mm in the reference system of the bump. A DC motor was used to rotate the cam up to 15,000RPMs in order to generate pressure per­turbations up to 500Hz. The rotating speed was monitored using an optical encoder located directly on the shaft of the motor. Rotating speed fhctuations and time drift were measured under ±0.024% in the worst case. Furthermore, a TTL pulse generated by the motor was used as a reference signal during unsteady pressure measurements and Schlieren visualizations in order to cor­relate both measuring techniques.

1.2 Measuring techniques

Steady state pressure measurements were performed using a 208-channels ’low speed’ data acquisition system. The scanners used feature a pressure range of ±100kPa relative to atmosphere with an accuracy of ±0.042% full scale. Taking into account the digital barometer, the overall accuracy for steady state pressure measurements is about ±43.5Pa. The sampling frequency and sampling time were respectively set to 10Hz and 200s in order to ’capture’ the lowest frequencies.

Additionally, a 32-channels high frequency data acquisition and storage sys­tem was used for unsteady pressure measurements. Accounting for the res­onance frequency of the capillarity pipes between the bump surface and the transducer, the sampling frequency was set to 8kHz with a low pass filter at 4kHz to avoid bias effects. Each channel was connected to a fast response Kulite transducer and individually programmed to fully use the 16bit AD con­version. A static calibration of all fast response transducers was performed prior and after the measurements in order to reduce the systematic error related to the drift of the sensitivity and offset coefficients. Furthermore, a dynamic calibration was performed on all pressure taps in order to estimate the damp­ing and time delay of propagating pressure waves through the capillarity tubes. The unsteady pressure measurements were thereafter corrected to account for the above estimated damping and phase-lag.

Finally, a conventional Schlieren system connected to a high speed CCD camera was used to monitor the shock motion throughout the whole test section height up to 8kHz. A special feature of the camera allows the display of the TTL signal position directly onto the pictures for referencing purpose during later post treatment. The sampling frequency and shutter speed of the camera were optimally set up depending on the perturbation frequency in order to ob­tain approximately 20 pictures per unsteady cycle (up to 500Hz). The spatial accuracy based on the camera resolution and optical system was estimated to be around ±0.33mm. However, it should be reminded that the processed im­age is an integration of density gradients throughout the channel’s width.

The test section was designed highly modular to be able to insert differ­ent test objects, so called ’bumps’, in a 100x120mm rectangular channel as sketched in figure 1(a). A continuous air supply is provided by a screw com­pressor driven by a 1MW electrical motor and capable of reaching a maximum mass ft>w up to 4.7 kg/s at 4 bar. A cooling system allows a temperature range from 30oC to 180oC. The adjustment of different valves also allows the exper­
imentalist to control independently the mass fbw and the pressure level in the

test section in the respective range of M* — n 1 — n й "n’1 — P“irU^d

1.87 104 – 1.57 106 with d = 0.26m, pair 10-5 m2/s.

1.1 Test object and instrumentation

The investigated test object consists of a long 2D bump, presented in figure 1(b), which can slide through the width of the test section using the traverse mechanism and inflated O-ring sealing system. The nozzle geometry thus consists of a 100mm wide and 120mm high flat channel with a 10.48mm max­imum thickness and 184mm long 2D bump on the lower wall. The beginning of the curvatures was chosen as the origin of the X-axis (x=0mm). The Y-axis and Z-axis were set to be aligned with the channels’s width and height respec­tively to form an orthogonal basis. The profile coordinates of the bump are presented in table A.1.

The bump is equipped with one row of 100 hot film sensors, and three stag­gered rows of 52 pressure taps each. The traverse mechanism fixed on the side window allows the displacement of both the pressure tap rows and the hot film sensors through the width of the channel. As a result, by sliding the 2D bump and successively position each rows of pressure taps at the same location in the channel will provide a spatial resolution measurements of 1.5mm for pressure measurements. Unsteady pressure measurements were performed using fast response Kulite transducers glued in protective pipes. Each pipe was designed with a locking device so that it could be inserted in any of the already instru­mented pressure hole located underneath the sliding 2D bump.

Olivier Bron1,2, Pascal Ferrand1, and Torsten H. Fransson2

bron@energy. kth. se, Pascal. Ferrand@ec-lyon. fr, fransson@energykth. se

1L. M.F. A., Ecole Centrale, Lyon, France

2

Heat and Power Technology, Royal Institute of Technology, Stockholm, Sweden

Abstract A prerequisite for aeroelastic stability prediction in turbomachines is the un­derstanding of the flictuating aerodynamic forces acting on the blades. Un­steady transonic fbws are complex because of mutual interactions between trav­elling pressure waves, outlet disturbances, shock motion, and fluctuating turbu­lent boundary layers. Complex phenomena appear in the shock/boundary layer region and produce phase lags and high time harmonics, which can give a signif­icant contribution to the overall unsteady lift and moment, and therefore affect flutter boundaries, cause large local stresses, or even severely damage the turbo­machine.

This paper is concerned with the understanding of phenomena associated with travelling waves in non-uniform transonic flows and how they affect the unsteady pressure distribution on the surface as well as the far field radiated sound. In similitude with turbomachines potential interaction, the emphasis was put on the unsteady interaction of upstream propagating acoustic waves with an oscillating shock in a 2D nozzle ft>w. Both numerical and experimental studies are carried out and compared with each other. Results showed that the unsteady pressure distribution results from the superposition of upstream and downstream propa­gating pressure waves, which are partly reflected or absorbed by the oscillating shock. Beside, the phase angle shift underneath the shock location was found to linearly increase with the perturbation frequency, which can be critical regarding aeroelastic stability since it might have a significant impact on the phase angle of the overall aerodynamic force acting on the blade and shift the aerodynamic damping from stable to exciting.

Keywords: Unsteady flrw, shock motion, Shock Boundary Layer Interaction, Nozzle ft>w

463

K. C. Hall et al. (eds.),

Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines, 463-481. © 2006 Springer. Printed in the Netherlands.

Introduction

Transonic fbws about streamlined bodies are strongly affected, particularly near the shock location, by unsteady excitations. Experimental and computa­tional studies [1,2] have shown that the unsteady pressure distribution along the surface of an airfoil or a cascade blade in unsteady transonic flow exhibits a significant bulge near the shock location. Tijdeman and Seebass [3] reported that the unsteady pressure bulge and its phase variation resulted from non­linear interaction between the mean and unsteady flows. This non-linear in­teraction causes a shift in the shock location, which produces the observed large bulge in the unsteady pressure distribution. Studies [4] on choked flutter have shown that, in unsteady transonic flows around a single airfoil, the shock motion, and thus the pressure distribution along the surface, can be critical re­garding to the self-exciting oscillations of the airfoil. It was also shown that the mean flow gradients are of high importance regarding the time response of the unsteady pressure distribution on the airfoil surface. Beside, numeri­cal computations [5] pointed out that the exact location of the transition point could strongly affect the prediction of stall flutter. Further studies [6] sug­gested that this sharp rise in the unsteady pressure distribution was due to the near sonic condition, and that the near-sonic velocity acts as a barrier they identified as acoustic blockage preventing acoustic disturbances from propa­gating upstream in a similar way to the shock in transonic ft>ws. A transonic convergent-divergent nozzle experimentally investigated by Ott et al [7] was thereafter used as a model to investigate the non-linear acoustic blockage. An­alytical and numerical computations [8, 9, 10, 11] were then carried out to analyze and quantify the upstream and downstream propagation of acoustic disturbances in the nozzle.

Similarly, in order to focus the present analysis on essential features, the investigation has been carried out in a simple geometry such as a 2D conver­gent divergent nozzle. Special influences of leading and trailing edges, and interblade row region interactions are therefore avoided.

The described optimized time discretization has been implemented and tested in an internal GE code. The main features of this code [Holmes et al., 1997] can be summarized as follows:

Multiblock structured meshes

Dual-time stepping for unsteady fbw simulations

Phase lag boundary conditions

Multigrid and local time stepping for convergence acceleration

Adaptive 2nd/4th order (JST) dissipation

k — и model for turbulence modeling

2. Numerical results

The first calculation considered is the propagation of an entropy wave in a periodic duct. The flow is subsonic and inviscid. The initial flow is uniform. At the inlet plane, an entropy wave is specified that convects downstream with fre­quency such that the expected wave length is approximately six times smaller than the length of the duct. At the exit, non-refecting boundary conditions are applied that allow all the outgoing waves to leave the computational domain. The calculations are performed with non-optimized and optimized second or­der temporal discretizations. The number of time steps per unsteady period is varied from 10 to 90. We also perform a calculation with an excessively large number of time steps per period to obtain a solution independent of the size of the time step. This solution is used to compute errors for the calculations with larger time steps. All inner iteration loops for all calculations are fully converged. The grid size is 250 by 50 which is sufficiently fine to resolve the specified wave. Figure 3 shows the density distribution along the duct for the both optimized and non-optimized schemes. Figures 4 and 5 show the ampli­tude and phase errors at the points located at the middle and the exit of the duct. These plots demonstrate that in order to achieve a certain level of the amplitude error one needs approximately twice as many time steps per period for a calculation with non-optimized discretization compared to the optimized one. To maintain the same phase error the number of time steps per period can be reduced by 15% when the optimized discretization is used.

Next, we consider a wake/blade row type calculation where the blade row is a rotor of a three-dimensional high pressure compressor. The wake is modeled by specifying a vorticity wave at the inlet. Again, calculations are performed

AERODYNAMICS

For simplicity, the analysis is performed for the one-dimensional wave equa­tion given by

(1)

Applying the dual-time stepping technique, a fictitious time, т, is introduced, and equation (1) is rewritten in the form

or

The derivative with the respect to the real time, t, is discretized as

/dun+1 aiun+1 + a,2Un + азип 1 + a^vl1 2 + … + am+2u" m

Ы = At ’ ()

where At is the physical time step, and the superscript n denotes the solution at physical time t = nAt. Equation (3) is marched in fictitious time, т, to reach a pseudo-steady state, which advances the solution forward in time from t = nAt to t = (n + 1)At. In order to speed up the convergence of the residual R*(u) to zero, acceleration techniques like local time stepping and multigridding are usually incorporated.

It is a common practice [Melson et al., 1993] to use a three-point backward formula to discretize the time derivative term, du/dt, i. e.

rdun+l 3u"+1 — 4u" + u"-1

Ы = 2ДЇ ’ ()

which is second order accurate in time and unconditionally stable. It can be shown that discretizations higher than second order accurate are only condi­tionally stable which makes them not practical. To improve the quality of the discretization (5) an optimization technique for constructing low-dissipation and low-dispersion schemes [Tam and Webb, 1993,Hu et al., 1994] is applied. Consider a Fourier representation of the solution, u, given by

u(r, t,x) = ^ ui (t )exp[i(u t — hi x)], (6)

і

where і = л/^І. Substituting (4) and (6) into (1) yields an approximation of the dispersion relation for the wave equation (1). This is given by

^(ai+a2 exp(—iu)At)+ct3 exp(—2iujAt)+a4 exp(—Зга; Af)+…) — ick = 0,

t (7)

or

u* — ck = 0, (8)

where u* = —i[a1 + a2 exp(—iuAt) + a3 exp(—2iuAt) + a4 exp(—3iuAt) + …]/At is a numerical approximation of the exact frequency, u. Equation (6) can be presented in the form

u(T, t,x) = ^ ul (T)exp(iui t)exp(—Re[iu*x/c]) exp(—ilm[iu* x/c]).

(9)

From equation (9) the dissipation (amplitude) and dispersion (phase) error are estimated by the equations

tampl = Re[iU*At — iuAt], (10)

{phase = 1m[iu*At — iuAt]. (11)

Optimized schemes can be constructed by introducing more points to the dis­cretization stencil (4) than the minimum number of points required to maintain the order of accuracy. These extra points are used to minimize the dissipation and dispersion errors given by (10) and (11).

In this paper, a four point second order optimized scheme is constructed. The coefficients, ai, i = 1,4, for this scheme are determined from the ac­curacy condition given by

ai = 1.5 — a4

a2 = — 2 + 3a4 (12)

a3 = 0.5 — 3a4

that is obtained by expanding the solution, u, in equation (4) into a Taylor series and keeping all the terms up to the second order. The free parameter, a 4, is found subject to constraints on the amplitude and phase errors,

—0.001 < eampi < 0.002, шДt < 0.7

tphase < 0.005, uAt < 0.3 ()

where the limits are set to keep the dissipation and dispersion errors smaller than those of the non-optimized second order scheme. The resulting coeffi­cients are given in Table 1. Figure 1 shows the stability plots and Fig. 2 shows the dissipation and dispersion errors. Note, that if Re(iw*Д^ < 0, then the scheme is unstable (see equation (9)). While the non-optimized second order scheme is unconditionally stable, the non-optimized third order scheme is only conditionally stable. The optimized scheme is weakly unstable, however the level has been minimized, and the lower limit was chosen to avoid instability on a typical turbomachinery problem.

Mikhail Nyukhtikov

Moscow Institute of Physics and Technology Moscow, Russia

Natalia V. Smelova, Brian E. Mitchell, D. Graham Holmes

General Electric Global Research Center Niskayuna, NY USA

Abstract This paper presents an optimized discretization of the time derivative term for the dual-time stepping method. The proposed discretization is second order ac­curate and has a lower level of dissipation and dispersion errors than the conven­tional non-optimized second order discretization. Sample calculations demon­strate that the optimized scheme requires approximately 45-50% less time steps per unsteady cycle compared to the standard non-optimized scheme to resolve an unsteady fbw within a certain margin of amplitude error. The number of time steps per cycle can be reduced by 10-15% to keep the phase error less than a cer­tain level when the optimized scheme is used. Since time-accurate calculations are expensive, the proposed approach leads to significant savings of computa­tional time and resources.

1. Introduction

Flows in the turbomachinery environment are inherently unsteady. Physi­cal phenomena like vortex shedding, wake/blade row interaction, tip leakage etc. can be modeled correctly only by time-accurate non-linear methods. A number of such methods were developed over the past few years. One of them, the dual-time stepping method [Jameson, 1991], is widely used and is fa­vored for its convergence properties and its ease of implementation. It employs well-known convergence acceleration techniques like multigridding, false time marching, and residual smoothing. It also has been shown [Melson et al., 1993] that this method allows an implicit treatment of the real time derivative and this removes the upper stability limit from the size of the time step. However, the

449

K. C. Hall et al. (eds.),

Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines, 449-459. © 2006 Springer. Printed in the Netherlands.

consideration of the stability limit is not sufficient because the dissipation and dispersion errors also limit the size of the time step.

This paper focuses on the development of a new optimized time derivative discretization, which has a better dissipation and dispersion properties as com­pared to conventional non-optimized discretization. This new discretization allows larger time steps (hence, reduced computational time) for a given level of numerical error. The technique for constructing a low-dissipation and low – dispersion scheme [Tam and Webb, 1993, Hu et al., 1994] is based on intro­ducing more than the minimum required number of points to the discretization stencil. The coefficients for these extra points are determined by minimizing the dissipation and dispersion errors. Note that the introduction of extra points increases the memory requirements.

In this paper, a second order accurate discretization of the time derivative is considered. The optimized scheme with low dissipation and dispersion errors is constructed by introducing only one additional point. A detailed description of the optimization process is provided. The paper also presents test problems, which demonstrate the overall improvements and speedup.

The calculations were performed on a personal computer with two Intel Pentium IV processors (jobs run in serial mode) with a clock speed of 1.7GHz. The computational effort required for some of the test cases presented in this paper are shown in Table 1. In terms of computational time, it can be seen that the current scheme is very efficient. The memory requirements are large but not unreasonable for a modern personal computer. However, if the current scheme is to be extended to three-dimensions, a supercomputer or a cluster of PCs would be necessary to deliver the required memory.

The linearised solutions to the Standard Configuration 11 test case described above, using various flux schemes but all based on the same steady-state solu­tion (AUSMDV) are shown in Fig. 6. The fact that it was possible to obtain accurate solutions when the flux scheme used to calculate the steady-state so­lution is different to the scheme used by the unsteady linearised solution is a good demonstration of the robustness of GMRES with preconditioning. Hall and Clark [17] stated that for transonic fbws with shock capturing, the discre – tised perturbation equations should be a faithful linearisation of the discretised unsteady nonlinear equations used to compute the steady flow. Moreover, Hall & Clark could not obtain converged linearised solutions in the vicinity of a res-

onant interblade phase angle when pseudo-time-stepping was employed. The author also encountered similar difficulties when using pseudo-time-stepping. However when GMRES was employed it was possible to find solutions for all interblade phase angles. Note the aerodynamic dampings shown in Fig. 3 are calculated at one degree intervals.

3. Summary

A linearised Navier-Stokes fbw solver which includes the Spalart & All – maras turbulence model that is suitable for flutter investigations has been pre­sented. Various flux schemes were tested and EFM gave the best results for Euler calculations and AUSMDV gave the best results for viscous calculations. GMRES with preconditioning was used to solve the unsteady linearised flow equations in a robust and efficient manner. A three-dimensional version of this method is currently being developed.

References

S. Weber, M. F. Platzer: “A Navier-Stokes Analysis of the Stall Flutter Characteristics of the Buffum Cascade”. Journal of Turbomachinery. 122, 2000, pp. 769-776.

W. S. Clark, K. C. Hall: “A Time-Linearized Navier-Stokes Analysis of Stall Flutter”. Jour­nal of Turbomachinery. 122, 2000, pp. 467-476.

L. Sbardella, M. Imregun: “Linearised Unsteady Viscous Turbomachinery Flows Using Hybrid Grids”. Journal of Turbomachinery. 123, 2001, pp. 568-582.

P. R. Spalart, S. R. Allmaras: “A One-Equation Turbulence Model for Aerodynamic Flows”. In Proceedings of 30th Aerospace Sciences Meeting & Exhibit. 1992, AIAA-92-0439.

J. E. Bardina, P. G. Huang, T. J. Coakley: “Turbulence Modeling Validation, Testing, and Development”. NASA Technical Memorandum 110446, April 1997.

M. Giles: “Non-Refbcting Boundary Conditions for the Euler Calculations”. AIAA Jour­nal. 28, 1990, pp. 2050-2058.

I. A. Johnston: “Simulation of Flow Around Hypersonic Blunt-Nosed Vehicles for the Cal­ibration of Air Data Systems”. Ph. D. Thesis. Department of Mechanical Engineering. Uni­versity of Queensland. Australia. 1999.

P. L. Roe: “Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes”. Journal of Computational Physics. 43, 1981, pp. 357-372.

Y. Wada, M.-S. Liou: “A Flux Splitting Scheme with High-Resolution and Robustness for Discontinuities”. AIAA Paper 94-0083, Jan. 1994.

P. J. Petrie-Repar: “Numerical Simulation of Diaphragm Rupture”. Ph. D. Thesis, Univer­sity of Queensland, Australia, 1998.

D. I. Pullin: “Direct simulation methods for compressible inviscid ideal-gas fbw”. Journal of Computational Physics 34 pp. 231-244, 1979.

P. A. Jacobs: “Single-block Navier-Stokes Integrator” NASA CR-187613, ICASE Interim Report 18, 1991.

Y. Saad: “Iterative Methods for Sparse Linear Systems”. SIAM, 2003.

T. H. Fransson, J. M. Verdon: “Updated Report on Standard Configurations for Unsteady Flow Through Vibrating Axial-Flow Turbomachine Cascades”. Report. Royal Institute of Technology, Stockholm, Sweden. URL: http://www. egi. kth. se/ekv/stck.

T. H. Fransson, M. Jocker, A. Bolcs, P. Ott. “Viscous and Inviscid Linear/Nonlinear Cal­culations Versus Quasi 3D Experimental Cascade Data for a New Aeroelastic Turbine Standard Configuration”. Journal of Turbomachinery. 121, 1999, pp. 717-725.

D. R. Lindquist, M. B. Giles: “Validity of Linearized Unsteady Euler Equations with Shock Capturing”. AIAA Journal 32(1), pp. 46-53, January 1994.

K. C. Hall, W. S. Clark: “Calculation of Unsteady Linearized Euler Flows in Cascades Us­ing Harmonically Deforming Grids”. In Proceedings of Unsteady Aerodynamics, Aeroa – coustics and Aeroelasticity of Turbomachines and Propellers. Editor: H. M. Atassi. Univer­sity of Notre Dame. 1991.

Standard Configuration 11 [15] is a two-dimensional turbine cascade. An off-design transonic condition is examined which is characterized by a sep­arated flow region on the suction side with detachment immediately down­stream of the leading edge and re-attachment at approximately 30 percent chord. A shock is also present on the suction surface at approximately 80 percent chord. The infbw conditions are: в і = — 34.0o and inlet Reynolds number is 1.4 x 106. The isentropic Mach number at the exit M2 = 0.99. The imaginary unsteady pressure response due to harmonic bending normal to chord with ш* = 0.309 and a = 180o is shown in Fig. 5. Solutions are shown at two mesh resolutions (17518 and 70072 cells) with the height of the first grid line above the profile equal to h/c = 2.0 x 10-5 (y+ = 1.15) and h/c = 1.0 x 10-5 (y+ = 0.58) respectively. AUSMDV was used to calculate the flixes because the dissipative behavior of EFM in a boundary layer makes it unsuitable for viscous fbws [7]. AUSMDV is preferred to Roe’s scheme because it is less susceptible to noise near shocks at high grid resolutions (see Fig. 4). The agreement between the current solution and the previous numer­ical work [3] is good. Note that grid convergence was not achieved where the ft)w separated and at the shock. Experimental data [15] is also shown and the disparity on the suction surface is probably due to the failure of the two­dimensional flow model to model accurately the three-dimensional separated fbw.

Standard configuration 10 [14] is a two-dimensional compressor cascade of a modified NACA0006 profile at a stagger angle of 45 degrees with a gap to chord ratio of 1.0. The unsteady inviscid pressure response due to the blades harmonically pitching about their mid-chord position is sought.

Solutions for Case 1 (inlet Mach number M1 = 0.7, inlet fbw angle в і = 55.0o, reduced frequency = 0.5 based on chord and interblade phase angle

a = 0) are shown in Fig. 1. The solutions were calculated using Roe’s scheme. The comparison with the previous numerical work is very good and the current method appears to give the correct answer for this case.

0.25

Figure 1. Imaginary unsteady pressure coefficients for Standard Configuration 10 Case 1

Mi = 0.7, h = 55.0 , u> = 0.5 and a = 0

pressure surface

suction surface

0.25

Figure 2. Imaginary unsteady pressure coefficients for Standard Configuration 10 Case 6: Mi = 0.7, ві = 55.0o, = 1.5 and a = 90o

Solutions for Case 6 (M1 = 0.7, ві = 55.0o, w* = 0.5, a = 90o) are shown in Fig. 2. These solutions were also calculated using Roe’s scheme. This case is significantly more challenging than Case 1 because it is at a resonant condition at the downstream far-field as the sum of the acoustic waves here form planar waves which are travelling in a direction perpendicular to the axis. Conversely, for Case 1 where a = 0, planar waves are formed in both far-fields, which are travelling parallel to the axis, and a one-dimensional treatment is sufficient to ensure non-refection of waves at these boundaries. The difficulty of Case 6 is evident in the fact that there is no consensus between the previous solutions. The solutions are similar but one would expect a better agreement (as achieved for Case 1) for such a well defined problem. Solutions from the current method are shown at two different grid resolutions and also for an extended mesh which has the same grid resolution as the mesh with 12116 cells but with the far-field boundaries located two chord length from the profile as opposed to one chord length for the other meshes. Note that the solutions from the current method are independent of grid resolution and the location of the far-field boundaries. Also note that a higher grid resolution than Case 1 was required to achieve grid convergence for this case. The fact that the current solutions are independent of far-field boundary location suggests that the non-refecting boundary condition has been correctly implemented. Further verification of this can be seen in the good comparison of the aerodynamic

damping versus interblade phase angle with previous work shown in Fig. 3, particularly the prediction of the peaks at resonant conditions.

Solutions for Case 17 (M1 = 0.8, ві = 58.0o, u* = 0.5 and a = 0) are shown in Fig. 4. The inlet Mach number is higher for this case than the previous two cases and causes a shock to form on the suction surface at approximately x/c = 0.25. Solutions calculated with Roe’s scheme, AUSMDV and EFM at a high grid resolution (27261 cells) are shown. For each solution shown, the same flix scheme was used for the the steady-state and linear solution. The Roe solution exhibits unphysical peaks in the solution near the shock. This is probably due to lack of numerical dissipation at the high grid resolution. These peaks are still present in the AUSMDV solution but are significantly smaller. The peaks are not present in the EFM solution due to the highly dissipative nature of the scheme. The conclusion is made that for high resolution Euler calculations, EFM is the better choice because its solutions at high grid resolu­tion are less likely to exhibit unphysical noise and the extra dissipation of the scheme only has a small affect on the accuracy of the solution [10].

The shape of the shock impulse (unsteady pressure response due to motion of shock) predicted by Huff is wider with a lower peak than that predicted by the current method. This is because Huff used a pitching amplitude of a0 = 2.0o, and the current method uses a very small amplitude (a0 = 1.0 x 10-6 o) in order to calculate the linear response. An EFM solution at a lower grid res­olution (12216 cells) is also shown and it can be seen that grid convergence has not been achieved. However, the work done on the blade by the shock im­pulse does converge. The aerodynamic dampings calculated by EFM solutions on grids with 3029, 12216 and 27261 cells were 0.2122, 0.2344 and 0.2340 respectively. The validity of using linearised unsteady analysis for ft>ws with shocks for flutter investigations was recognized by Lindquist and Giles [16].