Three regularization models of the Navier-Stokes equations

We determine how the differences in the treatment of the subfilter-scale physics affect the properties of the flow for three closely related regularizations of Navier-Stokes. The consequences on the applicability of the regularizations as SGS models are also shown by examining their effects on superfilter-scale properties. Numerical solutions of the Clark-alpha model are compared to two previously employed regularizations, LANS-alpha and Leray-alpha (at Re ~ 3300, Taylor Re ~ 790) and to a DNS. We derive the Karman-Howarth equation for both the Clark-alpha and Leray-alpha models. We confirm one of two possible scalings resulting from this equation for Clark as well as its associated k^(-1) energy spectrum. At sub-filter scales, Clark-alpha possesses similar total dissipation and characteristic time to reach a statistical turbulent steady-state as Navier-Stokes, but exhibits greater intermittency. As a SGS model, Clark reproduces the energy spectrum and intermittency properties of the DNS. For the Leray model, increasing the filter width decreases the nonlinearity and the effective Re is substantially decreased. Even for the smallest value of alpha studied, Leray-alpha was inadequate as a SGS model. The LANS energy spectrum k^1, consistent with its so-called "rigid bodies," precludes a reproduction of the large-scale energy spectrum of the DNS at high Re while achieving a large reduction in resolution. However, that this same feature reduces its intermittency compared to Clark-alpha (which shares a similar Karman-Howarth equation). Clark is found to be the best approximation for reproducing the total dissipation rate and the energy spectrum at scales larger than alpha, whereas high-order intermittency properties for larger values of alpha are best reproduced by LANS-alpha.Comment: 21 pages, 8 figure

Optimal model parameters for multi-objective large-eddy simulations

A methodology is proposed for the assessment of error dynamics in large-eddy simulations. It is demonstrated that the optimization of model parameters with respect to one flow property can be obtained at the expense of the accuracy with which other flow properties are predicted. Therefore, an approach is introduced which allows to assess the total errors based on various flow properties simultaneously. We show that parameter settings exist, for which all monitored errors are "near optimal," and refer to such regions as "multi-objective optimal parameter regions." We focus on multi-objective errors that are obtained from weighted spectra, emphasizing both large- as well small-scale errors. These multi-objective optimal parameter regions depend strongly on the simulation Reynolds number and the resolution. At too coarse resolutions, no multi-objective optimal regions might exist as not all error-components might simultaneously be sufficiently small. The identification of multi-objective optimal parameter regions can be adopted to effectively compare different subgrid models. A comparison between large-eddy simulations using the Lilly-Smagorinsky model, the dynamic Smagorinsky model and a new Re-consistent eddy-viscosity model is made, which illustrates this. Based on the new methodology for error assessment the latter model is found to be the most accurate and robust among the selected subgrid models, in combination with the finite volume discretization used in the present study

Response maxima in time-modulated turbulence: Direct Numerical Simulations

The response of turbulent flow to time-modulated forcing is studied by direct numerical simulations of the Navier-Stokes equations. The large-scale forcing is modulated via periodic energy input variations at frequency $\omega$. The response is maximal for frequencies in the range of the inverse of the large eddy turnover time, confirming the mean-field predictions of von der Heydt, Grossmann and Lohse (Phys. Rev. E 67, 046308 (2003)). In accordance with the theory the response maximum shows only a small dependence on the Reynolds number and is also quite insensitive to the particular flow-quantity that is monitored, e.g., kinetic energy, dissipation-rate, or Taylor-Reynolds number. At sufficiently high frequencies the amplitude of the kinetic energy response decreases as $1/\omega$. For frequencies beyond the range of maximal response, a significant change in phase-shift relative to the time-modulated forcing is observed.Comment: submitted to Europhysics Letters (EPL), 8 pages, 8 Postscript figures, uses epl.cl

Non-local modulation of the energy cascade in broad-band forced turbulence

Classically, large-scale forced turbulence is characterized by a transfer of energy from large to small scales via nonlinear interactions. We have investigated the changes in this energy transfer process in broad-band forced turbulence where an additional perturbation of flow at smaller scales is introduced. The modulation of the energy dynamics via the introduction of forcing at smaller scales occurs not only in the forced region but also in a broad range of length-scales outside the forced bands due to non-local triad interactions. Broad-band forcing changes the energy distribution and energy transfer function in a characteristic manner leading to a significant modulation of the turbulence. We studied the changes in this transfer of energy when changing the strength and location of the small-scale forcing support. The energy content in the larger scales was observed to decrease, while the energy transport power for scales in between the large and small scale forcing regions was enhanced. This was investigated further in terms of the detailed transfer function between the triad contributions and observing the long-time statistics of the flow. The energy is transferred toward smaller scales not only by wavenumbers of similar size as in the case of large-scale forced turbulence, but by a much wider extent of scales that can be externally controlled.Comment: submitted to Phys. Rev. E, 15 pages, 18 figures, uses revtex4.cl

Leray and LANS-α\alpha modeling of turbulent mixing

Mathematical regularisation of the nonlinear terms in the Navier-Stokes equations provides a systematic approach to deriving subgrid closures for numerical simulations of turbulent flow. By construction, these subgrid closures imply existence and uniqueness of strong solutions to the corresponding modelled system of equations. We will consider the large eddy interpretation of two such mathematical regularisation principles, i.e., Leray and LANS$-\alpha$ regularisation. The Leray principle introduces a {\bfi smoothed transport velocity} as part of the regularised convective nonlinearity. The LANS$-\alpha$ principle extends the Leray formulation in a natural way in which a {\bfi filtered Kelvin circulation theorem}, incorporating the smoothed transport velocity, is explicitly satisfied. These regularisation principles give rise to implied subgrid closures which will be applied in large eddy simulation of turbulent mixing. Comparison with filtered direct numerical simulation data, and with predictions obtained from popular dynamic eddy-viscosity modelling, shows that these mathematical regularisation models are considerably more accurate, at a lower computational cost.Comment: 42 pages, 12 figure

Approximate deconvolution discretisation

A new strategy is presented for the construction of high-order spatial discretisations extracted from a lower-order basic discretisation. The key consideration is that any spatial discretisation of a derivative of a solution can be expressed as the exact differentiation of a corresponding ‘filtered’ solution. Hence, each numerical discretisation method may be directly linked to a unique spatial filter, expressing the truncation error of the basic method. By approximately deconvolving the implied filter of the basic numerical discretisation an augmented high-order method can be obtained. In fact, adopting a deconvolution of the implied filter of suitable higher order enables the formulation of a new spatial discretisation method of correspondingly higher order. This construction is illustrated for finite difference (FD) discretisation schemes, solving partial differential equations in fluid mechanics. Knowing the implied filter of the basic discretisation, one can derive a corresponding higher order method by approximately eliminating the implied spatial filter to a certain desired order. We use deconvolution to compensate for the implied filter. This corresponds to a ‘sharpening’ of numerical solution features before the application of the basic FD method. The combination will be referred to as Approximate Deconvolution Discretisation (ADD). The accuracy of the deconvolved FD scheme depends on the order of approximation of the deconvolution filter. We present the ‘sharpening’ of several well-known FD operators for first- and second-order derivatives and quantify the achieved accuracy in terms of the modified wavenumber spectrum. Examples include high-order extensions up to new schemes with spectral accuracy. The practicality of the deconvolved FD schemes is illustrated in various ways: (i) by investigation of exactly solvable advection and diffusion problems, (ii) by tracking the evolution of the numerical solution to the Taylor-Green vortex problem and (iii) by showing that ADD yields spectral accuracy for the Burgers equation and for double-jet flow of an incompressible fluid.</p

Accuracy control for large-eddy simulation of turbulent mixing - Integral length-scale approach

Turbulent flow at high Reynolds numbers is currently not accessible on the basis of direct numerical simulation (DNS) of the Navier-Stokes equations - the computational complexity is too high to allow DNS in most realistic flow conditions. Instead, Large-Eddy Simulation (LES) offers an alternative in which the focus is on capturing the larger dynamic scales of a problem. However, the fundamental closure problem in LES induced by spatial filtering of nonlinear terms, and the role of discretization errors in the numerical treatment of the LES equations, induce a principal uncertainty in any LES prediction. This uncertainty requires quantification and control. We investigate error control capabilities of the Integral Length-Scale Approximation (ILSA) and apply this modeling to transitional and turbulent mixing, focussing on the achieved reliability of LES as function of the grid resolution and ‘sub-filter activity’.</p

Flow and bubble statistics of turbulent bubble-laden downflow channel

Direct numerical simulations of fully developed turbulent channel downflow at bulk Re equal to 6300, loaded with monodisperse bubbles at gas volume fractions α=0.5% , α=2.5% and α=10 have been carried out. Bubble deformability, surface tension, as well as discontinuity in the material properties across the bubble interfaces are explicitly accounted for. A full-scale channel of size 4πH × 2H × 4πH/3 in terms of the channel half-width H containing a number of bubbles up to O(103) is considered. The statistical structure of the bubbles, the probability density function describing the bubble velocity and the liquid kinetic energy spectra have been determined. A close range preferential clustering of the bubbles was found with a maximum density independent of the gas volume fraction at a separation distance of about 2.2R, with R the bubble radius. Preferential horizontal alignment and a general tendency to repulsion is shown for separation distances smaller than 3R. At larger separation distances a close to random distribution is observed for α=2.5% and α=10%, while tendency to vertical alignment is observed for α=0.5% . The pdf of the bubble velocity fluctuations was found to be well approximated by a Gaussian distribution. The liquid kinetic energy spectra in the channel core do not show a marked -3  scaling, which was previously reported for homogeneous isotropic turbulence and pseudo-turbulence