Non-parallel Boundary Research Articles

Exact surface energy of the Hubbard model with nonparallel boundary magnetic fields

Exact surface energy of the Hubbard model with nonparallel boundary magnetic fields

Neural prediction model for transition onset of a boundary layer in presence of two-dimensional surface defects

Predicting the laminar to turbulent transition is an important aspect of computational fluid dynamics because of its impact on skin friction. Traditional transition prediction methods such as local stability theory or the parabolized stability equation method do not allow for the consideration of strongly non-parallel boundary layer flows, as in the presence of surface defects (bumps, steps, gaps, etc.). A neural network approach, based on an extensive database of two-dimensional incompressible boundary layer stability studies in the presence of gap-like surface defects, is used. These studies consist of linearized Navier–Stokes calculations and provide information on the effect of surface irregularity geometry and aerodynamic conditions on the transition to turbulence. The physical and geometrical parameters characterizing the defect and the flow are then provided to a neural network whose outputs inform about the effect of a given gap on the transition through the ${\rm \Delta} N$ method (where N represents the amplification of the boundary layer instabilities).

Inverse Scattering Transform for the Defocusing Manakov System with Non-Parallel Boundary Conditions at Infinity

Inverse Scattering Transform for the Defocusing Manakov System with Non-Parallel Boundary Conditions at Infinity

Surface energy of the one-dimensional supersymmetric [formula omitted] model with general integrable boundary terms in the antiferromagnetic sector

In this paper, we study the surface energy of the one-dimensional supersymmetric t−J model with nonparallel boundary magnetic fields, which is a typical U(1)-symmetry broken quantum integrable strongly correlated electron system. It is shown that at the ground state, the contribution of inhomogeneous term in the Bethe ansatz solution of eigenvalues of transfer matrix satisfies the finite size scaling law Lβ where β<0. Based on it, the physical quantities of the system in the thermodynamic limit are calculated. We obtain the patterns of Bethe roots, root densities, ground state energy and surface energy. We also find that there exist the stable boundary bound states if the boundary fields satisfy some constraints.

Wall Cooling Effect on High-Enthalpy Supersonic Modes over a Cone

High-enthalpy hypersonic boundary-layer transition plays an important role in many entry/descent vehicles. The aerothermodynamic performance of these vehicles strongly depends on the transition location on the surface. However, detailed transition flow physics in these chemically reacting boundary layers are poorly understood and transition estimates during the design phase rely heavily on empirically derived transition criteria such as . One of the most intriguing characteristics in hypersonic boundary layers is the presence of unstable supersonic modes, first identified in the 1990s. Due to a recent surge in hypersonic applications, there has been renewed interest in studying the flow physics of supersonic modes using either theory or direct numerical simulations. This paper investigates the rise of supersonic modes in a high-enthalpy hypersonic flow over a 5 deg half-angle cone at various wall temperatures using both quasi-parallel linear stability theory and linear parabolized stability equations (PSEs). It was found that supersonic modes exist in all wall temperature conditions, including the adiabatic wall case. A cooler wall temperature causes the second Mack mode to become an unstable supersonic mode naturally downstream of the upper-branch neutral location when the wall is sufficiently cooled. In terms of the integrated growth, the second mode is still the dominant mode. Nonetheless, supersonic modes can cause additional series of relatively weaker growth than that of the second mode beyond the peak amplitude location. According to the present linear PSE results, the formation of unstable supersonic modes is mainly associated with the synchronization of phase speed between the instability and acoustic waves in a nonparallel boundary layer, and not due to nonlinear modal interaction as suggested in the literature.

Local receptivity in non-parallel boundary layer

The research on boundary-layer receptivity is the key issue for the laminarturbulent transition prediction in fluid mechanics. Many of the previous studies for local receptivity are on the basis of the parallel flow assumption which cannot accurately reflect the real physics. To overcome this disadvantage, local receptivity in the non-parallel boundary layer is studied in this paper by the direct numerical simulation (DNS). The difference between the non-parallel and parallel boundary layers on local receptivity is investigated. In addition, the effects of the disturbance frequency, the roughness location, and the multiple roughness elements on receptivity are also determined. Besides, the relations of receptivity with the amplitude of free-stream turbulence (FST), with the roughness height, and with the roughness length are ascertained as well. The Tollmien- Schlichting (T-S) wave packets are excited in the non-parallel boundary layer under the interaction of the FST and the localized wall roughness. A group of T-S waves are separated by the fast Fourier transform. The obtained results are in accordance with Dietz’s measurements, Wu’s theoretical calculations, and the linear stability theory (LST).

Persistent currents in open spin chains

Nonparallel boundary magnetic fields can induce a longitudinal (spin) current in a quantum spin chain. We use functional approaches to formulate the spectral problem for the spin-1/2 Heisenberg model subject to a class of integrable boundary conditions in terms of an infinite hierarchy of nonlinear integral equations. From these equations, we compute finite-size corrections to the ground-state energy of the antiferromagnetic chain and the induced spin current for a certain range of boundary parameters.

Nonlinear Stability of Supersonic Nonparallel Boundary Layer Flows

This article studies the nonlinear evolution of disturbance waves in supersonic nonparallel boundary layer flows by using nonlinear parabolic stability equations (NPSE). An accurate numerical method is developed to solve the equations and march the NPSE in a stable manner. Through computation, are obtained the curves of amplitude and disturbance shape function of harmonic waves. Especially are demonstrated the physical characteristics of nonlinear stability of various harmonic waves, including instantaneous stream wise vortices, spanwise vortices and Λ structure etc, and are used to study and analyze the mechanism of the transition process. The calculated results have evidenced the effectiveness of the proposed NPSE method to research the nonlinear stability of the supersonic boundary layers.

On Nonlinear Evolution of C-type Instability in Nonparallel Boundary Layers

The process of evolution, especially that of nonlinear evolution, of C-type instability of laminar-turbulent flow transition in nonparallel boundary layers are studied by means of a newly developed method called parabolic stability equations (PSE). Initial conditions, which are very important for the nonlinear problem, are investigated by computing initial solution of the harmonic waves, modifying the mean-flow-distortion, and giving initial value of TS wave and its subharmonic waves at initial station by solving linear PSE. A numerical method with high-order accuracy are developed in the text, the key normalization conditions in the PSE are satisfied, and nonlinear PSE are solved efficiently and implemented stably by the spatial marching. It has been shown that the computed process of nonlinear evolution of C-type instability in Blasius boundary layer is in good agreement with the experimental results.

Parabolic approach to optimal perturbations in compressible boundary layers

Optimal perturbations in compressible, non-parallel boundary layers are considered here. The flows past a flat plate and past a sphere are analysed. The governing equations are derived from the linearized Navier-Stokes equations by employing a scaling that relies on the presence of streamwise vortices, which are well-known for being responsible for the ‘lift-up’ effect. Consequently, the energy norm of the inlet perturbation encompasses the wall-normal and spanwise velocity components only. The effect of different choices of the energy norm at the outlet is studied, testing full (all velocity components and temperature) and partial (streamwise velocity and temperature only) norms. Optimal perturbations are computed via an iterative algorithm completely derived in the discrete framework. The latter simplifies the derivation of the adjoint equations and the coupling conditions at the inlet and outlet.Results for the flat plate show that when the Reynolds number is of the order of , a significant difference in the energy growth is found between the cases of full and partial energy norms at the outlet. The effect of the wall temperature is in agreement with previous parallel-flow results, with cooling being a destabilizing factor for both flat plate and sphere. Flow divergence, which characterizes the boundary layer past the sphere, has significant effects on the transient growth phenomenon. In particular, an increase of the sphere radius leads to a larger transient growth, with stronger effects in the vicinity of the stagnation point. In the range of interesting values of the Reynolds number that are typical of wind tunnel tests and flight conditions for a sphere, no significant role is played by the wall-normal and streamwise velocity components at the outlet.

Casimir effect of the Maxwell-Chern-Simons field for two non-parallel lines boundary

Based on the Faddeev formalism of path-integral quantization for a constrained Hamiltonian system, the Casimir effect between two non-parallel lines in the (2+1)-dimensional space is calculated by using conformal mapping and Plana summation formula in the theory of complex variable function. Without introducing any cutoff of parameter, the finite analytical expression is obtained.

Nonlinear evolution analysis of T-S disturbance wave at finite amplitude in nonparallel boundary layers

The nonlinear evolution problem in nonparallel boundary layer stability was studied. The relative parabolized stability equations of nonlinear nonparallel boundary layer were derived. The developed numerical method, which is very effective, was used to study the nonlinear evolution of T-S disturbance wave at finite amplitudes. Solving nonlinear equations of different modes by using predictor-corrector and iterative approach, which is uncoupled between modes, improving computational accuracy by using high order compact differential scheme, satisfying normalization condition, determining tables of nonlinear terms at different modes, and implementing stably the spatial marching, were included in this method. With different initial amplitudes, the nonlinear evolution of T-S wave was studied. The nonlinear nonparallel results of examples compare with data of direct nemerical simulations (DNS) using full Navier-Stokes equations.

On the application of en-methods to three-dimensional boundary-layer flows

Extension of the e n -method from two-dimensional to three-dimensional boundary-layer flows has not been straightforward. Confusion has centred on whether to use temporal or spatial stability theories, conversion between the two approaches, and the choice of integration path. The aim of this study is to clarify the confusion about the direction and magnitude of maximum growth in convectively unstable three-dimensional non-parallel boundary layers. To this end, the time-asymptotic response of the boundary layer to an impulsive point excitation is considered. Since all frequencies and all wavenumbers are excited by an impulsive point source, the most amplified component of the response is equivalent to the result of maximizing the growth over arbitrary choices of harmonic point excitation; the standard e n -approach. The impulse response is calculated using a spatial steepest-descent method, which is distinct from the earlier Cebeci–Stewartson method. It is necessary to allow both time and spanwise distance to become complex during integration, but with the constraint that both are real at the end point. This method has been applied to the two-dimensional Blasius boundary layer, for which validation of the method is more straightforward, and also to a three-dimensional Falkner–Skan–Cooke (with non-zero pressure gradient and sweep) boundary layer. Dimensional frequencies and spanwise wavenumbers of propagating components are kept constant (although not necessarily real), as is physically relevant to steady flows with spatial inhomogeneity in the chordwise direction only. With this method a spatial approach is taken without having to make a priori choices about the value of disturbance frequency or wavenumber. Further, purely by choosing a downstream observation point, it is possible to find the maximum-amplitude component directly without having to calculate the entire impulse response (or wave packet). If the flow is susceptible to more than one convective instability mode, provided the modes are separated in the frequency–wavenumber space, separate n-factors can be calculated for each mode. Wave-packet propagation in the Ekman layer (a strictly parallel three-dimensional boundary layer) is also discussed to draw comparisons between the conditions for maximum growth in parallel and non-parallel boundary layers.

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.

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.

Expansion problems in the linear stability of boundary layer flows

Several problems in the linearized stability of boundary layers are examined. They are all treated as perturbations of constant coefficient differential operators. Spectral theory and spectral expansions are developed. Possible anomalies, which might arise for nonparallel boundary layer flows with nonzero transverse component at infinity are also handled.

On spatial and temporal stability in three dimensional fluid flows

A three-dimensional stability analysis is presented for an incompressible viscous non-parallel boundary layer flow over a rotating disk. The method of multiple scales is used to derive partial differential equations that describe the axial and cross-flow variations of the disturbance amplitude, phase and wave numbers. These equations are then employed to derive the expressions that relate the spatial and temporal stabilities. These equations are also used to obtain the expression that relates the spatial growth rate along a given direction to a spatial growth rate along any other direction. These relations are verified by numerical results, obtained for three-dimensional disturbances. The numerical results are presented in order to relate the spatial and temporal stability for different values of Reynolds number, wave number and frequency. It is shown that the predicted results are in excellent agreement with those of the present numerical calculations. Further, it is predicted for a quasiparallel flow that any spatial growth rate can be transformed into a temporal growth rate, while a temporal growth rate can be transformed into a spatial growth rate along any specified direction.

On the Non-linear Evolution of Görtler Vortices in Non-parallel Boundary Layers

The non-linear development of finite amplitude Görtler vortices in a non-parallel boundary layer on a curved wall is investigated using perturbation methods based on the smallness of e, the non-dimensional wavelength of the vortices. The crucial stage in the growth or decay of the vortices takes place in an interior viscous layer of thickness O(ε2) and length O(ε). In this region the downstream velocity component of the perturbation contains a mean flow correction of the same order of magnitude as the fundamental which is driving it. Moreover, these functions satisfy a pair of coupled non-linear partial differential equations which must be solved subject to some initial conditions imposed at a given downstream location. It is found that, depending on whether the boundary layer is more or less unstable downstream of this location, the initial disturbance either grows into a finite amplitude Görtler vortex or decays to zero. For the Blasius boundary layer on a concave wall it is found that Görtler vortices can only develop if the rate of increase of curvature of the wall is sufficiently large. In this case the finite amplitude solution which develops initially in an ε-neighbourhood of the position where the disturbance is introduced changes its structure further downstream. This structure is investigated at a distance O(εδ) (with 0< δ<1) downstream of the above ε-neighbourhood. In this régime the downstream fundamental velocity component has an elliptical profile over most of the flow field. However, in two thin boundary layers located symmetrically either side of the centre of the viscous layer the fundamental velocity component decays exponentially to zero. The locations of these layers are determined by an eigenvalue problem associated with the one-dimensional diffusion equation. The mean flow correction persists both sides of the boundary layer and ultimately decays exponentially to zero. This large amplitude motion is not sensitive to the imposed initial conditions and appears to be the ultimate state of any initial disturbance. However, in the initial stages of the growth of the vortex, some surprising flows are possible. For example, it is possible to set up a vortex flow similar to that observed by Wortmann (1969) which consists of a sequence of cells inclined at an angle to the vertical.