Single-pixel ensemble correlation algorithm for boundary measurement on axisymmetric boattail surface

e velocity profile. 
 Fig. 5. Boundary-layer profile from two algorithms 
4.2. Mean velocity fields 
 a) Re = 4.34 × 104 b) Re = 5.92 × 104 
 c) Re = 7.30 × 104 d) Re = 8.89 × 104 
 Fig. 6. Streamwise velocity fields on symmetric vertical plan at β = 20° 
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 The mean flow velocity in the vertical plane was shown in Fig. 6 for different flow 
conditions. The black dots show position of zero velocity streamline (dividing streamline). 
For all case, the flow is highly bent around the shoulder, which is affected by boattail 
geometry. A small separation bubble region is observed on the surface. Interestingly, the 
size of separation bubble decreases quickly with increasing Reynolds number from 
Re = 4.34 × 104 to Re = 8.89 × 104. At Reynolds number around Re = 8.89 × 104, separation 
bubble region becomes narrow and flow above the boattail is mainly affected by the 
geometry. It is expected that the separation bubble will be disappeared at higher Reynolds 
number or high Mach number conditions. The separation bubble flow is, therefore, a typical 
regime at low-speed conditions and was captured well by the single-pixel ensemble 
correlation algorithm. Note that previous study by Lavrukhin and Popovich [14] did not 
show a separation bubble for a wide range of Mach number conditions. 
4.3. Characteristics of separation and reattachment on the boattail surface 
 Fig. 7. Separation and reattachment positions on boattail surface at different Reynolds 
 number conditions (S is separation position, R is reattachment position) 
 Figure 7 shows separation and reattachment position on the boattail surface by 
PIV measurement and global luminescent oil film (GLOF) skin-friction measurement, 
which was obtained from previous study by Tran et al. [10]. The GLOF measurement 
captured skin-friction fields on the surface by a luminescent oil-film layer. The 
separation and reattachment positions by PIV measurement are determined by 
streamwise velocity along the boattail surface changing to negative and positive, 
respectively. The separation positions in both two methods show analogous results. At 
high Reynolds numbers, reattachment positions present similar results for two methods. 
However, at Reynolds number around Re = 4.34 × 104, results of both methods show 
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remarkably different. It can be explained that the movement of air near reattachment 
position at low speed (Re = 4.34 × 104) is sufficient small and the number of particles 
near the boattail surface is not enough to obtain good data for PIV measurement 
processing. Additionally, due to unsteady behavior, the reattachment is often formed a 
large region on the surface. 
4.4. Boundary-layer velocity profiles 
 Figure 8 shows the boundary-layer profile for different Reynolds numbers tested 
at x/Lb = -0.2 (6 mm before the shoulder). The velocity profiles are averaged from 
10 pixels surrounding measurement point in horizontal direction. Boundary-layer 
thickness δ is identified by a distance from wall surface to the position where 
streamwise velocity reaches to 95% free-stream velocity. The boundary-layer thickness 
is around δ = 2.8 mm and changes slightly for different flow conditions. 
 Fig. 8. Boundary measurement at different Reynolds number 
 As boundary-layer profiles are obtained, the displacement thickness δ*, 
momentum thickness θ and shape factor H can be calculated. Those parameters are 
shown by below equations: 
 *
 * u()()() z u z u z 
  1 dz ,  1 dz , H (3) 
 0 UUU 0 
 The laminar boundary layer is characterized by the shape factor around 
H = 2.59 (Blasius boundary layer), while the turbulent boundary layer is characterized 
by H = 1.3-1.4. 
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 Table 1 shows boundary-layer parameters at Reynolds number of Re = 4.34 × 104. 
Clearly, boundary layer is fully turbulent before shoulder, which is shown by a shape 
factor of around H = 1.3. 
 Tab. 1. Characteristics of boundary layer 
 *
 δ99/D δ /D θ/D H 
 0.0933 0.0180 0.0134 1.34 
 Figure 9 shows boundary-layer profiles at different positions on the boattail 
surfaces for two cases of Reynolds numbers Re = 4.34 × 104 and Re = 8.89 × 104. The 
black dashed line presents dividing streamline at Re = 4.34 × 104. Clearly, the thickness 
of separation bubble at low Reynolds number is very high, which can be observed 
clearly from boundary-layer profile. However, separation bubble becomes smaller at 
high Reynolds number and it is not clearly illustrated. The figure also indicates that the 
thickness of boundary layer increases largely on the rear part of boattail model. Clearly, 
increasing thickness of boundary layer leads to a decreasing suction behind the base. 
Consequently, base drag of boattail model decreases. 
 Fig. 9. Boundary profile at different positions on the boattail surface 
 The relative thickness of boundary layer at different positions was shown in the 
Fig. 10 for two Reynolds number of Re = 4.34 × 104 and Re = 8.89 × 104. The different 
boundary-layer thickness at x/Lb = -0.2 is small, as it was indicated before. However, 
boundary-layer thickness changes quickly near the shoulder and in the boattail surface. 
As the Reynolds number increases, the separation bubble becomes smaller and the 
thickness of boundary layer near the shoulder is reduced. In fact, the changes of 
boundary-layer thickness occurred before the shoulder, which is caused by increasing 
streamwise velocity. However, at x/Lb > 0.2, the thickness of boundary layer increases 
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with Reynolds number. Clearly, at high Reynolds number, the kinetic energy is 
remarkably lost on the boattail region and velocity recovery is lower. The high thickness 
of boundary layer near the base edge leads to a weaker near-wake and a decrease of base 
drag [15]. The results of boundary-layer profile also show some unsmooth changes near 
the base edge. It occurs from unperfected smooth of glass window, which uses to cover 
the test section of wind tunnel. To improve the results, further experiment should be 
conducted. However, this region is far from shoulder and does not affect our discussions. 
 Fig. 10. Boundary-layer thickness 
4.5. Skin-friction examination 
 For turbulent flow in a smooth wall and non-pressure gradient, a log-law region 
exists above the buffer layer. In this region, the velocity changes as a logarithmic 
function of distance to wall surface [16]. The existence of the logarithmic law allows 
estimation of wall shear stress of the model. In more details, relation among those 
parameters is shown as: 
 1
 u ln z C (4) 
 
 u zu
where u , z  are non-dimensional velocity and distance from the wall and 
 u 
 
u w is the friction velocity. 
  
 The empirical constants κ = 0.41 and C+ = 5.0 are selected for this study. Since 
boundary layer velocity was acquired from PIV measurement, Eq. (4) allows estimating 
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wall shear stress of the model with some offsets. Then, skin-friction coefficient Cfx is 
calculated by the below equation (5): 
 2
 CUfx 2 w / ( ) (5) 
 The results of logarithmic fitted lines are shown in Fig. 11 for different Reynolds 
number. Clearly, experimental data is fitted well in log-law region. The skin-friction 
coefficients are listed in Tab. 2. Skin friction reduces slightly when Reynolds 
number increases. 
 Table 2 also listed a simple estimation skin-friction coefficient using theoretical 
 1
formula c 0.0263 / Re 7 for a flat plate. As can be seen, a high consistency 
 fx, fp x 
between two measurements is obtained. The maximum difference between the skin-
friction coefficient estimated by the log-law method with the one by theoretical 
methodology is around 1% at Re = 7.30 × 104. One reason for this is from the high 
pressure gradient near the shoulder. 
 Fig. 11. Profiles of mean velocity for various Reynolds number 
 Tab. 2. Skin-friction coefficient at different Reynolds number 
 Reynolds number (×104) 4.34 5.92 7.30 8.89 
 -3
 Cfx (×10 ) 5.61 5.32 5.09 4.88 
 -3
 Cfx,fp (×10 ) 5.65 5.33 5.13 4.90 
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5. Conclusions 
 In this study, velocity fields on axisymmetric boattail model at different Reynolds 
numbers were measured experimentally using single-pixel ensemble correlation 
algorithm. Major conclusions of the study are as bellow: 
 - The single-pixel ensemble correlation algorithm improves remarkably results of 
velocity fields and resolution of boundary-layer profile near the model surface by 
comparison to cross-correlation algorithm. Additionally, the single-pixel ensemble 
correlation algorithm is able to obtain accurate results of separation and reattachment 
positions at high Reynolds number. The results from the method are sufficient for 
estimating skin friction in non-pressure gradient region. 
 - Flow fields on the boattail surface are characterized by a separation bubble. The 
flow with separation bubble is a typical regime at low-speed condition and it was 
captured well by single-pixel ensemble correlation algorithm. The size of separation 
bubble highly decreases with increasing Reynolds number. 
 - Boundary-layer profiles do not change much in the region of x/Lb < -0.2. 
However, increasing Reynolds number leads to a large decrease of boundary-layer 
thickness near the shoulder and increase of boundary-layer thickness in the reattachment 
region. Results of boundary-layer profile could be useful for further investigation of 
afterbody flow and drag control strategy. 
Acknowledgments 
 The authors would like to thank Professor Keisuke Asai and Professor Taku 
Nonomura at Department of Aerospace Engineering, Tohoku University in Japan for 
their support during the experimental process. 
 Additionally, PIV measurement is a very important technique for studying fluid 
mechanics. We would like to thank Le Quy Don Technical University, Hanoi, 
Vietnam if the university can help us to build a good wind tunnel with PIV 
measurement systems. 
References 
1. Mair, W. A. (1969). Reduction of base drag by boat-tailed afterbodies in low speed flow. 
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3. Mair, W. A. (1978). Drag-Reducing Techniques for Axisymmetric Bluff Bodies. in 
 Proceedings on the Symposium on Aerodynamic Drag Mechanisms of Bluff Bodies and 
 Road Vehicles, Edited by Sovran, G., Morel, T., Mason, W.T., General Motors Research 
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4. Viswanath, P. R. (1991). Flow management techniques for base and afterbody drag 
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5. Buresti, G., Iungo, G. V., Lombardi, G. (2007). Method for the drag reduction of bluff 
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 CRF and DIA, DDIA, 10-2007. 
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7. Mariotti, A. (2017). Axisymmetric bodies with fixed and free separation: Base-pressure and near-
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 of Reynolds number on flow behavior and pressure drag of axisymmetric conical boattails 
 in low-speed conditions. Experiments in Fluids, 60(3). 
10. Tran, T. H., Ambo, T., Chen, L., Nonomura, T. Asai, K. (2019). Flow filed and 
 aerodynamic force analysis of axisymmetric afterbodies under low-speed condition, 
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14. Lavrukhin, G. N., Popovich, K. F. (2009). Aero-gazadynamics of jet nozzles - flow around 
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15. Mariotti, A. and Buresti, G. (2013). Experimental investigation on the influence of 
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 ỨNG DỤNG THUẬT TOÁN TƯƠNG QUAN TOÀN PHẦN 
 CỦA TỪNG PIXEL ẢNH CHO VIỆC ĐO LỚP BIÊN 
 TRÊN BỀ MẶT ĐUÔI ĐỐI XỨNG 
 Tóm tắt: Phương pháp đo vận tốc bằng ảnh hạt (PIV) là một kỹ thuật quan trọng cho phân 
tích trường vận tốc. Tuy nhiên, với thuật toán tương quan toàn phần truyền thống, độ phân giải 
của trường vận tốc bị giới hạn và rất khó để đo được lớp biên bề mặt. Trong bài báo này, thuật 
toán tương quan toàn phần từng pixel ảnh được ứng dụng để phân tích dòng sát bề mặt của mô 
hình đuôi đối xứng. Các ảnh ban đầu được chụp bằng phương pháp thực nghiệm. Kết quả nghiên 
cứu chỉ ra rằng phương pháp mới cải thiện đáng kể độ phân giải của dòng chảy sát bề mặt vật và 
có thể sử dụng để đo lớp biên. Các đặc tính cụ thể của lớp biên đối xứng tại điều kiện dòng chảy 
khác nhau được thảo luận. Lớp biên tại vùng liên kết đuôi tàu và thân vật không thay đổi nhiều 
trong điều kiện dòng chảy khác nhau. Tuy nhiên, kích thước của vùng xoáy trên bề mặt giảm đáng 
kể khi tăng số Reynolds. Kết quả của nghiên cứu này có thể hữu ích cho các nghiên cứu mô phỏng 
số và thực nghiệm tiếp theo trong việc giảm lực cản của vật đối xứng. 
 Từ khoá: Thuật toán tương quan pixel ảnh; PIV; mô hình đuôi; lớp biên. 
 Received: 20/3/2020; Revised: 23/6/2020; Accepted for publication: 01/7/2020 
  
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