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Effect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows

Effect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows,10.1016/

Effect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows   (Citations: 53)
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In this work, we investigate the effect of the introduction of a stress diffusive term into the classical Oldroyd-B constitutive equation on the numerical stability of time-dependent viscoelastic flow calculations. The channel Poiseuille flow at Re ⪢ 1 and O(1) We is chosen as a test problem. Through a linear stability analysis, we demonstrate that the introduction of a small amount of (dimensionless) diffusivity, typically of the order 10−3, does not affect the critical eigenmodes of the viscoelastic Orr-Sommerfeld problem appreciably. However, a diffusive term of that magnitude is shown to have a significant influence on the singular eigenmodes of the classical Oldroyd-B model, associated with the continuum spectra. A finite amplitude perturbation is constructed as a linear superposition of the eigenvectors corresponding to the most unstable eigenvalues of the problem. This is superimposed on the steady Poiseuille flow solution to provide the initial conditions for time-dependent simulations. The numerical algorithm involves a fully spectral spatial discretization and a semi-implicit second order integration in time. For the Oldroyd-B fluid, depending on the magnitude of the initial perturbation, numerical instabilities set in at relatively short times while the components of the conformation tensor increase monotonically in magnitude with time. Introduction of a diffusive term into this model is shown to stabilize the calculations remarkably, and for a three-dimensional simulation with Re = 5000 and We = 1, no instabilities were observed even at very large times. The effect of the magnitude of the diffusivity on the stability and the flow dynamics is addressed through a direct comparison of the results with those obtained for the Oldroyd-B model.
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    • ...For a careful numerical study of the Oldroyd-B model with stress diffusion, we refer to the paper of Sureshkumar and Beris [44]; see also the paper of Bhave, Armstrong and Brown [9]...

    John W. Barrett. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-...

    • ...Sureshkumar and Beris [15] overcame these instabilities by introducing a stress diffusion term into the equation for the conformation tensor...
    • ...This was the main obstacle to early attempts to numerically simulate friction drag reduction [14] that was overcome by Beris and coworkers by introducing an artificial stress diffusion term on the right hand side of Eq. (10) [2,15]...

    T. Vaithianathanet al. An improved algorithm for simulating three-dimensional, viscoelastic t...

    • ...DNS studies (Sureshkumar et al. 1997; Ptasinski et al. 2003; Sureshkumar & Beris 1995;...

    Wei Liet al. Nonlinear traveling waves as a framework for understanding turbulent d...

    • ...This value of Sc, though artificially small, is greater than or of the same order of magnitude as that used in many DNS studies (Sureshkumar et al., 1997; Ptasinski et al., 2003; Sureshkumar and Beris, 1995; Sibilla and Baron, 2002)...

    Wei Liet al. Viscoelastic Nonlinear Traveling Waves and Drag Reduction in Plane Poi...

    • ...Stress diffusion would precisely attenuate large cross-stream gradients, but in order to affect numerical computations the diffusivity coefficient should be about four orders of magnitude larger [16]...
    • ...The idea that sub-grid scales may be generated by advection and the use of stress diffusion to dampen those out are not new, and go back to the theoretical study of El-Kareh and Leal [15] (where it is even speculated that the coupled momentumstress equations may be ill-posed without some amount of diffusivity), and the numerical study of Sureshkumar and Beris [16]...
    • ...stream-wise direction. The addition of isotropic diffusivity has been studied by Sureshkumar and Beris [16], which reported that the addition of small amount of (artificial) diffusivity has little effect on the regular, spatially extended eigenmodes, while completely changing the nature of the singular spectra; in fact, the singular spectra are destroyed and discrete spectra emerge instead...
    • ...The first observation, is that in agreement with [16] the singular spectrum has disappeared, and instead there is a nearly vertical finite array of discrete eigenvalues...

    Raz Kupfermanet al. On the linear stability of plane Couette flow for an Oldroyd-B fluid a...

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