ZHAO Meng-hua, CHEN Xiao-peng

School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi?an 710072, China, E-mail: zhaomenghua@mail.nwpu.edu.cn

(Received December 17, 2011, Revised May 31, 2012)

where (,)gxy is the light intensity distribution of the incident light. As can be seen, (,)Iij is the average value of the light intensity. Furthermore, the edge will be blurred due to the convolution of devices and the diffraction effect, as a low-pass filter. It is a challenging task to detect the “high frequency changes”through the sampling data. In the sub-pixel edge detection approach, the pixel-to-pixel (between neighboring ones) derivative function is calculated first for the gray scale, corresponding to the cross points in Fig.(4a). According to the Central Limit Theorem, the difference of the gray intensity profile across the edge (along the gray line in Fig.(4b)) is accurately described by the Gauss function[20]. To achieve a sub pixel accuracy, the Gaussian function is fitted to a set of gray differences along the movement direction (the solid line in Fig.4(b)), the fitted peak point is then taken to be located on the edge (Fig.4(a)). The details of the sub-pixel algorithm are described in Ref.[21].

A COMBINED DATA PROCESSING METHOD ON WATER IMPACT FORCE MEASUREMENT*

ZHAO Meng-hua, CHEN Xiao-peng

School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi?an 710072, China, E-mail: zhaomenghua@mail.nwpu.edu.cn

(Received December 17, 2011, Revised May 31, 2012)

A combined method is proposed to determine the water entry acceleration at a low impact velocity through image processing. The procedure includes: (1) a sequence of images for water impact are recorded by a high speed camera, (2) the sub-pixel image processing method is employed to calculate the displacement with an accuracy on the “sub-pixel” level, (3) the acceleration of the object is acquired by differentiating the displacement twice and with results being further filtered by a carefully designed low-pass Butterworth filter. A theoretically based analysis is conducted for designing the parameters of the low-pass filters. It is shown that the water entry can be regarded as a procedure with a slowly changing velocity. The method is validated with the standard sinusoidal motion and the water entry of a sphere. This approach could be considered as an auxiliary method during the early-stage study of the water entry, and it could be further applied to some complicated circumstances, like the water entry of spinning spheres.

water impact, force measurement, image processing, low-pass filter, sub-pixel edge detection

Introduction

The water entry is a complex phenomenon, which is widely related to the ship slamming, the impact on air-to-water vehicles[1]and the locomotion of water-walking creatures[2]. The process of the impacting would induce a flow around the solid body, while the aroused fluid will exert a counterforce against the structure in return. That may cause problems such as the abnormal readings of devices, the uncertainty of the trajectory control, or the damage of the hull. In 2007, Duez et al.[3]presented their meticulous observations on the wettability effects on the generation of the water entry cavity (or splashing). The motion of the contact line is believed to play key roles in the low speed impact. This novel observation provides a newperspective for the water impact and the micro-scale effects.

The earliest work on the theoretical determination of the water entry impact forces dates back to 1929 by Von Karman, who investigated the impact loads on seaplane floats during water landings by using certain fundamental principles including the conservation of momentum and a concept of added mass. Moghisi and Squire[4]analyzed the initial impact force of a sphere striking a horizontal liquid surface vertically at speed of 1 m/s-3 m/s and compared the results with the theoretical predictions. Miloh[5]investigated the water entry of rigid spheres and derived hydrodynamic impact loads upon it by employing the matched asymptotic method and the increment method. The three dimensional predictions and the coupling of fluid-elastic solid motions were explored in recent years[6,7]. As can be seen, the water entry mechanism is far from being understood fully, including the influences of specific materials[8,9].

A substantial amount of experimental work was devoted to validate the analytical and numerical models. Yettou et al.[10]performed the drop tests with symmetrical wedges. They investigated the influence of the drop height, the deadrise angle and the mass of the wedge. Backer et al.[11]conducted experimentswith acceleration sensors and high speed camera. The experimental results were compared with 3-D and axisymmetric predictions. Wang and Wang[12]presented a CFD method to numerically simulate the physical process of twin cylinders simultaneously entering water vertically at a constant speed and constructed an empirical representation involving all state parameters to be used in the water entry of multiple columnar engineering structures.

In recent years, the high speed camera and the relative image processing method have seen a rapid development, and were applied widely in experiments of fluid mechanics[13-17]. Truscott and Techet[16,17]analyzed the unsteady hydrodynamic force after the water impact by employing a cross-correlation technique and the spline fitting. However, our preliminary tests show that it is difficult to apply this technique to the impact procedure itself. Generally speaking, a measurement technique based on images would be more advantageous than a contact measurement in some aspects. In a contact measurement, the dimension of the objects should be large and the measuring system should be designed carefully before experiments. In some complicated circumstances, like the water entry of spinning objects, the contact measurements would be difficult to be carried out.

In this work, an approach of the acceleration measurement for the water entry is proposed based on images taken by the high speed camera. The approach involves the measurement of the displacement of the object with the subpixel edge detection method, the data processing and filtering. The method was validated by a forced sinusoidal motion. And the water entry results show good agreements with the previous experiments, the theoretical predictions and the CFD simulations. It could be considered as an auxiliary method in the early-stage study of the water entry.

Fig.1 Schematic diagram of experimental setup

1. Experimental details

1.1 Experimental facilities

Figure 1 is a schematic diagram of our experimental setup. An aluminum sphere is held by a magnetic holder (Fig.2), fixed at a certain height above a glass water tank with the dimensions of 0.68 m×0.68 m× 0.98 m (width×length×height). The sphere is to be released freely onto the water surface vertically under gravity without a rotating speed. The impact velocity VIis controlled by the release height. A uniform illumination is achieved by setting a plane diffuser between the tank and the light source: two DC Halogens (LSY220-1300) with the power of 1 300 W each. The whole water entry process is recorded by a high speed camera (Mega Speed MS75K) with a resolution of 504 ×504 pixels, and 2 000 frames-persecond (fps) of the capture speed. A NIKON lens (AF 85 mm f/1.8D) is mounted on the camera. The sphere used in the experiment is smooth-shaped, with the diameter of 0.05 m and the static contact angle of 87o.

Fig.2 Top view of release device. The sphere is initially held between the fixed and sliding plates. The motion of the sliding plate is controlled by an electromagnet

1.2 Experiment procedures and image processing

The ultimate goal of the study is to determine the impact force (or the acceleration) during the water entry according to the measured displacement(y(t)) from the video sequences. An easy way to achieve this is to calculate the second order derivative ofy(t) through its finite difference representation. Since the time interval (Δt) between two successive images is small, a tiny error of y(t) would be amplified by a factor of 1/Δt during the velocity calculation, this error would be further amplified and transferred forward to the calculated acceleration, resulting in unacceptable data[18]. In this work, a four-step data processing is adopted as follows:

(1) The image sequences of the falling are recorded by the high speed camera (as in Fig.2).

(2) The displacement y(t ) is calculated by the subpixel edge detection method, which will be discussed later in details.

(3) The velocity, v(t), is obtained by differentiating y(t)

Fig.3 Image sequence of water impact of aluminum sphere. VI=3m/s ,Δt=1ms

(4) A preliminary (rough) acceleration a is acquired by Eq.(1), where v(t) is obtained in Step 3, and then the final acceleration,af, is obtained by filtering a with a zero-phase low-pass Butterworth filter, as Eq.(2).

To capture the sphere, different parts on the perimeter of it were measured corresponding to the impact stage. At the beginning of the impact, due to the splash and the refraction, the image of the lower part of the sphere would be distorted (as shown in Fig.3), the points on the upper edge (the first gray rectangle box in Fig.3) are chosen to determine the sphere displacement (yu). After half of the sphere penetrates into the water, the points on the lower part of the perimeter (the last gray rectangle box in Fig.3) are chosen to calculate motion, the location of which is denoted by yl. Both yuand ylare the averaged values of 80 points, as shown later in Section 2.1. The trajectory of the sphere is then obtained according to yuand yl. The sub-pixel edge detection algorithm is employed to determine the location of each point. Through differentiating (twice) and filtering (Butterworth) operations, af(t ) are determined. During the experiments, our camera was set horizontally on the same level of the free surface (as in Fig.1) and the distance from the tank to camera was 3 m. Therefore, the refraction shift could be neglected, which is verified by the fact that the velocities obtained through yuand ylin a certain period of (overlapping) time during the impact are the same.

1.3 Sub-pixel edge detection

Videos clips are decomposed into sequences of grayscale matrices (images), with the grayscale value I(i,j) ranging from 0-255, where i and j denote the horizontal and vertical coordinates in the image, respectively. The image edge detection is to locate the critical gray value where the physical edge is supposed to be. The accuracy of the traditional algorithm, such as the Canny?s method, which is at the pixel level[14], can not meet the requirement of calculating the acceleration, as discussed in Section 1.2. In order to improve the positioning accuracy, the gray profile is extracted in a two-step procedure.

1.3.1 Rough localization

The edge points are extracted roughly using the Canny?s method[14,19], through which the localization of the sphere can be achieved on the pixel level.

Fig.4 Gray profile sampled and its comparison

1.3.2 Precise localization

Based on the rough edge points extracted in the above step, the Gauss function is used to fit the profile of the gray difference centered at the Canny?s edge along the direction of movement[20], the peak of theGauss function is taken to point to the edge, which is at the sub-pixel level according to the theoretical considerations as follows.

Fig.5 Sequence of sinusoidal motion of a spherical nose on FTM at 2 Hz, with the time interval between each snapshot being 50 ms

The CMOS is a light integral device which integrates the light intensity illuminated on its photo-surface of a certain area in fixed time intervals and outputs gray-scale values with the aid of an integrated circuit. The output gray value can be expressed by

where (,)gxy is the light intensity distribution of the incident light. As can be seen, (,)Iij is the average value of the light intensity. Furthermore, the edge will be blurred due to the convolution of devices and the diffraction effect, as a low-pass filter. It is a challenging task to detect the “high frequency changes”through the sampling data. In the sub-pixel edge detection approach, the pixel-to-pixel (between neighboring ones) derivative function is calculated first for the gray scale, corresponding to the cross points in Fig.(4a). According to the Central Limit Theorem, the difference of the gray intensity profile across the edge (along the gray line in Fig.(4b)) is accurately described by the Gauss function[20]. To achieve a sub pixel accuracy, the Gaussian function is fitted to a set of gray differences along the movement direction (the solid line in Fig.4(b)), the fitted peak point is then taken to be located on the edge (Fig.4(a)). The details of the sub-pixel algorithm are described in Ref.[21].

2. Results and discussions

2.1 Pre-testing and validity of the method

A Fatigue Testing Machine (FTM) is known to generate standard forced sinusoidal motions to test fatigue limits and lives of materials. Our method proposed above was verified by analyzing the standard sinusoidal motion: a spherical head was hold by the fixture of the Fatigue Testing Machine (Instron 8871, Instron) and a standard sinusoidal motion is imposed with the frequency of 2 Hz and the amplitude of 0.016 m, as shown in Fig.5. The motion was captured with the frame rate of 200 Hz.

To eliminate the random noise of the images, a set of edge points on the lowest part of the sphere perimeter (as shown by a gray box in the last image of Fig.5, as an example) were selected, and the averaged displacement of these points was used to calculate the location at time t (Fig.6(a)). displacement and the velocity are1.279×10-4m and

Fig.6 Comparison for displacement (D) and velocity (V). Max-deviation of D is 1.279×10-4m, max-deviation of Vis 0.0108 m/s

Figure 6 shows the maximumdeviations of the 0.0108 m/s, respectively, as compared to the standard sinusoid values. Although they are quantitatively good on this level, massive noises for the acceleration(a) are still resulted as shown in Figs.7(a) and 7(b).

According to the Fast Fourier Transformation (FFT) in Fig.7(b) the spectrum of a is contaminated with high frequency noises ranging from 20 Hz to 100 Hz, which implies that the systematic errors cannot be removed through the aforementioned pureimage processing for the acceleration. A simple way to deal with the high frequency noises is the use of low-pass filters. Here, the Butterworth filter is adopted to filter the velocity, with the low-pass band cutoff frequency of 18 Hz, and the stop band cutoff frequency of 25 Hz, and with the maximum attenuation of 0.5 dB in the pass band and the minimum attenuation of 20 dB in the stop band. Figure 8 is the filtered result, where two cycles of movement are plotted in Fig.8(a) for convenience. Figure 8(b) shows that most noise has been eliminated.

Fig.7 Analysis of acceleration (a) calculated by differencing velocity directly

For the sake of clarity, the errors of velocity and acceleration (defined as EOV and EOA respectively) are plotted in Fig.9.

Figure 9(a) indicates that the sub-pixel image processing method can be used to determine the velocity with a certain accuracy, but is not good enough to be used to obtain a reasonable acceleration. When a proper filter is adopted, the measured errors of the acceleration can be controlled within 10%. The conclusion of this section is that, through the differentiating operations, the measured errors of the raw data are amplified and a carefully designed filter should be applied to obtain an acceptable acceleration value, with some prior experiences.

2.2 Analysis of water impact theory

Our prior experience for the water impact is obtained from a theoretical analysis.

Fig.8 Analysis of acceleration (af) after filtering

Fig.9 Error analysis. Maximum error of velocity is 0.0108 m/s, 10.9% on relative error. Maximum error of rough acceleration is 3.5935 m/s2, 284.3% on relative error. Maximum error of filtered acceleration is 0.1295 m/s2, 10.2% on relative error

Fig.10 Theoretical model of water impact problem. A rigid sphere with mass M impacting free surface with the speedIV

where m is the added mass of the sphere during impact, V the instantaneous velocity,ρlthe density of water and Cdthe drag coefficient. R and VIare the radius of the sphere and the impact velocity, respectively.

Defining the mass coefficient of the sphere as μ=3M/4πρlR3, the added mass coefficient as λ= m/0.5πρlR3, the Froude number asFr=VI2/2Rg and b=Vt/R, the drag coefficient is expressed as

2.2.1 Water impact theory

Considering a rigid sphere with mass M impacting on a free surface, as shown in Fig.10, including various components: the buoyancy (FB), the impact force (FI) and the gravity force (Mg).

Fig.11 Single-Sided Amplitude Spectrum (SSAS) ofa(t ) under FP

In the current circumstances Fr＞10, so the last term of Eq.(7) can be neglected.

Furthermore, the added mass m can be approximated by half of the mass of the Lamb disc moving vertically in water

where r denotes the radius of the disc, which closely depends on the perpetration depth. The drag coefficient Cd(for the “Lamb-disc” model), therefore, is expressed as

Besides, Miloh[5]used an asymptotic analysis for early stages of the water entry of spheres at a constant velocity, taking into account the three-dimensional effects of a spherical body in calculating the added mass, and a correcting surface wetting factor for compensating the undisturbed free surface assumption. The impact coefficient is then obtained as

The Newton?s second law and the momentum conversation lead to:

where Cwis a correcting surface wetting factor,

2.2.2 Spectrum of theoretical prediction ofCd

The water impact stage is normally in the range of b=0-0.5, as is the scope of the current investigation. The frame rate of the high speed camera is 2 000 fps, which means too few useful position points in this duration for the spectrum analysis. So, extra points in the free fall stage (a=g) of the sphere are added before the impacting in both the experimental data and the theoretical predictions. The influences of the added point numbers (FP) on the theoretical predictions are calculated and shown in Figs.11(a) and 11(b), where, Model 1 is the “Free-Fall+Lamb model” and Model 2 is the “Free-Fall+Miloh model”.

Table 1 Effect of FP on η for VI=5m/s

Fig.12 SSAS of a(t) under VI

According to Table 1, we can find the acceleration “energy” is concentrated in the range of 0 Hz-500 Hz. In the table,ak≤500Hz, aN=1000Hz . It is indicated that FP has little effect on the energy distribution of a, which is quite important for the design of the filter. Further testing is carried out for variousVI?s (as shown in Fig.12). Table 2 shows the results of all tests where η is larger than 95%, which implies that the impact can be regarded as a procedure with a slowly changing velocity. In the range of the testing velocities, 500 Hz is a good threshold for the low frequency filter.

Table 2 Effect of VIon η with FP=12

From the above discussion, it can be concluded that although the water impact drag force contains many high frequency components, their contribution is small. Accordingly, the low-pass filter with carefully designed parameters can be applied to the experimental data.

Fig.13 Scatter diagram of spectrum of a for various entry speeds. Theη is defined the same as in Section 2.2

2.3 Analysis of spectrum of the measured a

As a comparison with the theoretical analysis, the spectrum of a calculated from experiments is shown in Fig.13, where a preliminary acceleration (Eq.(1)) isapplied with FP=12. Albeit for each VIwe can find peaks at f100Hz and f 700Hz approximately, which agrees with the theoretical predictions, the non-physical noise of the preliminary acceleration is great shown in Table 3 as discussed in Section 2.1. Thevalueofaobtainedby differencing directly varies unbelievably from 292.9 m/s2to-572.7 m/s2for VI=5m/s . That is why we should filter the experimental data further. It should be noted again that although some of the information will lose, the contribution of this part to the whole process is limited according to the discussion in Section 2.2. The process is described in detail as follows.

Table 3 Effect of V0on energy distribution of rough a(f )

Fig.14 A realistic impact acceleration (a) detected under the effect of BP. t=0 is the moment of contacting water,VI=4m/s

During the image processing, different parts on the perimeter of the sphere are measured as discussed in details in Section 1.2, the trajectory of the sphere is obtained, and through differencing and filtering operations, af(t ) could be calculated. Following the course in the above two sections, the parameters of the zero phase low-pass Butterworth filter are set as: the low-pass band cutoff frequency is 400 Hz, the stopband cutoff frequency is 600 Hz with the maximum attenuation of 0.5 dB in the passband and the minimum attenuation of 10 dB in the stopband.

Furthermore, since the duration of the impact theoretically is somehow uncertain, in a rough range (b＜0.5), we checked the a data with extra more points after the impact. The number of the extra points is denoted by BP. Figures 14(a) and 14(b) show that the curves converge well when BP≥4, andBP=4 is chosen as the working parameter in the present study.

2.4 The applications of the data processing

With various testing models, plenty of data and predictions for sphere impacting are available in literature. For spheres, a series of water entry experiments with various impact speeds ranging from 3.1 m/s-5.0 m/s are shown in Fig.15, with a similar pattern. At least 3 runs were carried out for each impact velocity. In Fig.15, the measured values are obtained according to Eqs.(4) and (5), and the theoretical ones are obtained according to Eqs.(8) and (9), respectively. Meanwhile, previous experimental data[4]and computational results by Fluent, are plotted in the figure.

Fig.15 Coefficient of drag measured by the proposed method, and several other results

The diagram reveals that the trend of Cdmeasured by the proposed method agrees with the theoretical model generally. At the initial stage of the impact (t→0), the measured impact force is non-zero. The reason could be that the exact contacting time is misread due to the limited capturing speed, besides, the high frequency components are eliminated with the low pass filter resulting in a gradual change in Cd. Itmay be expected that the elimination of high frequency components will cause a small discrepancies of Cdwhen b=0-0.1 as compared to other results. In the region b=0.1-0.25, where the peak impact force occurs, the resulting Cdseems to be closer to that in the Miloh?s model (the “×” marked line in Fig.15) and the Moghisi?s experimental result (the “+”marked line in Fig.15). Especially, the Miloh model takes the free surface deformation into consideration, while the Lamb's model ignores the fact. Therefore, it can be said that the Miloh?s prediction is more suitable for the hydrophilic projectile. It is also demonstrated that although we have left out the high frequency components of Cd, the major feature of the water entry is retained. In the region of b =0.3-0.5, the present data agree with the CFD and the Miloh?s predictions qualitatively.

As a summary, the drag coefficient Cdmeasured by the proposed subpixel image processing and filtering method agrees generally with the previous results from various published sources.

3. Conclusions

The paper proposes a combined method to determine the acceleration of a moving object. A four-step data procedure is applied: (1) the image capture, (2) the sphere displacement is determined by a subpixel method, (3) the velocity and the preliminary acceleration are calculated with the finite-difference method, (4) the preliminary acceleration is filtered to obtain the final result. Both the subpixel processing and the low pass filter are crucial for obtaining a reasonable acceleration result. The method is verified with the forced sinusoidal motion tests.

It is further applied to the impact measurement of the water entry of a sphere. According to the theoretical predictions and experimental results, the water entry could be regarded as a process with a slowly changing velocity. After adopting a carefully designed low pass filter, the measured Cdagrees well with the previous results qualitatively, albeit the high frequency components are omitted. The low pass filter for measuring the acceleration of the water entry according to the theoretical predictions is suggested with the low pass band cutoff frequency of 400 Hz, the stop band cutoff frequency of 600 Hz, the maximum attenuation of 0.5 dB in the pass band and the minimum attenuation of 10 dB in the stop band.

A further reflection shows that a non-contacting measuring will be more practical and efficient for complex water entry procedures, such as for spinning objects, non-vertical entry etc.. In these cases, we should pay more attention to the influences of the installed cable and the data gathering, if sensor-based systems are applied. One might choose a method of offline data gathering and an analysis system. However, that would require a large model to accommodate the power supplier, the sensors and the additional memory, etc.. In addition, according to what we know, this system will be expensive. With the image-based data procedure, it can be hoped that more kinematic information can be obtained besides the free surface evolution, at the same time, including velocity, acceleration and rotating speed. Meanwhile, the PIV measurement can be conducted almost without any additional equipment. Our future work will include to apply this method to more complex cases of water entry, to increase the precision by tracing more marker points on the object, and to conduct more comparison tests to check the accuracy of the present methods.

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10.1016/S1001-6058(11)60293-X

* Project supported by the National Natural Science Foundation of China (Grant No. 11172241), the Innovation Foundation of Aerospace Science and Technology of China and the National High Technology Research and Development Programs of China (863 Program, Grant No. 2012AA011803).

Biography: ZHAO Meng-hua (1988-), Male,

Master Candidate

CHEN Xiao-peng,

E-mail: xchen76@nwpu.edu.cn

DOI: 10.1016/S1001-6058(11)60294-1