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Material-element deformation in isotropic turbulence

Published online by Cambridge University Press:  26 April 2006

S. S. Girimaji  and
S. B. Pope
S. S. Girimaji
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
S. B. Pope
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
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Abstract

The evolution of infinitesimal material line and surface elements in homogeneous isotropic turbulence is studied using velocity-gradient data generated by direct numerical simulations (DNS). The mean growth rates of length ratio (l) and area ratio (A) of material elements are much smaller than previously estimated by Batchelor (1952) owing to the effects of vorticity and of non-persistent straining. The probability density functions (p.d.f.'s) of l/〈l〉 and A/〈A〉 do not attain stationarity as hypothesized by Batchelor (1952). It is shown analytically that the random variable l/〈l〉 cannot be stationary if the variance and integral timescale of the strain rate along a material line are non-zero and DNS data confirm that this is indeed the case. The application of the central limit theorem to the material element evolution equations suggests that the standardized variables $\hat{l}(\equiv (\ln l - \langle \ln l\rangle)/({\rm var} l)^{\frac{1}{2}})$ and Â(≡(ln A − 〈ln A〉)/(var A)½) should attain stationary distributions that are Gaussian for all Reynolds numbers. The p.d.f.s of $\hat{l}$ and  calculated from DNS data appear to attain stationary shapes that are independent of Reynolds number. The stationary values of the flatness factor and super-skewness of both $\hat{l}$ and  are in close agreement with those of a Gaussian distribution. Moreover, the mean and variance of ln l (and ln A) grow linearly in time (normalized by the Kolmogorov timescale, τη), at rates that are nearly independent of Reynolds number. The statistics of material volume-element deformation are also studied and are found to be nearly independent of Reynolds number. An initially spherical infinitesimal volume of fluid deforms into an ellipsoid. It is found that the largest and the smallest of the principal axes grow and shrink respectively, exponentially in time at comparable rates. Consequently, to conserve volume, the intermediate principal axis remains approximately constant.

The performance of the stochastic model of Girimaji & Pope (1990) for the velocity gradients is also studied. The model estimates of the growth rates of 〈ln l〉 and 〈ln A〉 are close to the DNS values. The growth rate of the variances are estimated by the model to within 17%. The stationary distributions of $\hat{l}$ and  obtained from the model agree very well with those calculated from DNS data. The model also performs well in calculating the statistics of material volume-element deformation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 220 , November 1990 , pp. 427 - 458
Copyright
© 1990 Cambridge University Press

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References

Ashurst, W. T., Kerstein, A. R., Kerr, R. M. & Gibson, C. H., 1987 Alignment of vorticity and scalar gradient in simulated Navier–Stokes turbulence. Phys. Fluids 30, 2343.Google Scholar
Batchelor, G. K.: 1952 The effect of homogeneous turbulence on material lines and surfaces. Proc. R. Soc. Lond. A 213, 349.Google Scholar
Batchelor, G. K. & Townsend, A. A., 1956 Turbulent diffusion. In Surveys in Mechanics (ed. G. K. Batchelor & R. M. Davies), p. 352. Cambridge University Press.
Cocke, W. J.: 1969 Turbulent hydrodynamic line stretching: consequences of isotropy. Phys. Fluids 12, 2448.Google Scholar
Drummond, I. T. & Münch, W. 1990a Turbulent stretching of material elements. J. Fluid Mech. 215, 45.Google Scholar
Drummond, I. T. & Münch, W. 1990b Distortion of line and surface elements in turbulent flows. J. Fluid Mech. (submitted).Google Scholar
Eswaran, V. & Pope, S. B., 1988 An examination of forcing in direct numerical simulations of turbulence. Comput. Fluids 16, 257.Google Scholar
Girimaji, S. S. & Pope, S. B., 1990 A diffusion model for velocity gradients in homogeneous, isotropic turbulence. Phys. Fluids A 2, 242.Google Scholar
Kraichnan, R. H.: 1974 Convection of a passive scalar by a quasi-uniform random straining field. J. Fluid Mech. 64, 737.Google Scholar
Lumley, J. L.: 1972 On the solution of equations describing small scale deformation. Symposia Mathemetica (Istituto Nazionale di Alta Matematica), vol. 9, pp. 315–334.Google Scholar
Monin, A. S. & Yaglom, A. M., 1981 Statistical Fluid Mechanics, vol. 2 (ed. J. L. Lumley), Section 24.5, p. 578. MIT Press.
Orszag, S. A.: 1970 Comments on Turbulent hydrodynamic line stretching: consequences of isotropy. Phys. Fluids 13, 2203.Google Scholar
Pope, S. B.: 1988 The evolution of surfaces in turbulence. Intl J. Engng Sci. 26, 445–469.Google Scholar
Pope, S. B., Yeung, P. K. & Girimaji, S. S., 1989 The curvature of material surfaces in isotropic turbulence. Phys Fluids A 1, 2010.Google Scholar
Rogallo, R. S.: 1981 Numerical experiments in homogeneous turbulence. NASA TM 81315.Google Scholar
Tennekes, H. & Lumley, J. L., 1975 A First Course in Turbulence. MIT Press.
Townsend, A. A.: 1951 The diffusion of heat sports in isotropic turbulence. Proc. R. Soc. Lond. A 209, 418.Google Scholar
Yeung, P. K., Girimaji, S. S. & Pope, S. B., 1990 Straining and scalar dissipation on material surfaces in turbulence: implications for flamelets. Combust. Flame 79, 340.Google Scholar
Yeung, P. K. & Pope, S. B., 1988 An algorithm for tracking fluid particles in numerical simulations of homogeneous turbulence. J. Comput. Phys. 79, 373.Google Scholar
Yeung, P. K. & Pope, S. B., 1989 Lagrangian statistics from direct numerical simulations of isotropic turbulence. J. Fluid Mech. 207, 531.Google Scholar