Evolution of a stratified rotating shear layer with horizontal shear. Part 2. Nonlinear evolution

2013 ◽  
Vol 732 ◽  
pp. 373-400 ◽  
Author(s):  
Eric Arobone ◽  
Sutanu Sarkar
Keyword(s):  
Shear Layer ◽  
Rotation Rate ◽  
Nonlinear Evolution ◽  
Mixing Layer ◽  
Vertical Mixing ◽  
Three Dimensional ◽  
Vertical Axis ◽  
Buoyancy Frequency ◽  
Omega 3 ◽  
Horizontal Shear

AbstractDirect numerical simulation is used to investigate the nonlinear evolution of a horizontally oriented mixing layer with uniform stable stratification and coordinate system rotation about the vertical axis. The important dimensional parameters governing inviscid dynamics are maximum shear $S(t)$, buoyancy frequency $N$, angular velocity of rotation $\Omega $ and characteristic shear thickness $L(t)$. The effect of rotation rate, $\Omega $, on the development of fluctuations in the shear layer is systematically studied in a regime of strong stratification. An instability mechanism, qualitatively distinct from the inertial instability, is found to deform columnar vortex cores in vertical planes for a strongly stratified rotating mixing layer. This mechanism emerges when centreline absolute vertical vorticity, $\langle {\omega }_{3} \rangle (t)+ 2\Omega $, is nearly zero as predicted by the linear stability analysis in Part 1 (J. Fluid. Mech., vol. 703, 2012, pp. 29–48). When the initial rotation rate is moderately anticyclonic, strong destabilization and a cascade to small scales is observed, consistent with prior studies involving horizontally sheared flow in the presence of rotation. Examination of enstrophy budgets in cases which are initially inertially unstable reveal the importance of baroclinic torque in maintaining lateral enstrophy fluctuations substantially beyond the time when the flow becomes inertially stable. The cyclonic stratified cases show weak nonlinearity in vortex dynamics. At high Reynolds number, despite the strong stratification, the flow exhibits three-dimensional, nonlinear dynamics and significant vertical mixing except for cases where the rotation is stabilizing.

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 Flow Kinematics in Variable-Height Rotating Cylinder Arrays

2016 ◽  
Vol 138 (11) ◽  
Author(s):  
Anna E. Craig ◽  
John O. Dabiri ◽  
Jeffrey R. Koseff
Keyword(s):  
Power Output ◽  
Rotation Rate ◽  
Three Dimensional ◽  
Vertical Axis ◽  
Streamwise Velocity ◽  
Vertical Flow ◽  
Mean Flow ◽  
Freestream Velocity ◽  
Cylinder Arrays ◽  
The Mean

Experimental data are presented for large arrays of rotating, variable-height cylinders in order to study the dependence of the three-dimensional mean flows on the height heterogeneity of the array. Elements in the examined arrays were spatially arranged in the same staggered paired configuration, and the heights of each element pair varied up to ±37.5% from the mean height (kept constant across all arrays), such that the arrays were vertically structured. Four vertical structuring configurations were examined at a nominal Reynolds number (based on freestream velocity and cylinder diameter) of 600 and nominal tip-speed ratios of 0, 2, and 4. It was found that the vertical structuring of the array could significantly alter the mean flow patterns. Most notably, a net vertical flow into the array from above was observed, which was augmented by the arrays' vertical structuring, showing a 75% increase from the lowest to highest vertical flows (as evaluated at the maximum element height, at a single rotation rate). This vertical flow into the arrays is of particular interest as it represents an additional mechanism by which high streamwise momentum can be transported from above the array down into the array. An evaluation of the streamwise momentum resource within the array indicates up to a 56% increase in the incoming streamwise velocity to the elements (from the lowest to highest ranking arrays, at a single rotation rate). These arrays of rotating cylinders may provide insight into the flow kinematics of arrays of vertical axis wind turbines (VAWTs). In a physical VAWT array, an increase in incoming streamwise flow velocity to a turbine corresponds to a (cubic) increase in the power output of the turbine. Thus, these results suggest a promising approach to increasing the power output of a VAWT array.

A numerical study of shear layer characteristics of low-speed transverse jets

2016 ◽  
Vol 790 ◽  
pp. 275-307 ◽  
Author(s):  
Prahladh S. Iyer ◽  
Krishnan Mahesh
Keyword(s):  
Velocity Profile ◽  
Shear Layer ◽  
Convective Instability ◽  
Numerical Study ◽  
Velocity Ratio ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
Dominant Frequencies

Direct numerical simulation (DNS) and dynamic mode decomposition (DMD) are used to study the shear layer characteristics of a jet in a crossflow. Experimental observations by Megerian et al. (J. Fluid Mech., vol. 593, 2007, pp. 93–129) at velocity ratios ($R=\overline{v}_{j}/u_{\infty }$) of 2 and 4 and Reynolds number ($Re=\overline{v}_{j}D/{\it\nu}$) of 2000 on the transition from absolute to convective instability of the upstream shear layer are reproduced. Point velocity spectra at different points along the shear layer show excellent agreement with experiments. The same frequency ($St=0.65$) is dominant along the length of the shear layer for $R=2$, whereas the dominant frequencies change along the shear layer for $R=4$. DMD of the full three-dimensional flow field is able to reproduce the dominant frequencies observed from DNS and shows that the shear layer modes are dominant for both the conditions simulated. The spatial modes obtained from DMD are used to study the nature of the shear layer instability. It is found that a counter-current mixing layer is obtained in the upstream shear layer. The corresponding mixing velocity ratio is obtained, and seen to delineate the two regimes of absolute or convective instability. The effect of the nozzle is evaluated by performing simulations without the nozzle while requiring the jet to have the same inlet velocity profile as that obtained at the nozzle exit in the simulations including the nozzle. The shear layer spectra show good agreement with the simulations including the nozzle. The effect of shear layer thickness is studied at a velocity ratio of 2 based on peak and mean jet velocity. The dominant frequencies and spatial shear layer modes from DNS/DMD are significantly altered by the jet exit velocity profile.

Evolution of a stratified rotating shear layer with horizontal shear. Part I. Linear stability

2012 ◽  
Vol 703 ◽  
pp. 29-48 ◽  
Author(s):  
Eric Arobone ◽  
Sutanu Sarkar
Keyword(s):  
Angular Velocity ◽  
Linear Stability ◽  
Growth Rates ◽  
Vertical Axis ◽  
Density Fluctuations ◽  
Buoyancy Frequency ◽  
Horizontal Shear ◽  
Homogeneous Fluid ◽  
Rapid Rotation ◽  
Graphic Dependence

AbstractLinear stability analysis is used to investigate instability mechanisms for a horizontally oriented hyperbolic tangent mixing layer with uniform stable stratification and coordinate system rotation about the vertical axis. The important parameters governing inviscid dynamics are maximum shear $S$, buoyancy frequency $N$, angular velocity of rotation $\Omega $ and characteristic shear thickness $L$. Growth rates associated with the most unstable modes are explored as a function of stratification strength $N/ S$ and rotation strength $2\Omega / S$. In the case of strong stratification, growth rates exhibit self-similarity of the form $\sigma ({k}_{1} L, S{k}_{3} L/ N, 2\Omega / S)$. In the case of rapid rotation we also observe self-similar scaling of growth rates with respect to the vertical wavenumber and rotation rate. The unstratified cases show $\sigma ({k}_{1} L, 2\vert \tilde {\Omega } \vert {k}_{3} L/ S)$ dependence while the strongly stratified cases show $\sigma ({k}_{1} L, 2\vert \tilde {\Omega } \vert {k}_{3} L/ N)$ dependence where $\tilde {\Omega } $ represents the difference between the angular velocity of rotation and least stable anticyclonic angular velocity, $\Omega = S/ 4$. Stratification was found to stabilize the inertial instability for weak anticyclonic rotation rates. Near the zero absolute vorticity state, stratification and rotation couple in a destabilizing manner increasing the range of unstable vertical wavenumbers associated with barotropic instability. In the case of rapid rotation, stratification prevents the stabilization of low ${k}_{1} $, high ${k}_{3} $ modes that occurs in a homogeneous fluid. The structure of certain unstable eigenmodes and the coupling between horizontal vorticity and density fluctuations are explored to explain how buoyancy stabilizes or destabilizes inertial and barotropic modes.

The role of coherent structures in bubble transport by turbulent shear flows

1994 ◽  
Vol 259 ◽  
pp. 219-240 ◽  
Author(s):  
K. J. Sene ◽  
J. C. R. Hunt ◽  
N. H. Thomas
Keyword(s):  
Shear Layer ◽  
Large Scale ◽  
Mixing Layer ◽  
Vertical Mixing ◽  
Companion Paper ◽  
Unsteady Motion ◽  
Turbulent Shear ◽  
Velocity Difference ◽  
Discrete Vortex ◽  
Drag Forces

Using Auton's force law for the unsteady motion of a spherical bubble in inhomogeneous unsteady flow, two key dimensionless groups are deduced which determine whether isolated vortices or shear-layer vortices can trap bubbles. These groups represent the ratio of inertial to buoyancy forces as a relaxation parameter [tcy ] = ΔU2/2gx and a trapping parameter [Gcy ] = ΔU/VT where ΔU is the velocity difference across the vortex or the shear layer, x is streamwise distance measured from the effective origin of the mixing layer and VT is the terminal slip speed of the bubble or particle. It is shown here that whilst buoyancy and drag forces can lead to bubbles moving in closed orbits in the vortex flows (either free or forced), only inertial forces result in convergent trajectories. Bubbles converge on the downflow side of the vortex at a location that depends on the inertial and lift forces. It is important to note that the latter have been omitted from many earlier studies.A discrete-vortex model is used to simulate the large-scale unsteady flows within horizontal and vertical mixing layers between streams with velocity difference ΔU. Trajectories of non-interacting small bubbles are computed using the general force law. In the horizontal mixing layer it is found that Γ needs to have a value of about 3 to trap about 50% of the bubbles if Π is about 0.5 and greater if Π is less. The pairing of vortices actually enhances their trapping of bubbles. In the vertical mixing layer bubbles are trapped mainly within the growing vortices but bubbles are concentrated on the downflow side of the vortices as Γ and Π increase. In a companion paper we show that lateral dispersion of bubbles can be approximately described by an advective diffusion equation with the diffusivity about equal to the eddy viscosity, i.e. rather less than the diffusivity of heat or other passive scalars.

Nonlinear evolution of spiral density waves generated by the instability of the shear layer in a rotating compressible fluid

1991 ◽  
Vol 233 ◽  
pp. 587-612 ◽  
Author(s):  
I. G. Shukhman
Keyword(s):  
Shear Layer ◽  
Compressible Fluid ◽  
Acoustic Waves ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Density Wave ◽  
Mixing Layer ◽  
Mean Flow ◽  
Spiral Density Waves ◽  
Corotation Radius

In a previous paper we considered the nonlinear stability of a cylindrical mixing layer in an incompressible fluid at large Reynolds numbers. Nonlinear evolution results in the formation of vortex structures in the vicinity of the corotation radius rc. This paper considers the same model but in a compressible fluid. A fundamental difference implied by the presence of compressibility is the possibility of the generation of disturbances which are no longer localized near the shear layer but embrace the entire region. These are acoustic waves generated in the region of corotation resonance and emitted into the periphery. In the r > rc region lines of equal density are trailing spirals. The nonlinear evolution of such disturbances is determined by redistribution of the mean flow inside the critical layer (CL). It is shown that only two possible types of CL, viscous and unsteady, can be realized here. For both types of these regimes, evolution equations describing the dynamics of a spiral density wave amplitude are obtained and their solutions analysed. It appears that at any values (provided that they are small enough) of initial supercriticality of the flow, an explosive growth of amplitude occurs which continues as long as values comparable with background ones are reached.

Turbulent wake behind side-by-side flat plates: computational study of interference effects

2018 ◽  
Vol 855 ◽  
pp. 1040-1073 ◽  
Author(s):  
Fatemeh H. Dadmarzi ◽  
Vagesh D. Narasimhamurthy ◽  
Helge I. Andersson ◽  
Bjørnar Pettersen
Keyword(s):  
Shear Layer ◽  
Computational Study ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Flat Plates ◽  
Layer Transition ◽  
Initial Flow ◽  
Plane Mixing Layer ◽  
Gap Flow ◽  
Dimensional Instability

The complex wake behind two side-by-side flat plates placed normal to the inflow direction has been explored in a direct numerical simulation study. Two gaps, $g=0.5d$ and $1.0d$ , were considered, both at a Reynolds number of 1000 based on the plate width $d$ and the inflow velocity. For gap ratio $g/d=0.5$ , the biased gap flow resulted in an asymmetric flow configuration consisting of a narrow wake with strong vortex shedding and a wide wake with no periodic near-wake shedding. Shear-layer transition vortices were observed in the wide wake, with characteristic frequency 0.6. For $g/d=1.0$ , two simulations were performed, started from a symmetric and an asymmetric initial flow field. A symmetric configuration of Kármán vortices resulted from the first simulation. Surprisingly, however, two different three-dimensional instability features were observed simultaneously along the span of the upper and lower plates. The spanwise wavelengths of these secondary streamwise vortices, formed in the braid regions of the primary Kármán vortices, were approximately $1d$ and $2d$ , respectively. The wake bursts into turbulence some $5d$ – $10d$ downstream. The second simulation resulted in an asymmetric wake configuration similar to the asymmetric wake found for the narrow gap $0.5d$ , with the appearance of shear-layer instabilities in the wide wake. The analogy between a plane mixing layer and the separated shear layer in the wide wake was examined. The shear-layer frequencies obtained were in close agreement with the frequency of the most amplified wave based on linear stability analysis of a plane mixing layer.

scholarly journals On the formation of axial corner vortices during spin-up in a cylinder of square cross-section

2015 ◽  
Vol 772 ◽  
pp. 246-271 ◽  
Author(s):  
R. J. Munro ◽  
R. E. Hewitt ◽  
M. R. Foster
Keyword(s):  
Boundary Layer ◽  
Cross Section ◽  
Time Scale ◽  
Rotation Rate ◽  
Rotation Axis ◽  
Three Dimensional ◽  
Angular Frequency ◽  
Navier Stokes ◽  
Buoyancy Frequency ◽  
Two Dimensional

We present experimental and theoretical results for the adjustment of a fluid (homogeneous or linearly stratified), which is initially rotating as a solid body with angular frequency ${\it\Omega}-{\rm\Delta}{\it\Omega}$, to a nonlinear increase ${\rm\Delta}{\it\Omega}$ in the angular frequency of all bounding surfaces. The fluid is contained in a cylinder of square cross-section which is aligned centrally along the rotation axis, and we focus on the $O(\mathit{Ro}^{-1}{\it\Omega}^{-1})$ time scale, where $\mathit{Ro}={\rm\Delta}{\it\Omega}/{\it\Omega}$ is the Rossby number. The flow development is shown to be dominated by unsteady separation of a viscous sidewall layer, leading to an eruption of vorticity that becomes trapped in the four vertical corners of the container. The longer-time evolution on the standard ‘spin-up’ time scale, $E^{-1/2}{\it\Omega}^{-1}$ (where $E$ is the associated Ekman number), has been described in detail for this geometry by Foster & Munro (J. Fluid Mech., vol. 712, 2012, pp. 7–40), but only for small changes in the container’s rotation rate (i.e. $\mathit{Ro}\ll 1$). In the linear case, for $\mathit{Ro}\ll E^{1/2}\ll 1$, there is no sidewall separation. In the present investigation we focus on the fully nonlinear problem, $\mathit{Ro}=O(1)$, for which the sidewall viscous layers are Prandtl boundary layers and (somewhat unusually) periodic around the container’s circumference. Some care is required in the corners of the container, but we show that the sidewall boundary layer breaks down (separates) shortly after an impulsive change in rotation rate. These theoretical boundary-layer results are compared with two-dimensional Navier–Stokes results which capture the eruption of vorticity, and these are in turn compared to laboratory observations and data. The experiments show that when the Burger number, $S=(N/{\it\Omega})^{2}$ (where $N$ is the buoyancy frequency), is relatively large – corresponding to a strongly stratified fluid – the flow remains (horizontally) two-dimensional on the $O(\mathit{Ro}^{-1}{\it\Omega}^{-1})$ time scale, and good quantitative predictions can be made by a two-dimensional theory. As $S$ was reduced in the experiments, three-dimensional effects were observed to become important in the core of each corner vortex, on this time scale, but only after the breakdown of the sidewall layers.

Rapid gravitational adjustment of horizontal shear flows

2013 ◽  
Vol 721 ◽  
pp. 86-117 ◽  
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.

Nonlinear properties of convection rolls in a horizontal layer rotating about a vertical axis

1979 ◽  
Vol 94 (4) ◽  
pp. 609-627 ◽  
Author(s):  
R. M. Clever ◽  
F. H. Busse
Keyword(s):  
Rotation Rate ◽  
Fluid Layer ◽  
Three Dimensional ◽  
Vertical Axis ◽  
Horizontal Layer ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Nonlinear Properties ◽  
Prandtl Numbers ◽  
The Galerkin Method

Steady finite amplitude two-dimensional solutions are obtained for the problem of convection in a horizontal fluid layer heated from below and rotating about its vertical axis. Rigid boundaries with prescribed constant temperatures are assumed and the solutions are obtained numerically by the Galerkin method. The existence of steady subcritical finite amplitude solutions is demonstrated for Prandtl numbers P < 1. A stability analysis of the finite amplitude solutions is performed by superimposing arbitrary three-dimensional disturbances. A strong reduction in the domain of stable rolls occurs as the rotation rate is increased. The reduction is most pronounced at low Prandtl numbers. The numerical analysis confirms the small amplitude results of Küppers & Lortz (1969) that all two-dimensional solutions become unstable when the dimensionless rotation rate Ω exceeds a value of about 27 at P ≃ ∞. A brief discussion is given of the three-dimensional time-dependent forms of convection which are realized at rotation rates exceeding the critical value.

Sign in / Sign up

Export Citation Format

Share Document