Early-period dynamics of an incompressible mixing layer

The evolution of three-dimensional disturbances in an incompressible mixing layer in an inviscid fluid is investigated as an initial-value problem. A Green's function approach is used to obtain a general space–time solution to the problem using a piecewise linear model for the basic flow, thereby making it possible to determine complete and closed-form analytical expressions for the variables with arbitrary input. Structure, kinetic energy, vorticity, and the evolution of material particles can be ascertained in detail. Moreover, these solutions represent the full three-dimensional disturbances that can grow exponentially or algebraically in time. For large time, the behaviour of these disturbances is dominated by the exponentially increasing discrete modes. For the early time, the behaviour is controlled by the algebraic variation due to the continuous spectrum. Contrary to Squire's theorem for normal mode analysis, the early-time behaviour indicates growth at comparable rates for all values of the wavenumbers and the initial growth of these disturbances is shown to rapidly increase. In particular, the disturbance kinetic energy can rise to a level approximately ten times its initial value before the exponentially growing normal mode prevails. As a result, the transient behaviour can trigger the roll-up of the mixing layer and its development into the well-known pattern that has been observed experimentally.

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Evolution of disturbances in stagnation-point flow

Journal of Fluid Mechanics ◽

10.1017/s0022112094004295 ◽

1994 ◽

Vol 270 ◽

pp. 331-348 ◽

Cited By ~ 13

Author(s):

W. O. Criminale ◽

T. L. Jackson ◽

D. G. Lasseigne

Keyword(s):

Stagnation Point ◽

Transverse Direction ◽

Normal Mode Analysis ◽

Three Dimensional ◽

Stagnation Point Flow ◽

Mode Analysis ◽

Inviscid Fluid ◽

Initial Value ◽

Three Dimensional Flow ◽

Stable Flow

The evolution of three-dimensional disturbances in an incompressible three-dimensional stagnation-point flow in an inviscid fluid is investigated. Since it is not possible to apply classical normal-mode analysis to the disturbance equations for the fully three-dimensional stagnation-point flow to obtain solutions, an initial-value problem is solved instead. The evolution of the disturbances provides the necessary information to determine stability and indeed the complete transient as well. It is found that when considering the disturbance energy, the planar stagnation-point flow, which is independent of one of the transverse coordinates, represents a neutrally stable flow whereas the fully three-dimensional flow is either stable or unstable, depending on whether the flow is away from or towards the stagnation point in the transverse direction that is neglected in the planar stagnation point.

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Transient growth in strongly stratified shear layers

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.552 ◽

2014 ◽

Vol 758 ◽

Cited By ~ 18

Author(s):

A. K. Kaminski ◽

C. P. Caulfield ◽

J. R. Taylor

Keyword(s):

Shear Layer ◽

Normal Mode ◽

Richardson Number ◽

Mixing Layer ◽

Three Dimensional ◽

Buoyancy Frequency ◽

Transient Growth ◽

Time Intervals ◽

Optimal Perturbation ◽

Target Time

AbstractWe investigate numerically transient linear growth of three-dimensional perturbations in a stratified shear layer to determine which perturbations optimize the growth of the total kinetic and potential energy over a range of finite target time intervals. The stratified shear layer has an initial parallel hyperbolic tangent velocity distribution with Reynolds number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}=U_0 h/\nu =1000$ and Prandtl number $\nu /\kappa =1$, where $\nu $ is the kinematic viscosity of the fluid and $\kappa $ is the diffusivity of the density. The initial stable buoyancy distribution has constant buoyancy frequency $N_0$, and we consider a range of flows with different bulk Richardson number ${\mathit{Ri}}_b=N_0^2h^2/U_0^2$, which also corresponds to the minimum gradient Richardson number ${\mathit{Ri}}_g(z)=N_0^2/(\mathrm{d}U/\mathrm{d} z)^2$ at the midpoint of the shear layer. For short target times, the optimal perturbations are inherently three-dimensional, while for sufficiently long target times and small ${\mathit{Ri}}_b$ the optimal perturbations are closely related to the normal-mode ‘Kelvin–Helmholtz’ (KH) instability, consistent with analogous calculations in an unstratified mixing layer recently reported by Arratia et al. (J. Fluid Mech., vol. 717, 2013, pp. 90–133). However, we demonstrate that non-trivial transient growth occurs even when the Richardson number is sufficiently high to stabilize all normal-mode instabilities, with the optimal perturbation exciting internal waves at some distance from the midpoint of the shear layer.

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On the “Early-Time” Evolution of Variables Relevant to Turbulence Models for the Rayleigh-Taylor Instability

ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting: Volume 1, Symposia – Parts A, B, and C ◽

10.1115/fedsm-icnmm2010-30556 ◽

2010 ◽

Author(s):

Bertrand Rollin ◽

Malcolm J. Andrews

Keyword(s):

Turbulence Model ◽

Initial Conditions ◽

Mixing Layer ◽

Three Dimensional ◽

Turbulence Models ◽

Early Time ◽

Variable Density ◽

Taylor Instability ◽

Turbulent Regime ◽

Rayleigh Taylor Instability

We present our progress toward setting initial conditions in variable density turbulence models. In particular, we concentrate our efforts on the BHR turbulence model [1] for turbulent Rayleigh-Taylor instability. Our approach is to predict profiles of relevant variables before fully turbulent regime and use them as initial conditions for the turbulence model. We use an idealized model of mixing between two interpenetrating fluids to define the initial profiles for the turbulence model variables. Velocities and volume fractions used in the idealized mixing model are obtained respectively from a set of ordinary differential equations modeling the growth of the Rayleigh-Taylor instability and from an idealization of the density profile in the mixing layer. A comparison between predicted profiles for the turbulence model variables and profiles of the variables obtained from low Atwood number three dimensional simulations show reasonable agreement.

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Normal Mode Analysis of Three-Dimensional Propagation Over a Small-Slope Cosine Shaped Hill

Journal of Computational Acoustics ◽

10.1142/s0218396x15500058 ◽

2015 ◽

Vol 23 (03) ◽

pp. 1550005 ◽

Cited By ~ 6

Author(s):

Megan S. Ballard ◽

Benjamin M. Goldsberry ◽

Marcia J. Isakson

Keyword(s):

Parabolic Equation ◽

Normal Mode ◽

Hybrid Model ◽

Mode Coupling ◽

Transmission Loss ◽

Normal Mode Analysis ◽

Three Dimensional ◽

Mode Analysis ◽

The Hill ◽

Horizontal Refraction

Three-dimensional propagation over an infinitely long cosine shaped hill is examined using an approximate normal mode/parabolic equation hybrid model that includes mode coupling in the out-going direction. The slope of the hill is relatively shallow, but it is significant enough to produce both mode-coupling and horizontal refraction effects. In the first part of the paper, the modeling approach is described, and the solution is compared to results obtained with a finite element method to evaluate the accuracy of the solution in light of assumptions made in formulating the model. Then the calculated transmission loss is interpreted in terms of a modal decomposition of the field, and the solution from the hybrid model is compared to adiabatic and N × 2D solutions to assess the relative importance of horizontal refraction and mode-coupling effects. An analysis using a horizontal ray trace is presented to explain differences in the modal interference pattern observed between the 3D and N × 2D solutions. The detailed discussion provides a thorough explanation of the observed 3D propagation effects and demonstrates the usefulness of the approximate normal mode/parabolic equation hybrid model as a tool to understand measured transmission loss in complex environments.

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Towards enhancing and delaying disturbances in free shear flows

Journal of Fluid Mechanics ◽

10.1017/s0022112095002898 ◽

1995 ◽

Vol 294 ◽

pp. 283-300 ◽

Cited By ~ 16

Author(s):

W. O. Criminale ◽

T. L. Jackson ◽

D. G. Lasseigne

Keyword(s):

Shear Flows ◽

Piecewise Linear ◽

Equations Of Motion ◽

Mixing Layer ◽

Early Time ◽

Governing Equations ◽

Initial Value ◽

Inviscid Incompressible Fluid ◽

The Family ◽

Long Time

The family of shear flows comprising the jet, wake, and the mixing layer are subjected to perturbations in an inviscid incompressible fluid. By modelling the basic mean flows as parallel with piecewise linear variations for the velocities, complete and general solutions to the linearized equations of motion can be obtained in closed form as functions of all space variables and time when posed as an initial-value problem. The results show that there is a continuous spectrum as well as the discrete spectrum that is more familiar in stability theory and therefore there can be both algebraic and exponential growth of disturbances in time. These bases make it feasible to consider control of such flows. To this end, the possibility of enhancing the disturbances in the mixing layer and delaying the onset in the jet and wake is investigated. It is found that growth of perturbations can be delayed to a considerable degree for the jet and the wake but, by comparison, cannot be enhanced in the mixing layer. By using moving coordinates, a method for demonstrating the predominant early and long time behaviour of disturbances in these flows is given for continuous velocity profiles. It is shown that the early time transients are always algebraic whereas the asymptotic limit is that of an exponential normal mode. Numerical treatment of the new governing equations confirm the conclusions reached by use of the piecewise linear basic models. Although not pursued here, feedback mechanisms designed for control of the flow could be devised using the results of this work.

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The influence of initial conditions on turbulent mixing due to Richtmyer–Meshkov instability

Journal of Fluid Mechanics ◽

10.1017/s0022112010000492 ◽

2010 ◽

Vol 654 ◽

pp. 99-139 ◽

Cited By ~ 112

Author(s):

B. THORNBER ◽

D. DRIKAKIS ◽

D. L. YOUNGS ◽

R. J. R. WILLIAMS

Keyword(s):

Kinetic Energy ◽

Volume Fraction ◽

Initial Conditions ◽

Mixing Layer ◽

Three Dimensional ◽

Layer Growth ◽

Decay Rates ◽

Fine Grid ◽

Eddy Simulation ◽

Energy Decay Rates

This paper investigates the influence of different three-dimensional multi-mode initial conditions on the rate of growth of a mixing layer initiated via a Richtmyer–Meshkov instability through a series of well-controlled numerical experiments. Results are presented for large-eddy simulation of narrowband and broadband perturbations at grid resolutions up to 3 × 109 points using two completely different numerical methods, and comparisons are made with theory and experiment. It is shown that the mixing-layer growth is strongly dependent on initial conditions, the narrowband case giving a power-law exponent θ ≈ 0.26 at low Atwood and θ ≈ 0.3 at high Atwood numbers. The broadband case uses a perturbation power spectrum of the form P(k) ∝ k−2 with a proposed theoretical growth rate of θ = 2/3. The numerical results confirm this; however, they highlight the necessity of a very fine grid to capture an appropriately broad range of initial scales. In addition, an analysis of the kinetic energy decay rates, fluctuating kinetic energy spectra, plane-averaged volume fraction profiles and mixing parameters is presented for each case.

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Rapid gravitational adjustment of horizontal shear flows

Journal of Fluid Mechanics ◽

10.1017/jfm.2013.41 ◽

2013 ◽

Vol 721 ◽

pp. 86-117 ◽

Cited By ~ 11

Author(s):

Brian L. White ◽

Karl R. Helfrich

Keyword(s):

Shear Flows ◽

Complex Dynamics ◽

Gravity Current ◽

Mixing Layer ◽

Three Dimensional ◽

Early Period ◽

Shear Instability ◽

Strong Interactions ◽

Horizontal Shear ◽

Mixing Rate

AbstractThe evolution of a horizontal shear layer in the presence of a horizontal density gradient is explored by three-dimensional numerical simulations. These flows exhibit characteristics of both free shear flows and gravity currents, but have complex dynamics due to strong interactions between the turbulent features of each. Vertical vortices produced by horizontal shear are tilted and stretched by the gravitational adjustment, rapidly enhancing vorticity. Shear intensification at frontal convergences produces high-wavenumber vertical vorticity and the slumping of the density interface produces horizontal Kelvin–Helmholtz vortices typical of a gravity current. The interaction between these instabilities promotes a rapid transition to three-dimensional turbulence. The flow development depends on the relative time scales of shear instability and gravitational adjustment, described by a parameter $\gamma $ (where the limits $\gamma \rightarrow \infty $ and $\gamma \rightarrow 0$ represent a pure gravity current and a pure mixing layer, respectively). The growth rate of three-dimensional instability and the mixing increase for smaller $\gamma $. When $\gamma $ is sufficiently small, there are two distinct regimes: an early period of during which the interface grows rapidly, followed by horizontal diffusive growth. Numerical results are consistent with field observations of tidal separation flows in the Haro Strait (Farmer, Pawlowicz & Jiang, Dyn. Atmos. Oceans., vol. 36, 2002, pp. 43–58), including the magnitude of downwelling vertical currents, horizontal scales of surface vortex features and mixing rate.

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RAMAN SPECTRA OF TITANIUM DI-, TRI-, AND TETRA-CHLORIDES

International Journal of Modern Physics B ◽

10.1142/s021797920100886x ◽

2001 ◽

Vol 15 (28n30) ◽

pp. 3865-3868 ◽

Cited By ~ 2

Author(s):

H. MIYAOKA ◽

T. KUZE ◽

H. SANO ◽

H. MORI ◽

G. MIZUTANI ◽

...

Keyword(s):

Force Constant ◽

Raman Spectra ◽

Normal Mode ◽

Raman Band ◽

Normal Mode Analysis ◽

Force Constants ◽

Mode Analysis ◽

Oxidation Number

We have obtained the Raman spectra of TiCl n (n= 2, 3, and 4). Assignments of the observed Raman bands were made by a normal mode analysis. The force constants were determined from the observed Raman band frequencies. We have found that the Ti-Cl stretching force constant increases as the oxidation number of the Ti species increases.

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