1. Introduction

The flow behind a gap between two side-by-side bluff bodies has long been studied by Bearman and Wadcock[1], Roshko et al.[2], Zdravkovich[3], Hayashi et al.[4], Kim and Durbin[5] and Miau et al.[6]. If the gap is very small, the flow after the gap demonstrates a bi-stable characteristic. In other words, the flow will stably deflect to either side of the bluff bodies. As the gap increases, the flow goes through a transition from the bi-stable to a uni-stable characteristic. In other words, the steady flow will not deflect to either side of the bluff bodies if the gap is wide enough or, more precisely, the uni-stable characteristic is attained. In the transition, the flow was found flip-flopping between the two bluff bodies in a random manner[5-6].

In general, the studies involving this subject were carried out on either cylindrical shapes or flat plates. As one of the pioneering work, Bearman and Wadcock[1] experimentally measured the pressures on two side-by-side circular cylinders at a Reynolds number of 2.5 x 10⁴. Later, Roshko et al.[2] also carried out a similar experiment by measuring the pressure distribution on two triangular cylinders but at a Reynolds number of approximately 2.0 x 10⁴. Both Bearman and Wadcock[1] and Roshko et al.[2] confirmed that the pressure measurements as well as drag coefficients fluctuate between two extreme values when the gap ratio is between 0.1 and 1.0. The gap ratio in these cases was defined as the gap width between the cylinders to the diameter of the cylinders. Years later, Kim and Durbin[5] showed the possibility of controlling the intensity of theflip-flopping flow at a Reynolds number of 3.5 x 10³ and a gap ratio of 0.75 using a splitter plate along the centreline of the gap or acoustic excitations. For flow through the gap between two side-by-side flat plates, Hayashi et al.[4] have found solid evidence, based on their studies corresponding to Reynolds numbers ranging from 0.6 x 10⁴ to 1.9 x 10⁴ and a gap ratio of 1.75, showing that the formation of flip-flopping flow is caused by the formation of shedding vortices trailing the flat plates. For the cases involving flat plate, the definition of gap ratio is similar to the above except replacing the cylinder diameter with the plate thickness. Along the same direction, Miau et al.[6] considered the cases whose gap ratios vary from 1.50 to 1.85 and found that the frequency of flip-flopping flow is proportional to the gap ratio.

The problem of fluid seepage through the cracked walls of pressure vessels has been examined quite extensively focusing on the effect of pressure on flow rate[7-10] and the propagation of crack[11-14], but the flow field associated to the opening has not been reported.

On the other hand, Wang et al.[15] numerically investigated the patterns of leakage flow through two labyrinth seals. They found that the turbulent kinetic energy reached its maximum at the clearance of the seal. The static pressure dropped rapidly when the fluid flows through the clearance.

In the extensive studies mentioned above[1-6], the configuration considered was confined to two bluff bodies placed side-by-side. A stream of flow with a known velocity was forced through the gap in between. With this, the flow phenomena related to Reynolds number and gap ratio were examined closely. Although the possible presence of a flip-flopping flow had made this topic interesting, the authors think that these configurations considered can be extended to cover more engineering applications. In this study, not only does the thickness of the side-by-side flat plates infinitely large but also the length of the gap is considered. In practice, this can be related to a solid wall of a container having a gap though it. If the length of the gap is small, this study simulates the water leakage of a thin wall vessel through a small crack. On the other hand, if the length of the gap is large, this study then simulates the seepage effect associated to a wall with remarkable thickness. For these reasons, it is the pressure difference in this study that drives the flow through the gap. In contrast to the previous studies, a Reynolds number associated to a particular case is unknown. Instead, Reynolds numbers defined based on the maximum velocity within the gap will be estimated for various pressure differences.

2. Formulation

Figure 1 shows the schematic of the problem. The rectangles connecting to the semicircles on both sides are the solid walls. The gap or opening between these walls is denoted by H. The thickness of the wall (i.e., the length of the gap) is represented by L. The height of these walls is fixed set to be H. Non-slip boundary condition is applied at these walls whereas a prescribed value for pressure is applied along the semicircles. The diameter (= 2R) of these semicircles is arbitrarily chosen as long as their presence will not force unrealistic flow patterns close to the gap. With the given height of the walls and gap width, the diameter of these semicircles is 3H. The only aspect ratio investigated in this study is the one related to the gap or opening which is defined as H/L. In this study, this aspect ratio ranges from 0.2, 0.5, 1.0, 2.0, 5.0, to 10.0.

Figure 1. Schematics of the problem

To focus on the flow phenomenon entering and leaving the gap, an incompressible steady-state laminar flow is considered here for simplicity. In near future, these assumptions will be removed so that this study will consider other influences. The conservation equations in the vector form being solved simultaneously are

(1)

(2)

where  is the water density, U is the velocity vector, p is the pressure, and τ is the stress tensor. By relating the stress tensor to the rate of strain of a Newtonian fluid, equation (2) can be expressed in the following form

(3)

By introducing the following dimensionless parameters,

(4)

(5)

Equations (1)-(3) can be expressed in the following forms

(6)

(7)

Clearly, the solutions of this problem depend heavily on an additional dimensionless parameter in the momentum equations. It is hereafter referred to as the modified Euler number defined as follows

(8)

In general, the Euler number is defined as the ratio of force due to the pressure difference to the inertial force. Since the velocity scale involved in the definition of the conventional Euler number is unknown at a priory in this study, the definition of modified Euler number will no doubt simplifies the formulation of this problem.

In addition to the modified Euler number, another parameter related to the configuration of this problem is investigated. This parameter is the aspect ratio of the opening, defined as H/L. Although this parameter does not appear explicitly in the governing equations, it has a remarkable influence on the implementation of the boundary conditions in the course of numerical solution.

Although eqns. (6)-(7) can be easily discretized in either the Cartesian or the cylindrical coordinates, this work did not attempt to do so because the configuration considered in this work involves both the coordinate systems. For this reason, not only are two sets of governing equations needed, but also the velocity and pressure must satisfy the conservation laws on the interface between the Cartesian and cylindrical coordinates. These additional requirements will surely complicate the method of solution. One remedy is to make use of a relatively simple body-fit coordinate system but the grid is not expected to yield good numerical results at the regions close to the four corners of the walls because of the skewness of the mesh at these locations. With the above concerns in mind, the current problem is solved using a commercial package FLUENT. Through GAMBIT, a carefully design mesh can be constructed much more easily and effectively. Once the grid system was ready, it was imported into the FLUENT solver. Depending on the Euler numbers and gap aspect ratios, a typical computational time for a case can range from 5 minutes to several hours. The mesh distribution used for current study is shown in Fig. 2. A grid refinement test has been performed and it was found that the density of mesh varies from case to case. As shown in Figs. 2(a) and (b), the numbers of elements are about 2100. However, for H/L = 10.0, the number of elements is more than 70,000. It is important to point out that further refinement of mesh did not improve the results much; instead it tremendously increased the computational time.

Using FLUENT, assigning boundary conditions is easy for this case because non-slip boundary conditions are applied at the surfaces of the solid walls and constant pressures are prescribed on the circular boundaries. Since it is the pressure difference that drives the flow, the gage pressure on the right circular boundary is conveniently taken as zero whereas the one on the left circular boundary varies according to the modified Euler number.

Figure 2. Distribution of grids for various H/L

3. Results and Discussions

The study of a problem involving very fundamental geometries yet significant in terms of engineering applications has been investigated using FLUENT. Since a problem of this fashion has never been investigated before in terms of pressure gradient, it is the responsibility of the present study to examine the feasibility of characterizing this problem using the modified Euler number defined in the previous section. Conventionally, Euler number represents the relative importance between the pressure effect and the inertia effect. In fact, not only is the characteristic velocity scale related to this problem not known at a priory, but also a representative velocity scale is simply not required for problems associated to leakage of highly pressurized tanks. For this reason, a more suitable parameter is proposed in this work. The modified Euler number represents the relative importance between the pressure and viscous effect. A reasonable range of modified Euler number corresponding to practical situations should fall within the order of 10⁸ to 10¹². A modified Euler number of 10⁸ can represent water leakage into atmosphere from a tank of 1.3 atm. through a hole of 1 mm. On the other hand, the leakage of a dense fluid from a container is associated a typical modified Euler number in the order of 10¹². For instance, water in a tank, whose inner pressure is about 14 atm., leaks into water through a crack of 1 cm wide produces a modified Euler number in the order of 10¹¹. Certainly, as the pressure in the tank or container decreases with time, the corresponding modified Euler number reduces accordingly. Since this work is intended to investigate the steady flow phenomena related to high pressure gradient, current work does not take into consideration of situations of that extreme.

Before studying the effects of modified Euler number on the flow patterns, it is perhaps the most important to examine the feasibility and reliability of the modified Euler number and also to ensure its independence of dimensional scale. Figure 3 presents the evidence proving that the approach used in this study is feasible. In Fig. 3(a), H = 0.1 cm whereas in Fig. 3(b) H = 1.0 cm. In order to come out with the same modified Euler number, the pressure in Fig. 3(a) is 100 times greater than that applied in Fig. 3(b). For ease of comparison, the parameters assigned for each case are listed in Table 1. Apparently, the flow patterns for a particular modified Euler number are almost the same regardless of the actual dimensional length scale involved in simulations.

In the simulations performed, steady state solutions were assumed attained if the relative errors for two consecutive iterations were less than 10-9. The flow field of most cases skewed either upward or downward even though the geometry is symmetrical. It is remarkable to point out that the gravitational effect is excluded in the calculation. The presence of bi-stable solutions discovered in this work is consistent with what has been reported[1-2]. Hence, the simplification of geometry by assuming an asymmetric boundary condition will lead to incorrect flow distributions for it will always produce a non-biased solution. Obviously, the plausibility of using such an artificial boundary condition requires much more serious thought and justification.

Table 1. Parameter settings for the cases shown in Fig. 3 Case (a) Case (b) Gap, H (cm) 1.0 10.0 Pressure difference (Pa) 1253.1456 12.531456 Density (kg/m3) 999 999 Viscosity (kg/m,s) 1.12x10-3 1.12x10-3 Modified Euler number 1.0x108 1.0x108

Figure 3. Distribution of grids for various H/L

Figure 4 shows the distributions of stream function for cases with the same modified Euler number equals to 1010 but different gap aspect ratios. When the gap aspect ratio is small, it means that the gap is very long, as depicted in Figs. 4(a) and (b). If the aspect ratio is small enough, the gap flow is in the uni-stable mode. In this case, there exist two recirculation cells developed close to the top and bottom inner corners of the walls. Also notice is that flow entrainment takes place on both side of the flow downstream. In general, the width of the jet remains almost the same throughout its coarse starting from the vicinity of gap inlet. As the gap aspect ratio increases, the bi-stable mode kicks in. At a gap aspect ratio of unity (Fig. 4(c)), the stream of flow randomly tilts towards one of the walls and as a result one of the recirculation cells disappears. At the spot where recirculation cell develops, the flow separates due to an abrupt change in flow direction. If the wall thickness is reduced by half, the gap flow tends to return to the uni-stable mode as can be seen in Fig. 4(d). Close observation proves that the gap flow for H/L = 2.0 still demonstrates a very ambiguous bi-stable nature. In this case, the presence of recirculation cell is no longer preserved. On the other hand, the tail of the gap flow seems to expand. If the gap aspect ratio is further increased to 10.0, we observe from Fig. 4(e) two strong streams of entrained water exist next to the walls. Not only that, the retrained water is so strong that it forms two recirculation cells of finite strength. It is interesting to discover that the velocity of the gap flow increases until a location where the flow begins to expand and divert. As the flow diverts, it slows down.

Figure 4. Distributions of stream function for Eu* = 1010 with various gap aspect ratios

In the study of gap flow after two bluff bodies, it was believed that the generation of vortex shedding after the bluff bodies cause the presence of bi-stable flow patterns[16]. In this study, it is believed that the vortex shedding (or wake) starts its development at the sharp corners of the inner walls where the flow separates. The growing of this vortex shedding contributes to the bi-stable characteristics of gap flow. As in Fig. 4(a), when the gap is long enough, the gap serves as a guide stabilizing the flow stream and damps out the vortices generated at the corners. If the gap is not long enough, one side of the vortices propagates downstream and forms a vortex street pushing the main water stream tilted to the opposite wall. If the gap is very short, the vortices developed at the corner leave the gap without any significant loss due to friction. Although this explains the phenomena discovered in Fig. 4, it awaits further investigation and justification.

Figure 5 illustrates the presence of bi-stable characteristics associated to the gap flow. To help better understand these figures, the values of the maximum and increment of the stream function are listed in Table 2. There are twenty contour lines in each figure. Based on Table 2, it is obvious that the magnitude of the mass flow rate increases by an order whenever the order of magnitude of modified Euler number increases by two.

Among the modified Euler numbers investigated in this study, all cases demonstrate a bi-stable mode except the case whose modified Euler number is 10¹¹. For the cases with bi-stable mode, the distributions of streamlines are almost identical regardless of the side the gap flow tilts. It is surprising to discover that a uni-stable mode is possible within the bi-stable regime. At this mode, it is found that the stream of the gap flow is compressed as it enters the gap. As it exits the gap, the flow expands gradually.

For the paragraph that follows, we will discuss the characteristics of the gap flow in terms of the maximum velocity found in the computational domain. The location of the maximum velocity can be clearly identified at the regime where the streamlines are the most compact. Clearly from Figs. 4 and 5, the location of the maximum velocity is not always located within the gap.

Table 2. Stream function values in Fig. 5 Eu* Ψmax ΔΨ 108 0.22 0.011 109 0.693 0.0347 1010 2.19 0.11 1011 6.9 0.345 1012 21.9 1.1

Figure 5. Distribution of stream functions with H/L = 1.0 but associated to different Eu*

Figure 6. Variation of maximum velocity with gap aspect ratio for different modified Euler numbers

Figure 7. Variation of maximum velocity with modified Euler number for different aspect ratios

As shown in Figure 6, it is clear that the dependence of maximum velocity on the gap aspect ratio can be separated into two distinctive regimes. For H/L ≦ 1.0, the maximum velocity is seemed to be only a linear function of the modified Euler number in the log-log scale. In the other regime where H/L ≧ 1.0, the maximum velocity is apparently a function of both the modified Euler number and the gap aspect ratio. Hence, it is reasonable to claim that there exists a threshold gap aspect ratio which approximately equals to 1.0. Based on our observations, it believe that the increase in maximum velocity for H/L ≧ 1.0 can be reflected through the location of the maximum velocity although a perfect explanation to this phenomena is still under seeking. For the cases we have studied, it follows the trend that if the maximum velocity lies within the gap then the magnitude of the maximum velocity is merely a function modified Euler number. As the gap aspect ratio increases beyond unity, the location of maximum velocity begins to shift outside of the gap and accompanying this phenomenon is the increase in maximum velocity.

Figure 7 shows the variation of maximum velocity in terms of the modified Euler number and gap aspect ratio. Notice that the modified Euler number is plotted in logarithmic scale. In the plot, the corresponding Reynolds numbers are also presented for convenience. The Reynolds number is defined based on the maximum velocity. As shown in the figure, a maximum velocity of 150 m/s is equivalent of approximately Re = 100,000. The maximum velocity collapses into a single line when the gap aspect ratio is less than or equal to unity. This indicates that for a given modified Euler number, the maximum velocity remains the same as long as the length of the gap (L) is greater than the width of the gap (H). Once the length of the gap is less than the width, the plot begins to deviate from that associated to H/L = 1.0. This is in agreement with the independence of maximum velocity on gap aspect ratio as depicted in Figure 6.

4. Conclusions

A numerical simulation of a pressurized fluid flowing through a gap has been successfully performed using FLUENT. It was found that the modified Euler number, defined as the ratio of the force due to pressure difference to the viscous force, is good enough to capture the characteristics of this problem. The flow patterns are almost identical for problems having the same modified Euler number but different dimensions. Also found is that there exists a threshold aspect ratio beyond which the maximum velocity occurred within the flow increases with the gap aspect ratio. If below the threshold aspect ratio, the maximum velocity is independent of the aspect ratio. Based on the present study, the threshold aspect ratio is expected to be approximately unity. Furthermore, the maximum velocity is proportional to the modified Euler number. The proportional constant is found to be a function of the gap aspect ratio if the aspect ratio is greater than the threshold value. Below the threshold aspect ratio, the maximum velocity is merely a function of the modified Euler number. Although this work has only investigated the flow through a gap due to a pressure difference in the laminar regime, it provides result interesting enough for further attentions.