Non-parallel Flow Research Articles

On the growth of waves in boundary layers: a non-parallel correction

The estimation of the growth of propagating instability waves in laminar boundary layers is considered when the Reynolds number is sufficiently large for the mean flow to deviate only slightly from a truly parallel flow. An approximate solution for the linear perturbation is sought in the form of a scaled solution of the related locally parallel flow problem. The amplitude scaling is chosen so as to satisfy the full linearized perturbation equations as closely as possible by making the mean-square deviation of the remainder a minimum. By re-arranging the terms in the equations so that some of the small correction terms arising from the non-parallel mean flow are contained in the ordinary differential equation (ODE) defining the quasi-parallel flow solution, a useful simplification is obtained for the scaling function. Then a modified Orr–Sommerfeld equation defines the base solution and the differential expression for the scaling that can be integrated forms a simple conservation relation.

On resistive magnetohydrodynamic equilibria of an axisymmetric toroidal plasma with flow

It is shown that the magnetohydrodynamic (MHD) equilibrium states of an axisymmetric toroidal plasma with finite resistivity and flows parallel to the magnetic field are governed by a second-order partial differential equation for the poloidal magnetic flux function ψ coupled with a Bernoulli-type equation for the plasma density (which are identical in form to the corresponding ideal MHD equilibrium equations) along with the relation Δ*ψ = Vcσ (here Δ* is the Grad–Schlüter–Shafranov operator, σ is the conductivity and Vc is the constant toroidal-loop voltage divided by 2π). In particular, for incompressible flows, the above-mentioned partial differential equation becomes elliptic and decouples from the Bernoulli equation [H. Tasso and G. N. Throumoulopoulos, Phys. Plasma5, 2378 (1998)]. For a conductivity of the form σ = σ(R, ψ) (where R is the distance from the axis of symmetry), several classes of analytic equilibria with incompressible flows can be constructed having qualitatively plausible σ profiles, i.e. profiles with σ taking a maximum value close to the magnetic axis and a minimum value on the plasma surface. For σ = σ(ψ), consideration of the relation Δ*ψ = Vc σ(ψ) in the vicinity of the magnetic axis leads then to a proof of the non-existence of either compressible or incompressible equilibria. This result can be extended to the more general case of non-parallel flows lying within the magnetic surfaces.

Receptivity of three-dimensional boundary-layers to localized wall roughness and suction

The receptivity of three-dimensional, incompressible, boundary-layer flows to localized roughness or suction is studied theoretically and numerically. Three-dimensional boundary-layer flows have a strong “nonparallel” component due to the curvature of the potential-flow streamlines. Our theoretical model extends the Fourier-transform methodology to nonparallel flows using a Taylor expansion of the laminar mean-flow at the location of the roughness. Both the near-field and the far-field solutions are contained in the model. Additionally, the use of MacLaurin expansions in the complex plane leads quickly to the receptivity factors of eigenmodes. The theoretical results are validated with solutions to the linearized Navier–Stokes equations efficiently obtained by replacing the actual wall forcing with a smoother, but equivalent, forcing. We find that the far-field response is proportional to both H̃(αe) and dH̃(αe)/dα, where H̃(α) is the Fourier transform of the forcing distribution, and αe is the eigenmode’s wave number. The receptivity coefficient for H̃(αe) decreases when nonparallelism is included in our models, while that for dH̃(αe)/dα increases.

Exact solutions of non-Newtonian fluid flows with prescribed vorticity

The equations of motion of a non-Newtonian second-grade fluid flow are highly nonlinear partial differential equations. For this reason, there exists only a limited number of exact solutions. Due to the complexity of the equations, inverse methods described by Nemenyi [1] have become attractive in the study of non-Newtonian fluids. In these methods, certain physical or geometrical properties of the flow field are assumed a priori. Lin and Tobak [2] studied steady plane viscous incompressible flows for a chosen vorticity function by decomposing the nonlinear fourth-order partial differential equation in the streamfunction. This excellent approach yielded two second-order linear partial differential equations in the streamfunction. Hui [3] used this approach to study unsteady plane viscous incompressible flows. During the past decade, there has been substantial interest in flows of viscoelastic liquids due to the occurrence of these flows in industrial processes. In this paper, we study the steady and unsteady incompressible viscous non-Newtonian second-grade fluid flows in which the vorticity is proportional to the streamfunction perturbed by a uniform stream. The solutions obtained are exact solutions and represent various non-parallel flows of second-grade fluids. The plan of this paper is as follows: In Sect. 2, the equations of motion of an unsteady plane incompressible second-grade fluid are given, and the vorticity function is assumed to be∇ 2 ψ=A(ψ−Ux−BUy 2). In Sect. 3, solutions for the steady flow are obtained. In Sect. 4, solutions for unsteady flow are obtained.

Control of global instability in a non-parallel near wake

The effect of base suction on a plane wake was found to produce significant changes in wake dynamics. The wake is produced by merging two boundary layers from the trailing edge of a splitter plate in a two-stream water tunnel. A threshold suction speed exists which is approximately equal to half of the free-stream velocity. If the suction speed is below the threshold, the wake flow is unstable. If the suction speed is above the threshold, the wake becomes stable and no vortex shedding is observed. In the present experiment, the suction technique can stabilize a wake at a maximum tested Reynolds number of 2000.The suction significantly reduces the length of the absolutely unstable region in the immediate vicinity of the trailing edge of the splitter plate and produces a non-parallel flow pattern, resulting in the breakdown of global instability. The global growth rate changes from positive (unstable flow) to negative (stable flow) at the suction speed equalling 0.46 of the free-stream velocity. The threshold suction speed can be accurately predicted by the global linear theory of Monkewitz et al. (1993) with a non-parallel flow correction.

Sliding over anisotropic beds

AbstractMany glacier beds are anisotropic, by which is meant that the dominant wavelengths are different in the two map-plane directions. A largely unexplored consequence of Nye-Kamb sliding theory is the fact that an anisotropic bed can produce a sliding velocity not parallel to the tangential traction vector. This has important consequences, since observations of non-parallel flow are often taken as indications that the shallow-ice approximation has broken down, whereas this need not be the case with an anisotropic bed.Mathematically, this effect can be incorporated through the use of a sliding tensor. The mathematical properties of this tensor are outlined, and the correct "invariant" for the sliding law, a quadratic form, is deduced. Nye-Kamb theory for anisotropic beds is discussed. Flow on the infinite plane and the properties of surface-topography diffusion are elucidated. The properties of kinematic waves and shock waves are discussed. Kinematic waves can have a lateral component. Numerical computations of ice-sheet flow on beds with anisotropic roughness are presented, with emphasis placed on how this affects divide-ridge structure It is suggested that cold-based ice sheets, which have an anisotropic bed affecting the shear layer, may also show non-parallelism of surface slope and velocity.

Fully nonlinear global modes in slowly varying flows

We study the existence of nonlinear solutions of the real Ginzburg–Landau amplitude equation, with varying coefficients when the solution is subject to a boundary condition at x=0. These solutions, called nonlinear global modes, are explicitly obtained from a matched asymptotic expansion when nonlinear effect dominates over the inhomogeneity. The dynamics of this model is believed to mimic the behavior of strongly nonlinear but weakly nonparallel basic flow (basic flow varying in the streamwise direction). For the model, we derive scaling laws for the amplitude of nonlinear global modes and for the position of the maximum that explain for the first time the experimental observations of Goujon-Durand et al. [Phys. Rev. E 50, 308 (1994)] and the numerical simulations of Zielinska and Wesfreid [Phys. Fluids 7, 1418 (1995)] of the wake behind bluff bodies.

Ideal magnetohydrodynamic equilibria with helical symmetry and incompressible flows

A recent study on axisymmetric ideal magnetohydrodynamic equilibria with incompressible flows [H. Tasso and G. N. Throumoulopoulos, Phys. Plasmas5, 2378 (1998)] is extended to the generic case of helically symmetric equilibria with incompressible flows. It is shown that the equilibrium states of the system under consideration are governed by an elliptic partial differential equation for the helical magnetic flux function containing five surface quantities along with a relation for the pressure. The above-mentioned equation can be transformed to one possessing a differential part identical in form to the corresponding static equilibrium equation, which is amenable to several classes of analytical solutions. In particular, equilibria with electric fields perpendicular to the magnetic surfaces and non-constant-Mach-number flows are constructed. Unlike the case in axisymmetric equilibria with isothermal magnetic surfaces, helically symmetric T = T(ψ) equilibria are overdetermined, i.e. in this case the equilibrium equations reduce to a set of eight ordinary differential equations with seven surface quantities. In addition, the non-existence is proved of incompressible helically symmetric equilibria with (a) purely helical flows and (b) non-parallel flows with isothermal magnetic surfaces and with the magnetic field modulus a surface quantity (omnigenous equilibria).

Axisymmetric propagating vortices in the flow between a stationary and a rotating disk enclosed by a cylinder

The destabilization of the stationary basic flow occurring between two disks enclosed by a cylinder is studied experimentally when the radius of the disks is large compared to the spacing. In the explored range of the cell aspect ratio, when one disk only is rotating, circular vortices propagating to the centre are observed above a critical angular velocity. These structures occur naturally but can also be forced by small modulations of the angular velocity of the disk. For each rotation rate the dispersion relation of the instability is experimentally reconstructed from visualizations and it is shown that this dispersion relation can be scaled by the boundary layer thickness measured over the disk at rest. The bifurcation is found to be of supercritical nature. The effect of the forcing amplitude is in favour of a linear convective nature of this instability of the non-parallel inward flow existing above the stationary disk. The most unstable temporal frequency is found to be about four times the frequency of the rotating disk. The evolution of the threshold of this primary instability is described for different aspect ratios of the cell. Finally, two sets of experiments made under transient conditions are presented: one in order to investigate further a possible convective/absolute transition for the instability, and the other to compare with the impulsive spin-down-to-rest experiments of Savas (1983).

Perturbed vortical layers and shear sheltering

New theoretical results and physical interpretations are presented concerning the interactions between different types of velocity fields that are separated by thin interfacial layers, where there are dynamically significant variations of vorticity across the layers and, in some cases within them. It is shown how, in different types of complex engineering and environmental flow, the strengths of these interactions vary from the weakest kind of superposition to those where they determine the flow structure, for example by mutual exclusion of velocity fields from the other region across the interface, or by local resonance near the interface. We focus here on the excluding kinds of interactions between, on the one hand, elongated and compact regions containing vortical flows and large variations in velocity, and on the other hand various kinds of weak perturbation in the surrounding external flow region: rotational, irrotational; time-varying, steady; large, small; coplanar, non-coplanar; non-diffusive, diffusive. It is shown how all these kinds of external disturbances can be wholly, or partially, ‘blocked’ at the interface with the vortical region, so that beyond a certain sheltering distance into the interior of this region the fluctuations can be very small. For the special case of quasi-parallel co-planar external straining motions outside non-directional shear flows, weak sheltering occurs if the mean velocity of the shear flow increases – otherwise the perturbations are amplified. For non-parallel flows, the sheltering effect can be greater when the vorticity is distributed in thin vortex sheets. The mechanism whereby the vortical flow induces ‘blocking’ and ‘shear-sheltering’ effects can be quantitatively explained in terms of the small adjustments of the vorticity in the vortical layers, and in some cases by the change in impulse of these layers. If the vorticity in the outer part of the vortical region is weak, it can be ‘stripped away’ by the external disturbances until the remaining vorticity is strong enough to ‘block’ the disturbances and shelter the inner flow of the vortical region. The mechanisms presented here appear to explain on the one hand some aspects of the observed robustness of vortical structures and jet or plume like shear flows in turbulent and geophysical flows, and on the other hand the levels of external perturbation needed to erode or breakdown turbulent shear flows.

Roughness-induced receptivity to crossflow vortices on a swept wing

The generation of stationary crossflow vortices on the 45 degree swept NLF (2)-0415 airfoil at the Arizona State University Unsteady Wind Tunnel (AIAA Paper No. 96-0184, 1996; Ph.D. thesis, Arizona State University, 1996) is analyzed using a finite Reynolds-number linear receptivity theory. The receptivity theory is based on the locally-parallel flow approximation and also neglects surface curvature. The vortices are excited by a spanwise periodic array of circular roughness elements located near the attachment line. The amplitudes of the crossflow vortices at the source location are computed and compared with the experimental results. In general, the agreement is remarkably good. This suggests that the combined effect of nonparallel flow and surface curvature on the receptivity is not significant for the range of conditions considered. The linear analysis captures the receptivity variations with Reynolds number, roughness geometry, and roughness location. When the roughness height is small, the initial amplitudes of the roughness-generated crossflow vortices are well predicted using the finite Reynolds-number receptivity theory. Nonlinear effects appeared to become significant only when the roughness height exceeds about 10% of the local boundary layer displacement thickness.

Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis

The eigensolution method, more flexible and efficient than previously used for global non-parallel flow stability analysis, is presented. The differential eigenvalue problem, resulting from linear stability equations of non-parallel flow is discretized with the penalty FEM. The large-dimensional algebraic eigenvalue problem is solved with the use of Subspace Iteration Method. Separation of the eigenvalues interesting for the flow stability analysis is performed with the inverse Cayley transformation. It is shown that certain barriers, resulting from very large dimensions of the eigenvalue problem and limiting the non-parallel flow stability method can be overcome with this approach. To demonstrate the algorithm, the stability of the flow around circular cylinder is analyzed.

Error Growth in Flows with Finite-Amplitude Waves or Coherent Structures

Abstract Explanations of error growth in atmospheric flows are often based on the extension of barotropic and baroclinic instabilities from steady parallel flows to weakly nonparallel and time-dependent flows. Consideration of simple flows with finite-amplitude waves, however, suggests an additional scenario for error growth: an initial error that changes the wave amplitude or the medium through which the wave propagates will alter the propagation of the wave and result in a growing phase error. This scenario is illustrated and generalized to other coherent structures through several examples for which analytic solutions are available. For a basic state of a barotropic Rossby wave, growing phase errors account for the most rapidly growing disturbances over time intervals long compared to the basic-state advective timescale; over shorter intervals, amplifications of phase errors are smaller than, but comparable to, the optimal amplification. The role of this mechanism in forecast error growth is less certa...

Elementary stratified flows with instability at large Richardson number

In contrast to the Miles–Howard theorem for inviscid steady shear flow in stably stratified fluids, explicit elementary time-periodic solutions of the Boussinesq equations are developed here which are unstable for arbitrarily large Richardson numbers. These elementary flows are parameterized through solutions of a nonlinear pendulum equation and involve spatially constant but temporally varying vorticity and density gradients which interact through advection and baroclinic vorticity production. Exact nonlinear solutions for arbitrary wave-like disturbances for these flows are developed here and Floquet theory combined with elementary numerical calculations is utilized to demonstrate instability at all large Richardson numbers. The dominant inviscid instability for these non-parallel flows is a purely two-dimensional parametric instability with twice the period of the elementary flow and persists for all Reynolds numbers and the wide range of Prandtl numbers, 1[les ]Pr[les ]200, investigated here. Similar elementary time-periodic solutions of the Boussinesq equations in a constant external strain field are developed here which reduce to uniform shear flows in one extreme limit and the time-periodic vortical flows in the other extreme limit. These flows are stable in a strict sense for large Richardson numbers; however there is transient large-amplitude non-normal behaviour which yields effective instability for a wide range of Richardson numbers. For example, suitable initial perturbations can amplify by at least a factor of fifty with exponential growth for short times for [Rscr ]i=1, [mid ]σ[mid ][les ]0.5 and [Rscr ]i=5, [mid ]σ[mid ][les ]0.1, with σ the amplitude of the external strain.

On the Convective and Absolute Nature of Instabilities in Finite Difference Numerical Simulations of Open Flows

In numerical simulations of unstable flows the absolute or convective nature of the instability can be modified by numerical effects. We introduce a convective/absolute analysis of the dispersion relations associated with discretized operators. This analysis leads to conditions on the discretization parameters in order to avoid numerical transition from absolute to convective instability and vice versa. In numerical simulations of non-parallel flows, local numerical transitions, of the kind described in this paper, could lead to the wrong global dynamics.

Axisymmetric ideal magnetohydrodynamic equilibria with incompressible flows

It is shown that the ideal magnetohydrodynamic (MHD) equilibrium states of an axisymmetric plasma with incompressible flows are governed by an elliptic partial differential equation for the poloidal magnetic flux function ψ containing five surface quantities along with a relation for the pressure. Exact equilibria are constructed including those with nonvanishing poloidal and toroidal flows and differentially varying radial electric fields. Unlike the case in cylindrical incompressible equilibria with isothermal magnetic surfaces which should have necessarily circular cross sections [G. N. Throumoulopoulos and H. Tasso, Phys. Plasmas 4, 1492 (1997)], no restriction appears on the shapes of the magnetic surfaces in the corresponding axisymmetric equilibria. The latter equilibria satisfy a set of six ordinary differential equations which for flows parallel to the magnetic field B can be solved semianalytically. In addition, it is proved the nonexistence of incompressible axisymmetric equilibria with (a) purely poloidal flows and (b) nonparallel flows with isothermal magnetic surfaces and |B|=|B|(ψ) (omnigenous equilibria).

Theoretical Study on Instability Mechanism of Jet-induced Sloshing. Model Development using Orr-Sommerfeld Equation Generalized for Non-Parallel Flow.

A theoretical model was developed to study the mechanism of free surface sloshing in a vessel induced by a steady vertical jet flow. In the model, jet deflection is calculated with eigen values of the generalized Orr-Sommerfeld equation which is applicable to slightly non-parallel jet. Instability criteria employed in the model are (1) resonace condition between sloshing and jet frequencies and (2) π phase relation between jet displacement at an inlet and global jet deflection. Numerical results of the mathematical model have shown good agreement with experimental ones, which justifies that the inherent instability of free jet itself and edge tone feedback are the main causes of the self-excited sloshing.

A low-order theory for stability of non-parallel boundary layer flows

As a sequel to the earlier analysis of Govindarajan and Narasimha, we formulate here the lowest‐order rational asymptotic theory capable of handling the linear stability of spatially developing two‐dimensional boundary layers. It is shown that a new ordinary differential equation, using similarity‐transformed variables in Falkner–Skan flows, provides such a theory correct upto (but not including) O ( R −2/3), where R is the local boundary layer thickness Reynolds number. The equation so derived differs from the Orr–Sommerfeld in two respects: the terms representing streamwise diffusion of vorticity are absent; but a new term for the advection of disturbance vorticity at the critical layer by the mean wall‐normal velocity was found necessary. Results from the present lowest‐order theory show reasonable agreement with the full O ( R −1) theory. Stability loops at different wall‐normal distances, in either theory, show certain peculiar characteristics that have not been reported so far but are demonstrated here to be necessary consequences of flow non‐parallelism.

Simulation techniques for spatially evolving instabilities in compressible flow over a flat plate

In this paper we present numerical techniques suitable for a direct numerical simulation in the spatial setting. We demonstrate the application to the simulation of compressible flat plate flow instabilities. We compare second and fourth order accurate spatial discretization schemes in combination with explicit multistage time stepping for the simulation of the 2D Navier-Stokes equations. We consider Mach numbers 0.5 and 4.5. In the vicinity of the outflow boundary, an efficient buffer domain treatment is introduced, which is suitable in conjunction with an explicit time integration scheme. This treatment requires only a short buffer domain to damp wave reflections at the outflow boundary. Results for the instability of Tollmien-Schlichting (T-S) waves are compared with two instability theories, linear stability theory (LST) and linear parabolized stability equations (PSE). The growth rates of T-S waves for parallel base flow at both Mach numbers compare well with LST results. Moreover, the growth rates of T-S waves for nonparallel base flow compare well with results obtained by solving the PSE at Mach number 0.5. The second order discretization scheme requires, however, considerably higher grid resolution than the fourth order method to achieve accurate results. High amplitude disturbances were also considered to activate nonlinear terms. The nonlinearity strongly affects the form of the T-S waves and the growth rate of the disturbances. The results obtained here support the use of these numerical techniques in flow simulations with increasing complexity such as flat plate flow simulations up to the turbulent regime and with separation regions in 3D. The results also encourage the use of perturbations derived from the compressible PSE as inlet perturbations for nonparallel flow.

Global Measures of Local Convective Instabilities

We examine the linear stability of the Ginzburg-Landau operator with spatially varying coefficients, which mimics strongly nonparallel open flows such as wakes, jets, and boundary layers. The streamwise non-normality of the global eigenmodes explains the observed large transient growths, classically interpreted in terms of local convective instability. The use of pseudospectra provides an exact measure of spatial amplification and aids in the determination of when entrance noise dominates the open-flow dynamics.