## Abstract

Boundary layer information local to three longitudinal positions has been characterized for a 122 cm long acrylic cylinder with hemispherical endcaps, via analysis of stereo particle image velocimetry (PIV) measurements made during laterally oscillating motions and for steady, axisymmetric flow. No obvious turbulent flow structures or indications of boundary layer separation were observed at nonzero advance speeds, and skin friction coefficients were subsequently estimated for magnitude relative to the dynamic pressure associated with an axial flow speed of 0.25 m/s. Computational fluid dynamics (CFD) analysis is performed in commercial software to accompany the experimental measurements. Local skin friction coefficients are estimated and deviate, on average, from analytical solutions for zero gradient axisymmetric flows by around 22% and from the accompanying numerical solutions by around 4%. The effects of intermittent boundary layer compression and turbulence inhibition are readily observed for the oscillating cylinder, although vortex shedding is observed for some large amplitude, high frequency cases.

## 1 Introduction

Incompressible, axial flow over circular cylinders and the associated effects of transverse surface curvature on boundary layer development have been considered mostly analytically, while few known experimental and computational studies have been performed. Attention is typically focused on either thin, laminar boundary layers (where thicknesses are much smaller than cylinder radius and thus resemble flat plate boundary layers) or fully turbulent flow, with or without transverse flow components, exhibiting little growth with increasing distance.

We presently consider regions of steady, axisymmetric flow for a straight, circular cylinder which remain mostly laminar but have boundary layer thickness similar to cylinder radius. Also considered are oscillatory transverse fluctuations due to lateral cylinder motion and their influence on boundary layer behavior. Such fluctuations may cause intermittent transitional structures, instability, and periodic boundary layer thinning. Of primary interest is estimating wall shear stresses and drag forces in relation to characteristic Keulegan–Carpenter numbers. Longitudinal pressure gradients are known to cause significant deviations from Blasius' flat plate solution for regions near the leading edge; therefore, quantifying this distinction is also of interest.

The choice of cylinder dimensions and flow conditions, which are described in detail in Sec. 2, was driven by the need for comparison with swimming conditions for a biomimetic unmanned underwater vehicle that imitates the swimming motions of eels. In particular, the inflow speeds and oscillation frequencies were chosen to match those used for swimming speeds and undulation frequencies of the robot. More details can be found in Potts [1] and Eastridge [2].

An early, important analytical study dates back to Blasius [3] who developed a similarity solution for uniform, zero pressure gradient, viscous flow over a flat plate at zero incidence. It is well-known that the axisymmetric boundary layer near the leading edge of a finite cylinder behaves much like the Blasius boundary layer due to its relative thinness compared to the cylinder's radius, though transverse curvature quickly becomes important. Researchers readily observe that boundary layers associated with cylinders are thinner and have larger skin friction coefficients than flat plates with equivalent Reynolds numbers. The transition from Blasius applicability to curvature importance is not well-defined, but it is usually analyzed in terms of the nondimensional flow parameter $\nu x/(Uoro2)$. This ratio provides a relationship between each relevant variable which include kinematic viscosity *ν*, axial coordinate *x*, steady external flow velocity *U _{o}*, and cylinder radius

*r*.

_{o}Historically, in seeking skin friction coefficients for axial flow over finite cylinders, theoretical developments have sought some “correction factor” to be added to Blasius' flat plate solution. The earliest known attempt at this was Young [4], but Glauert and Lighthill [5] claim the estimates over-predict. Glauert and Lighthill [5] also derive solutions for the skin friction coefficient using two methods, a Pohlhausen method and an asymptotic series method, but only the former is of interest here as it deals with small values of the flow parameter $\nu x/(Uoro2)<0.04$. Actually, since Glauert and Lighthill assume the boundary layer to have zero thickness at *x *=* *0, their Pohlhausen estimate only provides improvements near the 0.04 threshold. Perhaps the best approximations have been achieved by Seban and Bond [6] coupled with the subsequent numerical improvements provided by Kelly [7].

King [8] experimented with a cylinder subjected to transverse flow and fixed at 0 deg, 15 deg, 30 deg, and 45 deg yaw angles for the Reynolds number range 2000–20,000. Large yaw angles were seen to induce an axial flow component which causes a reduction of the cross-flow force, stabilizes the separated shear flow, and suppresses the formation of von Kármán vortices. It was determined that axial velocity components manifest significant axial flow in the cylinder's wake and that separation appears delayed for larger yaw angles.

Examples of more recent investigations involving flow over cylinders include Feymark et al. [9], Kumar [10], and Peter and De [11]. Feymark et al. [9] examined flow over a cylinder undergoing streamwise oscillations, while Peter and De [11] considered transverse oscillations near a fixed wall. Both of these studies employed numerical approaches and are interested in engineering applications related to vortex-induced vibrations. Kumar [10] experimentally investigated the wake structure of a rotationally oscillating cylinder situated in channels of various widths with significant inlet blockage effects. Note that each of these recent studies consider flow components that are primarily transverse to the cylinder's symmetry axis.

No investigations regarding laterally oscillating, finite cylinders with primarily axial flow components that remain laminar are presently known to the authors. To study the effects associated with these conditions, experiments have been performed to measure near-wall fluid velocities using particle image velocimetry (PIV), and analogous CFD simulations were executed to accompany and complement the PIV measurements.

## 2 Problem Description and Analysis Methodology

Zero incidence inflow is investigated for an acrylic cylinder with constant 50.8 mm (2″) diameter *D _{c}* and 121.92 cm (48″) length

*L*excluding a hemispherical endcap at each end. The two endcaps together increase

*L*by one diameter to form the overall wetted length. Experiments were performed at 69.5 cm centerline immersion which equates to 13.68 diameters or 0.57 lengths.

Two modes of oscillation in an *x*–*y* plane parallel with the ground are studied: pure periodic sway and pure periodic yaw about a vertical axis at the leading edge, *x *=* *0 (the interface of the hemispherical endcap and cylinder face). A summary of the desired data may be listed as follows:

Oscillatory motions with no longitudinal inflow

Constant inflow with no oscillatory motions (axisymmetric flow)

Constant inflow with oscillatory motions

*x*positive aft,

*y*to the right when facing forward, and

*z*positive upward. A schematic of the flow problems is shown in Fig. 1, and a test matrix may be populated with the actual experimental parameters of interest as shown in Table 1. The Reynolds number for these tests is defined in terms of the forward (mean inflow) speed and axial location of interest along the test article, assuming $\nu =1.1386\xd710\u22126m2/s$

Mean inflow speeds | U_{o} | 0.25m/s, 0.5 m/s |

Measurement locations | x | $L/4,\u2009L/2,\u20093L/4$ |

Oscillation periods | T | 2.7 s, 4 s |

Periodic sway amplitude | Y | 0.1524 m |

Periodic yaw amplitude | Θ | 10 deg |

Mean inflow speeds | U_{o} | 0.25m/s, 0.5 m/s |

Measurement locations | x | $L/4,\u2009L/2,\u20093L/4$ |

Oscillation periods | T | 2.7 s, 4 s |

Periodic sway amplitude | Y | 0.1524 m |

Periodic yaw amplitude | Θ | 10 deg |

At *x *=* L*, these values would be $ReL=267,697$ for $Uo=0.25\u2009m/s$ and 535,394 for $Uo=0.50\u2009m/s$. The Reynolds numbers at each of the three experimental measurement locations for each of the inflow speeds are given in Table 2.

$Rex$ | KC | |||||
---|---|---|---|---|---|---|

0.25 m/s | 0.50 m/s | 4 s, sway | 4 s, yaw | 2.7 s, sway | 2.7 s, yaw | |

$L/4$ | 66,924 | 133,849 | 18.850 | 6.580 | 18.850 | 6.580 |

$L/2$ | 133,849 | 267,697 | 18.850 | 13.159 | 18.850 | 13.159 |

$3L/4$ | 200,773 | 401,546 | 18.850 | 19.739 | 18.850 | 19.739 |

$Rex$ | KC | |||||
---|---|---|---|---|---|---|

0.25 m/s | 0.50 m/s | 4 s, sway | 4 s, yaw | 2.7 s, sway | 2.7 s, yaw | |

$L/4$ | 66,924 | 133,849 | 18.850 | 6.580 | 18.850 | 6.580 |

$L/2$ | 133,849 | 267,697 | 18.850 | 13.159 | 18.850 | 13.159 |

$3L/4$ | 200,773 | 401,546 | 18.850 | 19.739 | 18.850 | 19.739 |

The Reynolds number, based strictly on an axial reference length and an axial velocity component, does not, however, thoroughly characterize the flow problem in terms of the complex and time-dependent relationship between inertial forces and viscous forces. For the present study, experimental measurements are made only for yaw angles of $\u03d1=0\u2009deg$ even for oscillating conditions. And since the test article's aspect ratio is high ($L/Dc\u226b1$), thicknesses remain relatively thin, and the flow components of interest are primarily axial, a Reynolds number defined on length will be of greatest interest.

*U*. Here, since the fluid is quiescent but the test article has lateral motion,

_{m}*V*is used which represents the amplitude of the cylindrical test article's transverse velocity at the trailing edge, calculated according to the following relations:

which is typically employed to qualitatively represent the ratio of oscillatory flow inertial forces to mean flow forces. The Keulegan–Carpenter number is used here to correlate observed boundary layer behavior with relative sectional speeds. The quantity is evaluated for each flow condition of interest and tabulated along with Reynolds numbers in Table 2.

Mean inflow was provided by towing the test article through the University of New Orleans (UNO) Towing Tank (39 m long with 4.6 m $\xd7$ 2.1 m wetted cross section), and oscillatory motions were induced by a planar motion mechanism (PMM). The PMM was secured to the towing carriage rails, and the cylindrical test article was affixed to the excited armature of the PMM, thereby subjecting the cylinder to both the carriage forward speed and the armature's sway and yaw motions. Photographs of the PMM situated on the towing carriage and the cylinder–armature attachment can be seen in Fig. 2. Two 25.4 mm (1″) vertical aluminum struts were responsible for transmitting the PMM armature's movements to the cylinder and were located 6.35 cm (2.5″) and 45.72 cm (18″) from the trailing edge. At least two cycles of oscillatory motion were lapsed before triggering data collection to allow dissipation of startup transience.

*x*=

*0, is desired requiring the following calculation:*

where $y(t)=Y\u2009sin(\omega t)$ is the oscillatory sway motion, $\u03d1(t)=\Theta \u2009sin(\omega t)$ is the yaw motion, *T* is the oscillation period, and *d* measures the distance in *x* between the PMM's rotation axis and the cylinder's leading edge at *t *=* *0.

### 2.1 Particle Image Velocimetry Measurement and Processing.

Three-dimensional velocity data is desired in select *y*–*z* planes, i.e., transverse planes along *x* (see Table 1 and Fig. 3). Stereo particle image velocimetry (SPIV or, generally, PIV) was employed to capture this data using two cameras submerged to the same depth as the test article, with both cameras situated on the same side of the cylinder and straddling the laser sheet. Images were collected for both $+y$ and –*y* motion, because the cylinder itself prevents the PIV cameras from viewing seed particles on the side of the cylinder opposite the PIV module. It is of particular interest to analyze flow fields for vortices shed in the local wake, corresponding to images collected for cylinder travel in the $+y$-direction, i.e., away from the PIV torpedo. Assuming flow symmetry about the longitudinal axis for steady forward advancement permits data collection on only one side of the test article.

Fluid measurements were made using a TSI model 6800 SPIV system. Two TSI model 630059 PowerView 4MP CCD cameras with Tokina Macro 100 mm f/2.8 D manual focus lenses coupled with 1.4× and 2.0× Kenko Teleplus Pro 300 DGX magnifiers, employed in series, were used to observe displacements of silver-coated hollow glass spheres with mean diameter of 5 *μ*m and density of 1,000 kg/m^{3}. The PIV laser is a Quantel Evergreen, dual pulsed Nd:YAG with 532 nm wavelength, 200 mJ @ 15 Hz. PIV processing was performed in Insight 4G version 11.

A schematic showing a view of the light sheet plane from behind is given in Fig. 3. The laser light flashes in transverse *y*–*z* planes and intersects the cylinder perpendicularly from the left side for cylinder coordinates of *y *=* *0, $\u03d1=0$. The image plane coincides with the light sheet, and the region of interest is the lower left quadrant within the image plane (when viewed from behind) and outside the cylinder's boundary as illustrated explicitly as the grayed area in Fig. 3.

Stereo camera calibration was performed with a precisely manufactured aluminum board of 10.7 mm and 11.7 mm thickness (recessed and nonrecessed thickness, respectively) with 20 mm dot spacing. The object plane for this configuration was approximately 149.6 mm square. Specifics of PIV image interrogation and processing are summarize in Table 3. Various levels of additional postprocessing were performed to detect and remove invalid measurements and to compute flow quantities of interest. Greater resolution for boundary layer analysis could have been achieved with a smaller, more focused object plane, but local wake vortical structures would have been lost. The selected arrangement was configured as a compromise for reasonable evaluation of the near-wall effects as well as those outside the boundary layer.

Inter-rogation area | $64px/32px$ ($starting/final$) | |

Grid overlap | 50% | |

Vector spacing | 1.169 mm | |

Spatial resolution | 13.688 px/mm | |

Grid engine | Recursive Nyquist | |

Correlation engine | FFT correlator | |

Peak engine | Gaussian peak | |

Vector conditioning | Recursive with | |

Neighborhood size 2 | ||

Kernel radius 2 | ||

Gaussian sigma 0.8 |

Inter-rogation area | $64px/32px$ ($starting/final$) | |

Grid overlap | 50% | |

Vector spacing | 1.169 mm | |

Spatial resolution | 13.688 px/mm | |

Grid engine | Recursive Nyquist | |

Correlation engine | FFT correlator | |

Peak engine | Gaussian peak | |

Vector conditioning | Recursive with | |

Neighborhood size 2 | ||

Kernel radius 2 | ||

Gaussian sigma 0.8 |

A constraint for transverse flow components was based on motion of the test article's cross section of interest. For this, the sectional transverse velocity (calculated as in Eq. (3) but using *x* instead of *L* in the yaw expression) was used. Finally, the value of $\Delta t$ was adjusted for directional dependency on the test article's motion, i.e., a slightly smaller value of $\Delta t$ was used when observing the cross-section's wake caused by transverse oscillation.

for consistency and better comparison of relative magnitude between inflow speeds and oscillation amplitudes. Here, *U _{o}* is the inflow speed and

*V*is the amplitude of the transverse velocity defined in Eq. (3). Note that the normalization simplifies to $|v|=Uo$ for the nonoscillating cylinder.

A sample PIV image for the cylinder experiments is given in Fig. 4. The image sample was taken during an experiment for axisymmetric flow at $x=L/2$ and $Uo=0.5\u2009m/s$. The cylinder's cross section always appears in the raw images as an ellipse with constant major and minor axis dimensions due to the oblique viewpoint of each camera.

### 2.2 Experimental Error and Uncertainty.

Likely the greatest source of error—though it is regrettably difficult to quantify—inherent with the present experimental measurements arises from misalignment of the calibration target within the laser light sheet. Peterson et al. [13] indicate that well-designed stereo PIV experiments should not exceed 10% measurement error. The best and most specific error estimation may be obtained by comparing velocity measurements made without the test article in place with the known/actual undisturbed flow speed. Preemptive experiments made without the cylinder revealed disparities of 0.0040 to 0.0095 m/s for a carriage speed of 0.25 m/s. This yields 1.6 to 3.8% difference. Similar measurements for a carriage speed of 0.5 m/s resulted in disparities of 0.0091 to 0.0198 m/s, which yield 1.8 to 4% difference. In both cases, PIV measurements over-predicted the carriage speed. These estimates account for uncertainty in the out-of-plane displacement measurements which are known to produce confidence concerns in the desired velocity data.

Two types of global error are estimated during processing in Insight 4G: pixel-displacement errors and residual stereo disparity errors. The two-dimensional pixel-displacement errors are evaluated for each camera from cross-correlation using the peak ratio uncertainty method [14]. Residual stereo disparity error basically evaluates discrepancies between cameras via stereo calibration information. Average upper bounds were estimated using these approaches to be within approximately $\xb10.04\u2009m/s$ for worst-case scenarios, which represent a maximum of 8% difference from axial flow speeds.

Interrogation window sizes were necessarily increased to accommodate diminished seeding density; however, reduced spatial resolution smooths high velocity gradients. Peak magnitudes are generally retained except for issues with temporal averaging over $\Delta t$. But alas, it is the velocity gradients that are crucial in determining wall shear stress for friction coefficient estimations, and thus these nonquantifiable errors may propagate under-predictions into the resulting calculations.

### 2.3 Numerical Simulations.

The immersed body is simulated in a computational domain of size $5L\xd73L\xd7L$ cuboid, which is discretized with a structured mesh with 1.3 × 10^{6} nodes. The immersed body has a separate structured mesh with approximately 200,000 nodes that are more compactly spaced and which moves with the body to more accurately resolve the boundary layer. Near-wall grid spacing in the wall-normal direction was chosen for satisfactory boundary layer resolution using thickness prediction formulas for laminar boundary layers. A centerline cross section of these overlapping meshes is shown in Fig. 5.

Since the immersed body moves (oscillates) temporally within the domain, some form of dynamic meshing needs to be implemented to reciprocate the motion for the computational grid. To accomplish this and to eliminate the need for remeshing at every time step, an overset mesh has been employed which allows the background mesh (the cuboid) and the component mesh (surrounding the immersed body) to be combined into one complete domain. The working principle of overset meshing is to arrange a background grid and an overlapping, or intersecting, component grid which can move with an immersed body while the background mesh is undisturbed [15]. Motions for the body and component mesh were defined for each time step using a user-defined function. A no-slip wall boundary condition was imposed on the immersed body. The outer boundaries of the domain have the following boundary conditions: uniform velocity inlet flow at the inlet plane, slip wall conditions at the lateral boundaries, and a zero pressure outlet boundary at the outlet plane. The time periods for lateral oscillations were divided into 100 time steps, $\Delta t=T/100$. Principal solver settings are shown in Table 4.

Solver type | Pressure-based |

Pressure-velocity coupling | Coupled (SIMPLE) |

Gradient discretization | Least squares cell-based |

Pressure | Second‐order |

Momentum | Second‐order upwind |

Transient formulation | First‐order implicit |

Residual criteria | $10\u221210$ |

Solver type | Pressure-based |

Pressure-velocity coupling | Coupled (SIMPLE) |

Gradient discretization | Least squares cell-based |

Pressure | Second‐order |

Momentum | Second‐order upwind |

Transient formulation | First‐order implicit |

Residual criteria | $10\u221210$ |

The calculations were performed for ten oscillation cycles to ensure elimination of initial transient effects from the final results. For each time-step, 180 iterations were performed to drive the residual criteria below $10\u221210$ for a tight convergence. Srivastava [16] explains in greater detail the methodology behind these simulations.

## 3 Results

### 3.1 Axisymmetry Conditions.

Sample ensemble-averaged measurements of velocity magnitude are given as flooded contours with overlaid vectors (though the vectors are almost undetectable) for $Uo=0.25\u2009m/s$ and 0.5 m/s at $x=L/2$ in Fig. 6. Predictions provided by corresponding numerical simulations are likewise provided. The region representing the test article's cross section has been masked for clarity. A more complete listing of experimental results is provided in Eastridge [2].

Velocity and boundary layer information appears as expected overall for laminar, axisymmetric flow. It was observed, however, that boundary layer growth propagating downstream seems to displace the outer inviscid flow considerably more at $x=3L/4$ than the other two locations. It does not appear that this is due to the incipience of transition, as the behavior seems to resemble the shape of Blasius profiles. Rather, it is more likely that trailing edge effects become influential near the $3L/4$ region. It could also be due in part to disruptions caused by the suspending struts. Locations $L/4$ and $L/2$ do not suffer from these adverse effects due to being well upstream of the first strut. Experimental measurements were made on the underside of the test article for minimize disturbances caused by the struts, but regardless the impact of the upstream strut is expected to be noticeable at $3L/4$.

The “bumps” found near the outer edge of the boundary layer in the CFD results are artifacts of overset mesh interpolation. While visualization and qualitative observations are desirable, estimating wall shear stress is of greater interest, especially for the cases of axial flow over the nonoscillating cylinder. Therefore, the present results are satisfactory.

### 3.2 Unsteady Oscillatory Conditions.

A few examples of measured velocity magnitude contours for the oscillating cylinder are given in Fig. 7. Velocity fields for the nonadvancing, oscillating cylinder display increasingly disturbed external flow with lower oscillation periods. If only the first one or two ensemble samples were used to display these contours, likely the far-field would appear more quiescent. However the nature of this situation is such that the test article repeatedly passes through the stationary image plane thereby reentering regions of fluid already disturbed during previous cycles. Ensemble-averaging dampens these disturbances somewhat, but consistent results are difficult to achieve for this reason.

Generally, the oscillating cylinder, with mean inflow as well as without, displays laterally accelerated flow producing overall higher velocities near the body surface, whereas deceleration is noticed for the axisymmetric conditions. Regions of acceleration are largely produced by inertial components added by the test article and apparently not by viscous action of the boundary layer which would diffuse these higher velocities. Cases where transverse speed greatly exceeds inflow velocity exhibit flow separation and vortex shedding in the local wake, but higher inflow speeds appear to dampen these effects.

An example where a shed, free vortex is illustrated clearly is given by the contours corresponding to $x=3L/4$ for $Uo=0.5\u2009m/s$ in yaw with $T=2.7s$ (Fig. 7). For clarity, the vorticity field was calculated and plotted as shown in Fig. 8. The vortex appears as strongly positive circulation centered at approximately $y=35\u2009mm$ and $z=\u221225\u2009mm$. A radial band of strongly negative vorticity also appears near approximately $\phi =100\u2009deg\u2009$ which is due to high shear caused by accelerated flow. The wake field is disturbed with vortical structures that remained despite ensemble-averaging indicating repeated appearance. Near-body flow directions along angular offsets near $\phi =0\u2009deg$ and 180$\u2009deg$ are primarily radial while those near $\phi =90\u2009deg$ are largely angular, following intuitive streamlines for transverse flow.

Correlation between vortex shedding and Keulegan–Carpenter numbers was found to exist where vortex shedding increases proportionately with KC. Contrasting results for sway and yaw at $L/4$ and $Uo=0\u2009m/s$ indicates that the significantly larger Keulegan–Carpenter numbers correspond directly with greater disturbances in the local section's wake. According to Keulegan and Carpenter [12], for the figures listed in Table 2, sway conditions pose tremendously larger potential for energy loss overall for the test article. The low values of KC associated with yaw oscillations lead to almost negligible disturbances to the local flow field. Furthermore, since high transverse velocities, *V*, are seen closer to the trailing edge, increases in transverse drag and thus energy loss are naturally associated with larger *x*.

These initial conclusions provide crucial intuition regarding periodic energy loss, imparted to the water. Particular attention will be given to vortex formation and lateral acceleration of the local flow field, as these are the primary culprits of energy sinks for transversely oscillating cylindrical bodies. Correlation between KC and *x* is evident here for oscillatory yaw. Of primary concern is the quantification of local skin friction coefficients for verification of numerical simulations.

## 4 Estimation of Wall Shear Stress

Distances, *r*, measured in an outward normal direction beginning at the wall may be taken along radial lines extending from the body's cross-sectional center, $y=z=0$. Analyzing axial velocity along these lines at a variety of angular values, $\phi $, provides intuition into the relative axisymmetry, characteristic wall shear stresses, and vaguely (as usual) the boundary layer thickness. However, an arbitrary line extending radially from the body surface must be interpolated for velocity values due to the rectangular nature of grid points in the experimental velocity fields. This is done also for the numerical solution data so that analogous values of $\phi $ could be analyzed. A cubic interpolation scheme for structured or unstructured two-dimensional data was employed to evaluate the velocity at any desired point within the vector fields.

The results of this method for axisymmetric flow are plotted in Fig. 9 along with predictions from numerical simulations shown as solid lines. Note that the curves representing radial velocity profiles according to the numerical predictions actually represent the average profile of a set evaluated at $\phi \u2208[1\u2009deg,\u20093\u2009deg,\u2009\u2026\u2009,\u200989\u2009deg]$ measured counterclockwise from the –*y*-axis. Figure 10 provides results of this vector field interpolation method for a yawing cylinder at $x=3L/4$ and with 0.5 m/s inflow. $\phi \u2208[0\u2009deg,\u200945\u2009deg,\u2009\u2026\u2009,\u2009180\u2009deg]$ was used for the oscillating cylinder. Profiles at $x=L/4$ and $L/2$ have similar structure to those presented in Fig. 10 but with different near-wall gradients. Additional phases of the oscillatory motion are not presently captured; the cost of retesting is high and the scope of the present investigation would not justify those expenses. However, radial velocity profiles are expected to be strongly phase-dependent and would exhibit significantly different behavior at different increments of the oscillation period.

A sharp progression (radially) from high shear to low shear is prevalent for $x=L/4$ and even $L/2$; profiles at $3L/4$ reveal a much more gradual progression from high to low shear as observed in Fig. 9. Predictions from numerical simulations generally agree with experimental measurements, but the shear transition does not appear to dampen as quickly as the experimental flow. This could be due to strong influences from transverse curvature or perhaps adverse effects from the two vertical suspending struts causing the high shear within the boundary layer to diffuse.

was evaluated to fit data points within the linear sublayer of the boundary layer for optimal values of the constant *a* in the least-squares sense. Then, $\u2202u\u0302/\u2202r$ is simply the value of *a* after being rescaled for dimensionalized velocity by multiplication with the corresponding normalization constant (Eq. (8)). For the axisymmetric experimental profiles shown in Fig. 9, only the data points $u\u0302\u22640.6$ were considered for the linear fit. Results from these estimations are given along with numerical and analytical predictions in Fig. 11 and Table 5. Note that the ordinates in Fig. 11 are scaled up by a factor of 1000, in keeping with common practice, for enhanced clarity and legibility.

0.25 m/s | 0.5 m/s | |||||||
---|---|---|---|---|---|---|---|---|

x | $Cf,Blasius$ | $Cf,SBK$ | $Cf,\u2009exp\u2009$ | $Cf,num$ | $Cf,Blasius$ | $Cf,SBK$ | $Cf,\u2009exp\u2009$ | $Cf,num$ |

$L/4$ | 2.567 | 2.803 | 2.451 | 2.470 | 1.815 | 1.935 | 1.456 | 1.509 |

$L/2$ | 1.815 | 2.046 | 1.617 | 1.628 | 1.283 | 1.402 | 0.955 | 1.098 |

$3L/4$ | 1.482 | 1.709 | 1.448 | 1.404 | 1.048 | 1.165 | 0.904 | 0.877 |

x | $\delta Blasius$ | $\delta SBK$ | $\delta \u2009exp\u2009$ | $\delta num$ | $\delta Blasius$ | $\delta SBK$ | $\delta \u2009exp\u2009$ | $\delta num$ |

$L/4$ | 5.891 | 2.027 | 6.111 | 6.735 | 4.166 | 1.432 | 5.234 | 6.486 |

$L/2$ | 8.331 | 2.869 | 8.507 | 9.479 | 5.891 | 2.027 | 8.047 | 7.359 |

$3L/4$ | 10.204 | 3.519 | 16.373 | 10.103 | 7.215 | 2.483 | 14.362 | 9.230 |

0.25 m/s | 0.5 m/s | |||||||
---|---|---|---|---|---|---|---|---|

x | $Cf,Blasius$ | $Cf,SBK$ | $Cf,\u2009exp\u2009$ | $Cf,num$ | $Cf,Blasius$ | $Cf,SBK$ | $Cf,\u2009exp\u2009$ | $Cf,num$ |

$L/4$ | 2.567 | 2.803 | 2.451 | 2.470 | 1.815 | 1.935 | 1.456 | 1.509 |

$L/2$ | 1.815 | 2.046 | 1.617 | 1.628 | 1.283 | 1.402 | 0.955 | 1.098 |

$3L/4$ | 1.482 | 1.709 | 1.448 | 1.404 | 1.048 | 1.165 | 0.904 | 0.877 |

x | $\delta Blasius$ | $\delta SBK$ | $\delta \u2009exp\u2009$ | $\delta num$ | $\delta Blasius$ | $\delta SBK$ | $\delta \u2009exp\u2009$ | $\delta num$ |

$L/4$ | 5.891 | 2.027 | 6.111 | 6.735 | 4.166 | 1.432 | 5.234 | 6.486 |

$L/2$ | 8.331 | 2.869 | 8.507 | 9.479 | 5.891 | 2.027 | 8.047 | 7.359 |

$3L/4$ | 10.204 | 3.519 | 16.373 | 10.103 | 7.215 | 2.483 | 14.362 | 9.230 |

### 4.1 Comparison With Theoretical Predictions.

The similarity solution developed by Blasius [3] can be written as a distinct function of the flow parameter to which “corrections” for transverse curvature have been added by Young [4], Seban and Bond [6], Kelly [7], and Glauert and Lighthill [5], inter alios. Similarity formulas for boundary layer thickness may be written in a similar fashion which provide a relationship to the cylinder's physical cross-sectional area. For brevity, Seban–Bond–Kelly will be written here at various places, e.g., the plot legend in Fig. 11 and subscripts in Table 5, using the acronym SBK.

Each estimation of *C _{f}* from the experimental data is slightly smaller than the theoretically predicted values, with deviations from the Blasius prediction appearing in the range 2.3–34.3%, from the Seban–Bond–Kelly prediction 14.4–46.8%, and from the numerical prediction 0.7–15.0%. A complete set of these deviations are provided in Table 6. The numerical and experimental estimations obviously agree much better, with significantly lower deviations. Each curve in Fig. 11 agrees most closely at $x=L/2$ which supports the hypothesis that leading and trailing edge effects strongly influence boundary layer development. It would be useful to extend these results to measurements taken at $L/16,\u2009L/8,\u20097L/8$, and $15L/16$ for verification of the numerical results and to refine the spacing of data points in Fig. 11. Additionally, some PIV measurements along a vertical plane intersecting the cylinder's longitudinal axis would be useful for better streamwise characterization.

0.25 m/s | 0.5 m/s | |||||
---|---|---|---|---|---|---|

x | Blasius | SBK | Num. | Blasius | SBK | Num. |

$L/4$ | 4.7 | 14.4 | 0.8 | 24.7 | 32.9 | 3.6 |

$L/2$ | 12.2 | 26.5 | 0.7 | 34.3 | 46.8 | 15.0 |

$3L/4$ | 2.3 | 18.0 | 3.0 | 15.9 | 28.9 | 3.0 |

0.25 m/s | 0.5 m/s | |||||
---|---|---|---|---|---|---|

x | Blasius | SBK | Num. | Blasius | SBK | Num. |

$L/4$ | 4.7 | 14.4 | 0.8 | 24.7 | 32.9 | 3.6 |

$L/2$ | 12.2 | 26.5 | 0.7 | 34.3 | 46.8 | 15.0 |

$3L/4$ | 2.3 | 18.0 | 3.0 | 15.9 | 28.9 | 3.0 |

Differences between the theoretical and numerical solutions near the leading and trailing edges are also interesting. Near the leading edge, theoretical predictions suffer from a discontinuity for *x *=* *0 due to Reynolds number dependency in the denominator. Therefore, *C _{f}* predictions are unrealistically high. Additionally, theoretical predictions do not account for the disruptive nature of nonzero wall-normal velocity components induced by the hemispherical leading edge. Therefore, numerical predictions will automatically be lower than theoretical ones. At the trailing edge, the situation is different where theoretical descriptions under-predict

*C*. This is presumably due to the accelerative effects of the trailing edge which result in a gentle suction that compresses the boundary layer and increases near-wall shear.

_{f}A plot of the longitudinal, hydrodynamic pressure distributions according to the numerical simulations, taken at a single angular coordinate $\phi $, is given in Fig. 12. Necessary assumptions to achieve the analytical formulas of Young [4], Seban and Bond [6], Kelly [7], and Glauert and Lighthill [5] include zero pressure gradients everywhere along the cylinder surface within a tangential plane. It has already been established that, as observed experimentally and numerically, the boundary layer is not perfectly axisymmetric, and numerical predictions allude to the intuitive hypothesis that longitudinal pressure gradients are nonzero. Hemispherical endcaps at the leading and trailing edges establish radial flow components affecting momentum exchange within the boundary layer as well as displacement of outer flow streamlines downstream and upstream of their respective locations. These influences likely lead to the larger deviations at $x=L/4,\u20093L/4$ evident in Table 6.

A rudimentary procedure for estimating boundary layer thickness, *δ*_{99} was employed for the fixed (nonoscillating) cylinder. In each radial direction, criteria were set to evaluate the spatial point at which the axial velocity component achieves or exceeds 99% of the forward speed. The distance between the cylinder's edge, *r* = *R _{c}*, and the first measurement of velocity passing this criteria determines the value of

*δ*. Table 5 indicates the theoretical predictions of boundary layer thickness according to Blasius' flat plate formula and the Seban–Bond–Kelly results for a long, thin cylinder.

Note that Seban–Bond–Kelly calculate smaller thickness and higher wall shear stresses than Blasius' prediction. This is to be expected in cases where transverse surface curvature is present, streamwise pressure gradients are exactly zero, and the flow is perfectly axisymmetric. Indeed, their scaling “corrections” were calculated to account for these effects. Also, both of these theoretical formulations assume zero boundary layer thickness at the leading edge, *x *=* *0. These assumptions, with the exception of surface curvature for $x\u22650$, are not physically accurate for the present scenario. A hemispherical endcap exists which displaces the flow radially outward beginning at $x=\u2212Rc$. This initiates a boundary layer whose thickness is not insignificant at *x *=* *0 and, importantly, that has radial velocity components within the boundary layer itself. Additionally, the two struts suspending the test article in the flow disrupt axisymmetry for downstream measurements. These physical conditions obviously play substantial roles in the initiation of steeply positive pressure gradients near the leading edge, growth and behavior of the boundary layer, and the longitudinally and angularly varying wall shear stress, $\tau w(x,\phi )$.

### 4.2 Summary of Estimations for Oscillating Cylinder.

Characterizing velocity along radially protruding lines was done for both experimental and numerical data from the oscillating cylinder analyses as well. Rather than axial velocity alone, the magnitude using all three components was considered. Note that transverse wall shear manifests as both positive and negative, depending on the angular position and inflow velocity. This is evident by inspecting slope inflections at *r *=* *0 for different cases.

For the angular position $\phi =180\u2009deg$, disturbances appear relatively far from the boundary. The profile at this angle obviously corresponds to velocity data in the local section's wake, hence the relatively dramatic disturbances. Agreement in the asymptotic approach to the mean outer flow is displayed universally, and generally, wall behavior is comparable, especially for the lower values of $\phi $, i.e., where the flow has not yet separated. Using velocity magnitude, it is difficult to tell exactly where separation occurs, but intuitive inspection of the contours indicates separated flow, especially for the higher oscillation frequency, $T=2.7\u2009s$.

Discontinuities appear where gradients remain relatively large at approximately $r=50\u2009mm$ in the numerical profiles due to differences in interpolation of the background and component grids of the overset mesh when extracting the data from Fluent's postprocessor. The component grid was designed to extend 50 mm beyond the cylinder's edge, so this interface with the background grid causes trouble in the interpolation. No user control over this interface was available when exporting the data.

Differences in wall shear between results from $Uo=0.25\u2009m/s$ and $Uo=0.5\u2009m/s$ depend quite loosely on Keulegan–Carpenter numbers and more heavily on inflow magnitude. This confirms that low oscillation frequency influences wall shear relatively insignificantly compared to local axial velocities. Note that it is the squared outer flow velocity that is found in the denominator of Eq. (15) which will reduce the value of *C _{f}* for larger inflow and equal shear stress. Hence, it is the slope of the profile at the wall that should be carefully considered in evaluating the viscous effects on drag. The skin friction coefficient merely provides a nondimensional relationship between the wall shear stress and the characteristic dynamic pressure.

Estimates for local skin friction coefficients for the oscillating cylinder are given in Table 7. Note that each value is calculated according to Eq. (15) as usual but with the fixed value of $Uo=0.25\u2009m/s$ in the dynamic pressure calculation. This was done for consistency and to eliminate dynamic pressure errors for $Uo=0\u2009m/s$. Further note that more radial profiles were used in determining these estimates than were used for the nonoscillating cases: 2 deg increments of $\phi $ from 1 deg to 179 deg. The velocity profiles fluctuate more strongly with $\phi $ for the oscillating cylinder than in cases with only longitudinal inflow, so more angular offsets were considered to better characterize this dependence.

0 m/s | 0.25 m/s | 0.5 m/s | ||
---|---|---|---|---|

$L/4$ | 4 s, sway | 0.143 | 2.861 | 4.847 |

4 s, yaw | 0.090 | 2.841 | 5.186 | |

2.7 s, sway | −0.011 | 3.375 | 5.224 | |

2.7 s, yaw | 0.474 | 2.799 | 5.686 | |

$L/2$ | 4 s, sway | −0.249 | 2.737 | 4.736 |

4 s, yaw | −0.004 | 2.604 | 4.959 | |

2.7 s, sway | 0.362 | 2.985 | 5.180 | |

2.7 s, yaw | 0.175 | 2.591 | 5.537 | |

$3L/4$ | 4 s, sway | 0.303 | 3.633 | 6.235 |

4 s, yaw | 0.912 | 5.719 | 7.854 | |

2.7 s, sway | 0.428 | 3.853 | 6.448 | |

2.7 s, yaw | 0.516 | 4.605 | 7.626 |

0 m/s | 0.25 m/s | 0.5 m/s | ||
---|---|---|---|---|

$L/4$ | 4 s, sway | 0.143 | 2.861 | 4.847 |

4 s, yaw | 0.090 | 2.841 | 5.186 | |

2.7 s, sway | −0.011 | 3.375 | 5.224 | |

2.7 s, yaw | 0.474 | 2.799 | 5.686 | |

$L/2$ | 4 s, sway | −0.249 | 2.737 | 4.736 |

4 s, yaw | −0.004 | 2.604 | 4.959 | |

2.7 s, sway | 0.362 | 2.985 | 5.180 | |

2.7 s, yaw | 0.175 | 2.591 | 5.537 | |

$3L/4$ | 4 s, sway | 0.303 | 3.633 | 6.235 |

4 s, yaw | 0.912 | 5.719 | 7.854 | |

2.7 s, sway | 0.428 | 3.853 | 6.448 | |

2.7 s, yaw | 0.516 | 4.605 | 7.626 |

It is also observed that increases in *τ* due to boundary layer compression and thinning on the upstream side of the local cross section is more dramatic than the reduction of *τ* on the downstream side. Eastridge [2] presents a more complete listing of results that indicate this behavior. This leads to increases in local *C _{f}* while moving toward the trailing edge. And this is opposite the behavior of axisymmetric and flat plate boundary layers

*except near the transition region*. These findings agree with the hypothesis that undulatory swimming characteristics lead to local skin friction augmentation in comparison with steady-state flow past submerged, nonocillating bodies. This was the conclusion of Anderson, McGillis, and Grosenbaugh [17]; however, even though local skin friction may unwantingly increase, other benefits of the unsteady oscillations may be present, such as the reduction of overall body-bound circulation and mitigation of vortices shed into the wake.

## 5 Summary and Conclusions

Radial velocity profiles from the axisymmetric trials reveal Blasius-like behavior indicating primarily laminar flow conditions longitudinally. Good agreement was found between experimentally and numerically estimated friction coefficients, although analytical solutions predict higher values due to the zero pressure gradient assumption.

Further observations from the friction coefficient estimations listed in Table 7 regarding the oscillating cylinder experiments may also be made. First, forward carriage speed clearly influences the magnitude of local friction coefficients, *C _{f}*, more strongly than oscillation mode, period, or longitudinal measurement location. Additionally, slight increases in

*C*are generally seen for increasing oscillation frequency. Regarding specific modes of motion, it is apparent that yaw oscillations generate slightly higher shear stress than sway. Unlike results from axisymmetric conditions, friction coefficients increase with

_{f}*x*. Since no steep increases in

*C*along

_{f}*x*were observed, this seems to indicate that transitional flow is either not present or not significant in the resulting local friction coefficient. These increases are likely due to boundary layer compression and thinning, and consequentially higher

*τ*, caused by the oscillations. Finally, deviations from zero

*C*for $Uo=0\u2009m/s$ are probably due to the strongly unsteady nature of periodically moving through the same location without advancing.

_{f}Additionally, some artificial scaling is present for the column of highest outer flowrate, $Uo=0.5\u2009m/s$, because the denominator of Eq. (15) is smaller than it would be with the actual carriage speed substituted for *U _{o}*. Overall, comparing with the coefficients given in Table 5 for the nonoscillating cylinder, appreciably larger values are found to be due to transverse oscillations. No estimates based on the results of numerical simulations are yet prepared for comparison, but they should be evaluated in a similar manner as used presently.

## Acknowledgment

This work was possible thanks to Office of Naval Research Grant N00014-17-1-2099, titled “Investigation into the boundary layer of an anguilliform-like propulsor”. The authors would like to thank George Morrissey and Ryan Thiel for their assistance.

## Funding Data

Office of Naval Research (Award No. N00014-17-1-2099; Funder ID: 10.13039/100000006).

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.