Review: Prediction of Unexpected Fluid-Induced Vibration in Pipeline Network

Ryuhei Yamaguchi^*, Makoto Ohta

Institute of Fluid Science, Tohoku University, Sendai, Japan.

DOI: 10.4236/wjm.2022.122002                

Abstract

This review considers unexpected destructive disasters involving fluid power plants, such as nuclear electric power plants and fluid power plants. It specifically addresses the possibility of fluid vibration induced in a pipeline network of such a plant. The authors investigate the flow oscillation induced within a T-junction for laminar steady flow at a Reynolds number less than 10³ and clarify that there is a periodic fluid oscillation with a constant Strouhal number independent of several flow conditions. Generally, a nuclear electric power plant is constructed using straight pipes, elbows, and T-junctions. Indeed, a T-Junction is a basic fluid element of a pipeline network. The flow in a fluid power plant is turbulent. There are peculiar flow phenomena that occur at high Reynolds numbers, which are also seen in other flow situations; e.g., Kaman vortices are observed around a circular cylinder in low Reynolds numbers, around structures like bridges and downstream of islands in oceans. Although the flow situation of a T-junction and elbow in a fluid power plant, such as the fluid suddenly changing its flow direction is turbulent flow, the authors mention the possibility of the fluid-induced vibration of a pipeline network.

Keywords

Fluid-Induced Vibration, Flow Oscillation, Tube Flow, Laminar Steady Flow, Fluid Power Plant

1. Introduction

The Karman vortex was first reported by von Kármán (1911, 1912) [1] [2] and Benard (1908) [3] and. The collapse of a bridge spanning the Takoma Narrows (1940) was a typical destructive incident resulting from a Karman vortex [4]. Meanwhile, a thermocouple in the Monju fast breeder reactor was destroyed by fluid-induced vibration and a fine tube fractured in the steam evaporator of the second unit of the Mihama Nuclear Power Plant [5] [6].

There have been destructive accidents in which fluids have induced the vibration of engineering structures, such as bridges, pipeline networks, and fluid power plants [1] [2]. It is difficult for the engineer and designer to predict such unexpected destructive phenomena related to fluid vibration. There might be a few possibilities of unpredictable fluid-induced vibration.

Brown & Roshko [7], Tani [8], and Taneda [9] clarified that a high shear rate induces flow instability. Oscillation, self-sustaining oscillation, and flow instability have been studied experimentally for a cavity flow [10] [11] and flow through an axisymmetric stenosed channel [12]. Mori & Naganuma [13] clarified the coherent oscillation of incompressible turbulent flow over a rectangular cavity experimentally. Flow instability has also been observed in a nasal channel and for a cantilevered flexible plate [14] [15] [16]. However, this flow instability can appear as weak periodic oscillations rather than strong periodic oscillations. The vibration of turbulent fluid passing through an elbow has been analyzed by Keshtkar & Jafari numerically [17]. Researchers in the field of hemodynamics have extensively investigated flow instability induced inside cerebral aneurysms in terms of the growth and rupture of the aneurysms [18] - [23]. Indeed, although there is flow instability and turbulence within a cerebral aneurysm which the flow instability affects the rupture of an aneurysm.

Flow instability is a common interest in fluid mechanical engineering. There are several phenomena of unstable flow at the interface of two viscous fluids in a turbulent region. Awasthi et al. [24] investigated the Rayleigh-Taylor instability of two viscous fluids enclosed by two cylindrical coaxial surfaces considering mass and heat transfer across the interface. The Richtmyer-Meshkov instability has been investigated in numerical and experimental studies. Bernard et al. [25] experimentally investigated the primary features of flow developing after a shock acceleration and found that a well-studied method of flow visualization did not produce a flow pattern that matched the numerical model. Liu & Lu [26] investigated the flow instability of viscoelastic annular liquid jets in a radial electric field and found that the jets with reasonable parameters had greater instability. Bouzgarrou [27] investigated the development of a turbulent mixing zone induced by the Richtmyer-Meshkov instability and its subsequent evolution.

This review focuses on several phenomena of the destructive potential of fluid-induced vibration in the pipeline network of a fluid power plant.

2. Examples of Karman Vortex

Five examples of flow instability in internal and external flow are described.

2.1. Karman Vortices on Medium and Large Scales

The most aesthetically attractive phenomenon of flow instability in the field of fluid dynamics is the Karman vortex at a Reynolds number less than approximately 100 as shown in Figure 1 [9]. This vortex is generated by high-shear-rate

Figure 1. Karman vortex downstream of circular cylinder.

flow associated with the Kelvin-Helmholtz instability. The destructive collapse of a bridge spanning the Takoma Narrows Bridge, as shown in Figure 2, was directly related to this phenomenon [14]. As an example of the phenomenon at a large scale, Karman vortices appear around islands in the ocean, as shown in Figure 3(a) and Figure 4. The scale of Jeju Island and wind direction are shown in Figure 3(b) [28]. The Reynolds number is calculated to be on the order of 10¹¹ from a characteristic distance L = 30 km along the wind direction, an assumed wind velocity U = 50 m/s at an altitude of 5000 m, and a molecular kinematic viscosity ν_mole = 2.47 × 10⁻⁵ m²/s (−23˚C):

$Re=UL/n{\nu }_{mole}=50\times 30\times {10}^3/2.47\times {10}^{-5}=0.60\times {10}^{11}$.

This Reynolds number is extremely large in comparison with that of a circular cylinder. Generally, the eddy kinematic viscosity or turbulent kinematic viscosity ε_eddy is adopted in the meteorological field for huge-scale flow:

${\tau }_{xy}=\rho \left[\left|u^{\prime }\right|\lceil v^{\prime }\rceil \right]=\rho {\epsilon }_{eddy}\left(\text{d}U/\text{d}y\right)$,

where u', v', and ρ respectively denote the velocity fluctuations along the wind direction and perpendicular direction and the density at an altitude of 5000 m. When postulating to be U = 50 m/s and v = 5 m/s, the velocity fluctuation is assumed to be 10% and 20% of each velocity, velocity fluctuation will be u' = 5 m/s and v' = 1 m/s and density is ρ = 0.648 kg/m³. The eddy kinematic viscosity is thus obtained as ε_eddy = ρ[u'v']/ρ(dU/dy) = 3.24 m²/[0.648 kg/m³(10 m/s/10 m)] = 5 m²/s. The Reynolds number for this stream is thus calculated as Re = L (30 km) × U (50 m/s)/ε_eddy (5 m²/s) = 3 × 10⁵. The order of 10⁵ of this Reynolds number is comparable to those in the field of fluid engineering [29]

2.2. Flow Oscillation in a T-Junction

A T-junction at which a side-branch joins a trunk at a right angle is a fundamental fluid element of a pipeline network. In the past six decades, the pressure

Figure 2. Collapse of Takoma Narrows Bridge.

(a) (b)

Figure 3. (a) Karman vortex downstream of Jeju Island; (b) Scale of Jeju Island.

Figure 4. Karman vortex around Canarias Islands.

loss through a T-junction has been extensively investigated and established for turbulent flow at a Reynolds number on the order of 10⁵ in the field of engineering [29]. Furthermore, in the past four decades, the flow pattern and wall shear stress at a T-junction have been examined experimentally and numerically [30] [31] [32] for laminar steady flow in hemodynamics. Karino et al. investigated the flow pattern around a square-edged corner at the entrance of the side-branch of a T-junction using a microscope and showed how a helical flow in the side-branch results in recirculating flow [30]. Liepsch et al. [31] experimentally studied the flow structure within a T-junction for its relevance to hemodynamics. Using finite-volume and finite-element methods, Perktold et al. carried out three-dimensional calculations of the flow pattern in a branch for a bifurcation from the abdominal to renal artery [33]. Yamaguchi et al. [34] [35] [36] [37] [38] reported that a periodic oscillation generated in the side-branch of a T-junction at laminar steady flow of less than 10³. The Karman vortex and periodic oscillation in a T-junction are induced by similar flow instabilities.

2.3. Flow Instability in a T-Junction with Single Side-Branch

The present review focuses on fluid-induced vibration, primarily oscillation in a T-junction of a pipeline network, as shown in Figure 5. The inlet of the side-branch is a square-edged corner in the pipeline network of a fluid power plant. An open-end condition is applied at both outlets. The T-junction has a trunk radius R_T = 12.2 mm and side-branch radii R_S = 4, 5, 7, and 9 mm (primarily, R_S = 7 mm). The flow division ratios Q_S/Q_T of the flow rate to the side-branch from the trunk are 0.25, 0.33, and 0.50 [39]. S_i denotes the measurement section.

Figure 5. Schematic of T-junction and measurement sections S_i.

The kinematic viscosity of the working fluid (i.e., the saturated aqueous iodide solution) is ν = 1.58 × 10⁻⁶ m²/s [36]. The mean Reynolds numbers for the trunk and side-branch are primarily Re_T = 800 and Re_S = 700 (Q_S/Q_T = 0.50), respectively. Figure 6 presents the results of particle image velocimetry (PIV) and laser Doppler velocimetry (LDV) [39]. Arrows depict velocity vectors and closed circles depict the locations of oscillation measurements spaced at intervals of 0.2 mm on the median plane of the side-branch. Along the near wall of the side-branch, the flow separates, reverses upstream, and with returns downstream along the shearing separation layer. The velocity along the near wall is low within the separation region and high outside the shearing separation layer as shown in Figure 6. There is thus a large velocity difference between two fluid flows across the shearing separation layer; i.e., there is a high-shear-rate flow having an inflection point. This flow is strongly related to flow instability. Figure 7 shows the profile of the tangential velocity q parallel to the shearing separation layer measured by PIV at points P₁ and P₂ with clear oscillation in both flow division ratios. At both flow division ratios, the inflection point across this shearing separation layer is clearly indicated. The shear rates Δq/Δh at flow division ratios of Q_S/Q_T = 0.25 and 0.50 are 80 and 296 s⁻¹, respectively. These magnitudes of the shear rate in both flow division ratios are respectively three and six

(a)(b)

Figure 6. Velocity vector and oscillation point at median plane in side-branch. (a) Q_S/Q_T = 0.25; (b) Q_S/Q_T = 0.50.

(a)(b)

Figure 7. Tangential velocity profile along shearing separation layer. [(a) Q_S/Q_T = 0.25. (Shear rate downstream of side-branch = 24 s⁻¹), (b) Q_S/Q_T = 0.50. (Shear rate downstream of side-branch = 47 s⁻¹)]

times those (24and 47 s⁻¹) at the downstream wall of the side-branch. Figure 8 presents the spectral analysis of the axial velocity at P₁ and P₂ measured by LDV to confirm the flow instability. The top, middle, and bottom panels respectively show the temporal variation of the axial velocity, power spectrum, and auto-correlation. In the T-junction, the frequencies of f = 1.86 and 5.76 Hz are respectively Q_S/Q_T = 0.25 and 0.50 in section S₃. This frequency is the same value on the global median plane within the side-branch for each condition. Here, the diameter D_S = 14 mm, the mean velocity U_S = 0.08 m/s at Q_S/Q_T = 0.50, and the Reynolds number Re_S and Strouhal number St_S of the side-branch are 700 and 1.01 respectively.

There is another source of flow instability in the T-junction. Figure 9 presents the secondary velocity in three sections of the side-branch. In cross-section S₃ of the side-branch, the oscillation is strongly related to the high-shear-rate flow between two pairs of vortices with the same rotational sense in upper and lower half semi-circular sections. There is a small vortex at a lower left magnified circle in section S₃ for both flow division ratios. The mechanism underlying the fluid-induced oscillation is the high-shear-rate flow between two pairs of like-rotating vortices in section S₃ immediately downstream of the inlet of the side-branch. In particular, secondary flow immediately downstream of the upstream corner attracts the attention of the two pairs of vortices in the upper half circle at section S₃; i.e., a small vortex appears at the near wall and a large vortex

(a)(b)

Figure 8. Spectral analysis of axial velocity in side-branch at Re_T = 800. (a) Q_S/Q_T = 0.25 at point P₁ (x/R_S = 0.09, y/R_S = 3.04); (b) Q_S/Q_T = = 0.50 at point P₂ (x/R_S = −0.010, y/R_S = 3.04).

(a)(b)

Figure 9. Secondary velocity at three cross sections in side-branch. (a) Q_S/Q_T = 0.25; (b) Q_S/Q_T = 0.50.

occupies the global median plane. Naturally, the two vortices have the same direction of rotational and there is thus friction between two vortices; i.e., high-shear-rate induces flow instability. Figure 10 shows the velocity profile of vortex and shear rate for two division ratios in section S₃. There is a clear inflection in the velocity profile. The shear rates of 55 and 143 s⁻¹ at the inflection point for flow division ratios of Q_S/Q_T = 0.25 and 0.50 are twice those (27 and 47 s⁻¹) for the downstream wall of the side-branch as shown in Figure 10, respectively.

There is a clear inflection in the velocity profile. The shear rates of 55 and 143 s⁻¹ at the inflection point for flow division ratios of Q_S/Q_T = 0.25 and 0.50 are twice those (27 and 47 s⁻¹) for the downstream wall of the side-branch as shown in Figure 10, respectively. A periodic oscillation has been shown to be induced in the side-branch of a T-junction for steady laminar flow with Reynolds number in the above range (i.e., Re_S = 300 to 900) [34] [35] [36] [39]. The Strouhal number of approximately 1.03 with a standard deviation of ±0.26 based on quantities in the side-branch is regarded as independent of the Reynolds number, flow division ratio, and radius ratio of the side-branch to trunk as shown in Figure 11 (open black symbols for a single side-branch). In the T-junction, the mean velocity U_S and the diameter D_S of the side-branch are considered characteristic quantities because the periodic oscillation appears in the side-branch. The characteristic flow behavior of the periodic oscillation in the side-branch is a universally recognized phenomenon of the T-junction [34] [35] [36] [39]. At Re_T >1000 in the turbulent region, it is not possible to measure at a data rate much higher than the data rate of 3 kHz using the current LDV system (FSA-3500, TSI Co. LTD).

Figure 10. Secondary velocity and shear rate at boundary of two vortices in section S₃. (a) Q_S/Q_T = 0.25; (b) Q_S/Q_T = 0.50.

Figure 11. Side-branch Strouhal number St_S, versus side-branch Reynolds number Re_S including double side-branches. (Re_S = 2R_SU_S/ν and St_S = 2fR_S/U_S. Open symbol is T-junction with single side-branch. and denote concentric branch 1 and branch 2 in double side-branches, respectively. denotes eccentric branch 2) Mean value (1.03) ± S.D. (0.26).

2.4. Flow Instability in a T-Junction with Two Side-Branches

This section describes the flow instability for two axis-aligned or axis-skewed open-ended side-branches of T-junctions. It also describes the mechanism of fluid-induced vibration in a T-junction with two side-branches bifurcating from a trunk in steady laminar flow for a single-phase [37]. Figure 12(a) shows the schema and labeling of the cross-section of axis-aligned side-branches in a T-junction with two concentric side-branches. Figure 12(b) shows a schema of the top and side views of the skewed branches of a T-junction with two eccentric side-branches. The center of the skewed side-branches has a radial offset of 1 and 2 mm at side-branches 1 and 2 in Figure 12(b) respectively, and the median planes in side-branches 1 and 2 are tilted away from that of the trunk axis.

(a)(b)

Figure 12. Schema of concentric and eccentric double side-branches in T-junction. (a) Concentric side-branch; (b) Eccentric side-branch.

The measurement method is the same as that used for the T-junction with the single side-branch described above except for the working fluid. The working fluid is 53% aqueous glycerin with a refractive index of 1.41, a density of ρ = 1.13 g/cm³, and a kinematic viscosity of ν = 7.00 × 10⁻⁶ m²/s at 20˚C. The measurement was primarily carried out at Re_T = 2R_TU_T/ν = 800 for steady laminar flow in the trunk for both axis-aligned and axis-skewed T-junctions. The mean velocity in the trunk is U_T = 23 cm/s. The flow division ratio Q_S/Q_T is set at 0.25 for both side-branches 1 and 2. A spacing of Ls = 30 mm between the two side-branches is set in the experiment. This is the minimum spacing for which extreme fluid interference between the side-branches is expected; the inner and outer diameters of the side-branch are 14 and 20 mm, respectively. In LDV, the velocity is measured with a data sampling resolution of 1 μs every 2 ms for a is set at 0.25 for both side-branches 1 and 2. A spacing of Ls = 30 mm between the two side-branches is set in the experiment. This is the minimum spacing for which extreme fluid interference between the side-branches is expected; the inner and outer diameters of the side-branch are 14 and 20 mm, respectively. In LDV, the velocity is measured with a period of 40 s, giving a 20-k data set.

The velocity vector, shearing separation layer, and points of periodic oscillation on the median plane of the concentric side-branch are shown in Figure 13. Figure 14 presents the spectral analysis of the axial velocity at p₁ and p₂ in side-branches 1 and 2. The oscillation along the separation shearing layer in side-branch 2 appears clearer than that in side-branch 1 and has a frequency higher than that in side-branch 1.

Figure 15 presents the secondary flow in the three sections of the double-concentric side-branches 1 and 2. At least a pair of vortices appear in each section of both branches 1 and 2. Figure 16 presents the vortex velocity and a clear inflection point in section S₃ along the near wall. In the vortex velocity at the boundary in section S₃, the shear rate for side-branch 2 is 1.5 times that for the downstream side-branch. In the tangential velocity along with the separating shearing layer in section S₇, the shear rate for side-branch 2 is 1.5 times that for the downstream side-branch as shown in Figure 17.

Velocity vectors on the median plane of the eccentric double side-branches of the T-junction are shown in Figure 18. The velocity vectors indicated by the red arrows show the flow with clear oscillation. The tangential velocity along the shearing separation layer in section S₅ is depicted in Figure 19. Across this layer, the velocity profile has an inflection point and the shear rate for side-branch 2 is twice that for the single side-branch. Figure 20 and Figure 21 respectively present the spectral analysis and auto-correlation for the data of side-branch 2.

Figure 11 presents the Strouhal number St_S versus Reynolds number Re_S for the concentric and eccentric double side-branches and the single side-branch. The Strouhal number of 1.03 based on quantities of the side-branch is independent of the Reynolds number (Re_S = 230 to 400), flow division ratio, and radius ratio of the side-branch to trunk for the two side-branches. This order of

(a)(b)

Figure 13. Velocity vector and oscillation point at median plane of concentric side-branch. (a) Side-branch 1 (Q_S1/Q_T = 0.25); (b) Side-branch 2 (Q_S2/Q_T = 0.25).

Reynolds number 10⁵ is comparable to that in the field of fluid engineering. In other words, although the field of fluid engineering associates with the Karman vortex with molecular kinematic viscosity, the same feature is seen in meteorology.

(a)(b)

Figure 14. Spectral analysis of axial velocity at p₁ near section S₅ in double concentric side-branches. (a) Side-branch 1 at p₁; (b) Side-branch 2 at p₂.

(a)(b)

Figure 15. Secondary flow at three sections in double concentric side-branches. (a) Side-branch 1; (b) Side-branch 2.

Figure 16. Inflection point and shear rate at vortex boundary near section S₃ in double concentric side-branches. [Velocity gradient dv/dx at wall (x = R_S) of side-branch = 96.6 s⁻¹]. (a) Side-branch 1; (b) Side-branch 2.

Figure 17. Tangential velocity profile along shearing separation layer with inflection point near section S₇ in double concentric side-branches. [Velocity gradient dv/dx at tube wall (x = R_S) of side-branch = 96.6 s⁻¹]. (a) Side-branch 1; (b) Side-branch 2.

Figure 18. Velocity vector in eccentric side-branches in T-Junction at median plane.

Figure 19. Tangential velocity profile along shearing separation layer with inflection point near section at section S₅ in eccentric side-branches [Velocity gradient dv/dx at wall (x = R_S) of side branch = 97.9 s⁻¹]. (a) Side-branch 1; (b) Side-branch 2.

Figure 20. Spectral analysis in branch 2.

2.5. Karman Vortex Downstream of an Island in Ocean in Meteorology

Karman vortices are observed on a huge scale around islands in the ocean. Figure 3(a) and Figure 4 show such a Karman vortex as described in Section 2.1. The Reynolds number of this stream is estimated as follows.

$Re=L\left(30\text{km}\right)\times U\left(50\text{m}/\text{s}\right)/\epsilon \left(5{\text{m}}^{\text{2}}/\text{s}\right)=3\times {10}^5$.

This order of 10⁵ is comparable to that in the field of fluid engineering. In other words, although the field of fluid engineering associates with the Karman vortex with molecular kinematic viscosity, the same feature is seen in meteorology.

3. Prediction of Flow Instability in Fluid Power Plant

Sections 2.2 to 2.5 presented periodic oscillations in single and double side-branches and described how the Strouhal number is independent of the

Figure 21. Auto-correlation in branch 2.

Reynolds number, flow division ratio, and ratio of tube radii. This oscillation of flow is basically related to the high-shear-rate flow in a side-branch and the geometry of the inlet of the side-branch; i.e., a square-edged corner. T-junctions with square-edged corners are generally used in the pipeline network of a fluid power plant as shown in Figure 22. The connecting corner usually has a sharp square edge because a side-branch in a pipeline network is welded to the trunk at the construction site, as shown in Figure 23. There is a slight possibility of induced to be flow instability even in a turbulent flow. Here, we denote this flow instability induced in the side-branch with the high-shear-rate flow, that is, along with the shearing separation layer. The connecting geometry of the trunk and side branch should have rounded corners rather than sharp corners to suppress this high shear rate.

It will be difficult to predict another flow phenomenon from past research, because fluid dynamics phenomena are generally nonlinear. Recently, there have been many unex-incidents at fluid power plants, such as the destruction of a fine heat exchange tube. It will be difficult to predict another flow phenomenon from past researches because fluid dynamics phenomena are generally nonlinear.

Figure 22. Geometry of T-junction.

Figure 23. Construction site in welding pipeline.

Recently, there have been many unexpected incidents at fluid power plants, such as the destruction of a fine heat exchange tube in the Mihama Nuclear Power Plant in Japan, schematically shown in Figure 24 [6]. This incident resulted from fluid dynamic vibration accompanying a Karman vortex. Although a major cause of the vibration was the unsuitable installation of the fine steam tube on a supporting plate, the frequency of flow instability based on a Strouhal number St = 0.2 [40] is estimated to be f = USt/d = 5.2 × 0.2/0.02 = 52 Hz, where U = 0.52 m/s is temporally assumed to be the mean velocity in the steam generator, d = 0.02 m is the diameter of the heat exchanger fine tube, and Re = Ud/ν = 5.2 × 0.02/10⁻⁶ = 10⁵.

Another incident was the leakage of liquid sodium due to the destruction of a thermocouple in the Monju fast breeder reactor in Japan, schematically shown in Figure 25. Neither the designer nor the researcher could predict either incident before construction. In the Monju fast breeder reactor constructed using straight tubes, T-junctions, and elbows, presumably the fluid oscillation based on the current review is tried to estimate the fluid vibration in T-junction. Using data provided by Monju FBR Co. [2], the frequency of oscillation of flow instability in a T-junction can be predicted from the current Strouhal number St = 1.02, tube diameter D = 54 cm, kinematic viscosity of the working fluid (liquid sodium at a temperature of 400˚C) ν = 0.3323 × 10⁻⁶ m²/s, and mean velocity U = 5.2 m/s in working operation as [2]

Figure 24. Structure and destruction part of Mihama nuclear power plant.

Figure 25. Global schema of fast breeder reactor “MONJU”.

$f=USt/D=5.2\times 1.02/0.54=9.82\text{Hz}$

The Reynolds number for the side-branch is estimated as

$Re=DU/\nu =0.54\times 5.2/0.3323\times {10}^{-6}=8.45\times {10}^6$

Meanwhile, the vibration of flow around a thermocouple in the Monju fast breeder reactor has a frequency f = 52 Hz at a Strouhal number of 0.2 for a Karman vortex at Re = 3.3 × 10⁵.

Including the Karman vortex downstream of the island, the order of the Reynolds number is approximately 10⁵ - 10⁶. There is a slight possibility for fluid-induced vibration to occur in this range of the Reynolds number. This possibility should be considered by the designer of a fluid power plant.

Acknowledgements

This review was partially supported by JSPS Science Research Budgets #09450083, #12650180, #144550160 & #19K04163. The authors thank Edanz (https://jp.edanz.com/ac ) for editing a draft of this manuscript.

Nomenclature

f frequency of fluid vibration

n ordinate normal to the separation shearing layer

Q_S rate of flow through the side-branch

Q_T rate of flow through the trunk

q tangential velocity along the shearing separation layer

Re_S = 2R_SU_S/ν Reynolds number based on variables of the side-branch

Re_T = 2R_TU_T/ν Reynolds number based on variables of the trunk

R_S radius of the side-branch

R_T radius of the trunk

St_S = 2fR_S/U_S Strouhal number based on variables of the side-branch

U_S mean velocity in the side-branch

U_T mean velocity in the trunk

u transverse velocity in the side-branch

v axial velocity in the side-branch

ρ density of the fluid

ν molecular kinematic viscosity of the fluid

ε eddy kinematic viscosity

Conflicts of Interest

The authors have no conflict of interest.

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