Elsevier

Ocean Modelling

Volume 90, June 2015, Pages 16-28
Ocean Modelling

Virtual Special Issue: Gulf of Mexico Modelling-Lessons from the spill
Numerical simulations of turbulent thermal, bubble and hybrid plumes

https://doi.org/10.1016/j.ocemod.2015.03.007Get rights and content

Highlights

  • Turbulence resolving, large eddy simulations of multi-phase, deep-water plumes are conducted.

  • An efficient, computable model accounting for primary effects of a dispersed gas phase is derived.

  • The turbulence enhancement in two-phase plumes is quantified and explained.

Abstract

To understand the near-field dynamics of blowout plumes such as the one produced by the 2010 Deepwater Horizon oil spill in the Gulf of Mexico, the effects of gas bubbles on turbulent mixing and entrainment are studied via turbulence resolving simulations. We compare the evolution of three plumes where extremely large buoyancy anomalies are produced either thermally (single phase), solely by an imposed gas phase volume fraction, or by a combination of both buoyancy forcings. The plumes, with identical volume, momentum and buoyancy fluxes at the inlet, are released into an environment stratified with a constant temperature gradient. To clarify the first-order effects of dynamically active, dispersed bubbles, we employ a simple model which neglects the momentum of the gas phase while retaining bubble induced buoyancy in the seawater momentum equation. The gas phase is then distinguished by a single, measurable parameter, the slip velocity relative to that of the liquid phase. We find that bubbles, parameterized simply by a constant slip velocity, without any explicit assumptions of direct bubble induced turbulent production, significantly increase turbulent mixing in the plume in agreement with previous experimental results. Examination of mean momenta and turbulent kinetic energy budgets shows that the increased turbulence is due to direct modification of the mean profiles of both the momentum and the active scalar fields by the slipping gas phase. The narrowing of the active scalar field in the two-phase flow results in larger direct buoyancy production of turbulent energy at all vertical levels. The turbulence production is, however, primarily mechanical. At modest values of z/D, where the slip velocity is only a small fraction of the liquid phase velocity, slip stretches the mean vertical velocity field producing larger radial gradients and increased conversion of mean to turbulent energy. This first order effect, acting on the mean vertical velocity component and not directly on the turbulence, implies that even relatively small gas volume fractions significantly enhance turbulent mixing with respect to a single phase plume.

Introduction

This work, motivated by the 2010 Deepwater Horizon (DwH) incident in the northern Gulf of Mexico, focuses on modeling and simulating the near-field region of flows driven by the extreme buoyancy fluxes typical of deep water oil well blowouts (Camilli et al., 2010). Deepwater blowout plumes are formed by the sustained point release of a hot, multiphase wellhead mixture through the riser pipe. The resulting buoyancy flux, released into a stratified fluid, was thermally equivalent to B0=O(1GW) in the Deepwater Horizon case (McNutt et al., 2011). The magnitude of the buoyancy anomaly, sustained over a period of months, and the extremely large range of spatial scales involved (0.5 m source at 1500 m depth) implies that these multiphase plumes are far from realizable laboratory conditions (Lima Neto, 2012). The extremely large, sustained buoyancy fluxes also imply that the wellhead convection problem has the potential to dynamically alter the larger scale, rotating and stratified environment.

The DwH release underscored the need for rapid and informed response to such events. An integral part of that response is the prediction of both the distribution of hydrocarbon effluent throughout the water column and the eventual form of its expression at the ocean surface. Since the near-field turbulent entrainment plays a critical role in the evolution of the plume buoyancy, mass and momentum fluxes, understanding and modelling multiphase plume dynamics in the vicinity of the release is potentially critical to accurate prediction of the near-surface pollution transport problem over of the last mile, where the spill encounters the public. The inlet buoyancy flux generates a strongly turbulent buoyant plume in which sea water entrained from the environment is mixed with the plume core fluids as they move along the water column. The evolution of the plume is strongly affected by the level of turbulent mixing and entrainment in the near-field with the ‘entrainment coefficient’ being the main parameter in classic integral solution approaches (Morton et al., 1956). It is well-known that the presence of a gas phase changes the turbulent transport properties, with bubble plumes typically exhibiting larger turbulence levels (McDougall, 1978, Milgram, 1983). There are distinct differences in the along-column effluent distribution in single and multiphase cases (Asaeda, Imberger, 1993, Lemckert, Imberger, 1993). The effects of bubbles on turbulent dynamics are studied here by comparing plumes distinguished only by the phase distribution of the (fixed) inlet buoyancy flux, either thermal (single phase) or due to the presence of gas (multiphase).

Modeling in complete, accurate detail, the physicochemical dynamics of the high pressure release of hot, multicomponent, multiphase effluent into a stratified, turbulent ocean environment is beyond our present theoretical and computational abilities. As in the case of the DwH incident, both short and intermediate term prediction and analysis of any future event will rely on simplified models for which there is existing literature. How the validity of underlying assumptions and the values of explicit parametrizations in simplified models change when such models are scaled up from the laboratory to the environment requires careful examination of the many physical and chemical processes involved.

Given their environmental and industrial importance, buoyant plumes have been the focus of considerable study. Schmidt (1941) presented an earlier dimensional analysis of thermal plumes, and the problem has attracted the attention of Batchelor (1954), Morton et al. (1956), Turner (1973; 1969) and others. Many of these efforts represent the depth dependence bulk plume measure, like momentum and buoyancy flux, in terms of the basic system parameters. Turner discusses several laboratory experiments in support of the scaling laws. List and Imberger (1973) is an early reference on plumes in stratified environments, pointing out the creation of intrusion layers. McDougall (1978) performed perhaps the first experiments of two phase convective systems, discussing the presence of dual plumes. Classification of bubble plume experiments and observations appear in Lemckert and Imberger (1993), Socolofsky et al. (2002) and Socolofsky and Adams (2005). Classical modeling efforts directed at oil spill prediction have been posed as a Lagrangian problem, in which integral plume measures, like momentum flux and centerline trajectory were computed (Zheng and Yapa, 2003). These have been tested against laboratory and limited field experiments and have demonstrated skill. In all cases, the equations are closed by relating the turbulent radial flux to the characteristic mean vertical velocity via an entrainment coefficient (Morton et al., 1956). Enhanced turbulent mixing of vertical momentum due to multiphase physics is typically modelled by a larger momentum amplification factor, reducing the vertical momentum growth rate in comparison to single phase plumes.

The goal of the present work is to understand and quantify, in the context of turbulence resolving simulations, basic physical properties of plumes driven by the extreme values of the expected buoyancy flux in a deepwater blowout. This fills a significant gap in our present knowledge of deep spills, specifically the role played by the gas in increasing the turbulent mixing. The DwH blowout involved both oil and gas escaping from the well head (McNutt et al., 2011). As a first step, we isolate the effect of bubbles on the turbulence by considering idealized situations where admittedly important physicochemical phenomena (hydrate formation, dissolution or biodegradation) are neglected and the input buoyancy is due solely to thermal and gas contributions. Given that it is not possible in such incidents to determine the exact proportions of oil and gas involved at specific instances or periods, we explore distinctions in the turbulent behavior of plumes produced by varying the relative importance of the inlet gas buoyancy contribution. Such idealized cases are closer to the flow configuration used in most experimental studies of bubble plumes but differ significantly in the magnitude of the inlet buoyancy flux. Moreover, to our knowledge there are no experimental observations of the combined effects of thermal and gas sources. Laboratory experiments of bubble and oil plumes face a number of challenges. Usually the tank size is limited in the vertical direction and the reflection of the plume from the upper boundary can change the flow field significantly. Due to box-filling problems, limited tank size necessarily limits both the inlet oil flux and the overall time scales of experiments. To attain flows that are highly turbulent, nozzle sizes used in the laboratory are typically on the order of a few millimeters. Even then, the observation period for measurements is typically minutes (Brandvik, Johansen, Leirvik, Farooq, Daling, 2013, Johansen, Brandvik, Farooq, 2013). It is unclear how representative plumes from such small nozzles are for the real oceanic case. Finally, we are not aware of laboratory experiments of hybrid (bubble and oil) plumes and highly-parameterized models do not allow direct exploration of the details of the turbulent behavior in such cases.

Computationally, significant efforts have been devoted to the study of multiphase plumes at oceanographic scales in the context of CO2 sequestration (Alendal, Drange, 2001, Sato, Sato, 2002). These studies emphasize the effects of physico-chemical phenomena on overall plume behavior using standard eddy-viscosity and eddy-duffusivity models to account for turbulent transport. Characterization of the differences in the turbulent mixing between single and multiphase plumes requires explicit resolution of finer scale motions. This is particularly important for plumes where differences in near-field turbulent entrainment processes significantly influence far field behavior and, consequently, the overall distribution of effluents in the water column including the fraction of effluent reaching the ocean surface.

While complete spatial and temporal descriptions via direct numerical simulations are typically unfeasible even for laboratory scale plumes, large-eddy simulations (LES) of bubble plumes have been conducted (Buscaglia, Bombardelli, Garca, 2002, Deen, Solberg, Hjertager, 2001, Niceno, Dhotre, Deen, 2008) using finite-difference or finite-volume discretizations to explicitly resolve the largest, most energetic scales while parametrizing sub-grid scales by Smagorinsky models. In this work the approach is slightly different: instead of low-order spatial discretizations, we use high-order spectral element methods characterized by exponential convergence. In lieu of explicit Smagorinsky models, the specifics of which are still uncertain for bubbly flows (Dhotre et al., 2013), we adopt an alternate approach based on spectral filtering (Karamanos, Karniadakis, 2000, Koal, Stiller, Blackburn, 2012). Progressive filtering of the highest modes of the Legendre polynomial approximations is equivalent to a hyperviscosity subgrid-scale model whose influence is explicitly confined to near-grid scale processes (Boyd, 1998, Fischer, Mullen, 2001). This approach has been shown to maintain spectral accuracy with increasing resolution and provide resolution independent solutions. The computational efficiency afforded permits the explicit resolution of a larger range of spatial scales.

For large-scale plumes, adequate turbulence resolution, in reasonable time, on available computational resources requires both limiting the vertical extent of the domain and simplifying the full multiphase equation set. Starting from the conservation equations for multiphase systems, following (Sokolichin et al., 2004), we derive a simplified, Boussinesq model by neglecting the momentum carried by the gas which is much smaller than that of the liquid phase. Even with this simplification, open questions remain concerning the details of dispersed phase modelling of liquid-gas systems. Bubbles in a turbulent flow can be subjected to a wide variety of phenomena that may significantly affect the overall dynamics. Bubble breakup and coalescence, bubble oscillation and rotation and other bubble–bubble and bubble–liquid interactions have been explicitly modeled as bubble induced turbulence (Clift, Grace, Weber, 1978, Dhotre, Niceno, Smith, Simiano, 2009, Sato, Sadatomi, Sekoguchi, 1981, Tomiyama, 1998). Although there are numerical studies devoted to understanding the physics behind these processes (Esmaeeli, Tryggvason, 1998, Esmaeeli, Tryggvason, 1999, Fox, 2012, Kuwagi, Ozoe, 1999), proper parameterization and modeling remains unclear (Sokolichin et al., 2004). In a typical environmental blowout, with reasonably small gas volume fractions at the inlet, effects associated with individual bubbles are expected to be small given the levels of local shear produced by the large buoyancy flux at the inlet and the expected small bubble diameters (Lima Neto et al., 2008a). Accordingly, we argue that the most relevant parameter in the first order physics of dispersed bubbles is the slip velocity of the gas phase relative to the liquid. Here we simply set this parameter to a constant value consistent with that reported in experimental plumes (Lima Neto, Zhu, Rajaratnam, 2008, Simiano, 2005), numerical simulations (Dhotre et al., 2009) and estimations of the rise velocity for gas bubbles during the DwH spill (Socolofsky et al., 2011).

The current model, with relatively minor modifications, can be used for a wide range of oceanographic multiphase plume problems. In particular, and perhaps most importantly, oil phase buoyancy contributions can be explicitly divorced from the thermal field by considering a mixture model for the liquid phases. Physicochemical effects can be modeled via evolving distributions of slip velocities (Lima Neto et al., 2008b) and extensions to the study of multiple intrusions in the water column (Asaeda, Imberger, 1993, Socolofsky, Adams, 2003, Socolofsky, Adams, 2005) and rotating systems (Helfrich, 1994, Julien, Leff, McWilliams, Werne, 1999) are possible.

Section snippets

Multiphase plumes

Equations describing the evolution of a mixture of fluids, each member of which is denoted by the subscript k, are introduced. By expressing in the continuum limit the transport equations obtained using the Volume of Fluid approach, we avoid here the subtleties of working with discrete phases in deriving a set of partial differential equations (Buscaglia et al., 2002).

The mass conservation equation for phase k can be written as t(αkρk)+·(αkρku˜k)=0,where t is time and ρk, αk and u˜k=[u˜,v˜,w˜

Plume parameters

We consider three plumes differentiated only by the ratio of active scalar contributions to the inlet buoyancy flux: a purely thermal, a (almost) purely bubble and a hybrid case. In each case, the inlet buoyancy flux is fixed to the order of magnitude estimated for the DwH spill, BDwH = gΔρ0w0A/ρ0 = 0.37 m4 s−3, where D = 0.5 m is the source diameter, w0 = 1.3 m s−1 is the inlet liquid phase velocity and A = πD2/4 ≈ 0.2 m2 is the inlet cross section area. This is thermally equivalent to B0 =

Results

The plumes are forced by identical inlet momentum and buoyancy fluxes. Consequently, any observed differences between the evolution of single and multiphase plumes is entirely due to the slip velocity of the gas phase. In the present cases, the imposed slip velocity is only 18% of the injection velocity w0, and an even smaller percentage of the maximum vertical velocity produced by the large buoyancy flux. The comparisons, therefore, minimize the differences between the three cases; the three

Conclusions

It is well known that the presence of a gas in multiphase plumes increases turbulence mixing, entrainment and spreading rates with respect to single phase, thermal cases. Here, turbulence resolving computations are used to examine the mechanisms for this difference in dynamics and to quantify its extent in the context of deep water blowouts with inlet buoyancy fluxes much larger than those typically found in experiments. To clearly isolate the role of a dispersed gas phase on the dynamics, we

Acknowledgments

This research is supported by a grant from BP/The Gulf of Mexico Research Initiative. Additional computer resources have also been provided by City University of New York High Performance Computing Center under NSF Grants CNS-0855217, CNS-0958379 and ACI-1126113.

References (61)

  • KuwagiK. et al.

    Three-dimensional oscillation of bubbly flow in a vertical cylinder

    Int. J. Multiphase Flow

    (1999)
  • Lima NetoI.E. et al.

    Bubbly jets in stagnant water

    Int. J. Multiphase Flow

    (2008)
  • NicenoB. et al.

    One-equation sub-grid scale (sgs) modelling for euler-euler large eddy simulation (eeles) of dispersed bubbly flow

    Chem. Eng. Sci.

    (2008)
  • ÖzgökmenT.M. et al.

    Large eddy simulation of stratified mixing in a three-dimensional lock-exchange system

    Ocean Modell.

    (2009)
  • ÖzgökmenT.M. et al.

    Reynolds number dependence of mixing in a lock-exchange system from direct numerical and large eddy simulations

    Ocean Modell.

    (2009)
  • SatoY. et al.

    Momentum and heat transfer in two phase bubble flow, part 1 – theory

    Int. J. Multiphase Flow

    (1981)
  • SchwarzM.P. et al.

    Applicability of the standard k-ϵ turbulence model to gas-stirred baths

    Appl. Math. Model.

    (1988)
  • AlendalG. et al.

    Two-phase, near-field modeling of purposefully released CO2 in the ocean

    J. Geophys. Res.

    (2001)
  • AsaedaT. et al.

    Structure of bubble plumes in linearly stratified environments

    J. Fluid Mech.

    (1993)
  • BatchelorG.K.

    Heat convection and buoyancy effects in fluids

    Q. J. R. Meteorol. Soc.

    (1954)
  • CamilliR. et al.

    Tracking hydrocarbon plume transport and biodegradation at Deepwater Horizon

    Science

    (2010)
  • CardosoS.S.S. et al.

    Turbulent plumes with heterogeneous chemical reaction on the surface of small buoyant droplets

    J. Fluid Mech.

    (2009)
  • ChenC. et al.

    Turbulent Buoyant Jets: A Review of Experimental Data

    (1980)
  • CliftR. et al.

    Bubbles, Drops and Particles

    (1978)
  • Crespo-MedinaM. et al.

    The rise and fall of methanotrophy following a deepwater oil-well blowout

    Nat. Geosci.

    (2014)
  • DehaanC. et al.

    Deep cyclonic circulation in the Gulf of Mexico

    J. Phys. Oceanogr.

    (2005)
  • DevilleM.O. et al.

    High-Order Methods for Incompressible Fluid Flow

    (2002)
  • DhotreM.T. et al.

    Large eddy simulation for dispersed bubbly flows: a review

    Int. J. Chem. Eng.

    (2013)
  • DitmarsJ.D. et al.

    Analysis of air-bubble plumes

    Coastal Engng. Conf.

    (1974)
  • DrewD.A.

    Mathematical modelling of two-phase flow

    Annu. Rev. Fluid Mech.

    (1983)
  • Cited by (38)

    • Large-eddy simulation of enhanced mixing with buoyant plumes

      2022, Chemical Engineering Research and Design
      Citation Excerpt :

      Hence the high computational cost of IS limits its practical application (Yujie et al., 2012; Lu et al., 2005). EE approaches compute the continuous and dispersed phase in a common Eulerian framework, introducing a void fraction variable to simulate the interaction between the two (Li et al., 2009), and has been applied successfully to many multiphase flows (Fabregat et al., 2015; Dhotre et al., 2009). Recent investigations reveal that EE models can generally capture the dynamics of bubbly flow at the liquid level (Sánchez et al., 2018; Terashima et al., 2009; Li et al., 2009; Deen et al., 2004), while the lack of physical details of the bubble dynamics and liquid-bubble interaction should be considered, since the assumption that the dispersed phase can be represented by a unique field prevents EE models from tracking individual bubbles and their effect on the carrier phase (Hryb et al., 2009).

    • Hydrodynamics and dilution of an oil jet in crossflow: The role of small-scale motions from laboratory experiment and large eddy simulations

      2020, International Journal of Heat and Fluid Flow
      Citation Excerpt :

      Simulation studies of plumes have also gained interest. Fabregat et al. (2015, 2016)) used a hybrid approach of miscible fluid with a rise velocity for the bubbles to capture the behavior of the DWH plume. Yang et al. (2016) used Large Eddy Simulation (LES) to investigate the mechanisms of buoyancy and trap height in the DWH.

    • Rotating 2d point source plume models with application to Deepwater Horizon

      2017, Ocean Modelling
      Citation Excerpt :

      The vertical flow suppression caused by rotation results in the buoyant fluid escaping laterally and at a slant from the wellhead. Our 2d results agree with the azimuthally averaged 3d results on this point, although as argued in Fabregat et al. (2015), the slant consists of a precession about the wellhead. Last, there appears to be an effect of rotation on the turbulence near the wellhead.

    View all citing articles on Scopus
    View full text