Abstract
Probability density function (PDF) methods are a promising alternative to predicting the transport of solutes in groundwater under uncertainty. They make it possible to derive the evolution equations of the mean concentration and the concentration variance, used in moment methods. A mixing model, also known as a dissipation model, is essential for both methods. Finding a satisfactory mixing model is still an open question and due to the rather elaborate PDF methods, a difficult undertaking. Both the PDF equation and the concentration variance equation depend on the same mixing model. This connection is used to find and test an improved mixing model for the much easier to handle concentration variance. Subsequently, this mixing model is transferred to the PDF equation and tested. The newly proposed mixing model yields significantly improved results for both variance modelling and PDF modelling.
Authors
L. Schüler
N. Suciu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)
P. Knabner
S. Attinger
Keywords
PDF method; variance; solute transport; heterogeneity; mixing; global random walk
Cite this paper as:
L. Schüler, N. Suciu, P. Knabner, S. Attinger (2016), Building a Bridge from Moments to PDF’s: A new approach to finding PDF mixing models, arXiv:1602.01353 [physics.flu-dyn] 3 Feb 2016
References
About this paper
Journal
Arxiv.org
Publisher Name
DOI
Not available yet.
Print ISSN
Not available yet.
Online ISSN
Not available yet.
Google Scholar Profile
[1] WWAP, The United Nations World Water Development Report 4: Managing Water under Uncertainty and Risk, no. Vol. 1 in World Water Assessment Programme, Unesco, Paris, 2012.
[2] L. W. Gelhar, C. L. Axness, Three Dimensional Stochastic Analysis of Macrodispersion in Aquifers, Water Resour. Res. 19 (1) (1983) 161–180.
CrossRef (DOI)
[3] D. T. Burr, E. A. Sudicky, R. L. Naff, Nonreactive and reactive solute transport in three-dimensional heterogeneous porous media: Mean displacement, plume spreading, and uncertainty, Water Resour. Res. 30 (3) (1994) 791–815.
CrossRef (DOI)
[4] H. Tennekes, J. L. Lumley, A First Course in Turbulence, MIT Press, Cambridge, Massachusets, 1972.
[5] M. Dentz, H. Kinzelbach, S. Attinger, W. Kinzelbach, Temporal behavior of a solute cloud in a heterogeneous porous medium 1. Point-like injection, Water Resour. Res. 36 (12) (2000) 3591–3604.
CrossRef (DOI)
[6] R. Andricevic, Effects of local dispersion and sampling volume on the evolution of concentration fluctuations in aquifers, Water Resour. Res. 34 (5) (1998) 1115–1129.
CrossRef (DOI)
[7] V. Kapoor, L. W. Gelhar, Transport in three-dimensionally heterogeneous aquifers: 1. Dynamics of concentration fluctuations, Water Resour. Res. 30 (6) (1994) 1775–1788.
CrossRef (DOI)
[8] V. Kapoor, L. W. Gelhar, Transport in three-dimensionally heterogeneous aquifers 2. Predictions and observations of concentration fluctuations, Water Resour. Res. 30 (6) (1994) 1789–1801.
CrossRef (DOI)
[9] G. Dagan, Stochastic Modeling of Groundwater Flow by Unconditional and Conditional Probabilities 1.Conditional Simulation and the Direct Problem, Water Resour. Res. 18 (4) (1982) 813–833.
CrossRef (DOI)
[10] V. Kapoor, P. K. Kitanidis, Advection-diffusion in spatially random flows: Formulation of concentration covariance, Stoch. Hydrol. Hydraul. 11 (5) (1997) 397–422.
CrossRef (DOI)
[11] R. Andricevic, V. Cvetkovic, Evaluation of Risk from Contaminants Migrating by Groundwater, Water Resour. Res. 32 (3) (1996) 611–621.
CrossRef (DOI)
[12] F. P. J. de Barros, A. Fiori, A. Bellin, A simple closed-form solution for assessing concentration uncertainty, Water Resour. Res. 47 (12) (2011) 1–5.
CrossRef (DOI)
[13] V. Fiorotto, E. Caroni, Solute concentration statistics in heterogeneous aquifers for finite Peclet values, Transp. Porous Media 48 (3) (2002) 331–351.
CrossRef (DOI)
[14] V. Srzic, V. Cvetkovic, R. Andricevic, H. Gotovac, Impact of aquifer heterogeneity structure and local-scale dispersion on solute concentration uncertainty: Impact of Aquifer Heterogeneity on Concentration Uncertainty, Water Resour. Res. 49 (6) (2013) 3712–3728.
CrossRef (DOI)
[15] C. Celis, L. F. Figueira da Silva, Lagrangian Mixing Models for Turbulent Combustion: Review and Prospects, Flow, Turbul. Combust. 94 (3) (2015) 643–689.
CrossRef (DOI)
[16] D. W. Meyer, P. Jenny, H. A. Tchelepi, A joint velocity-concentration PDF method for tracer flow in heterogeneous porous media, Water Resour. Res. 46 (12) (2010) 1–17.
CrossRef (DOI)
[17] S. B. Pope, PDF Methods for Turbulent Reactive Flows, Prog. Energy Combust. Sci. 11 (2) (1985) 119–192.
CrossRef (DOI)
[18] R. O. Fox, Computational Models for Turbulent Reacting Flows, Cambridge Series in Chemical Engineering, Cambridge University Press, New York, 2003.
[19] N. Suciu, F. A. Radu, S. Attinger, L. Schuler, P. Knabner, A Fokker-Planck approach for probability distributions of species concentrations transported in heterogeneous media, J. Comput. Appl. Math. 289 (2015) 241–252.
CrossRef (DOI
[20] N. Suciu, L. Schuler, S. Attinger, C. Vamos, P. Knabner, Consistency issues in PDF methods, An. St. Univ. Ovidius Constanta, Ser. Mat. 23 (3) (2015)187–208.
CrossRef (DOI)
[21] N. Suciu, L. Schuler, S. Attinger, P. Knabner, Towards a filtered density function approach for reactive transport in groundwater, Adv. Water Resour.Accepted.
[22] J. Villermaux, J. C. Devillon, Representation de la coalescence et de la redispersion des domaines de segregation dans un fluide par un modele d’interaction phenomenologique., in: Proceedings of the 2nd International symposium on chemical reaction engineering, Elsevier New York, 1972, pp.1–13.
[23] C. Dopazo, E. E. O’Brien, An approach to autoignition of a turbulent mixture, Acta Astronaut. 1 (1974) 1239–1266.
CrossRef (DOI)
[24] P. J. Colucci, F. A. Jaberi, P. Givi, S. B. Pope, Filtered density function for large eddy simulation of turbulent reacting flows, Phys. Fluids 10 (2) (1998) 499–515.
CrossRef (DOI)
[25] V. Raman, H. Pitsch, A consistent LES/filtered-density function formulation for the simulation of turbulent flames with detailed chemistry, Proc. Combust. Inst. 31 (2) (2007) 1711–1719.
CrossRef (DOI
[26] P. P. Popov, S. B. Pope, Implicit and explicit schemes for mass consistency preservation in hybrid particle/finite-volume algorithms for turbulent reactive flows, J. Comput. Phys. 257 (2014) 352–373.
CrossRef (DOI)
[27] V. Sabel’nikov, M. Gorokhovski, N. Baricault, The extended IEM mixing model in the framework of the composition PDF approach: applications todiesel spray combustion, Combust. Theory Modell. 10 (1) (2006) 155–169.
CrossRef (DOI)
[28] W. Jones, A. Marquis, V. Prasad, LES of a turbulent premixed swirl burner using the Eulerian stochastic field method, Combust. Flame 159 (10) (2012) 3079–3095.
CrossRef (DOI)
[29] I. A. Dodoulas, S. Navarro-Martinez, Large Eddy Simulation of Premixed Turbulent Flames Using the Probability Density Function Approach, Flow, Turbul. Combust. 90 (3) (2013) 645–678.
CrossRef (DOI)
[30] N. Suciu, Diffusion in random velocity fields with applications to contaminant transport in groundwater, Adv. Water Resour. 69 (2014) 114–133.
CrossRef (DOI)
[31] M. Dentz, H. Kinzelbach, S. Attinger, W. Kinzelbach, Temporal behavior of a solute cloud in a heterogeneous porous medium 3. Numerical simulations, Water Resour. Res. 38 (7) (2002) 23–1–23–13.
CrossRef (DOI)
[32] C. Vamos, N. Suciu, H. Vereecken, Generalized random walk algorithm for the numerical modeling of complex diffusion processes, J. Comput. Phys. 186 (2) (2003) 527–544.
CrossRef (DOI)
[33] R. H. Kraichnan, Diffusion by a Random Velocity Field, Phys. Fluids 13 (1) (1970) 22–31.
CrossRef (DOI)
[34] F. Heße, V. Prykhod’ko, S. Schluter, S. Attinger, Generating random fields with a truncated power-law variogram. A comparison of several numerical methods with respect to accurary and reproduction of structural features., Environ. Model. Softw. 55 (2014) 32–48.
CrossRef (DOI)
[35] J. P. Eberhard, N. Suciu, C. Vamos, On the self-averaging of dispersion for transport in quasi-periodic random media, J. Phys. A: Math. Gen. 40 (4) (2007) 597.
CrossRef (DOI) URL http://iopscience.iop.org/1751-8121/40/4/002
[36] I. T. Drummond, S. Duane, R. R. Horgan, Scalar diffusion in simulated helical turbulence with molecular diffusivity, J. Fluid Mech. 138 (1984) 75–91.
CrossRef (DOI)
[37] N. Suciu, C. Vamos, J. Vanderborght, H. Hardelauf, H. Vereecken, Numerical Investigations on Ergodicity of Solute Transport in Heterogeneous Aquifers, Water Resour. Res. 42 (4) (2006) 1–17.
CrossRef (DOI)
[38] N. Suciu, F. A. Radu, A. Prechtel, F. Brunner, P. Knabner, A coupled finite element–global random walk approach to advection-dominated transport in porous media with random hydraulic conductivity, J. Comput. Appl. Math. 246 (2013) 27–37.
CrossRef (DOI)
[39] H. G. Im, T. S. Lund, J. H. Ferziger, Large eddy simulation of turbulent front propagation with dynamic subgrid models, Phys. Fluids 9 (12) (1997) 3826–3833.
CrossRef (DOI)
[40] C. D. Pierce, P. Moin, A dynamic model for subgrid-scale variance and dissipationrate of a conserved scalar, Phys. Fluids 10 (12) (1998) 3041–3044.
CrossRef (DOI)