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A similarity theory of locally homogeneous and isotropic turbulence generated by a Smagorinsky-type LES
- Andreas Muschinski
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- Journal:
- Journal of Fluid Mechanics / Volume 325 / 25 October 1996
- Published online by Cambridge University Press:
- 26 April 2006, pp. 239-260
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A Kolmogorov-type similarity theory of locally homogeneous and isotropic turbulence generated by a Smagorinsky-type large-eddy simulation (LES) at very large LES Reynolds numbers is developed and discussed. The underlying concept is that the LES equations may be considered equations of motion of specific hypothetical fully turbulent non-Newtonian fluids, called ‘LES fluids’. It is shown that the length scale ls = csδ, which scales the magnitude of the variable viscosity in a Smagorinsky-type LES, is the ‘Smagorinsky-fluid’ counterpart of Kolmogorov's dissipation length $\eta = v^{3/4}\epsilon^{-1/4}$ for a Newtonian fluid where ν is the kinematic viscosity and ε is the energy dissipation rate. While in a Newtonian fluid the viscosity is a material parameter and the length ν depends on ε, in a Smagorinsky fluid the length ls is a material parameter and the viscosity depends on ε. The Smagorinsky coefficient cs may be considered the reciprocal of a ‘microstructure Knudsen number’ of a Smagorinsky fluid. A combination of Lilly's (1967) cut-off model with two well-known spectral models for dissipation-range turbulence (Heisenberg 1948; Pao 1965) leads to models for the LES-generated Kolmogorov coefficient αLES as a function of cs. Both models predict an intrinsic overestimation of αLES for finite values of cs. For cs = 0.2 Heisenberg's and Pao's models provide αLES = 1.74 (16% overestimation) and αLES = 2.14 (43% overestimation), respectively, if limcs → (αLES) = 1.5 is ad hoc assumed. The predicted overestimation becomes negligible beyond about cs = 0.5. The requirement cs > 0.5 is equivalent to Δ < 2ls. A similar requirement, L < 2η where L is the wire length of hot-wire anemometers, has been recommended by experimentalists. The value of limcs → (αLES) for a Smagorinsky-type LES at very large LES Reynolds numbers is not predicted by the models and remains unknown. Two critical values of cs are identified. The first critical cs is Lilly's (1967) value, which indicates the cs below which finite-difference-approximation errors become important; the second critical cs is the value beyond which the Reynolds number similarity is violated.
Small-scale and large-scale intermittency in the nocturnal boundary layer and the residual layer
- ANDREAS MUSCHINSKI, ROD G. FREHLICH, BEN B. BALSLEY
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- Journal:
- Journal of Fluid Mechanics / Volume 515 / 25 September 2004
- Published online by Cambridge University Press:
- 09 September 2004, pp. 319-351
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In high Reynolds-number turbulence, local scalar turbulence structure parameters,$( C_{\theta }^{2}) _{r}$, local scalar variance dissipation rates, $\chi _{r}$, and local energy dissipation rates, $\varepsilon _{r}$, vary randomly in time and space. This variability, commonly referred to as intermittency, is known to increase with decreasing $r$, where $r$ is the linear dimension of the local averaging volume. Statistical relationships between $\chi _{r}$, $\varepsilon _{r}$, and $( C_{\theta }^{2}) _{r}$ are of practical interest, for example, in optical and radar remote sensing. Some of these relationships are studied here, both theoretically and on the basis of recent observations. Two models for the conditionally averaged local temperature structure parameter, $\langle( C_{\theta }^{2}) _{r}| \varepsilon _{r}\rangle $, are derived. The first model assumes that the joint probability density function (j.p.d.f.) of $\chi _{r}$ and $\varepsilon _{r}$ is bivariate lognormal and that the Obukhov–Corrsin relationship, $( C_{\theta }^{2}) _{r}=\gamma\varepsilon _{r}^{-1/3}\chi _{r}$, where $\gamma\,{=}\,1.6$, is locally valid. In the second model, small-scale intermittency is ignored and $C_{\theta }^{2}$ and $\varepsilon $ are treated traditionally, that is, as averages over many outer scale lengths, such that $C_{\theta }^{2}$ and $\varepsilon $ change only as a result of large-scale intermittency. Both models lead to power-law relationships of the form $\langle( C_{\theta }^{2}) _{r}| \varepsilon _{r}\rangle \,{=}\,c\hspace{0.03in}\varepsilon _{r}^{\delta}$, where $c$ is a constant. Both models make predictions for the value of the power-law exponent $\delta$. The first model leads to $\delta\,{=}\,\rho _{xy}\sigma _{y}/\sigma _{x}-1/3$, where $\sigma _{x}$ and $\sigma _{y}$ are the standard deviations of the {logarithms} of $\varepsilon _{r}$ and $\chi _{r}$, respectively, and $\rho _{xy}$ is the correlation coefficient of the logarithms of $\chi _{r}$ and $\varepsilon _{r}$. This model leads to $\delta\,{=}\,1/3$ if $\rho _{xy}\,{=}\,2/3$ and if $\sigma _{x}\,{=}\,\sigma _{y}$. The second model predicts $\delta\,{=}\,2/3$, regardless of whether (i) static stability and shear are statistically independent, or (ii) they are connected through a Richardson-number criterion. These theoretical predictions are compared to fine-wire measurements that were taken during the night of 20/21 October 1999, at altitudes of up to 500 m in the nocturnal boundary layer and the overlying residual layer above Kansas. The fine-wire sensors were moved up and down with the University of Colorado's Tethered Lifting System (TLS). The data were obtained during the Cooperative Atmosphere-Surface Exchange Study 1999 (CASES-99). An interesting side result is that the observed frequency spectra of the logarithms of $\varepsilon _{r}$ and $\chi _{r}$ are described well by an $f^{-1}$ law. A simple theoretical explanation is offered.