Second-order Velocity Moments Research Articles

Spectral characteristics of particle preferential concentration in turbulence computed by Eulerian and Lagrangian methods

This study conducts a comprehensive analysis of the Fourier spectra characteristics of particle preferential concentration in turbulent flows obtained by Eulerian and Lagrangian modelling approaches for different Stokes numbers. Particle preferential concentration is characterized by clusters and voids in the spatial particle distributions formed in turbulent flows and can significantly influence processes such as droplet coalescence, evaporation, and gas-particle reactions. The research primarily focuses on comparing the performance of two Eulerian models—one with and one without second-order velocity moments—and one Lagrangian model, which are used to predict particle dispersion. Key findings include the impact of artificial diffusion in Eulerian methods, the superiority of methods incorporating second-order velocity moments contributions at higher Stokes numbers, and the sensitivity of the Lagrangian method to Poisson statistics. Notably, the study reveals minimal variation in mean velocity spectra across different Stokes numbers, as opposed to the marked variations in particle concentration, momentum, and energy spectra. Grid resolution emerges as a crucial factor in enhancing spectral energy predictions in Eulerian methods. The research underscores the nuanced distinctions between Eulerian and Lagrangian methods in modelling preferential concentration, providing a detailed spectral comparison of particle concentration, velocity, momentum, and energy, highlighting the importance of method selection based on specific modelling needs.

Force methods for the two-relaxation-times lattice Boltzmann.

The two-relaxation-times collision benefits the steady lattice Boltzmann method by yielding viscosity-independent numerical errors. We present in an intuitive way how to incorporate popular force methods into the two-relaxation-times collision. We subsequently rewrite force methods into a generic equation to reveal commonalities and differences. We prove that force methods with a second-order velocity moment of the force break the viscosity independence. A force method with only a first-order velocity moment of the force averts this breakage. We validate our proof numerically.

Discrete effects on the forcing term for the lattice Boltzmann modeling of steady hydrodynamics

This work concerns with the clean inclusion of the forcing term in the lattice Boltzmann method (LBM) for the modeling of non-uniform body forces in steady hydrodynamics. The study is conducted for the two-relaxation-time (TRT) scheme. Here, we consider a simple, but yet sufficiently generic, flow configuration driven by a spatially varying body force based on which we derive the analytical solution of the forced LBM-TRT and compare two force strategies set by first- and second-order expansions. This procedure exactly establishes the macroscopic system satisfied by each force formulation at discrete level. The obtained theoretical results are further verified in two distinct benchmark channel flow problems. Overall, this study shows that the spatial discrete effects posed by the LBM modeling of the force term may come in through two sources. The first one is a defect inherent to LBM, arising from the non-local spatial discretization of the forcing term, and given by the discrete Laplacian of the body force. While it corrupts the discrete momentum balance with a non-linear viscosity dependent term in the single-relaxation-time schemes, this inconsistency is avoided with the TRT scheme, through its free-tunable relaxation degree of freedom Λ. The other error source is unique to the use of second-order force expansions, where its non-zero second-order velocity moment interferes with the discrete momentum balance through a spurious first-order derivative term, leading to several forms of inconsistency in the LBM steady solution. For simulations under the convective scaling it makes the LBM scheme a numerical representation of a differential system distinct from the physical one, whereas under the diffusive scaling it leads to viscosity-dependent numerical errors, which corrupt the otherwise consistent structure of TRT steady-state solutions. In contrast, the defects reported herein are absent with the first-order force expansion scheme operated within the scope of the LBM-TRT model for steady hydrodynamics.

Spatially-averaged flows over mobile rough beds: equations for the second-order velocity moments

ABSTRACTThe double-averaging methodology is used in this paper for deriving equations for the second-order velocity moments (i.e. turbulent and dispersive stresses) that emerge in the double-averaged momentum equation for incompressible Newtonian flows over mobile boundaries. The starting point in the derivation is the mass and momentum conservation equations for local (at a point) instantaneous variables that are up-scaled by employing temporal and spatial averaging. First, time-averaged conservation equations for mass, momentum, and turbulent stresses for mobile bed conditions are derived. Then, the double-averaged hydrodynamic equations obtained by spatial averaging the time-averaged equations are proposed. The derived second-order equations can serve as a basis for the construction of simplified mathematical and numerical models and for interpretation of experimental and simulation data when bed mobility is present. Potential applications include complex flow situations such as free-surface flows over vegetated or mobile sedimentary beds and flows through tidal and wind turbine arrays.

Wind-Tunnel Simulation of Weakly and Moderately Stable Atmospheric Boundary Layers

The simulation of horizontally homogeneous boundary layers that have characteristics of weakly and moderately stable atmospheric flow is investigated, where the well-established wind engineering practice of using ‘flow generators’ to provide a deep boundary layer is employed. Primary attention is given to the flow above the surface layer, in the absence of an overlying inversion, as assessed from first- and second-order moments of velocity and temperature. A uniform inlet temperature profile ahead of a deep layer, allowing initially neutral flow, results in the upper part of the boundary layer remaining neutral. A non-uniform inlet temperature profile is required but needs careful specification if odd characteristics are to be avoided, attributed to long-lasting effects inherent of stability, and to a reduced level of turbulent mixing. The first part of the wind-tunnel floor must not be cooled if turbulence quantities are to vary smoothly with height. Closely horizontally homogeneous flow is demonstrated, where profiles are comparable or closely comparable with atmospheric data in terms of local similarity and functions of normalized height. The ratio of boundary-layer height to surface Obukhov length, and the surface heat flux, are functions of the bulk Richardson number, independent of horizontal homogeneity. Surface heat flux rises to a maximum and then decreases.

On the rate of black hole binary mergers in galactic nuclei due to dynamical hardening

We assess the contribution of dynamical hardening by direct three-body scattering interactions to the rate of stellar-mass black hole binary (BHB) mergers in galactic nuclei. We derive an analytic model for the single-binary encounter rate in a nucleus with spherical and disk components hosting a super-massive black hole (SMBH). We determine the total number of encounters $N_{\rm GW}$ needed to harden a BHB to the point that inspiral due to gravitational wave emission occurs before the next three-body scattering event. This is done independently for both the spherical and disk components. Using a Monte Carlo approach, we refine our calculations for $N_{\rm GW}$ to include gravitational wave emission between scattering events. For astrophysically plausible models we find that typically $N_{\rm GW} \lesssim$ 10. We find two separate regimes for the efficient dynamical hardening of BHBs: (1) spherical star clusters with high central densities, low velocity dispersions and no significant Keplerian component; and (2) migration traps in disks around SMBHs lacking any significant spherical stellar component in the vicinity of the migration trap, which is expected due to effective orbital inclination reduction of any spherical population by the disk. We also find a weak correlation between the ratio of the second-order velocity moment to velocity dispersion in galactic nuclei and the rate of BHB mergers, where this ratio is a proxy for the ratio between the rotation- and dispersion-supported components. Because disks enforce planar interactions that are efficient in hardening BHBs, particularly in migration traps, they have high merger rates that can contribute significantly to the rate of BHB mergers detected by the advanced Laser Interferometer Gravitational-Wave Observatory.

Influence of the mixing parameter on the second order moments of velocity and concentration in Rayleigh–Taylor turbulence

The purpose of this paper is to highlight the existence of simple algebraic expressions linking the second order moments of velocity and concentration in Rayleigh–Taylor turbulence, in the Boussinesq limit. Focusing on the concentration variance, these relations allow to underline the influence of mixing on the remaining second order correlations, as well as on the growth rate of the mixing zone.

On turbulence characteristics at Ny-Ålesund–Svalbard

The unique data set collected at the climate change tower (CCT) installed in the Arctic site of Ny-Alesund, Svalbard, allows to investigate the peculiar features of the high-latitude planetary boundary layer under different stability conditions. The analysis uses four conventional Young anemometers and Vaisala thermo-hygrometers, alternated by three lined up sonic anemometers, installed on the CCT between 2 and 33.4 m a.g.l. The data are averaged over 10 min. The results presented here highlight the problematic behavior of the vertical mixing in quasi-neutral, low-wind conditions and the properties of the similarity scaling in unstable conditions, coupled with the evaluation of the scaled second-order moments of velocity and temperature. As far as stable cases are considered, the vertical profiles of the mean velocity and of the momentum and heat fluxes are shown, for high- and low-wind conditions and for traditional and upside–down cases, respectively.

Wall-resolved Large Eddy Simulation in grid-free 5 × 5 rod bundle of MATiS-H experiment

In the present paper the WALE model has been implemented in OpenFOAM code and applied for Large Eddy Simulation (LES) of two wall-bounded turbulent flow cases at moderately high Reynolds numbers, i.e. Re=O(104). Firstly, the LES using the WALE model is verified against the direct numerical simulation (DNS) of the turbulent channel flow at Reτ=1020. An appropriate mesh discretization for a given numerical method is determined and the underresolution effects on first and second order moments of velocity are investigated. Then, the obtained best practices are used for accurate wall-resolved LES of a turbulent flow in the 5×5 rod fuel bundle. The velocities and fluctuations are compared with Laser Doppler Velocimetry measurements of the MATiS-H experiment, which was performed in 2011 at KAERI, South Korea. Experiments and simulations are compared in the bare fuel bundle section without spacer grids. LES using WALE model is found as a pretty accurate tool for simulations of the flow through the rod bundle, if sufficiently fine mesh is applied. Mean velocity profiles are predicted with a remarkable accuracy, while the second-order statistics exhibit slightly larger discrepancies.

Nuclear star cluster formation in energy-space

In a virialized stellar system, the mean-square velocity is a direct tracer of the energy per unit mass of the system. Here, we exploit this to estimate and compare root-mean-square velocities for a large sample of nuclear star clusters and their host (late- or early-type) galaxies. Traditional observables, such as the radial surface brightness and second-order velocity moment profiles, are subject to short-term variations due to individual episodes of matter infall and/or star formation. The total mass, energy and angular momentum, on the other hand, are approximately conserved. Thus, the total energy and angular momentum more directly probe the formation of galaxies and their nuclear star clusters, by offering access to more fundamental properties of the nuclear cluster-galaxy system than traditional observables. We find that there is a strong correlation, in fact a near equality, between the root-mean-square velocity of a nuclear star cluster and that of its host. Thus, the energy per unit mass of a nuclear star cluster is always comparable to that of its host galaxy. We interpret this as evidence that nuclear star clusters do not form independently of their host galaxies, but rather that their formation and subsequent evolution are coupled. We discuss how our results can potentially be used to offer a clear and observationally testable prediction to distinguish between the different nuclear star cluster formation scenarios, and/or quantify their relative contributions.

Examination of the four-fifths law for longitudinal third-order moments in incompressible magnetohydrodynamic turbulence in a periodic box

The four-fifths law for third-order longitudinal moments is examined, using direct numerical simulation (DNS) data on three-dimensional (3D) forced incompressible magnetohydrodynamic (MHD) turbulence without a uniformly imposed magnetic field in a periodic box. The magnetic Prandtl number is set to one, and the number of grid points is 512(3). A generalized Kármán-Howarth-Kolmogorov equation for second-order velocity moments in isotropic MHD turbulence is extended to anisotropic MHD turbulence by means of a spherical average over the direction of r. Here, r is a separation vector. The viscous, forcing, anisotropic and nonstationary terms in the generalized equation are quantified. It is found that the influence of the anisotropic terms on the four-fifths law is negligible at small scales, compared to that of the viscous term. However, the influence of the directional anisotropy, which is measured by the departure of the third-order moments in a particular direction of r from the spherically averaged ones, on the four-fifths law is suggested to be substantial, at least in the case studied here.

Calculation of Heat Transfer Coefficients on a Flat Plate by Pseudo Direct Numerical Simulation of Turbulence

Pseudo direct numerical simulation applying the finite difference method is performed on a fully-developed turbulent flow through parallel flat walls heated by uniform flux. The study is aimed at examining the influence brought to the turbulent field calculation by differences in the level of spatial resolution obtainable in the streamwise and spanwise directions. All in all, satisfactory results are obtained for the mean profiles and second-order moments of velocity and temperature, as well as on the Nusselt number of various flows presenting-various Prandtl numbers. Differences in the level of spatial resolution prove to affect little the accuracy of calculated results.

General solution of the Jeans equations for triaxial galaxies with separable potentials

The Jeans equations relate the second-order velocity moments to the density and potential of a stellar system. For general three-dimensional stellar systems, there are three equations and six independent moments. By assuming that the potential is triaxial and of separable Stackel form, the mixed moments vanish in confocal ellipsoidal coordinates. Consequently, the three Jeans equations and three remaining non-vanishing moments form a closed system of three highly symmetric coupled first-order partial differential equations in three variables. These equations were first derived by Lynden-Bell, over 40 years ago, but have resisted solution by standard methods. We present the general solution here. We consider the two-dimensional limiting cases first. We solve their Jeans equations by a new method which superposes singular solutions. The singular solutions, which are new, are standard Riemann–Green functions. The resulting solutions of the Jeans equations give the second moments throughout the system in terms of prescribed boundary values of certain second moments. The two-dimensional solutions are applied to non-axisymmetric discs, oblate and prolate spheroids, and also to the scale-free triaxial limit. There are restrictions on the boundary conditions that we discuss in detail. We then extend the method of singular solutions to the triaxial case, and obtain a full solution, again in terms of prescribed boundary values of second moments. There are restrictions on these boundary values as well, but the boundary conditions can all be specified in a single plane. The general solution can be expressed in terms of complete (hyper)elliptic integrals, which can be evaluated in a straightforward way, and provides the full set of second moments that can support a triaxial density distribution in a separable triaxial potential.

Diffusion in Turbulent Pipe Flow Using a Stochastic Model.

A generalized Langevin equation developed by Haworth and Pope [Phys. Fluids, 29-2 (1986), 387] is applied to calculate the dispersion of a passive contaminant in turbulent pipe flow. The model coefficients in the equation are determined from an algebraic relation based on the consistency condition for the second-order moments of velocity, which includes the third-order moments. In the present model, the first-and second-order moments are used as input data, but the third-order moments are not inputted due to lack of reliable data. First, we confirmed numerically the consistency condition for a simulated velocity field. Second, the long-time diffusion was examined. The Eulerian velocity statistics show good agreement with the prescribed data up to second order. With regard to long-time diffusion, the longitudinal distributions of a cross-sectional mean concentration agree well with experiments. It is also found that the appropriate value for the Kolmogorov constant Co is 1.9 for this problem.

Dynamics of direct large-small scale couplings in coherently forced turbulence: concurrent physical- and Fourier-space views

As discussed in a recent paper by Brasseur & Wei (1994), scale interactions in fully developed turbulence are of two basic types in the Fourier-spectral view. The cascade of energy from large to small scales is embedded within ‘local-to-non-local’ triadic interactions separated in scale by a decade or less. ‘Distant’ triadic interactions between widely disparate scales transfer negligible energy between the largest and smallest scales, but directly modify the structure of the smallest scales in relationship to the structure of the energy-dominated large scales. Whereas cascading interactions tend to isotropize the small scales as energy moves through spectral shells from low to high wavenumbers, distant interactions redistribute energy within spectral shells in a manner that leads to anisotropic redistributions of small-scale energy and phase in response to anisotropic structure in the large scales. To study the role of long-range interactions in small-scale dynamics, Yeung & Brasseur (1991) carried out a numerical experiment in which the marginally distant triads were purposely stimulated through a coherent narrow-band anisotropic forcing at the large scales readily interpretable in both the Fourier- and physical-space views. It was found that, after one eddy turnover time, the smallest scales rapidly became anisotropic as a direct consequence of the marginally distant triadic group in a manner consistent with the distant triadic equations. Because these asymptotic equations apply in the infinite Reynolds number limit, Yeung & Brasseur argued that the observed long-range effects should be applicable also at high Reynolds numbers.We continue the analysis of forced simulations in this study, focusing (i) on the detailed three-dimensional restructuring of the small scales as predicted by the asymptotic triadic equations, and (ii) on the relationship between Fourier- and physical-space evolution during forcing. We show that the three-dimensional restructuring of small-scale energy and vorticity in Fourier space from large-scale forcing is predicted in some detail by the distant triadic equations. We find that during forcing the distant interactions alter small-scale structure in two ways: energy is redistributed anisotropically within high-wavenumber spectral shells, and phase correlations are established at the small scales by the distant interactions. In the numerical experiments, the long-range interactions create two pairs of localized volumes of concentrated energy in three-dimensional Fourier space at high wavenumbers in which the Fourier modes are phase coupled. Each pair of locally phase-correlated volumes of Fourier modes separately corresponds to aligned vortex tubes in physical space in two orthogonal directions. We show that the dynamics of distant interactions in creating small-scale anisotropy may be described in physical space by differential advection and distortion of small-scale vorticity by the coherent large-scale energy-containing eddies, producing anisotropic alignment of small-scale vortex tubes.Scaling arguments indicate a disparity in timescale between distant triadic interactions and energy-cascading local-to-non-local interactions which increases with scale separation. Consequently, the small scales respond to forcing initially through the distant interactions. However, as energy cascades from the large-scale to the small-scale Fourier modes, the stimulated distant interactions become embedded within a sea of local-to-non-local energy cascading interactions which reduce (but do not eliminate) small-scale anisotropy at later times. We find that whereas the small-scale structure is still anisotropic at these later times, the second-order velocity moment tensor is insensitive to this anisotropy. Third-order moments, on the other hand, do detect the anisotropy. We conclude that whereas a single statistical measure of anisotropy can be used to indicate the presence of anisotropy, a null result in that measure does not necessarily imply that the signal is isotropic. The results indicate that non-equilibrium non-stationary turbulence is particularly sensitive to long-range interactions and deviations from local isotropy.

Diffusion in Turbulent Pipe Flow Using the Stochastic Model.

A generalized Langevin equation developed by Haworth and Pope [Phys. Fluids, 29-2 (1986), 387] is applied to calculate the dispersion of a passive contaminant in turbulent pipe flow. The model coefficients in the equation are determined from an algebraic relation based on the consistency condition for the second-order moments of velocity, which includes the third-order moments. In the present model, the first- and second-order moments are used as the input data, but the third order moments are not specified due to the lack of reliable data. First, we checked on the consistency condition for the simulated velocity field. Second, the long time dispersion was examined. The Eulerian velocity statistics show good agreement with the specified data to the second order. With regard to long time dispersion, the longitudinal distributions of a cross-sectional mean concentration agree well with experiments. It is also found that the proper value of the Kolmogorov constant C0 is 1.9 for this problem.

Kinematics of velocity and vorticity correlations in turbulent flow

The kinematic problem of calculating second-order velocity moments from given values of the vorticity covariance is examined. Integral representation formulas for second-order velocity moments in terms of the two-point vorticity correlation tensor are derived. The special relationships existing between velocity moments in isotropic turbulence are expressed in terms of the integral formulas yielding several kinematic constraints on the two-point vorticity correlation tensor in isotropic turbulence. Numerical evaluation of these constraints suggests that a Gaussian curve may be the only form of the longitudinal velocity correlation coefficient which is consistent with the requirement of isotropy. It is shown that if this is the case, then a family of exact solutions to the decay of isotropic turbulence may be obtained which contains Batchelor’s final period solution as a special case. In addition, the computed results suggest a method of approximating the integral representation formulas in general turbulent shear flows.