1. IntroductionThe 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 VortexFive examples of flow instability in internal and external flow are described.
2.1. Karman Vortices on Medium and Large ScalesThe 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
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 νmole = 2.47 × 10−5 m2/s (−23˚C):
R e = U L / n ν m o l e = 50 × 30 × 10 3 / 2.47 × 10 − 5 = 0.60 × 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 εeddy is adopted in the meteorological field for huge-scale flow:
τ x y = ρ [ | u ′ | ⌈ v ′ ⌉ ] = ρ ε e d d y ( d U / d y ) ,
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 εeddy = 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)/εeddy (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-JunctionA 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
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-BranchThe 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 RT = 12.2 mm and side-branch radii RS = 4, 5, 7, and 9 mm (primarily, RS = 7 mm). The flow division ratios QS/QT of the flow rate to the side-branch from the trunk are 0.25, 0.33, and 0.50 [39]. Si denotes the measurement section.
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 ReT = 800 and ReS = 700 (QS/QT = 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 QS/QT = 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
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 QS/QT = 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 DS = 14 mm, the mean velocity US = 0.08 m/s at QS/QT = 0.50, and the Reynolds number ReS and Strouhal number StS 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
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 QS/QT = 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 QS/QT = 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., ReS = 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 US and the diameter DS 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 ReT >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).
2.4. Flow Instability in a T-Junction with Two Side-BranchesThis 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.
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 ReT = 2RTUT/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 UT = 23 cm/s. The flow division ratio QS/QT 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 StS versus Reynolds number ReS 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 (ReS = 230 to 400), flow division ratio, and radius ratio of the side-branch to trunk for the two side-branches. This order of
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.
2.5. Karman Vortex Downstream of an Island in Ocean in MeteorologyKarman 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.
R e = L ( 30 km ) × U ( 50 m / s ) / ε ( 5 m 2 / s ) = 3 × 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 PlantSections 2.2 to 2.5 presented periodic oscillations in single and double side-branches and described how the Strouhal number is independent of the
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.
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]
f = U S t / D = 5.2 × 1.02 / 0.54 = 9.82 Hz
The Reynolds number for the side-branch is estimated as
R e = D U / ν = 0.54 × 5.2 / 0.3323 × 10 − 6 = 8.45 × 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.
Nomenclaturef frequency of fluid vibration
n ordinate normal to the separation shearing layer
QS rate of flow through the side-branch
QT rate of flow through the trunk
q tangential velocity along the shearing separation layer
ReS = 2RSUS/ν Reynolds number based on variables of the side-branch
ReT = 2RTUT/ν Reynolds number based on variables of the trunk
RS radius of the side-branch
RT radius of the trunk
StS = 2fRS/US Strouhal number based on variables of the side-branch
US mean velocity in the side-branch
UT 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