MODELLING TWO-PHASE FLOW-EXCITED DAMPING AND FLUIDELASTIC INSTABILITY IN TUBE ARRAYS

This paper reports the results of an experimental study of the flow-induced vibration of a heat exchanger tube array subjected to two-phase cross-flow of refrigerant 11. The primary concern of the research was to develop a methodology for predicting the critical flow velocities for fluidelastic instability which better characterize the physics of two-phase flows. A new method is proposed for calculating the average fluid density and equivalent flow velocity of the two-phase fluid, using a newly developed void fraction model to account for the difference in velocity between the gas and liquid phases. Additionally, damping measurements in two-phase flow were made and compared with the data of other researchers who used a variety of modelling fluids. The results show that the two-phase damping follows a similar trend with respect to homogeneous void fraction, and when normalized, agree well with the data in the literature. The fluidelastic threshold data of several researchers who used a variety of fluids, is re-examined using the proposed void fraction model, and the results show a remarkable change in trend with flow regime. The data corresponding to the bubbly flow regime shows no significant deviation from the trend established by Connors' theory. However, the data corresponding to the intermittent flow regime show a significant decrease in stability which is nearly independent of the mass-damping parameter. It is believed that the velocity fluctuations that are inherent in the intermittent flow regime are responsible for tripping the instability, causing lower than expected stability of the bundle.     

Two-phase damping and interface surface area in tubes with vertical internal flow

Two-phase flow is common in the nuclear industry. It is a potential source of vibration in piping systems. In this paper, two-phase damping in the bubbly flow regime is related to the interface surface area and, therefore, to flow configuration. Experiments were performed with a vertical tube clamped at both ends. First, gas bubbles of controlled geometry were simulated with glass spheres let to settle in stagnant water. Second, air was injected in stagnant alcohol to generate a uniform and measurable bubble flow. In both cases, the two-phase damping ratio is correlated to the number of bubbles (or spheres). Two-phase damping is directly related to the interface surface area, based on a spherical bubble model. Further experiments were carried out on tubes with internal two-phase air–water flows. A strong dependence of two-phase damping on flow parameters in the bubbly flow regime is observed. A series of photographs attests to the fact that two-phase damping in bubbly flow increases for a larger number of bubbles, and for smaller bubbles. It is highest immediately prior to the transition from bubbly flow to slug or churn flow regimes. Beyond the transition, damping decreases. It is also shown that two-phase damping increases with the tube diameter.     

Damping of tubes due to internal two-phase flow

Two-phase internal flow is present in many piping system components. Although two-phase damping is known to be a significant constituent of the total damping, the energy dissipation mechanisms that govern two-phase damping are not well understood. In this paper, damping of three different clamped–clamped tubes subjected to two-phase air–water internal axial flow is investigated. Experimental data are reported, showing a strong dependence of two-phase damping on void fraction, flow velocity and flow regime. Data-points plotted on two-phase flow pattern maps indicate that damping is greater in a bubbly flow regime. The two-phase damping ratio reaches a maximum value at the highest void fraction before the transition to a churn flow regime. An analytical model that relates the two-phase damping ratio to the interface surface area is proposed. The model is based on rigid spherical bubbles in cubic elementary flow volumes. The analytical results are well correlated with the experiments.     

Two-phase damping for internal flow: Physical mechanism and effect of excitation parameters

Two-phase flow induced-vibration is a major concern for the nuclear industry. This paper provides experimental data on two-phase damping that is crucial to predict vibration effects in steam generators. An original test section consisting of a tube subjected to internal two-phase flow was built. The tube is supported by linear bearings and compression springs allowing it to slide in the direction transverse to the flow. An excitation system provides external sinusoidal force. The frequency and magnitude of the force are controlled through extension springs. Damping is extracted from the frequency response function of the system. It is found that two-phase damping depends on flow pattern and is fairly proportional to volumetric fraction for bubbly flow. Measurements are completed by the processing of high-speed videos which allow to characterize the transverse relative motion of the gas phase with respect to the tube for bubbly flow. It is shown that the bubble drag forces play a significant role in the dissipation mechanism of two-phase damping.     

Investigation on two-phase flow-induced vibrations of a piping structure with an elbow

The dynamic behaviors of a horizontal piping structure with an elbow due to the two-phase flow excitation are experimentally investigated. The effects of flow patterns and superficial velocities on the pressure pulsations and vibration responses are evaluated in detail. A strong partition coupling algorithm is used to calculate the flow-induced vibration (FIV) responses of the pipe, and the theoretical values agree well with the experimental results. It is found that the lateral and axial vibration responses of the bend pipe are related to the momentum flux of the two-phase flow, and the vibration amplitudes of the pipe increase with an increase in the liquid mass flux. The vertical vibration responses are strongly affected by the flow pattern, and the maximum response occurs in the transition region from the slug flow to the bubbly flow. Moreover, the standard deviation (STD) amplitudes of the pipe vibration in three directions increase with an increase in the gas flux for both the slug and bubbly flows. The blockage of liquid slugs at the elbow section is found to strengthen the vibration amplitude of the bend pipe, and the water-blocking phenomenon disappears as the superficial gas velocity increases.     

Turbulent flow structure and heat transfer in an inclined bubbly flow. Experimental and numerical investigation

The effect of channel inclination on the variation in the wall shear stress and the heat transfer in a two-phase bubbly flow in a rectangular channel is experimentally and numerically investigated. The wall friction was measured using the electrodiffusion method and the temperature was measured by tiny platinum resistance thermometers. The model is based on the system of RANS equations with account for the back influence of the bubbles on the flow characteristics. Flow turbulence is calculated according to the model of transport of the Reynolds stress tensor components. It is shown that in the gas-liquid flow the angle of the channel inclination to the horizon can have a considerable effect on the friction and the heat transfer. The greatest friction and heat transfer values correspond to the angles of channel inclination ranging from 30 to 50°. In the inclined two-phase bubbly flow the shear stress enhancement on the wall amounts to 30% and that of the heat transfer to 15%. A friction and heat transfer reduction to 10 and 25%, respectively, is noticed in near-horizontal flows.     

Fluidelastic instability study in a rotated triangular tube array subject to two-phase cross-flow. Part I: Fluid force measurements and time delay extraction

Fluidelastic instability is a key issue in steam generator tube bundles subjected to cross-flow. The extension to two-phase flow of the existing theoretical models, developed and tested mostly for single phase flow, is investigated in this paper. The time delay is one of the key parameter for modeling fluidelastic instability, especially the damping controlled mechanism. The direct measurement of the time delay between the tube motion and the fluid force faces certain difficulties in two-phase flow since the high turbulence due to the interaction of the two components of the flow may increase the randomness of the measured force. To overcome this difficulty, an innovative method for extracting the time delay inherent to the quasi-steady model for fluidelastic instability is proposed in this study.Firstly, experimental measurements of unsteady and quasi-static fluid forces (in the lift direction) acting on a tube subjected to air–water two-phase flow were conducted. The unsteady fluid forces were measured by exciting the tube using a linear motor. These forces were measured for a wide range of void fractions, flow velocities and excitation frequencies. The experimental results showed that the unsteady fluid forces could be represented as single valued function of the reduced flow velocity. It was also found that for a given frequency, the unsteady fluid force phase was weakly dependent on the void fraction for the range of flow velocities considered.The time delay was determined by equating the unsteady fluid forces with the quasi-steady forces. The results given by this innovative method of measuring the time delay in two-phase flow were consistent with theoretical expectations. The time delay could be expressed as a linear function of the convection time and the time delay parameter was determined for void fractions ranging from 60% to 90%.     

Study of two-phase flows in reduced gravity using ground based experiments

Experimental studies have been carried out to support the development of a framework of the two-fluid model along with an interfacial area transport equation applicable to reduced gravity two-phase flows. The experimental study simulates the reduced gravity condition in ground based facilities by using two immiscible liquids of similar density namely, water as the continuous phase and Therminol 59^® as the dispersed phase. We have acquired a total of eleven data sets in the bubbly flow and bubbly to slug flow transition regimes. These flow conditions have area-averaged void (volume) fractions ranging from 3 to 30% and channel Reynolds number for the continuous phase between 2,900 and 8,800. Flow visualization has been performed and a flow regime map developed which is compared with relevant bubbly to slug flow regime transition criteria. The comparison shows that the transition boundary is well predicted by the criterion based on critical void fraction. The value of the critical void fraction at transition was experimentally determined to be approximately 25%. In addition, important two-phase flow local parameters, including the void fraction, interfacial area concentration, droplet number frequency and droplet velocity, have been acquired at two axial locations using state-of-the-art multi-sensor conductivity probe. The radial profiles and axial development of the two-phase flow parameters show that the coalescence mechanism is enhanced by either increasing the continuous or dispersed phase Reynolds number. Evidence of turbulence induced particle interaction mechanism is highlighted. The data presented in this paper clearly show the marked differences in terms of bubble (droplet) size, phase distribution and phase interaction in two-phase flow between normal and reduced gravity conditions.     

AN EXPERIMENTAL STUDY ON THE FLUIDELASTIC FORCES ACTING ON A SQUARE TUBE BUNDLE IN TWO-PHASE CROSS-FLOW

A tube in a square tube bundle of P/D=1·42 was oscillated in the lift direction in air–water two-phase cross-flow, and fluidelastic forces acting on the oscillated tube were measured. First, the tube amplitude was fixed to 3 mm (=0·136 D), and added mass, damping, and stiffness coefficients were obtained as a function of two-phase mixture characteristics such as nondimensional gap velocity and void fraction. When reference mixture density and velocity were estimated, the drift–flux model, in which the relative velocity between the gas and liquid phases was estimated, generated better results than the homogeneous model. The added mass coefficient was obtained from quiescent two-phase flow as a function of void fraction. Using the added mass coefficient, the added stiffness coefficient converged to zero with decreasing nondimensional gap velocity. This overcame the contradiction in the added stiffness estimation without added mass, in which the added stiffness coefficient did not converge to zero with decreasing nondimensional gap velocity. Next, the effects of the vibration amplitude on the fluidelastic force coefficients were considered. When the tube amplitude was 3 mm (=0·136 D) or less, the equivalent added stiffness and damping coefficients were almost constant and nonlinearity was small. This showed the validity of the fluidelastic force coefficients obtained based on the data of amplitude of 3 mm. The linearity did not exist when the tube displacement amplitude was 4·5 mm (=0·205 D) or more; a remarkable nonlinearity appeared in the equivalent added damping coefficient. A method to estimate the limit-cycle amplitude of the fluidelastic vibration was proposed when only one tube in the tube bundle was able to vibrate in the lift direction. The amplitude could be obtained from the amplitude at which the equivalent added damping coefficient changed from negative to positive with increase in the tube amplitude.     

One-group interfacial area transport equation and its sink and source terms in narrow rectangular channel

The characteristics of two-phase flow in a narrow rectangular channel are expected to be different from those in other channel geometries, because of the significant restriction of the bubble shape which, consequently, may affect the heat removal by boiling under various operating conditions. The objective of this study is to develop an interfacial area transport equation with the sink and source terms being properly modeled for the gas–liquid two-phase flow in a narrow rectangular channel. By taking into account the crushed characteristics of the bubbles a new one-group interfacial area transport equation was derived for the two-phase flow in a narrow rectangular channel. The random collisions between bubbles and the impacts of turbulent eddies with bubbles were modeled for the bubble coalescence and breakup respectively in the two-phase flow in a narrow rectangular channel. The newly-developed one-group interfacial area transport equation with the derived sink and source terms was evaluated by using the area-averaged flow parameters of vertical upwardly-moving adiabatic air–water two-phase flows measured in a narrow rectangular channel with the gap of 0.993 mm and the width of 40.0 mm. The flow conditions of the data set covered spherical bubbly, crushed pancake bubbly, crushed cap-bubbly and crushed slug flow regimes and their superficial liquid velocity and the void fraction ranged from 0.214 m/s to 2.08 m/s and from 3.92% to 42.6%, respectively. Good agreement with the average relative deviation of 9.98% was obtained between the predicted and measured interfacial area concentrations in this study.     

Vortex shedding and fluidelastic instability in a normal square tube array excited by two-phase cross-flow

Laboratory experiments were conducted to determine the flow-induced vibration (FIV) response and fluidelastic stability threshold of a model heat exchanger tube bundle subjected to a cross-flow of refrigerant 11. The tube bundle consisted of a normal square array of 12 tubes with outer tube diameters of 7.11 mm and a pitch over diameter ratio of 1.485. The experiments were conducted in a flow-loop that was capable of generating single- and two-phase cross-flows over a variety of mass fluxes and void fractions. The primary intent of the research was to improve our understanding of the FIVs of heat exchanger tube arrays subjected to two-phase cross-flow. Of particular concern was the effect of array pattern geometry on fluidelastic instability. The experimental results are analysed and compared with existing data from the literature using various methods of parameter definition. Comparison of tube vibration response in liquid flow with previous results shows a similar occurrence of symmetric vortex shedding that validates the scale model approach in single-phase flow. It was found that the introduction of a small amount of bubbles in the flow disrupted the vortex shedding and thereby caused a significant reduction in streamwise vibration amplitude. The fluidelastic stability thresholds for the present array agree well with results from previous studies. Furthermore, a good collapse of the stability data from various investigations is obtained when the fluid density is defined using the slip model of Feenstra et al. and when an effective two-phase flow velocity is defined using the interfacial velocity model of Nakamura et al.     

Bubbly-to-cap bubbly flow transition in a long-26 m vertical large diameter pipe at low liquid flow rate

The concurrent upward two-phase flow of air and water in a long vertical large diameter pipe with an inner diameter (D) of 200 mm and a height (z) of 26 m (z/D = 130) was investigated experimentally at low superficial liquid velocities from 0.05009 to 0.3121 m/s and the superficial gas velocities from 0.01779 to 0.5069 m/s. The resultant void fractions range from 0.03579 to 0.4059. According to the observations using a high speed video camera, the flow regimes of bubbly, developing cap bubbly and fully-developed cap bubbly flows prevailed in the flows. The developing cap bubbly flow appeared as a flow regime transition from bubbly to fully-developed cap bubble flow in the vertical large diameter pipe. The developing cap bubbly flow changes gradually and lasts for a long time period and a wide axial region in the flow direction, in contrast to a sudden transition from bubbly to slug flows in a small diameter pipe. The analysis in this study showed that the flow regime transition depends not only on the void fraction but also on the axial distance in the flow and the pipe diameter. The axial flow development brings about the transition to happen in a lower void fraction flow and the increase of pipe diameter causes the transition to happen in a higher void fraction flow. The measured void fraction showed an N-shaped axial changing manner that the void fraction increases monotonously with axial position in the bubbly flow, decreases non-monotonously with axial position in the developing cap bubbly flow, and increases monotonously again with axial position in the fully-developed cap bubbly flow. The temporary void fraction decrease phenomenon in the transition region from bubbly to cap bubbly flow can be attributed to the formation of medium to large cap bubbles and their gradual growth into the maximum size of cap bubble and/or cluster of large cap bubbles in the developing cap bubbly flow. In order to predict the N-shaped axial void fraction changing behaviors in the flow regime transition from bubbly to cap bubbly flow, the existing 12 drift flux correlation sets for large diameter pipes are reviewed and their predictabilities are studied against the present experimental data. Although some drift flux correlation sets, such as those of Clark and Flemmer (1986) and Hibiki and Ishii (2003), can predict the present experimental data with reasonable average relative deviations, no drift flux correlation set for distribution parameter and drift velocity can give a reliable prediction for the observed N-shaped axial void fraction changing behaviors in the region from bubbly to cap bubbly flow in a vertical large diameter pipe.     

Interfacial area concentration in gas–liquid bubbly to churn flow regimes in large diameter pipes

This study performed a survey on existing correlations for interfacial area concentration (IAC) prediction and collected an IAC experimental database of two-phase flows taken under various flow conditions in large diameter pipes. Although some of these existing correlations were developed by partly using the IAC databases taken in the low-void-fraction two-phase flows in large diameter pipes, no correlation can satisfactorily predict the IAC in the two-phase flows changing from bubbly, cap bubbly to churn flow in the collected database of large diameter pipes. So this study presented a systematic way to predict the IAC for the bubbly-to-churn flows in large diameter pipes by categorizing bubbles into two groups (group 1: spherical or distorted bubble, group 2: cap bubble). A correlation was developed to predict the group 1 void fraction by using the void fraction for all bubble. The group 1 bubble IAC and bubble diameter were modeled by using the key parameters such as group 1 void fraction and bubble Reynolds number based on the analysis of Hibiki and Ishii (2001, 2002) using one-dimensional bubble number density and interfacial area transport equations. The correlations of IAC and bubble diameter for group 2 cap bubbles were developed by taking into account the characteristics of the representative bubbles among the group 2 bubbles and the comparison between a newly-derived drift velocity correlation for large diameter pipes and the existing drift velocity correlation of Kataoka and Ishii (1987) for large diameter pipes. The predictions from the newly-developed two-group IAC correlation were compared with the collected experimental data in gas–liquid bubbly to churn flow regimes in large diameter pipes and their mean absolute relative deviations were obtained to be 28.1%, 54.4% and 29.6% for group 1, group 2 and all bubbles respectively.     

Application of photobleaching molecular tagging velocimetry to turbulent bubbly flow in a square duct

To evaluate turbulence energy budget in bubbly flows, an image processing method in a photobleaching molecular tagging velocimetry is improved for accurate evaluation of velocity gradients. Turbulence properties in single-phase and two-phase dilute-bubbly flows in a square duct are measured using the improved method. As a result, the following conclusions are obtained: (1) The axial velocity and axial turbulent intensity measured by the present method agree well with those measured by laser Doppler velocimetry not only for the single-phase flow but also for the dilute-bubbly flow. (2) The present method can measure velocity components and velocity gradients in the vicinity of the wall, and therefore the present method is of great use in understanding the mechanism of turbulence generation and dissipation near the wall. (3) The present method can provide detailed information on turbulence structure such as turbulence kinetic energy budget. (4) Bubbles tend to increase not only the turbulence production but also the turbulence dissipation.     

Comparison of several models for multi-size bubbly flows on an adiabatic experiment

This paper deals with the modelling and numerical simulation of isothermal bubbly flows with multi-size bubbles. The study of isothermal bubbly flows without phase change is a first step towards the more general study of boiling bubbly flows. Here, we are interested in taking into account the features of such isothermal flow associated to the multiple sizes of the different bubbles simultaneously present inside the flow. With this aim, several approaches have been developed. In this paper, two of these approaches are described and their results are compared to experimental data, as well as to those of an older approach assuming a single average size of bubbles. These two approaches are (i) the moment density approach for which two different expressions for the bubble diameter distribution function are proposed and (ii) the multi-field approach. All the models are implemented into the NEPTUNE_CFD code and are compared to a test performed on the MTLOOP facility. These comparisons show their respective merits and shortcomings in their available state of development.     

Order and disorder in particle–liquid flows in a pipe

The spherical expanded polystyrene particle–oil two-phase flow in a vertical pipe was used to simulate the dispersed phase distribution in laminar bubbly flows. A three-dimensional particle image tracking technique was used to track the particles in the flow to study the ordered structure of dispersed phase distribution and its transition to disorder. The ordered structures behaved as particle strings aligned in the flow direction as induced by the flow shear. The structures were quite durable in high liquid velocity flows and dispersed gradually as the liquid velocity decreased. In lower velocity flows, the particles tended to form clusters in the horizontal direction, as predicted by potential theory for spherical bubbles rising in a quiescent inviscid liquid and as observed in experiments on non-shear bubbly water flows.     

Turbulent cascade modeling of single and bubbly two-phase turbulent flows

The analysis of turbulent two-phase flows requires closure models in order to perform reliable computational multiphase fluid dynamics (CMFD) analyses. A spectral turbulence cascade-transport model, which tracks the evolution of the turbulent kinetic energy from large to small liquid eddies, has been developed for the analysis of the homogeneous decay of isotropic single and bubbly two-phase turbulence. This model has been validated for the decay of homogeneous, isotropic single and two-phase bubbly flow turbulence for data having a 5 mm mean bubble diameter. The Reynolds number of the data based on bubble diameter and relative velocity is approximately 1400.     

DYNAMICS OF AN IN-LINE TUBE ARRAY SUBJECTED TO STEAM–WATER CROSS-FLOW. PART III: FLUIDELASTIC INSTABILITY TESTS AND COMPARISON WITH THEORY

This paper presents the results of comprehensive flow-induced vibration tests conducted on an in-line array in steam–water two-phase flow. The responses of three essentially isolated flexible cylinders at different depths within the array were simultaneously measured. The main test parameters were, ambient pressure (and saturation temperature), in the range 0·5–5·8 MPa, void fraction, 0·70–0·96, and phase flow velocity. Tests reported here were conducted simultaneously with the damping tests reported in Part I of this study. At the highest pressures (3·0 and 5·8 MPa), strong instabilities, in homogeneous flow akin to single-phase flow occurred. The test tube located in the central region of the array was the most susceptible to instability. This was attributed partly to reduced two-phase damping deep in the array, while differences in local fluid forces at different locations in the array are not ruled out. The flow at 0·5 MPa was a nonhomogeneous intermittent slug-type flow. Strong turbulence excitation obscured clear fluidelastic instability; intermittent instability was, however, ascertained. Stability boundary calculations were done using unsteady fluid forces presented in Part II of this series of papers. Results for the case of P=5·8 MPa show good agreement with the measured instability boundary.     

DYNAMICS OF AN IN-LINE TUBE ARRAY SUBJECTED TO STEAM–WATER CROSS-FLOW. PART II: UNSTEADY FLUID FORCES

In this paper, experimental results of unsteady fluid-force measurements are reported. Important deviations of the measured fluid forces from their single-phase flow counterparts were uncovered. Most importantly, the resulting force coefficients are not simple functions of the reduced flow velocity U/fD, as is the case for single-phase flow. Test results at 0·5 MPa challenge the basic assumption of the existence of a time-invariant linear transfer function between tube displacement and the resulting fluid forces. Time–frequency analysis using Wigner–Ville transforms shows that the phase difference between tube displacement and the fluid force (an indicator of stabilizing or destabilizing fluid effects) undergoes significant variation under what may be considered steady flow conditions. This variation may explain the previously reported phenomenon of intermittent fluidelastic instability in two-phase flows.     

Numerical simulation of streamwise fluidelastic instability of tube bundles subjected to two-phase cross flow

This work aims to develop and validate a numerical model to simulate the flow-structure interaction in tube bundles subjected to two-phase flow. The model utilizes a mixture multiphase module in which a drift flux formulation is used to account for the slip between the phases. Two methods of numerical flow-structure interaction are used to predict the onset of fluidelastic instability (FEI) in the streamwise direction for a two-phase air–water flow mixture in parallel triangular tube bundles. These models are the hybrid analytical-flow field model and the direct numerical flow/structure coupling model. This work investigates the effects of void fractions in the range of 20% to 80% and several pitch-to-diameter ratios (P/D) in the range of 1.3 to 1.7. The results of the fluidelastic forces and the stability threshold are validated against the experimental data available in the literature and show an excellent agreement. The streamwise FEI threshold shows a significant dependency on the pitch-to-diameter ratio while the void fraction exhibits a lesser effect. Generally, the stability threshold increases as the pitch-to-diameter ratio increases. The model that was developed paves the way for devising of more reliable prediction tools for FEI in steam generators.