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Notes: Abstract Interspecific competitive interactions among sessile epibenthos were studied by suspending PVC panels at Tomioka Bay, south Japan, for a maximum period of 16 mo. Interactions were monitored from photographs of a fixed area of the panels. Four panels were suspended during two different months in autumn 1991, and the development of the community was followed until December 1992. Altogether, 6511 interspecific overgrowth interactions were recorded, of which 37 resulted in standoffs and the rest in overgrowths. The competitive relationship observed in this sessile assemblage followed the pattern of a hierarchy with numerous backloops. Among the 36 species, belonging to the seven taxonomic groups encountered during the study, the colonial ascidian Didemnum moseleyi was recorded as the dominant species (with respect to competitive ability) while the barnacle Balanus trigonus was the weakest species. The month of panel exposure and whether or not the panel surface was shaded had a significant influence on the competitive ability of the sessile organisms. The order of hierarchy of the most dominant species changed with the month of panel submersion and its light conditions. Among the several abundant species tested, longer residence times were recorded for serpulid worms than for the colonial species. A significant, positive relationship was obtained between the areal cover of competitively dominant sessile organisms and the number of their interspecific interactions. From the short residence time of sessile organisms and the significant relationship between their areal cover and number of interspecific interactions, it is concluded that the interspecific interactions played important role in the species succession.

Notes: In the present work the investigation of Weinitschke et al. [Phys. Fluids A 2, 912 (1990)] on the bifurcation structure of stationary, two-dimensional solutions of the Darcy–Oberbeck–Boussinesq model equations, which governs the convection heat transfer in a porous medium, is extended. The effect of imposing a symmetry breaking geometrical perturbation, viz., a tilt, on the unfolding of the bifurcation structure, is investigated first. The symmetry breaking bifurcation points that are found at zero tilt are structurally unstable to even a slight degree of tilt, and they unfold into limit points that coalesce with the neighboring limit points as the degree of tilt is increased. Two such limit points disappear through the formation of a double limit point at very small angles of tilt. On the fold curve of such limit points are found origins of paths of Hopf points, also known as the B-point singularity. Several such B points are located on the fold curves of limit points. This is helpful in pointing to regions of parameter space, where interesting dynamical behavior with oscillatory and chaotic flow structures are possible. The dynamical behavior is explored through simulation of the governing time-dependent equations after suitable spatial discretization through the Arakawa scheme. Linear stability analysis of stationary solutions indicate that such solutions remain stable over an increasing range of Rayleigh number (Ra) as the degree of tilt is increased. The first limit point (L1), below which there is a unique solution, tends toward infinite Rayleigh number as the tilt approaches 45°. At 45° tilt, a reflective symmetry about the diagonal of a square cell is restored and the stationary solutions computed by the continuation method remain stable for Ra as high as 200 000. Results from the dynamic simulation confirm this behavior. For tilts of 5° and 10° stationary solutions lose stability at Rayleigh numbers in the range of 4000–5000. Periodic solutions interspersed with regions of chaotic behavior are observed as Ra is increased continuously. For 20°, however, periodic solutions begin only above Ra≈10 000.

Notes: The bifurcation structure of two-dimensional, pressure-driven flows through a rectangular duct that is rotating about an axis perpendicular to its own is examined at a fixed Ekman number (Ek=ν/b2Ω) of 0.01. The solution structure for flow through a square duct (aspect ratio γ=1) is determined for Rossby numbers (Ro=U/bΩ) in the range of 0–5 using a computational scheme based on the arclength continuation method. The structure is much more complicated than reported earlier by Kheshgi and Scriven [Phys. Fluids 28, 2968 (1985)]. The primary branch with two limit points in Rossby number and a hysteresis behavior between the two- and four-cell flow structure that was computed by Kheshgi and Scriven is confirmed. An additional symmetric solution branch, which is disconnected from the primary branch (or rather connected via an asymmetric solution branch), is found. This has a two-cell flow structure at one end, a four-cell flow structure at the other and three limit points are located on the path. Two asymmetric solution branches emanating from symmetry breaking bifurcation points are also found for a square duct. Thus even within a Rossby number range of 0–5 a much richer solutions structure is found with up to five solutions at Ro=5. An eigenvalue calculation indicates that all two-dimensional solutions develop some form of unstable mode by the time Ro is increased to 5.0. In particular, the four-cell solution becomes unstable to asymmetric perturbations as found in a related problem of flow through a curved duct. The paths of the singular points are tracked with respect to variation in the aspect ratio using the fold following algorithm. A transcritical point is found at an aspect ratio of 0.815 and below which the four-cell solution is no longer on the primary branch. When the channel cross section is tilted even slightly (1°) with respect to the axis of rotation, the bifurcation points unfold and the two-cell solution evolves smoothly as Rossby number is increased. The four-cell solutions then become genuinely disconnected from the primary branch. The uniqueness range in Rossby number increases with increasing tilt.

Notes: The development of complex velocity fields in curved ducts from an initially parabolic profile is studied using a three-dimensional numerical model of the parabolized Navier–Stokes equations. The velocity profiles are influenced strongly by a geometrical parameter Rc (the radius of curvature) and a dynamic parameter Dn (Dean number, Re/(Rc)1/2). For Rc 〈 10 and Dn up to 200, the velocity fields develop into the previously observed two- and four-cell solutions that are axially invariant and symmetric about the midplane. For Rc =100 and Dn〉125 oscillatory solutions develop which are periodic in the axial direction, but are asymmetric about the midplane. Increasing the Dean number over a narrow range results in a significant increase in the frequency of such oscillations. Grid sensitivity tests indicate that such oscillations are not a numerical artifact. Development of oscillatory solutions is delayed with decreasing radius of curvature. Thus for Rc =10, axially invariant two-dimensional solutions that retain the symmetry about the midplane could be obtained for Dn as high as 300. This trend is consistent with one of the earliest observations by Taylor [Proc. R. Soc. London Ser. A. 124, 243 (1929)] that steady, symmetric laminar flows can be observed over a larger range of Dean number in tightly coiled tubes. However, when an asymmetric perturbation is imposed at the inlet, oscillatory solutions develop even for low Rc, indicating that symmetric two-dimensional solutions are not stable to asymmetric perturbations, as indicated by Winters [K. W. Winters and R. C. G. Brindley (private communication)]. Numerical results are also presented for flow through curved ducts with periodic step changes in curvature.

Notes: The development of streamwise-oriented, symmetric, two-dimensional vortices in curved channels of large aspect ratios is studied near the threshold of the first centrifugal instability. The nonlinear equations of motion governing the two-dimensional, stationary flows are solved numerically over a range of parameter values. The dynamical parameter is the normalized streamwise pressure gradient defined as ε=[(∂p/∂x)−(∂p/∂x)c]/(∂p/∂x)c, where (∂p/∂x)c is the critical value for the infinite geometry. The development of interior cells and the selection of the wavelength as ε is gradually increased through zero is quite different from that observed in Taylor vortex flow. For pressure gradients of 0.5% and 1.0% (ε=0.005, 0.01) above critical, the interior cells begin to grow spontaneously and are strongest in the middle of the channel. Unlike the interior cells in Taylor vortex flow, they are only weakly coupled to the end cells. The end cells (or Ekman vortices) are also found to be anomalously long.As ε is increased further to 0.04 and 0.07, the amplitude and wavelength of the interior cells are more uniform. There is, however, a complex interaction between the Ekman vortices and their neighboring interior cells, often resulting in the formation of additional cells in that region. Next, a simple Ginzburg–Landau (GL) model is tested for weakly nonlinear, two-dimensional vortices. The coefficient in the steady form of this equation is evaluated for a wide parameter range using high accuracy calculations of infinite aspect ratio neutral stability curves. (When suitably normalized, neutral stability curves are found to vary only a little with radius ratio.) For large aspect ratio curved channels, predictions from the model are compared with results from numerical simulation. The variation with ε of vortex amplitude near the center of the duct is correctly predicted by the Ginzburg–Landau equation. For given ε, however, agreement of the spanwise variation of vortex amplitude and spacing between the numerical simulation and the model is not obtained. The development of consistent amplitude equations and boundary conditions that link the interior flow to the boundary is expected to be a challenging task.

Notes: The convective heat transfer in a homogeneous porous duct of rectangular cross section in a horizontal orientation is examined. The flow is modeled by the Darcy equation and an averaged, single-equation model is used for the heat transfer. The aspect ratio of the duct (γ=a/b) appears as the natural geometrical parameter and the Rayleigh number Ra, which is the product of the Grashof number and the Prandtl number, appears as the natural dynamical parameter. Uniqueness of the solution at low values of Ra is demonstrated. The complete structure of the symmetric and asymmetric stationary solutions is traced numerically up to values of Ra of about 10 000, using arclength continuation. The limit points and the symmetry breaking bifurcation points are calculated numerically by using the appropriate extended system formulations. The manner in which these singular points unfold is examined as the aspect ratio is varied over 0.6≤γ≤1.4. Determination of linear stability shows that branches of stationary solutions above a Ra of about 4100 are unstable to arbitrary perturbations. The origin of a curve of Hopf points on one of the fold curves is detected around Ra=6560, γ=1.365.

Notes: The multiplicity features and the secondary flow structure of the fully developed, laminar flow of a Newtonian fluid in a straight pipe that is rotating about an axis perpendicular to the pipe axis are examined. The governing equations of motion are solved numerically using the control volume method and the SIMPLE algorithm. The solution structure is governed by two dynamical parameters, Ekman number, Ek=ν/D2Ω and Rossby number, Ro=U/DΩ, where D is the pipe diameter, ν is kinematic viscosity, Ω is rotational speed, and U is velocity scale. Results are presented for a fixed Ekman number of Ek=0.01 and a range of Rossby numbers between 0 to 20. The primary solution branch begins as a unique solution at low Rossby numbers. Its secondary flow structure consists of two-cells. At higher values of Ro a hitherto unknown solution with a four-cell flow structure appears, which coexists with the two-cell flow structure over a range of Ro up to 20. Transient, two-dimensional simulations were carried out to determine the stability of the solutions to two-dimensional perturbations. The two-cell flow structure is stable to both symmetric and asymmetric perturbations. Four-cell flow structure is stable to symmetric perturbations and unstable to asymmetric perturbations, where it breaks down to a two-cell flow structure. © 1995 American Institute of Physics.

Notes: The two-dimensional, stationary form of the equations of motion governing the flow in loosely coiled ducts of equilateral, triangular cross section are solved using a finite volume discretization procedure. The primary solution branch consists of a flow with two large, symmetrical streamwise vortices which are generated by the centrifugal force. Earlier works have investigated the formation of additional Moffat vortices near the corner of such ducts, driven by the secondary flow. Such corner vortices tend to be extremely weak in strength. When the outer wall is flat and one corner of the triangle points inwards, centrifugal instability gives rise to the formation of an additional pair of vortices near the center of the outer wall. Such four-cell solutions appear as dual solutions at the same Dean number, Dn, and the additional vortices are much stronger than the Moffat vortices. This phenomenon is very similar to those observed in the Dean problem for square and circular cross section. The four-cell solutions are found to be unstable to asymmetric perturbations. When the orientation of the triangular duct is changed such that the inner wall is flat such additional vortices do not form near the outer region of the duct and a unique two-cell solution that remains stable is obtained for Dean numbers as high as 350.

Notes: Pressure driven flow of an incompressible Newtonian fluid in a spiral duct of square cross-section was studied both experimentally and numerically. The duct has a curvature ratio (Rc=R/a, where R is the radius of curvature and a is the duct dimension) of 15.1 at the inlet and spirals inwards for nine turns at a uniform rate. A one-component laser-Doppler anemometer was used to measure streamwise velocities. The flow development was determined for Dean number, Dn, of 100, 125, 150, 180 and 250, based on the radius at the flow inlet [Dn=Re/(Rc)1/2, where Re is the Reynolds number, vθ′a/ν]. Steady oscillations in the streamwise direction between 2-cell and 4-cell states, first predicted by Sankar et al. [Phys. Fluids 31, 1348 (1988)], were observed for Dean numbers between 139 and 240. No time dependent flow phenomena were observed. The experimental data are in very good agreement with the numerical simulations, which were based on the parabolized steady three-dimensional Navier–Stokes equations. The results are consistent with calculations by Winters [J. Fluid Mech. 180, 343 (1987)] that predict the existence of a region where no stable two-dimensional solutions exist. ©1996 American Institute of Physics.