Turbulent Mixing Length Models and Their Applications to Elementary Flow Configurations

  • Kolumban HutterEmail author
  • Yongqi Wang
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)


In comparison to Chap.  15, this chapter goes back one step by scrutinizing the early zeroth order closure relations as proposed by Prandtl, von Kármán and collaborators. The basis is Bossinesq’s (Mém. Prés. Div. Savant Acad. Sci. Paris, 23:46 [3]) ansatz for the shear stress in plane parallel flow, \(\tau _{12}\), which is postulated to be proportional to the corresponding averaged shear rate \(\partial v_{1}/\partial x_{2}\) with coefficient of proportionality \(\rho \varepsilon \), where \(\rho \) is the density and \(\varepsilon \) a kinematic turbulent viscosity or turbulent diffusivity \((\mathrm {m}^{2}\mathrm {s}^{-1}\)). In turbulence theory the flux terms of momentum, heat and suspended mass are all parameterized as gradient-type relations with turbulent diffusivities treated as constants. Prandtl realized from data collected in his institute that \(\varepsilon \) was not a constant but depended on his mixing length squared and the magnitude of the shear rate (Prandtl, ZAMM 5:136–139, [23]). This proposal was later improved (“Prandtl (1942), Abriss der Strömungslehre” Prandtl [25]) to amend the unsatisfactory agreement at positions where shear rates disappeared. The 1942-law is still local, which means that the Reynolds stress tensor at a spatial point depends on spatial velocity derivatives at the same position. Prandtl, in a second proposal of his 1942-paper suggested that the turbulent diffusivity should depend on the velocity difference at the points where the velocity of the turbulent path assumes maximum and minimum values. This proposal introduces some non-locality, and it yielded better agreement with data, but Prandtl left the non-gradient-type dependence in order to stay in conformity with Boussinesq. It does become neither apparent nor clear that Prandtl or the modelers at that time would have realized that non-local effects would be the cause for better agreement of the theoretical formulations with data. The proposal of complete nonlocal behavior of the Reynolds stress parameterization came in 1991 by P. Egolf and subsequent research articles during 20 years, in which also the local strain rate (\(=\)local velocity gradient) is replaced by a difference quotient. We motivate and explain the proposed Difference Quotient Turbulence Model (DQTM) and demonstrate that for standard two-dimensional configurations analyzed in this chapter its performance is superior to other zeroth order models.


Local/nonlocal turbulent stress closure Criticisms on zeroth order local stress closures Prandtl turbulent plane wake Axisymmetric isothermal jet Turbulent jet in parallel co-flow Plane Poiseuille flow 


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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.c/o Versuchsanstalt für Wasserbau, Hydrologie und GlaziologieETH ZürichZürichSwitzerland
  2. 2.Department of Mechanical EngineeringTechnische Universität DarmstadtDarmstadtGermany

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