Alessandro Bottaro

NOTE ON OPEN BOUNDARY CONDITIONS FOR ELLIPTIC FLOWS

Four different open (outflow) boundary conditions are compared and evaluated by using the incompressible PoiseuillelBenard flow in a two-dimensional rectangular duel as a test case. The conditions are: simple upwinding, linearly and quadratically weighted upwinding, and vanishing of a linearized convective derivative. The upwinding conditions generate significant reflection of outgoing waves back into the computational domain, while the convective condition presents little reflection. This last condition, which is a Sommerfeld-type radiation condition, is recommended for use at boundaries where a net outflow of fluid occurs.

Görtler Vortices with System Rotation

The steady primary instability of Görtler vortices developing along a curved Blasius boundary layer subject to spanwise system rotation is analysed through linear and nonlinear approaches, to clarify issues of vortex growth and wavelength selection, and to pave the way to further secondary instability studies.A linear marching stability analysis is carried out for a range of rotation numbers, to yield the (predictable) result that positive rotation, that is rotation in the sense of the basic flow, enhances the vortex development, while negative rotation dampens the vortices. Comparisons are also made with local, nonparallel linear stability results (Zebib and Bottaro, 1993) to demonstrate how the local theory overestimates vortex growth. The linear marching code is then used as a tool to predict wavelength selection of vortices, based on a criterion of maximum linear amplification.Nonlinear finite volume numerical simulations are performed for a series of spanwise wave numbers and rotation numbers. It is shown that energy growths of linear marching solutions coincide with those of nonlinear spatially developing flows up to fairly large disturbance amplitudes. The perturbation energy saturates at some downstream position at a level which seems to be independent of rotation, but that increases with the spanwise wavelength. Nonlinear simulations performed in a long (along the span) cross section, under conditions of random inflow disturbances, demonstrate that: (i) vortices are randomly spaced and at different stages of growth in each cross section; (ii) “upright” vortices are the exception in a universe of irregular structures; (iii) the average nonlinear wavelengths for different inlet random noises are close to those of maximum growth from the linear theory; (iv) perturbation energies decrease initially in a linear filtering phase (which does not depend on rotation, but is a function of the inlet noise distribution) until coherent patches of vorticity near the wall emerge and can be amplified by the instability mechanism; (v) the linear filter represents the receptivity of the flow: any random noise, no matter how strong, organizes itself linearly before subsequent growth can take place; (vi) the Görtler number, by itself, is not sufficient to define the state of development of a vortical flow, but should be coupled to a receptivity parameter; (vii) randomly excited Görtler vortices resemble and scale like coherent structures of turbulent boundary layers.

Görtler vortices promoted by wall roughness

Numerical experiments are conducted to investigate spatially developing Gortler vortices and the way in which wall roughness promotes their formation and growth. Several different types of walls are examined and their relative merits as vortex promoters assessed. The only disturbances of the flow are due to the rough wall; hence, at each downstream station the local field feels (1) the upstream flow distribution (produced by the upstream wall conditions) and (2) the local forcing at the wall. Rapid vortex formation and growth, like in the case of ribleted walls, can be qualitatively explained by the positive combination of these two effects; when the two influences on the local flow field compete, e.g. for randomly distributed wall roughness, the equations with the boundary conditions filter the disturbances over some streamwise length, function of the roughness amplitude, to create coherent patches of vorticity out of the random noise. These patches can then be amplified by the instability mechanism. If a thin rough strip is aligned along the span of an otherwise smooth wall to trip the boundary layer, the filtering region is shorter and growth of the vortices starts earlier. Also for the case of an isolated three-dimensional hump a rapid disturbance amplification is produced, but in this case the vortices remain confined and a very slow spanwise spreading of the perturbation occurs. In all naturally developing cases, where no specific wavelengths are explicity favored, the average spanwise wavelengths computed are very close to those of largest growth from the linear stability theory.

Onset and two-dimensional patterns of convection with strongly temperature-dependent viscosity

The onset of two‐dimensional convection with strongly temperature‐dependent viscosity has been considered for a fluid obeying an Arrhenius law. The critical Rayleigh number Rc and the basic features of the flow field at criticality have been identified based on a linear stability analysis. Convective flow patterns near and beyond criticality have been determined based on a direct numerical simulation. It is shown that, as the Rayleigh number R increases beyond Rc, steady rolls first emerge supercritically and that at sufficiently large values of R there is a secondary Hopf bifurcation corresponding to pulsating cells; the peculiar structure of the flow field in each case has been described.

Laminar swirling flow and vortex breakdown in a pipe

Abstract A problem of asymmetric swirling flow in a pipe has been analyzed by the finite volume numerical simulation of the three-dimensional incompressible Navier-Stokes equations expressed in cylindrical polar coordinates. The swirl is generated by one tangential inlet near the closed base of the duct. Because of the high value of the swirl number, S, a vortex breakdown forms near the base of the cylinder; it is of bubble-like shape. Downstream of the bubble the axial flow is at first wavy and finally tends toward the Poiseuille profile, while the swirl decays exponentially. At high enough axial Reynolds number, Re, the flow becomes time dependent.

A Time-Reversed Approach to the Study of Görtler Instabilities

The original analysis (by Gortler himself and others) of the curvature-excited streamwise vortices that have become known as Gortler vortices was based on a quasi-parallel extension of the theory of parallel centrifugal instabilities of the Taylor-Couette and Dean type. However, the possibility of applying a quasi-parallel analysis to Gortler vortices has always been questioned because of the difficulties in identifying a suitable scaling parameter that could allow the theory to be cast in the form of a proper asymptotic expansion continuable to all orders.

EFFECT OF SPANWISE SYSTEM ROTATION ON SPATIALLY DEVELOPING DEAN VORTICES

SUMMARY Three-dimensional spatially developing Navier-Stokes calculations are carried out to simulate the flow in a curved, rotating channel. The competition between centrifugal and Coriolis forces, expressed by the ratio of the Dean number to the rotation number, gives rise to a variety of possible instability modes characterized by the presence of streamwise vortices. Cases in which the force produced by system rotation enhances or opposes the centrifugal force are treated and the effect on the ensuing instability are analysed. Evidence for a generalized Eckhaus instability of rotating Dean vortices is presented.

Three-Dimensional Navier-Stokes Computations of Spatially Developing Görtler Vortices

The Gortler flow is studied through three-dimensional incompressible Navier-Stokes computations, that solve for the spatially developing flow. By the use of inflow-outflow boundary conditions (as opposed to the temporal approach) we allow in a natural way for the development of the boundary layer, we preserve the convective nature of the Gortler instability and we do not impose a priori streamwise wavelengths (such as the wavelength of the secondary instability). Furthermore, we consider more than one vortex pair in the cross-section so that vortex interaction mechanisms are not overlooked. This is likely to be an important requirement in the selection of the secondary instability mode. The results computed in the primary instability regime are in very good agreement with experiments. To trigger the secondary instability we introduce the equivalent of a “vibrating ribbon”. The secondary instability starts as a sinuous motion of the low speed streaks. The spanwise movement of the streaks is a feature that the Gortler flow shares with shear layer structures in near-wall turbulence.