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Wake Induced Lift Force Acting on a Single Bubble

후류에 의해 유도되어 단일 기포에 작용하는 양력

  • 이우람 (Wooram Lee, 한동대학교 일반대학원 첨단그린에너지환경학과)

초록/요약

Bubbly flow is an important phenomenon that occurs in various applications such as bubble drag reduction, bubble column reactor, and nuclear safety analysis. Instead of simulating the specific interface of bubbles, the methodology of utilizing models of the interfacial momentum transfer has been widely used in order to simulate the bubbly flow efficiently. Among the various forces experienced by bubbles, both database and physical model of the lift force acting on single bubbles at bubble Reynolds number Re > 103 are lacking; therefore this study tried to fill the knowledge gap. First, the lift force acting on single bubbles at 500 < Re < 8000 rising in a linear shear flow was experimentally measured. In order to produce the linear shear flow, the special water channel of utilizing the recirculation flow generated at the sharply bent channel with a right angle was used. The performance of the test facility was designed by the three-dimensional computational fluid dynamics analysis. Experimentally, the downward linear shear flow with a shear ratio of 0.3 s-1 and center speed of 0.1 m/s was confirmed by the particle image velocimetry. As a working fluid, air, and filtered water at 27°C were used. The shape and the path of bubbles, which were obtained by the shadowgraph method, were approximated to an oblate spheroid and an oblique straight line, respectively. The present experimental method was supported by a good agreement between both drag force and shape aspect ratio from the present experiment and those from both data and correlations in the literature. The present experiment resulted in a decreasing pattern of the lift coefficient CL as increasing of Re (CL ~ -2 at Re ~ 8000). Moreover, the present experimental data of CL showed continuity with the data in the literature at a lower Re. Second, the lift force acting on single bubbles at 400 < Re < 4000 rising in a linear shear flow was obtained from the numerical simulations computed by the open-source flow solver Basilisk. The simulation method was confirmed by comparing both drag force and shape aspect ratio of the present result with those of the database on single clean bubbles. It was newly shown that CL obtained from the present simulation agreed with CL from the present experiment within the standard deviation of the experimental data. Notably, the characteristics of the vortices around a bubble, such as a vorticity distribution, size, and shedding frequency, could be appropriately identified by the present numerical simulation. Third, based on the present data obtained from both experiments and simulation, the lift force acting on single high-Re bubbles was newly modeled. The conventional model on the lift force induced by the wake behind a bubble was revisited. In order to predict the circulation of the detached vortices from a bubble, the Lamb vortex model was utilized. In addition, based on the present simulation results, the vorticity at the bubble’s equator and diameter at the bubble’s equator are used as the characteristic vorticity and characteristic length, respectively. The newly derived model successfully described the present results obtained from both experiments and simulation. The lift force acting on single free rising bubbles was also modeled by the same principle and compared with both experimental data and simulation results. Remarkably, a good agreement between them was observed, which supports the appropriateness of the present model on the wake-induced lift force.

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초록/요약

기포 유동은 기포 마찰 저감, 기포 반응기, 원자로 안전 해석 등 많은 응용에서 나타나는 중요한 현상이다. 기포 유동을 효율적으로 모사하기 위해서는, 기포 경계면을 직접 모사하지 않고 기포가 받는 힘들에 대한 모델들을 사용하는 방법이 현재 널리 쓰이고 있다. 기포가 받는 여러 힘들 중 양력의 경우, 103 이상의 기포 레이놀즈수 Re 에서의 연구 자료가 적고, 물리적 모델이 아직 없기에, 본 연구에서 이를 보완하고자 하였다. 첫째, 500 < Re < 8000 조건의 기포가 선형전단유동에서 상승할 때 받는 양력을 실험적으로 측정하였다. 이를 위해, 직각으로 꺽이는 유로 코너에서 발생하는 재순환 유동을 활용한 특수한 유로를 사용하였다. 3차원 전산유동해석을 통해 장치의 성능을 설계하고, 측정부에서 속도 구배 ~0.3 s-1 와 유로 중심부 유속이 ~0.1 m/s인 하향 선형 전단 유동이 구현됨을 입자영상유속계 기법을 통해 실험적으로 확인하였다. 작동 유체로는 공기와 필터 처리된 27°C의 물이 쓰였다. 쉐도우그라피로 얻어진 기포의 모양은 납작한 타원체로, 궤적은 기울어진 직선으로 근사하였다. 기포가 받은 항력과 기포의 모양은 표면이 오염된 기포들에 대한 기존 연구 결과 및 경향과 잘 일치하여 본 연구 방법론의 타당성을 확인할 수 있었다. 양력 계수 CL은 Re가 증가할 수록 감소하되 (Re ~ 8000 에서 CL ~ -2), 기존에 알려진 작은 Re에서의 결과들과 연속성 있는 결과들을 새롭게 얻었다. 둘째, 오픈소스 유동해석 프로그램인 Basilisk를 이용하여 400 < Re < 4000 조건에서 기포가 받는 양력을 수치적으로 얻었다. 해석의 건전성은 표면이 깨끗한 경우의 항력과 기포 모양에 대한 알려진 결과들과 비교하여 확보하였다. CL은 본 실험 결과와 표준 편차 범위 내에서 잘 맞는 결과가 새롭게 얻어졌다. 특히, 본 수치해석을 통해 기포 주변의 와류의 최대 와도, 크기, 이탈 주파수 등을 적절하게 파악할 수 있었다. 셋째, 위 결과들을 토대로, 높은 Re 에서 기포가 받는 양력에 대한 모델을 새롭게 제시하였다. 기포 후류에 의해 유도되는 양력에 대한 기존 모델을 재고하되, 기포에서 이탈되는 와류의 순환을 예측하기 위해 Lamb vortex의 와도 분포 모델을 사용하였다. 수치해석 결과에 근거하여, 특성 와도값으로 기포 가장자리에서의 와도를, 특성 진동수로 기포의 모양 진동수를, 특성 크기로 기포의 장축 직경을 사용하였다. 새롭게 유도된 모델은 위 실험 및 수치해석 결과를 적절하게 설명하였다. 모델의 타당성 확인을 위해, 자유상승 기포가 받는 양력을 실험과 수치해석을 통해 새롭게 얻었으며, 모델과 잘 맞음을 확인하였다.

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목차

Ⅰ. Introduction 1
1.1 Lift force model in the simulation of bubbly flow 1
1.2 Modeling of the lift force acting on single bubbles 4
1.3 Experimental investigations 7
1.4 Numerical investigations 12
1.5 Objective and structure of the thesis 14
Ⅱ. Experiment of single high Reynolds number bubbles in a linear shear flow (500 < Re < 8000) 16
2.1 Linear shear flow generator 16
2.1.1 Design by preliminary numerical simulation 16
2.1.2 PIV measurement 22
2.2 Experimental procedure 27
2.2.1 Genaration of a bubble 27
2.2.2 Imaging of the trajectory of a bubble 29
2.2.3 Uncertainty analysis 32
2.3 Result and discussion 40
2.3.1 Drag and shape aspect ratio 40
2.3.2 Pattern of the transverse migration 44
2.3.3 Lift force 51
2.4 Concluding remarks 56
Ⅲ. Numerical simulation of single high Reynolds number bubbles in a linear shear flow (400 < Re < 4000) 57
3.1 Method 57
3.2 Result and discussion 61
3.2.1 Drag and shape aspect ratio 61
3.2.2 Oscillatory kinematics 67
3.2.3 Vortices 79
3.2.4 Lift force 86
3.3 Concluding remarks 88
Ⅳ. Estimation of the lift force induced by the wake instability of a bubble 89
4.1 Wake induced lift force acting on a free rising bubble 89
4.1.1 Derivation of the model 89
4.1.2 Comparison between the model and simulation results 95
4.1.3 Comparison between the model and experimental results 104
4.2 Wake induced lift force acting on a bubble in a linear shear flow 111
4.2.1 Derivation of the model 111
4.2.2 Validation of the model (quasi-steady dynamics) 114
4.2.3 Validation of the model (unsteady dynamics) 117
4.2.4 Low-Reynolds number model of the wake induced lift force 120
Ⅴ. Conclusion 123
Appendix A. Velocity potential of the uniform flow around an oblate spheroid 124
Appendix B. Vorticity at the equator of the oblate spheroidal bubble 126
Appendix C. Lamb vortex 127
Appendix D. Added-mass of an oblate spheroid 128
List of References 129
Acknowledgement 138

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