Dissertations / Theses on the topic 'Two-dimensional viscous flow'

The subject of this thesis is viscous fingering in Hele-Shaw cells, or Hele-Shaw flows. We look for insights into the fundamental mechanisms<br/>underlying the physics of interface dynamics, which we hope will exhibit some degree of universality. The aim is twofold: on the one hand we focus on the role of surface tension and viscosity contrast in the dynamics of fingering patterns. On the other hand we introduce a modification of the original problem and study the effects of a inhomogeneous gap between the<br/>plates of a Hele-Shaw cell.<br/><br/>A dynamical systems approach to competition of Saffman-Taylor (ST) fingers in a Hele-Shaw channel is developed. This is based on global analysis of the phase space flow of the ODE sets associated to the exact solutions of the problem without surface tension. A general proof of the existence of finite-time singularities for broad classes of solutions is given. The existence of a continuum of multifinger fixed points and its dynamical implications are discussed. We conclude that exact zero-surface tension solutions taken in a global sense as families of trajectories in phase space are unphysical because the multifinger fixed points are nonhyperbolic. Hyperbolicity (saddle-point structure) of multifinger fixed points is argued to be essential to the physically correct qualitative<br/>description of finger competition. The restoring of hyperbolicity by surface tension is proposed as the key point to formulate a generic Dynamical Solvability Scenario for interfacial pattern selection.<br/><br/>We study the singular effects of vanishingly small surface tension on the dynamics of finger competition in the Saffman-Taylor problem, using the asymptotic techniques developed by Tanveer and Siegel, and numerical computation, following the numerical scheme of Hou, Lowengrub, and<br/>Shelley. We demonstrate the dramatic effects of small surface tension on the late time evolution of two-finger configurations with respect to exact<br/>(non-singular) zero-surface tension solutions. The effect is present even when the zero surface tension solution has asymptotic behavior consistent with selection theory. Such singular effects therefore cannot be traced back to steady state selection theory, and imply a drastic global change in the structure of phase-space flow.<br/><br/>Finger competition with arbitrary viscosity contrast (or Atwood ratio) is<br/>studied by means of numerical computation. Two different types of dynamics<br/>are observed, depending on the value of the viscosity contrast an the<br/>initial condition. One of them exhibits finger competition and ends up in<br/>the ST finger. In opposition, the second dynamics does not exhibit finger<br/>competition and the long time dynamics seems attracted to bubble shaped<br/>solutions. An initial condition appropriate to study the ST finger basin<br/>of attraction is identified, and used to characterize its dependence on<br/>the viscosity contrast, obtaining that its size decreases for decreasing<br/>viscosity contrast, being very small for zero viscosity contrast. The ST<br/>finger is not the universal attractor for arbitrary viscosity contrast. An<br/>alternative class of attractors is identified as the set of Taylor-Saffman<br/>bubble solutions, and one important implication of this result is that the<br/>interface is strongly attracted to finite time pinchoff.<br/><br/>A nonlocal interface equation is derived for two-phase fluid flow, with<br/>arbitrary wettability and viscosity contrast c, in a model porous medium defined as a Hele-Shaw cell with random gap b. Fluctuations of both capillary and viscous pressure are explicitly related to the microscopic quenched disorder, yielding conserved, non-conserved and power-law correlated noise terms. Two length scales are identified that control the possible scaling regimes and which depend on capillary number Ca as l sub 1 = b sub zero (c Ca)(superindex -1/2) and l sub 2 = b sub zero/Ca. Exponents for forced fluid invasion are obtained from numerical simulation and compared with recent experiments,obtaining good partial agreement.

Etude numerique d'ecoulements plans permanents de stokes qui s'etablissent dans un canal rectiligne illimite sous l'action de sources de mouvement variees (1 ou 2 cylindres de rayons, emplacements et mouvements divers). A faible distance de ces sources, le fluide decolle des parois pour former une suite de cellules visqueuses dont on analyse la structure et les particularites geometriques. Calculs bases sur la methode des moindres carres

The formulation and implementation of a control volume finite-element method (CVFEM) for steady, two-dimensional, viscous compressible and incompressible internal flows is reported in this thesis. In the development of this method, a CVFEM for steady, quasi-one-dimensional, viscous compressible flows was also formulated. The proposed method is an equal-order colocated shock capturing formulation, which incorporates adaptive grid techniques to discretize calculation domains with triangular elements and provide local grid refinement in the vicinity of strong gradients. Polygonal control volumes are constructed around each node in the finite-element mesh, and discretized forms of the governing equations are obtained by deriving algebraic approximations to integral conservation equations for each control volume.<br>The proposed methods are formulated using the velocity components, pressure, and temperatures as the dependent variables: density is calculated from an equation of state. The pressure is interpolated linearly. The velocity components, temperature, and density are interpolated using flow-oriented upwind type functions. The interpolation functions for the velocity components and temperature include source-related terms that permit the interpolated variable to respond appropriately to the influence of a source term within an element. The source-related terms in the velocity functions include the appropriate pressure gradients; this feature allows the formulation of equal-order colocated methods valid for incompressible flows.<br>The discretized forms of the governing equations are solved using an iterative algorithm. In this algorithm, linearized forms of the coupled discretized momentum and continuity equations are solved simultaneously using an iterative line-by-line application of a new coupled equation line solver.<br>The proposed quasi-one- and two-dimensional CVFEM's are applied to several compressible fluid flow problems, and the solutions generated are compared with theoretical, numerical, and experimental results available in the literature. This comparison shows that the proposed CVFEM's can generate solutions that are in good agreement with the expected physical behaviour of compressible flows and also with the available results. The ability of adaptive grids to improve solution quality is also demonstrated. The results suggest further research is required in the evaluation and enhancement of the interpolation functions currently used in CVFEM's, and in the incorporation of other adaptive grid schemes into CVFEM formulations.

Etude du decollement de l'ecoulement et de la formation de cellules. Structure et caracteristiques geometriques de ces cellules. Analyse du champ hydrodynamique. Calcul numerique base sur l'ecriture des conditions des conditions aux limites par la methode des moindres carres. Mise au point d'une technique de visualisation par intermittence pendant de longues durees, utilisant les traceurs solides

Un fluide visqueux newtonien est introduit entre deux ellipses confocables dont les parois interne et externe peuvent être déplacées le long de leur circonférence. Une solution analytique en coordonnées elliptiques a été récemment obtenue à partir des hypothèses du régime de Stokes. Les sections de Poincaré, les points périodiques de l'écoulement et l'évolution d'une tache de traceur, montrent qu'un régime de chaos lagrangien avec de grandes surfaces potentielles de mélange peut se développer dans cette géométrie qui est également un échangeur de chaleur efficace. Le dispositif expérimental a été construit et testé. Les résultats des études théoriques et numériques ont été confrontés avec succès aux résultats expérimentaux par les méthodes suivantes: lignes de courant, déformation de taches de traceur fluorescent, sensibilité aux conditions initiales, test de nombreux profils de vitesses des parois, étude d'écoulements asymptotiques. L’analogie entre les résultats dérives de la géométrie à ellipses concentriques et de la géométrie à cercles excentrés donne des règles générales sur le comportement des mélangeurs bi-dimensionnels annulaires: importance de l'origine relative de l'énergie transférée au fluide, efficacité de l'addition de petits effets inertiels. Des applications directes importantes de cette recherche sont proposées: développement d'une méthode simple d'optimisation de mélangeurs par un choix judicieux de profils de vitesse élémentaires ; développement de nouveaux outils (sections de Poincaré et étude entropique) pour optimiser la conception de nouveaux mélangeurs

An inverse blade design method, applicable to 2D and 3D flow in turbomachinery blading is developed and is implemented for the design of 2D cascades in compressible viscous flow. The prescribed design quantities are either the pressure distributions on the blade suction and pressure surfaces or the blade pressure loading and its thickness distribution. The design scheme is based on a wall movement approach where the blade walls are modified based on a virtual velocity distribution that would make the current and target momentum fluxes balance on the blade suction and pressure surfaces. The virtual velocity is used to drive the blade geometry towards a steady state shape corresponding to the prescribed quantities. The design method is implemented in a consistent manner into the unsteady Reynolds Averaged Navier-Stokes (RANS) equations, where an arbitrary Lagrangian-Eulerian (ALE) formulation is used and the boundaries of the computational domain can move and deform in any prescribed time-varying fashion to accommodate the blade movement. A cell vertex finite volume method is used for discretizing the governing equations in space and, at each physical time level, integration in pseudotime is performed using an explicit Runge-Kutta scheme, where local time stepping and residual smoothing are used for convergence acceleration. For design calculations, which are inherently unsteady due to blade movement, the time accuracy of the solution is achieved by means of a dual time stepping scheme. An algebraic Baldwin-Lomax model is used for turbulence closure. The flow analysis method is applied to several test cases for steady state internal flow in linear cascades and the results are compared to numerical and experimental data available in the literature. The inverse design method is first validated for three different configurations, namely a parabolic cascade, a subsonic compressor cascade and a transonic impulse turbine cascade, where different choices of the prescribed design variables are used. The usefulness, robustness, accuracy, and flexibility of this inverse method are then demonstrated on the design of an ONERA transonic compressor cascade, a NACA transonic compressor cascade, a highly cambered DFVLR subsonic turbine cascade, and a VKI transonic turbine cascade geometries, which are typical of gas turbine blades used in modern gas turbine engines

碩士<br>國立臺灣大學<br>造船及海洋工程學研究所<br>87<br>The purpose of this paper is to compute viscous flow of two dimensional profiles in the range of high Reynolds numbers by solving RANS(Reynolds Averaged Navier-Stokes) equation numerically. The emphasis is put on determination of hydrodynamic characteristics and prediction of cavitation inception. The Reynolds stress is then determined with the model , so that the turbulence flow is completely solved. Results of different wing sections such as YS-920 and NACA 66(MOD) are compared with Shen's experiments to validate the present numerical method. The items of results include the cavitation inception diagram, lift coefficient and drag coefficient. This paper also explores into influence upon two dimensional profiles cavitation inception characteristics by changing important parameter such as various Reynolds numbers, camber ratio , thickness ratio , two dimensional profiles symmetry.

In this thesis, we study the behavior of viscous flow past bodies of different shapes. In Chapter 2, we construct a boundary-fitted, numerical grid around a rigid spheroid of various aspect ratios and solve numerically the Navier-Stokes equations in steady, axisymmetric form at various Reynolds numbers. In addition, we use these steady solutions as a base flow and perform a linear stability analysis to determine the critical Reynolds numbers above which the base flow becomes unstable. We are able to confirm the results of Natarajan and Acrivos [26] and extend them to more generalized body shapes. In Chapter 3, we solve the Navier-Stokes equations to investigate flows past an oblate ellipsoidal bubble of fixed shape, which is characterized by a free-slip boundary condition. We then compare our results with previous results by Dandy and Leal [6] and Blanco and Magnaudet [4] and use the computed steady solutions as the base flow to perform a linear stability analysis. We show that even with a free-slip boundary condition, if the body is sufficiently oblate, the flow can become unstable in a manner similar to that of flows past rigid bodies. In Chapter 4, we develop an alternative numerical method to compute steady flows past a deforming, axisymmetric bubble. A newly developed conformal grid generation method is applied. We show that our results are in good agreement with those of Ryskin and Leal [34], [35] and then extend some of their results to higher Reynolds number. In Chapter 5, we modify the method developed in Chapter 4 to compute steady flows past a symmetric, two-dimensional bubble. We show that the bubble deforms to an elliptical shape and that a wake can develop if the deformation of the bubble is sufficiently large.

碩士<br>國立臺灣大學<br>機械工程學研究所<br>101<br>In this thesis, we employ similarity transformation and elliptic functions to solve the two dimensional radial flow of viscous fluid between two inclined plane walls with an oscillating source. Before the solving process, we carry out a scaling analysis to identify the allowable pressure gradient and maximum velocity. Moreover, we use the AC-circuit analogy to obtain the flow impedance Zf. By determining the ratio of flow impedance in opposite flowing direction, physical characteristics of the flow field are investigated. We also calculate the average power from instantaneous pressure drop Δp and flow rate Q. Influences of angular frequency w, slendness h/R, half angle a and Reynolds number Re in the ratio of flow impedance and average power are studied. Through scaling analysis, we find that the pressure gradient across a diffuser must be positive when the angular frequency w is sufficiently small while the non-dimensional central velocity F0 is sufficiently large. As a result, there is a upper maximum of pressure drop Δp above which velocity solution cannot be found and we can only calculate pressure drop Δp for a given velocity oscillation. At low angular frequency w, there exists an optimal Reynolds number Re and half angle a that minimize the ratio of flow impedance. For Re < Reopt or a < aopt, the ratio of flow impedance decreases with Reynolds number Re or half angle a. For Re > Reopt or a > aopt, the ratio of flow impedance increases with Reynolds number Re or half angle a. In contrast, the ratio of flow impedance increases monotonically with both increasing half angle a and Reynolds number Re at high angular frequency w. As slendness h/R and angular frequency w augment (or Womersley number Wo), the effect of unsteady term amplifies. This leads to an increase in the ratio of flow impedance with both slendness h/R and angular frequency w. The influence of Womersley number Wo in ratio of flow impedance can be categorized into three regions: quasi steady region (Wo < 0.1), transition region (0.1 ≤ Wo ≤ 10) and unsteady region (Wo > 10). On the other hand, at low angular frequency w, there exists an optimal half angle a that minimize the average power. For a < aopt, the average power decreases with half angle a. For a > aopt, the average power increases with half angle a. In contrast, the average power increases monotonically with increasing half angle a at high angular frequency w. Moreover, for Reynolds number Re, the average power increases monotonically with increasing Reynolds number Re either high angular frequency w or low angular frequency w. As slendness h/R and angular frequency w augment (or Womersley number Wo), pressure drop Δp decreases with increasing slendness h/R while increases with increasing angular frequency w. This leads to a decrease in the average power with increasing slendness h/R and an increase in the average power with increasing angular frequency w.

An aerodynamic inverse design method for viscous flow over airfoils is presented. In this approach, pressure distribution on the airfoil surfaces are prescribed as design target and the airfoil geometry is modified so as to reach the desired shape. In the design method, the walls are assumed to be moving with a virtual wall velocity that would balance the current momentum flux with the target pressure distribution; this virtual wall velocity drives the airfoil geometry to the shape that would produce the target pressure distribution where it would asymptotically vanish. This method was extended to address multi-point design and multi-element airfoils, and to use the pressure distribution of the airfoil suction surface as design variable. The approach is consistent with the viscous flow assumption and is incorporated into the governing equations which are expressed in an Arbitrary Lagrangian-Eulerian form, and are solved in a time accurate fashion. A cell-vertex finite volume scheme of the Jameson type is used for spacial discretization and time integration is performed by dual time stepping. Baldwin-Lomax turbulence model is used for turbulence closure. The validation of this approach is carried out for NACA 4-digit and 5-digit airfoils and RAE 2822 airfoil in transonic flow regime. The redesign cases include NACA 5-digit and 4-digit airfoils, the latter design experiencing large separation, RAE 2822 airfoil in transonic regime, multi-element airfoil and a dual-point design case.