Owen R. Tutty

Recurrence of travelling waves in transitional pipe flow

The recent theoretical discovery of families of unstable travelling-wave solutions in pipe flow at Reynolds numbers lower than the transitional range, naturally raises the question of their relevance to the turbulent transition process. Here, a series of numerical experiments are conducted in which we look for the spatial signature of these travelling waves in transitionary flows. Working within a periodic pipe of 5D (diameters) length, we find that travelling waves with low wall shear stresses (lower branch solutions) are on a surface in phase space which separates initial conditions which uneventfully relaminarize and those which lead to a turbulent evolution. This dividing surface (a separatrix if turbulence is a sustained state) is then minimally the union of the stable manifolds of all these travelling waves. Evidence for recurrent travelling-wave visits is found in both 5D and 10D long periodic pipes, but only for those travelling waves with low-to-intermediate wall shear stress and for less than about 10 % of the time in turbulent flow at Re = 2400. Given this, it seems unlikely that the mean turbulent properties such as wall shear stress can be predicted as an expansion solely over the travelling waves in which their individual properties are appropriately weighted. Instead the onus is on isolating further dynamical structures such as periodic orbits and including them in any such expansion.The recent theoretical discovery of families of travelling wave solutions in pipe flow at Reynolds numbers lower than the transitional range naturally raises the question of their relevance to the turbulent transition process. Here a series of numerical experiments are conducted in which we look for the spatial signature of these travelling waves in transitionary flows. Working within a periodic pipe of 5D (diameters) length, we find that travelling waves with low wall shear stresses (lower branch solutions) are on a surface which separates initial conditions which uneventfully relaminarise and those which lead to a turbulent evolution. Evidence for recurrent travelling wave visits is found in both 5D and 10D long periodic pipes but only for those travelling waves with low-to-intermediate wall shear stress and for less than about 10% of the time in turbulent flow. Given this, it seems unlikely that the mean turbulent properties such as wall shear stress can be predicted as an expansion over the travelling waves in which their individual properties are appropriately weighted. Rather, further dynamical structures such as periodic orbits need to be isolated and included in any such expansion.

H∞ control of nonperiodic two-dimensional channel flow

This paper deals with finite-dimensional boundary control of the two-dimensional (2-D) flow between two infinite parallel planes. Surface transpiration along a few regularly spaced sections of the bottom wall is used to control the flow. Measurements from several discrete, suitably placed shear-stress sensors provide the feedback. Unlike other studies in this area, the flow is not assumed to be periodic, and spatially growing flows are considered. Using spatial discretization in the streamwise direction, frequency responses for a relevant part of the channel are obtained. A low-order model is fitted to these data and the modeling uncertainty is estimated. An H-infinity controller is designed to guarantee stability for the model set and to reduce the wall-shear stress at the channel wall. A nonlinear Navier-Stokes PDE solver was used to test the designs in the loop. The only assumption made in these simulations is that the flow is two dimensional. The results showed that, although the problem was linearized when designing the controller, the controller could significantly reduce fundamental 2-D disturbances in practice.

Flow along a long thin cylinder

Two different approaches have been used to calculate turblent flow along a long thin culinder where the flow is aligned with the cylinder. A boundary-layer code is used to predict the mean flow for very long cylinders (length to ratio of up to O(106)), with the effects of the turbulence estimated through a turbulence model. Detailed comparison with experimental results shows that the mean properties of the flow are predicted within experimental accuracy. The boundary-layer model predicts that, sufficiently far downstream, the surface shear stress will be (almost) constant. This is consistent with experimental results from long cylinders in the form of sonar arrays. A periodic Navier-Stokes problem is formulated, and solutions generated for the boundary-layer model and experiments. Strongly turbulent flow occurs only near the surface of the cylinder, with relatively weak turbulence over most of the boundary layer. For a thick boundary layer with the boundary-layer thickness much larger than the cylinder radius, the mean flow is effectively constant near the surface, in both temporal and spatial frameworks, while the outer flow continues to develop in time or space. Calculations of the circumferentially averaged surface pressure spectrum sho that, in physical terms, as the radius of the cylinder decreases, the surface noise from the turbulence increases, with the maximum noise at a Reynolds number of O(103). An increase in noise with a decrease in radius (Reynolds number) is consistent with experimental results.

Boundary layer flow on a long thin cylinder

The development of the boundary layer along a long thin cylinder aligned with the flow is considered. Numerical solutions are presented and compared with previous asymptotic results. Very near the leading edge the flow is given by the Blasius solution for a flat plate. However, there is soon a significant deviation from Blasius flow, with a thinner boundary layer and higher wall shear stress. Linear normal mode stability of the flow is investigated. It is found that for Reynolds numbers less than a critical value of 1060 the flow is unconditionally stable. Also, axisymmetric modes are only the fourth least stable modes for this problem, with the first three three-dimensional modes all having a lower critical Reynolds number. Further, for Reynolds numbers above the critical value, the flow is unstable only for a finite distance, and returns to stability sufficiently far downstream.

Iterative Learning Control for Improved Aerodynamic Load Performance of Wind Turbines With Smart Rotors

Currently, there is significant research into the inclusion of smart devices in wind turbine rotor blades, with the aim, in conjunction with collective and individual pitch control, of improving the aerodynamic performance of the rotors. The main objective is to reduce fatigue loads, although mitigating the effects of extreme loads is also of interest. The aerodynamic loads on a wind turbine blade have periodic and nonperiodic components, and the nature of these strongly suggests the application of iterative learning control. This paper employs a simple computational fluid dynamics model to represent flow past an airfoil and uses this to undertake a detailed investigation into the level of control possible by, as in other areas, combining iterative learning control with classical control action with emphasis on how performance can be effectively measured.

Steady nonlinear waves in diverging channel flow

An infinitely diverging channel with a line source of fluid at its vertex is a natural idealization of flow in a finite channel expansion. Motivated by numerical results obtained in an associated geometry (Tutty 1996), we show in this theoretical model that for certain channel semi-angles ? and Reynolds numbers Re := Q/2? (Q is the volume flux per unit length and ? the kinematic viscosity) a steady, spatially periodic, two-dimensional wave exists which appears spatially stable and hence plausibly realizable in the physical system. This spatial wave (or limit cycle) is born out of a heteroclinic bifurcation across the subcritical pitchfork arms which originate out of the well known Jeffery–Hamel bifurcation point at ? =?2(Re). These waves have been found over the range 5 ? Re ? 5000 and, significantly, exist for semi-angles ? beyond the point ?2 where Jeffery–Hamel theory has been shown to be mute. However, the limit of ??0 at finite Re is not reached and so these waves have no relevance to plane Poiseuille flow.

Robust Control of Linearized Poiseuille Flow

An approach to feedback control of linearized planar Poiseuille flow using H\infty control is developed. Surface transpiration is used to control the flow and point measurements of the wall shear stress are assumed to monitor its state. A high-but-finite dimensional model is obtained via a Galerkin procedure, and this model is approximated by a low dimensional one using Hankel-optimal model reduction. For the purposes of control design the flow is modeled as an interconnection of this low dimensional system and a perturbation, reflecting the uncertainty in the model. The goal of control design is to achieve robust stability (i.e. to stabilize any combination of the nominal plant and a feasible perturbation), and to satisfy certain performance requirements. Two different types of surface actuation are considered -- harmonic transpiration and a model of a pair of suction/blowing panels. It is found that the latter is more efficient in suppressing disturbances in terms of the control effort required.

Nonlinear development of flow in channels with non-parallel walls

In Jeffery–Hamel flow, the motion of a viscous incompressible fluid between rigid plane walls, unidirectional flow is impossible if the angle between the walls exceeds a critical value of 2α2 which depends on the Reynolds number. In this paper the nonlinear development of the flow near this critical value is studied through numerical solutions of the two-dimensional Navier–-Stokes equations for flow in divergent channels with piecewise straight walls. It is found that if the angle between the walls exceeds 2α2 then Jeffery–-Hamel flow does not occur, and the solution takes the form of a large-amplitude wave with eddies attached alternately to the upper and lower walls. When viewed in the appropriate coordinate system, far downstream the wave has constant wavelength and strength, although, physically, there is a linear increase in wavelength with distance downstream, i.e. the wavelength is proportional to the channel width. If the angle between the walls is less than 2α2, then the existence (or otherwise) of the wave depends on the conditions near the inlet, in particular the local geometry of the channel. Jeffery–-Hamel flow is obtained downstream of the inlet for angles well below 2α2, but close to but below the critical value, solutions have been obtained with the wave extending (infinitely) far downstream. The wavelengths obtained numerically were compared with those from linear theory with spatially developing steady modes. No agreement was found: the wavelengths from the steady Navier–-Stokes solutions are significantly larger than that predicted by the theory. However, in other important aspects the results of this study are consistent with those from previous studies of the development/existence of Jeffery–-Hamel flow, in particular as regards the importance of the upstream conditions and the subcritical nature of the spatial development of the flow near the critical boundary in the Reynolds number–wall angle parameter space.

Stochastic optimisation based control of boundary layer transition

Abstract Suction is one of the most promising techniques for delaying the transition from laminar to turbulent flow and hence reducing the drag force acting on an aircraft. However, in order to achieve an overall reduction in energy consumption and thereby operating costs, it is necessary to apply the suction in an efficient if not optimal manner. An investigation is conducted into the use of distributed surface suction with multiple suction panels where the suction distribution is optimised to satisfy a desired objective. Cost functions based on minimising the suction effort while maintaining transition in a fixed position and minimising the total energy consumption of the system are employed. Evolutionary methods, genetic algorithms (GA) and simulated annealing (SA), are used to perform the optimisation. These methods were successful in cases in which gradient based search techniques fail. Physically sensible suction distributions were produced. In general, better behaviour was found with the GA than with SA, which was prone to premature convergence.

Iterative learning control applied to a non-linear vortex panel model for improved aerodynamic load performance of wind turbines with smart rotors

The inclusion of smart devices in wind turbine rotor blades could, in conjunction with collective and individual pitch control, improve the aerodynamic performance of the rotors. This is currently an active area of research with the primary objective of reducing the fatigue loads but mitigating the effects of extreme loads is also of interest. The aerodynamic loads on a wind turbine blade contain periodic and non-periodic components and one approach is to consider the application of iterative learning control algorithms. In this paper, the control design is based on a simple, in relative terms, computational fluid dynamics model that uses non-linear wake effects to represent flow past an airfoil. A representation for the actuator dynamics is included to undertake a detailed investigation into the level of control possible and on how performance can be effectively measured.