Formation, development and instability of Foppl vortices and associated surface pressure distribution are investigated experimentally for a family of two dimensional circular cylinders in the Reynolds number range of 5 - 20,000 and the blockage ratio of 2 - 50%. In the beginning, design and constructional details of a glycerol-water solution tunnel used in the experimental programme is briefly described followed by an explanation of the models, pressure measuring instrumentation, and test procedures. An approach to the data reduction, so critical at low Reynolds number, is discussed and a new definition of the pressure coefficient, which promises to be less dependent on test facilities and pressure gradients, is explained. , Finally, the test data are analyzed as functions of the confinement condition and Reynolds number. The results suggest that influence of the Reynolds number on the surface pressure distribution is primarily confined to the range R[sub n] < 1200. However, for the model with the highest blockage ratio of 50%, the pressure continues to show Reynolds number dependency for R[sub n] as high as 3000. In general, effect of the Reynolds number is to increase the minimum as well as the wake pressures. On the other hand, the effect of an increase in the blockage ratio is just the opposite. The pressure profiles become extremely sensitive to the wall confinement at the lower end of the Reynolds number range under study. An extensive flow visualization study using dye injection in conjunction with still and high speed photography complements the test program. Photographs suggest that influence of the blockage is to retard, in terms of the Reynolds number, evolution of the near-wake. Location of the separating shear layers tends to move upstream with an increase in the Reynolds number, however, the wall confinement promotes downstream shift in the separation point, which can be as large as 25° for a blockage ratio of 501. The thesis suggests for future investigation several areas which are likely to be fruitful. In particular, aspect ratio effects in the presence of blockage are likely to be significant and should be studied in depth. / Applied Science, Faculty of / Mechanical Engineering, Department of / Graduate

In this dissertation, I introduce a vortex method that is generally applicable to any two-dimensional, incompressible flow with or without boundaries. This method is deterministic, accurate, convergent, naturally adaptive, geometry independent and fully localized. For viscous flows, the vorticity distribution of each vortex element must evolve in addition to following a Lagrangian trajectory. My method relies upon an idea called core spreading. Core spreading is inconsistent by itself, but I have corrected it with a deterministic process known as "vortex fission" where one "fat" vortex is replaced by several "thinner" ones. Also, I examine rigorously a method for merging many blobs into one. This process maintains smaller problem sizes thus boosting the efficiency of the vortex method. To prove that this corrected core spreading method will converge uniformly, I adapted a continuous formalism to this grid-free scheme. This convergence theory does not rely on any form of grid. I only examine the linear problem where the flow field is specified, and treat the full nonlinear problem as a perturbation of the linear problem. The estimated rate of convergence is demonstrated to be sharp in several examples. Boundary conditions are approximated indirectly. The boundary is decomposed into a collection of small linear segments. I solve the no-slip and no-normal flow conditions simultaneously by superimposing a potential flow and injecting vorticity from the boundary consistent with the unsteady Rayleigh problem. Finally, the ultimate test for this new method is to simulate the wall jet. The simulations produce a dipole instability along the wall as observed in water tank and wind tunnel experiments and predicted by linear stability analysis. Moreover, the wavelength and height of these simulations agree quantitatively with experimental observations.

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An approximate method for calculating the steady, incompressible, viscous flow over an airfoil, including regions of separated flow, is presented. The finite-difference equations resulting from an integral method for the laminar and turbulent boundary layers are solved simultaneously in a line-relaxation procedure with the equations for the outer, inviscid flow. These coupled equations allow direct interaction between the viscous and inviscid regions, thus eliminating the mathematical difficulties usually associated with separation. A distributed source on the upper surface of the airfoil provides an outflow to simulate the displaced boundary, and a distributed sink downstream of the trailing edge closes the wake. Computed results, which are compared with measurements for an NACA 4412 airfoil, are quite satisfactory for engineering purposes. / Ph. D.

In this work, an analysis of the Tollmien-Schlichting instability of laminar viscous flows is presented. For two-dimensional incompressible flow past a flat plate with porous suction strips, we use linear triple-deck, closed-form solutions for the mean flow to do a linear, parallel, spatial stability analysis. We develop a simple linear optimization scheme to determine the number, spacing, and mass-flow rate through the strips and conclude, surprisingly, that suction should be concentrated near the Branch I neutral point of the stability curve. We then verify the results of our optimization scheme with experimental data. We find that the theory correctly predicts the experimental results and conclude that the optimization scheme is reliable enough to replace the experiment as a tool in designing efficient strips configurations in so far as two-dimensional, incompressible flows are concerned. For axisymmetric incompressible flow past a body with porous strips, we develop linear triple deck, closed-form for the mean-flow quantities, solutions which account for upstream influence. These solutions are linear superpositions of the flow past the body without suction plus the perturbations due to the suction strips. The flow past the suctionless body is calculated using the Transition Analysis Program System (TAPS). Using these linear triple deck, closed-form solutions we then develop a simple linear optimization scheme to determine number, spacing, and mass flow rate through the strips on an axisymmetric body. At present, we are finishing the development of and documentation for a computer code for official distribution that will interface with TAPS and suggest efficient configurations using our theory. For compressible three-dimensional flow, we use the method of multiple scales to formulate the three-dimensional stability problem and determine the partial-differential equations governing variations of the amplitude and complex wavenumbers. We then propose a method for following one specific wave along its trajectory to ascertain the characteristics of the most unstable disturbance. Numerical results using the flow over the X-21 wing as calculated from the Kaups-Cebeci code will be published when they become available. / Ph. D.

This thesis presents the results obtained so far in an investigation concerning viscous effects in incompressible flows over airfoils. These effects are taken into account by assuming the existence of a boundary layer which interacts with an external inviscid flow. Numerical methods for solving the inviscid and boundary-layer flows are briefly described. The main objective of the investigation is the development of an interaction technique between both regions of flow. The method chosen for the interaction is the so-called semi-inverse procedure. This procedure is derived from a perturbation analysis of the linearized versions of the governing equations. The resulting method is subjected to a stability analysis, which shows that it will break down when used in conjunction with separated-flow boundary-layer solvers. The semi-inverse procedure is tested on several airfoils, using an attached-flow boundary-layer solver. Numerical results show that the method is sufficiently accurate for engineering purposes in the low-to-medium range of angles of attack, but its applicability is questionable when there is a large separation region. Finally, recommendations are made regarding future work to overcome the limitations of the present technique. / Master of Science

The numerical simulation of viscoelastic flows was studied in this work. In particular the effect of mesh refinement on the quality and the convergence of the finite element method was examined, as well as the differences that may be found by using several rheological models to describe the behaviour of the non-Newtonian fluids. The finite element simulation of viscoelastic fluid flows results in non-linear simultaneous equation systems that have to be solved iteratively. The iterations for all the viscoelastic models and most flow geometries have been found to diverge when the stress level or the elasticity of the flow increases above some certain level. The limit of convergence depends both on the mesh used for the discretization of the flow domain and on the rheological model. The limit usually decreases with mesh refinement. The effect of the mesh refinement on the convergence and the accuracy of the solution was studied here in two flow geometries: flow into an abrupt contraction (4/1 contraction ratio) and slit flow over a transverse slot. The penalty formulation of the finite element method (FEM) was used to numerically calculate the stress and the velocity fields in the flow domain using a number of coarse and fine meshes. Several rheological models were used, with their parameters chosen so that they would best fit a certain polystyrene melt. The solutions obtained were compared to results of g flow birefringence measurements and streamline photographs of the same material flowing under the same conditions that were simulated. The range of conditions that were covered by the calculations was shear stress at the die wall of 0-43 kPa, flow rates of 0-17 (mm³/ sec mm-width) and elasticity of 0-11 Deborah number. Even though oscillations in all numerical solutions were observed around the corners of the flow domain, it was found that the overall agreement of the numerical results with the experimental data was reasonable. The coarse meshes showed lower oscillations near the comers, but the accuracy of their predictions were poor. The limit of convergence for such meshes was the highest. Finer meshes on the other hand, showed higher oscillations near the comers and lower limit of convergence, but more accurate results away from the corner. It seems that the optimum mesh for an engineering calculation is an intermediate fine mesh that will give relatively high limits and reasonable accuracy. On the effect of the rheological model, it was found that the lower limit of convergence was given by the upper convected Maxwell model (UCM). The Leonov-like model also gave low limits. The Phan-Thien Tanner (P-T T) and the White-Metzner (W-M) models, on the other hand, showed quite higher limits in terms of the maximum stress levels and flow rates that they could handle. In terms of the quality of the solution inside the convergence range of each model, there is very little difference between the results of the models. In general, the Phan-Thien Tanner and the White-Metzner models show slightly better solutions. A possible reason for the better behaviour of these two models is believed to be the shear thinning viscosity and primary normal stress difference coefficient that these models are able to predict in simple flows. A few other characteristics of the two flows that were studied included the hole pressure, the entrance pressure loss and the presence of extensional fields around the contraction. It was found that the numerical method gave lower results for the hole pressure than the experimental data. Two models (W-M and UCM) gave a maximum in the entrance pressure loss and then a decrease towards negative values as the wall shear stress in the die increased. The P-T T and the Leonov-like models predicted a monotonic increase with the wall shear stress. Finally, there are two areas with strong elongational flow field in the contraction flow. One extends along the centerline of the die and the other lies along a line that starts from the reentrant comer and extends towards the upstream wall at an angle of around 45° (but depending on the flow rate). It is believed that this area is related to the natural entry angle, at which the viscoelastic fluid enters the contraction. / Ph. D. / incomplete_metadata

Unsteady two-dimensional laminar flow of an incompressible viscous fluid over a periodically grooved wall is investigated by numerical simulation using two independent finite-difference methods. One is the vorticity-stream function method, and the other involves the vorticity-velocity induction law formulation. The fluid motion is initiated impulsively from rest and is assumed to be spatially periodic in the streamwise direction. The flow field, which includes the time development of the shear layer and the recirculating flow in the zone of separation, is examined in detail during the transient phase to the steady-state condition. The analytical and numerical formulations, which include the implementation of the boundary conditions, are derived in detail. The generation of vorticity at the solid surfaces is modelled differently in the two approaches. This vorticity production plays an important role in determining the surface-pressure distribution and the drag coefficient. Characteristics of the transient solution for a moderate Reynolds number in the laminar range are presented. Included with the graphical results are the temporal development of the constant stream function contours, including the dividing contour between the zone of separation and the main flow, and the constant vorticity contours. These latter contours show the interactions of separated vortices. The flow is found to approach a steady-state condition comprising an undisturbed uniform flow, a nonuniform irrotational flow, a shear layer adjacent to the grooved wall, and a recirculating vortex flow in the groove. Results also include the time development of the surface shear stress, surface pressure, drag coefficient and several typical velocity profiles, which characterize the flow in the recirculating region. Comparisons of the results obtained by the two numerical methods are made during the major development of the flow. The results showing the general features of the flow development including the time development of the shear layer, free shear layer and recirculating vortex flow are in good agreement. However, a significant deviation does exist at early times for the distribution of surface pressure, which accordingly has noticeable effect on the drag coefficient. Nevertheless, the gap between the distributions of surface pressure and drag coefficients dies out gradually as time progresses. The form of the stream function and vorticity contours at the steady state agrees well with those obtained from a recent numerical investigation of the steady flow in grooved channels.

Chan Man-fai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 40-41). / Abstracts in English and Chinese. / Acknowledgement --- p.i / Abstract --- p.ii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Equation of motion and main results --- p.3 / Chapter 3 --- The space K(x) --- p.9 / Chapter 4 --- Proof of the main theorem --- p.17 / Chapter 4.1 --- The passage to the limit as ε →0 --- p.18 / Chapter 4.2 --- The passage to the limit as δ→ 0 --- p.26 / Chapter 4.3 --- Properties of the solution --- p.29 / Chapter 5 --- Conclusion and comments on future works --- p.36 / Appendix --- p.38 / Bibliography --- p.40