Spectral Closure Research Articles

Nonuniversal k−3 energy spectrum in stationary two-dimensional homogeneous turbulence

A spectral closure analysis and numerical simulations suggest that there may be a class of two-dimensional turbulence in which the energy spectrum E(k) scales with the wave number k like E(k)=Ak−3 in the enstrophy transfer range in accordance with the Kraichnan–Leith–Batchelor (KLB) spectrum, but the prefactor A is different from the KLB spectrum and depends in a nontrivial way on the flow conditions at large scales.

Complete Closure of the Cylindrical Harmonic Eigenfunction Expansion for the Lorentz Gauge-Free Space Magnetic Vector Potential Dyadic Green's Function

All three spectral sums in the cylindrical harmonic eigenfunction expansion of the Lorentz gauge-free space magnetic vector potential dyadic Green's function are closed. The specific techniques used to close the radial, azimuthal, and axial spectral sums are detailed. The magnetic vector potential dyadic Green's function spectral closure process is paralleled with that of the corresponding scalar Green's function eigenfunction expansion. Fundamental relationships between the scalar and dyadic spectral closures are noted. The cancellation of static and traveling wave pole contributions is noted in order to achieve reduced expressions at each stage of the spectral closure process. The techniques used here to close the spectral sums in the free space eigenfunction expansion are also applicable to certain scattering geometries when the method of scattering superposition is employed. Thus, the complexity of the dyadic Green's function computation may be reduced.

Testing a random phase approximation for bounded turbulent flow.

Tractable implementation of a spectral closure requires that the modal representation of the energy satisfy a restricted random phase approximation (RRPA). This condition is exactly satisfied when the statistical system is homogeneous and the basis functions are Fourier modes. In this case, the ensemble average of the spectral covariance diagonalizes, i.e., <c(k(1))c(k(2))>=delta(k(1)+k(2))<c(k(1))c(k(2))>, where c(k,t) is a Fourier coefficient in a Galerkin representation of the velocity field. However, for inhomogeneous statistical systems in which the Fourier system is inappropriate, the RRPA requires validation. We use direct numerical simulations (DNSs) of the Navier-Stokes and truncated Euler equations to test the degree to which the RRPA is satisfied when applied to a recent representation due to Turner (LANL Unclassified Report No. LA-UR-96-3257) of a bounded turbulent rectangular channel flow with free slip, stress free walls. It is shown that a complete test of the RRPA for a fully inhomogeneous DNS with N3 grid points actually requires N3+1 members in the ensemble. The "randomness" of the phase can be characterized by a probability density function (PDF) of the modulus of the normalized spectral covariance. Results reveal that for both the Navier-Stokes and Euler systems the PDF does not change in time as the turbulence decays, and that the PDF for the Euler system is virtually identical to the one produced from an ensemble of random fields. This result is consistent with the equipartition of energy for the Euler system, in which the RRPA becomes an exact result rather than an approximation as the number of realizations approaches N3+1. The slight differences observed between the PDF produced from the random fields and the one from the Navier-Stokes system are thus shown to be entirely a result of the presence of a finite viscosity. It is also shown that there is great variation between statistics computed over the ensemble and those for a single realization.

Spectral Large-Eddy Simulations and Vortex Dynamics in Turbulence.

We present a point of view of large-eddy simulations(LES)in Fourier space, where the eddy coefficients are expressed thanks to a two-point spectral closure of isotropic turbulence, the EDQNM theory. Returning to real space, this leads to models of the structure-function family(plain, selective or filtered). These models are applied with success to predict the statistical distributions and coherent-vortex dynamics for a wide variety of turbulent flows. In three-dimensional decaying isotropic turbulence, we confirm the existence of a κ4 infrared backscatter in the kinetic-energy spectrum, and predict a new κ2 law for the pressure spectrum in this range. In the mixing layer(temporal or spatial), we show how to manipulate the topology of Kelvin-Helmholtz vortices, from quasi two-dimensionality to helical pairing. The latter vortex organization is found in a backward-facing step just behind the step, and yields big staggered Λ-vortices which are carried away downstream. In a developed turbulent boundary layer, coherent vortices are hairpins generated above the low-speed streaks by a secondary Kelvin-Helmholtz instability. Afterwards, we consider LES of compressible turbulence, studied with Favre averages, and where the introduction of a macro-temperature and a macro-pressure simplifies greatly the problem. Finally, we show in rotating shear flows(free or wall bounded, axis of rotation in the spanwise direction)a universal behaviour of the mean velocity which becomes linear in certain anticyclonic regions.

Thirty Years of VLBI: Early Days, Successes, and Future

AbstractThe early days of VLBI are discussed, with emphasis on the fifteen experiments performed in the U.S. and Canada in 1967. Some key technical ideas are described that influenced the development of the technique, including digital fringe rotation, bandwidth synthesis, spectral phase referencing, phase closure and self-calibration, and water vapor radiometry.

Spectral closure for non-commutative rings

Spectral closure for non-commutative rings

Classical closure theory and Lam's interpretation of epsilon -renormalization group theory.

It is shown that Lam's formulation of renormalization group theory [Phys. Fluids A 4, 1007 (1992)] is essentially the physical space version of the spectral classical closure theory [Leslie and Quarini, J. Fluid Mech. 91, 65 (1979)].

Inertial transfers in the helical decomposition

The directions of the triadic energy transfers assumed in a previous paper [Phys. Fluids A 4, 350 (1992)] agree with the spectral closures, in a similarity range, if the exponent of the power-law energy spectrum is less than unity. The helical interactions showing a strong local energy transfer when the triad is nonlocal, sum up to a reverse cascade unless the spectrum falls off faster than a −7/3 power of the wave number. The energy cascades from each type of helical interaction are calculated for a −5/3 inertial range using the eddy damped quasinormal Markovian (EDQNM) model. One type of interaction is responsible for 86% of the cascade. The contributions of the two classes of helical interactions to the subgrid-scale eddy viscosity are presented, together with the contributions from the forward and reverse cascading interactions. An application of the assumption on the triadic energy transfers to turbulence under rapid rotation gives a simple argument for the tendency toward nonlinear two-dimensionalization of the flow.

A spectral closure for premixed turbulent combustion in the flamelet regime

Premixed turbulent combustion in the flamelet regime is analysed on the basis of a field equation. This equation describes the instantaneous flame contour as an isoscalar surface of the scalar field G(x,t). The field equation contains the laminar burning velocity sL as velocity scale and its extension includes the effect of flame stretch involving the Markstein length [Lscr ] as a characteristic lengthscale of the order of the flame thickness. The scalar G(x,t) plays a similar role for premixed flamelet combustion as the mixture fraction Z(x,t) in the theory of non-premixed flamelet combustion.Equations for the mean $\overline{G}$ and variance $\overline{G^{\prime 2}}$ are derived. Additional closure problems arise for the mean source terms in these equations. In order to understand the nature of these terms an ensemble of premixed flamelets with arbitrary initial conditions in constant-density homogeneous isotropic turbulence is considered. An equation for the two-point correlation $\overline{G^{\prime}({\boldmath x},t)G^{\prime}({\boldmath x}+{\boldmath r},t)}$ is derived. When this equation is transformed into spectral space, closure approximations based on the assumption of locality and on dimensional analysis are introduced. This leads to a linear equation for the scalar spectrum function Γ(k,t), which can be solved analytically. The solution Γ(k,t) is analysed by assuming a small-wavenumber cutoff at k0 = lT−1, where lT is the integral lengthscale of turbulence. There exists a $k^{-\frac{5}{3}}$ spectrum between lT and LG, where LG is the Gibson scale. At this scale turbulent fluctuations of the scalar field G(x,t) are kinematically restored by the smoothing effect of laminar flame propagation. A quantity called kinematic restoration ω is introduced, which plays a role similar to the scalar dissipation χ for diffusive scalars.By calculating the appropriate moments of Γ(k,t), an algebraic relation between ω, $\omega,\overline{G^{\prime}({\boldmath x},t)^2}$, the integral lengthscale lT and the viscous dissipation ε is derived. Furthermore, the scalar dissipation χ[Lscr ], based on the Markstein diffusivity [Dscr ][Lscr ] = sL [Lscr ], and the scalar-strain co-variance Σ[Lscr ] are related to ω. Dimensional analysis, again, leads to a closure of the main source term in the equation for the mean scalar $\overline{G}$. For the case of plane normal and oblique turbulent flames the turbulent burning velocity sT and the flame shape is calculated. In the absence of flame stretch the linear relation sT ∼ u′ is recovered. The flame brush thickness is of the order of the integral lengthscale. In the case of a V-shaped flame its increase with downstream position is calculated.

Modeling of isotropic reacting turbulence by a hybrid mapping‐EDQNM closure

AbstractA hybrid model is developed and implemented for predicting the limiting bound of the reactant conversion rate in an isotropic turbulent flow under the influence of a reaction of the type A + B Products. This model is based on the amplitude mapping closure of Kraichnan for the molecular mixing of a stochastically distributed scalar, and the eddy‐damped quasi‐normal Markovian (EDQNM) spectral closure for the two‐point scalar covariance. The results predicted by this model compare well with available experimental data in both gaseous and aqueous plug‐flow reactors, but point to the need for more detailed measurements in future experimental studies. With the implementation of the mapping closure, a simple analytical expression is obtained for the decay rate of the unmixedness. This expression is very convenient and is recommended for direct practical applications in the modeling and design of plug‐flow reactors.

Turbulent premixed combustion: A flamelet formulation and spectral analysis in theory and IC-engine experiments

A flamelet formulation for premixed combustion has been derived based on a scalar field equation describing the instantaneous flame front location in the turbulent flow field. This partial differential equation is equivalent to the local kinematic balance between fluid flow velocity, burning velocity and flame propagation velocity. If the burning velocity depends locally on flame stretch and flame front curvature, an additional length scale, the Markstein length is introduced into the problem. A spectral closure of the problem has been performed on the basis of a two-point correlation. The transformation of the scalar field equation into wavenumber-space leads to the definition to a spectrum function for the fluctuations of the turbulent flame front. The cut-off of the inertial subrange occurs at the Gibson scale.

Passive scalar transport in β-plane turbulence

Evaluation of spectral closure theory and direct numerical simulation are used to examine the eddy transport of a passive scalar in barotropic β-plane flow. When a large-scale gradient of scalar concentration is imposed, the implied scale separation between fixed background gradient and eddies supports the concept of ‘eddy diffusion’. The results can be cast in terms of an eddy diffusion tensor K, whose behaviour as a function of mean vorticity gradient β is examined. Earlier theoretical work by Holloway &amp; Kristmannsson (1984) is extended to include cases where strong vorticity–scalar correlations are observed, and corrected in order to restore random Galilean invariance.The anisotropy of eddy energy and the direct influence of Rossby wave propagation contribute to the overall anisotropy of K, The resulting suppression of meridional diffusivity Kyy, and enhancement of zonal diffusivity Kxx, with increased β is examined. The variation in simulation Kyy is closely reproduced in the closure equations. However, the increased Kxx is the result of zonal jets whose persistence is not accounted for in the statistical theory.

Radiative Effects on Turbulent Temperature Spectra and Budgets in the Planetary Boundary Layer

The effects of radiative energy transfer on turbulent temperature fields are studied, and preliminary estimates show the infrared “radiative dissipation” mechanism to be dominant. Spectral computations for the idealized homogeneous-isotropic case allow an explicit solution in terms of the radiative damping rate of harmonic temperature fluctuations. Using new data and methods, this rate is evaluated for realistic values of absorption and diffusion by water vapor, carbon dioxide and water droplets. Effective computations of radiative modifications in the temperature spectrum are then performed with the aid of several spectral transfer closures. Radiative dissipation is found to occur over a wide range of scales, and its global effect is shown to be of the same order as conductive dissipation when the turbulent energy dissipation rate is low. Approximate computations are then made for the case of the planetary boundary layer, using semiempirical representations of the spectral structure. The resulting change in the temperature variance budget is discussed, and evaluation formulas are given for an apparent radiative damping frequency to be used in numerical models. Radiative damping could be significant in nocturnal stable terrestrial cases and in certain planetary atmospheres.

A comparative assessment of spectral closures as applied to passive scalar diffusion

We compare - both analytically and numerically – two related spectral (≡ two-point) closures for the problem of the decay of temperature fluctuations convected by isotropic turbulence. The methods are the test-field model (TFM) (Kraichnan 1971; Newman & Herring 1979) and the eddy-damped quasinormal Markovian (ENQNM) approximation (Orszag 1974; Lesieur & Schertzer 1978). We show that EDQNM may be regarded as a rational approximation to, and simplification of, the TFM, except at small wavenumbers, where an additional eddy-dissipative term is needed to produce satisfactory results for the former. We consider three available methods for determining the relaxation timescales: (i) comparison with experiments, (ii) comparison with the direct-interaction approximation (DIA) in thermal equilibrium, and (iii) comparison with DIA at very small wavenumber, where it is believed to represent the dynamics properly. Comparison with both large Reynolds number and wind-tunnel Reynolds numbers is presented. For the latter, we discuss the relationship of the present theoretical results to the experiments of Warhaft & Lumley (1978) and Sreenivasan et al. (1980), and to the theoretical analysis of Corrsin (1964), Kerr & Nelkin (1980) and Antonopolos-Domis (1981).