Computational viscoelastic fluid

Computational Viscoelastic Fluid Mechanics and Numerical Studies of Turbulent Flows of Dilute Polymer Solutions... 

Computational Viscoelastic Fluid Mechanics and Numerical Studies... 

This chapter aims to present, in a concise way, the major elements and results from applications of computational viscoelastic fluid mechanics in numerical studies (DNS) of turbulent channel flows of homogeneous, dilute, polymer solutions under drag-reducing conditions. In the next section, we present a summary and outline of governing equations with emphasis on polymer modeling. In Section 1.3, we... 

Most importantly for computational viscoelastic fluid mechanics, most of the charmel DNS calculations are not performed for a constant flux (which would have naturally resulted in a constant bulk Reynolds number) but for a constant pressure drop per unit length that results in a constant zero shear rate friction Reynolds number. These runs lead to substantial variations in the (instantaneous and average) bulk Reynolds number from which the drag reduction needs to be estimated. Knowing roughly the relationship between the friction and the average bulk Reynolds number for a Newtonian fluid (from the experimentally determined and DNS confirmed empirical relationships for the skin friction factor - see, for example. Ref [34]), one can extract such a relationship that also takes into account the already mentioned (in Section 1.2) shear thinning effect in association with viscoelastic results [78]. 

The chapter by Beris and Housiadas ( Computational Viscoelastic Fluid Mechanics and Numerical Studies of Turbulent Flows of Dilute Polymer Solutions ) aims at resolving the famous long-standing problems of turbulent drag reduction. This contribution describes recent efforts and achievements in direct computations of near-wall turbulent flows of dilute polymer solutions and comparisons with experimental data. 

Keunings, R., 1989. Simulation of viscoelastic fluid flow. Tn Tucker, C. L. HI (ed.), Computer Modeling for Polymer Proces.sing, Chapter 9, Hanser Publishers, Munich, pp. 403-469. 

M. Kawahara and N. Takeuchi. Mixed finite element method for analysis of viscoelastic fluid flow. Comput. Fluids., 5 33, 1977. 

C. Braudo, A. Fortin, T. Coupez, Y. Demay, B. Vergnes, and J. F. Agassant, A Finite Element Method for Computing the Flow of Multi-mode Viscoelastic Fluids Comparison with Experiments, J. Non-Newt. Fluid Meek, 75, 1 (1998)... 

Fig. 7. Experiments and computations on the advection of a dye blob in an eccentric cylinder apparatus (from Niederkom and Ottino, 1993). Left Top, experiment using Newtonian fluid bottom, numerical simulation of same situation. Right Top, identical experiment as in upper left, but using viscoelastic fluid (We 0.06) bottom, numerical simulation at We = 0.04. 

The complexity of viscoelastic flows requires a multidisciplinary approach including modelling, computational and mathematical aspects. In this chapter we will restrict ourselves to the latter and briefly review the state of the art on the most basic mathematical questions that can be raised on differential models of viscoelastic fluids. We want to emphasize the intimate connections that exist between the theoretical issues discussed here and the modelling of complex polymer flows (see Part III) and their numerical simulations (see Chapter II.3). 

IIOJ. Baranger and D. Sandri, Finite element method for the approximation of viscoelastic fluid flow with a differential constitutive law. First European Computational Fluid Dynamics Conference, Bruxelles, 1992, C. Hirsch (ed.), Elsevier, Amsterdam, 1993, 1021-1025. 

If a discrete inf-sup inequality is satisfied by the velocity and pressure elements, then the discrete problem has a unique solution converging to the solution of the continuous one (see equation (27) of the chapter Mathematical Analysis of Differential models for Viscoelastic Fluids). This condition is referred to as the Brezzi-Babuska condition and can be checked element by element. Finite element methods for viscous flows are now well established and pairs of elements satisfying the Brezzi-Babuska condition are referenced (see [6]). Two strategies are used to compute the numerical solution of these equations involving velocity and pressure. 

R. Keunings, Simulation of Viscoelastic fluid flow, in Computer Modeling fra- Polymer Processing, C.L. Tucker HI (Ed.), Hanser Verlag (1989) p. 403. 

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