Jan O. Pralits

Instability and sensitivity of the flow around a rotating circular cylinder

The two-dimensional flow around a rotating circular cylinder is studied at Re = 100. The instability mechanisms for the first and second shedding modes are analysed. The region in the flow with a r ...

Adjoint-based optimization of steady suction for disturbance control in incompressible flows

The optimal distribution of steady suction needed to control the growth of single or multiple disturbances in quasi-three-dimensional incompressible boundary layers on a flat plate is investigated. The evolution of disturbances is analysed in the framework of the parabolized stability equations (PSE). A gradient-based optimization procedure is used and the gradients are evaluated using the adjoint of the parabolized stability equations (APSE) and the adjoint of the boundary layer equations (ABLE). The accuracy of the gradient is increased by introducing a stabilization procedure for the PSE. Results show that a suction peak appears in the upstream part of the suction region for optimal control of Tollmien–Schlichting (T–S) waves, steady streamwise streaks in a two-dimensional boundary layer and oblique waves in a quasi-three-dimensional boundary layer subject to an adverse pressure gradient. The mean flow modifications due to suction are shown to have a stabilizing effect similar to that of a favourable pressure gradient. It is also shown that the optimal suction distribution for the disturbance of interest reduces the growth rate of other perturbations. Results for control of a steady cross-flow mode in a three-dimensional boundary layer subject to a favourable pressure gradient show that not even large amounts of suction can completely stabilize the disturbance.

Sensitivity analysis using adjoint parabolized stability equations for compressible flows

An input/output framework is used to analyze the sensitivity of two- and three-dimensional disturbances in a compressible boundary layer for changes in wall and momentum forcing. The sensitivity is defined as the gradient of the kinetic disturbance energy at a given downstream position with respect to the forcing. The gradients are derived using the parabolized stability equations (PSE) and their adjoint (APSE). The adjoint equations are derived in a consistent way for a quasi-two-dimensional compressible flow in an orthogonal curvilinear coordinate system. The input/output framework provides a basis for optimal control studies. Analysis of two-dimensional boundary layers for Mach numbers between 0 and 1.2 show that wall and momentum forcing close to branch I of the neutral stability curve give the maximum magnitude of the gradient. Forcing at the wall gives the largest magnitude using the wall normal velocity component. In case of incompressible flow, the two-dimensional disturbances are the most sensitive ones to wall inhomogeneity. For compressible flow, the three-dimensional disturbances are the most sensitive ones. Further, it is shown that momentum forcing is most effectively done in the vicinity of the critical layer.

Shape optimization for delay of laminar-turbulent transition

A method using gradient-based optimization is introduced for the design of wing profiles with the aim of natural laminar flow, as well as minimum wave drag. The Euler equations of gasdynamics, the laminar boundary-layer equations for compressible flows on infinite swept wings, and the linear parabolized stability equations (PSE) are solved to analyze the evolution of convectively unstable disturbances. Laminar‐turbulent transition is assumed to be delayed by minimizing a measure of the disturbance kinetic energy of a chosen disturbance, which is computed using the PSE. The shape gradients of the disturbance kinetic energy are computed based on the solutions of the adjoints of the state equations just named. Numerical tests are carried out to optimize the RAE 2822 airfoil with the aim to delay simultaneously the transition, reduce the pressure drag coefficient, and maintain the coefficients of lift and pitch moments. Constraints are also applied on the geometry. Results show a reduction of the total amplification of a large number of disturbances, which is assumed to represent a delay of the transition in the boundary layer. Because delay of the transition implies reduction of the viscous drag, the present method enables shape optimization to perform viscous drag reduction.

Structural Sensitivity of the Finite-Amplitude Vortex Shedding Behind a Circular Cylinder

In this paper we study the structural sensitivity of the nonlinear periodic oscillation arising in the wake of a circular cylinder for Re47. The sensibility of the periodic state to a spatially localised feedback from velocity to force is analysed by performing a structural stability analysis of the problem. The sensitivity of the vortex shedding frequency is analysed by evaluating the adjoint eigenvectors of the Floquet transition operator. The product of the resulting neutral mode with the nonlinear periodic state is then used to localise the instability core. The results obtained with this new approach are then compared with those derived by Giannetti & Luchini [8]. An excellent agreement is found comparing the present results with the experimental data of Strykowski & Sreenivasan [7].

Optimization of steady suction for disturbance control on infinite swept wings

We present a theory for computing the optimal steady suction distribution to suppress convectively unstable disturbances in growing boundary layers on infinite swept wings. This work includes optimization based on minimizing the disturbance kinetic energy and the integral of the shape factor. Further, a suction distribution in a continuous control domain is compared to an approach using a number of discrete pressure chambers. In the latter case, the internal static pressures of these chambers are optimized. Optimality systems are derived using Lagrange multipliers. The corresponding optimality conditions are evaluated using the adjoint of the parabolized stability equations and the adjoint of the boundary layer equations. Results are presented for an airfoil designed for medium range commercial aircraft. We show that an optimal suction distribution based on a minimization of the integral of the shape factor is not always successful in the sense of delaying laminar-turbulent transition. It is also demonstrated that including different types of disturbances, e.g., Tollmien-Schlichting and cross-flow types, in the analysis may be crucial.

Phakic Iris-Fixated Intraocular Lens Placement in the Anterior Chamber: Effects on Aqueous Flow

PURPOSE Phakic intraocular lenses (pIOLs) are used for correcting vision; in this paper we investigate the fluid dynamical effects of an iris-fixated lens in the anterior chamber. In particular, we focus on changes in the wall shear stress (WSS) on the cornea and iris, which could be responsible for endothelial and pigment cell loss, respectively, and also on the possible increase of the intraocular pressure, which is known to correlate with the incidence of secondary glaucoma. METHODS We use a mathematical model to study fluid flow in the anterior chamber in the presence of a pIOL. The governing equations are solved numerically using the open source software OpenFOAM. We use an idealized standard geometry for the anterior chamber and a realistic geometric description of the pIOL. RESULTS We consider separately the main mechanisms that produce fluid flow in the anterior chamber. The numerical simulations allow us to obtain a detailed description of the velocity and pressure distribution in the anterior chamber, and indicated that implantation of the pIOL significantly modifies the fluid dynamics in the anterior chamber. However, lens implantation has negligible influence on the intraocular pressure and does not produce a significant increase of the shear stress on the cornea, while the shear stress on the iris, although increased, is not enough to cause detachment of cells. CONCLUSIONS We conclude that alterations in the fluid dynamics in the anterior chamber as a result of lens implantation are unlikely to be the cause of medical complications associated with its use.

Riccati-less optimal control of bluff-body wakes

In this paper we propose a new method to solve the optimal control problem in which the feedback matrix K is computed in an efficient way for complex flows, with large number of degrees of freedom, using an approach similar to adjoint-based control optimization. The idea is to consider the direct-adjoint system as an input-output problem where the input is given by the current state and the output is the control. Since the control has much smaller dimension than the state, the feedback matrix K can be efficiently obtained from the solution of the adjoint of the direct-adjoint system. It can further be shown using the symplectic product that the direct-adjoint system is self adjoint. As a consequence the new adjoint system is equivalent to the direct-adjoint system with suitable initial and terminal conditions. With this method the optimal control problem can be solved efficiently for any value of the control penalty l 2. Results are presented of this novel technique as applied to suppressing the vortex shedding behind a circular cylinder, and compared to the minimal-energy feedback control presented in [4].

Three-dimensional stability, receptivity and sensitivity of non-Newtonian flows inside open cavities

We investigate the stability properties of flows over an open square cavity for fluids with shear-dependent viscosity. Analysis is carried out in context of the linear theory using a normal-mode decomposition. The incompressible Cauchy equations, with a Carreau viscosity model, are discretized with a finite-element method. The characteristics of direct and adjoint eigenmodes are analyzed and discussed in order to understand the receptivity features of the flow. Furthermore, we identify the regions of the flow that are more sensitive to spatially localized feedback by building a spatial map obtained from the product between the direct and adjoint eigenfunctions. Analysis shows that the first global linear instability of the steady flow is a steady or unsteady three-dimensionl bifurcation depending on the value of the power-law index n. The instability mechanism is always located inside the cavity and the linear stability results suggest a strong connection with the classical lid-driven cavity problem.

Flow in the anterior chamber of the eye with an implanted iris-fixated artificial lens

Flow in the aqueous humour that fills the anterior chamber of the eye occurs in response to the production and drainage of the aqueous humour, and also due to buoyancy effects produced by thermal gradients. Phakic intraocular lenses are manufactured lenses that are surgically inserted in the eyes of patients to correct refractive errors. Their presence has a dramatic effect on the circulation of the aqueous humour, resulting a very different flow in the anterior chamber, the effects of which have not been extensively investigated. In this article we use a simplified mathematical model to analyse the flow, in order to assess the effect of the implanted lens on the pressure drop required to drive the flow and also on the wall shear stress experienced by the corneal endothelial cells and the cells of the iris. A high pressure drop could result in an increased risk of glaucoma, whilst raised shear stress on the cornea could result in a reduction in the density of endothelial cells there, and on the iris it could result in the detachment of pigment cells, which block the outflow of the eye, also leading to glaucoma. Our results confirm those of previous fully numerical studies, and show that, although the presence of the lens causes significant differences in the flow topology and direction, the typical magnitudes of the shear stress are not significantly changed from the natural case. Our semi-analytical solution allows us to perform a thorough study of the dependence of the results on the controlling parameters and also to understand the basic physical mechanisms underlying flow characteristics.