Nonparallel Effects Research Articles

Nonparallel-flow stability of an axisymmetric buoyant plume in a coflowing uniform stream

Nonparallel effects of the basic flow and disturbances are included in a linear stability study of an axisymmetric thermal plume in a coflowing uniform stream. On invoking a slowly varying approximation and undertaking a multiple scale analysis, a larger region of stability than that predicted by parallel-flow theory is obtained. The amplification rate of the perturbation depends on the flow quantity under consideration and so there is no unique boundary between stable and unstable perturbations. The calculated critical Reynolds numbers for the perturbation velocities are, however, of the same order of magnitude, the lowest being 32.9 for a pure plume, and they increase significantly as the strength of the external stream increases. There are two mechanisms for instability: one is shear driven, the other gravitational.

Nonlinear Critical Layers in Barotropic Stability

Abstract Applying the method of matched asymptotic expansions (MAE) to the shallow water equations on a rotating sphere, the structure of critical layers that occur in the linear and inviscid analysis of neutral disturbances of barotropic zonal flows is investigated, assuming that the critical layers are controlled by nonlinearity rather than viscosity or nonparallel flow effects. It turns out that nonlinearity is insufficient to resolve the critical layer singularity completely. It suffices however to connect linear and nondissipative solutions across critical latitudes.

Nonparallel Effects of Renin Inhibitor Treatment on Plasma Renin Activity and Angiotensins I and II in Hypertensive Subjects

Nonparallel effects of renin inhibitor treatment on plasma renin activity (PRA) and the plasma levels of angiotensins (ANG), as well as on blood pressure, have been observed in subjects with hypertension. This study addresses the possibility that renin inhibitors may show a high degree of plasma protein binding in vivo and that displacement of protein-bound inhibitor during the assay of PRA in vitro may lead to overestimation of renin inhibition. Indeed, with the ultrafiltration technique it was found that 96% of the novel renin inhibitor Ro 42-5892, when added to EDTA plasma, was bound to protein. The angiotensinase inhibitors phenylmethylsulfonyl fluoride (PMSF) and 8-hydroxy-quinoline sulfate (8-OHQ), which are currently used in PRA assays, caused a displacement of protein-bound inhibitor, thereby increasing its free concentration. This displacement was sufficient to explain the reduction in IC 50 of Ro 42-5892, which was seen in the PRA assay when PMSF and 8-OHQ were added to plasma. Such reductions in IC 50 were also seen with the renin inhibitors CGP 29-287, CGP 38-560A, and SR 43-845. When Ro 42-5892 was given, 1 mg/kg intravenously in 10 min, to subjects with hypertension, it appeared that plasma ANG I and II returned to baseline after 6-8 h, whereas PRA measured in the presence of PMSF and 8-OHQ was still suppressed. However, when PRA was measured without these angiotensinase inhibitors, the inhibition of PRA was parallel to the suppression of ANG I and II.(ABSTRACT TRUNCATED AT 250 WORDS)

Analysis of stability in a self-similar circular jet with consideration of the nonparallel effect

Analysis of stability in a self-similar circular jet with consideration of the nonparallel effect

Non-parallel stability of a flat-plate boundary layer using the complete Navier-Stokes equations

Non-parallel effects which are due to the growing boundary layer are investigated by direct numerical integration of the complete Navier-Stokes equations for incompressible flows. The problem formulation is spatial, i.e. disturbances may grow or decay in the downstream direction as in the physical experiments. In the past various non-parallel theories were published that differ considerably from each other in both approach and interpretation of the results. In this paper a detailed comparison of the Navier-Stokes calculation with the various non-parallel theories is provided. It is shown, that the good agreement of some of the theories with experiments is fortuitous and that the difference between experiments and theories concerning the branch I neutral location cannot be explained by non-parallel effects.

Görtler vortices in growing boundary layers: The leading edge receptivity problem, linear growth and the nonlinear breakdown stage

Görtler vortices are thought to be the cause of transition in many fluid flows of practical importance. In this paper a review of the different stages of vortex growth is given. In the linear regime nonparallel effects completely govern this growth and parallel flow theories do not capture the essential features of the development of the vortices. A detailed comparison between the parallel and nonparallel theories is given and it is shown that at small vortex wavelengths the parallel flow theories have some validity; otherwise nonparallel effects are dominant. New results for the receptivity problem for Gortler vortices are given; in particular vortices induced by free-stream perturbations impinging on the leading edge of the wall are considered. It is found that the most dangerous mode of this type can be isolated and its neutral curve is determined. This curve agrees very closely with the available experimental data. A discussion of the different regimes of growth of nonlinear vortices is also given. Again it is shown that, unless the vortex wavelength is small, nonparallel effects are dominant. Some new results for nonlinear vortices of O(l) wavelengths are given and compared with experimental observations. The agreement between theory and experiment is shown to be excellent up to the point where unsteady effects become important. For small wavelength vortices the nonlinear regime is of particular interest since a strongly nonlinear theory can be developed there. Here the vortices can be large enough to drive the mean state which then adjusts itself to make all modes neutral. The breakdown of this nonlinear state into a three-dimensional time dependent flow is also discussed.

The growth of Görtler vortices in compressible boundary layers

The linear instability of Gortler vortices in compressible boundary layers is considered. Using asymptotic methods in the high-wavenumber regime, it is shown that a growth-rate estimate can be found by solving a sequence of linear equations. The growth rate obtained in this way takes non-parallel effects into account and can be found much more easily than by ordinary differential equation eigenvalue calculations associated with parallel-flow theories.

On the first-mode instability in subsonic, supersonic or hypersonic boundary layers

The work, addressing subsonic, supersonic and hypersonic boundary-layer instability, is motivated by the need for more understanding of compressible transition at high global Reynolds numbers Re. In the supersonic case, the so-called ‘first modes’ of instability found/suggested by previous Orr-Sommerfeld computations can be identified as triple-deck oblique ones, directed outside the local wave-Mach-cone directions. Less oblique instability modes inside are not of Orr-Sommerfeld form since they are substantially affected by non-parallel flow effects. The maximum linear growth rates are determined for a range of supersonic and subsonic free-stream Mach numbers M∞, and comparisons are made with previous computations, showing fairly good agreement at moderate Mach numbers. A second mound of unstable wavenumbers and frequencies is also evident. In addition, the nonlinear version is set up and emphasized (with attention drawn to a recent paper by the author (1988a) showing the possibility of nonlinear break-up), and certain extremes are examined including those of transonic and hypersonic boundary layers. In the hypersonic limit a new regime is found (for many conditions, including the insulated plate), namely applies to the Orr-Sommerfeld approach. This is a very severe restriction in practice.

Flow-induced morphological instabilities: The rotating disc

The morphological stability of a rotating and solidifying disc is investigated under the assumption that δ, the thickness of the viscous boundary layer, is much larger than δ c, the thickness of the solute boundary layer. We find that axisymmetric disturbances with wavelengths comparable to δ respond to non-parallel flow effects and have stability characteristics quite different from disturbances in a parallel flow. These long waves are unstable because of the non-parallel flow and would decay without it. This analysis thus identifies a new mechanism of morphological change induced by flow.

Subharmonic resonance, pairing and shredding in the mixing layer

An instability-wave analysis is presented to describe the spatial evolution of a fundamental mode and its subharmonic on an inviscid parallel mixing layer. It incorporates explicitly the weakly nonlinear interaction between the two modes. The computational finding that the development of the subharmonic, leading eventually to pairing or shredding, crucially depends on its phase relation with the fundamental is fully confirmed. Furthermore it is shown that a critical fundamental amplitude has to be reached before the (spatial) subharmonic becomes phase locked with the fundamental and exhibits a modified growth rate. Then the analysis is exploited to explain the occurrence of amplitude modulations in ‘natural’ mixing layers and to estimate the width of the subharmonic spectral peaks. Also, the case of oblique subharmonic waves is briefly touched upon. In the last part, ways are explored to model non-parallel effects, i.e. to handle the saturation of the rapidly growing subharmonic. Using this wave description, the role of mode interaction in the ‘vortex pairing’ and ‘shredding’ process is assessed.

Insulin and other stimulants have nonparallel translational effects on protein synthesis.

Isolated pancreatic acini from streptozocin-induced diabetic rats were used to study the role of insulin on the synthesis of specific cellular proteins. When acini were incubated with 0-100 nM insulin for 2 h and then pulsed with [35S]methionine, a dose-dependent increase in [35S]methionine incorporation into total cellular proteins was observed. When acinar cell lysates were subjected to gel electrophoresis, 12 major newly synthesized protein bands were resolved. Insulin (100 nM) increased the incorporation of [35S]methionine into all bands but with significantly different rates, varying from 84 to 216% of control. Next, specific antibodies to amylase, trypsin, ribonuclease, myosin, and lactate dehydrogenase (LDH) were used to evaluate the biosynthesis of known proteins. Insulin stimulated labeled amino acid incorporation into amylase by 148% over control. Insulin stimulated the synthesis of trypsinogen to a similar degree, but ribonuclease synthesis showed a significantly smaller increase of 53% over control. Insulin stimulated myosin and LDH synthesis by 169 and 184%, respectively. A differential pattern of protein synthesis was also observed when acini were treated with two other stimulators of protein synthesis, cholecystokinin and hemin. Both of these stimulators had a reduced effect on ribonuclease synthesis compared with amylase and trypsinogen synthesis but failed to increase myosin synthesis. When the RNAs extracted from control acini and acini treated with 100 nM insulin were translated in vitro, the proteins synthesized were quantitatively similar. This study therefore indicates that insulin has translational effects on acinar protein synthesis, and these effects are nonparallel for various specific acinar cell proteins.

Insulin and other stimulants have nonparallel translational effects on protein synthesis

Insulin and other stimulants have nonparallel translational effects on protein synthesis

Three-dimensional instabilities in steady and unsteady non-parallel boundary layers, including effects of Tollmien-Schlichting disturbances and cross flow

Three-dimensional (3D) linear stability properties are considered for steady and unsteady 2D or 3D boundary layers with significant non-parallelism present. Two main examples of such non-parallel flows whose stability is of interest are, firstly, steady motion, over roughness elements, in cross flow, or in large-scale separation and, secondly, unsteady 2D Tollmien-Schlichting (TS) motion, with its associated question of secondary instabilities. A high-frequency stability analysis is presented here. It is found that, for 2DTS or steady boundary layers, there is a swing in the direction of maximum TS spatial growth rate, from 0° for parallel flow towards 64.68° away from the free-stream direction, as the nonparallel flow effects increase. These effects then depend principally on, and indeed are proportional to, the local slope of the boundary-layer displacement. Cross flow can also have a profound impact on TS instabilities. Further implications for higher-amplitude and/or fasterscale disturbances, their secondary instability, and nonlinear interactions, are also discussed.

An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating disc

The paper investigates high-Reynolds-number stationary instabilities in the boundary layer on a rotating disc. The investigation demonstrates that, in addition to the inviscid mode found by Gregory, Stuart & Walker ( Phil. Trans. R. Soc. Lond. A 248, 155 (1955)) at high Reynolds numbers, there is a stationary short-wavelength mode. This mode has its structure fixed by a balance between viscous and Coriolis forces and cannot be described by an inviscid theory. The asymptotic structure of the wave-number and orientation of this mode is obtained, and a similar analysis is given for the inviscid mode. The expansion procedure provides the capacity of taking non-parallel effects into account in a self-consistent manner. The inviscid solution of Gregory et al . is modified to take account of viscous effects. The expansion procedure used is again capable of taking non-parallel effects into account. The results obtained suggest why the inviscid approach of Gregory et al . should give a good approximation to the experimentally measured orientation of the vortices. The results also explain partly why the inviscid analysis should not give such a good approximation to the wavenumber of the vortices. The asymptotic analysis of both modes provides a starting point for the corresponding nonlinear problems.

On the development of large-sized short-scaled disturbances in boundary layers

For high Reynolds numbers the planar nonlinear stability properties of boundary layers, as governed first by the triple-deck interaction, are discussed theoretically. When a disturbance of relatively high frequency is introduced or as the distance downstream increases in the Blasius case the characteristic length scale shortens. An account is given of the possible progress of such a disturbance, from an initially small Tollmien-Schlichting state, through a weakly nonlinear first stage that does not interrupt the amplitude growth, and thence into a fully nonlinear second stage where the Benjamin-Ono equation comes into force. Still higher frequencies point to the full Euler equations holding, although significant non-parallel flow effects then emerge. In the Euler stage, however, and in the previous second stage, bursts of vorticity from the viscous sublayer closest to the solid surface are almost certain to occur. The relations with other basic flow problems, and with turbulence-modelling, are noted, as well as the need to consider three-dimensional motion. Finally, an Appendix deals with certain issues of practical importance arising with respect to non-parallel flow effects, for example in breakaway separations or flow over surface-mounted obstacles, and points out that the slope of the local displacement in the basic flow is then a dominant controlling feature of the short-scale instabilities.

A spectral theory for weakly nonlinear instabilities of slowly divergent shear flows

A general theory is proposed for the downstream evolution of nonlinearly interacting waves in a nonparallel shear flow. The waves are represented as Fourier integrals over the spectral range. A multiple scales expansion is employed to predict evolutions of the Fourier mode amplitudes A(x,ω)—where x is the streamwise coordinate and ω is the frequency—by assuming that both nonlinear and nonparallel effects can be expressed in terms of the same expansion parameter ε. The theory produces in the leading order the Orr–Sommerfeld equation, and in the next order a Landau-type integrodifferential equation for A. The linear part of the equation contains only the first-order spatial derivative, and the nonlinear part consists of wave–wave interactions integrated over the entire spectral range of A. The approach is very general in that no Reynolds number (Re) scaling is involved as Re is retained independent of ε throughout and that no Landau constant is involved so that each wave can grow or decay in space or time. The Landau-type equation is too intricate for analytic solution but is suitable for numerical solution. An asymptotic analysis for large x via the method of stationary phase reveals that the subharmonic components provide the dominant contribution to the nonlinear terms, consistent with experimental observations.

On the stability of an axisymmetric plume in a uniform stream

The linear stability equations for a round laminar thermal plume in a coflowing vertical stream have been solved numerically. Both symmetric and asymmetric disturbances have been considered for strengths of the forced flow varying between very weak and very strong. The parallel flow analysis confirms that the forced flow has a stabilizing effect. The upper branch of the neutral curve for sinuous disturbances is qualitatively like that of a round momentum jet. However, neither a critical Reynolds number nor a lower branch of the neutral curve was found. Non-parallel effects are discussed.

On the Effects of Nonparallelism, Three‐Dimensionality, and Mode Interaction in Nonlinear Boundary‐Layer Stability

A self‐consistent theoretical investigation is described for the nonlinear stability, and spatial development, of disturbances in a plane boundary layer subject to a number of three‐dimensional modes, their nonlinear interactions, and the effects of nonparallelism of the basic flow. For the largest weakly nonlinear disturbances considered, nonparallel‐flow effects appear to be negligible at first sight, and primary, secondary, and/or tertiary bifurcations, usually supercritical but not always so, can occur when two fundamental modes are present. As a result the flow downstream then always has three ultimate possibilities: a unique stable disturbed state, two or more possible stable states, or no stable state possible. It is here that the nonparallel‐flow effects exert their crucial influence. For nonparallelism comes into play significantly during the initial growth or decay of a disturbance, and that initial spatial development, from given initial conditions upstream, controls what happens subsequently as the disturbance increases. Thus in the first possibility above, the stable state is achieved through a smooth bifurcation, due to nonparallelism; in the second possibility the nonparallelism decides which stable state is attained (smoothly) from the initial conditions; and in the third possibility the nonparallel flow effects force the disturbance to terminate in a singular fashion. This singularity then leads to a fully nonlinear effect, locally on the boundary‐layer flow. More complicated interactions can arise if more than two three‐dimensional modes are present. The novel effect of the nonparallelism has a connection with related Navier‐Stokes calculations even at near‐critical Reynolds numbers.

The effect of heating on the stability of flow over concave walls

The effect of heating or cooling on the linear Görtler instability of the boundary layer flow over a concave wall is described using an asymptotic analysis for short-wavelength disturbances. The analysis takes account of all nonparallel flow effects in a uniform and systematic manner without making the quasiparallel flow approximation. The fluid viscosity is assumed constant while the perturbation in density is allowed to vary linearly with the perturbation in temperature. Both steady Taylor–Görtler disturbances and those that propagate downstream are considered. An analytic expression for the neutral curve is obtained. It is found that the critical disturbances are concentrated in thin viscous layers near the middle of the boundary layer; the precise location depending upon the basic velocity and temperature profiles. Heating the boundary layer destabilizes the short-wavelength disturbances and cooling does the reverse. While steady Taylor–Görtler disturbances are more unstable than those propagating downstream under the assumptions of the present analysis, the difference between the neutral stability boundaries for the two types of disturbances is decreased by heating at low Prandtl numbers and by cooling at moderate to high Prandtl numbers.

A critical layer dominated by non-parallel effects in a rotating barotropic flow

The critical layer of perturbations to slowly modulating strong rectified currents induced by vorticity sources in a quasi-geostrophic barotrophic flow is investigated. For r ≫ Re -3/2 and r ≫ δ 2 which correspond to realistic geophysical conditions, the critical layer is dominated by the advection of the perturbation’s vorticity by the weak transverse velocity of the basic state and not by the traditional viscous or nonlinear balance. The Ekman, Rossby and Reynolds numbers of the flow are E , Ro and Re respectively and r = E ½ / Ro , while δ is the parameter of nonlinearity. The width of the critical layer is O ( r 1/6 ) which can be quite thick for relevant flow conditions. This suggests that non-parallel effects play an important role in the dynamics of large scale geophysical flows.