Spectral-based simulations of particle-laden turbulent flows

In this paper, we discuss the application of spectral-based methods to simulation of particle-laden turbulent flows. The primary focus of the article is on the past and ongoing works by the authors. The particles are tracked in Lagrangian framework, while direct numerical simulation (DNS) or large-eddy simulation (LES) is used to describe the carrier-phase flow field. Two different spectral methods are considered, namely Fourier pseudo-spectral method and Chebyshev multidomain spectral method. The pseudo-spectral method is used for the simulation of homogeneous turbulence. DNS of both incompressible and compressible flows with one- and two-way couplings are reported. For LES of particle-laden flows, two new models, developed by the authors, account for the effect of sub-grid fluctuations on the dispersed phase. The Chebyshev multidomain method is employed for the works on inhomogeneous flows. A number of canonical flows are discussed, including flow past a square cylinder, channel flow and flow over backward-facing step. Ongoing research on particle-laden LES of inhomogeneous flows is briefly reported.     

A Comparison of Large Scale Extraction Methods in the Study of Annular Wake Flow

Large scale periodic structures can exist in selected flow fields. Examples are the Precessing Vortex Core in swirling flows, vortex shedding behind a cylinder or the wake of an annular jet. A number of techniques are available to extract these large scales from the turbulent fluctuations in the flow field. In this paper, an analysis is made of three such methods: Eulerian Time Filtering (ETF), Proper Orthogonal Decomposition (POD) and non-linear least-squares regression POD (NLSR-POD). The accuracy of the three different extraction methods is compared quantitatively with phase averaged data of an annular wake flow. This flow was chosen as a test case, since it is widely used in industrial applications, such as for example bluff-body burners. It was shown that all three methods were able to reconstruct the flow field with reasonable accuracy. These techniques are therefore applicable to a number of periodic flows. The big advantage of these extraction methods is that they require 20 times less experimental data compared to phase averaging. All three methods require more or less the same computational time and since the computational time is a few orders of magnitude lower than the measurement time, application of these techniques results in a very large reduction in the total time to obtain the flow field characteristics. This results in a significant reduction of time in the design process of such flows.     

Calculation of the momentum and heat transfer in turbulent gas-particle jet flows

The equations for the second moments of the dispersed-phase velocity and temperature fluctuations are used for calculating gas-suspension jet flows within the framework of the Euler approach. The advantages of introducing the equations for the second moments of the particle velocity fluctuations has previously been quite convincingly demonstrated with reference to the calculation of two-phase channel boundary flows [9–11]. The flows considered below have a low solid particle volume concentration, so that interparticle collisions can be neglected and, consequently, the stochastic motion of the particles is determined exclusively by their involvement in the fluctuating motion of the carrier flow. In addition to the equations for the turbulent energy of the gas and its dissipation, the calculation scheme includes the equations for the turbulent energy and turbulent heat transfer of the solid phase; however, the model constructed does not contain additional empirical constants associated with the presence of the particles in the flow.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 69–80, May–June, 1992.     

Fundamental equations of turbulent two-phase flow

A new set of Reynolds equations for predicting turbulent two-phase flows has been developed by means of Reynolds averaging method on the unsteady laminar equations of two-phase flow. These equations involve average terms of products of turbulent fluctuations in some physical parameters in a large degree. The interaction forces between two phases, the pressures for dispersed phase, extra stresses except for pressure and the expressions for energy interchange between two phases have been discussed.     

Experimental investigation of a maximum entropy assumption for acceleration terms within a poly‐disperse moment framework

Moment transport methods are being developed to model poly‐dispersed multiphase flows by transporting statistical moments of the particle size–velocity joint probability density function (JPDF). A common feature of these methods is the requirement to reproduce or approximate the form of the JPDF from the transported moments for calculation of body force terms and other source terms. This paper examines the application of a maximum entropy technique against phase Doppler anemometry data sets from an electrostatically charged kerosene spray and also an automotive pressure swirl atomizer. An assessment of which moments are required to reproduce the JPDFs using a maximum entropy assumption to a sufficient level of accuracy is made. It is found that it is possible to reproduce the JPDFs to a high level of accuracy using a large number of moments; however, this incurs large computational overheads. If the moments to be transported are chosen on the basis of physical reasoning (such as the relationship between size and velocity due to drag) it is possible to reduce the number of moments to those which would be conserved via balance equations. This permits an approximation to the JPDF commensurate with the closure level of the moment transport method and thus the closure model method is naturally scalable with the degree of information from available conservation equations. Copyright © 2008 John Wiley & Sons, Ltd.     

Volume averaging for the analysis of turbulent spray flows

Spray flow calculations are usually based upon equations that have been developed by averaging droplet properties locally throughout the flow field. Presently, standard procedure for LES (large-eddy simulations) is to average these averaged equations once again to filter the short-length-scale fluctuations. In this paper, the theoretical foundations for the averaged spray equations are examined; then the volume-averaging process for LES and the volume-averaging process for two-phase flows are unified for the analysis of turbulent, two-phase flows. Comments are provided on the relationship between the averaging volume and the computational-cell volume. This paper provides generality to the weighting-function choice in the averaging process and precision to the definition of the volume over which the averaging is performed. New flux terms that result from the averaging process and appear in the governing averaged partial differential equations are identified and their modelling is discussed. Situations are identified where sufficient stratification of properties on the scale smaller than the averaging volume leads to the significance of these quantities. Evolution equations for averaged entropy and averaged vorticity are developed. The relationship amongst the curl of the average gas-phase velocity, the average of the gas-phase-velocity curl, and the rotation of the discrete droplets or particles is established. The needs and challenges for sub-grid modelling to account for small-vortex/droplet interactions are presented. Applications to spray combustion are discussed.     

Determination of boundary conditions in extended thermodynamics via fluctuation theory

A system of equations from extended thermodynamics is proposed for the calculation of the temperature field in a rarefied gas at rest. Numerical simulations suggest that the temperature field in this case should exhibit boundary layers superposed on the classical Fourier solution. In order to derive such a field from the extended thermodynamics of moments, one needs boundary values for higher moments, which in practice cannot be assigned and controlled. We suggest that such boundary data emerge as mean values of thermal fluctuations and thus calculate them. The result agrees qualitatively with numerical simulations and it is quantitatively of the same order of magnitude.Received: 12 September 2003, Accepted: 9 October 2003, Published online: 23 January 2004 Correspondence to: E. Barbera     

Higher-order moments and spectra of velocity fluctuations in adverse-pressure-gradient turbulent boundary layer

The characteristics of turbulent boundary layer flows with adverse pressure gradients (APGs) differ significantly from those of canonical boundary layers. We have investigated the effects of an APG on the higher-order moments and spectra of velocity fluctuations. The local wavelet spectra reveal a large difference in energy-containing frequencies of streamwise and wall-normal components of turbulent velocities, which results in smaller Reynolds shear stress production. Moreover, an analysis of bispectra in the Fourier space has revealed that non-local interactions, consisting of streamwise fluctuating velocity with low frequency and wall-normal velocity with high frequency, occur in the APG flow. However, the small-scale motions are not affected by imposing an APG.     

A phenomenological and extended continuum approach for modelling non-equilibrium flows

This paper presents a new technique that combines Grad’s 13-moment equations (G13) with a phenomenological approach to rarefied gas flows. This combination and the proposed solution technique capture some important non-equilibrium phenomena that appear in the early continuum-transition flow regime. In contrast to the fully coupled 13-moment equation set, a significant advantage of the present solution technique is that it does not require extra boundary conditions explicitly; Grad’s equations for viscous stress and heat flux are used as constitutive relations for the conservation equations instead of being solved as equations of transport. The relative computational cost of this novel technique is low in comparison to other methods, such as fully coupled solutions involving many moments or discrete methods. In this study, the proposed numerical procedure is tested on a planar Couette flow case, and the results are compared to predictions obtained from the direct simulation Monte Carlo method. This test case highlights the presence of normal viscous stresses and tangential heat fluxes that arise from non-equilibrium phenomena, which cannot be captured by the Navier–Stokes–Fourier constitutive equations or phenomenological modifications.      

Fourier time spectral method for subsonic and transonic flows

The time accuracy of the exponentially accu-rate Fourier time spectral method (TSM) is examined and compared with a conventional 2nd-order backward differ-ence formula (BDF) method for periodic unsteady flows. In particular, detailed error analysis based on numerical com-putations is performed on the accuracy of resolving the local pressure coefficient and global integrated force coefficients for smooth subsonic and non-smooth transonic flows with moving shock waves on a pitching airfoil. For smooth sub-sonic flows, the Fourier TSM method offers a significant accuracy advantage over the BDF method for the predic-tion of both the local pressure coefficient and integrated force coefficients. For transonic flows where the motion of the discontinuous shock wave contributes significant higher-order harmonic contents to the local pressure fluctuations, a sufficient number of modes must be included before the Fourier TSM provides an advantage over the BDF method. The Fourier TSM, however, still offers better accuracy than the BDF method for integrated force coefficients even for transonic flows. A problem of non-symmetric solutions for symmetric periodic flows due to the use of odd numbers of intervals is uncovered and analyzed. A frequency-searching method is proposed for problems where the frequency is not known a priori. The method is tested on the vortex shedding problem of the flow over a circular cylinder.     

Semiempirical theory of turbulence of inhomogeneous flows with body forces

A closed system of equations for the second moments, determining the turbulent transfer in inhomogeneous flows with body forces, is obtained in the local equilibrium approximation. Taking diffusion and convective transfer into account does not present difficulties, since the expressions obtained for the terms containing the fluctuations of the body forces and transfer terms are preserved. The effect of the external forces is considered for two cases: turbulent motion in a stratified medium and conducting flows in a longitudinal magnetic field. In the latter case the external forces are divergenceless and they therefore do not affect the pressure fluctuations. A single-scale model, in which the transverse scale is taken as the scale in the boundary layer, is considered.Translated from Izvestiya Akademii Nauk SSSS, Mekhanika Zhidkosti i Gaza, No. 3, pp. 33–40, May–June, 1985.     

An analytical approach for the closure equations of gas–solid flows with inter-particle collisions

This work examines the effect of inter-particle collisions on the motion of solid particles in two-phase turbulent pipe and channel flows. Two mechanisms for the particle–particle collisions are considered, with and without friction sliding. Based on these collision mechanisms, the correlations of the various velocity components of colliding particles are obtained analytically by using an averaging procedure. This takes into account three collision coordinates, two angles and the distance between the centers of colliding particles. The various stress tensor components are obtained and then introduced in the mass, linear momentum and angular momentum equations of the dispersed phase. The current approach applies to particle–particle collisions that result from both the average velocity difference and the turbulent velocity fluctuations. In order to close the governing equations of the dispersed phase, the pseudo-viscosity coefficients are defined and determined by the time of duration of the inter-particle collision process. The model is general enough to apply to both polydisperse and monodisperse particulate systems and has been validated by comparisons with experimental data.     

Methods of near-wall pressure-fluctuation measurements in the presence of vibration

Near-wall pressure fluctuations in turbulent flows are of considerable interest in many engineering applications. We shall concentrate on a number of specific questions related to the resolution of components of wall pressure spectra. Our emphasis shall be on outstanding problems of turbulent pressure fluctuations in the presence of vibration. A study on the interaction of a transducer with wall vibration resulting from near-wall turbulent flows has been performed. Three methods are described for the study of spectral components of turbulent surface pressure in conditions of flow-induced vibration: the method of separation of turbulent and vibration signals; the method of a vibration-proof turbulent pressure transducer; and a modified method of vibration suppression. A method of low-frequency acoustic-noise suppression is also suggested.     

The influence of phase-averaging window size on the determination of turbulence quantities in unsteady turbulent flows

 The phase-averaging window size is shown to affect the measurement of phase-averaged turbulence quantities in unsteady turbulent flows. The flow turbulence is usually estimated on the assumption of quasi-constant flow velocity during the duration of the phase-averaging window. The calculated turbulence level then consists of two parts: one due to the turbulent velocity fluctuations and the other due to the changes in the mean flow velocity. This second part is shown to be directly proportional to the averaging window size. In order to determine the true turbulence the averaging window size has to be made as small as possible, especially if the unsteady flow exhibits large temporal gradients and the flow turbulence itself is small. Received: 9 April 1996/Acceped: 17 August 1996     

Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I—theory

This paper is concerned with the development of an improved second-order homogenization method incorporating field fluctuations for nonlinear composite materials. The idea is to combine the desirable features of two different, earlier methods making use of “linear comparison composites”, the properties of which are chosen optimally from suitably designed variational principles. The first method (Ponte Castañeda, J. Mech. Phys. Solids 39 (1991) 45) makes use of the “secant” moduli of the phases, evaluated at the second moments of the strain field over the phases, and delivers bounds, but these bounds are only exact to first-order in the heterogeneity contrast. The second method (Ponte Castañeda, J. Mech. Phys. Solids 44 (1996) 827) makes use of the “tangent” moduli, evaluated at the phase averages (or first moments) of the strain field, and yields estimates that are exact to second-order in the contrast, but that can violate the bounds in some special cases. These special cases turn out to correspond to situations, such as percolation phenomena, where field fluctuations, which are captured less accurately by the second-order method than by the bounds, become important. The new method delivers estimates that are exact to second-order in the contrast, making use of generalized secant moduli incorporating both first- and second-moment information, in such a way that the bounds are never violated. Some simple applications of the new theory are given in Part II of this work.     

Nondestructive testing using temporal phase evaluation in speckle interferometry

A novel nondestructive testing (NDT) method is reported in which temporal evolution of the speckles in speckle interferometry is used to measure large object deformations. The basic principle of the method is that continuous object movement introduces fluctuations in the phase of the speckle and is recorded as intensity modulation. Acquiring a large number of frames of the object motion, the phase data for the whole object deformation are then retrieved by the Fourier transformation technique. The method is capable of measuring more than 100 μm in-plane and out-of-plane deformation with speckle interferometry and more than 500 μm for speckle shearing interferometry. The authors discuss the NDT results obtained with the three methods and make some relative comparisons of each.     

Interfacial transfer in Tryggvason's method

Tryggvason's method computes two phase flows by smoothing the regions near the boundaries between the two fluids. This requires averaging of the fluid properties, which may differ by orders of magnitude and can lead to large errors. Traditional linear averaging produces a first order error that can be extremely large. In this paper, we show that for a number of simple test cases, averaging of the inverses of the properties can lead to a significant improvement in the quality of the results. Copyright © 2003 John Wiley & Sons, Ltd.     

Ensemble phase averaged equations for multiphase flows in porous media. Part 2: A general theory

Most models for multiphase flows in a porous medium are based on a straightforward extension of Darcy’s law, in which each fluid phase is driven by its own pressure gradient. The pressure difference between the phases is thought to be an effect of surface tension and is called capillary pressure. Independent of Darcy’s law, for liquid imbibition processes in a porous material, diffusion models are sometime used. In this paper, an ensemble phase averaging technique for continuous multiphase flows is applied to derive averaged equations and to examine the validity of the commonly used models. Closure for the averaged equations is quite complicated for general multiphase flows in a porous material. For flows with a small ratio of the characteristic length of the phase interfaces to the macroscopic length, the closure relations can be simplified significantly by an approximation with a second order error in this length ratio. This approximation reveals the information of the length scale separation obscured during an averaging process and leads to an equation system similar to Darcy’s law, but with additional terms. Based on interactions on phase interfaces, relations among closure quantities are studied.     

Investigation of bed shear stresses subject to external turbulence

The fluctuating bed shear stress has largely been investigated only for uniform channel flows and boundary layers. In practical engineering, the flow conditions are often modified due to the presence of various hydraulic structures. To what extent the modification affects the known characteristics of the bed shear stress is not clear. This study presents experimental results of the bed shear stress fluctuations, which are obviously subjected to external turbulence generated by superimposing artificial structures in the open channel flows. The statistical analysis of the measurements shows that the probability density function of the bed shear stress can be described by the lognormal function. Some associated relations concerning higher-order moments, skewness and kurtosis, which are derived from the lognormal function, are further compared with the experimental data. Physical implication of the skewed probability density distribution is finally discussed.