GRAVITY-DRIVEN MOTION OF A SWARM OF BUBBLES IN A VERTICAL PIPE

27th International Conference on Parallel Computational Fluid Dynamics Parallel CFD2015 GRAVITY-DRIVEN MOTION OF A SWARM OF BUBBLES IN A VERTICAL PIPE Néstor Balcázar, Oriol Lehmkuhl, Jesús Castro, Joaquim Rigola, Assensi Oliva Heat and Mass Transfer Technological Center (CTTC) Universidad Politécnica de Cataluña ETSEIAT, Colom 11, 08222 Terrassa (Barcelona), Spain e-mail: cttc@upc.edu, web page: http://www.cttc.upc.edu Termo Fluids, S.L. Avda Jacquard 97 1-E, 08222 Terrassa (Barcelona), Spain e-mail: termofluids@termofluids.com Key words: Two-phase flow, Level-set method, Bubbles, Droplets, Parallel CFD applications Abstract. In this work, Direct Numerical Simulations of gravity-driven motion of a swarm of bubbles in a vertical pipe are presented. These simulations were carried out in the contex of a multiple marker level-set methodology, which is integrated in a finitevolume framework on collocated unstructured grids. Each fluid particle is described by a separate level-set function, thus, different interfaces can be solved at the same control volume, avoiding the coalescence of the bubbles. Present simulations have been performed in a periodic vertical domain divided in 21 10 6 hexahedral control volumes, distributed in 2048 CPUs. Collective and individual behaviour of the bubbles are characterized and compared with previous results reported in the literature for other flow configurations. 1 INTRODUCTION Gravity-driven bubbly flows play an important role in many natural and industrial processes. Steam generators in nuclear plants, unit operations in chemical engineering such as distillation, absorption, extraction, heterogeneous catalysis and bubble reactors are only a few among a multitude of applications that involve the motion of bubble swarms [9]. These applications have motivated a large number of numerical and experimental investigations of bubble dynamics, however, despite those efforts, the current understanding of such flows and their predictive models are far from satisfactory, and many challenging problems still remain as pointed out in recent reviews [9, 13]. The development of parallel computation has promoted Direct Numerical Simulation (DNS) of the Navier-Stokes equations as another means of performing controlled experiments [14, 13], providing a good way to non-invasive measure of droplet and bubble flows, although computationally demanding. There are multiple methods for DNS of bubbly flows, for instance: the front tracking (FT) method [14], level set (LS) methods 1

[11, 10, 1, 2, 3], volume-of-fluid (VOF) methods [7], and hybrid VOF/LS methods [12]. In these methods, two-phase flow is treated as a single flow with the density and viscosity varying smoothly across the moving interface which is captured in an Eulerian framework (VOF, LS, VOF/LS) or in a Lagrangian framework (FT), moreover, although the idea behind these methods is similar, their numerical implementation may differ greatly. Forthepresent DNSstudy, amultiplemarker level-set methoddeployed in[2]isusedto perform simulations of gravity-driven motion of a swarm of bubbles in a periodic vertical pipe. The main advantage of this method is its capacity to avoid the numerical coalescence of the bubbles, which is an artifact of interface capturing methods such as VOF and LS approaches. Hence, bubbles are able to approach each other closely within the size of one grid cell, and can even collide, while the bubble volume is kept constant throughout the simulation [2]. To the best of the authors knowledge, there are not previous DNS studies of gravity-driven bubbly flows in vertical pipes, therefore this is the main motivation of the present research. This paper is organized in the following order: The governing equations and numerical methods are presented in Section 2. Section 3 is devoted to the discussion of the numerical results. Finally, concluding remarks are given in Section 4. 2 MATHEMATICAL FORMULATION AND NUMERICAL METHODS The Navier-Stokes equations for the dispersed fluid in Ω d and continuous fluid in Ω c can be combined into a set of equations in a global domain Ω = Ω d Ω c, with a singular source term for the surface tension force at the interface Γ: (ρv) t + (ρvv) = p+ µ ( v+( v) T) +(ρ ρ 0 )g+f σ v = 0 (1) ρ = ρ d H d +ρ c (1 H d ) µ = µ d H d +µ c (1 H d ) (2) wherevandpdenotethefluidvelocityandpressurefieldrespectively, ρisthefluiddensity, µ is the dynamic viscosity, g is the gravitational acceleration, f σ is the surface tension force, subscripts d and c are used for the dispersed and continuous fluids respectively, while H d the Heaviside step function that is one in Ω d and zero elsewhere. Because a periodic domain is used in the y axis direction, a force ρ 0 g is added to the Navier-Stokes equations [3], with ρ 0 = V 1 Ω (ρ Ω dφ d +ρ c (1 φ d ))dv. The conservative level-set method (CLS) deployed in [1] has been selected for interface capturing on unstructured meshes. Moreover, multiple level-set functions are used in order to avoid the coalescence of the bubbles, according to the work reported by [2], therefore, the interface of the i th fluid particle is defined as the 0.5 iso-surface of a regularized indicator function φ i, where i = 1,2,...,n d and n d is the total number of fluid particles contained by the dispersed phase. The i th interface transport equation can be written in conservative form provided the velocity field is solenoidal, v = 0, namely, φ i t + φ iv = 0 (3) 2

Furthermore, an additional re-initialization equation is introduced in order to keep a sharp and constant interface profile φ i τ + φ i(1 φ i )n i = ε φ i (4) This equation is advanced in pseudo-time τ up to steady state. It consists of a compressive term, φ i (1 φ i )n i τ=0, which forces the level-set function to be compressed onto the interface along the normal vector n i, and of a diffusion term ε φ i that ensure the profile remains of characteristic thickness ε. Geometrical information on the interface Γ i, such as normal vector n i or curvature κ i, is obtained through: n i (φ i ) = φ i φ i κ i (φ i ) = n i (5) Surface tension forces are calculated by the continuous surface force model [5], which has been extended to include multiple markers in the same grid cell [2], f σ = σ i κ i (φ i ) φ i (6) Finally, in order to avoid numerical instabilities at the interface, fluid properties in Eq. 2 are regularized by using a global level-set function H d = φ d, defined as φ d (x,t) = max{φ 1 (x,t),...,φ nd 1(x,t),φ nd (x,t)} (7) The governing equations have been discretized on a collocated unstructured grid arrangement by means of the finite-volume method, according to [1]. In order to avoid unphysical oscillations of the level-set function, a TVD Superbee limiter [1] is used to discretize the convective term of advection Eq. 3, while the compressive term of the reinitialization Eq. 4 is discretized by using a central difference scheme (CD). CD scheme is also used to discretize the convective term of momentum Eq. 1. A distance-weighted linear interpolation is used to find the cell-face values, while gradients are computed at cell centroids by using the least-squares method [1]. The velocity-pressure coupling has been solved by means of a classical fractional step projection method. The reader is referred to [1, 2, 3] for technical details on the discretization of the Navier-Stokes and Level-set equations on unstructured meshes that are beyond the scope of this paper. 3 NUMERICAL RESULTS AND DISCUSSION The multiple marker level-set method has been validated and verified in previous publications by performing simulations of single bubbles rising in an initially quiescent liquid [1, 3], drop collision with a fluid-fluid interface [2] and binary droplet collision with bouncing outcome [2]. For present simulations, the computational set-up is defined as a vertical cylinder bounded by a rigid wall, with gravity in the y direction. The size of the domain is (D Ω,H Ω ) = (5d,4d), where d is the initial bubble diameter, D Ω is the cylinder diameter 3

and H Ω is the cylinder height, as shown in Fig. 3a. The domain Ω is divided in 21 10 6 hexahedral control volumes distributed on 2048 CPUs, whereas the grid size is given by h = d/60. As initial condition, both bubbles and liquid are quiescent. Imposed boundary conditions are non-slip at the rigid wall and periodic on the streamwise (y-direction). In this way bubbles go out of the domain on the top side, and they come back in the domain again from the opposite side. Bubbles and droplets rising or falling freely in infinite media can be characterized by four dimensionless numbers: M gµ4 c ρ ρ 2 c σ3 Eo gd2 ρ c σ η ρ ρ c ρ d η µ µ c µ d (8) where, η ρ andη µ arerespectively thedensity andviscosity ratio; M isthemortonnumber; Eo is the Eötvös number; ρ = ρ c ρ d, specifies the density difference between the continuous and dispersed fluid phases, and d is the spherical volume equivalent diameter of the droplet, while the subscript d denotes the dispersed fluid phase and c the continuous fluid phase. Simulations presented in this work were performed at Eo = 3, M = 10 6, η ρ = 10 and η µ = 10, which correspond to deformable bubbles. Finally, in order to report the numerical results, the velocity of the bubble centroid, v i, and the Reynolds number, Re i, are calculated as follows: v i (t) = vφ Ω i(x,t)dv φ Ω i(x,t)dv Re i (t) = ρ cd v i e y µ c for i = 1,..,n d (9) where e y is a unit vector parallel to +y direction. In addition the average Reynolds number of the bubble swarm is calculated as Re d = 1 n d Re i (10) n d Once the velocity of each bubble has been calculated by Eq. 9, the trajectory of the bubble can be determined by: x i (t) = x 0 i + i=1 t 0 v i (t)dt (11) where x 0 i is the initial position of the i th bubble centroid. First, we consider the motion of two bubbles in an initially quiescent fluid where the bubbles are initially released with different configuration angles and initial centroidcentroid distance of 2d in a periodic vertical pipe. As the bubbles move upward due to the buoyancy effect, their interaction leads to the formation of complex trajectories, as is illustrated in Fig. 1. Figure 2a shows the dimensionless separation distance between the bubbles, s/d, versus the dimensionless time for the different initial configuration angles, θ 0 = {0 o,45 o,60 o,75 o,90 o }. It can be seen that two deformable bubbles repel each other, except for θ 0 = 90 o where the bubbles experience the drafting-kissing-tumbling (DKT) 4

Figure 1: Interaction of two deformable bubbles in a vertical pipe at Eo = 3, M = 10 6, η ρ = 10 and η µ = 10, for different configuration angles Figure 2: Interaction of two deformable bubbles at Eo = 3, M = 10 6, η ρ = 10 and η µ = 10 (a) The centroid-centroid distance between two bubbles versus dimensionless time. (b) The angle between two bubbles versus dimensionless time. phenomenon. Moreover, the bubbles tend to migrate to the wall, however a repulsion force avoids the bubble-wall collision. Figure 2b illustrates the time evolution of θ for different initial configuration angles θ 0. A decrease of θ is observed as the time advances, 5

Figure 3: Eo = 3, M = 10 6, η ρ = 10 and η µ = 10. (a) Bubble distribution and velocity vectors on the symmetry plane x y for t = {2,20} (b) e z v on the plane x y, t = 20. (c) Bubble trajectories integrated from t = 0 up to t = 28. therefore it indicates a torque action on the bubbles that tends to align them side-by-side. We now study a free bubble array of 16 fluid particles which are initially placed in the periodic vertical pipe following a random pattern. This corresponds to a dilute bubbly flow, with an overall volumetric fraction of α Ω = 5.33%, moreover, since both fluids are assumed to be incompressible and coalescence of the bubbles is not allowed, α is constant throughout the simulation, so that statistics can be obtained for a constant number of bubbles. Fig. 3a shows snapshots of the swarm of bubbles and their periodic images at the dimensionless time t = t(g/d) 1/2 = {2,20}, and the velocity vectors on the symmetry plane of the cylinder. Fig. 3b illustrates the voticity field e y v generated by both the wall and bubbles on the x y plane at z = 0 and t = 20. Close to the wall, the vorticity generated by bubbles and wall have opposite signs, moreover, its interaction produces a velocity reduction, which leads to a pressure increment due to the Bernoulli effect, and a repulsion force directed from the wall to the bubbles. Fig. 3c shows the bubble trajectories integrated from t = 0 up to t = 28, where it is observed that bubbles move not only in streamwise direction, both also in radial direction. Fig. 4a shows the Reynolds number versus time, defined for each bubble as Re i = ρ c d(v i e y )µ 1 c with i = 1,2,..,n d. Even though the individual bubble motion show a transient behaviour due the wake interaction, the whole swarm of bubbles reaches a steady state at approximately t = 4. Fig. 4b shows the radial position for each bubble centroid, which lead to the formation of a bubble curtain between the wall and the symmetry axis, produced by the alignment of the bubbles at approximately constant distance from the wall. When 6

Figure 4: (a) Time evolution of Re for each bubble, and averaged Re. (b) Time evolution of radial position for each bubble centroid. all bubbles are at approximately the same vertical position, the probability of collision increases and these interactions are addressed by the DKT behaviour, which lead to the formation of horizontal clusters, as shown Fig. 4a. Present results are consistent with simulations reported by [6] using the front-tracking method [13]. 4 CONCLUSIONS Direct Numerical Simulations of gravity-driven motion of a swarm of bubbles in a vertical channel have been performed. These numerical experiments demonstrate the ability of the multiple marker level-set method [2] to simulate bubbly flows without numerical coalescence. Regarding the interaction of two-bubbles in a vertical pipe, a repulsion force was observed except for the in-line configuration where the bubbles follows the DKT phenomenon. For the present flow conditions, it was observed that deformable bubbles do not collide with the wall, moreover bubble-bubble interactions in the swarm follows also the DKT behaviour which leads to the formation of horizontal clusters. 5 ACKNOWLEDGMENTS This work has been financially supported by the Ministerio de Economía y Competitividad, Secretaría de Estado de Investigación, Desarrollo e Innovación, Spain (ENE2011-28699, ENE2014-60577-R). The simulations reported in this work were carried out using computer time provided by PRACE (project 2014112666) through the MareNostrum III supercomputer based in Barcelona, Spain. REFERENCES [1] Balcázar, N., Jofre, L., Lehmkhul, O., Castro, J., Rigola, J., 2014. A finitevolume/level-set method for simulating two-phase flows on unstructured grids. In- 7

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