Velocity-Arc Length Profiles without Collision

The velocity magnitude of the target sphere in Fig. 4.10 is too small when compared to that of the impact sphere. Therefore, an enlarged -v s_{t t} diagram of it is displayed in Fig. 4.12. The

fluctuation observed here is speculated to result primarily from the target slow motion that does not advance sufficient distance in consecutive images to suppress the PTV error discussed in Eqn. (3.4).

. Also, due to the closeness of each curve, it is not easy to distinguish them from each other.

Hence, the curve obtained for each wall spacing is shown separately in Fig. 4.13 (a)—(h) in the order of displaying the trend of conversion along with an increase in wall spacing.

Fig. 4.12: Velocity history of the target pendulum across different wall spacings for  = 254cP.

returning swing. Since the path of return was not investigated in this study, observations will only be focused on the forward swing, which can be summarized below:

(1). As shown in Fig. 4.13 (a), the maximum forward-moving velocity seems to occur at the

“mid-point” of its forward trajectory from 19 to 21.4 mm for W/D = 1.2. In other words, the forward swing curve seems to be symmetrical about where the maximum occurs. As the wall spacing increases, the forward swing curve becomes more and more asymmetrical. The target sphere now travels a larger portion of the forward trajectory before decelerating, as shown in Fig. 4.13 (h). It is obvious that the resistance prevails the impulse when the maximum velocity occurs in the forward swing. The symmetry about the mid-point in Fig.

4.13 (a) indicates that the “average net impulse” in the pre-maximum section is equal in magnitude to the “average net resistance” in the post-maximum section so that they require the same length of traveling distances for doing opposite work to balance each other. One of the components of the resistance is the hydrodynamic resistance, which helps to prevail the impulse at a prior location. As the wall spacing increases, the hydrodynamic resistance is less augmented by the wall effect and thus must wait for the gravitational resistance to also gain enough significance to collectively prevail the impulse, so the forward swing trajectory in Fig. 4.13 (h) appears asymmetrical.

(2). The v s_t- _t profile in Fig. 4.13 (h) presents one more particular feature. A large section of the

pre-maximum segment seems linear, extending from about 19mm to about 21mm. The linearity signifies a constant rate of increase in velocity with respect to the arc-length position. This linear correlation gradually degrades with a decrease in the wall spacing. In Fig. 4.13 (a) with W/D =1.2, the forward swing curve is entirely convex.

(3). As the wall spacing increases, the maximum velocity developed by the target increases, as shown in Fig. 4.12 and 4.13. This phenomenon may be attributed to two main causes. One is the better energy conversion from the impact sphere of greater momentum, and the other is the less energy dissipation due to increased wall spacing. A preliminary study has been conducted to evaluate which cause has the more contribution in the following.

(a) (b)

(c) (d)

Fig. 4.13: Velocity histories of the target pendulum displayed one by one for each wall spacing. (a) W/D = 1.2. (b) W/D = 1.4. (c) W/D = 1.7. (d) W/D = 2. (e) W/D = 2.4. (f) W/D = 3. (g) W/D = 4. (h) W/D = 5.

As stated in the observation (3) above, an attempt to distinguish the influences of the lateral walls from that of the impact sphere is briefed here. Firstly, we tried various flow conditions to achieve nearly identical Re_m’s under W/D = 1.2and 5 as shown in Fig. 4.14. This task was accomplished by using _o = 15^o for W/D = 1.2 and _o = 11.75^o for W/D = 5.0, as indicated by the corresponding curves in Fig. 4.14. Since these two curves have nearly identical Re_m and the

wall effect with W/D = 5 has been evaluated to be negligible (as in an infinite fluid domain), we may infer that the deviation in the corresponding target sphere motions is due primarily to the maximal decrease in the wall effect.

(e) (f)

(g) (h)

Fig. 4.14: Velocity histories of pendulums across various release angles and wall spacings assigned to separate the wall effect from the energy transfer supplied by the impact pendulum.

A deviation in the curves developed by the target sphere is quantified by the difference between the intercept with the st - axis, namely where vt = 0, for each curve. In Fig. 4.15, the curve measured for W/D = 1.2D, _o = 15^o (the red line) intercepts vt = 0 at st  21.75mm; the intercept for that using W/D = 5, _o= 11.75^o (the magenta line) is about 22.8mm. The two intercepts differ

by 1.05 mm, which signifies the deviation in response to the maximal decrease in the wall effect.

Also, the intercept for the curve under W/D = 5, _o = 15^o (the cyan line) is as high as 23mm.

Because the experiments for W/D = 5, _o = 15^o and those for W/D = 1.2, _o = 15^o were

performed under the same angle of release but two extreme wall spacings, the deviation between their st-axis intercepts, namely 1.25mm, signifies the deviation in responses to both the maximal increase in the impact sphere momentum and the maximal decrease in the wall effect. Finally, the

by subtracting the wall-effect type from the mixed type, which results in a deviation of 0.2mm.

Thus, it may suggest that the variation in the responses of the target sphere is due largely to the variation in wall spacing in a fluid of high viscosity. The influence from the momentum variation of the impact sphere seems minor.

Fig. 4.15: Enlarged velocity histories of the target pendulum. The deviation between the red curve and the magenta curve is due to a maximal decrease in the wall effect, the deviation between the magenta curve and the cyan curve is due to a maximal increase in the impact sphere momentum, and that between the red curve and the cyan curve is due to a mix of the above.

4.3.3: Comparison with Respect to the Separation

In order to explore more about the dual-pendulum motion without collision, a new parameter called the arc-length distance of separation is now defined as

 

ti t i t i

s  s  R R s (4.19)

From the configuration shown in Fig.4.1, Ri + Rt is the horizontal distance between the two pendulum centers when in touch at their lowest positions. Note that st – si approximates the arc-length distance between the two centers and its minimum is (Ri + Rt). After Ri and Rt are subtracted, sti can be viewed as an approximate distance of the interstitial gap between the two sphere surfaces. This approximation is sensible on the fact that the string length L (536mm) greatly exceeds the sphere diameter (D=19mm, 3.5% of L). By definition, sti is never negative and is 0 when the two spheres are in contact.

Although sti is an approximate interstitial distance between the two spheres, it still helps to reveal some new features of this interaction. The data in Fig. 4.10 ( = 254, _o = 15^o) is re-plotted in the -v s_ti diagrams in Fig. 4.16 and 4.17 (a), whose curves start from the release at sti

= 140mm to the final contact at sti = 0mm. In particular, the curves for the target sphere are shown separately for the sake of clarity in Fig 4.17 (b), in which only the cases for W/D = 1.2 and 5 are presented and the color of the curve of W/D = 5 has been changed to blue. It is worth commenting that the fluctuations observed here is speculated to result primarily from the target slow motion that does not advance sufficient distance in consecutive images to suppress the PTV error discussed in Eqn. (3.4).

Some observations can be drawn:

(1). The target sphere is set in motion before real contact occurs and this is observed in all the examined wall spacing. In fact, this early onset of motion is present even in all the situations where a collision does occur.

(2). With an increase in wall spacing, the onset of target sphere motion is gradually delayed. The momentum of this system comes from the momentum of the impact sphere, and a transfer of momentum between the spheres is accomplished by the fluid in between. Under the smaller wall spacing, the interstitial fluid is greatly confined in the direction normal to the walls and thus, propelled by the impact sphere, it is more likely to be directed towards the target sphere. Hence, the target sphere would be set in motion at an earlier stage.

(3). With an increase in wall spacing, the vt - sti profile gradually changes from largely convex to largely concave. Although the target sphere receives the transferred momentum at an earlier stage under the smaller wall spacing, the impact sphere momentum is at the same time severely diminished by the augmented wall effect. When the two spheres later get closer to each other, the supplied momentum is insufficient to counter the resistance imparted on the target sphere. Therefore, the target sphere then begins decelerating and results in a convex vt

- sti profile. Contrastingly, under the larger wall spacing, more momentum is retained when the two spheres reach the vicinity of each other, and the efficiency of momentum transfer gets higher due to the shortened separation. Therefore, the already rising target sphere velocity is then further raised by the increasing impulse and the concavity in the profile

results.

(4). The target sphere with a delayed onset of motion under larger wall spacing achieves a higher positive maximum velocity.

(5). Note that the nearly vertical segment around sti = 0 for the W/D = 5 case in both Fig. 4.16 and 4.17 indicates a phased motion, meaning that the two spheres moved forward with a constant separation; while in this case, they are in contact. There might be a small nonzero interstitial gap but the current measuring technique cannot fully resolve it. Such in-touch phased motion is not observed under smaller wall spacing. This is due to the fact that, for larger wall spacing, the impact sphere has a higher chance to catch up with the target sphere before it has stopped its forward swing.

Fig. 4.16: Velocity-separation profiles of the impact pendulum for  = 254cP.

Fig. 4.17: Velocity-separation profiles of the target pendulum for  = 254cP. (a) The cases for all kinds of wall spacing. (b) The cases for W/D = 1.2 and 5.

4.3.4: Methodology in Deciding Coupling Distance

Following the previous observation, we would like to quantify how the two fully immersed

spheres interact with each other without physical contact. We propose to use the coupling distance,

tic

s , to signify at what interstitial distance that the target sphere is set into motion. .

Ideally, we can directly search the vt – sti diagram for the point where vt is first elevated from vt

= 0 to determine the coupling distance. However, the severe fluctuations observed in Fig. 4.17 (a) and (b) invalidate this simple approach. Instead, we employed changes in the location of the target sphere, st, from its original status to determine the coupling distance, which bypassed the problematic time differentiation of this slow motion. The st data obtained from the circular Hough

slight fluctuation at the beginning of the process persists as boxed, which closely correlates with the results of the error analysis presented in Section 3.2.3.

Fig. 4.18: Example of the position-separation profile of the target sphere for detecting the onset of its motion.

As discussed in Section 3.2.3 and Fig. 3.15, the most probable fluctuation of the error, Δ, was evaluated to be 0.2 pixels around some center of fluctuation. Its corresponding physical scale can be easily determined by multiplying it with the physical-to-image conversion ratio, D/d. Next, we want to determine the center of fluctuations. An evaluation scheme that adopts varying number of data points are now proposed. In practice, we consider the first N data points of the position (st) time sequence to compute its mean and standard deviation (STD). By increasing N, we are averaging over a greater extent of position data. For a ‘true’ static target motion, increase of N would help to average out the spurious fluctuation from image processing. Thus, a local minimum of STD would

displacement in st from the center of fluctuation never exceeds Δ in its physical scale, this

displacement is considered illusory due to fluctuation. The very target location that exceeds this threshold is employed to determine the desired coupling distance by the corresponding sti.

Needless to say, a reasonable mean of fluctuation plays a crucial role in the success of this methodology, which requires a more detailed discussion in the following. Firstly, the STD results obtained for the data shown in Fig. 4.18 using increasing N, or equivalently decreasing sti, are shown in Fig 4.19. The trend for STD is reasonable. Initially, the STD starts from 0 as N = 1 and it jumps to a peak and gradually steps down. This overshooting suggests that the mean is highly sensitive to data fluctuation when the sample size is low. With more samples under consideration, the STD becomes more stable and reaches a local minimum indicated by the arrow. It is this sample size that is thought to give the most reasonable mean. Apart from this, the following trend keeps quite stable for a long range. If we ever use a point in this range to determine the data extend of average, the result should be acceptable as well. Note that a local minimum does not always occur so early and so discernably but if it does occur, it is observed to always reside in the ‘stable’ range stated above.

Fig. 4.19: Example of the trend of STD used to determine the center of fluctuation. The very head section of this trend is enlarged in the rectangle. The data point indicated by the arrow is a local minimum, and its index is thought to be the data extent that gives the most reasonable mean.

After a mean of fluctuation at the beginning of the -s s_{t ti} profile is determined as the blue

horizontal line added to Fig. 4.20, the estimated error,  , is superimposed after unit conversion as the red line. The intersection of this elevated line with the profile should represent the onset of target motion with the uncertainty in locating the sphere taken into account. The corresponding interstitial distance, sti, of the intersection thus determines the coupling distance, s_ti^c.

Fig. 4.20: Center of fluctuation and the threshold imposed onto the trend of STD. The st-intercept of the horizontal blue line is the center of fluctuation, and the threshold (red line) is elevated from the blue line by an amount of Δ in its physical scale.

However, when the region about the intersection is enlarged and examined in Fig. 4.21, eight

cover a range of sti for 5mm and it is difficult to exactly determine the point of intersection when interfered by this fluctuation. Thus, we now set out to evaluate the data trend in a statistical manner.

In Fig. 4.21, we firstly select point A from which all the preceding points are below the elevated threshold, Similarly, point B is selected from which all the following points are above the threshold. All the points with their sti falling between those for A and B are chosen as a basic set (as boxed by the rectangle) for fitting a linear -s s_{t ti} line through them. In doing so, N-paired

downstream and upstream data points are included stepwisely for alternative linear fits. For example, with N = 1, the point A1 and B1 are included and then we continue for N = 2 to further include the points A2 and B2. Thus a sequence of slopes for the line fitted recursively to the expanded data points is generated. The corresponding sequence of STD is also computed. The same procedure introduced above in determining the mean of fluctuation is again applied here to determine the most feasible set of data points which gives the best linear fit. In some situations, a proper N can never be found and 5-paired downstream and upstream data points will be automatically appended to the basic set for a linear fit. The so-called best-fitted line then gives a specific intersection with the elevated threshold line, and the desired coupling distance is given by the sti - coordinate of the intersection.

Fig. 4.21: Determining the intersection of the position-separation profile (data points) and the threshold (horizontal line) in a statistical manner. The boxed data points are taken as a basic set for fitting a linear line through them.

4.3.5: Coupling Distance

As defined in Eqn. (4.19), Section 4.3.3, sti is not exactly the arc-length interstitial distance between the two spheres. A more precise measure is preferred in marking the coupling distance. We consider that the motion of the target sphere is expected to result primarily from the pressure field generated by the incoming pendulum motion and it takes place at the lowest position of the pendulum. Since the local motion is temporarily horizontal, a horizontal separation, xti, is adopted and is evaluated by the horizontal distance from the frontal face of the impact sphere to a vertical base plane tangent to the target sphere surface on the side facing the impact sphere. The schematic illustration is shown in Fig. 4.22. An x-y coordinate system is defined to the same origin, Oi, for the impact sphere. From Fig. 4.22, the following relation can be verified:

ti i

x   x (4.20)

direction of the x-axis is from the upstream to the downstream.

Fig. 4.22: Coupling distance and the rectangular coordinate system imposed. The origin of the coordinate system is again set at the lowest point of the impact pendulum.

Therefore, the coupling distance can be redefined as the value of xti when the onset of motion of the target sphere is detected by the same methodology presented above, with only the horizontal sti-axis of Fig. 4.17 – 4.21 being replaced by xti. The coupling distance will be denoted as x_ti^c and is represented as a function of the wall spacing in Fig. 4.23. In Fig. 4.23, different symbols indicate different liquid viscosity and the different line colors distinguish the release angles. There are three to six experiments conducted for a certain wall spacing, so the average of their coupling distance is employed to represent the general behavior shown for each wall spacing in Fig. 4.23 (a). If the median is employed, similar trend is obtained as shown in Fig. 4.23 (b). Some general phenomena are observed:

(1). As the wall spacing increases, the coupling distance gets smaller. This means that the impact sphere has negligible influence on the target sphere until it reaches to the close vicinity of the target. The decrease in x_ti^c may be attributed to weaker wall confining effect under

preserve mass conservation. Smaller impulse is thus imparted on the target sphere and hence a smaller x_ti^c is required to accumulate sufficient impulse for the target sphere to take off.

Contrastingly, greater impact sphere momentum is developed under greater wall spacing, which may introduce to the system higher impulse. However, this enhancement seems not as effective as that resulted from reduction of mass flux due to wall confinement. It requires more rigorous investigation to fully understand the interplay between the two mechanisms.

(2). The trend of how x_ti^c monotonically increases with diminishing W/D has slightly different format. For W/D >3, fairly linear relationship is observed nut more pronounced increase

occurs when W/D falls between 1.7 and 3.0. It is very interesting to observe that the rise of

tic

x seems slowed down for even lower W/D = 1.4 and 1.2. This later peculiar phenomenon may result from experimental errors or be attributed to some hidden interaction mechanism that requires more careful investigation.

(3). For a certain angle of release, as the viscosity decreases, the overall trend of the curve is lowered. That is, for each wall spacing, a lower viscosity results in a smaller coupling

(3). For a certain angle of release, as the viscosity decreases, the overall trend of the curve is lowered. That is, for each wall spacing, a lower viscosity results in a smaller coupling