×

Hermite-DG methods for pdf equations modelling particle transport and deposition in turbulent boundary layers. (English) Zbl 1245.76060

Summary: A novel methodology is presented for the numerical treatment of multi-dimensional pdf (probability density function) models used to study particle transport in turbulent boundary layers. A system of coupled Fokker-Planck type equations is constructed to describe the transport of phase-space conditioned moments of particle and fluid velocities, both streamwise and wall-normal. This system, unlike conventional moment-based transport equations, allows for an exact treatment of particle deposition at the flow boundary and provides an efficient way to handle the 5-dimensional phase-space domain. Moreover, the equations in the system are linear and can be solved in a sequential fashion; there is no closure problem to address.
A hybrid Hermite-Discontinuous Galerkin scheme is developed to treat the system. The choice of Hermite basis functions in combination with an iterative scaling approach permits the efficient computation of solutions to high accuracy. Results demonstrate the effectiveness of the methodology in resolving the extreme gradients characteristic of distributions near an absorbing boundary.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76T99 Multiphase and multicomponent flows
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76F25 Turbulent transport, mixing
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs

Software:

DistMesh
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Reeks, M., On a kinetic equation for the transport of particles in turbulent flows, Phys. Fluids, 3, 446-456 (1991) · Zbl 0719.76068
[2] Reeks, M., On the probability density function equation for particle dispersion in a uniform shear flow, J. Fluid Mech., 522, 263-302 (2005) · Zbl 1065.76193
[3] Buyevich, Y., Statistical hydrodynamics of disperse systems. Part 1: Physical background and general equations, J. Fluid Mech., 49, 489-507 (1971) · Zbl 0225.76053
[4] Buyevich, Y., Statistical hydrodynamics of disperse systems. Part 2: Solution of the kinetic equation for suspended particles, J. Fluid Mech., 52, 345-355 (1972) · Zbl 0232.76088
[5] Buyevich, Y., Statistical hydromechanics of disperse systems. Part 3: Pseudo-turbulent structure of homogeneous suspensions, J. Fluid Mech., 56, 313-336 (1972) · Zbl 0267.76080
[6] Zaichik, L.; Alipchenkov, V., Kinetic equation for the probability density function of velocity and temperature of particles in an inhomogeneous turbulent flow: analysis of flow in a shear layer, High Temp: Heat Mass Trans. Phys. Gas Dynam., 36, 596-606 (1998)
[7] Zaichik, L.; Alipchenkov, V.; Sinaiski, E., Particles in Turbulent Flows (2008), Wiley-VCH
[8] Zaichik, L.; Alipchenkov, V., Pair dispersion and preferential concentration of particles in isotropic turbulence, Phys. Fluids, 15, 1776-1787 (2003) · Zbl 1186.76596
[9] Simonin, O.; Deutsch, E.; Minier, J., Eulerian prediction of the fluid/particle correlated motion in turbulent two-phase flow, Appl. Sci. Res., 51, 275-283 (1993) · Zbl 0778.76098
[10] O. Simonin, Statistical and continuum modelling of turbulent reactive particulate flows. Part I: Theoretical derivation of dispersed phase Eulerian modelling from probability density function kinetic equation, von Kármán Institute for Fluid Dynamics: Lecture Series, Belgium, 2000.; O. Simonin, Statistical and continuum modelling of turbulent reactive particulate flows. Part I: Theoretical derivation of dispersed phase Eulerian modelling from probability density function kinetic equation, von Kármán Institute for Fluid Dynamics: Lecture Series, Belgium, 2000.
[11] P. Février, O. Simonin, Statistical and continuum modelling of turbulent reactive particulate flows. Part II: Application of a two-phase second-moment transport model for prediction of turbulent gas-particle flows, von Kármán Institute for Fluid Dynamics: Lecture Series, Belgium, 2000.; P. Février, O. Simonin, Statistical and continuum modelling of turbulent reactive particulate flows. Part II: Application of a two-phase second-moment transport model for prediction of turbulent gas-particle flows, von Kármán Institute for Fluid Dynamics: Lecture Series, Belgium, 2000.
[12] Minier, J.; Peirano, E., The pdf approach to turbulent polydispersed two-phase flow, Phys. Rep., 352, 1-214 (2001) · Zbl 0971.76039
[13] Pope, S., Lagrangian pdf methods for turbulent flows, Ann. Rev. Fluid Mech., 26, 23-63 (1994) · Zbl 0802.76033
[14] Pope, S., Stochastic Lagrangian models of velocity in homogeneous turbulent shear flow, Phys. Fluids, 14, 1696-1702 (2002) · Zbl 1185.76299
[15] Iliopoulos, I.; Hanratty, T., Turbulent dispersion in an non-homogeneous field, J. Fluid Mech., 392, 45-71 (1999) · Zbl 0947.76038
[16] Iliopoulos, I.; Hanratty, T., A non-Gaussian stochastic model to describe passive tracer dispersion and its comparison to a direct numerical simulation, Phys. Fluids, 16, 3006-3030 (2004) · Zbl 1186.76246
[17] Swailes, D.; Reeks, M., Particle deposition from a turbulent flow. I: A steady-state model for high-inertia particles, Phys. Fluids, 6, 3392-3403 (1994) · Zbl 0825.76866
[18] Devenish, B.; Swailes, D.; Sergeev, Y., A PDF model for dispersed particles with inelastic particle-wall collisions, Phys. Fluids, 11, 1858-1868 (1999) · Zbl 1147.76374
[19] S. Aguinaga, O. Simonin, J. Borée, V. Herbert, A simplified particle-turbulence interaction model: application to deposition modelling in turbulent boundary layer, in: ASME 2009 Fluids Eng. Div. Summer Meeting FEDSM, 2009, Vail, CO, USA.; S. Aguinaga, O. Simonin, J. Borée, V. Herbert, A simplified particle-turbulence interaction model: application to deposition modelling in turbulent boundary layer, in: ASME 2009 Fluids Eng. Div. Summer Meeting FEDSM, 2009, Vail, CO, USA.
[20] Wilson, J.; Thurtell, G.; Kidd, G., Numerical simulation of particle trajectories in inhomogeneous turbulence, Boundary-Lay. Meteorol., 21, 423 (1981), (Consecutive papers I-III)
[21] Thomson, D., Criteria for the selection of stochastic models of particle trajectories in turbulent flows, J. Fluid Mech., 180, 529-556 (1987) · Zbl 0644.76065
[22] Risken, H., The Fokker-Planck Equation (1996), Springer-Verlag: Springer-Verlag Berlin · Zbl 0866.60071
[23] Moser, R.; Kim, J.; Mansour, N., Direct numerical simulation of turbulent channel flow up to \(Re_τ=590\), Phys. Fluids, 11, 943-945 (1999) · Zbl 1147.76463
[24] Kim, J.; Moin, P.; Moser, R., Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech., 177, 133-166 (1987) · Zbl 0616.76071
[25] Kallio, G.; Reeks, M., A numerical simulation of particle deposition in turbulent boundary layers, Int. J. Multiphase Flow, 15, 433-446 (1989)
[26] P. van Dijk, D. Swailes, Spectral-DG methods for pdf equations modelling particle transport and deposition in turbulent boundary layers, in: 6th Int. Symp. on Multiphase Flow, Heat Mass Transfer and Energy Conversion, Xi’an, China.; P. van Dijk, D. Swailes, Spectral-DG methods for pdf equations modelling particle transport and deposition in turbulent boundary layers, in: 6th Int. Symp. on Multiphase Flow, Heat Mass Transfer and Energy Conversion, Xi’an, China. · Zbl 1245.76060
[27] Hesthaven, J.; Warburton, T., Nodal Discontinuous Galerkin Methods (2008), Springer · Zbl 1134.65068
[28] Boyd, P., Chebyshev and Fourier Spectral Methods (2001), Dover · Zbl 0994.65128
[29] Persson, P.; Strang, G., A simple mesh generator in Matlab, SIAM Rev., 46, 329-345 (2004) · Zbl 1061.65134
[30] Liu, B.; Agarwal, J., Experimental observation of aerosol deposition in turbulent flow, Aer. Sci., 5, 145-155 (1974)
[31] Marchioli, C.; Soldati, A.; Kuerten, J.; Arcen, B.; Tanière, A.; Goldensoph, G.; Squires, K.; Cargnelutti, M.; Portela, L., Statistics of particle dispersion in direct numerical simulations of wall-bounded turbulence: results of an international collaborative benchmark test, Int. J. Multiphase Flow, 34, 879-893 (2008)
[32] Swailes, D.; Darbyshire, K., A generalized Fokker-Planck equation for particle transport in random media, Physica A, 242, 38-48 (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.