Dept. of Theoretical and Applied Mechanics (1926-2006)
http://hdl.handle.net/2142/178
Buckling and post-buckling behavior of a cylindrical shell subjected to external pressure
http://hdl.handle.net/2142/18754
Buckling and post-buckling behavior of a cylindrical shell subjected to external pressure
Langhaar, Henry Louis, 1909-
Boresi, Arthur Peter, 1924-
In an earlier report (TAM Report No. 80), the authors considered the buckling and post-buckling behavior of an ideal elastic cylindrical shell loaded by uniform external pressure on its lateral surface, and by an axial compressive force. Assumptions were introduced which reduced the shell to a system with one degree of freedom. The present investigation is a generalization and a refinement of this theory. The shell is treated as a system with 21 degrees of freedom. By the imposition of constraints on the 21 generalized coordinates, various end conditions can be realized; for example, simply supported ends with flexible end
plates (no axial constraint), simply supported ends with rigid end plates, and
clamped ends. Also, effects of reinforcing rings have been incorporated in a more
general way than in TAM Report No. 80. The restrictive assumption that the centroidal axis of a ring coincides with the middle surface of the shell has been eliminated.
A pressure-deflection curve for an ideal cylindrical shell that is loaded by external pressure has the general form shown in Figure 1. The falling part of the curve (dotted in the figure) represents unstable equilibrium
configurations. Also, the continuation of line OE (dotted) represents unstable
unbuckled configurations. Actually, the shell snaps from some configuration A to another configuration B, as indicated by the dashed line in Figure 1. Theoretically, point A coincides with the maximum point E on lhe curve, but initial imperfections and accidental disturbances prevent the shell from reaching this point. Point E is the buckling pressure of the classical infinitesimal theory (called the "Euler crítical pressure", since Euler applied the
infinitesimal theory to columns). To some extent, point A is indeterminate, but it is presumably higher than the minimum point C unless the shell has excessive initial dents or lopsidedness. In TAM Report No. 80. a
hypothesis of Tsien was used to locate point A. In the present investigation, point A is not considered. Rather, attention is focused on the development of a theory that will determine the en tire load-deflection curve.
For short thick shells, such as the inter-ring bays of a submarine hull, the Euler critical pressures, determined by TAM Report No. 80, are too high, presumably because the assumption that the shell buckles without
incremental hoop strain is inadmissible in this range. The present report corrects this error. Numerical data on the Euler critical pressures of shells with simply supported ends and flexible end plates have been obtained with the aid of lhe Illiac, an electronic digital computer. The data are tabulated at the end of this reporto For short shells without rings, the buckling pressures are appreciably lower than those determined by von Mises' theory. The
numerical data for the Euler buckling pressures of sheUs with uniformly spaced
reinforcing rings are sufficiently extensive to permit interpolation to estimate effects of various ring sizes. Some exploratory numerical investigations of post-buckling behavior have been conducted with the Illiac. lt is not feasible, at the present time, to handle nonlinear equilibrium problems for systems with 18 degrees of freedom. Consequently, for the numerical work, some higher harmonics were discarded so that the system was reduaed to 7 degrees of freedom.
Even then, the numerical problem is formidable. The calculations were confined principally to the determination of the minimum point C on the post-buckling curve (Figure 1). The pressure at point C is the minimum pressure at which a buckled form can exist. It is found that the ordinate of point C, determined by TAM report No. 80, is somewhat too high. The two theories are compared by a table anq curves at the end of this report.
Buckling behavior
Sun, 01 Apr 1956 00:00:00 GMTSpatial signatures of retrograde spanwise vortices in wall turbulence
http://hdl.handle.net/2142/1712
Spatial signatures of retrograde spanwise vortices in wall turbulence
Natrajan, Vinay K.
Wu, Yanhua
Christensen, Kenneth T.
The spatial signatures of retrograde spanwise vortices in wall turbulence are assessed
from particle-image velocimetry measurements in the streamwise–wall-normal plane
of a zero-pressure-gradient turbulent boundary layer at Reτ ≡u∗δ/ν =2350. The
present results suggest that a proportion of retrograde spanwise vortices have a welldefined
spatial relationship with neighbouring prograde vortices. Two-point crosscorrelations
and conditionally averaged velocity fields given a retrograde vortex
reveal that such structures are typically oriented either upstream of and below or
downstream of and above a prograde core. While these pairings are consistent with
the typical-eddy patterns reported by Falco and co-workers, we offer an alternative
interpretation for a proportion of these retrograde/prograde pairs. In particular, the
arrangement of a retrograde spanwise vortex upstream of and below a prograde
core is also consistent with the spatial signature revealed if an omega-shaped hairpin
structure were sliced through its shoulder region by a fixed streamwise–wall-normal
measurement plane.
Vortices
Wall turbulence
Mon, 01 Jan 2007 00:00:00 GMTPopulation trends of spanwise vortices in wall turbulence
http://hdl.handle.net/2142/1711
Population trends of spanwise vortices in wall turbulence
Wu, Yanhua
Christensen, Kenneth T.
The present effort documents the population trends of prograde and retrograde
spanwise vortex cores in wall turbulence outside the buffer layer. Large ensembles
of instantaneous velocity fields are acquired by particle-image velocimetry in the
streamwise–wall-normal plane of both turbulent channel flow at Reτ ≡ u∗δ/ν = 570,
1185 and 1760 and a zero-pressure-gradient turbulent boundary layer at Reτ = 1400,
2350 and 3450. Substantial numbers of prograde spanwise vortices are found to
populate the inner boundary of the log layer of both flows and most of these vortices
have structural signatures consistent with the heads of hairpin vortices. In contrast,
retrograde vortices are most prominent at the outer edge of the log layer, often
nesting near clusters of prograde vortices. Appropriate Reynolds-number scalings for
outer- and inner-scaled population densities of prograde and retrograde vortices are
determined. However, the Reτ = 570 channel-flow case deviates from these scalings,
indicating that it suffers from low-Re effects. When the population densities are
recast in terms of fractions of resolved prograde and retrograde spanwise vortices,
similarity is observed for 100<y+ <0.8δ+ in channel flow and in both flows for
100<y+ <0.3δ+ over the Reτ range studied. The fraction of retrograde vortices
increases slightly with Reτ beyond the log layer in both flows, suggesting that they may
play an increasingly important role at higher Reynolds numbers. Finally, while the
overall prograde and retrograde population trends of channel flow and the boundary
layer show little difference for y <0.45δ, the retrograde populations differ considerably
beyond this point, highlighting the influence of the opposing wall in channel
flow.
Wall turbulence
Vortices
Sun, 01 Jan 2006 00:00:00 GMTNonlinear oscillatory convection in rotating mushy layers
http://hdl.handle.net/2142/1710
Nonlinear oscillatory convection in rotating mushy layers
Riahi, Daniel N.
We consider the problem of nonlinear oscillatory convection in a horizontal mushy
layer rotating about a vertical axis. Under a near-eutectic approximation and the
limit of large far-field temperature, we determine the stable and unstable oscillatory
solutions of the weakly nonlinear problem by using perturbation and stability analyses.
It was found that depending on the values of the parameters, supercritical simple
travelling modes of convection in the form of hexagons, squares, rectangles or rolls
can become stable and preferred, provided the value of the rotation parameter τ is
not too small and is below some value, which can depend on the other parameter
values. Each supercritical form of the oscillatory convection becomes subcritical as τ
increases beyond some value, and each subcritical form of the oscillatory convection
is unstable. In contrast to the non-rotating case, qualitative properties of the left travelling
modes of convection are different from those of the right-travelling modes,
and such qualitative difference is found to be due to the interactions between the
local solid fraction and the Coriolis term in the momentum-Darcy equation.
Convection
Sun, 01 Jan 2006 00:00:00 GMTOn nonlinear convection in mushy layers. Part 2: Mixed oscillatory and stationary modes of convection
http://hdl.handle.net/2142/988
On nonlinear convection in mushy layers. Part 2: Mixed oscillatory and stationary modes of convection
Riahi, Daniel N.
This paper presents part 2 of a study of nonlinear convection in horizontal mushy
layers during the solidification of binary alloys. Part 1 dealt with only the oscillatory
modes of convection (Riahi, J. Fluid Mech. vol. 467, 2002, pp. 331–359). In the present
paper we consider the particular range of parameters where the critical values of the
scaled Rayleigh number R for the onset of oscillatory and stationary convection are
close to each other, and we develop and analyse a nonlinear theory in such a parameter
regime which takes into account those mixed stationary and oscillatory modes of
convection with common wavenumber vectors. Under a near-eutectic approximation
and in the limit of large far-field temperature, we first determine a number of weakly
nonlinear solutions, and then the stability of these solutions is investigated. The most
interesting result is the preference for a mixed solution composed of standing and
stationary hexagonal modes over a relatively wide range of the parameter values and
for R just above its lowest subcritical value where convection is possible. Such a
preferred solution has properties mostly in agreement with the experimental results
due to Tait et al. (Nature, vol. 359, 1992, pp. 406–408) in the sense that the flow
is downward at the cell centres, upward at the cell boundaries and there is some
tendency for channel formation at the cell nodes.
Nonlinear convection
Thu, 01 Jan 2004 00:00:00 GMTOn nonlinear convection in mushy layers. Part 1: Oscillatory modes of convection. With Corrigendums
http://hdl.handle.net/2142/987
On nonlinear convection in mushy layers. Part 1: Oscillatory modes of convection. With Corrigendums
Riahi, Daniel N.
We consider the problem of nonlinear convection in horizontal mushy layers during
the solidification of binary alloys. We analyse the oscillatory modes of convection
in the form of two- and three-dimensional travelling and standing waves. Under a
near-eutectic approximation and the limit of large far-field temperature, we determine
the solutions to the nonlinear problem by using a perturbation technique, and the
stability of two- and three-dimensional solutions in the form of simple travelling
waves, general travelling waves and standing waves is investigated. The results of the
stability and the nonlinear analyses indicate that supercritical simple travelling rolls
are stable over most of the studied range of parameter values, while supercritical
standing rolls can be stable only over some small range of parameter values, where
the simple travelling rolls are unstable. The results of the investigation of the onset
of plume convection and chimney formation leading to the occurrence of freckles in
the alloy crystal indicate that the chimney of the plume can be generated internally
or near the lower boundary of the mushy layer. The roles of a Stefan number, a
permeability parameter and a concentration ratio on the flow instability in both linear
and nonlinear regimes are also determined.
Nonlinear convection
Tue, 01 Jan 2002 00:00:00 GMTWall-induced forces on a rigid sphere at finite Reynolds number
http://hdl.handle.net/2142/986
Wall-induced forces on a rigid sphere at finite Reynolds number
Zeng, Lanying
Balachandar, S.
Fischer, Paul
We perform direct numerical simulations of a rigid sphere translating parallel to
a flat wall in an otherwise quiescent ambient fluid. A spectral element method is
employed to perform the simulations with high accuracy. For Re<100, we observe
the lift coefficient to decrease with both Reynolds number and distance from the
wall. In this regime the present results are in good agreement with the low-Reynoldsnumber
theory of Vasseur & Cox (1977), with the recent experiments of Takemura &
Magnaudet (2003) and with the simulations of Kim et al. (1993). The most surprising
result from the present simulations is that the wall-induced lift coefficient increases
dramatically with increasing Re above about 100. Detailed analysis of the flow field
around the sphere suggests that this increase is due to an imperfect bifurcation
resulting in the formation of a double-threaded wake vortical structure. In addition
to a non-rotating sphere, we also simulate a freely rotating sphere in order to assess
the importance of free rotation on the translational motion of the sphere. We observe
the effect of sphere rotation on lift and drag forces to be small. We also explore the
effect of the wall on the onset of unsteadiness.
Fluid mechanics
Sat, 01 Jan 2005 00:00:00 GMTTwo-Phase Densification of Cohesive Granular Aggregates
http://hdl.handle.net/2142/985
Two-Phase Densification of Cohesive Granular Aggregates
Gioia, Gustavo
Cuitiño, Alberto M.
Zheng, S.
Uribe, T.
When poured into a container, cohesive granular materials form low-density, open granular aggregates. If pressure is applied to these aggregates, they densify by particle rearrangement. Here we introduce experimental and computational results suggesting that densification by particle rearrangement occurs in the form of a phase transition between two configurational phases of the aggregate. Then we show that the energy landscape associated with particle rearrangement is nonconvex and therefore consistent with our interpretation of the experimental and computational results. Our conclusions are relevant to many technological processes and natural phenomena.
Granular materials
Tue, 01 Jan 2002 00:00:00 GMTTurbulent friction in rough pipes and the energy spectrum of the phenomenological theory
http://hdl.handle.net/2142/984
Turbulent friction in rough pipes and the energy spectrum of the phenomenological theory
Gioia, Gustavo
Chakraborty, Pinaki
The classical experiments on turbulent friction in rough pipes were performed by Nikuradse in the
1930s. Seventy years later, they continue to defy theory. Here we model Nikuradse’s experiments using
the phenomenological theory of Kolmogo´rov, a theory that is widely thought to be applicable only to
highly idealized flows. Our results include both the empirical scalings of Blasius and Strickler and are
otherwise in minute qualitative agreement with the experiments; they suggest that the phenomenological
theory may be relevant to other flows of practical interest; and they unveil the existence of close ties
between two milestones of experimental and theoretical turbulence.
Turbulent flow
Turbulence theory
Sun, 01 Jan 2006 00:00:00 GMTTheory of detonation with an embedded sonic locus
http://hdl.handle.net/2142/983
Theory of detonation with an embedded sonic locus
Stewart, D. Scott
Kasimov, Aslan R.
A steady planar self-sustained detonation has a sonic surface in the reaction zone that
resides behind the lead shock. In this work we address the problem of generalizing sonic conditions
for a three-dimensional unsteady self-sustained detonation wave. The conditions are proposed to
be the characteristic compatibility conditions on the exceptional surface of the governing hyperbolic
system of reactive Euler equations. Two equations are derived that are necessary to determine the
motion of both the lead shock and the sonic surface. Detonation with an embedded sonic locus is
thus treated as a two-front phenomenon: a reaction zone whose domain of influence is bounded by
two surfaces, the lead shock surface and the trailing characteristic surface. The geometry of the
two surfaces plays an important role in the underlying dynamics. We also discuss how the sonic
conditions of detonation stability theory and detonation shock dynamics can be obtained as special
cases of the general sonic conditions.
chemically reacting flows
shocks and singularities
supersonic flows
transonic flows
Sat, 01 Jan 2005 00:00:00 GMT