Turbulent boundary layers on axially inclined cylinders. Part 1. Surface

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Part 1. Surface-pressure/velocity correlations. A.F. Heenan, J.F. Morrison. Abstract .... $0.2. $0.15. CAWP measurements. 27.5. 280. 2.3 · 104. 6.4 · 106. –. –. 548 ...
Experiments in Fluids 32 (2002) 547–557  Springer-Verlag 2002 DOI 10.1007/s00348-001-0384-5

Turbulent boundary layers on axially inclined cylinders. Part 1. Surface-pressure/velocity correlations A.F. Heenan, J.F. Morrison

547 Abstract The boundary layer on a cylinder with its axis at small inclinations of 0–6 to the freestream (an idealisation of ‘streamers’ used in underwater seismic surveys) has been studied experimentally by measurements involving surface pressure fluctuations and their correlation with the axial velocity. There is no evidence of vortex shedding at Reynolds numbers typical of streamers at operating conditions. The behaviour of the wall-pressure field is substantially altered by small incidence: correlation length scales decrease on the upstream side, but remain relatively unaltered on the downstream side. Attention is also paid to the axisymmetry of the flow by reference to axial velocity statistics of up to fourth order.

1 Introduction Acoustic streamers are important in sonar detection in either military applications or those connected with the seismic surveying of the ocean bed rich in mineral resources. They are long flexible hoses, usually with a foam core, so that they are neutrally buoyant. Hydrophones are placed every 100 m or so along the axis in order to measure pressure waves generated by a periodic pulse and part-reflected from the sea bed. Yet their signals are subject to extraneous contamination from a variety of sources, which are mentioned in Part 2 of this paper (Heenan and Morrison 2002). See also Dowling (1998, 1992) for details concerning the acoustic field generated by the turbulence

Received: 10 June 2000/Accepted: 4 July 2001

A.F. Heenan, J.F. Morrison (&) Department of Aeronautics Imperial College, London SW7 2BY, UK e-mail: [email protected] Present address: A.F. Heenan Department of Mechanical Engineering Queen’s University at Kingston Ontario K7L 3N6, Canada This work was supported by Schlumberger Geco-Prakla. We are indebted to the contract monitors, Simon Bittlestone and, more recently, Øyvind Hillesund for their support and interest. We acknowledge enlightening discussions with Prof. Ann Dowling and we are grateful to Prof. Alexander Smits who was host to JFM during his stay at the Department of Mechanical and Aerospace Engineering, Princeton University, when the major part of this paper was written. We are also grateful to one of the referees for helpful comments.

on long streamers. This paper concentrates on the effect of small incidence (0  a  6 ) on the boundary layer around a streamer, it being supposed that this is a source of some noise which makes eduction of the sonar signal difficult. In Part 2, however, we show that the circumferentially averaged wall-pressure fluctuation (important because hydrophones are mounted along the streamer axis) actually decreases with incidence and that lowfrequency noise can be attributed more plausibly to changes in hydrostatic pressure induced by vertical oscillation of the streamer several kilometres in length. In this paper, we show that static changes in the streamer orientation do not cause a change in flow species (e.g. boundary layer to mixing layer at separation): thus, even though the flow is substantially modified, as evidenced by correlation length scales, there is no vortex shedding. Flow visualization confirms this – see Heenan and Morrison (1999) for further details. A video of the flow visualisation is available from the second author. Several workers have addressed the issue of scaling in axisymmetric boundary layers (Afzal and Narasimha 1985; Luxton et al. 1984; Willmarth et al. 1976; Lueptow et al. 1985; Denli and Landweber 1979), but for a recent review, see Heenan and Morrison (1999) or Neves et al. (1994). In summary, the effect of transverse curvature (with radius, a) is to introduce an imposed length scale, usually expressed non-dimensionally by c ¼ d=a, in addition to the viscous length scale, m=us , and the boundary layer thickness, d. For the present work, estimates for c and aþ ¼ aum s are 2 and 2000, respectively. Other principal quantities and their equivalent values for a full-size streamer are given in Table 1. Although the models are much shorter than streamers (which are several km long, with connectors at approximately 100-m intervals), typical drag coefficients for the latter suggest a much lower Reynolds number (based on length) giving an effective length of 2–3 m only. This matches closely the length of models actually used. Physically, this effect can be explained by the results of flow visualization (see Heenan and Morrison 1999; Lueptow and Haritonidis 1987; Luxton et al. 1984) which show large structures of several diameters in size that repeatedly move across the cylinder, maintaining high levels of shear and vorticity. As c decreases, this effect becomes more pronounced. Reviewing available experimental data, Afzal and Narasimha (1985) suggest that, for curvature effects to be significant in both the inner and outer regions, c > 1 and a+ < 250. Note that the threshold for c is a lower limit for significant curvature effects, while that for a+ is an upper

Table 1. Comparison of principal model parameters with those for a typical streamer, a = 0

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Streamer Point measurements CAWP measurements

Ue ðms1 Þ

x/a

Rea

Rex

d*/a

h/a

2.5 27.5 27.5

103–105 120 280

105 5.4 · 104 2.3 · 104

108–1010 6.5 · 106 6.4 · 106

– 0.2 –

– 0.15 –

limit: thus, if a+ > 250, only the outer layer is affected. Other workers are in general agreement with these values (Neves et al. 1994). Of these two criteria, only the first is met for both the present model and a full-size streamer. Thus, we anticipate that the inner layer remains largely unaffected by transverse curvature. Even so, the structural changes in the turbulence are sufficient for the wall-pressure field to be altered substantially. In reviewing previous work concerning the effects of transverse curvature on the velocity field (Heenan and Morrison 1999), they can be summarised as follows: 1. mean velocity profiles do not exhibit similarity; 2. away from the surface, Reynolds stresses are lower for an axisymmetric boundary layer than for a planar one at the same Reynolds number; 3. close to the surface, Reynolds stresses and the wall shear stress are larger than those in an equivalent planar boundary layer; 4. there is a substantial shift away from the surface of the intermittency profile; 5. at large x/a, the boundary layer behaves more like an axisymmetric ‘point wake’ that has a mean velocity profile similar to that of a wake, except that the layer is sustained by a vorticity-generating singularity in the mean strain rate at its centre. The effects of transverse curvature on point-surface pressure fluctuations have also been dealt with by several workers (Willmarth and Yang 1970; Willmarth et al. 1976; Panton et al. 1980; Neves and Moin 1994; Snarski and Lueptow 1995; Nepomuceno and Lueptow 1996; Bokde et al. 1999). These may be summarised as follows (for a fuller description, see Heenan and Morrison 1999): 1. energy in wall-pressure fluctuations moves to smaller scales, the overall level decreasing with increased transverse curvature; 2. the rate of decay of streamwise correlations is quicker in the axisymmetric case; 3. the aspect ratio of contours of wall-pressure correlations is about unity – it is greater than 2 for a planar boundary layer; 4. the outer layer irrotational motion is more highly correlated with the wall pressure than the vortical motion. The emphasis of the present work is to extend these findings to the effects of small axial incidence by reporting measurements of the axial velocity and surface pressure fluctuations. This has received very little attention, in spite of its relevance to streamer behaviour. Bull and Dekkers (1993) established, at low Reynolds numbers, two regimes which depend primarily on incidence and are distinguished by the appearance, or not, of regular vortex shedding. Extrapolation of their results to the present

Reynolds number suggests that for the present experiment, vortex shedding is likely to appear only for a > 10 . Scott (1986) shows how even for incidences as small as a  1 , the boundary layer thickness can change by as much as 80% and 50% on the windward (upstream) and leeward (downstream) sides, respectively. The paper is organised as follows: in Sect. 2, we describe the models used. We also consider the degree of axisymmetry of the velocity field with reference to moments of up to fourth order. Section 3 provides results for point-surface pressure fluctuation measurements for small incidences, 0  a  6 and examines their correlation with the axial velocity. Details concerning special noise reduction techniques are dealt with in Part 2.

2 Experimental setup The experiments were performed in a 914 · 914-mm closed-circuit tunnel, chosen specifically for its quietness and low vibration. The freestream turbulence intensity is less than 0.05%. The working section is 4,880 m long, permitting the use of long models, of which there are two: one was used for measurements of point-surface pressure fluctuations and velocity; the other involves a specific module for measurements of circumferentially averaged wall-pressure fluctuations. The latter are reported in Part 2. The model for the present measurements comprises 60-mm-diameter plastic pipe, 4,100 mm in length. This diameter is similar to that of streamers and allowed a comparable operating Rea to be achieved. The radius of curvature was sufficiently large to mount the flat-faced pressure transducers flush with the surface of the model. The measurements were made 3,600 mm (120a) from the semi-ellipsoid nose of the model. At a ¼ 3 and 6, the ‘crossflow’ Reynolds number (based on cylinder radius and mean velocity vector normal to the surface) is 2,880 and 5,750, respectively. The maximum blockage produced by the model was about 3.5%. The model was suspended in the wind tunnel using lengths of 0.5-mm-diameter piano wire, the tension in which was adjusted using turnbuckles. Figure 1 shows the convention used for describing the orientation of the model and azimuthal locations on it. Note that h ¼ 0 corresponds to the centre of the windward surface and that h ¼ 180 is the centre of the leeward surface. x denotes the axial direction; y and z are orthogonal directions in the plane of the cylinder cross section, with their origins at the cylinder centre. Fluctuating pressure measurements were made using Endevco 8507C-2 piezo-resistive pressure transducers, which have a 0–2-psi. (0–0.14 · 105 Pa) operating range and a sensitivity of about 200 mV/psi. (0.029 mV/Pa ). All transducers were individually calibrated with limitations to the frequency response being set by the spatial resolu-

tion, the effects of which become apparent at about 15 kHz for the flow conditions used (Table 1). The Helmholtz resonance is quoted as 45 kHz. Measurements made by surface-mounted pressure transducers are prone to errors caused by misalignment of the pressure transducer with the local surface (Gaudet 1978); with the aid of a microscope, special attention was paid to ensuring that the transducers were always mounted as flush as possible. With the wind tunnel switched off, typical noise levels were about 10 mV rms, which is, in fact, about 25% of the rms signal (a signal-to-noise ratio of about 10 dB) at zero incidence. The gain-bandwidth product of the amplifier circuits was approximately 107 and thus, for the gain employed, the frequency response was virtually flat to the limits imposed by the spatial resolution of the pressure transducers. Axial velocity measurements were made using single constant-temperature hot wires (l=d  200) which were driven by anemometers of the University of Melbourne ‘Wombat’ design. A standard ‘King‘s’ law was fitted to the calibration data with a velocity exponent of 0.45, calibrations being performed both before and after each traverse. Effective linearisation was achieved by the use of a look-up table. Both velocity and pressure data were sampled (16-bit resolution) at a rate of 20 kHz and low-pass filtered at 10 kHz (to avoid aliasing). Sample Fig. 2a, b. Azimuthmal surface-pressure distributions: times were typically 15 s. See Heenan and Morrison (1999) a dynamic; b static for further details.

3 Results 3.1 Qualification of the velocity field Although the boundary layer was tripped at the end of the semi-ellipsoid leading edge, the favourable pressure gradient on the pressure surface can, in principle, lead to relaminarisation. Thus, the lower limit to the Reynolds number for ensuring a fully turbulent boundary layer was determined by observing, firstly, at what velocity turbulent spots first appeared in the velocity – and pressure – field time histories, and secondly, at what Reynolds number the time histories became fully turbulent. At a ¼ 6 (the worst case), transition occurred at Rex » 2.7 · 104. This is less than half the Rex at which the measurements were made. The boundary layer on the suction side is always fully turbulent. Figure 2a shows the effect of incidence on the dynamic pressure distribution around the cylinder obtained using four uncalibrated Preston tubes fixed at 90 intervals around the surface, 120a downstream of the leading edge.

The total pressure from each tube was recorded together with the local surface static pressure, the section then rotated 90 and the measurements were repeated. This process was carried out until the total pressure at each location was measured using all four Preston tubes. The results presented are the average of the four measurements normalised by the average dynamic pressure around the cylinder for that incidence, P. On the leeward side, where separation is possible, the dynamic pressure rises at incidence, while the static pressure (Fig. 2b) is relatively constant. Thus, the surface shear stress is always positive and the boundary layer is always fully attached, even at h ¼ 180, a ¼ 6, where the surface shear stress actually increases. Thus, there is no change in flow species. On the windward side, the static pressure rises, while the dynamic pressure falls slightly. The reduction in surface shear stress here therefore suggests a reduction in turbulence intensity. It is interesting to note that the dynamic pressure distribution is most axisymmetric when the cylinder is at + 0.75 incidence. This is because, at this angle, there is a compensating asymmetry introduced by the supporting-wire wakes. However, a ¼ 0 was kept as the datum condition.

Fig. 1. Orientation of models with respect to the freestream direction

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At a ¼ 0, there is also some variation in the azimuthal surface static pressure distribution, Cp(h) ¼ (p(h) ) pstatic)/(ptotal ) pstatic), where ptotal and pstatic are tunnel reference conditions. Changes in Cp are consistent with the variations in the velocity field caused by the backoff and the supporting wires. At incidence, the static pressure increases on the upstream side as the mean flow impinges on the cylinder. The static pressure then drops as the flow accelerates around the cylinder, before increasing again due to the lower mean velocities on the leeward side of the cylinder. Note that Cp (180) is bigger than (or equal to) zero, indicating again that the flow remains attached. On a practical note, the azimuthal change in hydrostatic pressure around a streamer operating under typical conditions is more than four times the pressure change at 6 due to incidence effects. Moments of up to fourth order of the velocity field along the axis of the cylinder were checked to assess the degree of axisymmetry (Figs. 3, 4, 5, 6). Results for a ¼ 3 and )6 are also presented. Note that measurements in only one half of the (y, z)-plane were made and symmetry about z ¼ 0 was assumed. The contours of turbulence intensity compare well with those of Lueptow and Haritonidis (1987). Note however that while the contour values remain largely unaltered at incidence, they are redistributed so that they are closer to the surface on the windward side, but further from it on the leeward side. The ‘twin-lobe’ appearance of the velocity field, although somewhat artificial, is completely consistent with the flow visualization studies of Heenan and Morrison (1999). At a ¼ 0, asymmetries apparent in the outer parts of the boundary layer are caused by two factors: firstly, the blockage caused by the backoff which accelerates the flow above the cylinder by about 0.5% and secondly, the wakes of the supporting wires (at h ¼ ± 600 and 1,800) which reduce the local freestream velocity by up to 5% while increasing the turbulence intensity by up to 10%. The asymmetry is worst at h ¼ 1,800, where there are several wires. The axial component of skewness, Su (Fig. 4) and flatness, Fu (Fig. 5), show no significant changes with incidence, suggesting again that there are no significant structural changes in the boundary layer. As might be expected, the largest departure from Gaussian behaviour occurs in the outer layer, where Su becomes large and negative and Fu becomes large and positive. Thus, although the shape and size of the boundary layer changes significantly, its structure evidently does not. Interestingly, estimates of Fu in the outer layer at all incidences are significantly larger than those in a planar boundary layer (Fernholz and Finley 1996). This behaviour may be the result of large eddies crossing the axis of the cylinder. These higher moments are very sensitive to slight disturbances at ± 600 and 1,800 caused by the supporting wires. In particular, at incidence, the wires at ± 600 fan out so that their blockage is greater. However, it should be remembered that the effects of the supporting wires are c

Fig. 3a, b. Mean velocity, U/Ue: a a ¼ 0; b a ¼ 3; c a ¼ 6. Contours, together with those in Figs. 4, 5, 6 are reflected about z¼0

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Fig. 4a, b. Axial intensity, u0 =Ue : legend as in Fig. 3

Fig. 5a, b. Axial skewness, Su: legend as in Fig. 3

only evident at distances of about 3a or between 1d and 3d from the cylinder surface.

3.2 Point-surface pressure fluctuation measurements Details of the noise cancellation technique are provided in Part 2. Figure 7 shows the (noise-cancelled) point-surface rms pressure distribution normalised by the freestream dynamic pressure, q. Its value of about 0.8% for a ¼ 0, h ¼ 0 is typical of axisymmetric flows (Willmarth and Yang 1970). At h ¼ 180, the effect of the supporting wires is again apparent: the surface dynamic pressure (Fig. 2a) is also slightly lower in this region. At incidence, rms levels increase by about 10% near h ¼ 0. The effect of the acceleration around the cylinder (both axially and azimuthally) is to reduce the rms pressure, with a further increase to another maximum at 180. This appears to be an indication of wake-like characteristics developing on the leeward side of the cylinder where the surface dynamic pressure also increases. An increase in both turbulence intensity and rms wall pressure accompanied by an increase in wall shear stress is typical of boundary layer behaviour. Figure 8 shows the wall-pressure spectral density, /ðf Þ, normalised by the freestream dynamic head, where

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fZ¼1 p2 ¼ /ð f Þdf : q2

ð1Þ

f ¼0

Spikes visible in the range 500 < f < 1,200 are the result of electrical noise: acoustic noise with a fundamental frequency of about 70 Hz and significant sub- and higher harmonics cancel quite well. However, spikes, particularly at 600, 900 and 1,200 Hz produced by the tunnel drive, vary from transducer to transducer and even from measurement to measurement. It is possible that the spikes are influenced by the orientation of the transducer relative to the electromagnetic field produced by the tunnel drive and therefore cancellation will be similarly dependent. Therefore, the direct subtraction of the backoff signal does not always cancel the electrical noise of the measurement transducer. In fact, the 600-Hz spike is sometimes made larger by the cancellation process. Moreover, despite scrupulous attention to shielding of both the transducers and their cabling, electrical noise was still dependent on the precise earthing

Fig. 6a, b. Axial flatness, Fu: legend as in Fig. 3

Fig. 7. Azimuthal rms pressure distribution

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Fig. 8. Pressure spectra, /ðf Þ

details. These were optimised by examining an oscilloscope trace before measurements were made. The spectra for a ¼ 0 are similar to those of Snarski and Lueptow (1995) or Nepomuceno and Lueptow (1997): there is no significant region in which /ðf Þ / f 1 , deducible by dimensional analysis for logarithmic-region contributions to the wall pressure and assuming a constant convection velocity (Bradshaw 1967). In addition, there is no range in which /ðf Þ / f 2 , as suggested by the data of Markovitz (as quoted by Dowling 1992) for a rigid cylinder

at zero incidence. The only possible range in which this might be so is f » 2–3 kHz, but, while these frequencies are well within the scope of the present measurements, they are certainly beyond those frequencies at which the pressure-bearing eddies are likely to appear. Most of the mean square energy is contained in the range 1–4 kHz (corresponding roughly to a range of length scales of 0.17a–0.67a). At incidence, the main energy-containing region at h ¼ 0 shifts to higher frequencies. The increase of energy and its shift to higher frequencies is largest for

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a ¼ 6 and consistent with a decrease in the length scale of the energy-containing length scales there. The increase in high-frequency energy is greater than the slight attenuation at low frequencies, leading to larger rms values. At larger values of h, the increase in high-frequency energy with incidence is not as great, so that at h ¼ 180, there is virtually no change in the spectra with incidence. This somewhat surprising result is further confirmation that the near-wall structure on the leeward side of the cylinder does not change character even at the largest incidence. Figure 9 shows the corresponding autocorrelations defined by

Rpp ¼

pðx; tÞpðx; t þ sÞ p2 ðx; tÞ

:

ð2Þ

The width of peak around zero time delay (sUe/a » 0.4) corresponds to the peak-energy frequency of about 2 kHz in Fig. 8. At zero incidence, there is a small but significant correlation at large time delay that is the result of the same large-scale activity that produces the high spectral densities at low frequency. Thus, the integral correlation length scale is of the order of several cylinder radii. On the pressure side of the cylinder, the boundary layer thickness at incidence is reduced, as is also the correlation length scale, as pressure sources move closer to the surface. The correlations also show negative minima of about )0.1 around sU¥/a » ± 0.1. These are similar to those found in Rpp for planar boundary layers and are indicative of the streamwise convection of spanwise vorticity. Thus, the primary effect of incidence on the windward side of the cylinder is to reduce the wake-like

Fig. 9. Wall-pressure autocorrelation, Rpp: 0 £ a £ 6

component of the boundary layer while increasing turbulence intensities and surface shear stress. Towards the leeward side, the main correlation peak widens, but no regions of negative correlation develop. As with the spectra, at h ¼ 180, there is very little change in the nature of the boundary layer with increasing incidence. Figures 10 and 11 show contours of space-time correlations between the wall pressure and axial velocity at h ¼ 0 and 180 respectively. The sense of the time delay is such that positive time delay is equivalent to downstream displacement of the hot wire relative to the wall-pressure transducer. At a ¼ 0, significant correlation between the pressure and velocity fields is confined to a region close to the surface ( 0.5a). Even so, the downstream inclination of the large pressure-producing structures is clearly evident. The region of positive correlation extends the full

height of the layer, although it is very weak beyond y/a » 0.5. The streamwise extent of the correlation is generally much greater than the vertical extent. For example, the 0.1 contour has a length-to-height ratio of about 10. This suggests that the structures affecting the boundary layer are long and thin. The results for the cylinders at incidence show dramatic changes in structure. The effect of incidence on the windward side is to remove the large structures, the vertical extent of the correlation diminishing as the boundary thickness decreases. On the leeward side, however, the contours remain relatively unchanged by incidence. The length-to-height ratios of the correlation have also significantly decreased (to about 5). The windward-side structures are clearly much smaller, both vertically, longitudinally and, we suggest, azimuthally too, probably

Fig. 10a–c. Wall-pressure/ axial-velocity correlation, Rpu, h ¼ 0; a a ¼ 0; b 3; c 6

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Fig. 11a–c. Wall-pressure/ axial-velocity correlation, Rpu, h ¼ 180; a a ¼ 0; b 3; c 6

might first appear, owing to the strong azimuthal shear arising from the pressure gradient around the cylinder circumference. So, for h » 180, the rms wall pressure changes quickly with h. However, for the range of incidences investigated, there is no vortex shedding that could generate pressure fluctuations of significant azimuthal extent. The rms pressure at h ¼ 180 does not change significantly with incidence. Thus, there is no change in flow species. The most likely position for cross-flow sep4 aration (where the mean azimuthal component of skin Discussion and conclusions At incidence, both the surface shear stress and the axial friction goes to zero) is at h » 100 and » 250. However, turbulence intensities decrease on the windward in these regions, the axial component of skin friction is (upstream) side. While the decrease in turbulence inten- positive and increasing towards h » 180. This result is sity is not readily apparent from the contours of Fig. 4, entirely consistent with the conclusion in Part 2: namely such a conclusion is certainly consistent. However, the rms that the circumferentially averaged wall pressure decreases pressure fluctuations increase slightly there. Behaviour on with increasing incidence. While the axial turbulence inthe leeward (downstream) side, where the surface shear tensity and higher moments remain much the same as the stress increases, appears to be more complicated than incidence increases, they develop a ‘twin-lobe’ appearance because the structures no longer travel along the long axis of the cylinder, but are sheared around its circumference. On the leeward side (h ¼ 180), the spatial extent of the region of finite positive correlation is roughly comparable to that for a ¼ 0, although at incidence the correlation is generally weaker, probably due to the fact that the mean shear has a significant azimuthal component in this region.

with maxima which move progressively away from the cylinder as the incidence increases. By a ¼ 6, they are about one diameter away from the cylinder surface. Estimates of displacement thickness and momentum thickness show about a five-fold increase here, while the shape factor is approximately constant at about 1.2. Due to a reduction in boundary layer thickness on the windward side, correlation length scales obtained from the wall-pressure autocorrelation and the wall-pressure/velocity correlation decrease with increasing incidence there. The corresponding length scales on the leeward side, however, remain relatively unchanged, the behaviour here generally involving the displacement of the large scales away from the cylinder, so reducing correlations involving the wall pressure. This corroborates the principal conclusion that there is no evidence of vortex shedding for the range of incidences investigated at Rea  105 typical of operational streamers.

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