Simulations for Microbubble Drag Reduction (MBDR) at High Reynolds Numbers

Simulations for Microbubble Drag Reduction (MBDR) at High Reynolds Numbers M.R. Maxey, S. Dong, J. Xu, and G.E. Karniadakis Division of Applied Mathematics, Brown University, Providence, RI maxey@cfm.brown.edu and {jin_xu, sdong, gk}@dam.brown.edu HPCMP Challenge Project DARPA 556C1L Abstract Simulations of turbulent channel flows at Re = 135 400, seeded with microbubbles, have been used to investigate the dynamics of microbubble drag reduction. The bubbles are found to exert an effective streamwise body force on the flow that retards the near-wall flow and results in a reduction in drag. Besides this density effect of the bubbles there is a suspension viscosity effect, and both depend on the local void fraction and bubble size. In addition, collisions or contacts between bubbles can lead to an enhanced dispersion at larger void fractions. These features are investigated for a range of conditions to guide the development of RANS models. 1. Introduction Since the early experiments by Madavan, et al. (1985a) it has been repeatedly shown that the injection of gas microbubbles into the near-wall region of a turbulent boundary layer will lead to a reduction in drag. Further experiments are summarized by Deutsch, et al. (003) and Sanders, et al. (003), covering separately rough-walled flows and very high Reynolds number flows. For rough walls, the levels of drag reduction with microbubbles are similar to those achieved with smooth walls at the same Reynolds number. In the large-scale experiments of Sanders, et al. (003), persistence of drag reduction was found to be a limiting factor and this was closely linked to the rate at which bubbles disperse away from the wall. Even though there is a substantial body of experimental data many fundamental questions remain about the underlying mechanisms for drag reduction. Various models have been proposed by Legner (1984), Madavan, et al. (1985b), and Marie (1987) that focus on the change in effective density and viscosity of the bubble-liquid mixture. Our earlier results Xu, et al. (00) using direct numerical simulations for small bubbles seeded in a turbulent channel flow were the first demonstrate drag reduction. Other simulation studies have been reported by Ferrante & Elghobashi, 004 and Lu, et al. 004. Many factors undoubtedly contribute to the drag reduction process including bubble deformation, or evolution of the bubble sizes by splitting or coalescence, as well as the issues of void fraction, bubble size and method of injection. Our present goal is to focus on a specific set of issues in the context of microbubbles in a turbulent channel flow. We consider bubbles of fixed radius a, with a size range (in wall variables) of a + = 13.5 0, that remain spherical under the influence of surface tension and respond as essentially rigid particles through the effects of surfactant contamination, which is common for small bubbles. The equilibrium sizes of the bubbles observed in the experiments are 100 300 microns in diameter, and are larger than the buffer layer scales while small compared to the boundary layer thickness. In seawater, the equilibrium bubble sizes tend to be somewhat smaller and coalescence of bubbles is inhibited. We summarize here results at Re = 135 400 for the density effect and suspension viscosity effect arising from the microbubbles, and the rates of bubble dispersion at finite void fractions. These results are obtained for a periodic channel flow. We also report on the recent development of a spatially evolving channel flow to investigate the effects of bubbles injected from an upstream source.. Simulation Methods As in Xu, et al. (00), the coupled two-phase flow dynamics of the micro-bubbles and the turbulence is simulated by the force-coupling method (FCM), described in Maxey & Patel, (001), Lomholt, et al. (00) and Lomholt & Maxey, (003). Fluid is assumed to fill the whole flow domain, including the volume occupied by the bubbles. The presence of each bubble is then represented by a finite force monopole and a force dipole that generates a body force distribution f(x, t) on the fluid. This transmits the resultant force of the bubbles on the

flow to the fluid. The volumetric velocity field u(x, t) is incompressible and satisfies Du Dt p u fx, t, (1) where μ is the fluid viscosity and p is the pressure. The body force due to the presence of N B bubbles is N B n n, n n fi t Fi t Gij t x n1 j x x Y x Y, () where Y (n) is the position of the n th pherical bubble and F (n) is the force this exerts on the fluid. The force monopole for each bubble is determined by the function (x) which is specified as a Gaussian function 3/ x exp x / (3) and the length scale is set in terms of the bubble radius a as a /. The velocity of each bubble V (n) (t) is found by forming a local average of the fluid velocity over the region occupied by the bubble as n 3 V t u x, t xy n t d x. (4) From this the motion of each bubble is computed as n d Y =V n n t V C V n W dt. (5) n The additional term V C is the bubble-collision velocity and represents the effect of short-term contact forces n between bubbles and V W similarly represents the effect of bubble contact with a wall. It is important to include the collision or contact effects in order to maintain the bubble void fraction. Details of these short-range interactions are given by Dance, et al. (004). If m B and m F denote the mass of a bubble and the mass of displaced fluid, the force of the bubble acting on the fluid is n n dv F mb mf g. (6) dt This force is the sum of the net external force due to buoyancy of the bubble and the excess inertia of the bubble over the corresponding volume of displaced fluid. n The symmetric part of the force dipole G ij, the stresslet term, is set through an iterative procedure to ensure that the strain-rate acting on a bubble, spatially averaged on the bubble scale, is zero. The envelope length scale associated with is defined as a / 6, see Lomholt & Maxey, (003). 1/3 A spectral/hp element method (Karniadakis & Sherwin, 1999) has been used to solve for the primitive variables u, p in the Navier-Stokes equations for the channel with rigid walls at x = ±h. Periodic boundary conditions are applied in the other two directions with dimensions L 1 L 3. A uniform mean pressure gradient dp/dx 1 is applied in the streamwise direction and adjusted continuously to ensure that a constant volume flow rate is maintained. The bulk velocity is kept constant, U B = /3. At Re = 135 and 19, the domain size is with a numerical resolution of 18 97 18. Dealiassing schemes and over-integration are used to evaluate the nonlinear terms. At Re * = 380 the domain size is and the resolution is 56 41 56. 3. Periodic Channel Results 3.1. Density Effect We consider first the density effect of the microbubbles on the flow. This is achieved by retaining the force monopole term in () but setting the dipole term to zero. Results for simulations at Re * = 19 are shown in Figure 1 for a flow seeded with bubbles of radius a = 0.0675, or a + = 13.5 with an average void fraction of 4.%. The bubbles are initially seeded as two concentrated layers near each wall that then disperse under the action of the turbulence. The mean velocity gradient at the two walls drops over time and for the case shown the drag is reduced by about 7%. This is typical of the level of drag reduction seen in these relatively low Reynolds number flows. The bubbles disperse quickly during the initial phase and there is evidence (described later) that bubble-bubble collisions play a significant role in this process. Figure 1. Bubble concentrations at t = 0.5, 15, 35 and mean wall shear, at each wall, for Re = 19 and 4.% void fraction: no bubbles (solid), bubbles (dashed) Similar results are seen at Re * = 380, as shown in Figure for bubbles of radius a + = 19. In one instance the

bubbles were seeded near each wall in multiple layers to give an average total void fraction of 8.4%. In the other, the bubbles were dispersed randomly throughout the flow with a uniform distribution and an average void fraction of 1.73%. With near-wall seeding, the void fraction close to the wall drops from 8% at t = 10 to 18% at t = 0. The initially stronger drag reduction of 7 8% eventually diminishes over time as the bubbles disperse, while with the uniform seeding a lower level of 4% drag reduction is sustained. At the lower Reynolds number Re * = 19, bubbles of radius a + = 19 produced only a very weak reduction in drag. In general, the profiles for the turbulent velocity fluctuations such as u 1 show reductions, as does the Reynolds shear stress u u. Both are reduced at levels 1 consistent with the degree of friction. reduction in the skin Figure. Mean wall shear, at each wall, for bubbles of radius a + = 19 at Re = 380: solid line, near-wall seeding at void fraction 8.4%; dotted line, uniform random seeding at void fraction 1.73%; dashed line, no bubbles average value Figure 3. Force profiles for 1 (solid line), (dashed line) and 3 (dotted line) at t = 0 for Re = 380, bubble radius a + = 19 at void fraction 1.7% The displacement of fluid by the bubbles plays an important part in the drag reduction process. The average effect of the bubble phase on the mean momentum can be interpreted, using (1), as being equivalent to a streamwise body force 1. Statistics for 1 show that it is strongly negative close to the walls and then is positive in the core region, with the bubbles effectively slowing the fluid close to the wall. In contrast, a flow seeded with solid particles, denser than the fluid, produces the opposite effect and there is an increase in drag. Results for 1 are shown in Figure 3 for the uniformly seeded flow at Re * = 380, given at t = 0. The integral of 1 across the channel is zero in steady state. Where bubbles are seeded near the walls and then disperse the overall force is positive and contributes to a lower pressure gradient. The peak negative values near each wall are consistent with a density weighted scaling, f1 c u 1u. (7) x The body force acts in opposition to the gradient of the usual Reynolds shear stress. There is a body force component in the crossstream direction that is balanced on average by a pressure gradient effect. This is linked to a weak turbophoretic effect that tends to enhance the drift of bubbles from the wall. Specific features such as the degree of correlation of bubbles with the turbulence and their spatial distribution is dependent on the bubble size a +. 3.. Dipole Effect These results have been computed using only the force monopole term in (). If the bubbles are treated as rigid spherical inclusions due to the effects of surface tension and surfactant contamination, then a symmetric dipole term G ij should be considered too. For a laminar flow this additional stresslet term leads to an enhanced viscous dissipation and increase in the effective viscosity of a random suspension. Even for a neutrally buoyant particle moving with the surrounding fluid, any local rate of strain in the flow will be deflected around the particle. As shown by Lomholt & Maxey, (003), in a viscous Stokes flow the FCM scheme gives the usual Stokes- Einstein estimate for the suspension viscosity for the dilute limit as μ e = μ(1 +.5c). Even for a clean spherical bubble in a viscous flow, with free-slip boundary conditions, there is an increase in the effective viscosity and but with deformable, non-spherical bubbles it is possible to have a reduced viscosity. Tests with the present simulations using 800 neutrally buoyant spherical particles, radius a/h = 0.1, in a laminar Poiseuille flow at a

flow Reynolds number of unity verified the Stokes- Einstein result. For an average void fraction of 4.% the increase in drag is 10%. We have obtained results for a turbulent channel flow at Re = 135, where the flow is seeded with 800 bubbles of radius a/h = 0.1 that initially are evenly distributed in two layers at x /h = ±0.85. As reported by Xu, et al. (00), with the monopole term (density effect) only, there is a 6% reduction in drag. With both terms included, there is an increase in the overall skin friction of about 8%. There is a small increase in the Reynolds shear stress but mostly the stresslet term (force dipole) has its strongest effect in the near-wall region where the viscous shear stresses are large. Results are shown in Figure 4 for a turbulent channel flow at Re = 135, where the flow is seeded with 60 bubbles of radius a/h =0.5 at x /h = 0.66. The void fraction decreases from an initial peak value of 8% to around 1% at t = 60. There is an initial increase in the drag but this eventually drops again to either no net change in the drag or possibly a marginal reduction in the drag. For bubbles of this size there is a negligible change in the drag with just the monopole term (density effect). With both effects included, the Reynolds shear stress now shows a significant reduction between 0.8 < x < 0.4 with the peak value reduced by one third. Figure 4. Bubble concentration and mean wall shear for Re = 135 and 5% void fraction: no bubbles (solid), bubbles (dashed) Whether the dipole terms, or suspension viscosity effect, increases or reduces drag depends on the size and location of the bubbles as well as the Reynolds number. Smaller bubbles, in the viscous wall-region, will contribute more dissipation locally and as in a laminar flow will tend to increase drag. At higher Reynolds numbers and with larger bubbles, the increase in dissipation also acts to damp the turbulence in the buffer layer and beyond, and so reduce the Reynolds stresses. Results at higher Reynolds numbers suggest that this damping of the Reynolds stresses is more significant. 3.3. Bubble Dispersion A common underlying feature is that there must be a sufficient concentration of bubbles near the wall to achieve drag reduction. Bubble dispersion eventually leads to a loss of persistence of drag reduction so it is important to understand the factors contributing to the bubble flux. Turbulence is an obvious factor and for low void fractions this is the dominant mechanism. Bubblebubble contacts or collisions though become important as the void fraction increases. Bubble-bubble contacts are usually considered in the context of bubble coalescence, where the merger of bubbles depends on the rate of bubble collisions and the duration of contact as well as physical and chemical properties of the bubble surface and liquid. The issues of modeling bubble coalescence in RANS simulations is discussed by Kunz, et al. (005). Small bubbles in sea-water do not readily coalesce and contact between bubbles due to the turbulence will often result in simple collisions. With the present simulations we are able to evaluate the separate contribution of the turbulence and bubble collisions to the flux of bubbles cv using (5) with N 1 B n n n n cv mf V t VC VW x Y. (8) n 1 The contributions on the right are due to the turbulent flow, bubble-bubble collisions and bubble-wall collisions respectively. Profiles for these different terms are shown in Figure 5, taken at t = 15 for the same conditions as shown in Figure 1. The magnitude of the turbulent flux scales overall with the average void fraction C 0 =4.% and u *, and is a function of the concentration gradient and the local turbulence levels. Adjacent to the wall there is a counterbalance between bubble-bubble collisions and bubble-wall collisions, but further from the wall the bubble-bubble collisions contribute to the overall dispersion flux of the bubbles. Even at a local concentration of 8 10%, this contribution is comparable to the turbulent flux and increases with c.

Figure 5. Flux profiles at t = 15 for Re = 19, bubble radius a + = 13.5 at void fraction 4.% We have investigated the effects of bubble-bubble collisions for both the channel flow and for homogeneous turbulence to verify the underlying scaling relations. Collisions or contacts between bubbles (or particles) are not normally considered as a significant factor affecting turbulent dispersion in dispersed two-phase flow. However our results over a range of Reynolds numbers show that collisions are an important factor in the initial bubble dispersion and this has been verified in the comparison of RANS simulations with experiments, Kunz, et al. (005). same viscosity value is used corresponding to a Reynolds number Re = 19 based on the friction velocity of the periodic channel (or the inlet of spatial channel) and the channel half width. The spatial channel code employs an efficient two-level parallelization scheme (Dong & Karniadakis, 004) to achieve level distribution of computational effort with increasing number of processors as we expand the domain size and bubble void fractions. It can efficiently scale to over 1000 processors as demonstrated by Dong & Karniadakis, (004). Bubbles of radius a/h = 0.0675 are introduced into the channel from two slits in the spanwise direction, located near the walls at x/h =1.0, and y/h = ±0.9. The total number of bubbles in the flow domain is maintained at 4900, with a bubble void fraction 4.%. This is achieved by introducing a new bubble at a random location along the slits once a bubble convects out of domain. Bubble and flow statistics are collected by averaging along the spanwise direction and over time after the statistically stationary state is reached. Figure 6 visualizes all the bubbles in the spatial channel with a side view (Figure 6a) and a perspective view (Figure 6b). As the bubbles travel downstream, they disperse toward the middle of the channel due to the interaction with turbulence, bubble-wall collisions, and bubble-bubble collisions. As a result, the two near-wall regions occupied by bubbles grow along the streamwise direction and show a trend to merge around the outlet with the current channel dimensions. Bubble distribution shows streaking structures (Figure 6b), reminiscent of the streamwise streaks of the turbulence. 4. Spatially Evolving Turbulent Channel Results Issues of microbubble dispersion are further investigated with the development of a spatially-evolving turbulent channel flow code. This will provide closer comparisons with experiments and RANS simulations since stable long-term averages of turbulence and bubble statistics can be formed. We employ a spatially-solving channel with dimensions 1h in streamwise x-direction, h in wallnormal y-direction, and h in the spanwise z-direction, where h is half width of the channel. No slip conditions are applied at channel walls, and periodic conditions are imposed in the spanwise direction. An outflow condition is applied at the channel outlet. The instantaneous inlet flow field is taken from a fixed cross-stream plane of a periodic turbulent channel flow that is performed side by side. The periodic channel and the spatial channel are compatible in geometry (same dimensions and resolutions in both spanwise and wall-normal directions), and the Figure 6. Visualizations of bubbles in a spatiallydeveloping turbulent channel flow: (a) side view; (b) perspective view The presence of bubbles has a notable effect on the wall drag. In Figure 7a we plot the distribution of wall drag force along the streamwise direction. In the section of the channel preceding the bubble seeding location, a high drag level is observed. On the other hand, the

section of channel seeded with bubbles shows a decreased wall drag level, about a 5% reduction, which is comparable to the levels of drag reduction observed in periodic channel. Figure 7. a) Wall friction drag distribution and b) Bubble concentration profiles at several streamwise locations Figure 7b shows profiles of bubble concentration at several streamwise locations, providing a quantitative measure of the bubble distribution. The peak concentration of bubbles decreases from 0.47 at the seeding location to about 0.1 at the outlet. A second weaker peak in bubble concentration develops downstream away from the seeding location. 5. Further Results With resources provided from HPCMP, we have developed new high Reynolds number databases for channel flows at Re * = 630 and 930. These base flows will be used as inflow conditions for the new spatial channel flow simulations and will allow us to explore much higher Reynolds number conditions. Simulations at Re = 1000 have only recently been possible, and we have found the use here of a spectral element formulation provides an efficient balance of high, near-wall spatial resolution, and restrictions on time-step size. In a separate study, we have investigated the effects of a steady, streamwise force acting on a single-phase turbulent channel flow. The steady controlling force has a distribution similar to the time-averaged force produced by the presence of microbubbles in the turbulent channel flow shown in 3. Our goal here was to understand better the influence of this force density and the controlling parameters of the amplitude of the negative force at each wall and the length scale of the near-wall variation. Sustained drag reduction is achieved if the controlling force acts within a distance of about twenty wall units. By varying the amplitude, the skin friction can be reduced by as much as 70% by this constant streamwise forcing. This was tested for Reynolds numbers Re = 135 630 and the larger drag reduction is achieved at higher Re *. If the forcing is spread over a larger length scale there is only a transient reduction in skin friction. 6. Concluding Remarks The goals of our work have been to test different physical mechanisms involved in MBDR, and to develop physics-based models for RANS simulations. Using direct numerical simulations of turbulent channel flow we have investigated the separate contributions of the density effect and the enhanced suspension viscosity on the dynamics of microbubble drag reduction. Within the size range considered and the Reynolds numbers Re * = 135 380 we find that the density effect will lead to reductions in skin friction of about 10%. Larger levels of drag reduction, for a broader range of bubble size, are found for higher Reynolds numbers. At low Reynolds numbers the suspension viscosity effect will tend to increase skin friction, but will dampen Reynolds stresses at higher Reynolds numbers. In addition we have indicated the importance of bubble-bubble collisions to bubble dispersion, especially near the wall where local concentrations are large. 7. Significance to DoD Turbulent skin-friction is the major component of the drag force on a surface ship and achieving drag reduction would reduce fuel requirements and increase payload capacity. Gas injection in the form of microbubbles has been demonstrated experimentally as a method for drag reduction. The present work is part of a project to develop predictive engineering RANS models of microbubble drag reduction that can be used in the transition from small scale test systems to full scale applications of ship hydrodynamics. Systems Used ARSC, NAVO, ARL Acknowledgements This work was supported by a challenge grant of HPC resources from the Department of Defense High Performance Computing Modernization Program. Support by DARPA under the Friction Drag Reduction Program (ATO) is gratefully acknowledged. (The content of this paper does not necessarily reflect government policy and no official endorsement should be inferred.)

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