Abstract
When Lagrangian stochastic models for turbulent dispersion are applied to complex atmospheric flows, some type of ad hoc intervention is almost always necessary to eliminate unphysical behaviour in the numerical solution. Here we discuss numerical strategies for solving the non-linear Langevin-based particle velocity evolution equation that eliminate such unphysical behaviour in both Reynolds-averaged and large-eddy simulation applications. Extremely large or ‘rogue’ particle velocities are caused when the numerical integration scheme becomes unstable. Such instabilities can be eliminated by using a sufficiently small integration timestep, or in cases where the required timestep is unrealistically small, an unconditionally stable implicit integration scheme can be used. When the generalized anisotropic turbulence model is used, it is critical that the input velocity covariance tensor be realizable, otherwise unphysical behaviour can become problematic regardless of the integration scheme or size of the timestep. A method is presented to ensure realizability, and thus eliminate such behaviour. It was also found that the numerical accuracy of the integration scheme determined the degree to which the second law of thermodynamics or ‘well-mixed condition’ was satisfied. Perhaps more importantly, it also determined the degree to which modelled Eulerian particle velocity statistics matched the specified Eulerian distributions (which is the ultimate goal of the numerical solution). It is recommended that future models be verified by not only checking the well-mixed condition, but perhaps more importantly by checking that computed Eulerian statistics match the Eulerian statistics specified as inputs.
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References
Bailey BN, Stoll R, Pardyjak ER, Mahaffee WF (2014) Effect of canopy architecture on vertical transport of massless particles. Atmos Environ 95:480–489
Du S (1997) Universality of the Lagrangian velocity structure function constant (\({\cal C}_0\)) across different kinds of turbulence. Boundary-Layer Meteorol 83:207–219
Du Vachat R (1977) Realizability inequalities in turbulent flows. Phys Fluids 20:551–556
Hairer E, Wanner G (1996) Solving ordinary differential equations II: stiff and differential-algebraic problems, 2nd edn. Springer, Berlin, 614 pp
Kim J, Moin P, Moser R (1987) Turbulence statistics in fully developed channel flow at low Reynolds number. J Fluid Mech 177:133–166
Kloeden PE, Platen E (1992) Higher-order implicit strong numerical schemes for stochastic differential equations. J Stat Phys 66:283–314
Lamba H (2003) An adaptive timestepping algorithm for stochastic differential equations. J Comput Appl Math 161:417–430
Langevin P (1908) Sur la théorie du mouvement Brownein. C R Acad Sci (Paris) 146:530–533
Legg BJ, Raupach MR (1982) Markov-chain simulation of particle dispersion in inhomogeneous flows: the mean drift velocity induced by a gradient in Eulerian velocity variance. Boundary-Layer Meteorol 24:3–13
Leveque RJ (2007) Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems. Society for Industrial and Applied Mathematics, Philadelphia, PA, 357 pp
Lin J, Brunner D, Gerbig C, Stohl A, Luhar A, Webley P (eds) (2013) Lagrangian modeling of the atmosphere. American Geophysical Union, Washington, DC, 349 pp
Lin JC (2013) How can we satisfy the well-mixed criterion in highly inhomogenous flows? A practical approach. In: Lin J, Brunner D, Gerbig C, Stohl A, Luhar A, Webley P (eds) Lagrangian modeling of the atmosphere. American Geophysical Union, Washington, DC, pp 59–69
Luhar AK, Britter RE (1989) A random walk model for dispersion in inhomogeneous turbulence in a convective boundary layer. Atmos Environ 23:1911–1924
Mansour NN, Kim J, Moin P (1988) Reynolds-stress and dissipation-rate budgets in a turbulent channel flow. J Fluid Mech 194:15–44
Mason PJ, Callen NS (1986) On the magnitude of the subgrid-scale eddy coefficient in large-eddy simulations of turbulent channel flow. J Fluid Mech 162:439–462
Mauthner S (1998) Step size control in the numerical solution of stochastic differential equations. J Comput Appl Math 100:93–109
Meneveau C, O’Neil J (1994) Scaling laws of the dissipation rate of turbulent subgrid-scale kinetic energy. Phys Rev E 49:2866–2874
Pope SB (1987) Consistency conditions for randomwalk models of turbulent dispersion. Phys Fluids 30:2374–2379
Porté-Agel F, Meneveau C, Parlange MB (2000) A scale-dependent dynamic model for large-eddy simulations: application to a neutral atmospheric boundary layer. J Fluid Mech 415:261–284
Postma JV (2015) Timestep buffering to preserve the well-mixed condition in Lagrangian stochastic simulations. Boundary-Layer Meteorol 156:15–36
Postma JV, Yee E, Wilson JD (2012) First-order inconsistencies caused by rogue trajectories. Boundary-Layer Meteorol 144:431–439
Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes: the art of scientific computing. Cambridge University Press, Cambridge, U.K., 1256 pp
Rodean HC (1991) The universal constant for the Lagrangian structure function. Phys Fluids A 3:1479–1480
Rodean HC (1996) Stochastic Lagrangian models of turbulent diffusion. American Meteorological Society, Boston, MA, 84 pp
Sagaut P (2002) Large eddy simulation for incompressible flows: an introduction, 3rd edn. Springer, Berlin, 585 pp
Sawford BL (1986) Generalized random forcing in randomwalk turbulent dispersion models. Phys Fluids 29:3582
Schumann U (1977) Realizability of Reynolds-stress turbulence models. Phys Fluids 20:721–725
Stoll R, Porté-Agel F (2006) Dynamic subgrid-scale models for momentum and scalar fluxes in large-eddy simulations of neutrally stratified atmospheric boundary layers over heterogeneous terrain. Water Resour Res 42(W01):409
Thomson DJ (1984) Random walk modelling of diffusion in inhomogeneous turbulence. Q J R Meteorol Soc 110:1107–1120
Thomson DJ (1987) Criteria for the selection of stochastic models of particle trajectories in turbulent flows. J Fluid Mech 180:529–556
Weil JC (1990) A diagnosis of the asymmetry in top-down and bottom-up diffusion using a Lagrangian stochastic model. J Atmos Sci 47:501–515
Wilson JD (2013) “Rogue velocities” in a Lagrangian stochastic model for idealized inhomogeneous turbulence. In: Lin J, Brunner D, Gerbig C, Stohl A, Luhar A, Webley P (eds) Lagrangian modeling of the atmosphere. American Geophysical Union, Washington, DC, pp 53–57
Wilson JD, Thurtell GW, Kidd GE (1981) Numerical simulation of particle trajectories in inhomogeneous turbulence, II: systems with variable turbulent velocity scale. Boundary-Layer Meteorol 21:423–441
Yee E, Wilson JD (2007) Instability in Lagrangian stochastic trajectory models, and a method for its cure. Boundary-Layer Meteorol 122:243–261
Yoshizawa A (1986) Statistical theory for compressible turbulent shear flows, with the application to subgrid modeling. Phys Fluids 29:2152
Acknowledgments
The author wishes to acknowledge fruitful discussions with Drs. Rob Stoll and Eric Pardyjak in formulating the ideas presented in this work. This research was supported by U.S. National Science Foundation Grants IDR CBET-PDM 113458 and AGS 1255662, and United States Department of Agriculture (USDA) Project 5358-22000-039-00D. The use, trade, firm, or corporation names in this publication are for information and convenience of the reader. Such use does not constitute an endorsement or approval by the USDA or the Agricultural Research Service of any product or service to the exclusion of others that may be suitable.
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Bailey, B.N. Numerical Considerations for Lagrangian Stochastic Dispersion Models: Eliminating Rogue Trajectories, and the Importance of Numerical Accuracy. Boundary-Layer Meteorol 162, 43–70 (2017). https://doi.org/10.1007/s10546-016-0181-6
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DOI: https://doi.org/10.1007/s10546-016-0181-6