Abstract
Even though applications of direct numerical simulations are on the rise, today the most usual method to solve turbulence problems is still to apply a closure scheme of a defined order. It is not the case that a rising order of a turbulence model is always related to a quality improvement. Even more, a conceptual advantage of applying a lowest order turbulence model is that it represents the analogous method to the procedure of introducing a constitutive equation which has brought success to many other areas of physics. First order turbulence models were developed in the 1920s and today seem to be outdated by newer and more sophisticated mathematical-physical closure schemes. However, with the new knowledge of fractal geometry and fractional dynamics, it is worthwhile to step back and reinvestigate these lowest order models. As a result of this and simultaneously introducing generalizations by multiscale analysis, the first order, nonlinear, nonlocal, and fractional Difference-Quotient Turbulence Model (DQTM) was developed. In this partial review article of work performed by the authors, by theoretical considerations and its applications to turbulent flow problems, evidence is given that the DQTM is the missing (apparent) constitutive equation of turbulent shear flows.
Highlights
Difference-Quotient Turbulence Model (DQTM) was developed. In this partial review article of work performed by the authors, by theoretical considerations and its applications to turbulent flow problems, evidence is given that the DQTM is the missing constitutive equation of turbulent shear flows
One of its main results is the formula for the dynamic viscosity, being μ = 1/3ρλumol which contains the product of a characteristic length λ * (* following conventions in the physical literature, we leave away average and root mean square signs), being the mean distance between two collisions of molecules, called mean free path length, and the characteristic flight velocity of the molecules umol *
We propose the introduction of a new nonlinear, nonlocal and fractional constitutive equation, which was discovered in April 1985 and is called the Difference-Quotient Turbulence Model (DQTM)
Summary
With the new constitutive equation of turbulence (5), the following elementary turWith the new constitutive equation of turbulence (5), the following elementary turbubulent shear flows have analytically been solved (where each example listed is accompalent shear flows have analytically been solved (where each example listed is accompanied nied by the corresponding reference where the flow problem was solved): the turbulent by the corresponding reference where the flow problem was solved): the turbulent flow flow in the wake behind a cylinder [24], the axisymmetric turbulent jet into a quiescent in the wake behind a cylinder [24], the axisymmetric turbulent jet into a quiescent sursurrounding [24,27]. The turbulent flow flow [28] and plane turbulent Poiseuille flow [29] and the turbulent flow paralparallel to a wall.
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