Lee waves in three-dimensional stratified flow

1984 ◽  
Vol 148 ◽  
pp. 97-108 ◽  
Author(s):  
G. S. Janowitz
Keyword(s):  
Stream Function ◽  
Displacement Field ◽  
Three Dimensional ◽  
Frequency Dispersion ◽  
Vertical Displacement ◽  
Function Solution ◽  
Dispersion Surface ◽  
Phase Calculation ◽  
Lee Waves ◽  
The Green Function

The effect of a shallow isolated topography on a linearly stratified, three-dimensional, initially uniform flow in the x-direction is considered. The Green-function solution for the velocity disturbance due to this topography, which is equivalent to that due to a dipole at the origin, is shown to be without swirl, i.e. the velocity disturbance lies strictly in planes passing through the x-axis. Thus this disturbance can be described in terms of a stream function. The asymptotic forms of the wavelike portion of the stream function and the vertical displacement field are obtained. The latter is in agreement with the limited versions due to Crapper (1959). The Gaussian curvature of the zero-frequency dispersion surface is obtained analytically as a step in the stationary-phase calculation. The model is extended to determine the vertical displacement field for an arbitrary shallow topography far downstream. For topographies that are even functions of x and y it is shown that the details of the topography affect the displacement field only in the vicinity of the x-axis. Elsewhere, the amplitude of the displacement is proportional to the net volume of the topography.

Random fiber network loaded by a point force

2021 ◽  
pp. 1-11
Author(s):  
Catalin Picu ◽  
Jacob Merson
Keyword(s):  
Green Function ◽  
Linear Elasticity ◽  
Displacement Field ◽  
Point Force ◽  
Fiber Network ◽  
Function Solution ◽  
Network Parameters ◽  
The Green Function ◽  
Nonlocal Nature

Abstract This article presents the displacement field produced by a point force acting on an athermal random fiber network (the Green function for the network). The problem is defined within the limits of linear elasticity and the field is obtained numerically for nonaffine networks characterized by various parameter sets. The classical Green function solution applies at distances from the point force larger than a threshold which is independent of the network parameters in the range studied. At smaller distances, the nonlocal nature of fiber interactions modifies the solution.

Contracting ducts of finite length

1951 ◽  
Vol 2 (4) ◽  
pp. 254-271 ◽  
Author(s):  
L. G. Whitehead ◽  
L. Y. Wu ◽  
M. H. L. Waters
Keyword(s):  
Stream Function ◽  
Finite Length ◽  
Velocity Potential ◽  
Three Dimensional ◽  
Relaxation Method ◽  
Two Dimensional ◽  
Dimensional Flow ◽  
Hodograph Plane ◽  
Two Dimensional Flow ◽  
Working Section

SummmaryA method of design is given for wind tunnel contractions for two-dimensional flow and for flow with axial symmetry. The two-dimensional designs are based on a boundary chosen in the hodograph plane for which the flow is found by the method of images. The three-dimensional method uses the velocity potential and the stream function of the two-dimensional flow as independent variables and the equation for the three-dimensional stream function is solved approximately. The accuracy of the approximate method is checked by comparison with a solution obtained by Southwell's relaxation method.In both the two and the three-dimensional designs the curved wall is of finite length with parallel sections upstream and downstream. The effects of the parallel parts of the channel on the rise of pressure near the wall at the start of the contraction and on the velocity distribution across the working section can therefore be estimated.

Obstacle drag in stratified flow

1990 ◽  
Vol 429 (1876) ◽  
pp. 119-140 ◽  
Keyword(s):  
Experimental Study ◽  
Linear Density ◽  
Three Dimensional ◽  
Inviscid Flow ◽  
Force Balance ◽  
Video Recordings ◽  
Lee Waves ◽  
Time Variations ◽  
Obstacle Height ◽  
Stable Linear

This paper describes an experimental study of the drag of two- and three-dimensional bluff obstacles of various cross-stream shapes when towed through a fluid having a stable, linear density gradient with Brunt-Vaisala frequency, N . Drag measurements were made directly using a force balance, and effects of obstacle blockage ( h / D , where h and D are the obstacle height and the fluid depth, respectively) and Reynolds number were effectively eliminated. It is shown that even in cases where the downstream lee waves and propagating columnar waves are of large amplitude, the variation of drag with the parameter K ( = ND /π U ) is qualitatively close to that implied by linear theories, with drag minima existing at integral values of K . Under certain conditions large, steady, periodic variations in drag occur. Simultaneous drag measurements and video recordings of the wakes show that this unsteadiness is linked directly with time-variations in the lee and columnar wave amplitudes. It is argued that there are, therefore, situations where the inviscid flow is always unsteady even for large times; the consequent implications for atmospheric motions are discussed.

Truss and Beam Finite Elements Revisited: A Derivation Based on Displacement-Field Decomposition

1996 ◽  
Vol 11 (4) ◽  
pp. 371-380 ◽  
Author(s):  
Alphose Zingoni
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Symmetry Group ◽  
Displacement Field ◽  
Three Dimensional ◽  
Beam Elements ◽  
Symmetry Properties ◽  
Mass Matrices ◽  
General Displacement

Where a finite element possesses symmetry properties, derivation of fundamental element matrices can be achieved more efficiently by decomposing the general displacement field into subspaces of the symmetry group describing the configuration of the element. In this paper, the procedure is illustrated by reference to the simple truss and beam elements, whose well-known consistent-mass matrices are obtained via the proposed method. However, the procedure is applicable to all one-, two- and three-dimensional finite elements, as long as the shape and node configuration of the element can be described by a specific symmetry group.

scholarly journals TSUNAMI GENERATION AND PROPAGATION

1972 ◽  
Vol 1 (13) ◽  
pp. 146
Author(s):  
Joseph L. Hammack ◽  
Frederic Raichlen
Keyword(s):  
Linear Theory ◽  
Source Region ◽  
Three Dimensional ◽  
Frequency Dispersion ◽  
Nonlinear Effects ◽  
Its Region ◽  
Experimental Results ◽  
Specific Class ◽  
Two Dimensional ◽  
Three Dimensional Models

A linear theory is presented for waves generated by an arbitrary bed deformation {in space and time) for a two-dimensional and a three -dimensional fluid domain of uniform depth. The resulting wave profile near the source is computed for both the two and three-dimensional models for a specific class of bed deformations; experimental results are presented for the two-dimensional model. The growth of nonlinear effects during wave propagation in an ocean of uniform depth and the corresponding limitations of the linear theory are investigated. A strategy is presented for determining wave behavior at large distances from the source where linear and nonlinear effects are of equal magnitude. The strategy is based on a matching technique which employs the linear theory in its region of applicability and an equation similar to that of Korteweg and deVries (KdV) in the region where nonlinearities are equal in magnitude to frequency dispersion. Comparison of the theoretical computations with the experimental results indicates that an equation of the KdV type is the proper model of wave behavior at large distances from the source region.

Consistent Asymptotic Expansion Multiscale Formulation for Heterogeneous Column Structure

2012 ◽  
Vol 134 (3) ◽  
Author(s):  
Dongdong Wang ◽  
Pinkang Xie ◽  
Lingming Fang
Keyword(s):  
Asymptotic Expansion ◽  
Displacement Field ◽  
Three Dimensional ◽  
Axial Direction ◽  
Unit Cell ◽  
Local Unit ◽  
Cell Problem ◽  
Element Discretization ◽  
Free Condition ◽  
Column Structure

A consistent asymptotic expansion multiscale formulation is presented for analysis of the heterogeneous column structure, which has three dimensional periodic reinforcements along the axial direction. The proposed formulation is based upon a new asymptotic expansion of the displacement field. This new multiscale displacement expansion has a three dimensional form, more specifically, it takes into account the axial periodic property but simultaneously keeps the cross section dimensions in the global scale. Thus, this formulation inherently reflects the characteristics of the column structure, i.e., the traction free condition on the circumferential surfaces. Subsequently, the global equilibrium problem and the local unit cell problem are consistently derived based upon the proposed asymptotic displacement field. It turns out that the global homogenized problem is the standard axial equilibrium equation, while the local unit cell problem is completely three dimensional which is subjected to the periodic boundary condition on axial surfaces as well as the traction free condition on circumferential surfaces of the unit cell. Thereafter, the variational formulation and finite element discretization of the unit cell problem are discussed. The effectiveness of the present formulation is illustrated by several numerical examples.

Forcing of oceanic mean flows by dissipating internal tides

2012 ◽  
Vol 708 ◽  
pp. 250-278 ◽  
Author(s):  
Nicolas Grisouard ◽  
Oliver Bühler
Keyword(s):  
Numerical Study ◽  
Three Dimensional ◽  
Wave Drag ◽  
Perturbation Series ◽  
Internal Tides ◽  
Time Averaging ◽  
Mean Flow ◽  
Wave Source ◽  
Lee Waves ◽  
Lee Wave

AbstractWe present a theoretical and numerical study of the effective mean force exerted on an oceanic mean flow due to the presence of small-amplitude internal waves that are forced by the oscillatory flow of a barotropic tide over undulating topography and are also subject to dissipation. This extends the classic lee-wave drag problem of atmospheric wave–mean interaction theory to a more complicated oceanographic setting, because now the steady lee waves are replaced by oscillatory internal tides and, most importantly, because now the three-dimensional oceanic mean flow is defined by time averaging over the fast tidal cycles rather than by the zonal averaging familiar from atmospheric theory. Although the details of our computation are quite different, we recover the main action-at-a-distance result from the atmospheric setting, namely that the effective mean force that is felt by the mean flow is located in regions of wave dissipation, and not necessarily near the topographic wave source. Specifically, we derive an explicit expression for the effective mean force at leading order using a perturbation series in small wave amplitude within the framework of generalized Lagrangian-mean theory, discuss in detail the range of situations in which a strong, secularly growing mean-flow response can be expected, and then compute the effective mean force numerically in a number of idealized examples with simple topographies.

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