The objective of this work is to determine how the addition of small amounts of polymer affects the sequence of hydrodynamic instabilities leading to turbulence in free shear layers. The FENE-P constitutive equation is chosen to describe the fluid rheology, and a Hartley transform based pseudo-spectral method is applied to the coupled, nonlinear system of partial differential equations governing the fluid vorticity and the FENE-P dumbbell configuration. Both 2D and 3D simulations are performed using a time-developing mixing layer model. 2D simulations of the roll-up instability show that the viscoelastic vortices become flattened relative to their Newtonian counterparts if the elasticity number, E (the ratio of the fluid relaxation time to the characteristic time for vorticity diffusion), and the maximum dumbbell extensibility, b, are sufficiently large. Spanwise perturbations are introduced in the 3D simulations, and exponential growth rates for the resulting instability are computed. The viscoelastic vortices can either be more or less stable to 3D perturbations than the Newtonian vortices, depending on the values of E and b. The primary mechanism responsible for the differences in the Newtonian and viscoelastic 3D growth rates is a distortion to the 2D vortex shape. These observations suggest a plausible route through which polymers inhibit small scale turbulence.
[Cc.02] Three-Dimensional Instability of Viscoelastic Elliptic Vortices
H. Haj-Hariri (U. of Virginia), G. M. Homsy (Stanford U.)
An analysis of the three dimensional instability of two-dimensional viscoelastic elliptical flows is presented, extending the inviscid analysis of Bayly^1 to include both viscous and elastic effects. The problem is governed by three parameters: E, a geometric parameter related to the ellipticity; R\!e, a wavenumber-based Reynolds number; and De, the Deborah number based on the period of the base flow. New modes and mechanisms of instability are discovered. The flow is generally susceptible to instabilities in the form of propagating plane waves with a rotating wavevector, the tip of which traces an ellipse of the same eccentricity as the flow, but with the major and minor axes interchanged. Whereas a necessary condition for purely-inertial instability is that the wavevector has a nonvanishing component along the vortex axis, the viscoelastic modes of instability are most prominent when their wavevectors do vanish along this axis. Our analysis delineates the region of parameter space of (E,R\!e,De) for which the new instability exists. An oscillator equation of the Mathieu type is shown to embody the essential features of the secular viscoelastic instability. The cause of the instability is a buckling of the `compressed' polymers as they are perturbed transversely during the correct phase of the passage of the rotating plane wave. \small ^1~Bayly, B. J.,Phys.\ Rev.\ Lett., 57(17):2160--2163, 1986
[Cc.03] Mode selection in liquid-gas shear layer instability.
L. Raynal, E. Villermaux, E.J. Hopfinger (LEGI/IMG-CNRS, BP53X 38041 Grenoble, France)
The primary instability of a liquid-gas mixing layer is investigated. The frequency, wavelength and convection velocity of the primary unstable two-dimensional waves are measured on a plane mixing layer set-up for broad ranges of liquid and gas velocities. Particular attention is paid to the liquid physical properties (e.g. surface tension, viscosity) and to the gas stream initial velocity profile so that the characteristic parameters of the instability are precisely identified. The convection velocity, U_c, is found to be in good agreement with a relation used for gas shear layers with density differences, U_c=\sqrt\rho_1\,U_1+\sqrt\rho_2\,U_2øver\sqrt\rho_1+\sqrt\rho_ 2 where the subscripts 1 and 2 correspond to the liquid and to the gas respectively. Whatever the gas exit conditions may be, i.e. laminar or turbulent, the instability wavelength is found to be proportional to the vorticity thickness of the initial gas stream velocity profile, \delta_ømega, and the instability frequency, f, measured in a fixed frame, is given by f\proptoU_cøver\delta_ømega. Liquid viscosity and surface tension play no role. A linear stability analysis is performed for velocity and density profiles similar to the experimental ones, and the computed eigenvalues are found to be in good agreement with the observed ones within numerical accuracy: f\delta_ømegaøverU_c\simeq 8\times10^-3. In particular, the limiting trends for f are, for laminar and turbulent exit conditions, f\sim U_2^3/2 and f\sim U_2^7/4, respectively.
[Cc.04] Perturbed vs. plain plane Couette flow
Dwight Barkley (University of Warwick, UK), Laurette Tuckerman (LIMSI-CNRS)
Motivated by recent experiments by Bottin et al.(S. Bottin, O. Dauchot, and F. Daviaud, submitted to J. Fluid Mech.), we have performed a computational linear and nonlinear stability analysis of plane Couette flow perturbed by a small ribbon at mid-gap. The ribbon is infinitesimal in the streamwise direction, has height approximately one-tenth of the cross-channel dimension, and is infinite in the spanwise direction. The 2D steady flow in this geometry differs from plane Couette flow by the presence of a small counter-rotating region immediately surrounding the ribbon, as in the finite amplitude solutions to plane Couette flow computed by Cherhabili and Ehrenstein(A. Cherhabili and U. Ehrenstein, Eur. J. Mech. B/Fluids 14), 677 (1995). and Coughlin(K. Coughlin, submitted to J. Fluid Mech.). This 2D steady flow is, unlike unperturbed plane Couette flow, linearly unstable at Re \approx 230 to a three-dimensional perturbation whose spanwise wavelength \lambda = 4.8 h agrees with that observed experimentally. Nonlinear stability computations show that bifurcation is subcritical. Full three-dimensional simulations evolve towards a flow containing streamwise vortices, as seen experimentally.
[Cc.05] Direct Numerical Simulation of Two--Dimensional Incompressible Laminar Flow over Stratford Ramp
H.L. Zhang, H.F. Fasel (University of Arizona)
In recent years, Direct Numerical Simulation (DNS) has been extensively applied in transition research of wall bounded shear flows. However, due to the limited computer resources, most of the applications were limited to simple geometries like boundary layers on a flat plate. When the geometry becomes more complex (such as the flow over a curved wall), the transition processes will be more complicated. In particular, the effect of the pressure gradient due to wall curvature and the effect of wall curvature by itself are likely to have a pronounced influence on the transition mechanisms.
In the present study, an efficient DNS code (NST3D) developed in our research group for a Cartesian grid was extended to allow for wall curvature effects. A special orthogonal grid system was generated and the associated vorticity-velocity formulation was derived in order to keep the curvilinear form of the governing equations as close to the Cartesian form as possible. Therefore, the efficiency of the original code was essentially maintained, and the CPU time of flow simulations for a curved wall was comparable to that for a flat plate. Using the new code, the two-dimensional laminar flow over a Stratford ramp was calculated. The results agree qualitatively with experiments. In addition, the effect of controlled forcing of the flow was investigated. The results of these simulations provide suggestions for control of transition as well as separation.
[Cc.06] The Nonlinear Stability of a \tanh^3y Mixing Layer
Simal Saujani, Roland Mallier (University of Western Ontario)
Using a nonequilibrium, nonlinear critical layer, we consider the nonlinear evolution of a two-dimensional disturbance to a \tanh^3y mixing layer. The analysis is inviscid and incompressible. The distiguishing feature of this flow is that the first two derivatives of the base velocity both vanish at the critical layer, and as a result of this, the singularities at the critical layer are far worse than those normally encountered in nonlinear critical layer theory. Further, this flow has three neutral modes, each of which is singular at the critical layer and none of which appears to be a stability boundary. As a consequence of this, the evolution of the disturbance is governed by a set of equations which are more nonlinear than those derived by Goldstein amp; Leib (1988) for the more usual \tanh y mixing layer. These equations consist of a set of coupled nonlinear PDE's together with jump conditions across the critical layer.
[Cc.07] Finite amplitude rotational waves in viscous shear flows
W.R.C. Phillips (University of Illinois at Urbana-Champaign)
Growing finite-amplitude initially spanwise-independent two-dimensional rotational waves and their nonlinear interaction with unidirectional viscous shear flows of various strengths are considered. Both primary and secondary instabilities are studied, but only secondary instabilities are permitted to vary in the spanwise direction. A generalized Lagrangian-mean formulation is employed to describe wave-mean interactions and a separate theory is constructed to account for the back effect of the developing mean flow on the wave field. Viscosity is seen to significantly complicate calculation of the back effect. The primary instability is seen to act as a platform for, and catalyst to, secondary instabilities. The analysis leads to an eigenvalue problem for the initial growth of the secondary instability; this being a generalization of the eigenvalue problem constructed by Craik for inviscid neutral waves. Two inviscid secondary instability mechanisms to longitudinal vortex form are observed: the first has as its basis the Craik-Leibovich type-2 mechanism. The second, which is as yet unproven, requires that both the wave- and flow-field distort in concert at all levels of shear. Both mechanisms excite exponential growth on a convective rather than diffusive scale in the presence of neutral waves, but growing waves alter that growth rate.
[Cc.08] Inviscid Instability Characteristics of Curved Wake-Dominated Free Shear Layers
Mei Zhuang (Michigan State University)
The instability characteristics of both uniform and non-uniform density curved incompressible free mixing layers, with basic velocity profiles having wake components, is studied using linear inviscid stability theory in spatial formulation. The disturbance equation is solved numerically using a shooting technique. Both two- and three-dimensional disturbances are considered. Results indicate that the existence of the curvature affects the instability characteristics of both the shear layer and the wake modes. The oblique wake mode can become stronger than its corresponding 2-D wake mode due to the curvature effects. The instability behaviors of the oblique wake modes also vary significantly with the introduction of the curvature effects. It is found that for a curved incompressible mixing layer the effects of the density variation across the layer on the instability characteristics of both the shear layer and the wake modes are similar to those of a plane mixing layer.
[Cc.09] Spanwise Periodic Mean Flow Generated by Nonlinear Interaction Between Oblique Instability Waves
Sang Soo Lee (NYMA, Inc.)
The nonlinear interaction of instability waves may first occur within a thin critical layer where the basic mean velocity is equal to the phase speed of the instability wave. The resonant-triad interaction in an adverse-pressure-gradient boundary layer was considered. The resonant-triad is composed of a single two-dimensional mode and a pair of subharmonic oblique modes with equal and opposite spanwise wavenumbers. The frequency and streamwise wave number of the oblique modes are nearly equal to one halves of those of the two-dimensional mode. Our interest is in the case where the nonlinear interactions arise from the continued downstream growth of a triad of initially linear instability waves. The nonlinear effects in the critical layer produce a mean flow correction term which is periodic in the spanwise direction with the spanwise wavenumber equal to twice of that of the oblique modes. This mean flow distortion is very large in the sense that its magnitude is equal to that of the oblique waves.
[Cc.10] Viscous and inviscid instabilities of flow along a streamwise corner
Scott Parker, S. Balachandar (University of Illinois)
Here we consider the stability of flow along a streamwise corner formed by the intersection of two large flat plates held perpendiclar to each other. Solutions for the steady laminar mean flow have been obtained for zero and non-zero streamwise pressure gradient. The stability of the mean flow is examined using linear stability analysis and a novel eigensolver has been developed to solve the resulting eigenvalue problem. Stability results indicate that the entire spectrum of modes of a Blasius boundary layer is active in the corner layer. The effect of the corner on the two dimensional Blasius viscous instability is to superpose a spanwise standing wave pattern and decrease the growth rate. The growth rate for oblique disturbances is decreased for outgoing modes and enhanced for incoming modes. The instability at zero pressure gradient is dominated by viscous modes; however, an inviscid corner mode is also observed. At zero pressure gradient this inviscid mode remains stable at very high Reynolds number, but the critical Reynolds number rapidly decreases with even a small adverse streamwise pressure gradient and the inviscid mode becomes the dominant one.