50
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REPORT 455 LD
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ADVISORY GROUP FOR AERONAUTICAL RESEARCH AND DEVELOPMENT
64 RUE DE VARENNE, PARIS VII
10 ■■■* ■
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REPORT 455
V,
PROPERTIES OF THE
FLUCTUATING WALL-PRESSURE FIELD
OF A TURBULENT BOUNDARY LAYER
by
M. K. BULL
^i^SS2Sl|
APRIL 1963
DDC
L1 MAR 111964
TISIA B
NORTH ATLANTIC TREATY ORGANIZATION
nraff
1 ^Sltftj'
REPORT 455
NORTH ATLANTIC TREATY ORGANIZATION
ADVISORY GROUP FOR AERONAUTICAL RESEARCH AND DEVELOPMENT
PROPERTIES OP THE FLUCTUATING WALL-PRESSURE FIELD
OF A TURBULENT BOUNDARY LAYER
by
M.K. Bull
This Report is one In the Series 448-469 inclusive, presenting papers, with discussions, given at the AGARD Specialists' Meeting on 'The Mechanism of Noise Generation in Turbulent Plow" at the Training Center for Experimental Aerodynamics, Rhode-Saint-Genöse,
Belgium, 1-5 April 1963, sponsored by the AGARD Fluid Dynamics Panel
SUMMARY
- The results of measurements of various statistical properties of the fluctuating wall-pressure field associated with turbulent subsonic boundary layer flow in conditions covering a range of values of boundary layer thickness and flow speed are given. The measured quantities include overall rms pressures, frequency spectra, and longitudinal and lateral space-time correlations in both broad and narrow frequency bands. Some experimental values of space-time correlation between wall-pressure fluctuations and turbulent velocity fluctuations at various positions in the boundary layer are also presented.
These experimental results and some of their implications on the structure of the wall-pressure field and the nature of its convection and decay are discussed.
SOMMAIRE
On donne les rdsultats de mesures de diverses propri^täs statistiques du champ de pression de paroi variable associd a 1' ^coulement des couches limitrophes turbulentes subsoniques dans des conditions englobant une plage de valeurs d' ^paisseurs de couches limitrophes et de vitesse d' ^coulement. Les quantitds mesur^es sont entre autres les pressions totales racine de moyenne des carr^s, les spectres de frequence et les conflations espace-temps longitudinales et laterales tant dans les baudes de frequences larges qu' ^troites. On präsente aussi quelques valeurs expdrimentales de correlation espace-temps entre les fluctuations de pression k la paroi et les fluctuations de vitesse de turbulence k diverses positions dans la couche limitrophe.
On dlscute ces rdsultats experimentaux et certaines de leurs implications sur la structure du champ de pression ä la paroi et la nature de sa convection et sa decroissance.
532.526.4 3b2f:3b2ela
11
CONTENTS
SUMMARY
SOMMAIRE
LIST OF FIGURES
NOTATION
1. INTRODUCTION
2. THE WIND TUNNEL
3. INSTRUMENTATION
4. THE FLOW UNDER INVESTIGATION AND ASSOCIATED EXPERIMENTAL CONDITIONS 4.1 Boundary Layer Characteristics 4.2 Sources of Extraneous Pressure Signals
5. ROOT MEAN SQUARE AND FREQUENCY POWER SPECTRUM OF THE WALL PRESSURE FLUCTUATIONS
6. OVERALL SPACE-TIME CORRELATIONS 6.1 Space Correlations 6.2 Space-Time Correlations and Convection Velocity
7. NARROW-BAND SPACE-TIME CORRELATIONS AND CONVECTION VELOCITIES
8. PRESSURE-VELOCITY CORRELATIONS
REFERENCES
FIGURES
DISCUSSION
DISTRIBUTION
Page
ii
ii
iv
vi
1
1
2
7
10
12
13
33
ill
LIST OF FIGURES
Page
Pig. 1 Variation of displacement thickness, momentum thickness and form parameter of the boundary layer (M0 = 0.3) 13
Pig. 2 Variation of displacement thickness, momentum thickness and form parameter of the boundary layer (M0 = 0.5) 14
Pig. 3 Variation of Reynolds number and skin friction along the test section (M0 = 0.3) 15
Fig. 4 Variation of Reynolds number and skin friction along the test section (M0 =0.5) 16
Boundary layer velocity profiles (M0 = 0. 5) 17
Variation of the rms value of wall-pressure fluctuations along the test section 18
Frequency spectrum of wall-pressure fluctuations 19
Longitudinal space-time correlations of the wall-pressure field 20
Longitudinal space correlation of the wall-pressure field 21
Lateral space correlation of the wall-pressure field 22
Space correlations of the wall-pressure field at various angles to the flow direction 23
Fig.12 Contours of constant space correlation of the wall-pressure field 24
Fig.13 Peaks of curves of longitudinal space-time correlation 25
Fig.14 Variation of convection time with spatial separation 26
Pig.15 Variation of convection velocity with spatial separation 27
Pig.16 Amplitude of narrjw band longitudinal space-time correlations of the wall-pressure field 28
Pig. 17 Amplitude of narrow band lateral space-time correlations of the wall-pressure field 29
Pig.18 Convection velocities of the wall-pressure field from narrow band measurements 30
Fig. 5
Pig. 6
Pig. 7
Fig. 8
Fig. 9
Pig.10
Pig. 11
iv
Page
Fig. 19 Space time correlations of the wall-pressure and longitudinal velocity component
Pig.20 General arrangement of 9 in x 6 in boundary layer wind tunnel
31
32
cf
d
H
Mo
P
P'
«o
«pp^l ■t3 .r)
^p^i A .T)
Ree
V^ .£* r)
S7^ £3 T)
NOTATION
skin friction coefficient = T0/q0
diameter of pressure transducer
boundary layer form parameter = 8*/ö
free stream Mach number
fluctuating component of static pressure on the wall
root mean square wall pressure = ( p2 >
free stream dynamic pressure = )4p0Uo
wall pressure covariance (Equation (2))
covariance of narrow band wall pressure signals
Reynolds number = ü^/v
wall-pressure correlation coefficient (Equation (4))
correlation coefficient of narrow band wall-pressure signals
Rpui(^i'^2'^3'r^ pressure-velocity correlation coefficient (Equation (14))
u. fluctuating velocity component in Xj direction
u[ root mean square velocity fluctuation in stream direction = <u'>
Ü mean velocity in flow direction
U average convection velocity
U' instantaneous convection velocity
U„(&)) convection velocity derived from narrow band correlation c measurements
U0 mean flow velocity outside boundary layer
Ü friction velocity = ^(TQ/PQ)
x co-ordinate in the direction of mean flow
x co-ordinate normal to wall
vi
x3 co-ordinate transverse to flow
y distance normal to wall
8* displacement thickness of boundary layer
<i (w) power spectral density of wall-pressure fluctuations
^nnCb ,.f,,w) spectral function (Equation (5))
T time delay
rc convection time
r wall shear stress
6 momentan thickness of boundary layer
u circular frequency
p free stream density
v kx atic viscosity
^r^2'^3 separation distances in XJ.X^XJ directions
PROPERTIES OF THE FLUCTUATING WALL-PRESSURE FIELD OF A
TURBULENT BOUNDARY LAYER
M.K. Bull*
1. INTRODUCTION
Wall pressure fluctuations produced by a turbulent boundary layer have been the subject of many investigations. Early experimental work was, on the whole, confined to measurement of the root mean square and frequency spectrum of the wall pressure, and in many cases the instrumentation available did not give an adequate coverage of the band-width of the fluctuations. More recently the development of miniature pressure transducers and the extensive use of correlation techniques has led to a much more detailed examination of the wall pressure field, and its relation to turbulent velocity fluctuations in the boundary layer.
The experimental results presented in this Report show that space-time correlations of the overall pressure signals, supplemented by correlations in narrow frequency bands, can provide a considerable amount of information on details of the structure of the pressure field. Additional information on the relation between the wall pressure field and velocity fluctuations in the boundary layer is provided by correlating the outputs of a pressure transducer in the wall and a hot-wire anemometer in the boundary layer.
2. THE WIND TUNNEL
The experimental data which are presented in this Report were obtained from measurements made on the turbulent boundary layer developed on one of the ground and polished 9-inch walls of the subsonic test section of the 9 inch x 6 inch boundary layer tunnel in the University of Southampton,
This wind tunnel is of the induced flow non-return type driven by the injection of compressed air downstream of the working sections. It has two working sections, a subsonic section 10 feet long followed by a supersonic section 6 feet long, both of which have rectangular cross-sections nominally 9 inches x 6 inches. The working sections are of fairly massive steel construction, are mechanically independent, and carried on anti-vibration mountings. The subsonic section is slightly divergent to compensate for boundary layer growth and so that there is no pressure gradient along it. Instrumentation has to be mounted in 6-inch diameter plugs which fit ports in one of the 9-inch walls. The general arrangement of the tunnel is shown in Figure 20.
To keep the extraneous sound field in the test section to a minimum, injector and diffuser are heavily sound-proofed and in the present experiments the tunnel was always run in a choked condition so that injector and diffuser noise were not propagatfid internally into the test section. The diffuser outlet is outside the laboratory building.
*bepartment of Aeronautics, The i niversity of Southampton, Southampton, England
3. INSTRUMENTATION
All measurements of fluctuating pressures were made with piezoelectric transducers set flush in the wind tunnel wall. The pressure-sensitive elements were lead zirconate- titanate disks, 0.030 inch in diameter.
The pressure transducers were mounted rigidly in their instrument plugs, the mass of the side-plates of the wind tunnel being relied upon to keep vibration of the instruments down to an acceptable value.
The electrical output of the transducer was fed to a pre-amplifier with high input impedance and the signal then further amplified for direct spectral analysis (Bruel and Kjäer ^-octave Audio Frequency Spectrometer, Type 2111) or for recording on magnetic tape (Ampex Model PR-100). All correlation values were obtained from recorded signals processed by the correlation equipment described in Reference 1.
Calibration of a transducer was effected by first mounting it in the wall of a shock tube and checking the overall linearity by means of the response to the passage of a shock wave across its face. A detailed low frequency response was then obtained by calibration against a standard condenser microphone, the two instruments being mounted in a small cavity together with a moving coil hearing aid which was used as the calibration signal generator. With the latter procedure the transducer could be calibrated in its experimental mounting. The calibration was normally checked before and after each tunnel run when root mean square pressure and spectral density measurements were being made.
Velocity fluctuations were measured by a constant temperature hot-wire anemometer (Disa Type 55 A 01). The hot-wires were tungsten, 0.0002 inch in diameter. The wires were welded to their supports and subsequently copper plated leaving an unplated working length of 0.06 inch. The hot wire probes were carried on supports of aerofoil section which could be traversed normal to the tunnel wall.
4. THE FLOW UNDER INVESTIGATION AND ASSOCIATED EXPERIMENTAL CONDITIONS
4.1 Boundary Layer Characteristics
Measurements of wall pressure fluctuations were made for two free stream Mach numbers, M0 = 0.3 and 0.5 . The variation of displacement thickness, S* , and momentum thickness, B , of the boundary layer along the working section for these two flow conditions is shown in Figures 1 and 2. These Figures also show the variation of the form parameter H = d*/6 . The corresponding values of Reynolds number and skin friction are given by Figures 3 and 4.
Skin friction values were derived from the slopes of the measured velocity profiles using the law of the wall,
U 1 /yUX — = -In — + B (1)
with K = 0.40 .
Velocity profiles were obtained from Pitot tube and wall static pressure measurements. For both flow Mach numbers a satisfactory collapse of velocity profile data from various stations along the test section was obtained in velocity defect form in terms of the derived skin friction values. The plot of (U0 - U)/UT versus y/8 for M0 = 0.5 is shown in Figure 5.
The intensity of the longitudinal component of residual turbulence in the free stream outside the boundary lajrer was measured only for M0 = 0.5 . The value found was u'/U0 = 2.0 x 10"3 with virtually no variation along the length of the working section.
4.2 Sources of Extraneous Pressure Signals
The two effects most likely to give rise to interference with the wall pressure measurements are vibration of the working section of the wind tunnel and the presence of a sound field in the working section.
Measurements of the spectrum of the output of the transducer when it was mounted in the tunnel wall but blanked off from the flow showed that vibration effects were sufficiently small to be neglected.
On the other hand the sound field in the working section was not negligible under all measuring conditions. The intensity of this field in the free stream was measured by means of a faired %-inch condenser microphone. It was established that the introduction of the microphone itself produced no significant increase in the internal sound intensity by means of a monitor microphone located flush with the wall of the settling chamber, upstream of the contraction leading to the working section (it had been found, previously, that acoustic disturbances in the working section produced by the introduction of measuring apparatus such as Pitot tubes and hot-wire probes could be quite readily detected by a microphone in this position). The sound intensity registered by the faired microphone therefore represented the internal sound field of the empty tunnel plus possibly some small increase due to the disturbance effect of the microphone itself.
The sound pressure level in the test section was found to increase with increasing distance downstreajn for both M0 = 0. 3 and 0.5 , the increase being about 5 decibels from upstream to downstream end of the working section. For most test conditions the mean square acoustic pressure was less than 4 per cent of the mean square wall pressure, although in the worst case, which occurred at the downstream end of the working section at M0 = 0.5 , this figure approached 9 per cent.
The microphone signal had a broad band character, the spectral density being highest at low frequencies and falling off with increasing frequency. Because of the high spectral density of this background field at low frequencies, the spectral density measurements of the pressure field were rejected for frequencies less than 300 cycles/sec. This low frequency cut-off amounts to a Strouhal number limitation of 0.01 to 0.06 depending on the flow conditions. The recorded signals for correlation purposes were also cut off at this frequency by making use of the response characteristics of the tape recorder. The bandwidth of recorded signals was 300 to 30,000 cycles/sec.
5. ROOT MEAN SQUARE AND FREQIENCY POWER SPECTRUM OF THE WALL PRESSURE FLUCTUATIONS
The statistical properties of the wall-pressure field to be discussed are all particular values of the covariance
Qpp^.^.T) = (HXj.Xj.t) pUj + ^ X3+<f3, t + T) > (2)
(where < > denotes a statistical mean value) or of spectral functions derived from it.
Equation (2) can be normalised by dividing it by the mean square pressure
(PV = (P2 ) = Qpp(0,0,0) (3)
to give the correlation coefficient
Rpp(.V3,r) = :^-^- ■ (4)
Th e Fourier transfonc of (LD with respect to time yields the spectral function
<J>nJ^-£*-") ^pp^l'^3 llDT {£v£3.r)e-lar dr (5)
the inverse relation being
V^.^) V^V^)eia,T dw ■ (6)
The frequency power spectral density ^(w) of the pressure fluctuations, as measured by a wave-analyser, is given by
Q (0,0,T)cosajTdT. (7)
It was hoped that measurements of root mean square pressure at various stations along the working section at the two free stream Mach numbers M0 = 0.3 and 0.5 would give an indication of the effect of Reynolds number on this parameter. However, even with the small transducer elements used, for which the ratio of transducer diameter to boundary layer thickness was in the range 0.17 < d/S* < 0.52 , the real effects of Reynolds number seem to have been obscured by the effects of transducer resolution.
The measured values of p'/q0 and P'/T0 are shown in Figure 6. At both flow Mach numbers the value of p'/q0 was found to increase as the measuring station was moved downstream along the test section. Since the corresponding value of skin friction coefficient Cf decreases (Figs. 3 and 4), there is an even more rapid increase in the ratio of rms pressure to wall shear stress p'/7"0 . In fact the value of P'/TQ
increases by about 40 per cent for a Reynolds number change by a factor of about 4. In contrast to this behaviour the values of p'/V0 found by various previous investigators suggest a decrease with increasing Reynolds number (see for example the summary given in Reference 2) although the trend is by no means well established, largely because of the diversity of experimental conditions. For pipe flow Corcos3
finds a slow decrease of p'A« with increasing Reynolds number.
The data for the frequency power spectrum of the pressure, expressed in non-dimensional form, «^ ((i^ü0/q0S* , in terms of the dynamic pressure and velocity of the stream outside the boundary layer and displacement thickness, are shown in Figure 7. They collapse quite well at low and intermediate frequencies, wS*/U0 < 1 , but at higher frequencies there is a spread of values, consistent with an increasing attenuation (due to finite transducer size) with increasing d/S* . It is fairly clear, therefore, that a large part of the observed increase in p'A"0 with Reynolds number is due to the parallel decrease in size of the transducer relative to the scale of the pressure field.
However, if we assume that where the measured spectra fall on a single non-dimensional curve ((JJS*/ü0 < 1) the effects of finite size of transducer are negligible, there remains an indication that over this region spectral density expressed in terms of wall shear stress in the form </> (<iJ)U0/qgS* increases with Reynolds number although at a considerably smaller rate than suggested by the rms pressure measurements.
At M0 = 0. 5 the value of v'/Ta at the smallest experimental value of d/S*
(namely 0.17) was 2.68 whilst at M0 = 0.3 the smallest value of d/8* was 0.16 and the corresponding p'/T. was 2.53.
6. OVERALL SPACE-TIME CORRELATIONS
Overall space-time correlations R-.C^J.^J.T) were measured for various transducer separations along lines making angles of 0°, 30°, 60° and 90° with the mean flow direction at M0 = 0.3 and 0° and 90° at M0 = 0.5 . The flow conditions for these measurements are summarised in Table I. A typical set of curves of the longitudinal space-time correlation, R
DD(^1.0.T) . as a function of non-dimensional time delay
Ü0r/S* , for various values of ^/S*, are shown in Figure 8.
6.1 Space Correlations
The values of the longitudinal space correlation, RppCfj.O.O) , for a variety of flow conditions are shown in Figure 9 as a function of the non-dimensional spatial separation ^./S* . The points can be reasonably well represented by a single curve. The longitudinal correlation is negative for spatial separations greater than about 3.9S* , while the lateral correlation, R„T,(0,^,, 0) , shown in Figure 10, remains positive at the largest value of syo* investigated. Correlations along lines at
6
angles of 30° and 60° to the free stream flow direction for M0 = 0,3 (Pig. 11) have intermediate forms.
TABLE I
% 8»
(inches) Reg H
0.3 0.081 10,100 1.36
0.3 0.149 19,400 1.31
0.5 0.057 10,900 1.39
0.5 0.126 24,300 1.37
0.5 0.173 33,800 1.36
From the space correlation curves for the four angles to the flow (Pig.11), the correlation pattern of the pressure field can be mapped out as a set of isocorrelation contours. Pigure 12 shows the resulting pattern. If the long tail of the lateral correlation curve is accepted, the scale of the pressure field in the lateral direction Is considerably greater than the longitudinal scale, and this is clearly shown by the Isocorrelation contours of the field. The integral lateral scale.
Rpp(%^3 is about three times as great as the longitudinal scale
R (^1,010)df1 . At large separations the accuracy of determination of the J o correlation coefficients is not high and the results are more likely to be influenced by extraneous effects such as the background sound field in the wind tunnel, so the disparity in scales may not be quite as large as indicated above. However, the results certainly indicate anisotropy of the pressure field with the lateral scale exceeding the longitudinal.
6.2 Space-Time Correlations and Convection Velocity
If we take the position of the centre of a pressure eddy after a lapse of time T
to be that value of ^ for which the correlation with the original signal R (^,0,7-)
Is a maximum at the Instant r then U0 = ^,/T represents an average velocity of the eddy over the period r . This same velocity U„ can equally well be derived from the envelope of the curves of R-p^.O.T) versus T for various values of ^j = constant. A typical set of these curves is shown in Figure 13. The convection velocity is now given by
_ ^ (8)
where r0 is the time delay for which the curve f. = constant touches the envelope.
The Instantaneous convection velocity, U' , of the pressure field is given by the slope of the ?1 versus TC curve
(9)
This distinction between average and instantaneous velocities has to be made since it is found that the relation between ^ and rc is not linear, as Figure 14 shows. In fact, as will be seen from Figure 15, the convection velocity increases markedly with increasing spatial separation from a value less than 0.6 U0 at very small separations to a value in excess of 0.8U0 at large separations. The asymptotic values of Üo/Uo found in the present experiments are 0.53 and 0.825 for small and large separations respectively.
If the convection velocity is determined in the same way from correlations for which £3 t 0 it is found that the initial rise with increasing spatial separation is even more rapid than for the £3
= 0 case (Pig. 15).
A feature of the space-time correlation curves with increasing ^1 is that, in addition to the maximum value of correlation coefficient becoming smaller, the curves quickly lose their sharp peaks and become much broader (for example. Figure 8). This indicates that the spectrum of the correlation-producing wave-number components is becoming progressively curtailed at the high wave-number end, and further that each particular wave-number component loses coherence in the time taken for it to travel a distance which is broadly proportional to its wave-length.
This behaviour considered in conjunction with the observed variation in convection velocity indicates that the high wave-number (anall-scale) components of the pressure field are convected downstream slowly at speeds of the order of 0.5U0 and rapidly lose their coherence while the low wave-number (large-scale) components are convected rapidly at speeds typically of the order of 0.8 to 0.9U0 and lose coherence quite slowly.
7. NARROW-BAND SPACE-TIME CORRELATIONS AND CONVECTION VELOCITIES
More detailed information about the convection velocities and rate of loss of coherence of the various components of the pressure field can be obtained from measurements of correlations of the pressure signals after they have been passed through
narrow-band filters, a method first applied to wall pressure measurements by Harrison
If the covariance of the pressure signals in a narrow frequency band centred on <u is denoted by ^p(^v^3,T) , defined in the same way as QLjt;^g3,r) (Equation (2)) for the broad-band signal, then we can define a correlation coefficient
O^.^) VP(0'0'0>
which is the narrow-band analogue of Equation (4).
(10)
It can oe shown that
Q^.^.r) l^pp(0.0,w)|
cos((W + a) (11)
4> being the (complex) spectral function given by Equation (5). a, the phase angle of 4>™(£,,£Ta) . results from the convection of the pressure components in the frequency band being considered. Convection velocities could be detennined from the envelope of the major peaks of the curves of R" versus time delay for various constant values of spatial separation, in the same way as for the overall pressure correlations, but it is somewthat more convenient to find convection velocities, U0(w) , from the time delays corresponding to the major peaks themselves, i.e. given by
üC(«) (12)
The values of convection velocity obtained by the two methods will not differ very greatly.
The way in which a particular frequency component loses coherence during the convection process is represented by the behaviour with increasing spatial separation of the amplitude of narrow-band correlation coefficient l^pp^^fg.^l/U (0,0,^)1 which will be denoted by \KpP(£i,£3,
T)\- Experimental results for the variation of amplitude of the longitudinal narrow-band correlation, |Ril(^r
1, 0,T)|, are shown in Figure 16; they can be quite accurately represented as a unique function of a^r
1/Uc(ai) except at low values of this parameter. This type of dependence is untenable at low values of (JJ^/U^OJ) since 111^(^,0,r)| must equal unity at ^ = 0 for all w , but will not equal unity for (U = 0 at non-zero values of ^ . The deviation from co£jüe(coy dependence at low values of OJ^/U^W) might be expected to occur at higher values of ^y/Vc(co) the higher the value of ^/S* , and this expectation is confirmed by the trends indicated in Figure 16. In Reference 5 an expression for |R" (^^O.r)! was derived for a pressure field which was a slowly varying function of time in a convected reference frame. IR^L^J.O.T)] was found to be a function of i^/S* only, and not frequency dependent. This conclusion, which seems to have been largely a consequence of the particular forms assumed for the correlation functions, becomes increasingly at variance with the present experimental results and those of References 3 and 4 as the values of w^/ü (OJ) increases, but it may still provide a reasonable representation of the behaviour of the narrow-band correlations at low frequencies where w^/U^w) dependence is no longer valid.
Tbe fact that the amplitudes of the longitudinal narrow-band correlations behave as a function of coi; /U (u) over a wide range of values of this parameter indicates that, in the main, the loss of coherence of a frequency component occurs in a time which is Inversely proportional to its frequency, or, identifying frequency components with convected wave number components by means of the relation
a = kjiyoi), (13)
that a wave number component loses coherence in travelling a distance proportional to its wave-length.
The amplitude of the lateral filter band correlations, |B^,(0,f3,T)| , in the same way, tends to be a unique function of Cü^3/[JC(Cü) except for small values of o^3/vc(ci)) (Pig.17), indicating that the lateral coherence of a given longitudinal wave number component is also proportional to its wave-length. As in the longitudinal case the deviations from dependence on a)f3/ü (OJ) only are greater the larger the value of
Values of the ratio of convection velocity to free-stream velocity are shown in Figure 18 as a function of ü£*/VC(Cü) . As well as a frequency dependence the values of U„(<i))/ü„ show a variation with spatial separation of the measuring points, similar to that observed in the case of the overall convection velocity (Pig.15). A spatial dependence was also found in the low and high frequency measurements reported by Willmarth and Wooldridge in Reference 2.
Figure 18 shows that the increase in ü0(w)/IJ0 with spatial separation occurs mainly for 0 < jf./S* < 5 . At greater values of £/§* there is very little further convection velocity Increase.
It has been assumed on several occasions in this Report that fixed point frequencies and longitudinal wave-numbers are interchangeable. It is known that the time scale of the pressure fluctuations in a frame of reference moving with the pressure eddies is very much greater than the time scale in a fixed reference frame, so that the relation given by Equation (13) can be expected to be a good approximation. However, to further identify a given fixed point frequency with one wave-number component (or a very narrow band of wave number components) rather than with a wide range of kj all of which are associated with the same value of k1Uc(k1) requires that the convection velocity should be a unique function of kj . This is equivalent to saying that Uc(6j) should be a unique function of &JS»/UC(üJ) . Figure 18 shows that this condition is not precisely satisfied - in fact if this figure is interpreted as Uc(k1)/Uc versus k^* then there will be an uncertainty in U„(k1)/U0 associated with a given value of k,
CIO 1
which is small at small k^* but becomes of the order of ± 10 per cent at large kjS* . But, although there is a range of convection velocities associated with a particular value of kj , the indications are that it is confined within fairly narrow limits.
The results of the correlation measurements in narrow frequency bands therefore indicate that the pressure field can be regarded as made up by the superposition of longitudinal wave number components with each of which can be associated a reasonably well-defined convection speed or limited range of convection speeds. The various components are coherent over lateral distances which are proportional to their
■
10
wave-lengths and retain coherence In the flow direction for times proportional to the times taken for them to be convected distances equal to their wave-lengths.
This conclusion is consistent with the observation of the overall space-time correlation characteristics of the pressure field. It explains the change in character of the correlation with increasing time delay and spatial separation, the variation of convection velocity with increasing spatial separation and the different rates of variation of convection velocity with £1 for different values of ^3 .
The fact that a more or less unique convection velocity can be assigned to a particular wave number component, suggests that by and large the pressure fluctuations of a particular longitudinal wave number originate from a region of the boundary layer where the mean flow velocity is equal to the observed convection velocity. If longitudinal wave number components kj are associated with physical scales this leads to the conclusion that small-scale pressure fluctuations originate from boundary layer turbulence quite close to the wall while the large-scale fluctuations are associated with motion in the outer regions of the layer. This conclusion might be expected to apply with greater accuracy to the small-scale than to large-scale fluctuations, since the correspondence between the longitudinal wave number kj and physical scale tends to lose precision at small values of k1 ; this is borne out by the fact that the convection velocity of the overall field at large separations does not greatly exceed 0.8U and that even in narrow frequency bands Uc(a)) does not exceed 0.9U0 for small kj , indicating that the sources of fluctuation with low k1 cannot be so accurately located, but are more widely dispersed throughout the boundary layer, their convection velocities representing an average of a fairly wide range of values.
8. PRESSURE-VELOCITY CORRELATIONS
In an effort to determine how different regions of the boundary layer contribute to the wall pressure field, measurements of the correlation between the wall pressure fluctuations and turbulent velocity fluctuations at various positions in the layer were begun. It was the intention to investigate the correlation between p and nl
and between p and u2 , but because of experimental difficulties which are briefly referred to below values of the former correlation only have been obtained.
The coefficient of correlation between p and u1 is defined as
Rpui^i.^.^.-O = < p(xi,x2,xrt) p(x1 + fl,x2 + ^2,Xg + £3,t + T) >
< p2(X1,X2,X3) )'< U2(X1 + ^vXi + ^Xg + £3) ^ (14)
Values of Rpujör,,^,,0,T) have been obtained for 0.4 < ^2/8* < 4.4 and 0 < ^,/S* < 12.3 for flow conditions M0 = 0.5 and S* = 0.126 inch.
Considerable difficulty was experienced in eliminating the interference effect of the hot-wire probe on the pressure field, even when the wire was located downstream of the pressure transducer. By careful re-design of the hot-wire probe the interference was considerably reduced, to a state where it amounted to only 1 or 2 per cent increase in the output of the pressure transducer. Under these conditions the main effect was an increase in pressure spectral density at low frequencies,
11
wS'/U,, < 0.14 , and the pressure and velocity signals were therefore passed through high-pass filters which cut out frequencies below this value.
Typical results for various values of £2/b* and £,/&* = 3.88 are shown in Figure 19. The correlation curves are asymmetrical, the correlation being zero at the centre of the disturbance which is correlated with the wall pressure. The greatest magnitude reached by the coefficient on either side of its zero-crossing was found to be greatest for small values of £./$* and ^2/8* and to decrease slowly as these separation distances increased. The slow decrease towards the outer edge of the boundary layer does not necessarily imply that the wall pressure receives comparable contributions from all parts of the boundary layer since the intensity of velocity fluctuation falls off rapidly as the distance from the wall, ^2/S* , increases. A more realistic indication of the contributions of various parts of the layer to the wall pressure can probably be obtained by considering the variation of ujRpUl across the layer. Since the value of uj/ü0 varies from about 0.10 in the inner region of the boundary layer to the residual free-stream value, which in this case was about 2 x 10"3 , the value of ujRpUl falls off quite rapidly with increasing distance from the wall. In relation to its value at about one displacement thickness from the wall, the value at ^2/S* = 5 is about 10 per cent and at the outer edge of the layer only 1 or 2 per cent, supporting the view that the inner region of the boundary layer is mainly responsible for the wall pressure fluctuations.
The measurements also show that the longitudinal velocity disturbance at a particular distance from the wall, which is correlated with the wall pressure, is convected at the speed of the local mean flow.
12
-
REFERENCES
1. Allcock, G.A. et alii
A General Purpose Analogue Computer for the Analysis of Random Noise Signals. University of Southampton Report AASU 205, March 1962.
2. Willmarth, W.W. Woodridge, C.E.
Measurements of the Fluctuating Pressure at the Wall Beneath a Thick Turbulent Boundary Layer. University of Michigan Report 0290-1-T, Apr. 1962.
3. Corcos, G.M.
4. Harrison,
Pressure Fluctuations in Shear Flows. California Report, July 1962.
University of
Pressure Fluctuations on the Wall Adjacent to a Turbulent Boundary Layer. David Taylor Model Basin Report 1260, Dec. 1958.
5. Bull, M.K. Space-Time Correlations of the Boundary Layer Pressure Field in Narrow Frequency Bands. University of Southampton Report AASU 200, Dec. 1961.
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DISCUSSION
Comment by P. Davies
Dr. Curie of Southampton has Investigated the effect of discarding parts of a flat spectrum and shows the error of doing so is not large. This is not of course true when discrete frequencies form a large proportion of the overall signal power.
Comment by J.S. Serafini
I would like to clarify and emphasize the points I made after the paper by W.W. Willmarth. Of importance in the filtering here is not only what is the cut-off frequency, but how sharply the rate of cut-off takes place. What is your rate of cut-off of your filter, Dr. Bull?
Author's reply
In obtaining the pressure-velocity correlations the low-frequency cut-off for ajS,/V0< 0.14 was sharp, from memory 12 db/octave .
Comment by J.S. Serafini
If indeed your cut-off is quite sharp, note that in so doing you have removed not just increased spectral level due to interference, but you also have removed the 'non-interfered' level in this filtered range of the frequency. Thus, it is not entirely clear to me precisely under what grounds one can neglect the fact that in removing the 'interference effect' by sharply filtering the spectra one also has changed the basic character of the fluctuations to be correlated.
Author's reply
It is true that filtering removed the true signal as well as the interference effect but the contribution to the pressure energy from frequencies for which coS*/\}Q < 0.14 amounts to only 3 or 4 per cent of the total, and I believe that the effect on the measured correlations is small.
M. Strasberg
Dr. Serafini' s point, that the contribution of the low frequencies to the cross correlation may be greater than their contribution to the mean-square signal itself, is well taken. This may be stated in a more definite way be noting that the cross correlation is the Fourier transform of the spectral density, viz.
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R12(T) = i rPlz(&))cosüf + Q12(w)sincJr Idw
where P and Q12 are the real and imaginary parts of the cross-spectral density. If the signals are filtered with high-pass filters having a cut-off frequency coc ,
34
^
then the integration is performed only from a)0 to oo . But the contribution to the integral of P12 and Q12 from zero to ajc may be a larger portion of the total integral than the contribution of PJJM to the mean-square signal.
Author's reply
Errors could certainly be introduced in this way, but it was found that when the interference effect itself was small, the effect of filtering was also small, producing changes in correlation coefficient of the order of 0.01 or 0.02 when the interference signal produced about 2 per cent increase of rms pressure - only in the case of large interference effects did filtering give rise to large changes in correlation coefficient. This reinforces my belief that the effect of filtering in this case has been to remove the interference signal without significantly affecting the correlation of the real pressure signal with the velocity.
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