Transactions of the Korean Nuclear Society Virtual Spring Meeting July 9-10, 2020 Modeling of the wake-induced lift force acting on an unbounded bubble at arbitrary Reynolds number Wooram Lee a , Jae-Young Lee b* a Institute of Advanced Machine Technology, Handong Global University b School of Mechanical and Control Engineering, Handong Global University * Corresponding author: jylee7@handong.edu 1. Introduction normal vector at the bubble’s interface, S B indicates bubble’s surface, V L indicates volume of the liquid In the safety analysis of a nuclear reactor, the outside of the bubble, Ω is the angular velocity vector Eulerian two-fluid method with interfacial momentum of the body, and e T is the unit vector that is the transfer models has been widely utilized due to its translational component of U B,A . In comparison with Eq. computational efficiency compared to the method that (3.14) of Magnaudet [3], all quantities in Eq. (1) are fully resolves the complex interfaces of the two-phase dimensional ones, and the contribution from the outside mixture. However, current models of the lift, wall-lift, wall is ignored by assuming single bubbles sufficiently and turbulent dispersion force experienced by bubbles far from a wall. Closed terms in the right-hand side of are not sufficiently universal to predict the lateral void Eq. (1) is related to the inertia, and the surface integral fraction distribution at high- Re condition [1], where Re is mainly related to the viscous contribution of the drag is bubble Reynolds number Re = ρU R d / μ , d is the force. We are interested in the wake-induced force F wake volume equivalent diameter of the bubble ( V = π d 3 /6, V that results in the horizontal translational motion of a is the volume of the bubble), U R = | U R | = | U B - U L | is the bubble. magnitude of the bubble’s relative velocity ( U B is the { } ( ) bubble velocity, and U L is the velocity of the ( ) ò × = r w + w ´ × - e F U e U dV (2) T wake B L T L A , undisturbed liquid flow taken at the bubble center), and V L ρ and μ are the density and dynamic viscosity of the liquid, respectively. With this background, Lee and Lee Fig. 1 shows the present idealized concept of the [2] reported experimental measurement results of the wake-induced-zigzag motion of a free rising bubble in a lift force coefficient C L at 440 < Re < 7200. Lee and two-dimensional viewpoint. Assuming instantaneous Lee [2] also suggested a physical model that can be planar symmetry of the wake, flow around the bubble used to effectively estimate C L at high- Re . near the symmetry plane can be approximated to such In this abstract, the C L model of Lee and Lee [2] is circumstances. The thick dashed line indicates the derived from Eq. (3.14) of Magnaudet [3] to illustrate a vortex-ring like vortex structure at the bubble equator. general picture of the wake-induced dynamics The cross point and dot point indicate vorticity vectors experienced by single bubbles in both unbounded and at the xy -plane directing to the positive z -direction and bounded cases. Moreover, the C L model’s prediction the negative z-direction, respectively. results are compared with both experimental and numerical data of the C L reported by Lee [4]. L e L U e U 2. Derivation of the wake-induced lift force U e U The force acting on a body moving in a fluid at rest with a fixed shape F H can be expressed by Eq. (3.14) of Magnaudet [3], which is given as follows. y x ( ) æ × ö × d A U e F ( ) = - × W B W ´ × T H e + A U ç ÷ r T R dt è ø (a) (b) (c) { } ( ) ( ) ò + w + w ´ × - U U U dV (1) Fig. 1. Generation of the wake-induced lift force acting on a B L B A , L A , V 0 L zigzagging bubble 2 ( ) ( ) ò - - ´ w - W × U U 2 n dS L A , B A , Re By approximating the vortex at the surface of the S B bubble to axisymmetric, the contribution of ω B to lift where U B,A and U L,A are the auxiliary unit velocity of the force can be ignored. When the wake becomes unstable, body and the auxiliary irrotational velocity field, a hairpin vortex is detached from one side, as illustrated respectively. ω and ω B are the free vorticity and bound in Fig. 1. The vortex detached from the right side of the vorticity, respectively. ω B only exists at the surface of bubble in Fig. 1 can be approximately expressed as the the bubble. A is the added mass tensor, n is the unit negative z directional line vortex Γδ ( x - x V ) δ ( y - y V )(- k ),
Transactions of the Korean Nuclear Society Virtual Spring Meeting July 9-10, 2020 ( ) where Γ is the magnitude of the circulation, ( x V , y V ) is 3/2 c c - 2 5/3 2 1 U = the position of the detached vortex, and δ ( x ) is the Dirac w (5) R ( ) E 1/2 r - delta function. c 2 1 c - c 2 - sec 1 In Eq. (2), U L is the liquid velocity at ( x V , y V ) and represents the velocity of the convected vortex. U L ( x V , Applying Eqs. (4) and (5) to 4 ρlΓ 0 ω 0 X with l = 2 b y V ) may be similar to the body’s relative velocity right and X = b /2, Lee and Lee [2] finally derived the wake- after the detachment of the vortex and will be induced contribution of the lift coefficient C L,wake as continuously slowed down. U L,A is determined by the follows. kinematical boundary condition at the bubble interface, ( ) ( ) 3/4 3/2 c 4/3 + c 2 c 2 - 1 1 p U L,A ∙ n = e T ∙ n , and | U L,A | approaches to 0 at far from 24 = - (6) C Oh the body, i.e., U L,A decays as r -3 . Therefore, it can be ( ) L wake , 3/4 1/2 2 c 2 - 1 c - c 2 - sec 1 approximated that U L ≈ U R e V and U L,A << 1 at ( x V , y V ), where e V is the unit vector directed to the vortex where Oh is Ohnesorge number Oh = μ /( ρσd ) 1/2 . The convection. Then Eq. (2) can be simplified as follows. miswritten constant of proportionality of [2] is corrected here. In order to describe the positive C L at { } ò × » r Gd d - ´ × e F ( ) ( )( x y k ) ( U e ) e dV intermediate Re , Lee and Lee [2] added 0.5 that is the T wake R V T V L (3) added mass coefficient of spherical body to C L,wake é ò ù = rG - ´ × U dz ( k e ) e ë û based on the good agreement between the model and R V T l their data. If the line integral is equal to l and e V = - U R /| U R |, = + C 0.5 C F L,wake becomes ρlΓU R also derived by de Vries et al. [5] (7) L L wake , in the case of a free rising bubble. By approximating the vorticity at the center of the Fig. 2 shows the prediction results of Eq. (7) by viscous vortex detached from the bubble to the vorticity curves in the case of single air bubbles in water flow at at the bubble’s equator ω E , Γ can be approximated to d > 1 mm. Both results obtained by approximating 4 π ( μ / ρ ) t V ω E [2, 4], where t V is the vortex shedding quasi-steady rising of bubbles in experimental cases [2] period. In the case of clean bubbles, Veldhuis et al. [7] and numerical cases [4] are also compared with them. experimentally showed that the t V is the same as 1/ f (2,0) , The experimental data were obtained by using where f (2,0) is the (2,0) mode frequency of the bubble contaminated bubbles at 26.7℃ (440 < Re < 7200), and shape oscillation. the numerical data were obtained from clean bubbles at 29.9℃ (400 < Re < 4000). For χ , (1 + 0.21 Eo 0.58 ) and 1/2 æ ö (1.8 + 0.036 Eo 1.1 ) are used for the experimental case, 1/2 c æ s ö 2 1 16 2 ç ÷ = and numerical case, respectively [2, 4], where Eo is f (4) ç ÷ ( ) ç ( ) ÷ 2,0 p 3/2 r 3 2 r c + è ø 2 Eötvös number Eo = ρgd 2 / σ , and g is the gravitational 1 è ø acceleration. At d < 4 mm, l = b /2 is applied to represent experimental observation [5]. where χ = b / a is the shape aspect ratio, a and b are the lengths of the minor and major semi-axes of the spheroidal shape of the bubble, respectively. σ is the liquid surface tension, and r is d /2. Until this stage, the derivation of F L,wake is applicable to unbounded single bubbles at arbitrary Re conditions. 3. Comparison with data (unbounded single high- Re bubbles in a linear shear flow) C L [-] Accounting for the spatial variation of ΓU R in the case of single unbounded bubbles in a linear shear flow, the time-averaged lift force acting on the bubble can be approximated to 4 ρlΓ 0 ω 0 X . Γ 0 is Γ evaluated with U R at the center of the bubble, ω 0 is the shear ratio of the flow, and X is the horizontal distance from the bubble center to consider the location of the detached vortices from the bubble [2]. At high- Re condition, ω E can be derived from the potential theory as follows [6]. Fig. 2. Validation of the model on the wake-induced contribution to the lift coefficient at high- Re condition [2, 4]
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