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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.
On constructing realizable, conservative mixed scalar equations using the eddy-damped quasi-normal Markovian theory
- MARK ULITSKY, LANCE R. COLLINS
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- Journal:
- Journal of Fluid Mechanics / Volume 412 / 10 June 2000
- Published online by Cambridge University Press:
- 10 June 2000, pp. 303-329
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The eddy-damped quasi-normal Markovian (EDQNM) turbulence theory has been applied to the covariance spectrum of two passive isotropic scalars with different diffusivities in stationary isotropic turbulence. A rigorous application of EDQNM, which introduces no new modelling assumptions or constants, is shown to yield a covariance spectrum that violates the Cauchy–Schwartz inequality over some of the wavenumbers. One approach to this problem is to derive a model based on a stochastic differential equation, as its presence guarantees realizability. For an isotropic scalar, it is possible to construct a Langevin equation for the Fourier transform of the scalar concentrations that is consistent with each EDQNM scalar autocorrelation spectrum. The Langevin equations can then be used to construct a model for the covariance spectrum that is realizable. However, the resulting covariance transfer term does not properly conserve the scalar covariance, and so the model is still not satisfactory. The problem can be traced to the Markovianization step, which leads to the presence of the scalar diffusivities in the transfer functions in an unphysical fashion. A simple fix is described which reconciles the two approaches and gives conservative, realizable results for all time.
Next, we apply the EDQNM theory to a more general system involving the mixing of anisotropic scalars. Anisotropy in this case results from a uniform mean gradient of the two scalar concentrations in one direction. As with the isotropic scalars, direct application of the EDQNM closure results in a covariance spectrum that violates the Cauchy–Schwartz inequality; however, in this case it is not as simple to construct a Langevin model that reproduces all of the spectral interactions that result from the EDQNM procedure. Nevertheless, we show that the same modification of the inverse time scale as is done for the isotropic scalar results in an anisotropic scalar covariance spectrum that is realizable for all times.