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Abstract

The drag reducing efficiency of the outer-layer vertical blades, which were first devised by Hutchins (2003), have been demonstrated by the recent towing tank measurements. From the drag measurement of flat plate with various vertical blades arrays by Park et al. (2011), a maximum 9.6% of reduction of total drag was achieved. The scale of blade geometry is found to be weakly correlated with outer variable of boundary layer. The drag reduction of 2.8% has been also confirmed by the model ship test by An et al. (2014). With a view to enabling the identification of drag reduction mechanism of the outer-layer vertical blades, detailed flow field measurements have been performed using 2D time resolved PIV in this study. It is found that the skin friction reduction effect is varied according to the spanwise position, with 2.73% and 7.95% drag reduction in the blade plane and the blade-in-between plane, respectively. The influence of vertical blades array upon the characteristics of the turbulent coherent structures was analyzed by POD method. It is observed that the vortical structures are cut and deformed by blades array and the skin frictional reduction is closely associated with the subsequent evolution of turbulent structures.

1. Introduction

The reduction of frictional drag of turbulent boundary layer is of great importance for the fuel economy of ship. Along with the development of hull form optimization technique, the wave-making resistance is less than 20% of the total drag of most modern ships. Therefore, the advantage from the reduction of the remaining frictional drag would be enormous. The fuel consumption of global ocean shipping in 2003 was estimated 2.1 billion barrel/year [1], which corresponds to approximately 200 billion US$/year. Thus, 10% reduction of frictional drag, with the propulsive power being estimated to be 90% of total power consumption, would lead to saving of approximately 14 billion US$/year.

The skin frictional drag is closely associated with the coherent structures, for example, hairpin vortices in the turbulent boundary layer flow. Various control strategies toward the attenuation of the drag-inducing flow structure have been proposed during several decades. From the viewpoint of reliability, the passive techniques such as riblet [2, 3], compliant coating [4], and LEBU (Large Eddy BreakUp device) are more suitable for the marine application. Hefner et al. [5] conducted the experiments with LEBU to reduce the skin friction downstream of the LEBU devices and achieved 24% of drag reduction compared to undisturbed flat plate levels. The LEBU devices directly interact with and change the large eddy structures, thus interrupting the production loop and reducing the bursting events causing surface stress.

Recently, Hutchins [6] used the array of thin vertical plates in the turbulent boundary layer. This array chops off large structures, thereby disconnecting the link between outer and inner structures. The height and the spanwise packing (the spacing between each plate) were varied to find optimal values. Maximum skin friction reduction amounted to 30%. However, these results do not necessarily imply the usefulness of this device in real application. This is because only the reduction of local skin friction downstream of the device was quantified. The device drag associated with the momentum deficit was not investigated in detail. In case of LEBU, the device drag usually exceeds the reduced skin friction, thereby severely restricting the applicability. A couple of towing tank measurements of a flat plate and a ship model with blades array have been conducted to assess the total drag reduction capability. Park et al. [7] showed a 9.6% reduction of total drag for flat plate. For a KVLCC ship model, a 2.8% total drag reduction has been reported by An et al. [8]. In both studies, the drag reduction efficiency appeared to be correlated with the outer scaling based on the boundary layer thickness. This implies that the present outer-layer vertical blades array is more plausible in terms of the applications to such high Reynolds number flows as the flow around ship hull.

With a view to enabling the identification of drag reduction mechanism, a detailed flow field measurements have been performed using 2D time resolved PIV in this study. The time-mean velocity profiles and turbulence quantities are compared between the baseline case and the blade case. The influence of vertical blades array upon the turbulent coherent structures is scrutinized in xy-planes as well as xz-planes. The POD analyses based on the unsteady flow field in both planes are employed to substantiate the changes of the coherent structures due to the vertical blades array.

2. Experimental Methods

2.1. PIV Measurement Setup

The PIV measurement in this study was performed in the circulating water tunnel displayed in Figure 1. The test section is a 2-dimensional channel with the cross section of 0.4 m (width) × 0.16 m (height). Water flow in the test section is driven by a centrifugal pump. The flow speed is controlled by adjusting the rotating speed of the pump by the inverter. A magnetic flow meter was employed to monitor the flow rate. In this study, the average flow velocity U_M, which is defined by dividing volume flow rate by the cross sectional area, was set to 0.534 m/s.

Figure 1:

Circulating water tunnel and outer-layer vertical blades installed in the test section.

The 2D time-resolved PIV system (Dantec Dynamics) consisted of high repetition rate Nd:YAG laser, high-speed CMOS camera, and synchronizer. The illuminating laser was a Lee diode-pumped Nd:YAG laser (LDP-100MQG) with output wavelength of 532 nm, variable repetition rate from 10 Hz to 20 kHz, and pulse energy of 11 mJ. The high-speed camera was a 10 bit resolution NanoSense Mk. III CMOS camera with a maximum frame rate of 1040 Hz, a pixel resolution of 1,280 × 1,024, and internal flash memory of 2 GByte. This memory capacity allowed successive acquisition of 500 frame pairs with the maximum frame resolution of 1,280 × 1,024 pixels. Hollow glass beads with a diameter of 10 μm had been added in the water reservoir prior to the measurement.

Profile of time-mean velocity for the baseline, undisturbed fully-developed channel flow was measured by the PIV system. The parameters of the baseline channel flow are listed in Table 1. The local wall shear stress τ_w was calculated by using the CPT (Computational Preston Tube) method from the mean velocity profile, which was first introduced by Nitsche et al. [9]. This method is basically to fit the measured velocity profile onto the canonical velocity profile in turbulent boundary layer of Szablewski [10] as follows:

u^{+} = \int_{0}^{y^{+}} ((2 (1 + K_{3} y^{+}) d y^{+}) \times (1 + \{1 + 4 {(K_{1} y^{+})}^{2} (1 + K_{3} y^{+}) \times [1 - \exp ((- y^{+} \sqrt{1 + K_{3} y^{+}}) {{\times {(K_{2})}^{- 1})]}^{2}\}^{0.5})}^{- 1}) .

(1)

Here, K₁ corresponds to the von Karman constant, K₂ to the van Driest damping factor, and K₃ = (ν/ρu_τ³)(dp/dx) being the dimensionless pressure parameter. Compared with the Clauser plot method, this method is not affected by subjective selection of the extent for the logarithmic region, thereby giving more robust estimation of τ_w. This was verified as a useful tool to estimate the skin friction in a wide variety of nonequilibrium turbulent boundary layer flows [11].

Table 1:

Parameters for the baseline case without blades array.

U_M (m/s)	u_τ (m/s)	δ (mm)	Re _τ	C_f × 10³

0.534	0.0252	80.0	2,104	4.440

Figure 1 also demonstrates the outer-layer blades installed in the test section of the circulating water tunnel. Here, the height and the spanwise packing were set to 20 mm and 12 mm, respectively. These values correspond to the nondimensional heights of h/δ = 0.177 (h⁺ = 385) and nondimensional spanwise packings of z/δ = 0.106 (z⁺ = 231) based on the half-channel height δ and the friction velocity u_τ for the undisturbed baseline case. Although the nondimensional height of h/δ = 0.177 is less than the optimal range found in the preceding studies of Park et al. [7] and An et al. [8], the height of the blade is high enough to extend to the outer-layer flow, thereby affecting the flow field.

The PIV measurements were performed in two measurement plane setups, xy-planes (0 ≤ x/h ≤ 32, z = 0, 6 mm) and xz-planes (0 ≤ x/h ≤ 32, y = 1, 4, 9 mm). The field of view had dimensions of 90 mm by 75 mm, with the plane oriented parallel to the mean flow direction, and this yielded 78 by 60 velocity vectors after processing with 50% overlap. 4,000 PIV realizations were used to compute the mean velocity profile. The analysis of PIV measurement uncertainty described in Scarano and Riethmuller [12] was employed based on the formula $ε_{μ} = Z_{C} σ / μ \sqrt{N}$ , where Z_C is the confidence coefficients μ and σ are the time-mean and the standard deviation of the measured velocity. The uncertainty for the time-mean streamwise velocity U was estimated to be 7% using a 95% confidence interval.

As depicted in Figure 2, the xy-planes with z = 0 mm correspond to the in-blade plane, while the xy-planes with z = 6 mm being the midblade plane. It is worthwhile to mention that the flow behind the blades array would exhibit a significant three-dimensionality, that is, the change of flow field depending on the spanwise location relative to blade. Therefore, the two spanwise locations are selected to investigate such three-dimensionality. The xz-planes were located at three heights from the wall, y = 1 mm (y⁺ = 20), y = 4 mm (y⁺ = 80), and y = 9 mm (y⁺ = 180). Thus, these heights were set for the investigation in the inner layer (y⁺ = 20) and outer layer (y⁺ = 80 and 180).

Figure 2:

Measurement domain and schematic diagram of PIV.

2.2. POD Analysis Method

The POD is a well-known technique determining an optimal basis for the reconstruction of a data set. Since introduced by Karhunen [13], this technique has been extensively employed for the extraction and identification of the coherent structures [14]. The basis function obtained from POD analysis of a spatial function represents a dominant structure. For a spatiotemporal velocity field $u (\vec{x}, t)$ , POD determines orthonormal functions $φ_{j} (\vec{x})$ , j = 1, 2, …, such that the projection of the velocity field onto the first n functions $\hat{u} (\vec{x}, t) = \sum_{j = 1}^{n} a_{j} (t) φ_{j} (\vec{x})$ minimizes the square error of the projection E, defined by $E = 〈{∥u (\vec{x}, t) - \sum_{j = 1}^{n} a_{j} (t) φ_{j} (\vec{x})∥}^{2}〉$ . The functions $φ_{j} (\vec{x})$ are obtained by solving the integral equation

\int R (\vec{x}, {\vec{x}}^{*}) φ ({\vec{x}}^{*}) d {\vec{x}}^{*} = λ φ (\vec{x}),

(2)

where $R (\vec{x}, {\vec{x}}^{*}) = 〈u (\vec{x}) u ({\vec{x}}^{*})〉$ is the autocorrelation of the velocity. The above equation is again an eigenvalue problem with the integration variable being ${\vec{x}}^{*}$ . Solving this equation gives n eigenfunctions $φ_{j} (\vec{x})$ , j = 1,2, …, n. For a function with two-dimensional spatial dependence, direct numerical calculation of the above integral equation requires considerable amount of calculation time. The method of snapshots proposed by Sirovich [15] leads to a dramatic saving in computational effort in computing the eigenfunctions. In this study, the method of snapshots has been employed.

3. Results

3.1. Time-Mean Statistics and Unsteady PIV Measurement Results

From the time-mean velocity profiles measured in xy-planes, the local wall shear stress τ_w and friction velocity $u_{τ} = \sqrt{τ_{w} / ρ}$ are estimated as a function of x, the downstream distance from the trailing edge of the blade. The local shear stress τ_w(x) is then nondimensionalized as a local skin frictional coefficient C_f(x) = τ_w(x)/0.5ρU_m². Figure 3 presents the streamwise development in two spanwise measurement locations. For comparison, the baseline skin friction value without blades is designated as a horizontal solid line. It is found that the Z06 plane (midblade plane) is noted predominantly by skin friction reduction, whilst the Z00 plane (in-blade plane) shows local skin friction augmentation regions. The average skin friction reduction rates are calculated as 7.95% for the Z06 plane and 2.73% for the Z00 plane, respectively. Also it is revealed that the streamwise local skin friction distributions at two spanwise locations exhibit negative correlation. It is worth mentioning that momentum deficit is expected downstream of blade in Z00 plane, whilst the Z06 plane is not associated with it. Despite the higher momentum in Z06 plane, the skin friction becomes smaller in the average. This suggests that the skin friction reduction is mainly attributable to the constraint of spanwise motion of the coherent structures between the blades, which is expected in the Z06 plane.

Figure 3:

Streamwise development of local skin friction coefficient.

Figures 4 and 5 show the profiles of time mean streamwise velocity in the Z00 and Z06 planes, respectively. The velocity profiles in the Z00 plane (Figure 4) exhibit hollows near the edge of the blade y = 20 mm, which almost disappear at x/h = 5. These are associated with the wake of the blade. On the other hand, such hollow is not observed from the velocity profiles in the Z06 plane (Figure 5). The profiles of the streamwise turbulence intensities $\sqrt{\bar{u^{' 2}}} / U_{m}$ in the Z00 and Z06 planes are plotted in Figures 6 and 7, respectively. In Z00 plane (Figure 6(a)), the streamwise turbulence intensity becomes larger than the baseline case just after the blade x/h = 0.5. This is caused by the vortices in blade wake. The increase of turbulent energy, however, does not persist in further downstream region and the streamwise turbulence intensity becomes slightly smaller than the baseline case. Similarly, in Z06 plane (Figure 7(a)), the streamwise turbulence intensity shows increase over the baseline case and then subsequent decrease downstream. The wall-normal turbulence intensity, plotted in Figures 6(b) and 7(b), presents similar behavior with more discernible reduction from the baseline case. The most significant reduction is observable in the Reynolds stress in Figures 6(c) and 7(c). It is notable that the Reynolds stress becomes minimum near the edge of the blade y/h = 1.0 in the Z00 plane, whilst there exists a local maximum of the Reynolds stress at y/h = 1.0 in the Z06 plane. The decrease in the Reynolds stress implies the attenuation of the turbulence activities, which is in support of the skin friction reduction.

Figure 4:

Mean velocity profiles measured in Z00 plane; (a) dimensional plot, (b) nondimensional plot.

Figure 5:

Mean velocity profiles measured in Z06 plane; (a) dimensional plot, (b) nondimensional plot.

Figure 6:

Turbulence intensity profiles in Z00 plane; (a) $\sqrt{\bar{u^{' 2}}} / U_{m}$ , (b) $\sqrt{\bar{v^{' 2}}} / U_{m}$ , and (c) $- \bar{u^{'} v^{'}} / U_{m}^{2}$ .

Figure 7:

Turbulence intensity profiles in Z06 plane; (a) $\sqrt{\bar{u^{' 2}}} / U_{m}$ , (b) $\sqrt{\bar{v^{' 2}}} / U_{m}$ , and (c) $- \bar{u^{'} v^{'}} / U_{m}^{2}$ .

Figure 8 displays the plots of instantaneous velocity vectors and the contours of spanwise vorticity ω_z for the baseline case and the blade case in Z06 plane in comparison. Here, the volumetric mean velocity U_M has been subtracted from the streamwise velocity to exhibit coherent structures convecting downstream. Figures 8(a) and 8(b) are both clearly characterized by such features of coherent structures as the concentrated spanwise vorticity in the shear layer, ejection/sweep motion, and so forth. There is hardly found a qualitative change in the coherent structures observed in xy plane due to the presence of blade.

Figure 8:

Vector plot and spanwise vorticity plot in xy-plane; (a) without blade, (b) with blade (Z06 plane).

Figure 9 through Figure 11 illustrate the instantaneous flow field viewed from the above, that is, in the xz plane at varying distance from the wall. Here, the velocity vector plots of (u^', w^') and contour plots of the streamwise velocity fluctuations u^' are given for baseline case and blade case. The red-colored contours designates high-speed streamwise velocity region, while blue ones correspond to low speed regions. These plots enable the comparison of flow structures at respective flow region. Figure 9 compares the flow field at y = 1 mm (y⁺ = 20) which corresponds to the buffer layer. The baseline case in Figure 9(a) is spotted with red (high-speed) streaks and blue (low-speed) streaks, which is a clear indication of the near wall turbulent flow features. The blade case in Figure 9(b) shows essentially similar characteristics as those in Figure 9(a) with some minor change of extended low speed streaks. From this observation, it is suggested that the near-wall turbulent structures are seldom changed by the vertical blades. This is consistent that the spanwise distance between blades in this case is over 200 in wall unit, which is wider than the spanwise spacing of the near wall streaky structures.

Figure 9:

Vector plot and contour plot of streamwise velocity fluctuation in xz-plane at y = 1 mm (y⁺ = 20); (a) without blade, (b) with blade.

The instantaneous flow structure observed in the xz-plane of the outer layer (y⁺ = 80) is compared in Figure 10. It is first found that the streaks are grown both in length and width in the baseline case in Figure 10(a). Adrian et al. [16] described the coherent structure as the nested packet of hairpin vortices, not evenly distributed individual hairpins. The extended streak is attributable to the packet of hairpin vortices. On the contrary, the blade case in Figure 10(b) exhibits contours which are torn apart. It is conjectured that the presence of blade interrupts the growth of the large scale turbulent structures primarily in the outer layer. The suppression of the coherent structure growth in outer layer by the blade becomes even more pronounced in Figure 11 at y = 9 mm (y⁺ = 180). Along with the growth of the streak, there is found a significant spanwise velocity in the baseline case (in Figure 11(a)). However, the flow field for the blade case in Figure 11(b) remains unchanged from that observed in inner layers, hardly showing any significant spanwise motion. In summary, the present vertical blades array is found to suppress the growth of outer-layer turbulent coherent structures by shredding them and blocking the spanwise momentum transfer in the outer layer.

Figure 10:

Vector plot and contour plot of streamwise velocity fluctuation in xz-plane at y = 4 mm (y⁺ = 80); (a) without blade, (b) with blade.

Figure 11:

Vector plot and contour plot of streamwise velocity fluctuation in xz-plane at y = 9 mm (y⁺ = 180); (a) without blade, (b) with blade.

3.2. POD Analysis Results

The eigenvalue of each mode in POD analysis results represents the energy share of corresponding mode. Figure 12 displays eigenvalues and cumulative energy sum of baseline flow in comparison with those for the blade cases (xy-Z00, xy-Z06). The energy shares of the 1st and 2nd mode for the baseline flow appear to be 15.8% and 7.6%, respectively. The cumulative sum of energy up to 9th mode is 43.7% of the total energy. In the case of xy-Z00, the 1st and 2nd modes take 10.4% and 8.2% and the cumulative sum up to 9th modes contains 39.5% of the total energy. This implies that the less amount of energy is occupied by the lower order modes for the blade case. On the other hand, lower order modes become more dominant in the midblade plane (xy-Z06); 27.0% for the 1st mode and cumulative sum up to 9th mode being 53.6%. From these lower order energy distributions, it can be stated that the evolution of coherent structures is interrupted by the blades in the blade plane, whilst it is promoted in the midblade plane. It is worthwhile to mention that the present POD analysis was performed for the initial flow region just downstream of the vertical blades, that is, 0 ≤ x/h ≤ 4. Therefore, the promotion of coherent structures for Z06 plane is responsible for the initial skin friction increase observed on Figure 3. In the meantime, in the case of xz-plane observations shown in Figure 12(b), cumulative energy sum of blade case (xz-Y09) is lower than that of baseline flow. This implies that the evolution of coherent structures is generally impeded by the presence of the vertical blades array.

Figure 12:

Eigenvalue versus eigenmode; (a) xy plane, (b) xz plane.

Figures 13 –15 show time histories of POD coefficients a_j(t) and POD modes up to 6th lower order in the case of baseline flow, xy-Z00 and xy-Z06 cases, respectively. The 3rd through 6th modes in xy-Z00 case in Figure 14 become different from those for the baseline flow in Figure 13. This is consistent with the change of coherent structure due to the interaction of flow between the blades.

Figure 13:

POD coefficient and POD mode vector of baseline case in xy plane, baseline case; (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th mode, (e) 5th mode, and (f) 6th mode.

Figure 14:

POD coefficient and POD mode vector of blade case in xy plane, blade case in Z00 plane; (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th mode, (e) 5th mode, and (f) 6th mode.

Figure 15:

POD coefficient and POD mode vector of blade case in xy plane, blade case in Z06 plane; (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th mode, (e) 5th mode, and (f) 6th mode.

Figures 16 and 17 compare POD modes of baseline flow with those of blade case (xz-Y09) observed in xz-plane. A closer inspection indicates that the spanwise velocity component of the POD modes are suppressed for the blade case (xz-Y09) compared with the baseline case. This again manifests the skin friction reduction mechanism of vertical plates array, the constriction of spanwise motion, and consequent attenuation of coherent structures of the flow.

Figure 16:

POD coefficient and POD mode vector of baseline case in xz-plane, baseline case; (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th mode, (e) 5th mode, and (f) 6th mode.

Figure 17:

POD coefficient and POD mode vector of baseline case in xz-plane, blade case; (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th mode, (e) 5th mode, and (f) 6th mode.

4. Conclusions

In this study, an experimental investigation has been conducted to investigate the drag reduction mechanism of the outer-layer vertical blades array using a time-resolved 2D PIV. Turbulent flow modification effect by blades array has been revealed from the unsteady flow field measurement results from the PIV. The POD (Proper Orthogonal Decomposition) analyses based on the unsteady flow field in both planes are employed to substantiate the changes of the coherent structures due to the vertical blades array. The skin frictional reduction effect exhibited different behaviors at different spanwise location; the blade plane (Z00 plane) and the blade-in-between plane (Z06) showed 2.73% and 7.95% drag reduction effect, respectively. Decrease in the turbulence quantities, particularly the reduction of Reynolds stress, was noted for the blade case. Whilst the turbulent flow field measured in xy plane remained unchanged, those measured in xz planes significant changes in the outer layer. The instantaneous flow field and the POD modes indicated that the spanwise momentum transfer and consequent growth in the outer layer are hindered by the blades array, thereby attenuating the coherent structure of turbulent flows. In the previous study of Park et al. [7], the outer scaling is found to give better collapse of drag reduction efficiency C_F/C_F0. This observation is in support of the outer-scaling of drag reduction effect found in Park et al. [7].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Footnotes

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government (MSIP) through GCRC-SOP (no. 2011-0030013) and Industrial Strategic Technology Development Program (Grant no. 10038606) funded by the Ministry of Trade, Industry and Energy (MOTIE, Korea).

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