Journal of Guangdong University of Technology ›› 2012, Vol. 29 ›› Issue (2): 1-6.doi: doi:10.3969/j.issn.1007-7162.2012.02.001

• Feature Articles •     Next Articles

Multiscale Mixing Norms for Scalar Mixing and Transport in Fluids

  

  1. Department of Mathematics, Zhejiang University，Hangzhou 310027，China
  • Online:2012-06-25 Published:2012-06-25

HTML

PDF (PC)

2339

Abstract: The mixing and transport of scalar fields in fluids are fundamental phenomena in nature and in engineering.  However, there have been various measures and models for quantifying mixing.  In recent years, signficant research efforts have been dedicated to the study of a mutiscale mixing measure related to Sobolev norms. It reviewed some common mixing measures, applied them and corresponding mathematical approaches to a test problem, and considered the optimal control problem for scalar mixing. Besides, it explored the possibility and advantanges of using multicale mixing norms as the unified mixing measure from both analytical and computational perspectives. The research shows that this will provide a theoretical and numerical research framework to evaluate, predict and control scalar mixing and transport in practical applications.

Key words: Mixing and transport; optimal control; fluid mechanics; partial differential equations

［1］Aref H.Stirring by chaotic advection［J］. J Fluid Mech,1984,143:1-21.

［2］ Ottino J M.  The kinematics of mixing: Stretching, chaos, and transport［M］. U K：Cambridge University Press,1989. 

［3］ Adalsteinsson D, Camassa R, Harenberg S,et al.  Optimal mixing enhancement in 3，d pipe flow［J］.  Geophys Monogr Ser,2011,195:257-262. 

［4］ Mezi  c ' I, Loire S, Fonoberov V A,et al.  A new mixing diagnostic and gulf oil spill movement［J］.Science,2010,330(6003):486-489.

［5］ Avellaneda M, A Majda J.  Stieltjes integral representation and effective diffusivity bounds for turbulent transport［J］.  Phys Rev Lett,1989,62:753-755.

［6］ Batchelor G K.  Diffusion in a field of homogeneous turbulence  I Eulerian Analysis［J］.  Aust J Sci Res,1949,2:437-450.

［7］ Falkovich G, Gawdzki K, Vergassola M.  Particles and fields in turbulent flows［J］.  Siam J  Appl Math,1994,54:333-408. 

［8］ Warhaft Z.  Passive scalars in turbulent flows［J］.  Ann Rev  Fluid Mech,2000,32:203-240. 

［9］ 刘亚明，柳朝晖，贺铸，等.  湍流被动标量研究的最新远展［J］.  力学进展, 2005,35(4):549-558.

           Liu Yaming， Liu Zhaohui，He Zhu，et al.Recent progress instatistics of turbulent passive scalar［J］.Advances in Mechanics,2005,35(4):549-558.



［10］ Keating S R, Kramer P R, Smith K S.  Homogenization and mixing measures for a replenishing passive scalar field［J］.Phys Fluids,2010,22:75-105. 

［11］ Lin Z,Boová K,Doering C R.  Models and measures of mixing and effective diffusion［J］.  Discr Cont Dyn Sys,2010,28:259-274. 

［12］ Majda A J, Kramer P R.  Simplified models for turbulent diffusion: Theory, numerical modelling and physical phenomena［J］. Phys Reports,1999,314(4/5):237-574.

［13］ Plasting S, Young W R.  A bound on scalar variance for the advection diffusion equation［J］.J Fluid Mech, 2006,552:289-298. 

［14］ Shaw T A, Thiffeault J L, Doering C R. Stirring up trouble: Multi scale mixing measures for steady scalar sources［J］. Physica D,2007,231(2):143-164. 

［15］ Thiffeault J L, Doering C R, Gibbon J D.  A bound on scalar variance for the advectiondiffusion equation［J］. J Fluid Mech,2004,521:105-114. 

［16］ Camassa R, Lin Z, McLaughlin R M.  Evolution of the probability measure for the majda model: New invariant measures and breathing PDFs［J］.  J Stat Phys,2008,130(2):343-371.

［17］ Mi J.  Correlation between non，gaussian statistics of a scalar and its dissipation rate［J］.  Phys Rev E,2006,74(1):016301. 

［18］ Eckart C.  An analysis of the stirring and mixing processes in incompressible fluids［J］.  J Mar Res,1948,7:265-275.

［19］ Thiffeault J L.  Using multiscale norms to quantify mixing and transport［J］.  Nonlinearity,2012,84(3):R1R44.

［20］ Mathew  G, Mezic＇SymbolQCpI, Petzold L.  A multiscale measure for mixing［J］.  Physica D, 2005,211(1/2):23-46.

［21］ Doering C R, Thiffeault J L.  Multiscale mixing efficiencies for steady sources［J］.  Phys Rev E,2006,74(2):025301(R). 

［22］ Camassa R, Lin Z, McLaughlin R M.  The exact evolution of the scalar variance in pipe and channel flows［J］. Commun Math Sci,2010,8(2):601-626. 

［23］ Cortelezzi L, Adrover A, Giona M.  Feasibility, efficiency and transportability of short，horizon optimal mixing protocols［J］.  J Fluid Mech,2008,597:199=231. 

［24］ Liu  W J.  Mixing enhancement by optimal flow advection［J］.  SIAM J Control and Opt,2008,47:624-638. 

［25］  Mathew G, Mezic＇I, Grivopoulos S,et al.  Optimal control of mixing in stokes fluid flows［J］.  J Fluid Mech,2007,580:261-281. 

［26］ Lin Z, Doering C R, Thiffeault J L.  An optimal stirring strategy for passive scalar mixing［J］.  J Fluid Mech,2011,675:465-476.
No related articles found!
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright © 2017 Journal of Guangdong University of Technology
Add：729 East Dongfeng RD., Guangzhou PRC. Postcode: 510090
Tel：020-37626653,020-37626139,020-37626396
Email：xbzrb@gdut.edu.cn
Supported:Beijing Magtech