Heat Transfer Coefficients

L q

к T*2 — Tw

The heat transfer on the blade surfaces is expressed by the Nusselt number

Similar to the blade pressure distribution the unsteady effects are less obvi­ous in the NGV. There, the most significant phenomena are taking place on the suction side close to the leading edge. The ejecting cooling ft>w interacts with the main flow, triggering time dependent separations of the main flow immedi­ately behind the NGV leading edge. The obvious discontinuity at around 50% normalized axial distance (x/lax = 0.5) on the NGV suction surface is caused by the connection of a very fine grid to the relatively coarse surrounding grid. The high gradients of the quite sensitive Nusselt number are smeared out on the coarser grid, causing a discontinuity if plotted along the blade surface.

The overall level of the Nusselt number along the uncooled rotor blade sur­face is by far smaller compared to the cooled NGV. Unsteady effects are domi­nant throughout the entire blade passage (Fig. 7). The range of the time depen­dent Nusselt number can reach more than three times the level of the steady or time averaged calculation questioning the reliability of steady heat transfer calculations in multistage configurations.

The hot streaks of uncooled flow and the cooling jets emerging from the NGV enter the rotor passage in an alternating way (Fig. 8). In cases where relatively cool air from the jets impinges on the rotor blade surface the Nusselt number changes its sign, indicating a heat flux from the rotor into the flow (Fig.7).

Blade Pressure Distribution

The blade pressure distribution, given as isentropic Mach number (Fig. 4) in the NGV at 50% span compares the results of the steady and unsteady results of both the source term approach and the fully discretized cooling holes as well as experiments.

Quite interestingly, although the unsteady results are fhctuating within a hardly visible range, the time average deviates significantly from the steady calculation performed by using a mixing plane approach. The differences oc­cur mainly in three areas.

First, all the pressure peaks around the emerging cooling jets are by far more dominant in the unsteady calculation than in the steady results. Here, any in – flience from the downstream rotor can be excluded since the location of the cooling holes is upstream of the sonic throat. The pressure peaks are partic­ularly significant in case of the fully meshed cooling holes, and less obvious in the source term results. These pressure over – and undershoots originate in a quasi stagnation of the main ft>w immediately in front of the cooling jet. After a severe deceleration, the main flow is forced around the cooling jet re­sulting in a strong acceleration. In such a case the cooling jet behaves very much like a solid obstacle in the flow, characteristic for cylindrical cooling holes (Hildebrandt, Ganzert, Fottner (2000)). Strong interactions between the emerging cooling jets and the main flow occur. These interactions lead to a complicated system of vortices (Vogel (1997)), which are prone to self-excited unsteadiness.

The second region of interest is around the exits of the second row of cooling holes located at the pressure side at around (x/lax = 0.5). The cooling holes on the pressure side are arranged in two double rows. In the steady calculation, a strong peak occurs, which corresponds to the first set of holes of the second rows, while the effects from the second part of the double row is barely visible. In contrast, the time accurate solution produces the dominant velocity peak

Normalized Axial Distance

Figure 5. Blade Pressure Distribution Rotor 50% Span

around the position of the second set of cooling holes in the double row. The unsteadily computed jets of the first row are apparently by far stronger than their counterparts from the steady solution. The strong peak visible for the first cooling hole row on the suction side gives also evidence to this. Consequently, the stronger unsteady jet of the first line of holes forces the main fbw away from the blade surface, which results in a much less severe interaction between the main ft>w and the jets emerging from the second line of holes. Again this effect is by far less pronounced, but still detectable in case of the source term approach. Here, the cooling jets are always weaker than in case of the fully discretized holes. The steady source term calculation hardly shows any sign of the cooling jets in the isentropic Mach number distribution.

Third, the second row of cooling holes on the suction side have the most visible effect on the main ft>w, recognizable by a strong pressure under – and overshoot. The location (x/lax = 0.7) is close to the peak Mach-Number of the main flow. Hence, the jets are emerging into a region of low pressure, result­ing in a high local blowing rate. The succeeding shock (x/lax = 0.75) is less pronounced in the unsteady time-averaged calculation. The unsteady shock fluctuations are smeared out by the time-averaging. Since there are hardly any differences between source term approach and the discretized cooling holes, it is obvious that this phenomena is not connected to any film cooling effects.

The blade pressure on the rotor surface is given for all the unsteady time steps, the unsteady time average and the steady computation (Fig. 5). Natu­rally, the time dependent flictuations inside the rotor are by far more dominant, forced by the impinging wakes from the upstream NGV. The differences be­tween the time averaged and the steady results is largest at the rotor leading edge. It is this region, which suffers most from the numerical simplifications necessary for a mixing plane approach. The range of the time dependent flic – tuations is large throughout nearly the complete blade. However, approaching the trailing edge, the fluctuations are damped out, showing hardly any influence on the rotor exit Mach number.

Computational Performance

All computations were carried out on a single processor PC at 1800 MHz, running under LINUX. Starting from a steady state solution the unsteady com­putation took about 18 times to pass the rotor leading edge behind the NGV trailing edge in order to achieve a satisfactory periodical behaviour. The un­steady mass fbw was taken as a convergence criteria (Fig.3). The total CPU time was in the order of 20 days, requiring about 1 GB of RAM. The overall level of convergence was slightly fhctuating around three orders of magnitude reduction in the total RMS residual.

The unsteady calculations were carried out using the domain scaling tech­nique. The rotor pitch was brought from 60 to 64 blades, allowing to mesh two rotor blades with the same periodicity as one NGV pitch. For convergence acceleration dual time stepping was used. The rotor turning was resolved by 32 discrete angular positions for one rotor pitch.

3. Comparison Full Discretization/Source Term Approach

Apart from the human effort of meshing 120 additional cooling holes, the source term approach requires considerably less computational resources. The larger RAM requirements are obvious, considering the higher number of grid cells and blocks. In addition, the CPU time increases over-proportionally since the coupling between the main fbw and the cooling jets is much stronger in case of the fully discretized approach. Here, convergence is slowed down due to the slow propagation from the main fbw through the holes into the plenum.

4. Results

Numerical Boundary Conditions

These types of inlet and exit boundary conditions are typical for turboma­chinery cases. There was some uncertainty about the specification of the wall boundary conditions. As a best possible assumption, the thermal wall bound­ary conditions had been set to a constant wall temperature inside the entire NGV as well as on the rotor blade surface and hub. All other walls within the domain were treated as adiabatic. Considering the very short measurement times (approx. 500ms) this simplification seems justified.

2. Computational Grid

The numerical domain was discretized using a structured multi-block grid. Compared to an unstructured tetrahedral approach structured grids usually pro­vide a higher numerical accuracy. Consequently, emphasis was laid on a high grid quality in order to minimize numerical errors, particularly inside the cool­ing holes and their immediate vicinity. The grid in these regions is locally highly refined. This high level of refinement would have led to an overall number of grid points, far beyond any reasonable limits. In order to reduce the problem size coarser grid blocks are located around the highly resolved grid regions. The coarse and fine grid areas are connected by means of a non­congruent block-to-block connection using a fully conservative interpolation technique. The application of this technique in film cooling configurations had been described by Hildebrandt (2001).

Around the blades as well as in the front and rear plenum and inside the cooling holes HOH-topologies had been applied (Fig.1, Fig. 2). The grid is composed of 651 grid blocks with a total number of 2.1 Mio. Grid points.

About 75% of the grid points are located in the immediate vicinity of the cool­ing holes. The refined areas around the rows of cooling holes are visible in Fig.2. These areas are resolved about four times finer in each spatial direction than the surrounding regions of the main ft>w.

The non-dimensional wall distance y+ varies typically around 1 and 2, de­pending on the local fbw conditions. The laminar sub-layer, important for any prediction of wall shear stress or heat transfer, is therefore well captured.

Figure 3. Mass Flow Convergence History

Table 3. Resource requirements

Source Term

Full Discretization

Iterations for full convergence



Grid points






Relative CPU time



Relative RAM



The MT-1 Single Stage HP Turbine

The MT-1 single stage HP-turbine, which had been investigated in the present study, is described in detail in (Kluge et. al. (2003)). Table 1 summarizes some basic geometrical and aerodynamic specifications of the design data of the TATEF turbine stage.

In order to carry out unsteady CFD simulations with an acceptable computa­tional effort the domain scaling method had been applied. There, it is desirable to obtain a small common integer factor as a blade number ratio between NGV and rotor. The original blade number of the rotor had been increased from 60 to 64 enabling to perform a time-dependent periodic computation with one stator passage and two rotor passages meshed. Usually the error, which results from changing the solidity, is acceptable, if the change in blade pitch is less than 10%, which is the case herein.

Table 2. Numerical Boundary Conditions


Inlet, NGV









Inlet, Front cavity





axial into Plenum




Inlet, rear cavity





axial into Plenum





P2 (rad. eq.)

142.100 Hub


Walls: all NGV, rotor hub & blade

Tw imposed

288.5 /333


Walls: all other


Computational Method

Within the frame of the presented computations a commercial CFD systems has been employed. FINE/Turbo, developed by NUMECA Int. S. A (NU – MECA (2002)), is a specialized CFD package for all sort of turbomachinery applications. The package includes grid generation, the flow solver and a post processing software. All program modules are embedded into a turbomachin­ery specific environment.

The numerical scheme solves the 3D Reynolds-averaged Navier-Stokes equa­tions (RANS) on general structured non-orthogonal multi-block grids. The flexibility of the structured grids is greatly enhanced by use of so-called "Full Non Matching Connections", a technique, which allows to arbitrarily connect grids block of different grid topologies or point numbers to each other.

The numerical algorithm incorporated into FINE/Turbo is an explicit four stage Runge-Kutta scheme (Jameson and Baker (1984)). A variety of conver­gence acceleration techniques are employed, such as implicit residual smooth­ing, dual time stepping and full multigrid. Space integration is performed us-

Table 1. Design Data of the MT-1 Turbine


Blade Number NGV / Rotor


32 / 60, 64*


Mass Flow, Inlet



[ kg/s ]

Rotational Speed




Exit Mach Number




Reynolds Number




Gas-to-Wall Temperature Ratio


ing a second order cell-centered finite volume discretization with second and fourth order artificial dissipation. Coarse grid calculations can be carried out in an automatic way on every coarser grid level.

A number of turbulence models are available within FINE/Turbo. In the scope of the present work the algebraic turbulence model of Baldwin and Lo­max (1978) has been chosen. All solid walls have been treated as fully tur­bulent. The authors are well aware that a simple turbulence model and the assumption of fully turbulent boundary layers cannot capture sufficiently ac­curate the quite complex turbulent structures typical for film cooling. With the main objectives of this study in mind, comparing a fully discretized film cooling geometry with a source term approach, the use of a somewhat sim­pler model seemed justified and effective. Moreover, new experimental data suggest (Ardey (1998)) that in film cooling simulations the use of any eddy viscosity turbulence model is questionable due to the extreme anisotropic na­ture of turbulence in these cases.


Th. Hildebrandt, J. Ettrich

NUMECA Ingenieurburo, D-90530 Wendelstein Thomas. Hildebrandt@numeca. de

M. Kluge, M. Swoboda, A. Keskin, F. Haselbach, H.-P. Schiffer

ROLLS ROYCE Deutschland, Eschenweg 11, D-15287 Dahlewitz, Germany Marius. Swoboda@rolls-royce. com

Abstract Every modern high-pressure turbine needs a highly sophisticated cooling sys­tem. The most frequently used cooling method of to date is film cooling, char­acterized by a high degree of interaction between the main fbw and the cooling flaw. Therefore the effects of film cooling have to be taken into account in the aerodynamic design of film cooled high-pressure turbines.

Using modern commercial turbomachinery oriented CFD-methods, the mod­eling of film cooling holes can be achieved by various numerical methods of dif­ferent complexity. The so-called source term modeling is fast and easy to apply, but cannot provide very detailed ft>w information. In contrast, the discretization of every single cooling hole represents a very complex approach, but provides more in-depth information about the cooling pattern. The efforts of full-scale modeling need to be balanced against the more detailed and accurate results. In addition to the complex geometries of film cooled turbines, the flow phenomena are highly unsteady, thus requiring a CPU intensive time dependent numerical approach.

The present paper is focused on a detailed investigation of the unsteady flow field in a film cooled high-pressure turbine stage. An unsteady 3D Navier-Stokes calculation is applied to the entire stage configuration including a full discretiza­tion of all the cooling holes.


M = Blowing rate

v = Velocity (m/s)


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

Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines, 533-549. © 2006 Springer. Printed in the Netherlands.

p = Pressure (Pa)

Ma = Mach Number

Re = Reynolds Number

p = Density (kg/m3)


c = cooling

2 = Inlet, exit conditions

t = total

is = isentropic


NGV = Nozzle Guide Vane

1. Introduction

In order to obtain maximum thermodynamic cycle efficiency a high temper­ature level is required in the high pressure (HP) turbines of modern environ­mentally friendly gas turbines. The temperature level there is usually by far higher than the maximum allowable temperature of even the most advanced materials. Therefore, every modern HP turbine needs a sophisticated cooling system. From a variety of available cooling methods film cooling emerged as today’s standard cooling method. Relatively cool compressor air is injected through numerous holes and slots on the blade and endwall surfaces of a HP – turbine. Apart from the desired inflience of the injected cooling air on the heat transfer coefficients of the blade and endwall surfaces, the cooling jets have a considerable effect on the main fbw as well (Benz (1994), Hildebrandt et. al. (2001), Vogel (1997)). As a consequence, the effects of film cooling have to be taken into account in the aerodynamic design of a HP turbine.

Modern commercial Navier-Stokes solvers provide the designer in the turbo­machinery environment with a variety of options to simulate the flow inside the blade passage of a film-cooled turbine. The CFD modeling of film cooling holes can be achieved by various numerical methods of different complexity. The numerical technique of source term modeling is the fastest and least com­plex method to introduce the effects of film cooling into a 3D Navier-Stokes calculation of a turbine. This method is computationally least expensive and easy to apply, making it well suitable for the fast turn-around times, which are required in the modern design processes. The cooling ft>w is taken into ac­count by a distribution of various sources of mass, momentum and energy on the blade and endwall surfaces. In contrast, the full modeling of every single cooling hole represents the most complex approach. Using this method every cooling hole, including the cooling air plenum is discretized. Obviously, turn­
around times and engineering efforts are by far higher if compared to the source term method. The reward of applying this method to a film-cooled turbine is a high amount of very detailed ft>w information.

The complex ft)w phenomena of film cooling are apparently time dependent themselves, and additionally, highly inflienced by the unsteady rotor-stator in­teraction of the adjacent blade rows. The impinging wakes of a preceding blade row are periodically altering the local cooling efficiency along the blade sur­faces of the succeeding turbine rotor. Vice versa, the circumferentially chang­ing backpressure induced by a succeeding blade row can lead to considerable fluctuations in blade pressure distribution and shock location. The local blow­ing rate given by

is a function of the local velocity ratio, hence depending strongly on the pressure gradient between the plenum and the local ejection position on the blade surface. Therefore, a periodically flictuating blade pressure distribu­tion leads directly to an equivalently flictuating local film cooling efficiency. Therefore Unsteadiness is crucial if the focus is on very detailed cooling fbw phenomena.

The present paper is focused on a detailed investigation of an unsteady flow field in a film cooled high-pressure turbine stage. The flow is simulated using an unsteady 3D Navier-Stokes calculation of the entire turbine stage of a noz­zle guide vane and rotor configuration including a full modeling of all single cooling holes.


Figure 2 shows a typical example of the measured tubing transfer function for a tubing length used in the measurement of unsteady pressures in an os­cillating cascade. In this case a slight amplification can be seen over the fre­quency range of interest, indicating a resonant peak at a higher frequency. The phase distortion is more significant due to the importance of the relative phase of surface pressure flictuations and the vibration of the blade.

Figure 2. Transfer Function of the measurement system for the blade flitter case (brass tube, 180mm x 1mm Portex tubing and connector)

Figure 3 shows the transfer function for a single tube of a 5-hole probe used to make measurements in the wake of a bluff body exhibiting vortex shedding. Small tube diameters near the probe head and a longer tubing length results in a system in which viscous attenuation dominates over any resonant effects.

Figure 3. Transfer Function of the measurement system for the vortex shedding case (5 hole probe, 0.75mm Portex tubing and connector)

Figure 4 shows the effectiveness of the transfer function correction method in reconstructing an original reference signal from a distorted one. The tubing system of Fig. 3 was subjected to a 100Hz saw waveform using the transfer function measurement apparatus. Significant phase lag and attenuation rela­tive to the reference signal is clearly apparent in the uncorrected signal and the increased attenuation of higher harmonics alters the waveform shape. The pre­viously measured transfer function was then used to infer the original signal and this is labeled “corrected” in Fig. 4. This can be seen to closely match the original reference signal.

Figure 4. Effect of transfer function correction with single hole of a 5-hole probe (100Hz saw wave)

The requirement for miniaturization of pneumatic probes makes the use of off-board transducers particularly attractive, however, traditionally this has been assumed to limit the probe to steady-state measurements only. By us­ing transfer function correction, it is possible to use a conventional pneumatic probe to make time-accurate measurements. To validate the use of transfer function correction for probe measurements, the 5-hole probe used above was mounted adjacent to a single element hot-wire probe in the wake of a bluff body exhibiting vortex shedding at frequency of 58 Hz. The agreement be­tween the hot-wire and pneumatic probe with transfer function correction was similar to the level of agreement between two hot-wire probes at the same spacing in the same fbw. Further details can be found in Sims-Williams and Dominy (1998b).

Because probes are generally used to make measurements at different loca­tions in the ft>w-field sequentially, some form of synchronization is required in order to obtain instantaneous ft>w-field data. In cases where the unsteadi­ness is imposed externally (eg: forced vibration), or where it is coupled with some mechanical oscillation (eg: aeroelasticity), this may be accomplished us­ing triggered sampling from the mechanical motion. For cases of self-excited aerodynamic unsteadiness, this is more difficult. The unsteady reconstruction technique of Sims-Williams and Dominy (2000) uses a signal from a station­ary reference probe, and a complex convolution in the frequency domain, to effectively synchronize probe measurements made sequentially. This provides a more robust determination of relative phase than simply using triggered sam­pling, and this makes the technique appropriate even for weakly periodic fbw – fields. Figure 5 shows the instantaneous vorticity field in the wake of a ‘Gurney Flap” high lift device on the trailing edge of an inverted airfoil. By producing a series of these images vortex shedding can be clearly observed.

Figure 5. Instantaneous non-dimensional vorticity in the wake of a Gurney Flap

Unlike other methods of unsteady ft>w-field measurement, the use of a pressure probe allows the observation of static and stagnation pressure, as well as velocity. Figure 6 illustrates the instantaneous stagnation pressure field corresponding to Fig. 5. An issue of interest regarding the understand – ing/interpretation of unsteady results is the decoupling between stagnation pressure (the measure of loss for steady flow only) and entropy (the measure of loss in general). This has been observed computationally for a LP turbine cascade subject to incoming unsteady wakes (He, 1992, 1996) and has been observed computationally and experimentally adjacent to the wake of bluff bodies exhibiting vortex shedding (Sims-Williams and Dominy 1998b). In Fig. 6, packets of stagnation pressure deficit corresponding to the shed vortices can be observed, but importantly, it is also possible to see regions where the stagnation pressure coefficient is greater than unity. As discussed above, in an unsteady flow, instantaneous stagnation pressure and entropy become uncou­pled. The frequency of the shedding in this case was approximately 300Hz. Further details of this work on Gurney flap vortex shedding may be found in Sims-Williams et al. (1999) and Sims-Williams (2001).

The upper limit on the frequency response, which can be obtained for multi­hole probes using transfer-function correction, is restricted both by the level of correction required (which results in a deterioration in signal to noise ratio), and by time required for the flow around the head of the probe to develop (since the assumed sensitivity of the probe is based on a steady-state calibration).

Figure 6. Instantaneous stagnation pressure coefficient in the wake of a Gurney Flap

For typical multi-hole probes used in low-speed applications, these two factors both suggest a similar upper limit in the region of 1000Hz.


Bell, D. L. and He, L., 2000, Three-Dimensional Unsteady Flow for an Oscillating Turbine Blade and the Infhence of Tip Leakage, Journal of Turbomachinery, Vol. 122, pp. 93-101.

Buffum, D. H. and Fleeter, S., 1993, Wind Tunnel Wall Effects in a Linear Oscillating Cascade, Journal of Turbomachinery, Vol. 115, pp. 147-156.

Bolcs, A. and Korbacher, H., 1993, Periodicity and Repetitivity of Unsteady Measurements of an Annular Turbine Cascade at off design Flow Conditions, ASME 93-GT-107.

Carta, F. O. and St. Hilaire, A. O., 1978, Experimentally Determined Stability Parameters of a Subsonic Cascade Oscillating Near Stall, Journal of Engineering for Power, Vol. 100,

pp. 111-120.

Fleeter, S., Novick, A. S., Riffel, R. E. and Caruthers, J. E., 1977, An Experimental Deter­mination of the Unsteady Aerodynamics in a Controlled Oscillating Cascade, Journal of Engineering for Power, Vol. 99, pp. 88-96.

Fransson, T. H., 1990, Analysis of Experimental Time-Dependent Blade Surface Pressures from an Oscillating Turbine Cascade with the Inflience-Coefficient Technique, ASME 90- GT-225.

Frey, K. K. and Fleeter, S., 2001, Oscillating Airfoil Aerodynamics of a Rotating Compres­sor Blade Row, Journal of Propulsion and Power, Vol. 17, pp. 232-239.

He, L., 1992, Stagnation Pressure-Entropy Decoupling on a High Load LP Turbine Cas­cade, Unpublished work, Whittle Laboratory, Cambridge University.

He, L., 1996, Time-marching Calculations of Unsteady Flows, Blade Row Interaction and Flutter, Unsteady Flows in Turbomachines, Lecture Series 1996-05, von Karman Institute for Fluid Dynamics, Brussels, Belgium.

He, L. and Denton, J. D., 1991, An Experiment on Unsteady Flow Over an Oscillating Airfoil, ASME paper 91-GT-181.

Hooper, J. D. and Musgrove, A. R., 1991, Multi-Hole Pressure Probes for the Determina­tion of the Total Velocity Vector in Turbulent Single-Phase Flow, 4th International Sym­posium Transport Phenomena in Heat and Mass Transfer, The University of New South Wales, Sydney, Australia, ed. JA Reizes, July, 1991.

Irwin, H. P.A. H., Cooper, K. R. and Girard, R., 1979, Correction of Distortion Effects Caused by Tubing Systems in Measurements of Fluctuating Pressures, Journal of Indus­trial Aerodynamics, Vol. 5, pp. 93-107.

Manwaring, S. R., Rabe, D. C., Lorence C. B. and Wadia, A. R., 1997, Inlet Distortion Gen­erated Forced Response of a Low-Aspect-Ratio Transonic Fan, Journal of Turbomachin­ery, Vol. 119, pp. 665-676.

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Sims-Williams, D. B., 2001, Self-Excited Aerodynamic Unsteadiness Associated with Pas­senger Cars, PhD Thesis, School of Engineering, University of Durham, Durham.

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Implementation Issues

A block diagram of the apparatus used in measurements of TTF of a static pressure tapping and the pneumatic tubing is presented in Fig. 1.

A swept sine wave is generated which covers the range of frequencies of interest, and this is fed to an audio amplifier and loudspeaker. For the blade flitter case, the frequency range used was 0.1 Hz to 50 Hz, with a sweep period 0.75 second when logging sets of 2048 samples at 800 Hz. The loudspeaker produces pressure fluctuations with roughly the same wave forms as the input voltage. The loudspeaker is connected to a small cavity via a short rubber tube

to isolate mechanical vibrations. A reference pressure transducer is directly connected to the small cavity and used to record the pressure inside the cavity. A static pressure tapping used in the unsteady pressure measurement (0.3 mm diameter for blade flitter case) is also connected to the cavity. A length of plas­tic tube is used to connect the static pressure tapping with the other (off-board) pressure transducer as would be done for the aerodynamic measurements.

In the blade flitter case, the reference transducer (type: Sensym 113LP01d – PCB, -1-+1 mbar range) uses the ambient pressure as a reference, and the test transducer (type: Sensym 142C01D, 0-1 psi range) uses the total pressure of the setting chamber of the wind tunnel as a reference, which is the same as that in unsteady pressure measurements. The tubing system includes the trans­ducer’s internal volume, the connector, the Portex plastic tubing, and the brass tube with six static tappings – the tapping style for the blade flitter case.

The definition used to calculate the complex transfer function is:

1 M

M j=i

where M is the number of sets used to average TF(f). The Fourier coefficients A(f) and B(f) are defined above.

In order to obtain smooth transfer function desired for correcting pressure signals, M, can be greater than 20. A Hanning window function is used to reduce the effect of the finite data length, which has been found to improve the quality of the results.

Theory of Tubing Transfer Function Approach

The tubing transfer function approach presented in this paper is based on a technique originally employed for wall pressure measurements in wind engi­neering by Irwin et al. (1979). This technique was subsequently applied for multi-hole probe measurements by Sims-Williams and Dominy (1998a) and by Hooper and Musgrove (1991).

The unsteady pressure signal propagates from the pressure tapping to the off-board pressure transducer via the tubing between them. The signal can be amplified by resonance effects at particular frequencies and will be attenuated by viscous effects at higher frequencies. There will also be a time-lag for the pressure signal to reach the transducer which will result in an increasing phase

offset at higher frequencies. This frequency-dependent tubing response can be characterized by a transfer function. Once the transfer function of a given tubing system is known, then it is possible to correct for the tubing distortion. This technique requires that the system obeys the principal of linear superpo­sition so that an unsteady signal can be decomposed into multiple frequency components, and this has been confirmed.

To utilize this approach, the tubing transfer function of the pressure measur­ing system must be known in advance, and this can be obtained experimentally. A test unsteady pressure signal including a range of frequencies is recorded by a reference pressure transducer directly and by another pressure transducer via a tubing length used for actual unsteady pressure measurements. Fast Fourier Transforms (FFTs) of both the undistorted and distorted signals are computed. The complex tubing system transfer function TF(f) is expressed as:

ТПЛ = Щ (1)

The corrected coefficients A (f) are then transformed back to the time domain using an inverse FFT in order to obtain a corrected pressure signal with the effect of tubing distortion eliminated. Both amplitude and phase distortions are removed, the latter being essential if multiple simultaneous signals are to be compared.