Skip to main content Accessibility help

Wake behind a three-dimensional dry transom stern. Part 2. Analysis and modelling of incompressible highly variable density turbulence

  • Kelli Hendrickson (a1) and Dick K.-P. Yue (a1)


We analyse the turbulence characteristics and consider the closure modelling of the air entraining flow in the wake of three-dimensional, rectangular dry transom sterns obtained using high-resolution implicit large eddy simulations (iLES) (Hendrickson et al., J. Fluid Mech., vol. 875, 2019, pp. 854–883). Our focus is the incompressible highly variable density turbulence (IHVDT) in the near surface mixed-phase region ${\mathcal{R}}$ behind the stern. We characterize the turbulence statistics in ${\mathcal{R}}$ and determine it to be highly anisotropic due to quasi-steady wave breaking. Using unconditioned Reynolds decomposition for our analysis, we show that the turbulent mass flux (TMF) is important in IHVDT for the production of turbulent kinetic energy and is as relevant to the mean momentum equations as the Reynolds stresses. We develop a simple, regional explicit algebraic closure model for the TMF based on a functional relationship between the fluxes and tensor flow quantities. A priori tests of the model show mean density gradients and buoyancy effects are the main driving parameters for predicting the turbulent mass flux and the model is capable of capturing the highly localized nature of the TMF in ${\mathcal{R}}$ .


Corresponding author

Email address for correspondence:


Hide All
Aliod, R. & Dopazo, C. 1990 A statistically conditioned averaging formalism for deriving two-phase flow equations. Part. Part. Syst. Charact. 7 (1-4), 191202.10.1002/ppsc.19900070133
Aspden, A., Nikiforkakis, N., Dalziel, S. & Bell, J. B. 2008 Analysis of implicit LES methods. Commun. Appl. Maths Comput. Sci. 3 (1), 103126.10.2140/camcos.2008.3.103
Balachandar, S. & Eaton, J. K. 2010 Turbulent dispersed multiphase flow. Annu. Rev. Fluid Mech. 42 (1), 111133.10.1146/annurev.fluid.010908.165243
Baldy, S. 1993 A generation-dispersion model of ambient and transient bubbles in the close vicinity of breaking waves. J. Geophys. Res. 98 (C10), 1827718293.10.1029/93JC01627
Banerjee, S., Krahl, R., Durst, F. & Zenger, C. 2007 Presentation of anisotropy properties of turbulence, invariants versus eigenvalue approaches. J. Turbul. 8, N32.10.1080/14685240701506896
Brocchini, M. & Peregrine, D. H. 2001 The dynamics of strong turbulence at free surfaces. Part 2. Free-surface boundary conditions. J. Fluid Mech. 449, 255290.10.1017/S0022112001006024
Bunner, B. & Tryggvason, G. 2002a Dynamics of homogeneous bubbly flows. Part 1. Rise velocity and microstructure of the bubbles. J. Fluid Mech. 466, 1752.10.1017/S0022112002001179
Bunner, B. & Tryggvason, G. 2002b Dynamics of homogeneous bubbly flows. Part 2. Velocity fluctuations. J. Fluid Mech. 466, 5384.10.1017/S0022112002001180
Chachereau, Y. & Chanson, H. 2011 Bubbly flow measurements in hydraulic jumps with small inflow Froude numbers. Intl J. Multiphase Flow 37 (6), 555564.10.1016/j.ijmultiphaseflow.2011.03.012
Chassaing, P., Antonia, R. A., Anselmet, F., Joly, L. & Sarkar, S. 2002 Variable Density Fluid Turbulence. Kluwer Academic.10.1007/978-94-017-0075-7
Dakos, T. & Gibson, M. M. 1987 On modelling the pressure terms of the scalar flux equations. In Turbulent Shear Flows 5, pp. 718. Springer.10.1007/978-3-642-71435-1_2
Daly, B. J. & Harlow, F. H. 1970 Transport equations in turbulence. Phys. Fluids 13 (11), 26342649.10.1063/1.1692845
Deane, G. B. & Stokes, M. D. 2002 Scale dependence of bubble creation mechanisms in breaking waves. Nature 418 (6900), 839844.10.1038/nature00967
Dimotakis, P. E. 2005 Turbulent mixing. Annu. Rev. Fluid Mech. 37 (1), 329356.10.1146/annurev.fluid.36.050802.122015
Dodd, M. S. & Ferrante, A. 2016 On the interaction of Taylor length scale size droplets and isotropic turbulence. J. Fluid Mech. 806, 356412.10.1017/jfm.2016.550
Drazen, D. A., Fullerton, A. M., Fu, T. C., Beale, K. L. C., O’Shea, T. T., Brucker, K. A., Dommermuth, D. G., Wyatt, D. C., Bhushan, S., Carrica, P. M. et al. 2010 A comparison of model-scale experimental measurements and computational predictions for a large transom-stern wave. In Proceedings 28th Symp. on Naval Ship Hydrodynamics. Pasadena, California. US Office of Naval Research.
Drew, D. A. 1983 Mathematical modeling of two-phase flow. Annu. Rev. Fluid Mech. 15 (1), 261291.10.1146/annurev.fl.15.010183.001401
Duranti, S. & Pittaluga, F. 2000 Navier–Stokes prediction of internal flows with a three-equation turbulence model. AIAA J. 38 (6), 10981100.10.2514/2.1075
Emory, M. & Iaccarino, G.2014 Visualizing turbulence anisotropy in the spatial domain with componentality contours. Annual Research Briefs. Center for Turbulence Research.
Favre, A. 1969 Statistical equations of turbulent gases. Probl. Hydrodyn. Contin. Mech. 231266.
Friedberg, R. & Cameron, J. E. 1970 Test of the Monte Carlo method: fast simulation of a small ising lattice. J. Chem. Phys. 52 (12), 60496058.10.1063/1.1672907
Fu, T. C., Fullerton, A. M., Terrill, E. J. & Lada, G. 2006 Measurements of the wave fields around the R/V Athena I. In Proceedings 26th Symp. on Naval Ship Hydrodynamics. Strategic Analysis, Inc.
Gui, L., Longo, J. & Stern, F. 2001 Towing tank PIV measurement system, data and uncertainty assessment for DTMB Model 5512. Exp. Fluids 31 (3), 336346.10.1007/s003480100293
He, X., Zhang, R., Chen, S. & Doolen, G. D. 1999 On the three-dimensional Rayleigh–Taylor instability. Phys. Fluids 11 (5), 11431152.10.1063/1.869984
Hendrickson, K., Weymouth, G., Yu, X. & Yue, D. K.-P. 2019 Wake behind a three-dimensional dry transom stern. Part 1. Flow structure and large-scale air entrainment. J. Fluid Mech. 875, 854883.10.1017/jfm.2019.505
Jovanovic, J. 2004 The Statistical Dynamics of Turbulence. Springer.10.1007/978-3-662-10411-8
Kang, C., Zhang, W., Gu, Y. & Mao, N. 2017 Bubble size and flow characteristics of bubbly flow downstream of a ventilated cylinder. Chem. Engng Res. Des. 122, 263272.10.1016/j.cherd.2017.04.019
Karn, A., Shao, A., Arndt, R. E. A. & Hong, J. 2016 Bubble coalescence and breakup in turbulent bubbly wake of a ventilated hydrofoil. Exp. Therm. Fluid Sci. 70, 397407.
Kim, W. J., Van, S. H. & Kim, D. H. 2001 Measurement of flows around modern commercial ship models. Exp. Fluids 31 (5), 567578.10.1007/s003480100332
Lance, M. & Bataille, J. 1991 Turbulence in the liquid phase of a uniform bubbly air–water flow. J. Fluid Mech. 222, 95118.10.1017/S0022112091001015
Larsson, L., Stern, F. & Betram, V. 2003 Benchmarking of computational fluid dynamics for ship flows: the Gothenburg 2000 Workshop. J. Ship Res. 47 (1), 6381.
Larsson, L., Stern, F. & Visonneau, M.(Eds) 2013 Numerical Ship Hydrodynamics: An Assessment of the Gothenburg 2010 Workshop. Springer.
Launder, B. E. 1975 Heat and mass transport. In Turbulence (ed. Bradshaw, P.), pp. 231287. Springer.10.1007/3540088644_13
List, E. J. 1982 Turbulent jets and plumes. Annu. Rev. Fluid Mech. 14 (1), 189212.10.1146/annurev.fl.14.010182.001201
Livescu, D. & Ristorcelli, J. R. 2007 Buoyancy-driven variable-density turbulence. J. Fluid Mech. 591, 4371.10.1017/S0022112007008270
Livescu, D. & Ristorcelli, J. R. 2008 Variable-density mixing in buoyancy-driven turbulence. J. Fluid Mech. 605, 145180.10.1017/S0022112008001481
Lumley, J. L. & Newman, G. R. 1977 The return to isotropy of homogeneous turbulence. J. Fluid Mech. 82 (1), 161178.10.1017/S0022112077000585
Ma, G., Shi, F. & Kirby, J. T. 2011a A polydisperse two-fluid model for surf zone bubble simulation. J. Geophys. Res. 116, C05010.10.1029/2010JC006667
Ma, J., Oberai, A. A., Hyman, M. C., Drew, D. A. Jr & Lahey, R. T. L. 2011b Two-fluid modeling of bubbly flows around surface ships using a phenomenological subgrid air entrainment model. Comput. Fluids 52, 5057.10.1016/j.compfluid.2011.08.015
Martínez-Legazpi, P., Rodríguez-Rodríguez, J., Marugán-Cruz, C. & Lasheras, J. C. 2013 Plunging to spilling transition in corner surface waves in the wake of a partially submerged vertical plate. Exp. Fluids 54 (1), 111.
Morales, J. J., Nuevo, M. J. & Rull, L. F. 1990 Statistical error methods in computer simulations. J. Comput. Phys. 89 (2), 432438.10.1016/0021-9991(90)90151-P
Mortazavi, M., Le Chenadec, V., Moin, P. & Mani, A. 2016 Direct numerical simulation of a turbulent hydraulic jump: turbulence statistics and air entrainment. J. Fluid Mech. 797, 6094.10.1017/jfm.2016.230
Mudde, R. F. 2005 Gravity-driven bubbly flows. Annu. Rev. Fluid Mech. 37 (1), 393423.10.1146/annurev.fluid.37.061903.175803
Olivieri, A., Pistani, F., Avanzini, A., Stern, F. & Penna, R.2001 Towing tank experiments of resistance, sinkage and trim, boundary layer, wake, and free surface flow around a naval combatant INSEAN 2340 model. Tech. Rep. Iowa Univ Iowa City Coll of Engineering.
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.10.1017/CBO9780511840531
Ratcliffe, T. 1998 Validation of free surface Reynolds averaged Navier–Stokes and potential flow codes. In Proceedings 22nd Symp. on Naval Ship Hydrodynamics, pp. 964980. National Academy Press.
Rouse, H., Bhoota, B. V. & Hsu, E.-Y. 1949 Design of channel expansions. Proc. ASCE 75 (9), 13691385.
Sarkar, S. & Lakshmanan, B. 1991 Application of a Reynolds stress turbulence model to the compressible shear layer. AIAA J. 29 (5), 743749.10.2514/3.10649
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Physica D 12 (1), 318.
Shen, L., Zhang, C. & Yue, D. K.-P. 2002 Free-surface turbulent wake behind towed ship models: experimental measurements, stability analyses and direct numerical simulations. J. Fluid Mech. 469, 89120.10.1017/S0022112002001684
Shih, T.-H., Lumley, J. L. & Janicka, J. 1987 Second-order modelling of a variable-density mixing layer. J. Fluid Mech. 180, 93116.10.1017/S0022112087001745
Sotiropoulos, F. & Patel, V. C. 1995 Application of Reynolds-stress transport models to stern and wake flows. J. Ship Res. 39 (4), 263283.
Taulbee, D. & VanOsdol, J.1991 Modeling turbulent compressible flows – the mass fluctuating velocity and squared density. AIAA Paper 91-524, pp. 1–9.
Terrill, E. J. & Fu, T. C. 2008 At-sea measurements for ship hydromechanics. In Proceedings of 27th Symp. on Naval Ship Hydrodynamics, vol. 1. US Office of Naval Research.
Wei, X., Zhang, J. & Zhou, L. 2004 A new algebraic mass flux model for simulating turbulent mixing in swirling flow. Numer. Heat Tr. B-Fund. 45 (3), 283300.10.1080/1040779049025383
Weymouth, G. D. & Yue, D. K.-P. 2010 Conservative volume-of-fluid method for free-surface simulations on cartesian-grids. J. Comput. Phys. 229 (8), 28532865.10.1016/
Weymouth, G. D. & Yue, D. K.-P. 2011 Boundary data immersion method for Cartesian-grid simulations of fluid-body interaction problems. J. Comput. Phys. 230 (16), 62336247.10.1016/
Yoshizawa, A., Liou, W. W., Yokoi, N. & Shih, T.-H. 1997 Modeling compressible effects on the Reynolds stress using Markovianized two-scale method. Phys. Fluids 9 (10), 30243036.10.1063/1.869412
Young, Y. L., Harwood, C. M., Miguel Montero, F., Ward, J. C. & Ceccio, S. L. 2017 Ventilation of lifting bodies: review of the physics and discussion of scaling effects. Appl. Mech. Rev. 69 (1), 010801–010801–38.
Younis, B. A., Speziale, C. G. & Clark, T. T. 2005 A rational model for the turbulent scalar fluxes. Proc. R. Soc. Lond. A 461 (2054), 575594.10.1098/rspa.2004.1380
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed