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Passive separation control using a self-adaptive hairy coating Julien Favier, Antoine Dauptain, Davide Basso, Alessandro Bottaro To cite this version: Julien Favier, Antoine Dauptain, Davide Basso, Alessandro

Passive separation control using a self-adaptive hairy coating Julien Favier, Antoine Dauptain, Davide Basso, Alessandro Bottaro To cite this version: Julien Favier, Antoine Dauptain, Davide Basso, Alessandro Bottaro. Passive separation control using a self-adaptive hairy coating. Journal of Fluid Mechanics, Cambridge University Press (CUP), 29, 627, pp hal HAL Id: hal Submitted on Oct 24 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Passive separation control using a self-adaptive hairy coating JULIEN FAVIER, ANTOINE DAUPTAIN, DAVIDE BASSO AND ALESSANDRO BOTTARO DICAT, Universita di Genova, Via Montallegro, 645, Genova, Italy A model of hairy medium is developed using a homogenized approach, and the fluid flow around a circular cylinder partially coated with hair is analyzed by means of numerical simulations. The capability of this coating to adapt to the surrounding flow is investigated, and its benefits are discussed in the context of separation control. This fluid-structure interaction problem is solved with a partitioned approach, based on the direct resolution of the Navier-Stokes equations together with a non-linear set of equations describing the dynamics of the coating. A volume force, arising from the presence of a cluster of hair, provides the link between the fluid and the structure problems. For the structure part, a subset of reference elements approximates the whole layer. The dynamics of these elements is governed by a set of equations based on the inertia, elasticity, interaction and losses effects of articulated rods. The configuration chosen is that of the two-dimensional flow past a circular cylinder at Re = 2, a simple and well documented test case. Aerodynamics performances quantified by the Strouhal number, the drag and the maximum lift in the laminar unsteady regime are modified by the presence of the coating. A set of parameters corresponding to a realistic coating (length of elements, porosity, rigidity) is found, yielding an average drag reduction of 5% and a decrease of lift fluctuations by about 4%, associated to a stabilization of the wake.. Introduction The manipulation of fluid flows to bring about performance enhancements on air/water vehicles is a topic of growing interest in the fluid mechanics community. Besides the highly stimulating and fundamental problems raised by the control of the non-linear Navier-Stokes equations, flow control has a tremendous economical and ecological impact on society (see Gad-el-Hak (2) for a detailed survey). In this context, it is particularly worthwhile to analyze swimming and flying animals, in order to import novel ideas into technological applications. Not surprisingly many efficient locomotion techniques are found in Nature, as they have survived the tests of evolution over millions of years and reached a high level of adaptation. One interesting example is represented by the feathers over the wings of birds. Even though it is difficult to monitor their dynamics due to the animal s rapid motion, they are believed to play a crucial role in the aerodynamics of birds. As mentioned in the excellent review on biological surface by Bechert et al. (997) and Meyer et al. (27), the pop-up of feathers observed on snapshots and movies of landing birds is probably relevant for the control of flow separation. Several drag-reducing biological surfaces inspired by aquatic animals have also shown their efficiency: riblets are inspired by the skin of sharks(bechert& Bartenwerfer 989; Luchini et al. 99) and allow to reduce the shear stress compared to a smooth surface; they have been Present address: CERFACS, 42 Avenue Gaspard Coriolis, 357 Toulouse Cedex, France. 2 J. Favier, A. Dauptain, D. Basso, A. Bottaro successfully tested on large airplanes (Viswanath 22), although in-service application appears to be prevented by the need to replace the riblet film every two or three years; the presence of bumps on whale flippers can delay stall and thus enhance hydrodynamics manoeuvrability performances (van Nierop et al. 28); the release of trapped air bubbles from the skin of a penguin appears to have an effect on the reduction of skin friction (Xu et al. 22). By looking at this short list it may appear that straightforward mimicry of nature might lead to novel and efficient technological applications. The task is however not so straightforward. In-depth understanding of physical mechanisms is required to manufacture efficient actuators since a biological skin is meant to handle multiple functions: for example the presence of mucus on the skin of fish may protect it against parasites and infections, and has a drag reducing function as well. Thus, direct imitation of the skin of fish in the effort to minimize drag might yield a sub-optimal solution, since the skin performs many other functions. The so-called Gray paradox of the compliant skins of dolphins is a striking example of the difficulty to mimick a biological surface. It was believed that the impressive swimming ability of dolphins was due to the compliance of their skin, able to delay transition to turbulence and/or maintain a laminar boundary layer on the surface of the dolphin s body. Many studies were inspired by the original observations of Gray & Sand (936), starting with the theories of Benjamin (96) and Landahl (962), and later with the analyses of disturbances developing in boundary layers over compliant plates (Carpenter & Garrad 985, 986). It is now clear that Gray s premises were flawed, as mentioned by Fish & Lauder(26) and confirmed recently by Hœpffner et al.(28); the latter authors have shown that compliance can yield very large transient disturbance amplifications compared to smooth surfaces, potentially dangerous for the onset of turbulence. Fish & Lauder (26) have demonstrated that the drag reduction observed on dolphins is linked mostly to behavioral functions of the animal, mainly related to its breathing habits. Coming back to the presence and function of feathers on the wings of birds, we aim here at making progress in understanding the effect of the feathers (or similar protuberances) on the aerodynamic performances. All birds have six different types of feathers covering their body, performing different tasks during flight. They are adapted to flight conditions, and used for many purposes, including to shape the wings, insulate and protect the animal s skin. This type of system is then clearly very complex to model, but the property of interest here is the ability of the wings to adapt to the surrounding flow to influence the aerodynamics (cf. figure ). The assumption that the raising of feathers during birds landing phases plays a role in the increase of the lift coefficient of the wing has to be demonstrated. It is probable that this pop-up is not coincidental, but is due to a self-adaptation of birds wings to the separated flow during landing, in order to control it. Outstanding questions are then: do the feathers act like classical slats on commercial airplanes wings which locally increase the angle of attack? Do they behave like vortex generators stabilizing the recirculation zone by redistributing energy? Is it more of a slowing down effect due to the suddenly popped-up porous fence or another effect affecting the stability of the boundary layer towards separation? The physical mechanism is not clearly identified so far: although the impact on the flow is as yet undefined, we believe, along with Bechert et al. (997), that the phenomenon is worth studying since there are indications that these small feathers are important for the flight control of birds at high lift conditions during landing. Indeed, the control of the recirculation zone would explain the amazing manoeuvrability aptitudes of birds, experiencing high angles of attack with a perfect wing stability. Incidentally it has already Passive separation control using a self-adaptive hairy coating 3 Figure. Raising of birds feathers, observed during landing phases. Left: snapshot of a pelican just before landing (thus gliding flight). Right: pop-up of feathers observed on the upper-side of the wings during the landing of an egret (courtesy of J R Compton, been found that a static porous layer can be used as a mean of boundary layer separation control (Bruneau & Mortazavi 28). The study of the flow past hairy coatings finds many applications: for example in the study of thick bundles of immersed vegetation (Sukhodolova et al. 24) and windexposed plants (De Langre 28) in strong interaction with the surrounding fluid flows. Another possible application of porous fuzzy coatings is found in the realm of sports: for example the felt of a tennis ball plays an important role on the aerodynamics of the ball (Mehta & Pallis 2) and new techniques of digital imaging have recently been implemented by Steele et al. (26) to properly assess the quality of the textile surface roughness, predict ball performances and develop acceptable wear limits. Finally, new concepts of sensors and actuators for flow control are based on tiny rod-like elements, whose deflection provides a measure of the wall shear stress (Brücker et al. 25; Große & Schröder 28). In this paper we build a simplified model of hairy coating, with the following featherlike characteristics: porous, since fluid can flow through the feathers; the non-homogenous character of the coating formed by the different types of feathers, more or less packed, is taken into account through a density parameter, non-isotropic, as fluid is oriented along a specific direction as it enters the layer, just like in realistic feathers, compliant, since the layer can bend and deform according to the surrounding flow. Such properties are those which appear to us as the most important in modeling birds feathers. The possibility of shape adaptation of this wall coating is tested and analysed on a classical and academic configuration of separated flow: the motion around a twodimensional circular cylinder at Reynolds number Re = 2. The numerical framework is presented to clearly illustrate the simulation procedures relative to fluid and structure parts, the assumptions on which the model is built, its perspectives of further applications and limitations. As this domain of investigation is naturally related to the studies of flows through arrays of fibers, we will base our work on experimental, theoretical and numerical results on such configurations (Howells 998; Koch & Ladd 997). Various models of different orders of approximation are built for the drag per unit length, as a function of the density of fibers, in Howells (998). Different organizations of fibers (parallel, random) are assessed and estimates are made to take into account the effects of finite length, 4 J. Favier, A. Dauptain, D. Basso, A. Bottaro Γ f Fluid Fluid-Solid Γ h Solid Γ s Figure 2. The three zones of the computational domain (not to scale). curvature and neighbouring fibers interactions, leading to results in good agreement with experiments. These theoretical developements will be useful to derive an expression of the volume force used in this article. Schematically, the domain of study is decomposed into three zones corresponding to a solid body, a surrounding fluid in motion and a mixed fluid-solid portion representing the hairy coating. In figure 2 the fluid area is included between the fixed boundary (Γ f ) of the fluid domain, and the fixed boundary (Γ s ) of the cylinder. The hairy layerbetween Γ s and the moving boundary Γ h is in interaction with the fluid. There is no mass exchange between fluid and solid domains and the temperature is assumed to be constant and uniform throughout. The first sections of the article are dedicated to a description of the numerical treatment of the fluid and structure parts, and how the two-way coupling between the two is achieved. The application that follows refers to the control of the unsteady wake and it illustrates the potential of the approach. In the following the elements forming the coating will be referred to as pillars, hair, cilia, beams or fibers, always to mean the same thing. 2. Fluid domain 2.. Equations The simulation of the unsteady flow around a cylinder of diameter D is performed by solving the discrete version of the incompressible Navier-Stokes equations in a twodimensional periodic domain. The equations are given below, with U Eulerian velocity, p pressure, µ dynamic viscosity, ρ density and F a volume force: ρ[ U t +(U. )U] = p+µ 2 U+F; U =. (2.) A sketch of the domain over which equations (2.) have been discretized is provided in figure 3. The Reynolds number is defined as Re = ρ U D, with U the free µ stream velocity. The volume force F in (2.) is decomposed into three contributions F = F c +F b +F h : (a) F c is introduced to account for the presence of the solid cylinder, i.e. it renders Passive separation control using a self-adaptive hairy coating 5 y-periodic Hairy layer Buffer zone Cylinder M c M h x-periodic x-periodic M b D/2 r Y X b X y-periodic Figure 3. Computational domain of the fluid problem for the flow around a coated cylinder of diameter D. One cell over ten is represented on the mesh. The immersed boundary method is used for the solid cylinder through the function M c, the hairy layer through the function M h and a buffer zone through the function M b. equal to zero the fluid velocity inside the circular obstacle. This volume force is computed using the immersed boundary method described in Peskin (22), i.e. ( t ) F c = M c α c ( U)dt+β c ( U). (2.2) t M c is a non-dimensional scalar field equal to one inside the cylinder, zero outside, as shown in figure 3. Appropriate values of the positive constants α c and β c are found to be respectively and 6/ t, with t the time step of the computations. With this set of parameters, the velocity within the cylinder section is always such that V U / U dxdy 5, where the volume of integration V is the volume of the cylinder per unit depth. (b) A buffer zone of thickness X b is imposed with a volume force F b to damp the unsteady structures in the wake of the cylinder, before they reach the end of the domain. Since the domain is periodic, this buffer volume force is also used to ensure that the inflow speed is equal to U on the left-hand side of the domain (figure 3): ( t ) F b = M b α b (U U)dt+β b (U U). (2.3) t M b is equal to one inside the buffer layer, zero outside. Here α b and β b are set to.8 and 5/ t, such that the velocity at the exit of the buffer zone in a control volume V 2 of thickness D is V 2 ( U / U )dxdy 5. (c) The hairy layer is imposed with a force F h, evaluated as the drag force past a cluster of tiny beams of various density and orientation (cf. 3.). F h vanishes strictly outside the volume occupied by the coating. For the three immersed boundary domains, it is necessary to smooth the edges of the filter functions by using a progressive interpolation. A hyperbolic tangent function is used on M b for the buffer zone, and a distributed interpolation approach is employed for the cylinder and the hairy layer (M c and M h ), following the methodology described in Dauptain et al. (28) Resolution and convergence To solve (2.), a finite difference formulation is used on a regular cartesian mesh made up by 8 4 cells in a 4D 2D domain; we have ensured that this resolution yields grid-converged results for the flow past a cylinder. Staggered flow variables are 6 J. Favier, A. Dauptain, D. Basso, A. Bottaro y-periodic. Fluid Solid x-periodic Fluid Solid x-periodic E. x x y-periodic Fluid... x Figure 4. Spatial convergence of the fluid solver. Left: Domain geometry with fluid and solid parts non aligned with the mesh. Right: Norm of the error versus grid spacing. used. The solver uses the explicit Adams-Bashforth scheme for the convective part, and the semi-implicit Crank-Nicolson method for the viscous part. The Poisson equation for the pressure and the implicit step are treated by the conjugate gradient method (due to the periodic boundary conditions the matrices involved are symmetric and positive definite). This method is second order in time and space. Moreover, the periodicity of the domain and the use of the immersed boundary method allow straightforward and accurate computations of the energy balance terms and of aerodynamic loads. The time scales of fluid and structures phenomena are comparable; a restrictive condition on the time step of the simulation is imposed to make sure that such phenomena are properly captured. To validate the immersed boundary method, a convergence study is performed on a two dimensional Poiseuille flow (figure 4, left frame). The domain is a periodic square, and the walls are not aligned with the mesh, with an inclination of 45 o. The error E plotted in figure 4 (right frame) is the norm of the difference between the theoretical profile U th and the velocity U on a cross section of the duct at the computational nodes i =,...,N: E = N ( ) U th 2. i U i (2.4) NU max i= Grids ranging from 2 2 to 5 5 are tested to check the global order of the solver. Figure 4 demonstrates second-order convergence of the spatial resolutions for grids finer than As far as the laminar flow past a cylinder is concerned, the lift and drag coefficients found in the literature (He et al. 2; Bergmann et al. 25) are well reproduced (see 6.) and this is sufficient evidence for the solver to be considered suitable for the present investigation Communications with the structure part The link between the fluid and the structure problems is done via the volume force F h, either expressed in the fluid discretization space (F h ij ) or in the structure discretization space (F h k ). The state variables of the fluid equations (U, p and F) are discretized in the space of dimensions N x N y (U ij, p ij and F ij ). On the other hand, the dynamics of Passive separation control using a self-adaptive hairy coating 7 the hairy layer is described via the angular positions θ k of each reference element, with k =,...,N c (cf. 3.2), corresponding to a discretization in a space of dimension N c. 3. Hairy domain The coating is a dense cluster of hair and is described with a homogenized approach, as a non-isotropic, compliant layer of variable porosity. The motion in time of the layer is modelled by a set of non-linear equations derived from the dynamical equilibrium of the system. The coupling with the fluid part is described hereafter. 3.. Homogenized drag model The interaction of the hairy medium with the flow is taken into account with an estimate of the drag force past the cluster of hair sketched in figure 5a. This force per unit volume F h is assumed to be decomposed into a normal and a tangential component: F h t estimated as the drag force past a cluster of very long thin cylinders aligned with the flow, F h n approximated by the drag force past a random cluster of cylinders orthogonal to the flow. Inordertoevaluatethesecomponents,weintroducethepackingdensityφ = V hair /V layer, ratio of the volume occupied by the hair (solid) over the total sampling volume. This quantity varies continuously between (no cilia) and (solid) inside the layer. Another variable is defined inside the layer, the unit orientation vector d, characterizing the direction of each element of the coating. Both of these variables are schematically shown in figure 5b. In the fixed reference frame of the cylinder, we can define for any point P belo

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