Turbulence is ever-present: from flows in engineering, such as a wake behind a submarine, the boundary layer over an aircraft wing, and the swirl in an internal combustion engine, to flows in nature, such as convection in lakes, riffles on rivers and ocean currents. Turbulence can be found in flows at relatively small scales, such as blood flow in arteries and while mixing cream in a morning coffee, to flows at astrophysical scales, for instance, in accretion disks around stars or black holes. Because of their ubiquitous nature, progress in science and technology often hinges on progress in research on turbulent flows. In many situations described above, two lines of inquiry are of most interest. In the first direction, we are interested in quantities that are the ``net'' outcome of a fluid system, i.e., bulk quantities or global mean quantities such as drag force, rate of energy dissipation, mass, momentum and heat transport and mixing rate, which are usually long-time and volume averages and therefore depend only on the system's input parameters such as viscosity and diffusivity of the fluid, characteristics velocity scale, domain shape. The second direction, which is complementary to the first one, is the study of different structures in turbulent flows, for example, quantifying the range of scales and the energy distribution through this range in turbulent flows. In this thesis, we study a few problems that are related to and inspired by these two directions of questioning. While working on a problem, we always try to incorporate different perspectives: engineering, physics and mathematics. It is our intention to work at the interface of physical and mathematical fluid dynamics, as there appears to be great potential for an exchange of ideas that can eventually benefit both fields. On the one hand, having knowledge of various phenomenological theories from the physics literature gives one the advantage in tackling the various pressing problems considered in the mathematical community. On the other hand, putting various theoretical predictions on a rigorous mathematical footing can allow us to gain a deeper understanding of the physical mechanism/phenomenon. In accordance with this theme, below we describe the problems considered in this thesis, which is divided into two parts.In the first part of the thesis, we are interested in quantifying bulk properties of turbulent flows, such as energy dissipation, drag force, heat and mass transfer. We obtain rigorous bounds on these quantities using a well-known technique known as the background method. We consider four problems in the first part: (1) uniform flow past a flat plate, (2) pressure-driven flow in a helical pipe, (3) Taylor--Couette flow (flow between two independently rotating concentric cylinders), and (4) internally heated convection. In the flat plate study, we show that the energy dissipation rate for uniform flow past a flat plate remains bounded. This is the first and only example so far of an external flow problem (flow past a body) where such a bound has been established. In the second and third problems, we derive bound on mean quantities such as friction factor, volume flow rate, energy dissipation, torque on the cylinder and angular momentum transport not just as a function of the principal flow parameter, the Reynolds number, but more importantly, as a function of the geometry of the domain (i.e., curvature and torsion in the case of helical pipe flow and the ratio of the radii of two cylinders in Taylor--Couette flow). These studies are motivated by several engineering applications where the geometry of the domain plays an important role. In the fourth study, we consider the problem of convection between two solid boundaries driven by a source of internal heating and derive a bound on the mean vertical heat flux, an inquiry that is motivated, for example, by convection in the earth's mantle and the sun's radiative zone. In the second part of the thesis, we are concerned with designing incompressible flows that possess some specific desired properties. The first problem in this direction is related to the optimal heat transport from a hot to a cold wall using a flow whose enstrophy is bounded by a given constant. The bound on the enstrophy can also be thought of as a bound on the power supply needed to generate this flow (using a body-force in the momentum equation) Navier--Stokes system. An upper bound on the heat transfer that scales as 1/3-power of the power supply had formally been derived previously, but whether a flow exists that transports heat at that rate remained an outstanding question. For this problem, we design three-dimensional branching flows to prove that the corresponding heat transfer saturates this known upper bound, which then establishes the exact asymptotic behavior of the optimal heat transport between two plates. Beyond the mathematical proof, our method also reveals why three-dimensional branching flows are so efficient in transferring heat. Finally, in the second part, we study a problem related to the nonuniqueness of flow maps in an ODE system for the class of velocity fields that are divergence-free and belong to Sobolev space $W^{1, p}$. We reprove and improve the known result that had been previously established using the method of convex integration. Our goal for this problem is simple: provide explicit constructions and use them to gain insights into the exact mechanism of the nonuniqueness of solutions of the ODE and the PDEs, transport and continuity equation with the same vector field. Beyond proving the nonuniqueness results, we anticipate that such explicit constructions will be helpful for designing velocity fields in the convection-diffusion equation or the body-forced Navier--Stokes equation to demonstrate the phenomenon of anomalous dissipation, an intrinsic characteristic of turbulent flows.