scholarly journals Transient perturbation growth in time-dependent mixing layers

2013 ◽  
Vol 717 ◽  
pp. 90-133 ◽  
Author(s):  
C. Arratia ◽  
C. P. Caulfield ◽  
J.-M. Chomaz
Keyword(s):  
Velocity Distribution ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Base Flow ◽  
Time Dependent ◽  
Time Interval ◽  
Start Time ◽  
Time Intervals ◽  
Optimal Perturbation ◽  
2D Flow

AbstractWe investigate numerically the transient linear growth of three-dimensional (3D) perturbations in a homogeneous time-evolving mixing layer in order to identify which perturbations are optimal in terms of their kinetic energy gain over a finite, predetermined time interval. We model the mixing layer with an initial parallel velocity distribution $\mathbi{U}(y)= {U}_{0} \tanh (y/ d)\mathbi{e}_{x}$ with Reynolds number $Re= {U}_{0} d/ \nu = 1000$, where $\nu $ is the kinematic viscosity of the fluid. We consider a range of time intervals on both a constant ‘frozen’ base flow and a time-dependent two-dimensional (2D) flow associated with the growth and nonlinear saturation of two wavelengths of the most-unstable eigenmode of linear theory of the initial parallel velocity distribution, which rolls up into two classical Rayleigh instabilities commonly referred to as Kelvin–Helmholtz (KH) billows, which eventually pair to form a larger vortex. For short times, the most-amplified perturbations on the frozen $\tanh $ profile are inherently 3D, and are most appropriately described as oblique wave ‘OL’ perturbations which grow through a combination of the Orr and lift-up mechanisms, while for longer times, the optimal perturbations are 2D and similar to the KH normal mode, with a slight enhancement of gain. For the time-evolving KH base flow, OL perturbations continue to dominate over sufficiently short time intervals. However, for longer time intervals which involve substantial evolution of the primary KH billows, two broad classes of inherently 3D linear optimal perturbation arise, associated at low wavenumbers with the well-known core-centred elliptical translative instability, and at higher wavenumbers with the braid-centred hyperbolic instability. The hyperbolic perturbation is relatively inefficient in exploiting the gain of the OL perturbations, and so only dominates the smaller wavenumber (ultimately) core-centred perturbations when the time evolution of the base flow or the start time of the optimization interval does not allow the OL perturbations much opportunity to grow. When the OL perturbations can grow, they initially grow in the braid, and then trigger an elliptical core-centred perturbation by a strong coupling with the primary KH billow. If the optimization time interval includes pairing of the primary billows, the secondary elliptical perturbations are strongly suppressed during the pairing event, due to the significant disruption of the primary billow cores during pairing.

Transient growth in strongly stratified shear layers

2014 ◽  
Vol 758 ◽  
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.

scholarly journals Non-Modal Three-Dimensional Optimal Perturbation Growth in Thermally Stratified Mixing Layers

Fluids ◽  
2021 ◽  
Vol 6 (1) ◽  
pp. 37
Author(s):  
Helena Vitoshkin ◽  
Alexander Gelfgat
Keyword(s):  
Mixing Layer ◽  
Three Dimensional ◽  
Stable Stratification ◽  
Base Flow ◽  
Transient Growth ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Dimensional Growth ◽  
Layer Flow ◽  
Perturbation Growth

A non-modal transient disturbances growth in a stably stratified mixing layer flow is studied numerically. The model accounts for a density gradient within a shear region, implying a heavier layer at the bottom. Numerical analysis of non-modal stability is followed by a full three-dimensional direct numerical simulation (DNS) with the optimally perturbed base flow. It is found that the transient growth of two-dimensional disturbances diminishes with the strengthening of stratification, while three-dimensional disturbances cause significant non-modal growth, even for a strong, stable stratification. This non-modal growth is governed mainly by the Holmboe modes and does not necessarily weaken with the increase of the Richardson number. The optimal perturbation consists of two waves traveling in opposite directions. Compared to the two-dimensional transient growth, the three-dimensional growth is found to be larger, taking place at shorter times. The non-modal growth is observed in linearly stable regimes and, in slightly linearly supercritical regimes, is steeper than that defined by the most unstable eigenmode. The DNS analysis confirms the presence of the structures determined by the transient growth analysis.

ON THE REGULARITY OF THREE-DIMENSIONAL ROTATING EULER–BOUSSINESQ EQUATIONS

1999 ◽  
Vol 09 (07) ◽  
pp. 1089-1121 ◽  
Author(s):  
A. BABIN ◽  
A. MAHALOV ◽  
B. NICOLAENKO
Keyword(s):  
Initial Data ◽  
Three Dimensional ◽  
Fluid Flows ◽  
Boussinesq Equations ◽  
Stable Stratification ◽  
Primitive Equations ◽  
Time Interval ◽  
Regular Solutions ◽  
Time Intervals ◽  
Long Time

The 3-D rotating Boussinesq equations (the "primitive" equations of geophysical fluid flows) are analyzed in the asymptotic limit of strong stable stratification. The resolution of resonances and a nonstandard small divisor problem are the basis for error estimates for such fast singular oscillating limits. Existence on infinite time intervals of regular solutions to the viscous 3-D "primitive" equations is proven for initial data in Hα, α≥ 3/4. Existence on a long-time interval T*of regular solutions to the 3-D inviscid equations is proven for initial data in Hα, α > 5/2 (T*→∞ as the frequency of gravity waves →∞).

Long-Time Solutions to Heat-Conduction Transients with Time-Dependent Inputs

1980 ◽  
Vol 102 (1) ◽  
pp. 115-120 ◽  
Author(s):  
H. T. Ceylan ◽  
G. E. Myers
Keyword(s):  
Heat Conduction ◽  
Lower Cost ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Linear Functions ◽  
Time Dependent ◽  
Time Interval ◽  
Piecewise Linear Functions ◽  
One Dimensional ◽  
Long Time

An economical method for obtaining long-time solutions to one, two, or three-dimensional heat-conduction transients with time-dependent forcing functions is presented. The conduction problem is spatially discretized by finite differences or by finite elements to obtain a system of first-order ordinary differential equations. The time-dependent input functions are each approximated by continuous, piecewise-linear functions each having the same uniform time interval. A set of response coefficients is generated by which a long-time solution can be carried out with a considerably lower cost than for conventional methods. A one-dimensional illustrative example is included.

Evolution of stream wise vortices and generation of small-scale motion in a plane mixing layer

1991 ◽  
Vol 231 ◽  
pp. 257-301 ◽  
Author(s):  
K. J. Nygaard ◽  
A. Glezer
Keyword(s):  
Surface Film ◽  
Mixing Layer ◽  
Excitation Frequency ◽  
Three Dimensional ◽  
Streamwise Velocity ◽  
Base Flow ◽  
Small Scale ◽  
Streamwise Vortices ◽  
Plane Mixing Layer ◽  
Scale Motion

The evolution of streamwise vortices in a plane mixing layer and their role in the generation of small-scale three-dimensional motion are studied in a closed-return water facility. Spanwise-periodic streamwise vortices are excited by a time-harmonic wavetrain with span wise-periodic amplitude variations synthesized by a mosaic of 32 surface film heaters flush-mounted on the flow partition. For a given excitation frequency, virtually any span wise wavelength synthesizable by the heating mosaic can be excited and can lead to the formation of streamwise vortices before the rollup of the primary vortices is completed. The onset of streamwise vortices is accompanied by significant distortion in the transverse distribution of the streamwise velocity component. The presence of inflexion points, absent in corresponding velocity distributions of the unforced flow, suggests the formation of locally unstable regions of large shear in which broadband perturbations already present in the base flow undergo rapid amplification, followed by breakdown to small-scale motion. Furthermore, as a result of spanwise-non-uniform excitation the cores of the primary vortices are significantly altered. The three-dimensional features of the streamwise vortices and their interaction with the base flow are inferred from surfaces of r.m.s. velocity fluctuations and an approximation to cross-stream vorticity using three-dimensional single component velocity data. The striking enhancement of small-scale motion and the spatial modification of its distribution, both induced by the streamwise vortices, can be related to the onset of the mixing transition.

On finite amplitude oscillations in laminar mixing layers

1967 ◽  
Vol 29 (3) ◽  
pp. 417-440 ◽  
Author(s):  
J. T. Stuart
Keyword(s):  
Equilibrium State ◽  
Stability Theory ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Time Dependent ◽  
Present Calculation ◽  
Periodic Perturbations ◽  
Linearized Stability ◽  
Non Linear

In the first part of the paper, a mixing layer of tanhyform is considered, and twodimensional solutions of the non-linear inviscid equations are found representing periodic perturbations from the neutral wave of linearized stability theory. To second order in amplitude the solutions are equivalent to the equilibrium state calculated by Schade (1964), who studied the development of perturbations in time and found an evolution towards that equilibrium state. The present calculation, however, is taken to fourth-order in amplitude. It is noted that the solutions presented in this paper are regular, even though viscosity is ignored; and the relationships to the singular (if inviscid) time-dependent solutions of Schade are explained. Such regular, inviscid solutions have been found only for odd velocity profiles, such as tanhy.Although the details of the velocity distributions are not of the form observed experimentally, it is shown that the amplitude ratios of fundamental and first harmonic, for a given absolute amplitude, are comparable with those observed.In part 2 some exact non-linear solutions are presented of the inviscid, incompressible equations of fluid flow in two or three spatial dimensions. They illustrate the flows of part 1, since they are periodic in one co-ordinate (x), have a shear in another (y) and are independent of the third. Included, as two-dimensional cases, are (i) the tanh y velocity distribution for a flow wholly in the x-direction, (ii) the well-known solution for the flow due to a set of point vortices equi-spaced on the axis, and (iii) an example of linearized hydrodynamic (Orr-Sommerfeld) stability theory. The flows may involve concentrations of vorticity. In three-dimensional cases the z component of velocity is even iny, whereas the x component is odd. Consequently, the class of flows represents, in general, small or large periodic perturbations from a skewed shear layer. Time-dependent solutions, representing waves travelling in the x direction may be obtained by translation of axes.

scholarly journals Some Results for a Time Interval Approach to Field Theory and Gravitation

2021 ◽  
Author(s):  
Harmen Henricus Hollestelle
Keyword(s):  
Field Theory ◽  
Space Time ◽  
Time Dependent ◽  
Field Energy ◽  
Time Interval ◽  
Time Intervals ◽  
Metric Tensor ◽  
Zero Mass ◽  
Interval Approach ◽  
Wave Emission

This paper consists of two parts. In part I some new relations for a field theory with time intervals are derived. One concept of field theory evaluated is complementarity, another is field operators both defined within a time interval description. Part II includes specific results and commentary. Discussed are time interval dependent wave propagation surfaces for star source emission waves and derived is a metric propagation surface area requirement. The results allow to consider one same field that like gravitation within General Relativity applies to both non zero and zero mass. The associated field energy is space time dependent for non zero mass, and is related to a space time dependent metric tensor for zero mass wave particles. Defined is internal energy transfer where wave particle numbers increase linearly and mass and momentum diminish, decrease inversely with the distance from the wave emission source. The commentary are applications related to cosmological overall volume and temperature dependence.

The stability of the variable-density Kelvin–Helmholtz billow

2008 ◽  
Vol 612 ◽  
pp. 237-260 ◽  
Author(s):  
JÉRÔME FONTANE ◽  
LAURENT JOLY
Keyword(s):  
Stability Analysis ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Variable Density ◽  
Base Flow ◽  
Shear Instability ◽  
Numerical Continuation ◽  
Small Scale ◽  
Two Dimensional ◽  
Two Fluids

We perform a three-dimensional stability analysis of the Kelvin–Helmholtz (KH) billow, developing in a shear layer between two fluids with different density. We begin with two-dimensional simulations of the temporally evolving mixing layer, yielding the unsteady base flow fields. The Reynolds number is 1500 while the Schmidt and Froude numbers are infinite. Then exponentially unstable modes are extracted from a linear stability analysis performed at the saturation of the primary mode kinetic energy. The spectrum of the least stable modes exhibits two main classes. The first class comprises three-dimensional core-centred and braid-centred modes already present in the homogeneous case. The baroclinic vorticity concentration in the braid lying on the light side of the KH billow turns the flow into a sharp vorticity ridge holding high shear levels. The hyperbolic modes benefit from the enhanced level of shear in the braid whereas elliptic modes remain quite insensitive to the modifications of the base flow. In the second class, we found typical two-dimensional modes resulting from a shear instability of the curved vorticity-enhanced braid. For a density contrast of 0.5, the wavelength of the two-dimensional instability is about ten times shorter than that of the primary wave. Its amplification rate competes well against those of the hyperbolic three-dimensional modes. The vorticity-enhanced braid thus becomes the preferred location for the development of secondary instabilities. This stands as the key feature of the transition of the variable-density mixing layer. We carry out a fully resolved numerical continuation of the nonlinear development of the two-dimensional braid-mode. Secondary roll-ups due to a small-scale Kelvin–Helmholtz mechanism are promoted by the underlying strain field and develop rapidly in the compression part of the braid. Originally analysed by Reinoud et al. (Phys. Fluids, vol. 12, 2000, p. 2489) from two-dimensional non-viscous numerical simulations, this instability is shown to substantially increase the mixing.

Some Results for a Time Interval Approach to Field Theory and Gravitation

2021 ◽  
Author(s):  
Harmen Henricus Hollestelle
Keyword(s):  
Field Theory ◽  
Space Time ◽  
Time Dependent ◽  
Field Energy ◽  
Time Interval ◽  
Time Intervals ◽  
Metric Tensor ◽  
Zero Mass ◽  
Interval Approach ◽  
Wave Emission

This paper consists of two parts. In part I some new relations for a field theory with time intervals are derived. One concept of field theory evaluated is complementarity, another is operators however both defined within a time interval description. Part II includes specific results and commentary. Discussed are time interval dependent wave propagation surfaces for star source emission waves and derived is a metric surface area requirement for propagation surfaces. The results allow to consider one field that like gravitation within General Relativity applies to both non zero and zero mass. The associated field energy is space time dependent for non zero mass, and is related to a space time dependent metric tensor for zero mass wave particles. Defined is internal energy transfer where wave particle numbers increase linearly and mass and momentum diminish, decrease inversely with the distance from the wave emission source. The commentary are applications related to cosmological overall volume and temperature dependence.

Time, Equilibrium, and General Relativity

2020 ◽  
Author(s):  
Harmen Henricus Hollestelle
Keyword(s):  
General Relativity ◽  
Radiation Energy ◽  
Entropy Change ◽  
Good Method ◽  
Time Dependent ◽  
Time Interval ◽  
Step Method ◽  
Time Intervals ◽  
Mean Velocity ◽  
The Past

Considered is “time as an interval” including time from the past and from the future, in contrast to time as a moment. Equilibrium as the basis for a description of changing properties in physics is understood to depend on the “mean velocity theorem”, while a “time” of equilibrium resembles a center of weight. This turns out to be a good method to derive properties for any function of time t including space coordinates q(t) and expressions for the time dependent Hamiltonian. Introduced are derivatives depending on time intervals instead of time moments and with these a new relation between the Lagrangian L and the Hamiltonian H. As an application introduced is a step by step method to integrate stationary state “local” time interval measurements to beyond “locality” in General Relativity. Because of limits on the measures of the resulting time intervals and their asymmetry, this allows for a probabilistic interpretation of quantities that have these intervals as time domain in QM. Their asymmetry also questions the time reversal symmetry of GR. Another application of time intervals is the discussion of the measurement of starlight radiation energy and QM wave packet collapse as an example of a time dependent Hamiltonian. Finally a relation between starlight frequency, metric and space- and time intervals is found. Discussed is how finite and asymmetric time intervals correspond to time dependent H and symmetric infinite time intervals to a time independent H. From there, in cosmological perspective, finite time intervals can help to describe how entropy change could relate to dark energy.

Sign in / Sign up

Export Citation Format

Share Document