Dendritic solidification is a multiscale problem in which the interplay between physical mechanisms on different length scales determines the growth kinetics and, thus, the solidified material’s thermophysical properties. The Dendritic Needle Network (DNN) model for binary alloys was developed to bridge the gap between well-established modeling methods, such as the phase-field (PF) method, operating on the scale of the dendritic tip radius, and larger scale methods, such as cellular automaton (CA) methods which operate on a scale several orders of magnitude larger. Within the DNN model, dendritic needle-like thin branches growing in a hierarchical network are represented as paraboloids, such that the growth dynamics of such a network are entirely described by the tip radii and tip velocities of its needles. Earlier implementations of the DNN model have been verified in numerous studies and were shown to successfully predict important length scales, such as the primary dendritic arm spacing. 

Buoyancy-induced advective currents in the liquid melt, unavoidable under Earth-based experimental conditions, alter the dendritic growth dynamics substantially and are often responsible for defect formation. Within this thesis, the DNN model is extended to incorporate fluid flow via the implementation of a Navier-Stokes (NS) solver for two important cases: (i) three-dimensional dendritic growth under isothermal conditions and (ii) two-dimensional dendritic growth in a temperature gradient, i.e., directional solidification. The new three-dimensional NS implementation is verified by comparison to several fluid mechanics benchmarks. All current versions of the DNN model are implemented in the CUDA programming language for parallelization on Nvidia graphics processing units (GPUs).

We present several applications, each giving more insight into different aspects of dendritic solidification. A comparison between 2D and 3D morphologies and growth dynamics of equiaxed grains under a forced inflow confirms that 3D simulations are indispensable for quantitative comparison to experimental data, especially when fluid flow is involved. Spacing selection in several binary alloys is studied in 2D and 3D, with and without fluid flow, comparing to PF and experimental results. Experimentally observed oscillatory growth in nickel-based superalloys is investigated via the 2D DNN model for directional solidification. Finally, preliminary results of a quantitative comparison between PF, DNN, and the grain envelope model (GEM) based on two benchmark cases are presented.