ALBERT

Notes: The time evolution of the Loitsianski integral at high-Reynolds numbers is determined by computing an ensemble average of a large number of independent large-eddy simulations of decaying isotropic turbulence. It is found that the Loitsianski integral becomes proportional to tγ at large times and that γ ≈0.25. The present simulations illustrate the efficient use of massively parallel computers for simulating large ensembles of turbulent flows.

Notes: Direct numerical simulations of decaying two-dimensional turbulence in a fluid of large extent are performed primarily to ascertain the asymptotic decay laws of the energy and enstrophy. It is determined that a critical Reynolds number Rc exists such that for initial Reynolds numbers with R(0)〈Rc final period of decay solutions result, whereas for R(0)〉Rc the flow field evolves with increasing Reynolds number. Exactly at R(0)=Rc, the turbulence evolves with constant Reynolds number and the energy decays as t−1 and the enstrophy as t−2. A t−2 decay law for the enstrophy was originally predicted by Batchelor for large Reynolds numbers [Phys. Fluids Suppl. II, 12, 233 (1969)]. Numerical simulations are then performed for a wide range of initial Reynolds numbers with R(0)〉Rc to study whether a universal power-law decay for the energy and enstrophy exist as t→∞. Different scaling laws are observed for R(0) moderately larger than Rc. When R(0) becomes sufficiently large so that the energy remains essentially constant, the enstrophy decays at large times as approximately t−0.8. © 1997 American Institute of Physics.

Notes: If a random and statistically homogeneous density distribution is created in a fluid, a random motion of the fluid is subsequently generated by buoyancy forces. This motion is resisted by viscous stresses, while the density variation is smoothed by molecular diffusion of the relevant scalar property of the fluid. Furthermore, advective mixing generates smaller-scale components of the scalar quantity, increasing the rate of smoothing. If the Reynolds number of the motion is sufficiently large, the motion becomes turbulent in the ordinary sense. Such turbulence may be said to be generated by an "active'' conserved scalar quantity. To elucidate the nature of this buoyancy-driven turbulence, we consider an infinite fluid that is initially at rest everywhere, with a given distribution of the conserved scalar quantity. In order to have a well-defined initial state specified by a manageably small number of parameters, the initial distribution of the scalar is assumed to be statistically homogeneous and isotropic, and to be characterized by a single length L, and a measure of the magnitude of the scalar variations, say, the initial root-mean-square scalar fluctuation θ0. The other parameters on which the field of buoyancy-driven turbulence depends are the kinematic viscosity ν, the diffusivity of the conserved scalar quantity D, and the gravitational constant g.Dimensionless variables are constructed by choosing L as the unit of length and [L/gθ0]1/2 as the unit of time. The choice of these units introduces two dimensionless parameters into the problem, a Reynolds number R=[gθ0L3]1/2/ν and a Schmidt number σ=ν/D. Since analytical treatment of the problem is limited to the special cases where the nonlinear interactions may be neglected (e.g., R and σR(very-much-less-than)1), the primary means of inquiry is a numerical simulation of the flow field. We have performed numerical simulations (using a subgrid model) which sample the entire parameter space of the flow. Of particular interest is the asymptotic behavior of the flow as R and σR increase. One observes that a competition arises between the increase in magnitude of the fluid velocity according to the linear equations and the nonlinear generation of small-scale density and velocity fluctuations. The linear analysis predicts that, in dimensionless units, the time tm at which the mean-square velocity fluctuations reach a maximum, as well as the value of this maximum 〈uiui〉m, increases without limit as R and σR increase. In contrast, the results of the numerical simulation show that tm and 〈uiui〉m become independent of R and σR for sufficiently large values of these parameters. The linear amplification of the flow is thus checked by nonlinear effects. Buoyancy-generated turbulence at high Reynolds number has a period of growth from an initial state of rest, lives for a time in which there is a burst of activity, and then dies in consequence of this activity. This example of self-induced mixing can be given a quantitative description which may be useful in other contexts.

Notes: By simple analytical and large-eddy simulations, the time evolution of the kinetic energy and scalar variance in decaying isotropic turbulence transporting passive scalars are determined. The evolution of a passive scalar field with and without a uniform mean gradient is considered. First, similarity states of the flow during the final period of decay are discussed. Exact analytical solutions may be obtained, and these depend only on the form of the energy and scalar-variance spectra at low wave numbers, and the molecular transport coefficients. The solutions for a passive scalar field with mean-scalar gradient are of special interest, and we find that the scalar variance may grow or decay asymptotically in the final period, depending on the initial velocity distribution. Second, similarity states of the flow at high Reynolds and Péclet numbers are considered. Here it is assumed that the solutions also depend on the low-wave-number spectral coefficients, but not on the molecular transport coefficients. This results in a nonlinear dependence of the kinetic energy and scalar variance on the spectral coefficients, in contrast to the final period results. The analytical results obtained may be exact when the similarity solutions depend only on spectral coefficients that are time invariant. The present analysis also leads directly to a similarity state for a passive scalar field with uniform mean scalar gradient. Last, large-eddy simulations of the flow field are performed to test the theoretical results. Asymptotic similarity states at large times in the simulations are obtained and found to be in good agreement with predictions of the analysis. Several dimensionless quantities are also determined, which compare favorably to earlier experimental results. An argument for the inertial subrange scaling of the scalar-flux spectrum is presented, which yields a spectrum proportional to the scalar gradient and decaying as k−7/3. This result is partially supported by the small-scale statistics of the large-eddy simulations.

Notes: The decay of a homogeneous turbulence generated by an axisymmetric distribution of random impulsive forces acting at the initial instant is studied by means of large-eddy simulations. The impulsive forces may be either parallel or perpendicular to the symmetry axis. For impulsive forces, which result in a k4 low wave number energy spectrum of the turbulence, it is determined that the flow approaches isotropy on all scales of motion at long times, provided the Reynolds number is large. However, for the type of impulsive forces originally proposed by Saffman [J. Fluid Mech. 27, 581 (1967)], in which a k2 low wave number energy spectrum is produced, the turbulence approaches isotropy only at the smallest scales, and remains significantly anisotropic at the largest and energy-containing scales. Nevertheless, a similarity state of the flow field establishes itself asymptotically, in which the kinetic energy per unit mass of the turbulence decays as t−6/5. © 1995 American Institute of Physics.

Notes: The mixing of a passive scalar field by turbulence that is generated by buoyancy forces acting on an initial random density field is considered. Various asymptotic similarity states of the passive scalar field with and without a uniform mean passive scalar gradient are determined by dimensional arguments based on exact or near invariants of the density and passive scalar fields. The results of large-eddy numerical simulations are shown to support the derived scaling laws. The large-eddy simulations also demonstrate the different mixing properties of an active and passive scalar field. © 1995 American Institute of Physics.

Notes: The similarity form of the scalar-variance spectrum at high Schmidt numbers is investigated for nonstationary turbulence. Theoretical arguments show that Batchelor scaling may apply only at high Reynolds numbers. At low Reynolds numbers, Batchlor scaling is not possible unless the turbulence is stationary or the enstrophy decays asymptotically as t−2. When this latter condition is satisfied, it is shown from an analysis using both the Batchelor and Kraichnan models for the scalar-variance transfer spectrum that the k−1 power law in the viscous-convective subrange is modified. Results of direct numerical simulations of high Schmidt number passive scalar transport in stationary and decaying two-dimensional turbulence are compared to the theoretical analysis. For stationary turbulence, Batchelor scaling is shown to collapse the spectra at different Schmidt numbers and a k−1 viscous-convective subrange is observed. The Kraichnan model is shown to accurately predict the simulation spectrum. For nonstationary turbulence decaying at constant Reynolds number for which the enstrophy decays as t−2, scalar fields for different Schmidt numbers are simulated in situations with and without a uniform mean scalar gradient. The Kraichnan model is again shown to predict the spectra in these cases with different anomalous exponents in the viscous-convective subrange. © 1998 American Institute of Physics.