scholarly journals Stability of three-dimensional columnar convection in a porous medium

2017 ◽  
Vol 829 ◽  
pp. 89-111 ◽  
Author(s):  
Duncan R. Hewitt ◽  
John R. Lister
Keyword(s):  
Porous Medium ◽  
Aspect Ratio ◽  
Large Scale ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Base Flow ◽  
Apparent Difference ◽  
Exchange Flow ◽  
Two Dimensional ◽  
The Difference

The stability of steady convective exchange flow with a rectangular planform in an unbounded three-dimensional porous medium is explored. The base flow comprises a balance between vertical advection with amplitude $A$ in interleaving rectangular columns with aspect ratio $\unicode[STIX]{x1D709}\leqslant 1$ and horizontal diffusion between the columns. Columnar flow with a square planform ($\unicode[STIX]{x1D709}=1$) is found to be weakly unstable to a large-scale perturbation of the background temperature gradient, irrespective of $A$, but to have no stronger instability on the scale of the columns. This result provides a stark contrast to two-dimensional columnar flow (Hewitt et al., J. Fluid Mech., vol. 737, 2013, pp. 205–231), which, as $A$ is increased, is increasingly unstable to a perturbation on the scale of the columnar wavelength. For rectangular planforms with $\unicode[STIX]{x1D709}<1$, a critical aspect ratio is identified, below which a perturbation on the scale of the columns is the fastest growing mode, as in two dimensions. Scalings for the growth rate and the structure of this mode are identified, and are explained by means of an asymptotic expansion in the limit $\unicode[STIX]{x1D709}\rightarrow 0$. The difference between the stabilities of two-dimensional and three-dimensional exchange flow provides a potential explanation for the apparent difference in dominant horizontal scale observed in direct numerical simulations of two-dimensional and three-dimensional statistically steady ‘Rayleigh–Darcy’ convection at high Rayleigh numbers.

scholarly journals Vorticity forces on an impulsively started finite plate

2012 ◽  
Vol 694 ◽  
pp. 464-492 ◽  
Author(s):  
Jian-Jhih Lee ◽  
Cheng-Ta Hsieh ◽  
Chien C. Chang ◽  
Chin-Chou Chu
Keyword(s):  
Aspect Ratio ◽  
Three Dimensional ◽  
Longitudinal Component ◽  
Spanwise Direction ◽  
Two Dimensional ◽  
Force Analysis ◽  
Finite Plate ◽  
Dimensional Flow ◽  
Two Dimensional Flow ◽  
The Difference

AbstractIn this study, we consider various contributions to the forces on an impulsively started finite plate from the perspective of a diagnostic vorticity force theory. The wing plate has an aspect ratio (AR) between 1 and 3, and is placed at low and high angles of attack ($\ensuremath{\alpha} \leq $ and ${\gt }2{0}^{\ensuremath{\circ} } $), while the Reynolds number is either 100 or 300. The theory enables us to quantify the contributions to the forces exerted on the plate in terms of all of the fluid elements with non-zero vorticity, such as in the tip vortices (TiVs), leading- and trailing-edge vortices (LEV and TEV) as well on the plate surface. This line of force analysis has been pursued for two-dimensional flow in our previous studies. In contrast to the pressure force analysis (PFA), the vorticity force analysis (VFA) reveals new salient features in its applications to three-dimensional flow by examining sectional force contributions along the spanwise direction. In particular, at a large aspect ratio ($\mathit{AR}= 3$), the force distributions of PFA and VFA show close agreements with each other in the middle sections, while at a lower aspect ratio ($\mathit{AR}= 1$), the force distribution of PFA is substantially larger than that of VFA in most of the sections. The difference is compensated for by the contributions partly by the edge sections and mainly by the vortices in the outer regions. Further investigation is made fruitful by decomposing the vorticity into the spanwise (longitudinal) component (the only one in two-dimensional flow) and the other two orthogonal (transverse) components. The relative importance of the force contributions credited to the transverse components in the entire flow regions as well as in the two outer regions signifies the three-dimensional nature of the flow over a finite plate. The interplay between the LEV and the TiVs at various time stages is shown to play a key role in distinguishing the force contributions for the plate with a smaller aspect ratio and that with a larger aspect ratio. The present VFA provides a better perspective for flow control by relating the forces directly to the various sources of vorticity (or vortex structures) on or near the wing plate.

A theory of the jet flap in three dimensions

1959 ◽  
Vol 251 (1266) ◽  
pp. 407-425 ◽  
Keyword(s):  
Aspect Ratio ◽  
Leading Edge ◽  
Elliptic Cylinder ◽  
Interpolation Formula ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Induced Drag ◽  
Aspect Ratios ◽  
The Difference

The treatment of two-dimensional jet-flapped wings in incompressible flow by the methods of thin-aerofoil theory given by one of the authors (Spence 1956) has been extended to the case of a thin wing of finite aspect ratio which possesses a deflected jet sheet of zero thickness emerging with a small angular deflexion at its trailing edge. The restriction is imposed that the streamlines of the jet flow lie in planes perpendicular to the wing-span, transverse momentum-transport being thus excluded. As in classical theories, the downwash field is assumed to arise from elementary horseshoe vortices proportional in strength to the local lift distribution; a new feature is the ability of the sheet formed by these elements to sustain a pressure difference on account of the longitudinal flux of momentum of the jet within it. The induced downwash w i ( x, y ) in the plane z = 0 at intermediate distances x from the wing cannot be calculated, and is therefore replaced by an interpolation formula having the correct values U ∞ α i at the wing and U ∞ α i ∞ far downstream. To ensure that the errors so introduced are small the aspect ratio A must be large, its permissible minimum increasing with the jet momentum coefficient C J . Two methods of interpolating to w i ( x, y ), both of which lead within close limits to the same expression for C L , are discussed. They are chosen so as to allow the two-dimensional equation for loading, whose solution is known, to be used to calculate the loading in a streamwise section in three dimensions. The spanwise variation of loading could be calculated for arbitrary planforms and jet-momentum distributions, but the present paper is confined to the case in which the loading and downwash distribution depend only on x/c , where x measures distance from the leading edge and c ( y ) is the local chord. This is shown to require both c and the jet-momentum flux per unit span to be elliptically distributed, the deflexion τ and incidence a being constant over the span. The relation between the coefficients of induced drag and lift is then C D i = C (2) L /( πA + 2 C J ) induced drag being defined as the difference between the thrust and the (constant) flux of momentum in the jet. (The interpolations for induced downwash are not used in deriving this relation.) The ratio of C L to the value C (2) L which it would have in two dimensions is C L / C (2) L = { A + (2/π) C J } / { A + 2/π ) (∂ C (2) L /∂α) - 2(1+ σ)}, where ∂ C (2) L /∂α is the two-dimensional derivative of lift with respect to incidence, a known function of C J and σ = 1 — α i /(½ α i ∞ ). An expression for σ is found in terms of known quantities by equating the induced drag calculated from the detailed forces on the wing to that given above. The results have been compared with experimental measurements made on an 8:1 elliptic cylinder of rectangular planform at aspect ratios 2⋅75, 6⋅8 and infinity. Remarkably close agreement with observed values of C L is obtained in all cases, and the difference C D — C Di = C D 0 , say, between the total- and induced-drag coefficients, is virtually independent of the aspect ratio. C D 0 represents the effects of the Reynolds number, section shape and jet configuration, which are excluded from the present theory.

Rheology of a dilute two-dimensional suspension of vesicles

2010 ◽  
Vol 653 ◽  
pp. 489-518 ◽  
Author(s):  
GIOVANNI GHIGLIOTTI ◽  
THIERRY BIBEN ◽  
CHAOUQI MISBAH
Keyword(s):  
Velocity Field ◽  
Large Scale ◽  
Free Boundary Problems ◽  
Numerical Study ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Boundary Integral ◽  
Function Technique ◽  
Two Dimensional ◽  
Vesicle Membrane

The rheology of a dilute two-dimensional suspension of vesicles (closed bags of a lipid bilayer membrane) is studied by numerical simulations. The numerical methods used are based on the boundary integral formulation (Green's function technique) and the phase field approach, which has become a quite popular and powerful tool for the numerical study of free-boundary problems. The imposed flow is an unbounded linear shear. The goal of the present study is to elucidate the link between the rheology of vesicle suspensions and the microscopic dynamics of the constituent particles (tank-treading and tumbling motions). A comparison with emulsion rheology reveals the central role played by the membrane. In particular, at low viscosity ratio λ (defined as the viscosity of the internal fluid over that of the ambient one), the effective viscosity decreases with λ, while the opposite trend is exhibited by emulsions, according to the classical Taylor result. This fact is explained by considering the velocity field of the ambient fluid. The area-incompressibility of the vesicle membrane modifies the surrounding velocity field in a quite different manner than what a drop does. The overall numerical results in two dimensions are in reasonable agreement with the three-dimensional analytical theory derived recently in the small deformation limit (quasi-spherical shapes). The finding that the simulations in two dimensions capture the essential features of the three-dimensional rheology opens the way for extensive and large-scale simulations for semi-dilute and concentrated vesicle suspensions. We discuss some peculiar effects exhibited by the instantaneous viscosity in the tumbling regime of vesicles. Finally, the rheology is found to be relatively insensitive to shear rate.

Theory of ultrasonic attenuation in one and two dimensional metals

1976 ◽  
Vol 54 (14) ◽  
pp. 1454-1460 ◽  
Author(s):  
T. Tiedje ◽  
R. R. Haering
Keyword(s):  
Ultrasonic Attenuation ◽  
Three Dimensional ◽  
Electronic Systems ◽  
Two Dimensional ◽  
Temperature Dependent ◽  
One Dimensional ◽  
The Difference

The theory of ultrasonic attenuation in metals is extended so that it applies to quasi one and two dimensional electronic systems. It is shown that the attenuation in such systems differs significantly from the well-known results for three dimensional systems. The difference is particularly marked for one dimensional systems, for which the attenuation is shown to be strongly temperature dependent.

Perception of Risk as Related to Choice in a Two-Dimensional Risk Situation

1978 ◽  
Vol 43 (3_suppl) ◽  
pp. 1059-1062 ◽  
Author(s):  
John W. Dickson
Keyword(s):  
Risky Choice ◽  
Expected Value ◽  
Two Dimensions ◽  
Choice Situation ◽  
Two Dimensional ◽  
Objective Risk ◽  
Subjective Risk ◽  
Risk Situation ◽  
The Difference ◽  
Two Alternatives

A risky choice was created by manipulating two dimensions of risk for 21 managers attending a conference. The first dimension varied risk by altering the difference in expected value between two alternatives of widely differing variance. The second dimension varied the expectancy of achieving a particular outcome. Whereas choice was significantly related to both dimensions of risk, it was not significantly related to estimates of the subjective risk inherent in the choice situation. It appears that subjective risk does not mediate between objective risk and choice.

scholarly journals The Two Dimensional Representation of Three Dimensional Interferometer Measurements

1979 ◽  
Vol 49 ◽  
pp. 19-26
Author(s):  
R.H. Frater
Keyword(s):  
Three Dimensional ◽  
General Purpose ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Dimensional Representation ◽  
General Purpose Computer ◽  
Efficient Processing ◽  
Purpose Computer

SummaryA convolution technique for the reduction of three dimensional interferometer measurements to two dimensions is described. With the addition of relatively simple hardware to a general purpose computer the technique allows fast, efficient processing of three dimensional data.

scholarly journals Surface, size and topological effects for some nematic equilibria on rectangular domains

2020 ◽  
Vol 25 (5) ◽  
pp. 1101-1123 ◽  
Author(s):  
Lidong Fang ◽  
Apala Majumdar ◽  
Lei Zhang
Keyword(s):  
Aspect Ratio ◽  
Point Defects ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Variable Formula ◽  
Limiting Profile ◽  
Switching Dynamics ◽  
Line Defects ◽  
Carry Over

We study nematic equilibria on rectangular domains, in a reduced two-dimensional Landau–de Gennes framework. These reduced equilibria carry over to the three-dimensional framework at a special temperature. There is one essential model variable, [Formula: see text], which is a geometry-dependent and material-dependent variable. We compute the limiting profiles exactly in two distinguished limits: the [Formula: see text] 0 limit relevant for macroscopic domains and the [Formula: see text] limit relevant for nanoscale domains. The limiting profile has line defects near the shorter edges in the [Formula: see text] limit, whereas we observe fractional point defects in the [Formula: see text] 0 limit. The analytical studies are complemented by some bifurcation diagrams for these reduced equilibria as a function of [Formula: see text] and the rectangular aspect ratio. We also introduce the concept of ‘non-trivial’ topologies and study the relaxation of non-trivial topologies to trivial topologies mediated via point and line defects, with potential consequences for non-equilibrium phenomena and switching dynamics.

scholarly journals Effects of morphology in controlling propagation of density currents in a reservoir using uncalibrated three-dimensional hydrodynamic modeling

2020 ◽  
Author(s):  
Behnam Zamani ◽  
Manfred Koch ◽  
Ben R. Hodges
Keyword(s):  
Hydrodynamic Model ◽  
Large Scale ◽  
Three Dimensional ◽  
Density Current ◽  
Density Currents ◽  
Large Reservoir ◽  
Bottom Water Temperature ◽  
The Difference ◽  
Basin Morphology ◽  
Southwest Iran

In this study, effects of basin morphology are shown to affect density current hydrodynamics of a large reservoir using a three-dimensional (3D) hydrodynamic model that is validated (but not calibrated) with in situ observational data. The AEM3D hydrodynamic model was applied for 5-month simulations during winter and spring flooding for the Maroon reservoir in southwest Iran, where available observations indicated that large-scale density currents had previously occurred. The model results were validated with near-bottom water temperature measurements that were previously collected at five locations in the reservoir. The Maroon reservoir consists of upper and lower basins that are connected by a deep and narrow canyon. Analyses of simulations show that the canyon strongly affects density current propagation and the resulting differing limnological characteristics of the two basins. The evolution of the Wedderburn Number, Lake Number, and Schmidt stability number are shown to be different in the two basins, and the difference is attributable to the morphological separation by the canyon. Investigation of the background potential energy (BPE) changes along the length of the canyon indicated that a density front passes through the upper section of the canyon but is smoothed into simple filling of the lower basin. The separable dynamics of the basins has implications for the complexity of models needed for representing both water quality and sedimentation.

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