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1.
The Darcy Model with the Boussinesq approximation is used to study natural convection in a shallow porous layer, with variable
permeability, filled with a binary fluid. The permeability of the medium is assumed to vary exponentially with the depth of
the layer. The two horizontal walls of the cavity are subject to constant fluxes of heat and solute while the two vertical
ones are impermeable and adiabatic. The governing parameters for the problem are the thermal Rayleigh number, R
T, the Lewis number, Le, the buoyancy ratio, φ, the aspect ratio of the cavity, A, the normalized porosity, ε, the variable permeability constant, c, and parameter a defining double-diffusive convection (a = 0) or Soret induced convection (a = 1). For convection in an infinite layer, an analytical solution of the steady form of the governing equations is obtained
on the basis of the parallel flow approximation. The onset of supercritical convection, or subcritical, convection are predicted by the present theory. A linear stability analysis of the parallel flow model is conducted and the
critical Rayleigh number for the onset of Hopf’s bifurcation is predicted numerically. Numerical solutions of the full governing
equations are found to be in excellent agreement with the analytical predictions. 相似文献
2.
The effect of canopy roughness density on the constitutive components of the dispersive stresses 总被引:1,自引:0,他引:1
How to represent the effects of variable canopy morphology on turbulence remains a fundamental challenge yet to be confronted.
Planar averaging over some minimal area can be applied to average-out this sort of spatial variability in the time-averaged
mean momentum balance. Because of the multiply connected air-spaces, spatial averaging gives rise to covariance or dispersive
stress terms that are produced by the spatial correlations of the time-averaged quantities. These terms are “unclosed” and
require parameterization, which to date remains lacking due to the absence of data. Here, flume experiments were conducted
to quantify the magnitude and sign of the dispersive stresses for a cylindric canopy where the rod density was varied but
the individual rod dimensions (rod height h
c
and rod diameter d
r) remained the same. Quadrant analysis was used to explore the genesis of their spatial coherency inside the canopy for a
wide range of rod densities. When compared to the conventional turbulent stresses, these dispersive stresses can be significant
in the lowest layers of sparse canopies. For dense canopies, the dispersive terms remain negligible when compared to the conventional
momentum fluxes at all the canopy levels consistent with previous experiments in vegetated and urban canopies. It was also
shown that the spatial locations contributing most to the dispersive terms were in the immediate vicinity downstream of the
rods. In the deeper layers of sparse canopies, these positions contributed large and negative stresses, but in the upper levels
of the canopy, they contributed large but positive stresses. Because the longitudinal velocity spatial perturbation behind the rods is negative, the switch in sign in these stresses was connected with the sign of the vertical velocity spatial
perturbation Simplified scaling arguments, using a reduced mean continuity equation and the vertical mean momentum balance for the flow
field near the rods, offer clues as to why in much of the lower canopy levels (about 0.75 h
c
) while in the upper canopy levels. 相似文献
3.
4.
Forced convective heat transfer coefficients and friction factors for flow of water in microchannels with a rectangular cross
section were measured. An integrated microsystem consisting of five microchannels on one side and a localized heater and seven
polysilicon temperature sensors along the selected channels on the other side was fabricated using a double-polished-prime
silicon wafer. For the microchannels tested, the friction factor constant
obtained are values between 53.7 and 60.4, which are close to the theoretical value from a correlation for macroscopic dimension,
56.9 for D
h
= 100 μm. The heat transfer coefficients obtained by measuring the wall temperature along the micro channels were linearly
dependent on the wall temperature, in turn, the heat transfer mechanism is strongly dependent on the fluid properties such
as viscosity. The measured Nusselt number in the laminar flow regime tested could be correlated by which is quite different from the constant value obtained in macrochannels. 相似文献
5.
Momentum and heat transfer from an asymmetrically confined circular cylinder in a plane channel 总被引:2,自引:0,他引:2
Unsteady momentum and heat transfer from an asymmetrically confined circular cylinder in a plane channel is numerically investigated using FLUENT for the ranges of Reynolds numbers as 10≤Re≤500, of the blockage ratio as 0.1≤β≤0.4, and of the gap ratio as 0.125≤γ≤1 for a constant value of the Prandtl number of 0.744. The transition of the flow from steady to unsteady (characterized by critical Re) is determined as a function of γ and β. The effect of γ on the mean drag
and lift
coefficients, Strouhal number (St), and Nusselt number (Nu
w
) is studied. Critical Re was found to increase with decreasing γ for all values of β.
and St were found to increase with decreasing values of γ for fixed β and Re. The effect of decrease in γ on
was found to be negligible for all blockage ratios investigated. 相似文献
6.
The longitudinal steady-state control for going from hovering to small speed flight of a model insect is studied, using the method of computational fluid dynamics to compute the aerodynamic derivatives and the techniques based on the linear theories of stability and control for determining the non-zero equilibrium points. Morphological and certain kinematical data of droneflies are used for the model insect. A change in the mean stroke angle (δФ) results in a horizontal forward or backward flight; a change in the stroke amplitude (δФ) or a equal change in the down- and upstroke angles of attack (δα1) results in a vertical climb or decent; a proper combination of δФ and δФ controls (or δФ and δα1 controls) can give a flight of any (small) speed in any desired direction. 相似文献
7.
Joel Avrin 《Journal of Dynamics and Differential Equations》2008,20(2):479-518
We obtain attractor and inertial-manifold results for a class of 3D turbulent flow models on a periodic spatial domain in
which hyperviscous terms are added spectrally to the standard incompressible Navier–Stokes equations (NSE). Let P
m
be the projection onto the first m eigenspaces of A =−Δ, let μ and α be positive constants with α ≥3/2, and let Q
m
=I − P
m
, then we add to the NSE operators μ A
φ in a general family such that A
φ≥Q
m
A
α in the sense of quadratic forms. The models are motivated by characteristics of spectral eddy-viscosity (SEV) and spectral
vanishing viscosity (SVV) models. A distinguished class of our models adds extra hyperviscosity terms only to high wavenumbers
past a cutoff λ
m0
where m
0 ≤ m, so that for large enough m
0 the inertial-range wavenumbers see only standard NSE viscosity.
We first obtain estimates on the Hausdorff and fractal dimensions of the attractor (respectively and ). For a constant K
α on the order of unity we show if μ ≥ ν that and if μ ≤ ν that where ν is the standard viscosity coefficient, l
0 = λ1−1/2 represents characteristic macroscopic length, and is the Kolmogorov length scale, i.e. where is Kolmogorov’s mean rate of dissipation of energy in turbulent flow. All bracketed constants and K
α are dimensionless and scale-invariant. The estimate grows in m due to the term λ
m
/λ1 but at a rate lower than m
3/5, and the estimate grows in μ as the relative size of ν to μ. The exponent on is significantly less than the Landau–Lifschitz predicted value of 3. If we impose the condition , the estimates become for μ ≥ ν and for μ ≤ ν. This result holds independently of α, with K
α and c
α independent of m. In an SVV example μ ≥ ν, and for μ ≤ ν aspects of SEV theory and observation suggest setting for 1/c within α orders of magnitude of unity, giving the estimate where c
α is within an order of magnitude of unity. These choices give straight-up or nearly straight-up agreement with the Landau–Lifschitz
predictions for the number of degrees of freedom in 3D turbulent flow with m so large that (e.g. in the distinguished-class case for m
0 large enough) we would expect our solutions to be very good if not virtually indistinguishable approximants to standard NSE
solutions. We would expect lower choices of λ
m
(e.g. with a > 1) to still give good NSE approximation with lower powers on l
0/l
ε, showing the potential of the model to reduce the number of degrees of freedom needed in practical simulations. For the choice
, motivated by the Chapman–Enskog expansion in the case m = 0, the condition becomes , giving agreement with Landau–Lifschitz for smaller values of λ
m
then as above but still large enough to suggest good NSE approximation. Our final results establish the existence of a inertial
manifold for reasonably wide classes of the above models using the Foias/Sell/Temam theory. The first of these results obtains such
an of dimension N > m for the general class of operators A
φ if α > 5/2.
The special class of A
φ such that P
m
A
φ = 0 and Q
m
A
φ ≥ Q
m
A
α has a unique spectral-gap property which we can use whenever α ≥ 3/2 to show that we have an inertial manifold of dimension m if m is large enough. As a corollary, for most of the cases of the operators A
φ in the distinguished-class case that we expect will be typically used in practice we also obtain an , now of dimension m
0 for m
0 large enough, though under conditions requiring generally larger m
0 than the m in the special class. In both cases, for large enough m (respectively m
0), we have an inertial manifold for a system in which the inertial range essentially behaves according to standard NSE physics,
and in particular trajectories on are controlled by essentially NSE dynamics.
相似文献
8.
Kalabin et al. (Numer. Heat Transfer A 47, 621-631, 2005) studied the unsteady natural convection for the sinusoidal oscillating wall temperature on one side wall
and constant average temperature on the opposing side wall. The present article is on the unsteady natural convective heat
transfer in an inclined porous cavity with similar temperature boundary conditions as those of Kalabin et al. The inclined
angle of the cavity is varied from 0° to 80°. The flow field is modeled with the Brinkman-extended Darcy model. The combined effects
of inclination angle of the enclosure and oscillation frequency of wall temperature are studied for Ra* = 103, Da = 10−3, , and Pr=1. Some results are also obtained with the Darcy–Brinkman–Forchheimer model and Darcy’s law and are compared with the present
Brinkman-extended Darcy model. The maximal heat transfer rate is attained at the oscillating frequency f = 46.7π and the inclined angle . 相似文献
9.
We study the dynamics and regularity of level sets in solutions of the semilinear parabolic equation
where is a ring-shaped domain, a and μ are given positive constants, is the Heaviside maximal monotone graph: if s > 0, if s < 0. Such equations arise in climatology (the so-called Budyko energy balance model), as well as in other contexts such as
combustion. We show that under certain conditions on the initial data the level sets are n-dimensional hypersurfaces in the (x, t)-space and show that the dynamics of Γ
μ
is governed by a differential equation which generalizes the classical Darcy law in filtration theory. This differential
equation expresses the velocity of advancement of the level surface Γ
μ
through spatial derivatives of the solution u. Our approach is based on the introduction of a local set of Lagrangian coordinates: the equation is formally considered
as the mass balance law in the motion of a fluid and the passage to Lagrangian coordinates allows us to watch the trajectory
of each of the fluid particles. 相似文献
10.
An analysis is presented for the unsteady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric spheres rotating about a common axis of symmetry. A solution of the Navier-Stokes equations is obtained by employing an iterative technique. The solution is valid for small values of Reynolds numbers and acceleration parameters of the spheres. In applying the results of this analysis to a rotationally accelerating sphere, a virtual moment of intertia is introduced to account for the local inertia of the fluid.Nomenclature
R
i
radius of the inner sphere
-
R
o
radius of the outer sphere
-
radial coordinate
-
r
dimensionless radial coordinate,
-
meridional coordinate
-
azimuthal coordinate
-
time
-
t
dimensionless time,
- Re
i
instantaneous Reynolds number of the inner sphere,
i
R
k
2
/
- Re
o
instantaneous Reynolds number of the outer sphere,
o
R
o
2
/
-
radial velocity component
-
V
r
dimensionless radial velocity component,
-
meridional velocity component
- V
dimensionless meridional velocity component,
-
azimuthal velocity component
-
V
dimensionless azimuthal velocity component,
-
viscous torque
-
T
dimensionless viscous torque,
-
viscous torque at surface of inner sphere
-
T
i
dimensionless viscous torque at surface of inner sphere,
-
viscous torque at surface of outer sphere
-
T
o
dimensionless viscous torque at surface of outer sphere,
-
externally applied torque on inner sphere
-
T
p,i
dimensionless applied torque on inner sphere,
-
moment of inertia of inner sphere
-
Z
i
dimensionless moment of inertia of inner sphere,
-
virtual moment of inertia of inner sphere
-
Z
i,v
dimensionless virtual moment of inertia of inner sphere,
-
virtual moment of inertia of outer sphere
-
i
instantaneous angular velocity of the inner sphere
-
o
instantaneous angular velocity of the outer sphere
-
density of fluid
-
viscosity of fluid
-
kinematic viscosity of fluid,/
-
radius ratio,R
i/R
o
-
swirl function,
-
dimensionless swirl function,
-
stream function
-
dimensionless stream function,
-
i
acceleration parameter for the inner sphere,
-
o
acceleration parameter for the outer sphere,
-
shear stress
-
r
dimensionless shear stress,
相似文献
11.
Thierry Cazenave Flávio Dickstein Fred B. Weissler 《Journal of Dynamics and Differential Equations》2007,19(3):789-818
In this paper, we construct solutions u(t,x) of the heat equation on such that has nontrivial limit points in as t → ∞ for certain values of μ > 0 and β > 1/2. We also show the existence of solutions of this type for nonlinear heat equations.
相似文献
12.
F. Javadpour 《Transport in Porous Media》2009,79(1):87-105
Carbon dioxide (CO2) injections in geological formations are usually performed for enhanced hydrocarbon recovery in oil and gas reservoirs and
storage and sequestration in saline aquifers. Once CO2 is injected into the formation, it propagates in the porous rock by dispersion and convection. Chemical reactions between
brine ions and CO2 molecules and consequent reactions with mineral grains are also important processes. The dynamics of CO2 molecules in random porous media are modeled with a set of differential equations corresponding to pore scale and continuum
macroscale. On the pore scale, convective–dispersive equation is solved considering reactions on the inner boundaries in a
unit cell. A unit cell is the smallest portion of a porous media that can reproduce the porous media by repetition. Inner
boundaries in a unit cell are the surfaces of the mineral grains. Dispersion process at the pore scale is transformed into
continuum macroscale by adopting periodic boundary conditions for contiguous unit cells and applying Taylor-Aris dispersion
theory known as macrotransport theory. Using this theory, the discrete porous system changes into a continuum system within
which the propagation and interaction of CO2 molecules with fluid and solid matrix of the porous media are characterized by three position-independent macroscopic coefficients:
the mean velocity vector , dispersivity dyadic , and mean volumetric CO2 depletion coefficient . 相似文献
13.
Crack Initiation in Brittle Materials 总被引:1,自引:0,他引:1
Antonin Chambolle Alessandro Giacomini Marcello Ponsiglione 《Archive for Rational Mechanics and Analysis》2008,188(2):309-349
In this paper we study the crack initiation in a hyper-elastic body governed by a Griffith-type energy. We prove that, during
a load process through a time-dependent boundary datum of the type t → t
g(x) and in the absence of strong singularities (e.g., this is the case of homogeneous isotropic materials) the crack initiation
is brutal, that is, a big crack appears after a positive time t
i
> 0. Conversely, in the presence of a point x of strong singularity, a crack will depart from x at the initial time of loading and with zero velocity. We prove these facts for admissible cracks belonging to the large
class of closed one-dimensional sets with a finite number of connected components. The main tool we employ to address the
problem is a local minimality result for the functional where , k > 0 and f is a suitable Carathéodory function. We prove that if the uncracked configuration u of Ω relative to a boundary displacement ψ has at most uniformly weak singularities, then configurations (uΓ, Γ) with small enough are such that . 相似文献
14.
Philippe G. Ciarlet Liliana Gratie Cristinel Mardare 《Archive for Rational Mechanics and Analysis》2008,188(3):457-473
The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a
αβ
) of order two and a field of symmetric matrices (b
αβ
) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a simply connected open subset ω of , then there exists an immersion such that these fields are the first and second fundamental forms of the surface , and this surface is unique up to proper isometries in . The main purpose of this paper is to identify new compatibility conditions, expressed again in terms of the functions a
αβ
and b
αβ
, that likewise lead to a similar existence and uniqueness theorem. These conditions take the form of the matrix equation
where A
1 and A
2 are antisymmetric matrix fields of order three that are functions of the fields (a
αβ
) and (b
αβ
), the field (a
αβ
) appearing in particular through the square root U of the matrix field The main novelty in the proof of existence then lies in an explicit use of the rotation field R that appears in the polar factorization of the restriction to the unknown surface of the gradient of the canonical three-dimensional extension of the unknown immersion . In this sense, the present approach is more “geometrical” than the classical one. As in the recent extension of the fundamental
theorem of surface theory set out by S. Mardare [20–22], the unknown immersion is found in the present approach to exist in function spaces “with little regularity”, such as , p > 2. This work also constitutes a first step towards the mathematical justification of models for nonlinearly elastic shells
where rotation fields are introduced as bona fide unknowns. 相似文献
15.
Two-Phase Inertial Flow in Homogeneous Porous Media: A Theoretical Derivation of a Macroscopic Model
The purpose of this article is to derive a macroscopic model for a certain class of inertial two-phase, incompressible, Newtonian
fluid flow through homogenous porous media. Starting from the continuity and Navier–Stokes equations in each phase β and γ, the method of volume averaging is employed subjected to constraints that are explicitly provided to obtain the macroscopic
mass and momentum balance equations. These constraints are on the length- and time-scales, as well as, on some quantities
involving capillary, Weber and Reynolds numbers that define the class of two-phase flow under consideration. The resulting
macroscopic momentum equation relates the phase-averaged pressure gradient to the filtration or Darcy velocity in a coupled nonlinear form explicitly given by
or equivalently
In these equations, and are the inertial and coupling inertial correction tensors that are functions of flow-rates. The dominant and coupling permeability tensors and and the permeability and viscous drag tensors and are intrinsic and are those defined in the conventional manner as in (Whitaker, Chem Eng Sci 49:765–780, 1994) and (Lasseux
et al., Transport Porous Media 24(1):107–137, 1996). All these tensors can be determined from closure problems that are to
be solved using a spatially periodic model of a porous medium. The practical procedure to compute these tensors is provided. 相似文献
16.
Dr. M. Dow Dr. H. Nakamura Prof. Dr. G. I. N. Rozvany 《Archive of Applied Mechanics (Ingenieur Archiv)》1982,52(5):335-353
Summary The first part of this paper is concerned with the optimal design of spherical cupolas obeying the von Mises yield condition. Five different load combinations, which all include selfweight, are investigated. The second part of the paper deals with the optimal quadratic meridional shape of cupolas obeying the Tresca yield condition, considering selfweight plus the weight of a non-carrying uniform cover. It is established that at long spans some non-spherical Tresca cupolas are much more economical than spherical ones.
List of Symbols ak, bk, ck, Ak, Bk, Ck coefficients used in series solutions - A, B constants in the nondimensional equation of the meridional curve - normal component of the load per unit area of the middle surface - meridional and circumferential forces per unit width - radial pressure per unit area of the middle surface, - skin weight per unit area of the middle surface, - vertical external load per unit horizontal area, - base radius, - R radius of convergence - s - cupola thickness, - u, w subsidiary functions for quadratic cupolas - vertical component of the load per unit area of middle surface - resultant vertical force on a cupola segment - structural weight of cupola, - combined weight of cupola and skin, - distance from the axis of rotation, - vertical distance from the shell apex, - z auxiliary variable in series solutions - specific weight of structural material of cupola - radius of the middle surface, - uniaxial yield stress - meridional stress, - circumferential stress, - a, b, c, d, e subsidiary variables used in evaluating the meridional stress - auxiliary function used in series solutions This paper constitutes the third part of a study of shell optimization which was initiated and planned by the late Prof. W. Prager 相似文献
Optimale Kuppeln gleicher Festigkeit: Kugelschalen und axialsymmetrische Schalen
Übersicht Im ersten Teil dieser Arbeit wird der optimale Entwurf sphärischer Kuppeln behandelt, wobei die von Misessche Fließbewegung zugrunde gelegt wird. Fünf verschiedene Lastkombinationen werden untersucht. Der zweite Teil befaßt sich mit der optimalen quadratischen Form des Meridians von Kuppeln, die der Fließbedingung von Tresca folgen.
List of Symbols ak, bk, ck, Ak, Bk, Ck coefficients used in series solutions - A, B constants in the nondimensional equation of the meridional curve - normal component of the load per unit area of the middle surface - meridional and circumferential forces per unit width - radial pressure per unit area of the middle surface, - skin weight per unit area of the middle surface, - vertical external load per unit horizontal area, - base radius, - R radius of convergence - s - cupola thickness, - u, w subsidiary functions for quadratic cupolas - vertical component of the load per unit area of middle surface - resultant vertical force on a cupola segment - structural weight of cupola, - combined weight of cupola and skin, - distance from the axis of rotation, - vertical distance from the shell apex, - z auxiliary variable in series solutions - specific weight of structural material of cupola - radius of the middle surface, - uniaxial yield stress - meridional stress, - circumferential stress, - a, b, c, d, e subsidiary variables used in evaluating the meridional stress - auxiliary function used in series solutions This paper constitutes the third part of a study of shell optimization which was initiated and planned by the late Prof. W. Prager 相似文献
17.
18.
19.
When a porous melt layer saturated by liquid is solidified from above, convection often sets in due to buoyancy forces. In
this study, the onset of buoyancy-driven convection during time-dependent solidification is investigated by using the similarly
transformed disturbance equations. The thermal disturbance distribution of the solid phase is approximated by the WKB method
and effects of various parameters on the stability condition of the melt phase are analyzed theoretically. For the limiting
case of λ → 0 and finite k
r, the critical conditions approach asymptotically and . This study presenting a constant-temperature cooling model predicts greater instability and gives more unstable results
than those obtained from the constant solidification rate model. 相似文献
20.
The work presented is a wind tunnel study of the near wake region behind a hemisphere immersed in three different turbulent boundary layers. In particular, the effect of different boundary layer profiles on the generation and distribution of near wake vorticity and on the mean recirculation region is examined. Visualization of the flow around a hemisphere has been undertaken, using models in a water channel, in order to obtain qualitative information concerning the wake structure.List of symbols
C
p
pressure coefficient,
-
D
diameter of hemisphere
-
n
vortex shedding frequency
-
p
pressure on model surface
-
p
0
static pressure
-
Re
Reynolds number,
-
St
Strouhal number,
-
U, V, W
local mean velocity components
-
mean freestream velocity inX direction
-
U
*
shear velocity,
-
u, v, w
velocity fluctuations inX, Y andZ directions
-
X
Cartesian coordinate in longitudinal direction
-
Y
Cartesian coordinate in lateral direction
-
Z
Cartesian coordinate in direction perpendicular to the wall
- it*
boundary layer displacement thickness,
-
diameter of model surface roughness
-
elevation angleI
-
O
boundary layer momentum thickness,
-
w
wall shearing stress
-
dynamic viscosity of fluid
-
density of fluid
-
streamfunction
- x
longitudinal component of vorticity,
- y
lateral component of vorticity,
-
z
vertical component of vorticity,
This paper was presented at the Ninth symposium on turbulence, University of Missouri-Rolla, October 1–3, 1984 相似文献