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Numerical investigation and extension of quadrature-based moment methods for population balances
(2023)
Particulate systems can be described by a number density function (NDF) with respect to a vector of internal coordinates. The evolution of the NDF is governed by the typically high-dimensional population balance equation (PBE). A common approach to reduce the dimensionality of the problem is to solve only for a set of moments instead of the NDF. The derived system of moment equations, however, includes unclosed integral terms that still contain the unknown NDF. One way to close the system of moment equations is to approximate the unclosed integral terms using a Gaussian quadrature computed from the moments. The procedure of taking a set of moments to compute a Gaussian quadrature, which is, in turn, used to close the moment equations, is known as the quadrature method of moments (QMOM). It gave rise to an entire family of methods, the quadrature-based moment methods (QBMMs), which are the primary focus of this work. The presented research can be divided into three major parts. The first part involves the formulation of a common Lagrangian droplet breakup model for QBMMs and the numerical investigation with the QMOM as well as the more sophisticated extended QMOM (EQMOM). The results indicate that the approximations are reasonably accurate when at least six moment equations are solved, with the EQMOM providing no advantages for the investigated configurations. In the second part, a quadrature-based moment model for the effects of fluid turbulence on particle velocities is formulated. The resulting moment equations contain non-smooth integrands that are the source of large errors when using common QBMMs. As an alternative, the Gauss/anti-Gauss QMOM (GaG-QMOM) is proposed that uses the average of a Gaussian and an anti-Gaussian quadrature. Numerical studies show that the GaG-QMOM is able to significantly reduce the previously observed large errors. Another novelty proposed in this context is the modification of the second-order strong-stability preserving Runge-Kutta method to guarantee the preservation of moment realizability in the presence of phase-space diffusion. The third part is concerned with the numerical exploration of the core algorithm of most QBMMs in terms of performance and accuracy. The algorithm consists of, first, computing the recurrence coefficients of the orthogonal polynomials associated with a set of moments, second, solving a symmetric tridiagonal eigenvalue problem to obtain the quadrature nodes and weights, and third, evaluating the integral terms in the moment equations. The results indicate that the contribution of the first step to compute the recurrence coefficients from moments to the overall computational costs is negligible. Instead, the primary focus should be on the fast solution of the eigenvalue problem and, possibly, on the efficient implementation of the moment source term evaluation, which becomes important when second-order processes are considered.
The Taylor-Couette (TC) flow, the flow confined between two concentric independently rotating cylinders, is used as a perfect model to investigate shear flow over concave surfaces and one of the paradigmatic systems of the physics of fluids. In this thesis, an experimental investigation of the turbulent TC flow in a very wide gap geometry with a radius ratio 𝜂 = 0.1 is performed. The physical and dynamic behavior of the flow is studied in a geometry that has rarely been investigated before the current study, which makes this study unique. The study aims to understand the effect of curvature on the TC flow, particularly in cases where the circumferential length of the inner cylinder is smaller than the gap width. The flow is studied in the different rotation regimes: counter-rotating, co-rotating, and purely inner cylinder rotating regimes up to shear Reynolds numbers Re_s≤ 150000. The flow field has been qualitatively studied using visualization techniques. By probing the different flow parameters, familiar coherent TC flow patterns appear, in addition to newly observed patterns we assume only exist for very wide gap TC flows. For a more detailed quantitative study, a time-resolved velocity field measurement has been conducted using the High-speed Particle Image Velocimetry technique through the system end plate. The radial and azimuthal velocity components in the 2D horizontal plane are measured at different axial positions, in order to scan the axial variance of the flow. The recorded flow field is used to compute the angular momentum transport in terms of the quasi-Nusselt number (Nu_ω). The results show a maximum of Nu_ω for low counter-rotating rates of −0.011 ≤ μ_max ≤ −0.0077, which is associated with large-scale structures that span the entire gap. Moreover, the Nu_ω decreases for counter-rotation rates higher than μ_max until it reaches a minimum value and then tends to increase again for higher counter-rotation cases. The space-time behavior of the turbulent flow field for the high counter-rotating cases shows the existence of newly observed patterns next to the outer cylinder wall that propagates inward, enhancing the angular momentum transport and resulting in a second maximum in transport for higher counter-rotating rates. For the pure rotating inner cylinder, the momentum transport scaling with the shear rate Nu_ω ∼ Re_s^α has been studied, and it shows a transition in scaling to 𝛼 = −0.76 for all flows with Re_(s )≥ 25000. This new scaling reveals the transition of the flow from the classical turbulent regime to the ultimate one, where this transition is accompanied by a clear change in the flow behavior. Moreover, the flow in the co-rotating regime and particularly in the centrifugal stable regime (𝜇 > +0.01) is investigated. The Velocity measurements show the presence of disturbed flow near the inner cylinder, where the measured velocity profiles showed a clear deviation from those predicted by laminar flow for flows up to 𝜇 = +0.04.