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Two major problems in viscoelastic turbulence, the effects of polymers on the laminar-turbulent transition dynamics and the origin of the maximum drag reduction asymptote, can be both better understood in the regime near the margin of turbulence. In the first part of this thesis, direct numerical simulation trajectories initiated from the edge state are used to follow its unstable manifold into the turbulent basin. In Newtonian flow, the growth of turbulence starts with the intensification of velocity streaks and a sharp rise in the Reynolds shear stress. It is followed by a quick breakdown into high-intensity small-scale fluctuations before entering the core of turbulence. Adding drag-reducing polymers does not affect the initial growth of turbulence but stabilizes the primary streak-vortex structure, which help the flow circumvent the breakdown stage. Throughout the process, polymers act in reaction to the growing turbulence and do not drive the instability. This part not only reveals the transition dynamics into turbulence but also presents a comprehensive view of the bursting stage observed in the near-wall self-sustaining cycle, which starts as the flow leaves hibernating turbulence and is redirected towards the turbulent basin by the unstable manifold of the edge state. On the other hand, this thesis also discusses the effects of polymer addition on the laminar-turbulent transition in extended domains. Localized turbulent spot can be clearly observed in the large box, and this turbulent region will spread as well as tend to “split” but finally fill up the whole domain before it is separated. Polymers do not affect the flow dynamics until the burst. Similarly, vortex structures rapidly break down into small scales after the first bursting of Reynolds shear stress, but polymer additives depress this process. The thesis offers a clear and comprehensive overview of the transition into turbulence in the presence of drag-reducing polymers. Future work remains in two major directions. The first is to pinpoint the flow states responsible for the quantitative origin of the universal upper limit of drag reduction observed in experiments. The second is to determine the role, if any, of elasticity-driven instabilities in the transition. / Thesis / Master of Applied Science (MASc) / Turbulence exists everywhere and can be observed in most fluid flows occurring in nature. To reduce the energy consumption, frictional resistance in the turbulence must be considered in fluid transportation. It has been known since the 1940s that a small amount of long-chain polymer additives can dramatically reduce such drag. The mechanism of drag reduction has attracted extensive attention. Two problems of particular interest are the upper limit of drag reduction (termed maximum drag reduction) and the polymer effects on the laminar-turbulent transition. In this thesis, full transient trajectories from marginal turbulent states towards sustained turbulence in both Newtonian and polymeric flows are monitored by direct numerical simulations. It is observed that polymer additives do not affect the initial growth of turbulence but prevent flows from breaking into strong but small-scale fluctuations afterwards. In a more extended domain, turbulence starts as localized spots which spread across the channel. Adding polymers changes the dynamics of turbulence propagation as well. In addition to the aforementioned problems, this study also sheds lights on the so-called bursting events intermittent surges in turbulent activities observed in experiments.

The effects of contaminated supply air on the performance of a laminar proportional amplifier were experimentally investigated. The air supply was contaminated with oil vapor and particulate matter. Characteristic gain curves were obtained after each stage of contamination for various loading conditions. Photographs showing the location of contaminant deposits were taken. Two inlet geometries were studied: right-angle and straightthrough. The effects of maintaining a constant pressure and constant flow rate at the inlet throughout the duration of the tests were studied. Also, aspect ratio effects. were studied. Rapid deterioration of performance was evident with the right-angle entry due to inlet blockage. By milling a cavity in the bottom cover plate, the detrimental effects of inlet blockage were delayed. With constant pressure inlet conditions and straight-through geometry, significant buildup occurred in the nozzle region and downstream. Decrease in pressure recovery was linear with time. It was determined that null shift was caused by asymmetric buildup in either the nozzle region or splitter region. With constant flow inlet conditions, the damaging effects on performance were delayed for both inlet geometries. Also, for the straight-through inlet, the nozzle region was relatively clean when compared with the constant pressure inlet case. Null shift was found to be the result of asymmetric buildup in the downstream region. At lower aspect ratios, the damaging effects of contamination were more severe and occurred in less time. / Master of Science

A theoretical investigation of internal flow subjected to an axial variation of the external convection coefficient is presented. Since the variable boundary condition parameter causes the problem to become nonseparable, conventional techniques do not apply. Instead, the Green's function technique is used to convert the governing partial differential equations into a singular Volterra integral equation for the temperature of the fluid at the wall. The integral equation is resolved numerically by the trapezoid rule with the aid of a singularity subtraction procedure. The solution methodology is developed in terms of a fully turbulent flow which is shown to contain fully laminar and slug flow as special cases. Before examining the results generated by numerical solution of the integral equation, a thorough study is made of each of the building blocks required in the solution procedure. A comparison of the respective dimensionless velocity profiles and dimensionless total diffusivities for each of the flow models is presented. Next, an analysis of the eigenvalue problem for each flow model is presented with consideration given to the normalized eigenfunctions and the eigenvalues themselves. Finally, the singular nature of the Green's function is examined showing the effect of the parameters Ho, Re and Pr. The technique is applied to study the heat transfer from a finned tube. A parameter study is presented to examine the effects of the external finning and the flow model. The effect of external finning is examined through specific variations of the external convection coefficient, while the flow model is selected through the velocity profile and eddy diffusivity. In examining turbulent flow, the effects of the parameters, Re and Pr, are considered. / M.S.

The problem studied is the laminar flow over inclined, finite flat plates for moderately high Reynolds numbers in two and three dimensions. There are only few prior analyses, mainly for two dimensional flow, for this problem, and thus it was decided that it was worthwhile to study it now in great detail. The full Navier-Stokes equations were solved using a weak Galerkin formulation for the Finite Element Method with the pressure determined by a penalty approach. The influence of grid resolution, boundary conditions and size of the domain was studied. The true nature of the flow for different Reynolds numbers was also examined through steady and unsteady simulations of the two dimensional cases for 6600 â ¤ ReL â ¤18000. Results for the three dimensional flow over square plates at two angles of attack, a = 3.0 and 8.0 degrees for ReL = 100 are presented. The results are given in terms of skin friction and pressure coefficient variations along with flowfield visualization. Comparison between the two dimensional and three dimensional flow indicates the influence of the third coordinate to the flow. The analysis indicated that the two dimensional flow over a finite thick plate at 3.0 degrees angle of attack is steady up to Re = 12000. The solution for the upper surface is strongly influenced by the presence of a recirculation bubble at the leading edge. The slope of the lift curve for the 2D viscous flow is less than 2Ï , the result predicted by the thin wing theory. The solution for the three dimensional flow is strongly influenced by the the existence of the tip vortices. The slope of the lift curve for the 3D viscous flow is less than the one corresponding to the 2D flow. In addition, the effect of the aspect ratio on the lift does not agree with the inviscid lifting line theory prediction. / Master of Science

The accuracy of the compressible interacting boundary-layer computations is investigated and their limitations are established by comparison with solutions to the Navier-Stokes equations both for the mean flow profiles and for the stability characteristics. The instabilities of flows around smooth forward and backward facing steps are investigated. Results presented include the effect of computational grid refinement, geometrical parametres such as heights and slopes of steps, Mach number and Reynolds number on the mean flow as well as the stability characteristics. A proper grid should be chosen to predict accurately the mean flow profiles, including their first and second derivatives. The study has shown that the heights of the steps are more influential in triggering transition than their slopes. Increasing the Mach number reduces the growth rates and amplification factors but the presence of small separation bubbles, which increase in size with increasing Mach number, partially offset this benefit. / M.S.

A method of solving the equations for the three–dimensional, incompressible laminar and turbulent flow along the intersection of two planes at ninety-degrees has been developed. The Alternating Direction Implicit (ADI) finite-difference method was applied for both types of flow. The turbulent stresses in the corner region were modeled with an eddy-viscosity model which was obtained from mixing length theory. The method was compared with other types of solutions for the laminar case and good agreement was achieved. For the turbulent case, the method was compared with experimental data and good agreement was obtained. The three momentum equations were solved simultaneously and the continuity equation was used to verify the method. The method appeared to predict the velocity components correctly since the continuity equation residual approached zero as the solution proceeded from the leading edge in the mainstream flow direction. No analysis was presented for the convergence or stability of the finite-difference equations and no convergence or stability problems solved were encountered when the finite-difference equations were solved. The method predicted symmetry about the corner bisector in all cases and gave the expected u-velocity profile along the bisector for both the laminar and turbulent cases. / Ph. D.

The laminar boundary layer over a semi-infinite flat plate which is impulsively set in motion in an incompressible fluid and which has a simultaneous step change in surface temperature was studied. An approximate method was derived which can be used to determine the thermal boundary layer thickness as a function of the distance from the leading edge and of time. From the thermal boundary layer thickness the temperature of the fluid can be determined at any position in the boundary layer and at any time. The local Nusselt number can also be determined from the thermal boundary layer thickness. The approximate solution was compared with exact steady-state and infinite-plate solutions of the energy equation and with a finite-difference solution of the unsteady continuity, momentum and energy equations. Agreement between the solutions was close enough to indicate that the approximate solutions for the temperature in the boundary layer and for the Nusselt number approximate the actual situation with reasonable accuracy. / Master of Science

This paper deals with three problems, (1) laminar incompressible viscous flow past a cylinder and a sphere, (2) laminar incompressible viscous flow past a finite flat plate (second-order solutions), and (3) laminar viscous past a sphere at a high Mach number. These problems are solved by using an implicit finite-difference method. The first problem (flow past a sphere and a cylinder) involves the classical boundary-layer equations which are the first approximation to the Navier-Stokes equations in a region near to the body surface for high Reynolds number. The computational results were obtained for the distribution of velocity components in the boundary-layer, and the variation of skin-friction and displacement-thickness along the body. The second problem (second-order flow past a finite flat plate) involves the second-order boundary-layer equations which introduce only the effect of the displacement-thickness in the case of flow past a flat plate. An assumption is made that the displacement-thickness is constant in the wake behind the flat plate. The adequacy of this assumption is checked from solutions based on the calculated displacement-thickness in the wake. The wake behind the finite flat plate is assumed laminar, and its displacement-thickness and the velocity distribution are computed downstream, by using the implicit finite-difference method. In the third problem (high Mach number flow past a sphere), constant density is assumed in the shock layer. This is nearly true in the stagnation-point region, especially if the flow is hypersonic and the temperature of the sphere is nearly the same as the stagnation-temperature. It is also assumed that the shock is nearly spherical, even though it is not spherical as it is in the inviscid case. The numerical results will show that the assumption of a spherical shock will, however, nearly be true. This problem involves the solution of the complete Navier-Stokes equations. These equations are solved for various Reynolds numbers by two methods; namely the truncated series method and the implicit finite-difference method. The solutions by the implicit finite-difference method are in excellent accord with those obtained by the series solutions in the stagnation-point region. As the computation by the finite-difference method proceeds downstream, the deviation of the finite-difference solution from the series solution increases. This is due to the fact that the series is valid only around the stagnation-point, and is thus expected to give inaccurate solutions downstream. The finite-difference method has no such restrictions, however, and gives accurate results in the whole flow field. In conclusion, solutions by the implicit finite-difference method have proven not only to be accurate but also to be stable in all examples computed. / Ph. D.