Governing Equations and Polymer Modeling

The first among the governing equations used in DNS are the mass and momentum balances, which for an incompressible, isothermal, and constant polymer concentration flow in a channel are given in dimensionless form as 

Note that in viscoelastic turbulent flows, because of the shear thinning effect [3 5,79], we have to distinguish between two different types of wall units. One is based on the zero shear properties and the other, applicable for channel and boundary layer 

Quantity Computational scale Reference wall scale Actual wall scale Computational scale (fixed flux) 

Equations (1.1) and (1.2) are not closed because of the presence of the extra stress tensor x. Therefore, one more equation is required, which is provided by a viscoelastic constitutive model [35, 49, 79). According to nonequilibritun thermodynamics, the most thermodynamically consistent way to describe the constitutive model is in terms of internal (structural) variables for which separate evolution equations are to be described [49]. The simplest case is when a single, second-order 

On the right-hand side of the constitutive equation, Eq. (1.3), a diffusion term has been added, as proposed by Sureshkumar and Beris [81], so that in turbulent simulations the high wavenumber contributions of the conformation tensor do not diverge during the numerical integration of this equation in time. This parallels the introduction of a numerical diffusion term in any scalar advection equation (e.g., a concentration equation with negligible molecular diffusion) that is solved along with the flow equations under turbulent conditions [82]. In Eq. (1.3), Dq is the dimensionless numerical diffusivity [54-56]. The issue of the numerical diffusivity is further discussed in Sections 1.3.2 and 1.4.3. 

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