Basic-state Flow Research Articles

Transverse electric-thermal-fluid instabilities in an electromagnetic heat exchanger

Electromagnetic (EM) heat exchangers (HX) are critical components in power beaming applications where EM waves are radiated towards an EM HX, which then converts incident energy into heat or mechanical work. An EM HX consists of a lossy ceramic and a fluid flow that maintains thermal contact with and transfers heat from the ceramic. These materials have loss factors which increase with temperature, so that beyond a critical temperature thermal runaway can take place. Stable characteristic temperatures, which depend on the rate of energy removal from the system, during high-power EM heating of ceramic materials suggest that coolants would be in the gaseous phase. As a first step, we consider a model EM HX system consisting of a horizontal channel containing a viscous, dielectric fluid with a constant coefficient of thermal expansion, bounded from below by a grounded ceramic receiver of finite thickness. The system is subject to plane EM waves, propagating normally to the channel from above, and polarized in the same direction as a plane Poiseuille flow of the coolant. With the Boussinesq approximation, we calculate the base state solution of the system and then investigate the linear stability of this base-state. We find three modes of instability: thermal runaway, Rayleigh-Bénard convection, and a novel instability which we call the fringe-field instability, that takes place in the plane normal to base-state flow direction.

Nonlinear stability of Prats flow in Brinkman porous medium with Soret effect

The study aims at establishing the nonlinear stability theory for a uniform flow along a Brinkman porous layer with two impermeable and isothermal horizontal boundaries. Convection in the flow is initiated by the integrated effect of the buoyant forces in thermal and solutal fields. The cross-diffusion effect is also considered as the contributing factor towards the convective instability in the medium. The equations associated with the nonlinear analysis are derived with the help of the Energy method. The results obtained from the linear stability analysis are compared with those of nonlinear stability analysis and the conditions under which the two stability boundaries are widely separated, are discussed elaborately. It is observed that the stability boundary obtained using the nonlinear analysis is solely affected by the concentration gradient (Sa) and the Soret parameter (Sr). However, the concentration gradient is found to have a destabilizing effect on both linear as well as nonlinear stability boundaries. It is also observed that the oscillatory instability is seen only in the case of oblique rolls, whereas, the longitudinal rolls always occur in the stationary mode. For Sa<0, the basic state flow gets stabilized with the increasing values of the Soret parameter. The flow tends to stabilize with the increasing magnitude of throughflow in both forward as well as backward directions. However, an opposite trend is noticed for La>1 and Sa<0, for the smaller magnitude of throughflow (|Pe|<20).

Inertio–elastic instability of a vortex column

We analyse the instability of a vortex column in a dilute polymer solution at large ${{Re}}$ and ${{De}}$ with ${{El}} = {{De}}/{{Re}}$ , the elasticity number, being finite. Here, ${{Re}} = \varOmega _0 a^2/\nu$ and ${{De}} = \varOmega _0 \tau$ are, respectively, the Reynolds and Deborah numbers based on the core angular velocity ( $\varOmega _0$ ), the radius of the column ( $a$ ), the total (solvent plus polymer) kinematic viscosity ( $\nu = (\mu _s +\mu _p)/\rho$ with $\mu _s$ and $\mu _p$ being the solvent and polymer contributions to the viscosity) and the polymeric relaxation time ( $\tau$ ). The stability of small-amplitude perturbations in this distinguished limit is governed by the elastic Rayleigh equation whose spectrum is parameterized by ${E} = {{El}}(1-\beta )$ , $\beta$ being the ratio of the solvent to the solution viscosity. The neglect of the relaxation terms, in the said limit, implies that the polymer solution supports undamped elastic shear waves propagating relative to the base-state flow. Unlike the neutrally stable inviscid case, an instability of the vortex column arises for finite ${E}$ due to a pair of elastic shear waves being driven into a resonant interaction under the differential convection by the irrotational shearing flow outside the core. An asymptotic analysis for the Rankine profile shows the absence of an elastic threshold for this instability. The growth rate is $O(\varOmega _0)$ for order unity $E$ , although it becomes transcendentally small for ${E} \ll 1$ , being $O(\varOmega _0 {E}^2{\rm e}^{-1/{E}^{{1}/{2}}})$ . An accompanying numerical investigation shows that the instability persists for smooth monotonically decreasing vorticity profiles, provided the radial extent of the transition region (from the rotational core to the irrotational exterior) is less than a certain ${E}$ -dependent threshold.

Stability of two-layer flows past slippery surfaces. II. Inclined channels

In this paper, the linear stability characteristics of a two-layered liquid–liquid flow in an inclined channel with slippery walls are investigated. Previous studies on two-layered inclined channel flows have observed the presence of multiple base state flow profiles, two for countercurrent flow and up to three base states for co-current flow. The role of wall slip on the multiple base states associated with each holdup solution is analyzed here. Subsequently, a linear stability analysis, using a combination of a long-wave asymptotic analysis and finite wavenumber numerical calculation, is carried out with the slip boundary condition. Neutral stability boundaries are presented for each base state, with comparisons made with the previous results obtained for the no-slip boundary condition. It was found that the wall slip could have both stabilizing and destabilizing effects depending on the flow rates and the value of holdup—the location of an interface.

Axisymmetric pulse train solutions in narrow-gap spherical Couette flow

We numerically compute the flow induced in a spherical shell by fixing the outer sphere and rotating the inner one. The aspect ratio ϵ=(ro−ri)/ri is set at 0.04 and 0.02, and in each case the Reynolds number measuring the inner sphere’s rotation rate is increased to ∼10% beyond the first bifurcation from the basic state flow. For ϵ=0.04 the initial bifurcations are the same as in previous numerical work at ϵ=0.154, and result in steady one- and two-vortex states. Further bifurcations yield travelling wave solutions similar to previous analytic results valid in the ϵ→0 limit. For ϵ=0.02 the steady one-vortex state no longer exists, and the first bifurcation is directly to these travelling wave solutions, consisting of pulse trains of Taylor vortices travelling toward the equator from both hemispheres, and annihilating there in distinct phase-slip events. We explore these time-dependent solutions in detail, and find that they can be both equatorially symmetric and asymmetric, as well as periodic or quasi-periodic in time.

Subcritical instability of finite circular Couette flow with stationary inner cylinder

The presence of endwalls in Taylor–Couette flows has far reaching effects, leading to dynamics that are qualitatively different from those associated with the idealized situation involving infinitely long cylinders. This is well known in the classical situation where the inner cylinder is rotating and the outer cylinder is stationary. The effects of endwalls in the centrifugally stable situation with stationary inner cylinder and rotating outer cylinder have not been previously considered in detail. The meridional flows induced by the endwalls lead to the formation of a thin sidewall boundary layer on the inner cylinder wall if the endwalls are rotating, or on the outer cylinder wall if they are stationary. At sufficiently high Reynolds numbers (non-dimensional rotation rate of the outer cylinder), the sidewall boundary layer has concentrated shear, the pressure gradient in the azimuthal direction (which is the streamwise direction for the boundary layer flow) is zero (the flow is axisymmetric) and the boundary layer thickness is constant. At a critical Reynolds number, the sidewall boundary layer loses stability at a subcritical Hopf bifurcation, breaking the axisymmetry of the basic state flow, and for Reynolds numbers slightly above critical, the basic state is unstable to a packet of Hopf modes with azimuthal wavenumbers clustered about the critical wavenumber. The early time evolution of the critical Hopf mode is a rotating wave whose behaviour is analogous to a Tollmien–Schlichting wave. As the Hopf modes grow with time, nonlinear interactions lead to modulations in the waves, localization of the disturbances and the evolution of concentrated streamwise vortical streaks which become very intense via vortex stretching.

Linear instability of warm core, constant potential vorticity, eddies in a two‐layer ocean

Linear instability of warm‐core eddies of constant potential vorticity (PV) is studied in a two layer, finite depth, shallow‐water ocean. The basic state flow in the constant PV eddy that obeys the gradient balance cannot be described by explicit expressions and can be solved only numerically. The various cases of gradient balance are classified by constructing a canonical formulation that relates any PV value to a value of the angular velocity that has to prevail near the centre of the constant PV eddy. The growth rates of perturbations imposed on the basic state are calculated for a variety of values of the (constant) PV and the depth of the surrounding ocean. The growth‐rates, i.e. the eigenvalues, are calculated numerically by using a shooting to fitting point method which guarantees that the corresponding eigenfunctions are regular at all singular points. The maximal growth rates are contoured as functions of the PV and ocean depth for azimuthal wavenumbers 2 and 3 and the maximum of these growth rates is of the order of 1 day, which is similar to that of a solidly rotating eddy. The range of angular velocity and ocean depth where the constant PV eddy is unstable, however, is greatly reduced compared with that of a solidly rotating eddy. The instabilities found here are classified in terms of wave–wave interactions by comparing our results in each PV value with the known instabilities of the solidly rotating eddy with the same angular velocity. In the constant PV eddy the Baroclinic instability is filtered out and the range of angular velocity where the Hybrid instability exists is significantly reduced. All instabilities decay monotonically with the increase in ocean depth.

Linear Stability Analysis of Open-Channel Shear Flow Generated by Vegetation

A linear stability analysis of flow in an open-channel partially covered with vegetation was performed. The differential drag between vegetated zones and adjacent nonvegetated zones is known to induce a lateral gradient of the streamwise velocity. The velocity gradient may result in flow instability in the shear layer around the edge of the vegetated zone causing the generation of discrete horizontal vortices. We assume that the base state flow field before the occurrence of instability is characterized by turbulencewith a smaller length scale than the flow depth, which is mainly generated by the bottom friction. By introducing perturbations to the flow depth as well as the stream- wise and transverse velocities in the base state, the conditions required for perturbations to grow in time were studied over a wide range of (1) Froude number, (2) normalized nonvegetated zone width, and three other dimensionless parameters that represent the relative effect of (3) bed friction, (4) vegetation drag, and (5) subdepth eddy viscosity. All parameters were found to have positive and negative growth rates of perturbations within their respective evaluated ranges. The characteristic vortex shedding frequencies associated with the maximum growth rate was compared with those observed in experiments. Although the analysis that employs a base state set without the large scale lateral motions was shown to be capable of predicting the order of magnitude of the frequencies, there is a systematic discrepancy between the predicted and observed frequencies, which may be due to the limitation of linear stability analysis. DOI: 10.1061/(ASCE)HY.1943-7900 .0000822. © 2014 American Society of Civil Engineers.

Effect of elongational flow on ferrofuids under a magnetic field

To set up a mathematical model for the flow of complex magnetic fluids, noninteracting magnetic particles with a small volume or an even point size are typically assumed. Real ferrofluids, however, consist of a suspension of particles with a finite size in an almost ellipsoid shape as well as with particle-particle interactions that tend to form chains of various lengths. To come close to the realistic situation for ferrofluids, we investigate the effect of elongational flow incorporated by the symmetric part of the velocity gradient field tensor, which could be scaled by a so-called transport coefficient λ(2). Based on the hybrid finite-difference and Galerkin scheme, we study the flow of a ferrofluid in the gap between two concentric rotating cylinders subjected to either a transverse or an axial magnetic field with the transport coefficient. Under the influence of a transverse magnetic field with λ(2)=0, we show that basic state and centrifugal unstable flows are modified and are inherently three-dimensional helical flows that are either left-winding or right-winding in the sense of the azimuthal mode-2, which is in contrast to the generic cases. That is, classical modulated rotating waves rotate, but these flows do not. We find that under elongational flow (λ(2)≠0), the flow structure from basic state and centrifugal instability flows is modified and their azimuthal vorticity is linearly changed. In addition, we also show that the bifurcation threshold of the supercritical centrifugal unstable flows under a magnetic field depends linearly on the transport coefficient, but it does not affect the general stabilization effect of any magnetic field.

Interfacial instability of compressible slip flows in a microchannel

In microfluidics the instability of an interface between two fluids may be favorable or unfavorable. Such an instability for incompressible flows has been extensively studied in the literature; in this paper we extend it to compressible flows which slip on the solid boundary. A generalized Darcy's law taking into consideration the compressibility and velocity slip is obtained. Using a conformal-mapping method we derive an interface equation on a fixed domain, the solutions of which determine the moving interface. We also examine the linear stability of the base-state flow by perturbing the corresponding conformal map and show that the velocity slip has a stabilizing effect.

Zonostrophic Instability

Abstract Zonostrophic instability leads to the spontaneous emergence of zonal jets on a β plane from a jetless basic-state flow that is damped by bottom drag and driven by a random body force. Decomposing the barotropic vorticity equation into the zonal mean and eddy equations, and neglecting the eddy–eddy interactions, defines the quasilinear (QL) system. Numerical solution of the QL system shows zonal jets with length scales comparable to jets obtained by solving the nonlinear (NL) system. Starting with the QL system, one can construct a deterministic equation for the evolution of the two-point single-time correlation function of the vorticity, from which one can obtain the Reynolds stress that drives the zonal mean flow. This deterministic system has an exact nonlinear solution, which is an isotropic and homogenous eddy field with no jets. The authors characterize the linear stability of this jetless solution by calculating the critical stability curve in the parameter space and successfully comparing this analytic result with numerical solutions of the QL system. But the critical drag required for the onset of NL zonostrophic instability is sometimes a factor of 6 smaller than that for QL zonostrophic instability. Near the critical stability curve, the jet scale predicted by linear stability theory agrees with that obtained via QL numerics. But on reducing the drag, the emerging QL jets agree with the linear stability prediction at only short times. Subsequently jets merge with their neighbors until the flow matures into a state with jets that are significantly broader than the linear prediction but have spacing similar to NL jets.

A Computational Fluid Dynamics/Bulk-Flow Hybrid Method for Determining Rotordynamic Coefficients of Annular Gas Seals

This paper presents a new computational fluid dynamics (CFD)/bulk-flow hybrid method to determine the rotordynamic characteristics of annular gas seals. The method utilizes CFD analysis to evaluate the unperturbed base state flow, an averaging method to determine the base state bulk-flow variables, and a bulk-flow perturbation method to solve for the fluid forces acting on an eccentric, whirling rotor. In this study the hybrid method is applied to a hole-pattern seal geometry and compared with experimental data and numerical and analytical methods. The results of this study show that the dynamic coefficients predicted by the hybrid method agree well with the experimental data, producing results that are comparable with a full, three-dimensional, transient, whirling rotor CFD method. Additionally, the leakage rate predicted by the hybrid method is more agreeable with experiment than the other methods. The benefit of the present method is the ability to calculate accurate rotordynamic characteristics of annular seals that are comparable to results produced by full, transient CFD analyses with a simulation time on the order of bulk-flow analyses.

Dynamics of a climbing surfactant-laden film – I: Base-state flow

The dynamics of a surfactant-laden film climbing up an inclined plane is investigated through a two-dimensional (2-D), nonlinear evolution equation for the interface coupled to convective-diffusion equations for the surfactant, derived using lubrication theory. One-dimensional (1-D) solutions, representing the base-state flow, are investigated for constant flux and constant volume configurations; these flows are parameterised by capillarity, gravity, convection–diffusion ratios (represented by Péclét numbers at the surface and bulk), a solubility parameter, sorption kinetics constants, the number of surfactant monomers in a micelle, and the nonlinearity of the surfactant equation of state. In both configurations studied, a front develops spreading up the substrate against the direction of gravity whereby the leading edge of the front follows a power–law as a function of time. The effect of system parameters on the base-state flow is explored through an extensive parametric study, while the stability of the above-mentioned system to spanwise perturbations is the focus of Part II [1].

Spatial Growth of Perturbations in a Turbulent Baroclinic Jet

Abstract The spatial spreading of infinitesimal disturbances superposed on a turbulent baroclinic jet is explored. This configuration is representative of analysis errors in an idealized midlatitude storm track and the insight gained may be helpful to understand the spreading of forecast errors in numerical weather prediction models. This problem is explored through numerical experiments of a turbulent baroclinic jet that is perturbed with spatially localized disturbances. Solutions from a quasigeostrophic model for the disturbance fields are compared with those for a passive tracer to determine whether disturbances propagate faster than the basic-state flow. Results show that the disturbance spreading rate is sensitive to the structure of the initial disturbance. Disturbances that are localized in potential vorticity (PV) have far-field winds that allow the disturbance to travel downstream faster than disturbances that are initially localized in geopotential, which have no far-field wind. Near the jet, the spread of the disturbance field is observed to exceed the tracer field for PV-localized disturbances, but not for the geopotential-localized disturbances. Spreading rates faster than the flow for geopotential-localized disturbances are found to occur only for disturbances located off the jet axis. These results are compared with those for zonal and time-independent jets to qualitatively assess the effects of transience and nonlinearity. This comparison suggests that the average properties of localized perturbations to the turbulent jet can be decomposed into a superposition of dynamics associated with a time-independent parallel flow plus a “diffusion” process.

Tropical–Extratropical Teleconnections in Boreal Summer: Observed Interannual Variability*

Abstract Maximum covariance analysis is performed on the fields of boreal summer, tropical rainfall, and Northern Hemisphere (NH) 200-hPa height for the 62-yr period of record of 1948–2009. The leading mode, which appears preferentially in summers preceding the peak phases of the El Niño–Southern Oscillation (ENSO) cycle, involves a circumglobal teleconnection (CGT) pattern in the NH extratropical 200-hPa height field observed in association with Indian monsoon rainfall anomalies. The second mode, which tends to occur in summers following ENSO peak phases, involves a western Pacific–North America (WPNA) teleconnection pattern in the height field observed in association with western North Pacific summer monsoon rainfall anomalies. The CGT pattern is primarily a zonally oriented wave train along the westerly waveguide, while the WPNA pattern is a wave train emanating from the western Pacific monsoon trough and following a great circle. The CGT is accompanied by a pronounced tropical–extratropical seesaw in the zonally symmetric geopotential height and temperature fields, and the WPNA is observed in association with hemispherically uniform anomalies. These ENSO-related features modulate surface air temperature in both the tropics and extratropics. ENSO also affects the wave structure of the CGT and WPNA indirectly, by modulating the strengths of the Indian and western North Pacific monsoons. Linear barotropic mechanisms, including energy propagation and barotropic instability of the basic-state flow, also act to shape and maintain the CGT. The implications of these findings for seasonal prediction of the NH extratropical circulation are discussed.

Diagnosis of dynamic process over rainband of landfall typhoon

This paper introduces a new physical parameter — thermodynamic shear advection parameter combining the perturbation vertical component of convective vorticity vector with the coupling of horizontal divergence perturbation and vertical gradient of general potential temperature perturbation. For a heavy-rainfall event resulting from the landfall typhoon ‘Wipha’, the parameter is calculated by using National Centres for Enviromental Prediction/National Centre for Atmospheric Research global final analysis data. The results showed that the parameter corresponds to the observed 6 h accumulative rainband since it is capable of catching hold of the dynamic and thermodynamic disturbance in the lower troposphere over the observed rainband. Before the typhoon landed, the advection of the parameter by basic-state flow and the coupling of general potential temperature perturbation with curl of Coriolis force perturbation are the primary dynamic processes which are responsible for the local change of the parameter. After the typhoon landed, the disturbance is mainly driven by the combination of five primary dynamic processes. The advection of the parameter by basic-state flow was weakened after the typhoon landed.

Role of Subtropical Precipitation Anomalies in Maintaining the Summertime Meridional Teleconnection over the Western North Pacific and East Asia

Abstract The meridional teleconnection patterns over the western North Pacific and East Asia (WNP–EA) during summer have a predominant role in affecting East Asian climate on the interannual time scale. A well-known seesaw pattern of tropical–subtropical precipitation is associated with the meridional teleconnection, and the subtropical precipitation anomaly has been previously viewed as a result of anomalous circulations associated with the teleconnection. In this study, however, the authors suggest that subtropical precipitation anomalies, in turn, can significantly affect large-scale circulations and may be crucial for maintenance of the meridional teleconnection. Diagnosis by using observational and reanalysis data indicates that the meridional teleconnection patterns are clearer in summers when the subtropical rainfall anomalies are greater. The simulated results by a linear baroclinic model indicate that a subtropical heat source, which is equivalent to the diagnosed positive subtropical precipitation anomaly, induces zonally elongated zonal wind anomalies that resemble the diagnosed ones in both the upper and lower troposphere over the extratropical WNP–EA. The simulated results also indicate that the horizontal and vertical structures of circulation responses are insensitive to the locations and shapes of imposed subtropical heat anomalies, which implies the important role of basic flow in circulation responses. This study suggests that, for confidential dynamical seasonal forecasting in East Asia, general circulation models should be required to capture the features of interannual subtropical rainfall variability and basic-state flows in WNP–EA.

Theoretical Studies of Convectively Forced Mesoscale Flows in Three Dimensions. Part I: Uniform Basic-State Flow

Abstract Convectively forced mesoscale flows in three dimensions are theoretically investigated by examining the transient response of a stably stratified atmosphere to convective heating. Solutions for the equations governing small-amplitude perturbations in a uniform basic-state wind with specified convective heating are analytically obtained using the Green function method. In the surface pulse heating case, it is explicitly shown that the vertical displacement at the center of the 3D steady heating decreases as fast as t−1 for large t. Hence, unlike in two dimensions, the steady state is approached in three dimensions. In the finite-depth steady heating case, the perturbation vertical velocity field in the stationary mode shows a main updraft region extending over the heating layer and V-shaped upward and downward motions above and below the heating layer. Including the third dimension results in a stronger updraft at an early stage, a weaker compensating downward motion, and a weaker stationary gravity wave field in a quasi-steady state than in the case of two dimensions. An examination of flow response fields for various vertical structures of convective heating indicates that stationary gravity waves above the main updraft region become strong in intensity as the height of the maximum convective heating increases. In response to the transient heating, a main updraft region extending over the heating layer no longer appears at a dissipation stage of deep convection. Instead, alternating regions of upward and downward motion with an upstream phase tilt appear.

Instabilities with polyacrylamide solution in small and large aspect ratios Taylor-Couette systems

We have investigated the stability of viscoelastic polyacrylamide solution in Taylor-Couette system with different aspect ratios. The first instability modes observed in a Taylor-Couette system with Γ = 10 were TVF and WVF, as for Newtonian fluid. At higher Taylor numbers moving vortices occur, a wavy mode with non-stationary vortex size. In the Taylor-Couette system with Γ = 45.9 we note a coexistence of various instability modes. In addition to TVF, counterpropagating waves developed at the transition from the base state flow. At higher Taylor number values Taylor vortices of different sizes occurred. Reduced amplitude Wavy vortex flow has also been observed.

Energy-enstrophy stability of β-plane Kolmogorov flow with drag

We develop a nonlinear stability method, the energy-enstrophy (EZ) method, that is specialized to two-dimensional hydrodynamics and basic state flows consisting of a single Helmholtz eigenmode. The method is applied to a β-plane flow driven by a sinusoidal body force and retarded by drag with damping time scale μ−1. The standard energy method [H. Fukuta and Y. Murakami, J. Phys. Soc. Jpn. 64, 3725 (1995)] shows that the laminar solution is monotonically and globally stable in a certain portion of the (μ,β)-parameter space. The EZ method proves nonlinear stability in a larger portion of the (μ,β)-parameter space than does the energy method. Moreover, by penalizing high wavenumbers, the EZ method identifies a most strongly amplifying disturbance that is more physically realistic than that delivered by the energy method. Linear instability calculations are used to determine the region of the (μ,β)-parameter space where the flow is unstable to infinitesimal perturbations. There is only a small gap between the linearly unstable region and the nonlinearly stable region, and full numerical solutions show only small transient amplification in that gap.