scholarly journals Stress-gradient Coupling in Glacier Flow: IV. Effects of the “T” Term

1986 ◽  
Vol 32 (112) ◽  
pp. 342-349 ◽  
Author(s):  
Barclay Kamb ◽  
Keith A. Echelmeyer
Keyword(s):  
Shear Stress ◽  
Coupling Theory ◽  
Longitudinal Stress ◽  
Longitudinal Flow ◽  
Stress Gradients ◽  
Flow Coupling ◽  
Flow Response ◽  
Longitudinal Variations ◽  
Local Shear ◽  
Local Shear Stress

AbstractThe “T term” in the longitudinal stress-equilibrium equation for glacier mechanics, a double y-integral of ∂2τxy/∂x2 where x is a longitudinal coordinate and y is roughly normal to the ice surface, can be evaluated within the framework of longitudinal flow-coupling theory by linking the local shear stress τxy at any depth to the local shear stress τB at the base, which is determined by the theory. This approach leads to a modified longitudinal flow-coupling equation, in which the modifications deriving from the T term are as follows: 1. The longitudinal coupling length is increased by about 5%. 2. The asymmetry parameter σ is altered by a variable but small amount depending on longitudinal gradients in ice thickness h and surface slope α. 3. There is a significant direct modification of the influence of local h and α on flow, which represents a distinct “driving force” in glacier mechanics, whose origin is in pressure gradients linked to stress gradients of the type ∂τxy/∂x. For longitudinal variations in h, the “T force” varies as d2h/dx2 and results in an in-phase enhancement of the flow response to the variations in h, describable (for sinusoidal variations) by a wavelength-dependent enhancement factor. For longitudinal variations in α, the “force” varies as dα/dx and gives a phase-shifted flow response. Although the “T force” is not negligible, its actual effect on τB and on ice flow proves to be small, because it is attenuated by longitudinal stress coupling. The greatest effect is at shortest wavelengths (λ 2.5h), where the flow response to variations in h does not tend to zero as it would otherwise do because of longitudinal coupling, but instead, because of the effect of the “T force”, tends to a response about 4% of what would occur in the absence of longitudinal coupling. If an effect of this small size can be considered negligible, then the influence of the T term can be disregarded. It is then unnecessary to distinguish in glacier mechanics between two length scales for longitudinal averaging of τb, one over which the T term is negligible and one over which it is not.Longitudinal flow-coupling theory also provides a basis for evaluating the additional datum-state effects of the T term on the flow perturbations Δu that result from perturbations Δh and Δα from a datum state with longitudinal stress gradients. Although there are many small effects at the ~1% level, none of them seems to stand out significantly, and at the 10% level all can be neglected.The foregoing conclusions apply for long wavelengths λh. For short wavelengths (λ h), effects of the T term become important in longitudinal coupling, as will be shown in a later paper in this series.

scholarly journals Stress-gradient Coupling in Glacier Flow: IV. Effects of the “T” Term

1986 ◽  
Vol 32 (112) ◽  
pp. 342-349 ◽  
Author(s):  
Barclay Kamb ◽  
Keith A. Echelmeyer
Keyword(s):  
Shear Stress ◽  
Coupling Theory ◽  
Longitudinal Stress ◽  
Longitudinal Flow ◽  
Stress Gradients ◽  
Flow Coupling ◽  
Flow Response ◽  
Longitudinal Variations ◽  
Local Shear ◽  
Local Shear Stress

AbstractThe “Tterm” in the longitudinal stress-equilibrium equation for glacier mechanics, a doubley-integral of ∂2τxy/∂x2wherexis a longitudinal coordinate andyis roughly normal to the ice surface, can be evaluated within the framework of longitudinal flow-coupling theory by linking the local shear stressτxyat any depth to the local shear stressτBat the base, which is determined by the theory. This approach leads to a modified longitudinal flow-coupling equation, in which the modifications deriving from theTterm are as follows: 1. The longitudinal coupling lengthis increased by about 5%. 2. The asymmetry parameterσis altered by a variable but small amount depending on longitudinal gradients in ice thicknesshand surface slopeα. 3. There is a significant direct modification of the influence of localhandαon flow, which represents a distinct “driving force” in glacier mechanics, whose origin is in pressure gradients linked to stress gradients of the type ∂τxy/∂x.For longitudinal variations inh, the “Tforce” varies as d2h/dx2and results in an in-phase enhancement of the flow response to the variations inh, describable (for sinusoidal variations) by a wavelength-dependent enhancement factor. For longitudinal variations in α, the “force” varies as dα/dxand gives a phase-shifted flow response. Although the “Tforce” is not negligible, its actual effect onτBand on ice flow proves to be small, because it is attenuated by longitudinal stress coupling. The greatest effect is at shortest wavelengths (λ2.5h), where the flow response to variations inhdoes not tend to zero as it would otherwise do because of longitudinal coupling, but instead, because of the effect of the “Tforce”, tends to a response about 4% of what would occur in the absence of longitudinal coupling. If an effect of this small size can be considered negligible, then the influence of theTterm can be disregarded. It is then unnecessary to distinguish in glacier mechanics between two length scales for longitudinal averaging ofτb, one over which theTterm is negligible and one over which it is not.Longitudinal flow-coupling theory also provides a basis for evaluating the additional datum-state effects of theTterm on the flow perturbations Δuthat result from perturbations Δhand Δα from a datum state with longitudinal stress gradients. Although there are many small effects at the ~1% level, none of them seems to stand out significantly, and at the 10% level all can be neglected.The foregoing conclusions apply for long wavelengths λh.For short wavelengths (λh), effects of theTterm become important in longitudinal coupling, as will be shown in a later paper in this series.

scholarly journals Stress-Gradient Coupling in Glacier Flow: I. Longitudinal Averaging of the Influence of Ice Thickness and Surface Slope

1986 ◽  
Vol 32 (111) ◽  
pp. 267-284 ◽  
Author(s):  
Barclay Kamb ◽  
Keith A. Echelmeyer
Keyword(s):  
Weighting Function ◽  
Ice Thickness ◽  
Longitudinal Stress ◽  
Surface Slope ◽  
Longitudinal Flow ◽  
Stress Gradients ◽  
Flow Coupling ◽  
Exponential Weighting ◽  
Longitudinal Variations ◽  
Longitudinal Coordinate

AbstractFor a glacier flowing over a bed of longitudinally varying slope, the influence of longitudinal stress gradients on the flow is analyzed by means of a longitudinal flow-coupling equation derived from the “vertically” (cross-sectionally) integrated longitudinal stress equilibrium equation, by an extension of an approach originally developed by Budd (1968). Linearization of the flow-coupling equation, by treating the flow velocity u (“vertically” averaged), ice thickness h, and surface slope α in terms of small deviations Δu, Δh, and ∆α from overall average (datum) values uo, h0, and α0, results in a differential equation that can be solved by Green’s function methods, giving Δu(x) as a function of ∆h(x) and ∆α(x), x being the longitudinal coordinate. The result has the form of a longitudinal averaging integral of the influence of local h(x) and α(x) on the flow u(x): where the integration is over the length L of the glacier. The ∆ operator specified deviations from the datum state, and the term on which it operates, which is a function of the integration variable x′, represents the influence of local h(x′), α(x′), and channel-shape factor f(x′), at longitudinal coordinate x′, on the flow u at coordinate x, the influence being weighted by the “influence transfer function” exp (−|x′ − x|/ℓ) in the integral.The quantity ℓ that appears as the scale length in the exponential weighting function is called the longitudinal coupling length. It is determined by rheological parameters via the relationship , where n is the flow-law exponent, η the effective longitudinal viscosity, and η the effective shear viscosity of the ice profile, η is an average of the local effective viscosity η over the ice cross-section, and (η)–1 is an average of η−1 that gives strongly increased weight to values near the base. Theoretically, the coupling length ℓ is generally in the range one to three times the ice thickness for valley glaciers and four to ten times for ice sheets; for a glacier in surge, it is even longer, ℓ ~ 12h. It is distinctly longer for non-linear (n = 3) than for linear rheology, so that the flow-coupling effects of longitudinal stress gradients are markedly greater for non-linear flow.The averaging integral indicates that the longitudinal variations in flow that occur under the influence of sinusoidal longitudinal variations in h or α, with wavelength λ, are attenuated by the factor 1/(1 + (2πℓ/λ)2) relative to what they would be without longitudinal coupling. The short, intermediate, and long scales of glacier motion (Raymond, 1980), over which the longitudinal flow variations are strongly, partially, and little attenuated, are for λ ≲ 2ℓ , 2ℓ ≲ λ ≲ 20ℓ, and λ ≳ 20ℓ.For practical glacier-flow calculations, the exponential weighting function can be approximated by a symmetrical triangular averaging window of length 4ℓ, called the longitudinal averaging length. The traditional rectangular window is a poor approximation. Because of the exponential weighting, the local surface slope has an appreciable though muted effect on the local flow, which is clearly seen in field examples, contrary to what would result from a rectangular averaging window.Tested with field data for Variegated Glacier, Alaska, and Blue Glacier, Washington, the longitudinal averaging theory is able to account semi-quantitatively for the observed longitudinal variations in flow of these glaciers and for the representation of flow in terms of “effective surface slope” values. Exceptions occur where the flow is augmented by large contributions from basal sliding in the ice fall and terminal zone of Blue Glacier and in the reach of surge initiation in Variegated Glacier. The averaging length 4l that gives the best agreement between calculated and observed flow pattern is 2.5 km for Variegated Glacier and 1.8 km for Blue Glacier, corresponding to ℓ/h ≈ 2 in both cases.If ℓ varies with x, but not too rapidly, the exponential weighting function remains a fairly good approximation to the exact Green’s function of the differential equation for longitudinal flow coupling; in this approximation, ℓ in the averaging integral is ℓ(x) but is not a function of x′. Effects of longitudinal variation of J are probably important near the glacier terminus and head, and near ice falls.The longitudinal averaging formulation can also be used to express the local basal shear stress in terms of longitudinal variations in the local “slope stress” with the mediation of longitudinal stress gradients.

scholarly journals Stress-Gradient Coupling in Glacier Flow: I. Longitudinal Averaging of the Influence of Ice Thickness and Surface Slope

1986 ◽  
Vol 32 (111) ◽  
pp. 267-284 ◽  
Author(s):  
Barclay Kamb ◽  
Keith A. Echelmeyer
Keyword(s):  
Weighting Function ◽  
Ice Thickness ◽  
Longitudinal Stress ◽  
Surface Slope ◽  
Longitudinal Flow ◽  
Stress Gradients ◽  
Flow Coupling ◽  
Exponential Weighting ◽  
Longitudinal Variations ◽  
Longitudinal Coordinate

AbstractFor a glacier flowing over a bed of longitudinally varying slope, the influence of longitudinal stress gradients on the flow is analyzed by means of a longitudinal flow-coupling equation derived from the “vertically” (cross-sectionally) integrated longitudinal stress equilibrium equation, by an extension of an approach originally developed by Budd (1968). Linearization of the flow-coupling equation, by treating the flow velocityu(“vertically” averaged), ice thicknessh, and surface slope α in terms of small deviations Δu, Δh, and ∆α from overall average (datum) valuesuo,h0, andα0, results in a differential equation that can be solved by Green’s function methods, giving Δu(x) as a function of∆h(x) and ∆α(x),xbeing the longitudinal coordinate. The result has the form of a longitudinal averaging integral of the influence of localh(x) and α(x) on the flowu(x):where the integration is over the lengthLof the glacier. The ∆ operator specified deviations from the datum state, and the term on which it operates, which is a function of the integration variablex′, represents the influence of localh(x′),α(x′), and channel-shape factorf(x′), at longitudinal coordinatex′, on the flowuat coordinatex, the influence being weighted by the “influence transfer function” exp (−|x′ −x|/ℓ) in the integral.The quantityℓthat appears as the scale length in the exponential weighting function is called thelongitudinal coupling length. It is determined by rheological parameters via the relationship, wherenis the flow-law exponent,ηthe effective longitudinal viscosity, andηthe effective shear viscosity of the ice profile,ηis an average of the local effective viscosityηover the ice cross-section, and (η)–1is an average of η−1that gives strongly increased weight to values near the base. Theoretically, the coupling lengthℓis generally in the range one to three times the ice thickness for valley glaciers and four to ten times for ice sheets; for a glacier in surge, it is even longer,ℓ ~ 12h. It is distinctly longer for non-linear (n = 3) than for linear rheology, so that the flow-coupling effects of longitudinal stress gradients are markedly greater for non-linear flow.The averaging integral indicates that the longitudinal variations in flow that occur under the influence of sinusoidal longitudinal variations inhor α, with wavelength λ, are attenuated by the factor 1/(1 + (2πℓ/λ)2) relative to what they would be without longitudinal coupling. The short, intermediate, and long scales of glacier motion (Raymond, 1980), over which the longitudinal flow variations are strongly, partially, and little attenuated, are for λ ≲ 2ℓ , 2ℓ ≲ λ ≲ 20ℓ, and λ ≳ 20ℓ.For practical glacier-flow calculations, the exponential weighting function can be approximated by a symmetrical triangular averaging window of length 4ℓ, called thelongitudinal averaging length. The traditional rectangular window is a poor approximation. Because of the exponential weighting, the local surface slope has an appreciable though muted effect on the local flow, which is clearly seen in field examples, contrary to what would result from a rectangular averaging window.Tested with field data for Variegated Glacier, Alaska, and Blue Glacier, Washington, the longitudinal averaging theory is able to account semi-quantitatively for the observed longitudinal variations in flow of these glaciers and for the representation of flow in terms of “effective surface slope” values. Exceptions occur where the flow is augmented by large contributions from basal sliding in the ice fall and terminal zone of Blue Glacier and in the reach of surge initiation in Variegated Glacier. The averaging length4lthat gives the best agreement between calculated and observed flow pattern is 2.5 km for Variegated Glacier and 1.8 km for Blue Glacier, corresponding toℓ/h≈ 2 in both cases.Ifℓvaries withx, but not too rapidly, the exponential weighting function remains a fairly good approximation to the exact Green’s function of the differential equation for longitudinal flow coupling; in this approximation,ℓin the averaging integral isℓ(x) but is not a function ofx′. Effects of longitudinal variation of J are probably important near the glacier terminus and head, and near ice falls.The longitudinal averaging formulation can also be used to express the local basal shear stress in terms of longitudinal variations in the local “slope stress” with the mediation of longitudinal stress gradients.

scholarly journals Longitudinal Stress Gradients and Basal Shear Stress of a Temperate Valley Glacier

1979 ◽  
Vol 24 (90) ◽  
pp. 507-508 ◽  
Author(s):  
Robert Bindschadler
Keyword(s):  
Shear Stress ◽  
Strain Rate ◽  
Field Data ◽  
Numerical Models ◽  
Lateral Strain ◽  
Longitudinal Stress ◽  
Base Shear ◽  
Stress Gradients ◽  
Valley Glacier ◽  
Image Position

AbstractFor the first time field data from a temperate valley glacier, the Variegated Glacier, are used to investigate the behavior of longitudinal stress gradients predicted by the relation(1)whereHis the local depth, andysandybare the surface and bed elevations respectively. This equation is similar to one derived by Budd (1970) for plane strain-rate, to evaluate the importance of longitudinal stress gradients, but a shape factorfis included to account approximately for lateral strain-rate gradients. Predictive numerical models of valley glaciers require the local base shear stress to be known as accurately as possible. It has been argued on theoretical grounds that whenTis averaged over distances of more than five to ten times the depth, this term is negligible. At larger averaging scales, 2Gcan then be considered a correction to the simple geometric expression of base stress due to the presence of longitudinal stress gradients. Field data of velocity and geometry are used to evaluate the terms of Equation (1), whereτband 2Gare estimated asandat intervals of 100 m,Usis the measured surface center-line velocity,Aandnare the flow-law parameters, andis the surface longitudinal strain-rate. The expression for 2Gis an approximation proposed by Budd (1970).

Roughness Influence on Turbulent Flow Through Annular Seals

1994 ◽  
Vol 116 (2) ◽  
pp. 321-328 ◽  
Author(s):  
Victor Lucas ◽  
Sterian Danaila ◽  
Olivier Bonneau ◽  
Jean Freˆne
Keyword(s):  
Surface Roughness ◽  
Shear Stress ◽  
Wall Shear Stress ◽  
Turbulent Flow ◽  
Dynamic Characteristics ◽  
Reynolds Equation ◽  
Mixing Length ◽  
Local Shear ◽  
Wall Shear ◽  
Local Shear Stress

This paper deals with an analysis of turbulent flow in annular seals with rough surfaces. In this approach, our objectives are to develop a model of turbulence including surface roughness and to quantify the influence of surface roughness on turbulent flow. In this paper, in order to simplify the analysis, the inertial effects are neglected. These effects will be taken into account in a subsequent work. Consequently, this study is based on the solution of Reynolds equation. Turbulent flow is solved using Prandtl’s turbulent model with Van Driest’s mixing length expression. In Van Driest’s model, the mixing length depends on wall shear stress. However there are many numerical problems in evaluating this wall shear stress. Therefore, the goal of this work has been to use the local shear stress in the Van Driest’s model. This derived from the work of Elrod and Ng concerning Reichardt’s mixing length. The mixing length expression is then modified to introduce roughness effects. Then, the momentum equations are solved to evaluate the circumferential and axial velocity distributions as well as the turbulent viscosity μ1 (Boussinesq’s hypothesis) within the film. The coefficients of turbulence kx and kz, occurring in the generalized Reynolds’ equation, are then calculated as functions of the flow parameters. Reynolds’ equation is solved by using a finite centered difference method. Dynamic characteristics are calculated by exciting the system numerically, with displacement and velocity perturbations. The model of Van Driest using local shear stress and function of roughness has been compared (for smooth seals) to the Elrod and Ng theory. Some numerical results of the static and dynamic characteristics of a rough seal (with the same roughness on the rotor as on the stator) are presented. These results show the influence of roughness on the dynamic behavior of the shaft.

Comparison of void fraction change and local shear stress fluctuation in channel flow

2018 ◽  
Vol 2018 (0) ◽  
pp. OS3-5
Author(s):  
Hayato NAKAMURA ◽  
Satoshi OGAMI ◽  
Yoshihiko OISHI ◽  
Hideki KAWAI ◽  
Yuichi MURAI
Keyword(s):  
Shear Stress ◽  
Channel Flow ◽  
Void Fraction ◽  
Stress Fluctuation ◽  
Local Shear ◽  
Local Shear Stress

scholarly journals The Brachial Artery Remodels to Maintain Local Shear Stress Despite the Presence of Cardiovascular Risk Factors

2009 ◽  
Vol 29 (4) ◽  
pp. 606-612 ◽  
Author(s):  
William B. Chung ◽  
Naomi M. Hamburg ◽  
Monika Holbrook ◽  
Sherene M. Shenouda ◽  
Mustali M. Dohadwala ◽  
...  
Keyword(s):  
Risk Factors ◽  
Shear Stress ◽  
Cardiovascular Risk ◽  
Cardiovascular Risk Factors ◽  
Brachial Artery ◽  
Local Shear ◽  
Local Shear Stress
Sign in / Sign up

Export Citation Format

Share Document