published_or_final_version / Civil Engineering / Master / Master of Philosophy

published_or_final_version / Mechanical Engineering / Master / Master of Philosophy

voir conclusion, pp. 151-154 du pdf

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The control of stresses and liquid pressure surges in pipes is an important problem in the design of hydraulic pipe networks. The method of characteristics has been used to solve the transient stresses and pressures in liquid-filled piping systems. The friction force is included in the equations of motion for the fluid and the pipe wall. The maximum pressure and maximum stress at any point along the length of the pipe are evaluated for the entire simulation time. A nonlinear search technique has been developed using the simplex method. The optimal valve closure is sought, that will minimize the maximum pressure and/or stresses. A continuous optimal valve closure policy is specified using spline functions. Numerical examples are presented showing the reduction of the dynamic pressure and the dynamic stress from linear valve closure to optimal valve closure for a simple pipeline and a complex pipeline. Also, a method for choosing the shortest time of closure which will keep the stresses below specified allowable stresses is presented.

The nonlinear interactions between a fundamental instability mode and both its harmonics and the changing mean flow are studied using the weakly nonlinear stability theory of Stuart and Watson, and numerical solutions of coupled nonlinear partial differential equations. The first part of this work focuses on incompressible cold (or isothermal; constant temperature throughout) mixing layers, and for these, the first and second Landau constants are calculated as functions of wavenumber and Reynolds number. It is found that the dominant contribution to the Landau constants arises from the mean flow changes and not from the higher harmonics. In order to establish the range of validity of the weakly nonlinear theory, the weakly nonlinear and numerical solutions are compared and the limitation of each is discussed. At small amplitudes and at low-to-moderate Reynolds numbers, the two results compare well in describing the saturation of the fundamental, the distortion of the mean flow, and the initial stages of vorticity roll-up. At larger amplitudes, the interaction between the fundamental, second harmonic, and the mean flow is strongly nonlinear and the numerical solution predicts flow oscillations, whereas the weakly nonlinear theory yields saturation. Beyond the region of exponential growth, the instability waves evolve into a periodic array of vortices. In the second part of this work, the weakly nonlinear theory is extended to heated (or nonisothermal mean temperature distribution) subsonic round jets where quadratic and cubic nonlinear interactions are present, and the Landau constants also depend on jet temperature ratio, Mach number and azimuthal mode number. Under exponential growth and nonlinear saturation, it is found that heating and compressibility suppress the growth of instability waves, that the first azimuthal mode is the dominant instability mode, and that the weakly nonlinear solution describes the early stages of the roll-up of an axisymmetric shear layer. The receptivity of a typical jet flow to pulse type input disturbances is also studied by solving the initial value problem and then examining the behavior of the long-time solution. The excitation produces a wave packet which consists of a few oscillations and is convected downstream by the mean flow. The magnitude of the disturbance in the jet depends on the location of the excitation and there is an optimum position at which little energy input will produce large perturbations. It is found that in order to generate the largest perturbations at any point in the jet, the disturbance should be deposited into the flow at a point where the phase velocity of the most amplified wave equals the fluid velocity (of the base flow).

Unsteady heat transfer by natural convection in a closed square cavity is investigated numerically. A new finite-volume approach is developed and applied to the two-dimensional continuity, vorticity, and energy equations. The variation of the field variables is approximated by bi-quadratic interpolation formulas over the space occupied by the finite volume and the region surrounding it. These are used in the integral conservation laws for energy, vorticity and mass. The convective transport is modelled using a new upstream-weighting approach which uses volume averages for the vorticity and the energy transported across the boundaries of the finite volume. The weighting is dependent on the skewness of the velocity field to the surfaces of the finite volume as well as its strength. It is adaptive to local flow conditions. The velocities are obtained from the application of the velocity induction law. Use is made of an image system for the free vorticity of fluid. In this way, the no-penetration condition is enforced at the cavity boundaries, but at the same time it may allow a slip condition to exist. This is not permitted in a viscous flow analysis, and the slip velocity is reduced to zero by the production of free vorticity at the boundaries. Two test cases are treated which have exact solutions. The first is not new and involves a rotating shaft. The errors are less than.06% for this case. The second case is new and involves convection past a source and sink. The maximum error is 2.3%. For both test cases, the maximum error occurs at moderate values of the cell Peclet number and diminishes at the extreme low and high values. The time-development of the profiles of the vorticity, horizontal velocity, and temperature is examined at different locations within the cavity for Rayleigh numbers equal to 10³, 10⁴, and 10⁵. For these calculations, a 21 x 21 grid was used. The flow is found to approach a steady-state condition. The steady-state results are compared with a benchmark solution. In general, the agreement is excellent. The discrepancy is found to be less than 2% for the vast majority of the results for this relatively coarse grid.

We have studied the stationary solutions to the two-dimensional Euler's equation. A highly accurate scheme, based on boundary integral techniques was used in investigating these steady-state configurations. Bifurcation analysis on the solution of a uniform vortex patch in an externally applied strain field, yield new non-elliptical steady-state solutions apart from the elliptical structures reported by Moore & Saffman. The elliptical solutions correspond to the points on the primary solution branch and the non-elliptical solutions correspond to points on the bifurcation branches. We also observe the presence of a turning point indicating the finite resistance of these uniform vortices. Some of these new solutions suggest the possibility of coalescence between neighboring vortices. This leads to a new problem of considering a vortex pair in a strain field and computing their steady-state solutions. Numerical computations suggest that this guess is indeed correct, as we see the solution branch corresponding to the vortex pair intersect the bifurcation branch of the single vortex at a unique strain rate. Furthermore, looking at the profiles on the other bifurcation branches, it appears that merger of neighboring vortices is a recurring phenomenon.

Thermal and fingering convection in a horizontal porous layer underlying a fluid layer was studied using linear stability analysis, experiment (for the thermal convection case only), and nonlinear simulation. For the thermal convection case, the linear analysis shows that when the fluid layer is thin, convection is largely confined to the porous layer. When the fluid layer thickness exceeds 15% of the porous layer thickness, convection is localized in the fluid layer and the critical wavelength is dramatically reduced. Experimental investigations were then conducted in a test box 24 cm x 12 cm x 4 cm high to substantiate the predictions. The ratio of the thickness of the fluid layer to that of the porous layer, d, varied from 0 to 1. The results were in good agreement with predictions. To investigate supercritical convection, a nonlinear computational study was carried out. It was found that for d ≤ 0.13, the Nusselt number increases sharply with the thermal Rayleigh number, whereas at larger values of d, the increase is more moderate. Heat transfer rates predicted for d = 0.1 and 0.2 are in good agreement with the experimental results. For salt-finger convection at R(m) ≤ 1, the critical value of the solute Rayleigh number R(sm) decreases as d increases; the convection is unicellular. For 5 ≤ R(m) ≤ 10, the critical R(sm) initially decreases with d, and then remains almost constant for larger values of d; multicellular convection prevails at high d. For 20 ≤ R(m) ≤ 50, the critical R(sm) first decreases and then increases as d increases from 0 to 0.1. When d > 0.1, the critical R(sm) decreases slowly with d and remains almost constant for d ≥ 0.4. In the nonlinear computations for R(m) = 1, periodic convection sets in at a value of R(sm) between ten and eleven times the critical value. For the case of R(m) = 50, an aperiodic oscillation occurs when R(sm) is between four and five times the critical value. For the superposed layer cases d = 1 and 0.5, the convection characteristics are similar to those of thermal convection when R(m) = 0.01. For R(m) = 1, it was found that the onset of salt-finger convection is oscillatory. For R(m) = 50, the nonlinear code failed to obtain satisfactory results.