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 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.

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 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.

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.

A first order theory for Newtonian flow over two-dimensional airfoils

1987 ◽  
Vol 91 (910) ◽  
pp. 471-478 ◽  
Author(s):  
Hamdi T. Hemdan
Keyword(s):  
Order Theory ◽  
Two Dimensional ◽  
Small Disturbance ◽  
Newtonian Flow ◽  
Approximate Methods ◽  
First Order ◽  
Specific Heats ◽  
First Order Theory ◽  
Free Stream Mach Number ◽  
Recent Formulation

Summary A simple closed-form formula for the coefficient of surface pressure, Cp is given in this paper. The formula is based on first and second approximations of the full problem and can be used for the Newtonian flow past two-dimensional thin airfoils at small angles of attack and with attached shock waves. It thus extends Cole's zero order theory to the effects of non-zero (γ – 1) and finite values of Mx where γ is the ratio of the specific heats of the gas and Mx is the free stream Mach number. The analysis presented relies on a previous recent formulation of the hypersonic small disturbance theory which has proved to be more advantageous than the hypersonic small disturbance theory. The results are compared with other approximate methods and the agreement is found to be generally good.

Numerical Calculation of Second-Order Wave Resistance

1977 ◽  
Vol 21 (02) ◽  
pp. 94-106
Author(s):  
Young S. Hong
Keyword(s):  
Numerical Calculation ◽  
Steady Motion ◽  
Order Theory ◽  
Iteration Scheme ◽  
Wave Resistance ◽  
Second Order ◽  
Lagrangian Coordinates ◽  
Restricted Range ◽  
First Order ◽  
First Order Theory

The wave resistance due to the steady motion of a ship was formulated in Lagrangian coordinates by Wehausen [1].2 By introduction of an iteration scheme the solutions for the first order and second order3 were obtained. The draft/length ratio was assumed small in order to simplify numerical computation. In this work Wehausen's formulas are used to compute the resistance numerically. A few models are selected and the wave resistance is calculated. These results are compared with other methods and experiments. Generally speaking, the second-order resistance shows better agreement with experiment than first-order theory in only a restricted range of Froude number, say 0.25 to 0.35, and even here not uniformly. For larger Froude numbers it underestimates seriously.

Directed fluid sheets

1976 ◽  
Vol 347 (1651) ◽  
pp. 447-473 ◽  
Keyword(s):  
Water Waves ◽  
Three Dimensional ◽  
Steady Motion ◽  
Finite Depth ◽  
Free Surface Flows ◽  
Inviscid Fluid ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
Inviscid Fluids ◽  
Two Dimensional Flows

This paper is concerned mainly with incompressible inviscid fluid sheets but the incompressible linearly viscous fluid sheet is also considered. Our development is based on a direct formulation using the two dimensional theory of directed media called Cosserat surfaces . The first part of the paper deals with the formulation of appropriate nonlinear equations (which may include the effects of gravity and surface tension) governing the two dimensional motion of incompressible inviscid media for two categories, namely those ( a ) for two dimensional flows confined to a plane perpendicular to a specified direction and ( b ) for propagation of fairly long waves in a stream of variable initial depth. The latter development is a generalization of an earlier direct formulation of a theory of water waves when the fixed bottom of the stream is level (Green, Laws & Naghdi 1974). In the second part of the paper, special attention is given to a demonstration of the relevance and applicability of the present direct formulation to a variety of two dimensional problems of inviscid fluid sheets. These include, among others, the steady motion of a class of two-dimensional flows in a stream of finite depth in which the bed of the stream may change from one constant level to another, the related problem of hydraulic jumps, and a class of exact solutions which characterize the main features of the time-dependent free surface flows in the three dimensional theory of incompressible inviscid fluids.

scholarly journals Experiments on the motion of solid bodies in rotating fluids

1923 ◽  
Vol 104 (725) ◽  
pp. 213-218 ◽  
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Steady Motion ◽  
The Body ◽  
Two Dimensional ◽  
Experimental Conditions ◽  
Axis Of Rotation ◽  
Rotating Liquid ◽  
Steady Motions ◽  
Solid Bodies

Some years ago it was pointed out by Prof. Proudman that all slow steady motions of a rotating liquid must be two-dimensional. If the motion is produced by moving a cylindrical object slowly through the liquid in such a way that its axis remains parallel to the axis of rotation, or if a two-dimensional motion is conceived as already existing, it seems clear that it will remain two-dimensional. If a slow three-dimensional motion is produced, then it cannot be a steady one. On the other hand, if an attempt is made to produce a slow steady motion by moving a three-dimensional body with a small uniform velocity (relative to axes which rotate with the fluid) three possibilities present themselves:— ( a ) The motion in the liquid may never become steady, however long the body goes on moving. ( b ) The motion may be steady but it may not be small in the neighbourhood of the body. ( c ) The motion may be steady and two-dimensional. In considering these three possibilities it seems very unlikely that ( a ) will be the true one. In an infinite rotating fluid the disturbance produced by starting the motion of the body might go on spreading out for ever and steady motion might never be attained, but if the body were moved steadily in a direction at right angles to the axis of rotation, and if the fluid were contained between parallel planes also perpendicular to the axis of rotation, it seems very improbable that no steady motion satisfying the equations of motion could be attained. There is more chance that ( b ) may be true. A class of mathematical expressions representing the steady motion of a sphere along the axis of a rotating liquid has been obtained. This solution of the problem breaks down when the velocity of the sphere becomes indefinitely small, in the sense that it represents a motion which does not decrease as the velocity of the sphere decreases. It seems unlikely that such a motion would be produced under experimental conditions.

scholarly journals Most general theory of 3d gravity: covariant phase space, dual diffeomorphisms, and more

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Marc Geiller ◽  
Christophe Goeller ◽  
Nelson Merino
Keyword(s):  
Phase Space ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Teleparallel Gravity ◽  
Massive Gravity ◽  
Simons Theory ◽  
First Order ◽  
First Order Theory ◽  
Dimensional Gravity ◽  
Finite Distance

Abstract We show that the phase space of three-dimensional gravity contains two layers of dualities: between diffeomorphisms and a notion of “dual diffeomorphisms” on the one hand, and between first order curvature and torsion on the other hand. This is most elegantly revealed and understood when studying the most general Lorentz-invariant first order theory in connection and triad variables, described by the so-called Mielke-Baekler Lagrangian. By analyzing the quasi-local symmetries of this theory in the covariant phase space formalism, we show that in each sector of the torsion/curvature duality there exists a well-defined notion of dual diffeomorphism, which furthermore follows uniquely from the Sugawara construction. Together with the usual diffeomorphisms, these duals form at finite distance, without any boundary conditions, and for any sign of the cosmological constant, a centreless double Virasoro algebra which in the flat case reduces to the BMS3 algebra. These algebras can then be centrally-extended via the twisted Sugawara construction. This shows that the celebrated results about asymptotic symmetry algebras are actually generic features of three-dimensional gravity at any finite distance. They are however only revealed when working in first order connection and triad variables, and a priori inaccessible from Chern-Simons theory. As a bonus, we study the second order equations of motion of the Mielke-Baekler model, as well as the on-shell Lagrangian. This reveals the duality between Riemannian metric and teleparallel gravity, and a new candidate theory for three-dimensional massive gravity which we call teleparallel topologically massive gravity.

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