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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.
Application of ODT to turbulent combustion problems in incompressible and compressible regimes
(2016)
The one-dimensional turbulence (ODT) model is applied to a reactant - to - product counterflow configuration as well as to a shock tube configuration in non-reactive flow and in deflagration and detonation regimes. The model employed herein solves conservation equations for momentum, energy, and species on a one dimensional (1D) domain corresponding to the line spanning the domain between nozzle orifice centers in the counterflow configuration and corresponding to the tube length in the shock tube configuration. The effects of turbulent mixing are modeled via a stochastic process, while the Kolmogorov and reactive length and time scales are explicitly resolved.
In the counterflow configuration, comparisons between model and DNS results for spatial mean and root-mean-square (RMS) velocity, temperature, and major and minor species profiles are shown. The ODT approach shows qualitatively and quantitatively reasonable agreement with the DNS data. Scatter plots and statistics conditioned on temperature are also compared for heat release rate and all species. ODT is able to capture the range of results depicted by DNS. However, conditional statistics show signs of underignition.
To carry out the shock tube simulations, the ODT methodology is extended to include an efficient compressible implementation and a model for capturing shock-induced turbulence is presented. The necessary algorithmic changes to include compressibility effects are highlighted and the model for capturing shock-turbulence interaction is presented. To validate the compressible solver, results for Sod’s shock tube problem are compared against a finite volume Riemann solver. To validate the model for shock-turbulence interaction, comparisons for a non-reactive and a reactive case are presented. First, results of a shock traveling from light (air) to heavy (SF6) with reshock have been simulated to match mixing width growth data of experiments and turbulent kinetic energy results from LES. Then, for one-step chemistry calibrated to represent an acetylene/air mixture, the interaction of a shock wave with an expanding flame front is simulated, and results with 2D simulation (2D-sim) data for flame brush formation and ensuing deflagration-to-detonation transitions (DDT) are compared. Results for the Sod shock tube comparison show that the shock speed and profile are captured accurately. Results for the nonreactive shock-reshock problem show that interface growth at all simulated Mach numbers is captured accurately and that the turbulent kinetic energy agrees in order of magnitude with LES data. The reactive shock tube results show that the flame brush thickness compares well to 2D-sim data and that the approximate location and timing of the DDT can be captured. The known sensitivity of DDT characteristics to details of individual Wow realizations, seen also in ODT, implies that model agreement can be quantified only by comparing Wow ensembles, which are presently unavailable other than in an ODT run-to-run sensitivity study that is reported herein.
Investigations of high Reynolds number pipe flow is up to now a great challenge due to the complex mechanisms which appear in pipe flow turbulence. Hence, suitable experimental facilities are necessary to resolve turbulent dynamics and therewith to provide the knowledge for the understanding of such a simple shear flow. For this reason the recent thesis deals with conceptual design and setup of a new high Reynolds number pipe test facility further on named CoLaPipe - Cottbus Large Pipe. It also comprises first investigations on pipe flow obtained from the new CoLaPipe, which can be classified into 1.)calibration measurements to put the facility into service and 2.)continuative measurements to provide experimental results helping to understand pipe flow.
The first results within the CoLaPipe show that this new experimental facility is suitable to investigate turbulence at high Reynolds numbers, where this conclusion can be drawn from intensive investigations on the development length of the flow either for natural and artificial transition. From further experiments on the evaluation of the wall friction velocity using different estimation methods great difficulties and variations in the calculated values are obtained. These deviations are directly related to the scaling behavior of the mean and fluctuating velocity, which is also shown within this thesis and intensively discussed.
Among the discussion of the setup of the new CoLaPipe and the first experimental results this thesis contains a broad literature review with the focus on high and very high Reynolds numbers. Nevertheless, pipe flow at low and moderate Reynolds numbers is described as well.