3 results
Polymer-laden homogeneous shear-driven turbulent flow: a model for polymer drag reduction
- ASHISH ROBERT, T. VAITHIANATHAN, LANCE R. COLLINS, JAMES G. BRASSEUR
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- Journal:
- Journal of Fluid Mechanics / Volume 657 / 25 August 2010
- Published online by Cambridge University Press:
- 28 June 2010, pp. 189-226
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Drag reduction (DR) under a turbulent boundary layer implies the suppression of turbulent momentum flux to the wall, a large-eddy phenomenon. Our hypothesis is that the essential mechanisms by which dilute concentrations of long-chain polymer molecules reduce momentum flux involve only the interactions among turbulent velocity fluctuations, polymer molecules and mean shear. Experiments indicate that these interactions dominate in a polymer-active ‘elastic layer’ outside the viscous sublayer and below a Newtonian inertial layer in a polymer-laden turbulent boundary layer. We investigate our hypothesis by modelling the suppression of momentum flux with direct numerical simulation (DNS) of homogeneous turbulent shear flow (HTSF) and the finite extensible nonlinear elastic with Peterlin approximation (FENE-P) model for polymer stress. The polymer conformation tensor equation was solved using a new hyperbolic algorithm with no artificial diffusion. We report here on the equilibrium state with fixed mean shear rate S, where progressive increases in non-dimensional polymer relaxation time WeS (shear Weissenberg number) or concentration parameter 1 − β produced progressive reductions in Reynolds shear stress, turbulence kinetic energy and turbulence dissipation rate, concurrent with increasing polymer stress and elastic potential energy. The changes in statistical variables underlying polymer DR with 1 − β, WeS, %DR and polymer-induced changes to spectra are similar to experiments in channel and pipe flows and show that the experimentally measured increase in normalized streamwise velocity variance is an indirect consequence of DR that is true only at lower DR. Comparison of polymer stretch and elastic potential energy budgets with channel flow DNS showed qualitative correspondence when distance from the wall was correlated to WeS. As WeS increased, the homogeneous shear flow displayed low-DR, high-DR and maximum-DR (MDR) regimes, similar to experiments, with each regime displaying distinctly different polymer–turbulence physics. The suppression of turbulent momentum flux arises from the suppression of vertical velocity fluctuations primarily by polymer-induced suppression of slow pressure–strain rate correlations. In the high-Weissenberg-number MDR-like limit, the polymer nearly completely blocks Newtonian inter-component energy transfer to vertical velocity fluctuations and turbulence is maintained by the polymer contribution to pressure–strain rate. Our analysis from HTSF with the FENE-P representation of polymer stress and its comparisons with experimental and DNS studies of wall-bounded polymer–turbulence supports our central hypothesis that the essential mechanisms underlying polymer DR lie directly in the suppression of momentum flux by polymer–turbulence interactions in the presence of mean shear and indirectly in the presence of the wall as the shear-generating mechanism.
Polymer mixing in shear-driven turbulence
- T. VAITHIANATHAN, ASHISH ROBERT, JAMES G. BRASSEUR, LANCE R. COLLINS
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- Journal:
- Journal of Fluid Mechanics / Volume 585 / 25 August 2007
- Published online by Cambridge University Press:
- 07 August 2007, pp. 487-497
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We investigate numerically the influence of polymer mixing on shear-driven turbulence. Of particular interest is the suppression of mixing that accompanies drag reduction with dilute polymer solutions. The simulations use the finite extensible nonlinear elastic model with the Peterlin closure (FENE-P) to describe the polymer stresses in the momentum equation, with polymer concentration allowed to vary in space and time. A thin slab of concentrated polymer was placed in an initially Newtonian homogeneous turbulent shear flow on a plane perpendicular to the mean velocity gradient, and allowed to mix in the gradient direction while actively altering the turbulence. The initially higher concentration of polymer near the centreplane suppressed production of turbulent kinetic energy and Reynolds stress in that region, while turbulence outside the polymer-rich region remained shear-dominated Newtonian turbulence. The rate of mixing in the shear direction was severely damped by the action of the polymer compared to a passive scalar in the corresponding Newtonian turbulent shear flow. This, in part, was a result of the same damping of vertical velocity fluctuations by the polymer that leads to the suppression of momentum flux. However, the cross-correlation between the polymer concentration and vertical velocity fluctuations was also suppressed, indicating that the explanation for the reduction in polymer mixing involves both the suppression of vertical velocity fluctuations and an alteration of turbulence structure by the polymer–turbulence interactions.
A spectral study of differential diffusion of passive scalars in isotropic turbulence
- MARK ULITSKY, T. VAITHIANATHAN, LANCE R. COLLINS
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- Journal:
- Journal of Fluid Mechanics / Volume 460 / 10 June 2002
- Published online by Cambridge University Press:
- 25 June 2002, pp. 1-38
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In a companion paper, Ulitsky & Collins (2000) applied the eddy-damped quasi-normal Markovian (EDQNM) turbulence theory to the mixing of two inert passive scalars with different diffusivities in stationary isotropic turbulence. Their paper showed that a rigorous application of the EDQNM approximation leads to a scalar covariance spectrum that violates the Cauchy–Schwartz inequality over a range of wavenumbers. The violation results from the improper functionality of the inverse diffusive time scales that arise from the Markovianization of the time evolution of the triple correlations. The modified inverse time scale they proposed eliminates this problem and allows meaningful predictions of the scalar covariance spectrum both with and without a uniform mean gradient.
This study uses the modified EDQNM model to investigate the spectral dynamics of differential diffusion. Consistent with recent DNS results by Yeung (1996), we observe that whereas spectral transfer is predominantly from low to high wavenumbers, spectral incoherence, being of molecular origin, originates at high wavenumbers and is transferred in the opposite direction by the advective terms. Quantitative comparisons between the EDQNM model and the DNS show good agreement. In addition, the model is shown to give excellent estimates for the dissipation coefficient defined by Yeung (1998).
We show that the EDQNM scalar covariance spectrum for two scalars with different molecular diffusivities can be approximated by the EDQNM autocorrelation spectrum for a scalar with molecular diffusivity equal to the arithmetic mean of the first two scalars. The result is exact for the case of an isotropic scalar and is shown to be a very good approximation for the scalar with a uniform mean gradient. We then exploit this relationship to derive a simple formula for the correlation coefficient of two differentially diffusing scalars as a function of their two Schmidt numbers and the turbulent Reynolds number. A comparison of the formula with the EDQNM model shows the model predicts the correct Reynolds number scaling and qualitatively predicts the dependence on the Schmidt numbers.
To investigate the degree of local versus non-local transfer of the scalar covariance spectrum, we divided the energy spectrum into three ranges corresponding to the energy-containing eddies, the inertial range, and the dissipation range. Then, by conditioning the scalar transfer on the energy contained within each of the three ranges, we have determined that the transfer process is dominated first by local interactions (local transfer) followed by non-local interactions leading to local transfer. Non-local interactions leading to non-local transfer are found to be significant at the higher wavenumbers. This result has important implications for defining simpler spectral models that potentially can be applied to more complex engineering flows.