<p dir="ltr"> This article-based dissertation provides an account of two distinct classes of expansive astrophysical outflows and techniques to interpret their observations using numerical modeling. The primary purpose of this dissertation is to provide an extensive description of the research projects I undertook during my tenure as a Graduate Research Assistant, under the guidance of my advisor Prof. Paul Duffell.</p><p dir="ltr">Chapter 1 provides a brief introduction to numerical hydrodynamics and techniques of modeling expanding flows numerically. I also introduce the aforementioned classes of astrophysical outflows, namely relativistic jets from Active Galactic Nuclei (AGN), and supernova remnants (SNRs). I provide a general overview of the theoretical picture, and the general strategy used in this work to model them.</p><p dir="ltr">Chapter 2 describes my investigation on the connection of kiloparsec scale AGN jet properties to their intrinsic parameters and surroundings, based on an article published in The Astrophysical Journal. Using a suite of over 40 relativistic hydrodynamic jet models, we find that the dynamics of relativistic jets can be described in terms of only two parameters, the jet to ambient medium energy density ratio, and the jet opening angle. The former is found to strongly control the Fanaroff-Riley (FR) morphological dichotomy, which was previously thought to be tied to the magnitude of the jet luminosity. We also suggest a purely hydrodynamical origin of bright spots observed in some AGN jets. Our models were tested against and found to be consistent with the observations of the jets in M87 and Cygnus A.</p><p dir="ltr">In chapter 3, I present my moving-mesh hydrodynamics code Sprout, also described in an article published in The Astrophysical Journal Supplements. Sprout solves the equations of ideal hydrodynamics on an expanding Cartesian mesh. The expanding mesh can follow fluid outflows for several orders of magnitude with very little numerical diffusion. This allows Sprout to capture expanding flows with very high dynamic range. Sprout is thus particularly suitable for studying expanding outflows such as supernova remnants and active galactic nuclei. Relative to other moving mesh codes, the simple mesh structure in Sprout is also convenient for implementing additional physics or algorithms. I discuss many code tests that were performed to test the accuracy and performance of the numerical scheme.</p><p dir="ltr">Chapter 4 details my study of hydrodynamic instabilities in supernova remnants (SNRs) as they expand against the circumstellar medium (CSM). This is based on an article published in The Astrophysical Journal. A suite of 3D hydrodynamical SNR models, generated using my hydro code \sprout, was used to study the impact of the stellar ejecta density profile and seed anisotropies in the ejecta and the CSM on formation of turbulent structures in the SNRs. We found that most of the turbulent power in these models resides at a typical angular mode or scale that is determined by the ejecta density structure. It was also found that clumps or anisotropies in either the ejecta or CSM do not imprint upon these turbulence structures unless they are massive and form large-scale coherent structures.</p><p dir="ltr">In chapter 5, I discuss the implementation of a technique to measure anisotropies in observed SNRs just using 2D high-resolution images. This technique is calibrated using 3D hydro SNR models and synthetic images derived from them. As seen in Chapter 4, we find a similar dominant angular scale of turbulent structures dictated by the ejecta density structure. Both the 3D models and the synthetic images yield the same value of this scale, which validates the image analysis technique used in this work. As an example of how this technique can be applied to observations, we analyze observations of a known supernova remnant (Tycho's SNR) and compare with our models. Our technique picks out the angular scale of Tycho's fleece-like structures and also agrees with the small-scale power seen in Tycho.</p><p dir="ltr">PhChapter 6 summarizes the results, conclusions, and future prospects of all the research work described so far. It is followed by a bibliography, my curriculum vita, and a list of publications.</p>