Measurements in the near-wall region of a boundary

J. Fluid Mech. (2010), vol. 664, pp. 33–50. doi:10.1017/S0022112010003770

c Cambridge University Press 2010


33

Measurements in the near-wall region of a boundary layer over a wall with large transverse curvature M. H. K R A N E1 , L. M. G R E G A2 1 2 3

AND

T. W E I3 †

Applied Research Laboratory, Pennsylvania State University, State College, PA 16804, USA

Department of Mechanical Engineering, The College of New Jersey, Ewing, NJ 08628, USA

Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA

(Received 22 August 2008; revised 2 July 2010; accepted 2 July 2010; first published online 26 October 2010)

Measurements of the near-wall velocity field of the flow over cylinders aligned with a uniform flow are presented. The broader objective of this investigation was to quantify and understand the role of transverse curvature in the limit as cylinder diameter approaches zero. The specific goal was to begin with a turbulent boundary layer over a larger radius cylinder and see what happens as the radius is reduced. Spatially and temporally resolved digital particle image velocimetry (DPIV) measurements were made on three different radius cylinders, 0.14 cm 6 a 6 3.05 cm, extending along the length of a large free-surface water tunnel. Mean and fluctuating profiles are presented at a fixed streamwise location and free-stream speed. For the first time, spatially resolved measurements were made very close to the wall, permitting direct determination of wall shear stress, i.e. uτ , from near-wall velocity profiles. The measurements revealed a region close to the wall for small radii where the mean streamwise velocity profile is inflectional. This has significant implications on assumptions regarding what happens in the limit of a vanishing cylinder radius. Key words: free shear layers, hydrodynamic noise, turbulent boundary layers

1. Introduction 1.1. Problem statement The fundamental question addressed in this paper is whether or not it is possible to sustain a turbulent boundary layer (TBL) along an axisymmetric body in the limit as body radius a tends to zero. The technological driver is whether or not very small radius piezoelectric cylinders can be used for hydroacoustic sensing applications. That is, is there a radius below which turbulent fluctuations, and most importantly, turbulent pressure fluctuations, are either negligibly small or nonexistent? If this were so, then the need for compliant cladding would be eliminated. The independent parameter space for axisymmetric boundary layer flows includes not only cylinder radius, a, but free-stream velocity U∞ and axial position along the cylinder z as well. These parameters couple in a nonlinearly coupled way such that it is † Email address for correspondence: [email protected]

34

M. H. Krane, L. M. Grega and T. Wei dθ

R + dr

R

Figure 1. Cross-sectional cut of a circular cylinder showing the effects of a vanishing radius on the cross-stream transport. Flow is out of the page. The cylinder radius is ‘R’ and an infinitely thin annulus of thickness dr is shown.

not obvious if a low-Reynolds-number boundary layer around a large radius cylinder at large z is dynamically similar to a high-Reynolds-number boundary layer for small a and z. Adequate mapping of this parameter space is a tremendous challenge to both computationalists and experimentalists. The specific focus of this study was to examine the effect of reducing cylinder radius on a low-Reynolds-number TBL at a fixed streamwise speed and location. The approach was to develop a very low Reynolds number, albeit turbulent, boundary layer over a large diameter cylinder and to examine if and how the flow characteristics change with decreasing diameter. 1.2. Axisymmetric boundary layers at vanishing radii: a thought experiment From a fundamental perspective, the existence of turbulence over a no-slip boundary with a vanishing surface area is quite interesting. The plausibility of relaminarization lies in the high transverse curvature close to the cylinder surface for small radii. To see this, consider the cylinder with radius, a, in figure 1. An infinitesimal fluid element, part of a thin annular ring, has been positioned on the cylinder surface with differential height and angle, dr and dθ, respectively. Flow is into or out of the page. For large radii, i.e. a approaching infinity, the flow becomes that over a flat plate. For sufficiently high Reynolds numbers, turbulence ensues because viscous transport by itself is insufficient to maintain mass, momentum and energy balances in the flow. However, as the body radius vanishes, the ratio of the outer arclength, (r + dr)dθ, to the inner arclength, r dθ, becomes infinitely large. In this case, the surface area infinitesimally far away from the cylinder boundary, across which mass momentum and energy are transported, is infinitely larger than the surface of the cylinder. Consequently, one could argue that the boundary layer over an axisymmetric body with zero radius would not require turbulence in order to balance transport in the wall-normal direction. An alternative way of envisioning this is from the vantage point of the TBL structure. For a canonical two-dimensional flat-plate boundary layer, turbulent streak

Measurements in the near-wall region of a boundary layer

35

spacing is approximately 100 viscous lengths. Streaks have been simply defined as the ‘footprints’ of hairpin vortex structures whose legs are roughly 100 viscous length scales apart. These hairpins are thought to be integral to the Reynolds stress and turbulent kinetic energy production as the vortex heads are violently ejected away from the wall. The feasibility of hairpin vortices forming over or around a cylinder with diminishing radius clearly becomes problematic. One can imagine that the dynamics must change significantly when the spanwise spacing of turbulent streaks becomes of the same order or larger than the entire transverse dimension, i.e. circumference of the body. This thought experiment leads to the conclusion that for zero radius, it is unlikely that a TBL would develop. It certainly indicates that turbulence, if it were to exist, must be very different from that of a canonical TBL. For large radii, however, turbulence is most definitely present. The focus of the present work lies in trying to develop an understanding of what happens in between these two limits. And as noted earlier, the specific focus is on the effects of a reduced radius on a low-Reynoldsnumber TBL.

1.3. Literature review One of the key motivators for the axisymmetric TBL research has been the need to understand how boundary layers develop on axial bodies of revolution, such as fuselages or submarines. For this reason, there is a body of literature on flow past low aspect ratio bodies with variable radius. For example, the reader is referred to Patel, Nakayama & Damian (1974). There has also been work done on what happens to the flow when the bodies become misaligned. A study on flow past axisymmetric bodies at small angles of attack may be found in Heenan & Morrison (2002). Advances in towed sensor technologies, however, have motivated the study of very high aspect ratio towed cylinders where the boundary layer thickness becomes significantly larger than the cylinder radius. Some of the more noteworthy earlier papers include those by Rao (1967, 1974), Cebeci (1970), Rao & Keshavan (1972), Bradshaw & Patel (1973), Patel et al. (1974), Patel (1974) and Willmarth, Winkel & Sharma (1976). Much of the work identified above addressed issues such as being able to predict drag, or establishing scaling laws for the mean velocity profile. Limitations in both experimental and computational capabilities necessitated integral approaches to data processing and analysis. Following the approaches developed by Clauser (1956) and Coles (1956) for flat-plate TBLs, Rao (1967) and Patel (1974) focused on developing integral momentum based frameworks for extracting critical engineering information from a rigorous scientific foundation. Patel (1973) developed a theoretical framework attempting to unify turbulent boundary layer scaling close to the wall. His arguments began by noting that for a flat-plate TBL, the wall shear stress, µ∂U/∂y, is constant across some region very close to the wall. This is simply an alternative way of saying that the velocity profile very close to the wall is linear. Now, because the plate is flat, the shear force per unit length is also constant at any distance from the wall within the linear region. Mathematically, this can be expressed as b µ∂U/∂y = constant,

(1.1)

36

M. H. Krane, L. M. Grega and T. Wei

where b is some fixed spanwise dimension. That is, for any distance y from the wall, as long as the velocity profile is linear, then the shear force per unit length will also be invariant in y. The assumption was then made that for an axisymmetric body with finite curvature, the shear force per unit length, 2πrµ∂U/∂r, will also be uniform close to the wall. In order for this to happen, however, the shear stress, µ∂U/∂r, must vary as 1/r. This relation can be directly integrated to obtain an expression for the near-wall velocity profile. In the outer part of the near-wall region, where the mean shear force per unit length is still uniform, Reynolds stresses begin to dominate. For the flat-plate TBL, Reynolds stresses have typically been related to the mean velocity gradient through a mixing length model. It was natural, then, for Rao (1967, 1974) to use a flat-plate mixing length model for the axisymmetric TBL as well. In these studies, the mixing length model was predicated on the flat-plate TBL assumption that the wall limits the sizes of eddies, i.e. the mixing length is proportional to distance from the wall. This results in a log–log velocity profile. It presumes the boundary layer over an axisymmetric body will be fuller than for a flat plate and will become increasingly so with decreasing radius. This work by Patel (1973, 1974) and Rao (1967, 1974) led to a hypothesis that for a dimensionless cylinder radius, a + , smaller than 28, the boundary layer will relaminarize. Alternatively, Lueptow & Haritonidis (1987) postulated that the wall is less and less able to affect eddy size in the limit as body radius tends towards zero. They therefore argued that the mixing length should be proportional to the local boundary layer thickness. This results in a log-law profile similar to that found in a planar boundary layer. Lueptow & Haritonidis (1987) showed mean velocity profile data that were consistent with their analysis. The difficulties in obtaining spatially and temporally resolved data very close to the cylinder surface continue to this day. As a result, there continues to be important research being done using traditional integral approaches. Within the past few years, Cipolla & Keith (2003a,b) and Keith, Cipolla & Furey (2009), for instance, used the integral momentum equations and drag measurements coupled with wall pressure measurements to develop an understanding of the character of the TBL forming over towed cylinders which were very long and very small in radius. There have, of course, been a number of excellent studies done for axisymmetric TBLs involving highly resolved turbulence measurements. These include Willmarth et al. (1976), Luxton, Bull & Rajagopolan (1984) and a series of studies by Lueptow and collaborators, including Lueptow, Leehey & Stellinger (1985), Lueptow & Leehey (1986), Lueptow & Haritonidis (1987) and Lueptow & Jackson (1991). However, perhaps the most comprehensive data set to date may be found in a two-part numerical paper by Neves, Moin & Moser (1994a,b). This paper is discussed later in this section. Table 1 provides a comparison of the flow conditions of the present study with those of Luxton et al. (1984), Lueptow et al. (1985) and Neves et al. (1994a). In turn, Luxton et al. (1984) provide a thorough benchmarking to earlier measurements including those of Willmarth et al. (1976). From table 1, one can see that, at least at a casual glance, the present experiments are in the same range as the works cited above. It is critically important to note, however, that there are subtle but very significant differences between the present study and those in table 1. In many respects, the closest comparison to the present work is Luxton et al. (1984). The two studies

37

Measurements in the near-wall region of a boundary layer Authors Lueptow et al. (1985) Lueptow et al. (1985) Lueptow et al. (1985) Luxton et al. (1984) Luxton et al. (1984) Luxton et al. (1984) Neves et al. (1994a) Neves et al. (1994a) Present paper Present paper Present paper

δ/a

x/a

Rea

Rex

a+

13.4 24.4 33.0 41.6 39.1 2.1 5 11 7.95 4.02 0.60

407 2440 4271 400–5640 400–5640 400–5640 n/a n/a 1286 416 59.0

154 154 154 125 315 4470 311 674 266 817 5795

62 680 375 770 657 752 50 000–705 000 126 000–1 776 600 1 788 000–25 210 800 n/a n/a 340 000 340 000 340 000

n/a n/a n/a 12.5 25.1 250 21 43 16.8 46.3 268

Table 1. Comparison of axisymmetric TBL parameters used by key references and the present experiments. Note that for the present experiments, the displacement thickness has been used as a conservative estimate of the boundary layer thickness, δ.

occupy roughly the same parameter space in terms of boundary layer thickness, Reynolds numbers and dimensionless cylinder radius. Within the limits of resolution of the Luxton et al. (1984) experiments, there appears to be good agreement between the data sets. This study, however, contains mean and fluctuating data in the axial and radial directions, while Luxton et al. (1984) report only streamwise velocity data. Furthermore, the present experiments have a much higher resolution close to the cylinder. Lueptow et al. (1985) reported measurements with thicker boundary layers and lower Re a than this study. The critical distinction is that the explicit objective of Lueptow et al. (1985) was to generate a thick TBL. As such, they employed a turbulent boundary layer trip whose size was on the order of the cylinder diameter. No trip was used in the present study, as this would be as counterproductive to the purposes of this study (i.e. to gain insight into whether or not the boundary layer relaminarizes with a decreasing cylinder radius) as it was necessary in Lueptow’s investigation. The key point here is that even though Lueptow’s measurements are nominally at a lower cylinder radius Reynolds number, the use of a boundary layer trip would significantly alter the flow. This creates a clear distinction between their work and the present study. In general, the use of hot wires and pitot tubes in all of the earlier studies had limited near-wall spatial resolution because of probe interference, especially on the smallest radius cylinders. In addition, a precise determination of the hot-wire distance from the wall suffered from bias errors, as described by Willmarth et al. (1976). As shown in § 3.1, wall effects on hot wires also affected the near-wall measurements of Lueptow et al. (1985). Neves et al. (1994a,b) conducted a DNS study of axisymmetric TBLs in a very similar parameter range. The significant difference here was that the computational study of Neves et al. (1994a,b) was for a fully developed annular pipe flow, where the present study was for a growing boundary layer. While the ratio of the outer to inner radius of their computational domain was large enough that the inner boundary layer could be examined independently of the outer boundary layer, the critical issue for this study is that the flow in the numerical experiment was necessarily driven by an axial pressure gradient. As shown in the results and discussion section, the imposition

38

M. H. Krane, L. M. Grega and T. Wei

of a pressure gradient necessitates maximum shear at the inner wall, a finding that is counter to the boundary layer measurements presented in this paper. In summary, not only have these experiments strong similarities but also distinct differences from key studies done by other researchers. Because the studies of Willmarth et al. (1976), Luxton et al. (1984), Lueptow et al. (1985) and Neves et al. (1994a,b) were focused on thick TBLs, the experiments, both physical and numerical, were designed to generate fully turbulent boundary layers. The objective of this study was, in a sense, the opposite. That is, to characterize what happens when the cylinder radius decreases towards zero. The focus of this study was to determine whether or not the boundary layer was turbulent with decreasing radius. In contrast, the focus of the earlier studies was to generate turbulence over as small a radius cylinder as possible and characterize the flow in that state. 2. Apparatus and methods 2.1. Water tunnel facility Experiments were conducted in the large free-surface water tunnel described by Smith (1992) and Grega et al. (1995). The facility consisted of an upstream end tank, settling chamber, contraction, glass test section, downstream end tank and pumps. The size of the test section was 610 cm × 122 cm × 57.2 cm. The maximum flow rate was 1500 l min−1 , corresponding to a free-stream speed of ∼30 cm s−1 when the test section was full. The flow was uniform within ±2 % across the cross-section and turbulence intensities were