Non-parallel Flow Research Articles

Evolution of a localized vortex in plane nonparallel viscous flows with constant velocity shear. I. Hyperbolic flow

The framework of the linear theory is employed to study the evolution of an initial compact vortical disturbance in unbounded plane nonparallel viscous incompressible flows with constant velocity gradients. Two types of such flows are known to be possible: hyperbolical and elliptical (as well as an intermediate case of the well-studied parallel Couette flow). The results presented here are obtained for a hyperbolical flow. (Results concerning the elliptical flow are to be issued in a separate publication.) This paper is a development of earlier work by R. R. Lagnado, N. Phan-Thien, and L. G. Leal [Phys. Fluids 27, 1094 (1984)] studying the stability of a hyperbolical flow relative to the simplest perturbations in the form of plane waves with a time-dependent wave vector. The dynamics of vortex intensity is investigated as well as the evolution of its geometrical form and orientation. The results are discussed in the context of the problem of hairpin vortex formation.

A non-dimensional criterion and its proof for transient flow caused by fire in ventilation network

Application of 1D Transient Flow Model to a network analysis finds that the fluids in two branches (which are connected in parallel and contain no drive source) may no longer flow in a same direction while the network flow state changes from being steady into being unsteady due to a fire in the ventilation network. This phenomenon can be referred to as non-parallel flows in parallel branches. A non-dimensional criterion ξ is deduced theoretically. It is proved that for any two branches which are connected in parallel and contain no drive source, Non-parallel flows can be expected in the branches only if ξ is not equal to unity. If ξ is equal to unity, the non-dimensional flow velocities in the two branches will always be equal to each other. If ξ is not equal to unity, a prediction can be made on which one of the non-dimensional velocities in the two parallel branches will change more rapidly.

Nonlinear growth (and breakdown) of disturbances in developing Hagen Poiseuille flow

The effect of imposing disturbances on developing Hagen Poiseuille flow is investigated for large Reynolds numbers. A formal large Reynolds number analysis is used throughout, which leads to a parabolic approximation (to the Navier-Stokes equations). In this inlet region [which is long O(Reynoldsnumber)×pipe radius], assuming uniform/near uniform inlet flow conditions, the boundary layers develop on the walls of the pipe, that eventually merge. Boundary layers are known to be susceptible to three-dimensional eigensolutions that grow algebraically in the streamwise direction. In this study, nonaxisymmetric disturbances are triggered through (i) the imposition of the aforementioned eigenstates at the pipe inlet and (ii) forcing the azimuthal velocity on the pipe wall. Fully nonlinear, steady disturbances are considered in detail; if the disturbance amplitude is sufficiently large, a solution “breakdown” is observed (associated with a rapid growth of the radial and azimuthal velocity components close to lines of symmetry). This appears to be linked to reversals in both the azimuthal and radial velocity components, suggesting a possible mechanism for flow transition. Some comparison is also made with the analogous effect on planar (Blasius-type) boundary layers. The analysis is wholly rational, with (in particular) the highly nonlinear and nonparallel flow effects being treated in an entirely consistent manner.

Separation of variables in the hydrodynamic stability equations

The problem of variable separation in the linear stability equations, which govern the disturbance behaviour in viscous incompressible fluid flows, is discussed. The so-called direct approach, in which a form of the 'ansatz' for a solution with separated variables as well as a form of reduced ODEs are postulated from the beginning, is applied. The results of application of the method are the new coordinate systems and the most general forms of basic flows, which permit the postulated form of separation of variables. Then the basic flows are specified by the requirement that they themselves satisfy the Navier–Stokes equations. Calculations are made for the (1+3)-dimensional disturbance equations written in Cartesian and cylindrical coordinates. The fluid dynamics interpretation and stability properties of some classes of the exact solutions of the Navier–Stokes equations, defined as flows for which the stability analysis can be reduced via separation of variables to the eigenvalue problems of ordinary differential equations, are discussed. The eigenvalue problems are solved numerically with the help of the spectral collocation method based on Chebyshev polynomials. For some classes of perturbations, the eigenvalue problems can be solved analytically. Those unique examples of exact (explicit) solution of the nonparallel unsteady flow stability problems provide a very useful test for numerical methods of solution of eigenvalue problems, and for methods used in the hydrodynamic stability theory, in general.

Perturbation Growth in Baroclinic Waves

Abstract Floquet theory is applied to the stability of time-periodic, nonparallel shear flows consisting of a baroclinic jet plus a neutral wave. This configuration is chosen as an idealized representation of baroclinic waves in a storm track, and the stability analysis may be helpful for understanding generic properties of the growth of forecast errors in such regions. Two useful attributes of Floquet theory relevant to this problem are that the period-average mode growth rate is norm independent, and the t→ ∞ stability limit is determined by the stability over one period. Exponentially growing Floquet modes are found for arbitrarily small departures from parallel flows. Approximately 70% of Floquet-mode growth in energy is due to barotropic conversion, with the remainder due to zonal heat flux. Floquet-mode growth rates increase linearly with neutral wave amplitude (i.e., the “waviness” of the jet) and also increase with neutral wave wavelength. Growth rates for meridionally localized jets are approximately 40% smaller than for comparable cases with linear vertical shear (the Eady jet). Singular vectors for these flows converge to the leading Floquet mode over one basic-state period, and the leading instantaneous optimal mode closely resembles the leading Floquet mode. Initial-value problems demonstrate that the periodic basic states are absolutely unstable, with Floquet modes spreading faster than the basic-state flow both upstream and downstream of an initially localized disturbance. This behavior dominates the convective instability of parallel-flow jets when the neutral baroclinic wave amplitude exceeds a threshold value of about 8–10 K. This result suggests that forecast errors in a storm track may spread faster, and affect upstream locations, for sufficiently wavy jets.

The role of instability waves in predicting jet noise

There is a continuing debate about the role of linear instability waves in the prediction of jet noise. Parallel mean flow models, such as the one proposed by Lilley, usually neglect these waves because they cause the solution to become infinite. The result is then non-causal and can, therefore, be quite different from the more realistic causal solution, especially for the chaotic flows being considered here. The present paper solves the relevant acoustic equations for a non-parallel mean flow by using a vector Green's function approach and requiring that the mean flow be weakly non-parallel, i.e. requiring that the spread rate be small. It demonstrates that linear instability waves must be accounted for in order to construct a proper causal solution to the jet noise problem. Recent experimental results (e.g. see Tam, Golebiowski & Seiner 1996) show that the small-angle spectra radiated by supersonic jets are significantly different from those radiated at larger angles (say, at 90°) and even exhibit dissimilar frequency scalings (i.e. they scale with Helmholtz number as opposed to Strouhal number). The present solution is (among other things) able to explain this rather puzzling experimental result.

On the domain of validity of the near-parallel combined stability analysis for the 2D intermediate and far bluff body wake

MSC (2000) 76D25, 76M45 Local parallelism is a representation scheme of flow instability, which is founded on the assumption of local uniformity of the flow in the streamwise direction, the global disuniformity being given by the slow variations of the local values of the equilibrium flow solution. Since the remarkable numerical experiment by Triantafyllou and Karniadakis [46] proved that the details of the flow separation from the body generating the wake can be disregarded in wake-stability analyses and since the intermediate and far regions of the basic wake flow are actually physical systems which show a slow streamwise evolution, it is here considered – as a first level of extension of the near-parallel analysis – the hypothesis that this evolution influences the instability characteristics mainly through a slow spatial modulation of the instability wave. To determine the domain of validity of the assumption of local parallellism for the two dimensional bluff body wake, a non local analysis of the combined stability of the intermediate and far field regions of the flow has been carried out using the method of multiple spatial scales. The analysis takes into account nonparallel flow effects, such as the streamwise variations of the mean flow, of the disturbance profile function and its longitudinal modulation and of the wavenumber and spatial growth rate. The nonparallel effects are introduced through a basic fiel d–aN avier-Stokes expansion solution that represents the intermediate and far regions of the wake – which involves the streamwise and transverse momentum components as functions of both the coordinates. After the introduction of the slow streamwise variable, this leads to a zero order parametric Orr-Sommerfeld eigenvalue problem and to a first order nonparallel correction, which is represented by a non homogeneous Orr-Sommerfeld equation. The first order correction results in remarkable variations of perturbation wave numbers and spatial growth rates. These variations are distributed in the intermediate region of the field, their intensity increase getting closer to the body up to a level which yields the break-up of the near-parallellism assumption. The break-up point is placed well downstream to the region occupied by the standing vortices. The theory allows one to determine the distance from the body beyond which the local parallel analysis is fully valid. The continuous spectrum of the parametric Orr-Sommerfeld combined eigenvalue problem is also determined. It has been shown that, according to the Briggs criterium, the continuum cannot give rise to unstable single modes.

A minimal composite theory for stability of non-parallel compressible boundary-layer flow

A new “minimal composite” theory that extends the approach of Govindarajan and Narasimha [1] is proposed here for 2D non-parallel compressible boundary-layer stability subject to 3D disturbances. The mean profiles are obtained from Horton’s analysis, which provides a good approximation for a large range of Prandtl numbers at non-zero pressure gradients. In the lowest order, all effects of order lower than O(R-2/3) anywhere in the boundary-layer are included, R being the local boundary-layer Reynolds number; the resulting non-parallel formulation yields a set of four ordinary differential equations, as compared to the five coupled equations of classical parallel flow theory of Mack [2]. The largest effect on stability of flow non-parallelism is found to be due to the wall-normal advection of velocity and temperature disturbance quantities by the mean flow. The present theory shows that the bulk viscosity, invariably included in compressible stability theories, is irrelevant at the lowest order. In comparison with the “full” [O(R-1)] non-parallel theory, the present theory is marginally better than the parallel flow theory.

Hysteresis and precession of a swirling jet normal to a wall.

Interaction of a swirling jet with a no-slip surface has striking features of fundamental and practical interest. Different flow states and transitions among them occur at the same conditions in combustors, vortex tubes, and tornadoes. The jet axis can undergo precession and bending in combustors; this precession enhances large-scale mixing and reduces emissions of NOx. To explore the mechanisms of these phenomena, we address conically similar swirling jets normal to a wall. In addition to the Serrin model of tornadolike flows, a new model is developed where the flow is singularity free on the axis. New analytical and numerical solutions of the Navier-Stokes equations explain occurrence of multiple states and show that hysteresis is a common feature of wall-normal vortices or swirling jets no matter where sources of motion are located. Then we study the jet stability with the aid of a new approach accounting for deceleration and nonparallelism of the base flow. An appropriate transformation of variables reduces the stability problem for this strongly nonparallel flow to a set of ordinary differential equations. A particular flow whose stability is studied in detail is a half-line vortex normal to a rigid plane-a model of a tornado and of a swirling jet issuing from a nozzle in a combustor. Helical counter-rotating disturbances appear to be first growing as Reynolds number increases. Disturbance frequency changes its sign along the neutral curve while the wave number remains positive. Short disturbance waves propagate downstream and long waves propagate upstream. This helical instability causes bending of the vortex axis and its precession-the effects observed in technological flows and in tornadoes.

Optimal Disturbances in Compressible Boundary Layers

The problem of transient growth in compressible boundary layers is considered within the scope of partial differential equations including the nonparallel flow effects. As follows from previous investigations, the optimal disturbances correspond to steady counter-rotating streamwise vortices. The corresponding scaling of the perturbations leads to the governing equations for a Gortler-type of instability, with the Gortler number equal to zero. The iteration procedure employs back and forth marching solutions of the adjoint and original systems of equations

Optimal Disturbances in Compressible Boundary Layers

The problem of transient growth in compressible boundary layers is considered within the scope of partial differential equations including the nonparallel flow effects. As follows from previous investigations, the optimal disturbances correspond to steady counter-rotating streamwise vortices. The corresponding scaling of the perturbations leads to the governing equations for a Gortler-type of instability, with the Gortler number equal to zero. The iteration procedure employs back and forth marching solutions of the adjoint and original systems of equations

The Spatial Stability of Natural Convection Flow on Inclined Plates

The spatial stability of a natural convection flow on upward-facing, heated, inclined plates is revisited. The eigenvalue problem is solved numerically employing two methods: the collocation method with Chebyshev polynomials and the fourth-order Runge-Kutta method. Two modes, traveling waves and stationary longitudinal vortices, are considered. Previous theoretical models indicated that nonparallel effects of the mean flow are significant for the vortex instability mode, but most of them ignored the fact that the eigenfunctions are dependent on the streamwise coordinate as well. In the present work, the method of multiple scales is applied to take the nonparallel flow effects into consideration. The results demonstrate the stabilizing character of the nonparallel flow effects. The vortex instability mode is also considered within the scope of partial differential equations. The results demonstrate dependence of the neutral point on the initial conditions but, farther downstream, the results collapse onto one curve. The marching method is compared with the quasi-parallel normal mode analysis and with theoretical results including correction to nonparallel flow effects. The marching method provides better agreement of theoretical and experimental growth rates.

Marginal and weakly nonlinear stability in spatially developing flows

This work is devoted to revealing the essence of near-critical phenomena in nonlinear problems with nonparallel effects. As a generalization of the well-known concept of linear stability in Fourier space for a parallel basic state, we introduce a new concept valid for nonparallel flows as well. The new picture allows one to demonstrate the possible singular limit to the parallel case. Also, on its basis we derive a weakly nonlinear model valid near criticality. The damped Kuramoto-Sivashinsky equation with variable coefficients is used to illustrate the application of the theory.

Comparison of blood particle deposition models for non-parallel flow domains

Adhesions of monocytes and platelets to a vascular surface, particularly in regions of flow stagnation, recirculation, and reattachment, are a significant initial event in a broad spectrum of particle–wall interactions that significantly influence the formation of stenotic lesions and mural thrombi. A number of approximations are available for the simulation of both monocyte and platelet interactions with the vascular surface. For the simulation of blood particle adhesion, this study hypothesizes that: (a) the discrete element approach, which accounts for finite particle size and inertia, is advantageous in the context of non-parallel flow domains including stagnation, recirculation, and reattachment; and (b) the likelihood for particle deposition may be effectively approximated as being non-linearly proportional to local particle concentration, residence time, and wall proximity. Models such as wall shear stress correlations, the multicomponent mixture approach, and Lagrangian particle tracking with and without hydrodynamic particle–wall interactions were evaluated. Quantitative performance of the selected models was established by comparisons to available experimental data sets for non-parallel axisymmetric suspension flows of monocytes and platelets. Factors including the convective-diffusive transport of particles, finite particle size and inertia, as well as near-wall hydrodynamic interactions were found to significantly influence blood particle deposition. Of the models studied, the near-wall residence time approach was found to be a particularly effective indicator for the deposition of monocytes ( r 2=0.74) and platelets ( r 2=0.57), given that nano-scale physical and biochemical effects must be greatly approximated in computational simulations involving relatively large-scale geometries and complex flow fields.

The Opposing Effect of Viscous Dissipation Allows for a Parallel Free Convection Boundary-Layer Flow Along a Cold Vertical Flat Plate

External free convection boundary-layer flows are usually treated by neglecting the effect of viscous dissipation. This assumption always results in a non-parallel flow, besides a strong parallel component also a weak transversal component of the (steady) velocity field occurs. The present paper shows, however, that the weak opposing effect of the buoyancy forces due to heat release by viscous dissipation, can give rise along a cold vertical plate adjacent to a fluid saturated porous medium to a strictly parallel steady free convection flow. This boundary-layer flow is described by an algebraically decaying exact analytical solution of the basic balance equations.

Generation of collimated jets by a point source of heat and gravity

New solutions of the Boussinesq equations describe the onset of convection as well as the development of collimated bipolar jets near a point source of both heat and gravity. Stability, bifurcation, and asymptotic analyses of these solutions reveal details of jet formation. Convection (with l cells) evolves from the rest state at the Rayleigh number Ra = Racr = (l − 1)l(l + 1)(l + 2). Bipolar (l = 2) flow emerges at Ra = 24 via a transcritical bifurcation: Re = 7(24 − Ra)/(6 + 4Pr), where Re is a convection amplitude (dimensionless velocity on the symmetry axis) and Pr is the Prandtl number. This flow is unstable for small positive values of Re but becomes stable as Re exceeds some threshold value. The high-Re stable flow emerges from the rest state and returns to the rest state via hysteretic transitions with changing Ra. Stable convection attains high speeds for small Pr (typical of electrically conducting media, e.g. in cosmic jets). Convection saturates due to negative ‘feedback’: the flow mixes hot and cold fluids thus decreasing the buoyancy force that drives the flow. This ‘feedback’ weakens with decreasing Pr, resulting in the development of high-speed convection with a collimated jet on the axis. If swirl is imposed on the equatorial plane, the jet velocity decreases. With increasing swirl, the jet becomes annular and then develops flow reversal on the axis. Transforming the stability problem of this strongly non-parallel flow to ordinary differential equations, we find that the jet is stable and derive an amplitude equation governing the hysteretic transitions between steady states. The results obtained are discussed in the context of geophysical and astrophysical flows.

Inertia effects on non-parallel thermal instability of natural convection flow over horizontal and inclined plates in porous media

The inertia effect on the onset of thermal instability in natural convection flow over heated horizontal and inclined flat plates embedded in fluid-saturated porous media is analyzed. The linear non-parallel flow model is employed in the instability analysis, which takes into account the streamwise variation as well as the transverse variation of the disturbance amplitude functions. The set of partial differential equations for the disturbance amplitude functions are converted to a system of homogeneous linear ordinary differential equations with homogeneous boundary conditions by the local non-similarity method. The resulting eigenvalue problem is then solved by an implicit finite-difference method. Representative neutral stability curves and critical Rayleigh numbers are presented. It has been found that as the angle of inclination relative to the horizontal increases, the surface heat transfer rate increases, whereas the flow becomes more stable to the vortex mode of instability. Also, as the inertia effect, expressed in terms of Forchheimer number, Fr, increases, the heat transfer rate decreases, but the flow becomes more stable. It is demonstrated that the non-parallel flow model predicts a more stable flow than the parallel flow model.

Influence of the Prandtl number on the location of recirculation eddies in thermocapillary flows

Two-dimensional thermocapillary-driven flow in a horizontal cavity of large aspect ratio is considered. Of particular interest is the asymptotic matching between the known parallel core flow solution and the non-parallel flow near the ends of the cavity. The influence of the end regions on the core flow is described by the superposition of spatial disturbances to the core flow solution. The stability analysis of this perturbed state leads to an eigenvalue problem for the complex wave number of the disturbances. The end flow structures are related to the nature of the lowest eigenvalue: when purely real it describes a boundary layer regime whereas a non-vanishing imaginary part reveals the existence of periodic structures known as recirculation rolls or eddies. It is found that, depending on the Prandtl number value, Pr, recirculation eddies can exist either near the hot wall ( Pr≫1) or the cold wall ( Pr≪1). Our results provide a direct interpretation to the different behaviors observed in previous experimental and numerical studies.

Higher order stability effects in a natural convection boundary layer over a vertical heated wall

The stability of a laminar boundary layer flow under natural convection on a vertical isothermally heated wall is studied analytically. The analysis is performed by using two different two-dimensional linear models: (1) The non-parallel flow model in which the steady mean flow as well as the disturbance amplitude functions can change in the streamwise direction; (2) The parallel flow model in which the effects of the mean flow and disturbance changes in the streamwise direction are neglected. The linear non-parallel stability analysis is based on the so-called parabolised stability equations (PSEs) which have been successfully applied to the stability analysis of forced convection boundary layers. In this study the PSE equations are applied to natural convection boundary layers in order to show the difference between parallel and non-parallel stability analysis. A second part of this study deals with the effects of variable properties, which are always present in natural convection flows. They are analysed by an extended version of the Orr–Sommerfeld equation (EOSE).

The Estimation of Regional Precipitation Recycling. Part II: A New Recycling Model

Abstract A new two-dimensional analytical recycling model is presented. The model may be thought of as a generalization of the existing (mostly Budyko type) models considered in Part I, which incorporates the additional effects of inhomogeneity of the fluxes and the nonparallel flow effects. The model results are given by a general analytical formula for the recycling ratio distribution. This formula may be applied both for theoretical studies of the influence of regional parameters and flow fields on precipitation recycling and for estimating the recycling ratio from observational data. To study the effects of different flow fields in precipitation recycling a decomposition of the two-dimensional flow field into components of pure flow types is used. In addition to the exact results, simple analytical formulas giving lower bounds of the estimates by the new model are obtained for some basic flow fields. The estimates produced by the model are compared with those given by existing recycling models, and th...