scholarly journals Approximate Ad Hoc Parametric Solutions for Nonlinear First-Order PDEs Governing Two-Dimensional Steady Vector Fields

2010 ◽  
Vol 2010 ◽  
pp. 1-23
Author(s):  
M. P. Markakis
Keyword(s):  
Vector Fields ◽  
Ad Hoc ◽  
Three Dimensional ◽  
Solid Surfaces ◽  
Two Dimensional ◽  
Nonlinear Odes ◽  
First Order ◽  
Parabolic Geometry ◽  
Abel Differential Equation ◽  
Analytical Tools

Through a suitable ad hoc assumption, a nonlinear PDE governing a three-dimensional weak, irrotational, steady vector field is reduced to a system of two nonlinear ODEs: the first of which corresponds to the two-dimensional case, while the second involves also the third field component. By using several analytical tools as well as linear approximations based on the weakness of the field, the first equation is transformed to an Abel differential equation which is solved parametrically. Thus, we obtain the two components of the field as explicit functions of a parameter. The derived solution is applied to the two-dimensional small perturbation frictionless flow past solid surfaces with either sinusoidal or parabolic geometry, where the plane velocities are evaluated over the body's surface in the case of a subsonic flow.

On the Hysteresis of Adsorption on Solid Surfaces

1952 ◽  
Vol 5 (2) ◽  
pp. 288
Author(s):  
RG Wylie
Keyword(s):  
Phase Transitions ◽  
Solid Surface ◽  
Three Dimensional ◽  
Solid Surfaces ◽  
Two Dimensional ◽  
Capillary Effects ◽  
First Order ◽  
Adsorption Of Gases ◽  
First Order Phase ◽  
Hysteresis Phenomena

Hysteresis phenomena associated with the adsorption of gases on solid surfaces are usually explained in terms of three-dimensional capillary effects or with more or less unspecific reference to phase transitions. It is shown that hysteresis effects are to be expected when two dimensional phase transitions occur on solids. In the connection, the thermodynamic equation governing the equilibrium of small, incompressible two-dimensional phases is derived. Such phases can form on an imperfect solid surface in an irreversible manner and, as calculation shows, can contribute significantly to the hysteresis of adsorption. In some cases the phase change may be responsible for the whole effect. The diffuseness of first-order phase transitions may be due to the same mechanism.

Upstream motions in stratified flow

1988 ◽  
Vol 187 ◽  
pp. 487-506 ◽  
Author(s):  
I. P. Castro ◽  
W. H. Snyder
Keyword(s):  
Body Height ◽  
Three Dimensional ◽  
Time Dependent ◽  
Experimental Measurements ◽  
Two Dimensional ◽  
Towing Tank ◽  
First Order ◽  
Density Perturbations ◽  
Small Obstacle ◽  
Obstacle Height

In this paper experimental measurements of the time-dependent velocity and density perturbations upstream of obstacles towed through linearly stratified fluid are presented. Attention is concentrated on two-dimensional obstacles which generate turbulent separated wakes at Froude numbers, based on velocity and body height, of less than 0.5. The form of the upstream columnar modes is shown to be largely that of first-order unattenuating disturbances, which have little resemblance to the perturbations described by small-obstacle-height theories. For two-dimensional obstacles the disturbances are similar to those found by Wei, Kao & Pao (1975) and it is shown that provided a suitable obstacle drag coefficient is specified, the lowest-order modes (at least) are quantitatively consistent with the results of the Oseen inviscid model.Discussion of some results of similar measurements upstream of three-dimensional obstacles, the importance of towing tank endwalls and the relevance of the Foster & Saffman (1970) theory for the limit of zero Froude number is also included.

scholarly journals Steady-State Three-Dimensional Ice Flow over an Undulating Base: First-Order Theory with Linear Ice Rheology

1987 ◽  
Vol 33 (114) ◽  
pp. 177-185 ◽  
Author(s):  
Niels Reeh
Keyword(s):  
Three Dimensional ◽  
Steady Motion ◽  
Simple Shear Flow ◽  
Finite Width ◽  
Two Dimensional ◽  
Internal Layers ◽  
Ice Flow ◽  
Depth Variation ◽  
First Order ◽  
First Order Theory

AbstractThe problem of ice flow over threedimensional basal irregularities is studied by considering the steady motion of a fluid with a linear constitutive equation over sine-shaped basal undulations. The undisturbed flow is simple shear flow with constant depth. Using the ratio of the amplitude of the basal undulations to the ice thickness as perturbation parameter, equations to the first order for the velocity and pressure perturbations are set up and solved.The study shows that when the widths of the basal undulations are larger than 2–3 times their lengths, the finite width of the undulations has only a minor influence on the flow, which to a good approximation may be considered two-dimensional. However, as the ratio between the longitudinal and the transverse wavelengthL/Wincreases, the three-dimensional flow effects becomes substantial. If, for example, the ratio ofLtoWexceeds 3, surface amplitudes are reduced by more than one order of magnitude as compared to the two-dimensional case. TheL/Wratio also influences the depth variation of the amplitudes of internal layers and the depth variation of perturbation velocities and strain-rates. With increasingL/Wratio, the changes of these quantities are concentrated in a near-bottom layer of decreasing thickness. Furthermore, it is shown, that the azimuth of the velocity vector may change by up to 10° between the surface and the base of the ice sheet, and that significant transverse flow may occur at depth without manifesting itself at the surface to any significant degree.

scholarly journals Thermomechanically coupled modelling for land-terminating glaciers: a comparison of two-dimensional, first-order and three-dimensional, full-Stokes approaches

2015 ◽  
Vol 61 (228) ◽  
pp. 702-712 ◽  
Author(s):  
Tong Zhang ◽  
Lili Ju ◽  
Wei Leng ◽  
Stephen Price ◽  
Max Gunzburger
Keyword(s):  
Initial Conditions ◽  
Three Dimensional ◽  
Integration Time ◽  
Two Dimensional ◽  
Shape Factors ◽  
Coupled Modelling ◽  
First Order ◽  
Glacier Changes ◽  
Basal Sliding ◽  
Long Time

AbstractFor many regions, glacier inaccessibility results in sparse geometric datasets for use as model initial conditions (e.g. along the central flowline only). In these cases, two-dimensional (2-D) flowline models are often used to study glacier dynamics. Here we systematically investigate the applicability of a 2-D, first-order Stokes approximation flowline model (FLM), modified by shape factors, for the simulation of land-terminating glaciers by comparing it with a 3-D, ‘full’-Stokes ice-flow model (FSM). Based on steady-state and transient, thermomechanically uncoupled and coupled computational experiments, we explore the sensitivities of the FLM and FSM to ice geometry, temperature and forward model integration time. We find that, compared to the FSM, the FLM generally produces slower horizontal velocities, due to simplifications inherent to the FLM and to the underestimation of the shape factor. For polythermal glaciers, those with temperate ice zones, or when basal sliding is important, we find significant differences between simulation results when using the FLM versus the FSM. Over time, initially small differences between the FLM and FSM become much larger, particularly near cold/temperate ice transition surfaces. Long time integrations further increase small initial differences between the two models. We conclude that the FLM should be applied with caution when modelling glacier changes under a warming climate or over long periods of time.

scholarly journals Steady-State Three-Dimensional Ice Flow over an Undulating Base: First-Order Theory with Linear Ice Rheology

1987 ◽  
Vol 33 (114) ◽  
pp. 177-185 ◽  
Author(s):  
Niels Reeh
Keyword(s):  
Three Dimensional ◽  
Steady Motion ◽  
Simple Shear Flow ◽  
Finite Width ◽  
Two Dimensional ◽  
Internal Layers ◽  
Ice Flow ◽  
Depth Variation ◽  
First Order ◽  
First Order Theory

AbstractThe problem of ice flow over threedimensional basal irregularities is studied by considering the steady motion of a fluid with a linear constitutive equation over sine-shaped basal undulations. The undisturbed flow is simple shear flow with constant depth. Using the ratio of the amplitude of the basal undulations to the ice thickness as perturbation parameter, equations to the first order for the velocity and pressure perturbations are set up and solved.The study shows that when the widths of the basal undulations are larger than 2–3 times their lengths, the finite width of the undulations has only a minor influence on the flow, which to a good approximation may be considered two-dimensional. However, as the ratio between the longitudinal and the transverse wavelength L/W increases, the three-dimensional flow effects becomes substantial. If, for example, the ratio of L to W exceeds 3, surface amplitudes are reduced by more than one order of magnitude as compared to the two-dimensional case. The L/W ratio also influences the depth variation of the amplitudes of internal layers and the depth variation of perturbation velocities and strain-rates. With increasing L/W ratio, the changes of these quantities are concentrated in a near-bottom layer of decreasing thickness. Furthermore, it is shown, that the azimuth of the velocity vector may change by up to 10° between the surface and the base of the ice sheet, and that significant transverse flow may occur at depth without manifesting itself at the surface to any significant degree.

An Educational Two-Dimensional Interactive Dynamic Grid Generator

1996 ◽  
Vol 24 (4) ◽  
pp. 279-290 ◽  
Author(s):  
M. Darwish ◽  
H. Diab ◽  
F. Moukalled
Keyword(s):  
Vector Fields ◽  
Three Dimensional ◽  
Object Oriented ◽  
Object Oriented Programming ◽  
Two Dimensional ◽  
Coordinate Systems ◽  
Curvilinear Grids ◽  
Windows Applications ◽  
Interactive Dynamic ◽  
Dimensional Domain

This paper describes IDGG, an Interactive Dynamic Grid Generator, for use as an educational tool by students studying computational fluid dynamics. The package is a Windows applications and runs on IBM PC, or compatible, computers. It is written in Pascal and built using object-oriented programming. The computer program allows the user to generate boundary-fitted curvilinear grids in any two-dimensional domain. The procedure adopted requires the user to perform the transformation step by step allowing him/her to easily grasp the concept of boundary-fitted coordinate systems. In addition, IDGG may be used by CFD researchers to display results graphically in the form of vector fields, contours, and two- and three-dimensional plots. The examples provided show the effectiveness of the package as a teaching aid.

Large-scale vortex structures in turbulent wakes behind bluff bodies. Part 1. Vortex formation processes

1987 ◽  
Vol 174 ◽  
pp. 233-270 ◽  
Author(s):  
A. E. Perry ◽  
T. R. Steiner
Keyword(s):  
Large Scale ◽  
Vector Fields ◽  
Vortex Formation ◽  
Three Dimensional ◽  
Point Theory ◽  
Shear Stresses ◽  
Bluff Bodies ◽  
Two Dimensional ◽  
Mean Flow ◽  
Turbulent Wakes

An investigation of turbulent wakes was conducted and phase-averaged velocity vector fields are presented, as well as phase-averaged and global Reynolds normal and shear stresses. The topology of the phase-averaged velocity fields is discussed in terms of critical point theory. Here in Part 1, the vortex formation process in the cavity region of several nominally two-dimensional bluff bodies is investigated and described using phase-averaged streamlines where the measurements were made in a nominal plane of symmetry. It was found that the flows encountered were always three-dimensional and that the mean-flow patterns in the cavity region were quite different from those expected using classical two-dimensional assumptions.

scholarly journals NONORIENTABLE MANIFOLDS IN THREE-DIMENSIONAL VECTOR FIELDS

2003 ◽  
Vol 13 (03) ◽  
pp. 553-570 ◽  
Author(s):  
HINKE M. OSINGA
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Three Dimensional ◽  
Period Doubling ◽  
Building Blocks ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Three Dimensional Vector Field

It is well known that a nonorientable manifold in a three-dimensional vector field is topologically equivalent to a Möbius strip. The most frequently used example is the unstable manifold of a periodic orbit that just lost its stability in a period-doubling bifurcation. However, there are not many explicit studies in the literature in the context of dynamical systems, and so far only qualitative sketches could be given as illustrations. We give an overview of the possible bifurcations in three-dimensional vector fields that create nonorientable manifolds. We mainly focus on nonorientable manifolds of periodic orbits, because they are the key building blocks. This is illustrated with invariant manifolds of three-dimensional vector fields that arise from applications. These manifolds were computed with a new algorithm for computing two-dimensional manifolds.

A New CFD-Perturbation Model for the Rotordynamics of Incompressible Flow Seals

2000 ◽  
Author(s):  
Namhyo Kim ◽  
David L. Rhode
Keyword(s):  
Incompressible Flow ◽  
Ad Hoc ◽  
Present Model ◽  
Three Dimensional ◽  
Radial Clearance ◽  
Shear Stresses ◽  
Perturbation Model ◽  
First Order ◽  
Primary Advantage ◽  
Rotordynamic Forces

A new, quasi-three-dimensional perturbation model is developed for the computation of rotordynamic forces for all incompressible flow seals with an axisymmetric geometry. The model assumes a small circular whirl motion of the rotor around the stator center. By avoiding the complicated perturbation coordinate transformation, the perturbation solution directly accounts for the disturbance of the wall shear stresses that are caused by the whirl motion of the rotor. The primary advantage of the present model is that it can immediately be applied to any radial-clearance seal of axisymmetric geometry without ad hoc adjustments. It was found from computations that include the upstream chamber that the magnitude of the first-order variables at the seal inlet are much larger and abruptly changing than was previously assumed.

Stability Analysis Of Cracks Propagating In Three Dimensional Solids

1995 ◽  
Vol 409 ◽  
Author(s):  
H. Larralde ◽  
A. A. Al-Falou ◽  
R. C. Ball
Keyword(s):  
Perturbation Theory ◽  
Stability Analysis ◽  
Fracture Surface ◽  
Three Dimensional ◽  
Type I ◽  
Two Dimensional ◽  
Singular Terms ◽  
First Order ◽  
Left Behind ◽  
First Order Perturbation

AbstractWe present a theory for the morphology of the fracture surface left behind by slowly propagating cracks in linear, isotropic and homogeneous three dimensional solids. Our results are based on first order perturbation theory of the equations of elasticity for cracks whose shape is slightly perturbed from planar. For cracks propagating under pure type I loading we find that all perturbation modes are linearly stable, from which we can predict the roughness of the fracture surface induced by fluctuations in the material. We compare our results with the classical results for cracks propagating in two dimensional systems, and discuss the effects in the three dimensional analysis which result from taking into account contributions from non-singular terms of the stress field, as well as the effects arising from finite speeds of crack propagation.

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