On the evolution of large scale structures in three-dimensional mixing layers

In this paper, several mathematical models for the large scale structures in some special kinds of mixing layers, which might be practically useful for enhancing the mixing, are proposed. First, the linear growth rate of the large scale structures in the mixing layers was calculated. Then, using the much improved weakly non-linear theory, combined with the energy method, the non-linear evolution of large scale structures in two special mixing layer configurations is calculated. One of the mixing layers has equal magnitudes of the upstream velocity vectors, while the angles between the velocity vectors and the trailing edge were π/2-φ and π/2+φ, respectively. The other mixing layer was generated by a splitter-plate with a 45-degree-sweep trailing edge. The project supported by the National Natural Science Foundation of China (19642001) and Deutsche Forschungsgemeinschaft (DFG)     

The evolution of nonlinear waves in active media

An evolution equation that describes the behaviour of single waves in active media is derived. The basic mathematical model corresponds to pulse transmission in nerve fibers according to the hyperbolic telegraph equation. A numerical experiment is carried out in which the evolution equation is solved by the pseudospectral method and the corresponding stationary wave equation by the standard Runge-Kutta method. The evolution equation has a stationary solution in the form of an unsymmetric solitary wave with a refractive tail. The numerical simulation of the process gives physically admissible results, namely, the suitable wave profile and the existence of the threshold and asymptotic values. The situation analyzed here in an active medium with energy influx is in a certain sense similar to the formation of solitary waves in a conservative medium.     

Formation and evolution of the streamwise vortices in mixing layers

The evolution of the three-dimensional time-developing mixing layer is simulated numerically using the pseudo-spectral method. The initial perturbations used in this study consisted of the two-dimensional fundamental wave and the streamwise-invariant three-dimensional disturbance. A comparison of the formations of the streamwise vortices with different amplitude functions for three-dimensional disturbances is made. In one case the results are similar to that of Rogers & Moser^[1], whereas a different way in which the quadrupole forms and sudden expansion of the rib are observed in another case. The simulation also confirms that the stretching by the forming roller rather than Rayleigh centrifugal instability is responsible for the formation of the rib. Finally, numerical flow visualization results are presented. The project supported by the Zhejiang Province Natural Science Special Fund for Youth Scientistis' Cultivation.     

The evolution of free-disturbances in a two-dimensional nonlinear critical layer

This paper investigates the evolution of (relatively) long Rayleigh waves on an inflectional two-dimensional boundary layer such as may occur when a flow encounters a small surface mounted obstacle. Under the assumption that the flow remains essentially two-dimensional a coupled set of evolution equations are derived that describe the nonlinear growth of an essentially arbitrary (2D) disturbance to the base flow. Numerical solutions are presented for a representative initial condition.     

High-resolution measurements of two-dimensional instabilities and turbulence transition in plane mixing layers

 High-resolution two-dimensional (2D) measurements on a large plane mixing layer provide new quantitative information of its spatial and temporal evolution to turbulence. Periodic acoustic excitation with three frequencies was used to stabilize the fundamental instability of the mixing layer (roll-up) and its first and second subharmonics (vortex pairings). Phase-locked velocity measurements of the time evolution in 2D space (x, y, t) reveal accurate spatially resolved primary (2D) instabilities of the mixing layer and turbulence transition. The measurements unveil new quantitative details of the initial Kelvin–Helmholtz waves and their spatial and temporal evolution into vortex shedding and the effect of the second subharmonic on the first vortex pairing. The second-subharmonic effect hastens alternate first pairings of the rollers, with the result that pairing is completed at two downstream locations. The pairings that occur closer to the knife-edge are more organized (laminar) than those occurring farther downstream (transitional). This effect is corroborated using Taylor’s hypothesis to compute the vorticity distributions from the measured velocity field and a pseudo-spectral simulation of the temporal evolution of the mixing layer. Received: 26 March 1998/Accepted: 2 March 1999     

Localized nonlinear, soliton-like waves in two-dimensional anharmonic lattices

We discuss here nano-scale size localized wave excitations, which are intrinsic localized traveling modes in two-dimensional anharmonic crystal lattice systems. In particular, using different initial conditions of coordinates and momenta we search for the longest lasting excitations in triangular lattices. As most stable and longest lasting unaltered appear quasi-one-dimensional Toda-like solitons running in rectilinear chains along the main crystallographic axes of such lattices. Furthermore, by following the trace of high energetic excitations like in “bubble chamber” methodology (or in scanning tunneling microscopy) we show how such localized nonlinear waves appearing spontaneously in heated systems can be detected and followed in space-time.     

Theory of two-dimensional nonlinear waves in liquid covered by ice

The aim is to develop a method of Hamiltonian formalism for the waves in the liquid beneath an ice sheet and on that basis to construct a systematic nonlinear theory. Attention is concentrated on the investigation of the essentially two-dimensional effects whose properties depend to a large extent on the stresses in the ice.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 125–133, July–August, 1991.     

On two-dimensional linear waves in Blasius boundary layer over viscoelastic layers

This work concerns the direct numerical simulation of small-amplitude two-dimensional ribbon-excited waves in Blasius boundary layer over viscoelastic compliant layers of finite length. A vorticity-streamfunction formulation is used, which assures divergence-free solutions for the evolving flow fields. Waves in the compliant panels are governed by the viscoelastic Navier's equations. The study shows that Tollmien–Schlichting (TS) waves and compliance-induced flow instability (CIFI) waves that are predicted by linear stability theory frequently coexist on viscoelastic layers of finite length. In general, the behaviour of the waves is consistent with the predictions of linear stability theory. The edges of the compliant panels, where abrupt changes in wall property occur, are an important source of waves when they are subjected to periodic excitation by the flow. The numerical results indicate that the non-parallel effect of boundary-layer growth is destabilizing on the TS instability. It is further demonstrated that viscoelastic layers with suitable properties are able to reduce the amplification of the TS waves, and that high levels of material damping are effective in controlling the propagating CIFI.     

On steady three-dimensional deep water weakly nonlinear gravity waves

Various kinds of steady weakly nonlinear gravity waves are examined. Corrections to the linear phase speed and the direction of modulation are calculated.     

Vortex structure simulation for supersonic mixing layers using nonlinear PSE method

The method of nonlinear parabolized stability equations (PSE) is applied in the simulation of vortex structures in compressible mixing layer. The spatially-evolving unstable waves, which dominate the vortex structure, are investigated through spatial marching method. The instantaneous flow field is obtained by adding the harmonic waves to basic flow. The results show that T-S waves do not keep growing exponentially as the linear evolution, the energy transfer to high order harmonic modes, and that finally all harmonic modes get saturated due to nonlinear interaction. The mean flow distortion induced by the nonlinear interaction between the harmonic modes and their conjugate harmonic ones, makes great change of the average flow and increases the thickness of mixing layer. PSE methods can well capture the two- and three-dimensional large scale nonlinear vortex structures in mixing layers such as vortex roll-up, vortex pairing, and Λ vortex.     

The evolution laws of dilatational spherical and cylindrical weak nonlinear shock waves in elastic non-conductors

The theory of singular surfaces yields a set of coupled evolution equations for the shock amplitude and the amplitudes of the higher order discontinuities which accompany the shock. To solve these equations, we use perturbation methods with a perturbation parameter characterising the initial shock amplitude. It is shown that for decaying shock waves, if the accompanying second order discontinuity is of order one, the straightforward perturbation procedure yields uniformly valid solutions, but if the accompanying second order discontinuity is of order , the method of multiple scales is needed in order to render the perturbation solutions uniformly valid with respect to the distance of travel. We also construct shock wave solutions from modulated simple wave solutions which are obtained with the aid ofHunter & Keller's Weakly Nonlinear Geometrical Optics method. The two approaches give exactly the same results within their common range of validity. The explicit evolution laws thus obtained enable us to see clearly how weak nonlinear curved shock waves are attenuated because of the effects of geometry and material nonlinearity, and on what length scale these effects are most pronounced. Communicated by C. C. Wang     

One-dimensional nonlinear induced dynamics of acoustic waves in a finite three-dimensional domain

The problem of the time-dependent viscous compressible gas flow excited by a small external time-dependent space-and time-periodic force is considered within the framework of the Navier-Stokes equations on a finite interval with periodic boundary conditions. The investigation is carried out numerically for a periodicity interval L, divided by the viscous length, from 10² to 2 × 10³ and external force amplitudes from 10^?4 to 0.1. The nonlinear dynamics of the wave processes are investigated within the framework of this problem. It is shown that nonlinear steady-state oscillations with sharp variation of the quantities in space and time develop when L is greater than or of the order of 10³. This leads to the onset of a continuous spectrum.     

Large-eddy simulation of nonlinear evolution and breakdown to turbulence in high-speed boundary layers

The nonlinear evolution and laminar-turbulent breakdown of a boundary-layer flow along a cylinder at Mach 4.5 is investigated with large-eddy temporal simulation. The results are validated using the direct numerical simulation data of Pruett and Zang. The structure of the flow during the transition process is studied in terms of the vorticity field. The subgrid scales are modeled dynamically, where the model coefficients are determined as part of the solution from the local resolved field. In the numerical simulation the dynamic-model coefficients are obtained by using both the strain-rate contraction of Germano et al. and the least-squares contraction of Lilly; they produced some differences in the details of the vorticity structure inside the transition region. A new dynamic model that utilizes the second-order velocity structure function is used to parametrize the small-scale field. The evolution to turbulence is successfully simulated with dynamic subgrid-scale modeling at least in terms of average quantities as well as vorticity fields. This is achieved with one-sixth of the grid resolution used in direct numerical simulation.This work was sponsored by the Theoretical Flow Physics Branch of the Fluid Mechanics Division of NASA Langley Research Center under Contract NAS1-19320.     

The structure and dynamics of reacting plane mixing layers

The reacting two-dimensional plane mixing layer has been studied in two configurations: a rearward facing step and a two-stream mixing layer. Observations have been made of the steady state behavior, and the unsteady behavior when the flow was forced by a specific acoustic frequency. The steady behavior of the mean properties of the reacting flows is similar to that of non-reacting free shear flows except for the global effects of thermodynamic property changes. The structure of these flows is qualitatively similar to that of non-reacting flows. Vortices form by the two-dimensional Kelvin-Helmholtz instability and grow by subharmonic combination until the mixing layer interacts with the walls. Entrainment is dominated by the two-dimensional vortex motion. Three-dimensional instabilities give rise to secondary vortices which are coherent over several Kelvin-Helmholtz structures and dominate the fine scale mixing process. The mixing transition corresponds to a loss of coherence in the layer. Unsteady behavior occurs when there are resonant interactions with the Kelvin-Helmholtz instability or the instability associated with the recirculation vortex in the rearward facing step flow. Modeling efforts are reported which show promise of simulating the essential features of plane mixing layers.A version of this paper was presented at the ASME Winter Annual Meeting of 1984 and printed in AMD-Vol. 66     

On the nonlinear development of two-dimensional and three-dimensional perturbations in the presence of Rayleigh-Taylor instability

The nonlinear evolution of two-dimensional and three-dimensional perturbations of finite amplitude in the presence of Rayleigh-Taylor instability is investigated. It is assumed that the problem is one of potential flow. The solution is constructed by the Fourier method [1]. In the two-dimensional case the conformal mapping method [2, 3] is employed, which makes it possible to consider the strongly nonlinear stages of development of the perturbations, including the formation of surfaces with multiply valued dependence of the variables in Cartesian coordinates. The construction of the mappings reduces to the solution of the Hilbert problem, which is given in the form of Schwartz integrals [4]. Explicit expressions for these integrals [5], obtained with the aid of Fourier series, are employed. Effective computational algorithms are developed and a series of numerical investigations is carried out. Inter alia, a destabilizing effect of the short-wave components is detected, the regularizing action of the surface tension is demonstrated, and the characteristic times of nonlinear development of the perturbations and the characteristic spectral distributions are found. The role of three-dimensional effects, characterized by a decrease in the rate of development of perturbations, is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 38–46, March–April, 1985.In conclusion, the authors wish to express their deep gratitude to Yu. B. Rabinovich and S. L. Petrov whose research was used in compiling the calculation program and, moreover, Yu. M. Shtempler for discussing methodilogy and the numerical results.     

The two-dimensional nonlinear orthotropic plate

Free University Berlin     

Comparison of three-dimensional and two-dimensional statistical mechanics of shear layers for flow between two parallel plates

It is shown that the averaged velocity profiles predicted by statistical mechanics of point vortices and statistical mechanics of vortex lines are practically indistinguishable for a shear flow between two parallel walls.     

Certain properties of mixing layers

We are examining the properties of solutions of the Falkner-Skan equation in the limiting case when the parameter in the equation approaches zero. Two types of boundary conditions are formulated. The first type corresponds to flow in a symmetric wake. The second corresponds to flow about a plate. The results of calculations and an asymptotic analysis make it possible to conclude that the transition from one type of boundary conditions to another involves a sharp change in the position of the mixing layer.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 45–54, May–June, 1986.