Numerical investigation of the design of transition delay devices for a Blasius boundary layer

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UNIVERSITÁ DI PISA

Dipartimento di Ingegneria Civile e Industriale

Sezione Ingegneria Aerospaziale

Tesi di Laurea Magistrale

NUMERICAL INVESTIGATION OF THE DESIGN OF

TRANSITION DELAY DEVICES FOR A BLASIUS

BOUNDARY LAYER

Relatori Allievo

Prof. Ing. Simone Camarri Alessio Nannini

Ing. Lorenzo Siconolfi Ing. Alessandro Mariotti

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Abstract

A numerical investigation on transition delay devices for a Blasius boundary layer is presented. Recent studies have highlighted the advantages of using streamwise streaks generated by an array of appropriate counter-rotating vortices, placed out-side the boundary layer in transition delay control methods. Also Direct Numerical Simulations (DNS) have been carried out to evaluate the effect of idealised pairs of Rankine vortices on the Blasius boundary layer transition, but no attention has not been paid to the devices which can generate such vortices.

This thesis yet is focused on the design of physical winglets which have to gen-erate patterns of streamwise vortices which have been shown to be successful in delaying transition. To generate these configurations, where streawise vortice are arranged in a periodic pattern, a series of various devices with different geome-tries, such as delta-shaped or rectangular winglets, has been analysed. At first a numerical low-cost computational campaign has been developed to identify the geometries of the devices which allow to generate a flow approximating the refer-ence cases. Once the referrefer-ence configurations have been recreated, the effect of the devices on the evolution of the boundary layer has been investigated by DNS. In this way it has been possible to study the far-wake evolution of the vortices, pay-ing attention to the streaks generated and the effects they produce on the Blasius boundary layer. Finally, a comparison with the reference cases has been proposed to evaluate the effective agreement between the flow generated by the different devices tested in this work and the numerical reference configurations.

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List of Figures

2.1 Streak generation method by means cylindrical elements of surface

roughness. Fransson et al. (2004) . . . 5

3.1 Experimental set-up . . . 10

3.2 C01 set-up scheme and reference dimensions . . . 11

3.3 Characteristics of the vortex generated by C01 . . . 11

3.4 Neutral stability curve for C01 compared with the Blasius one . . . 12

3.5 Free parameters for numerical simulations . . . 13

3.6 Streamwise streak amplitude evolution of the cases C01, E2 and E1 for Batchelor (dashed blue line) and Rankine (solid blue line) vortices 14 3.7 Sketch of DNS simulations domains . . . 15

3.8 Computational domain and boundary conditions . . . 15

3.9 Computational domain scheme . . . 17

4.1 Characteristic dimensions of a delta winglet (only half winglet is shown) . . . 20

4.2 Schemes of computational domain reduction . . . 21

4.3 Velocity fields generated by winglet C3 . . . 23

4.4 X-vorticity field and circulation trend for winglet C3 . . . 24

4.5 Circulation curve for C5 and C6 . . . 25

4.6 Circulation trends for thickness t = 0.1mm, cases C9(a), C10(b) and C11(c) . . . 27

4.7 Sketch of half gapped winglets . . . 28

4.8 Velocity fields generated by the gapped delta winglet G1 . . . 28 4.9 X-vorticity field and circulation trend for the counter-clockwise vortex 29

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4.10 Characteristic dimensions of the trapezoidal winglet . . . 29

4.11 Velocity fields generated by T1 . . . 30

4.12 X-vorticity field and circulation trend for case T1 . . . 31

4.13 Circulation trend for T2 case . . . 31

4.14 Scheme of EMVG1 configuration . . . 32

4.15 Velocity fields generated by EMVG1 . . . 32

4.16 X-vorticity field and circulation trend for counter-clockwise vortex generated by EMVG1 . . . 33

4.17 Characteristic dimensions of EMVG2 . . . 33

4.18 Characteristic dimensions of rectangular winglet . . . 34

4.19 Velocity fields generated by R1 . . . 35

4.20 X-vorticity field and circulation trend of R1 . . . 35

5.1 Streamwise streak amplitude evolution of the case R3 and the refer-ence case E1 calculated for Batchelor and Rankine vortices . . . 40

5.2 Cross-sections of the streamwise velocity component, normalized with U∞, calculated for case R3 . . . 41

5.3 Cross-sections of the streamwise velocity component, normalized with U∞, calculated for case E1 . . . 42

5.4 X-velocity perturbation fields for case R3 at three subsequent cross-sections . . . 42

5.5 X-velocity perturbation fields for case E1 at three subsequent cross-sections . . . 43

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List of Tables

2.1 Comparison of streaks amplitude between cylindrical elements and

MVGs array . . . 7

3.1 E1 and E2 simulation set-up . . . 14

3.2 Geometrical dimensions of the computational domain . . . 16

3.3 Parameters of the CFD domain . . . 17

4.1 Mesh elements size for "domain reduction" simulations . . . 21

4.2 Geometrical dimensions of the computational domain . . . 22

4.3 Mesh features of all analysed grids . . . 22

4.4 Forces and circulation evaluation . . . 23

4.5 Analysed cases for different values of α . . . 24

4.6 Chord dimensions for the analysed cases . . . 25

4.7 Analysed cases for delta-shaped devices . . . 26

4.8 Trapezoidal winglet dimensions for analysed cases . . . 30

4.9 Characteristic dimensions of EMVG1 . . . 31

4.10 Rectangular winglets R1 and R2 set-up . . . 34

4.11 Rectangular winglet R3 set-up . . . 36

4.12 Circulation values for delta-shaped winglets . . . 37

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Chapter 1

Introduction

Drag reduction has always been one of the most interesting topics in aerodynamic research field and industrial environment. When considering the flow over a sur-face, a boundary layer develops due to the fluid viscosity with the disadvantage to bring about skin-friction drag. The drag quantification is linked to its skin-friction coefficient cf whose value depends on the Reynolds number (Re), the surface rough-ness and the external disturbance level. In aerodynamic bodies, since the flow is attached over the whole surface and the wakes are thin, the friction drag represents a significant part of the overall drag. The boundary-layer can be either laminar or turbulent, but a turbulent one causes a larger friction drag than a laminar one, due to the difference in cf. In many applications, for increasing Re, this difference can amount to an order of magnitude. This explains why there is a huge interest in transition to turbulent delay for aerodynamic bodies, particularly by passive methods. These methods in fact do not need an additional energy supply, which reduces the effective gain of the mechanism. Studies observed in literature, con-cerning transition in boundary layer, highlighted that the presence of high velocity streaks, i.e. regions of velocities higher than the average surrounding field inside a boundary layer, can lead to a stabilization of the Tollmien-Schlicting (TS) waves. These waves are the main reason of the transition in boundary layer when external disturbances are sufficient low, as it happens for example for a wing in cruise con-ditions. Various approaches are presented in literature to generate these streaks by means of devices properly dimensioned. Some of these methods are listed in

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Introduction following chapters.

Vortex generators were used in the past only to control the boundary-layer separa-tion, but it has been proved that the operating mechanism of these devices, if care-fully designed, can be twofold, i.e. both transition and separation are delayed or even prevented. Previous investigations showed for example that miniature-vortex generators (MVGs) can modulate the laminar boundary layer in the direction or-thogonal to the base flow and parallel to the surface. This modulation generates an additional term in the perturbation energy equation, which counteracts the wall-normal production term and stabilize the flow. In classical MVGs method these devices are directly placed on the surface and their shape optimisation is linked with the boundary-layer thickness and the Reynolds number. Moreover the incoming flow direction could invalidate the method, and the efficiency in flow modulation is rapidly decreasing in the streamwise direction, due to a high viscous damping effect acting on the vortices generated inside the boundary layer. This can lead to the introduction of more than one array of MVGs. In order to make this mechanism independent of Reynolds number and to reduce the damping effect on the streaks generated by the MVGs, a method where counter-rotating vortices are placed outside the boundary-layer has been proposed. This also makes the vor-tices more stable and with a lower decay rate. On the basis of this new proposed method, a series of DNS simulations has been carried out in order to identify possi-ble external vortices configurations which can lead to a transition delay. Idealised Rankine and Batchelor vortices have been studied by means DNS simulations, evaluating their wake evolution and their effects on Blasius boundary layer tran-sition. As inflow condition for these DNS simulations, the sum of the velocity field induced by idealised vortices and a fully developed Blasius profile, has been used. The characteristics of these idealised vortices were based on an experimental reference configuration studied by Shahinfar et al.(2013), that will be presented in detail in §3.1. The vortices generated by this experimental configuration, named C01, in fact allowed a remarkable transition delay for a Blasius boundary layer. The most interesting configurations obtained from the DNS mentioned above have been used in this thesis as reference cases. The purpose of the present work is in fact to develop a numerical low-computational-cost campaign to design realistic devices, such as delta winglets or rectangular devices, properly dimensioned, to

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Introduction

recreate the numerical reference set-up of idealised vortices obtained as results of the previously mentioned study. Once a proper configuration of idealized vortices has been recreated by the designed devices, the same procedure applied for the DNS simulations of idealised vortices, briefly explained before, has been used to characterize the controlled boundary layer. The results of this analysis have been finally compared with the reference cases to demonstrate an agreement between the cases studied in this work and the reference configuration results.

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Chapter 2

Passive methods for transition

delay

2.1 Streaks generation previous studies

The study of streaks generation and of their development in streamwise direction in laminar boundary layers has a great importance both for industrial applications and for fundamental research. Streamwise streaks represent in fact the main fac-tor in most bypass transition scenarios. These streaks are generated by a particle transport mechanism, known in literature as lift-up mechanism (see Landahl 1980), which implies that a streamwise perturbation, due to the transfer of low- and high-speed momentum through the shear layer, is induced. When the magnitude of this streamwise perturbation becomes significant, mean velocity in the boundary layer field become unstable by inflectional instability (see e.g. Andersson et al. 2001) leading to the so-called by-pass transition. A secondary mode may grow on the primary instability, and for high enough amplitudes it will break down into a tur-bulent pattern. The first experimental works by Kachanov et Tararykin (1987) and by Westin et al. (1994) with more recent numerical linear stability analyses (see Cossu et Brandt 2002, 2004) have shown that the presence of stable stream-wise streaks, i.e. streaks with sufficiently low amplitude, can have a stabilizing effect on low-amplitude Tollmien-Schlichting (TS) waves. TS waves characterize

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Passive methods for transition delay

the initial stage of the transition scenario in the case of low amplitude external disturbances and they can be predicted by local linearized stability theory. The above referenced works inspired a long-term research campaign with the purpose of developing a passive control method, i.e. not requiring an extra energy contri-bution for flow control, for viscous drag reduction on aerodynamic bodies. This research activity has aimed at obtaining drag reduction through the extension of laminar region as much as possible, i.e. by delaying the onset of transition to tur-bulence in the boundary layer. Fransson et al. (2004) in one of their first attempts, used small cylindrical roughness elements, whose scheme is shown in figure 2.1; in this way they were able to generate stable streaks up to 12% of the free stream velocity (U∞).

Figure 2.1: Streak generation method by means cylindrical elements of surface rough-ness. Fransson et al. (2004)

In the dedicated experiments (see Fransson et al.(2005)) a damping effect of the growth of TS waves was highlighted. It was observed that an increase in the streak amplitude was directly linked with a reduction of the growth rate of the TS waves. It was demonstrated in Fransson et al. (2006) that the generated streaks are ef-ficient in reducing the growth of the TS waves, leading in some of the considered cases to a sensible delay of transition to turbulence. In many practical applica-tions, streaks are generated on a flat surface without the need for any roughness. Indeed, the onset of the bypass transition scenarios, and the consequent growth of the disturbances amplitude, could be initiated by the presence of turbulence in the free stream flow, which can lead to unsteady streamwise streaks. Even though these unsteady streaks act by stabilizing the boundary layer, they can lead to break down to turbulence at a subcritical Reynolds number by the secondary instability

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Passive methods for transition delay

acting on the streak. If this happens the stabilizing control effect will cease to exist and the control will trigger transition instead of delaying it. Therefore one must be able to generate high-amplitude laminar and steady streamwise streaks by means of some devices that only weakly interacts with free stream disturbances. One example is documented in 1962 by Tani &Komoda, where these authors used small wings located outside the boundary layer to generate steady streamwise vortices to form small amplitude (∼ 7% of U∞) streaks.

Most articles or scientific publications present in literature, instead, show many techniques to generate steady boundary layer streaks by means of vortex gener-ators periodically placed on the surface. For instance, White (2002) investigated the transient growth of disturbances of small-amplitude steady streaks generated by using a periodic spanwise array of roughness elements of circular section and small height, obtaining a maximum amplitude of the streaks that was below 4% of U∞; instead, Bakchinov et al. (1995) generated the streaks by using an array of solid rectangular rods reaching a streak amplitude around 18% of U∞.

These works inspired Fransson et al. (2004,2005), who were trying to generate high-amplitude stable streaks for TS wave control. These authors were able to generate stable streaks up to 12% of U∞by using a similar roughness element con-figuration where the streaks were generated by the lift-up mechanism, previously mentioned and explained. In their work they showed that there are two different physical mechanisms whose effects have to be compared in order to understand which one dominates the transition: the perturbation induced by the vortices pro-duced by the incoming vorticity upstream of the roughness elements and the wake generated past them. For the considered geometry, i.e. using circular roughness elements, the limitation in the roughness height so as to generate steady and stable streaks of large amplitudes was not due to an instability of the streaks but, rather, to a near-wake instability associated with the generating device (e.g. vortex shed-ding). More recently Miniature Vortex Generator (MVG) pairs mounted on a plate have been used instead of roughness elements, and they have been investigated in Fransson, Talamelli (2012). In this way it was possible to generate directly steady counter-rotating vortices inside the boundary layer capable of setting up the de-sired streaks through the well-known lift-up mechanism.

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of U∞) respect to surface-roughness cylinders as show in figure 2.1.

Max streak amplitude

Surface Roughness ∼ 12%

MVGs ∼ 32%

Table 2.1: Comparison of streaks amplitude between cylindrical elements and MVGs array

To obtain a successful passive control method we need to grant robustness and persistence of the modulated laminar boundary layer. This means that the gener-ated streaks have to be stable, in the sense that no trace of secondary instability on the streaks must be present, and that the streaks, whose efficiency decay in downstream direction, may be regenerated to reinforce the stabilizing mechanism without causing destructive vortex interaction and breakdown to turbulence as studied before by Pearcey (1961) and confirmed experimentally by Frasson et al. (2012).

2.2 Streaks generation by means of external

vor-tices

The studies on boundary-layer control techniques discussed in the previous sec-tion, demonstrate that the use of vortex generators placed directly on the surface

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allows to obtain a good speed modulation inside the boundary layer and a re-markable transition delay. These devices can produce stable streaks, but they also present some disadvantages. First, the MVG’s maximum height is dependent to the Reynolds number, which determines the boundary-layer thickness. In addi-tion, the streaks generated by this method have a high streamwise decay rate. This means that one or more additional downstream MVGs array usually occurs. Finally, the MVGs presence create an unstable region in their near-wake in which some modes will be excited.

In order to overcome these problems, the idea of generating external vortices to induce streaks in the boundary layer has been investigated, with reference to the experimental study of Tani &Komoda (1962). This allows to solve the problem of downstream streak efficiency and to size the devices independently from the bound-ary layer thickness. In fact, the streaks generated by external vortices reaches a higher maximum amplitude and larger streamwise extension respect to the MVGs case, where the dissipative effect of the boundary layer is predominant.

In previous numerical investigations, dedicated to the use of external vortices, the device which generates such vortices has never been simulated, but only its idealized wake has been studied in order to identify particular wake configurations which can lead to a transition delay. The Blasius boundary layer and ideal vortices fields have been superimposed and used as inflow condition of the computational domain. More detailed information about the flow configuration will be presented in the following section of this work.

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Chapter 3

Flow configuration and

numerical set-up

The purpose of the first part of this chapter §3.1 is to show the experimental ref-erence configuration C01 described in Shahinfar et al., whose characteristics are presented in Fig.3.2. This particular configuration has been chosen as reference case because it has been highlighted that the C01 vortices have a high stabilising effect. In §3.2 a preliminary numerical study based on the use of external vortices to induce transition delay in the boundary layer is presented. In §3.3 the com-putational domain and the numerical set-up for the DNS simulations have been reported. Finally, in §3.4 of this chapter all numerical tools used in this work are briefly presented.

3.1 Experimental reference configuration

First of all it has been necessary to determine a reference case among all the experi-mental data available of a research campaign carried out at KHT Royal Institute of Technology. Here the experimental reference case set-up used to study the MGVs effects on transition delay in boundary layer is briefly represented in Fig.3.1. The flow domain is divided into four regions. In region (I) a two-dimensional laminar boundary-layer develops, while in (II) TS waves, indicated by the black and white

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Flow configuration and numerical set-up

pattern perpendicular to the main stream, are generated by means of blowing and suction through a spanwise slot in the plate located at xT S. In region (III) an array of MVGs is placed at xM V Gs and the high and low speed streaks are generated. In region (IV) the amplitude of the streaky base flow has finally decayed and the base flow found in region (I) will be recovered, unless the streaks breakdown to turbulence. The distances xT S = 190 mm and xM V Gs = 222 mm are calculated from the leading edge of the flat plate. The C01 case has been chosen as reference

Figure 3.1: Experimental set-up

one and its characteristic dimensions are directly highlighted in Fig.3.2. In the sketch the coordinates system (x, y, z) is introduced with the corresponding mean velocity components (U, Y, Z) and perturbation velocity components (u, v, w). The distance between the neighbouring MVGs pairs is D, the MVG height is h and the freestream velocity is U∞. In previous works, the longitudinal vorticity distribu-tion for one of the counter-rotating vortices generated by a pair of MVGs has been calculated and even the circulation associated to such vortex has been evaluated. These results are reported in Fig.3.3, where the values of the parameters, which identify such configuration, have been highlighted. In Fig.3.3(a) the distance be-tween the counter-rotating vortices generated by a MVGs pair is lv = 3.2 mm, in Fig.3.3(b) instead the circulation and the core radius of the vortex are shown. The values of these two parameters are equal to Γ0 = 9 · 10−4 m2/s and r0 = 0.8 mm. To highlight the properties of C01 as transition delay device, reason why it has

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Flow configuration and numerical set-up

Case h D d U∞ RexM V G

(mm) (mm) (mm) (m/s) (–)

C01 1.3 13.0 3.25 7.7 1.71 · 105

Figure 3.2: C01 set-up scheme and reference dimensions

Figure 3.3: Characteristics of the vortex generated by C01

been selected as reference case, the stability curve of C01 is reported in Fig.3.4, where the region B shows the delay to higher Reynolds numbers of the onset of the transition to turbulence zone with respect to the uncontrolled Blasius bound-ary layer (dashed blue line). The region A is linked instead at the disturbance generated by the physical presence of the MVGs inside the boundary layer.

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Figure 3.4: Neutral stability curve for C01 compared with the Blasius one

3.2 External vortices reference configurations

On the basis of the theory presented in §2.2 a preliminary numerical investigation has been carried out (see Lisanti (2013)), where idealised wakes of Rankine vortices have been studied to define configurations which could lead to a transition delay. In order to obtain suitable external vortices configurations to ensure good char-acteristics of the streaks, a series of free parameters have been considered. These parameters are shown in Fig.3.5 and explained in the following list:

• r0 = core radius of the vortex

• hv = distance of the core centres from the flat plate

• lv = distance between the cores centres of the same couple • Γ0 = global vortices circulation

• Xv = distance of the vortices, evaluated from the leading edge of the flat plate

Starting from the characteristics of the experimental reference vortex obtained in case C01, the free parameters available have been modified. All the numerical sim-ulations have been carried out in air with kinematic viscosity ν ' 1.46 · 10−5m2_/s

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Figure 3.5: Free parameters for numerical simulations

and the Reynolds number was kept constant as experimental experience Reh = U∞·h

ν = 658.62. All the geometrical and physical parameters have been made no-dimensional using the reference length href = 1.3mm, that we recall to be equal to the height of the experimental MVG, and the reference velocity U∞ = 7.7m/s as listed below: • r0 = _hr_ref0 • hv = _hhv ref • lv = _hlv ref • Γ0 = _U_∞Γ_·h0 ref • xst = _hXv ref

On the basis of this preliminary study, two particular configurations, whose char-acteristics are reported in Tab.3.1, have been identified. It has been demonstrated that these configurations ensure good features of the streaks intensity as reported in detail in Fig.3.6. In particular the case E1 is characterised by a circulation value equal to the experimental case C01, while the case E2 reaches a local maximum amplitude equal to the case C01.

These configurations reported in Tab.3.1 will be considered the reference cases we have to recreate in this work through ad hoc vortex generator devices.

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Figure 3.6: Streamwise streak amplitude evolution of the cases C01, E2 and E1 for Batchelor (dashed blue line) and Rankine (solid blue line) vortices

Case Reh r0 hv lv Γ0 xst E1 685.62 1.0 3.0 2.4 0.09 40 E2 685.62 1.0 3.0 2.4 0.06 40

Table 3.1: E1 and E2 simulation set-up

3.3 DNS simulations

In order to evaluate the evolution of the vortices generated by these designed devices, it has been decided to divide the simulations in two subsequent steps, in order to obtain a computational costs reduction. As first the fluid-dynamic analysis of the flow has been executed in the proximity of the device, in the second part instead the far-wake simulations have been carried out in order to evaluate the wake evolution.

The results of the first simulations calculated into the domain D1, whose di-mensions will be presented in Tab.3.2, will be extracted at a certain Xsection and used as inflow condition into the domain D2 where the far-wake simulations are carried out (see Fig.3.7).

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Figure 3.7: Sketch of DNS simulations domains

3.3.1 Near-wake simulations

The fluid-dynamic analysis for the near-wake simulations is carried out by using F luent R _{as CFD software. A steady laminar three-dimensional flow has been}

solved, a SIMPLE segregated algorithm has been used for pressure-velocity cou-pling and a second-order upwind scheme has been applied respectively for pressure and momentum discretizations. The computational domain and the boundary con-ditions applied are presented in Fig.3.8 and in Tab.3.2.

Figure 3.8: Computational domain and boundary conditions

Taking advantage of the symmetrical configuration of the problem the spanwise extension of the domain has been reduced. The procedure used to demonstrate

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l1 w1 h1 l2 w2 h2 65href 5href 8href 80href 5href 40href

Table 3.2: Geometrical dimensions of the computational domain

this assumption is reported in §4.1.1. As inflow condition a uniform velocity inlet U∞= 7.7m/s is introduced and a pressure outlet boundary condition is imposed at the end of the domain. We recall that U∞is the freestream velocity in the reference cases E1 and E2. A symmetry condition is applied on lateral and top surfaces and no-slip and no-penetration conditions are imposed on the device surface and on the flat plate. Two different subvolumes are created inside the domain to have a better control on the growth rates of the elements of the calculation grid. Most of the elements are placed next to the body, so as to have a good local grid resolution where the vortices are generated. The grid convergence of the results applied in near-wake simulations will be discussed in §4.1.2. The velocity fields generated by the tested winglets will be extracted at a Xsection = 10href past the device. For the most interesting case among those considered here, a simulation of the far-wake has been carried out. Before superimposing the field generated by the vortices to the Blasius velocity profile, it has been necessary to isolate the wake velocity deficit caused by the vortex, subtracting the freestream velocity component to the velocity field obtained from Fluent simulations. The field obtained from this procedure will be used as inflow condition of the far-wake computational domain.

3.3.2 Far-wake simulations

The far-wake simulations have been carried out with the numerical code NEK5000. The computational domain height, width and streamwise extension are represented in Tab.3.3. The streamwise extension ranges from Rex = 2.7 · 104 to Rex = 5.5 · 105 and the simulated domain has a spanwise dimension which allows to include only a single vortex of the couple.

The flow, at a freestream velocity of U∞ = 7.7m/s arrives on a flat plate as in Fig.3.2. The boundary conditions applied for the simulations are reported in Fig.3.9. At the inflow boundary the velocity field is imposed to be equal to the

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Figure 3.9: Computational domain scheme Parameters Values h 66 href w 5 href l 619 href Number of elements 1.01 · 105

Table 3.3: Parameters of the CFD domain

sum of a fully developed Blasius boundary layer with the flow field generated by the counter-rotating vortex created by the device. Symmetry conditions on the lateral surfaces allows instead to simulate only one vortex instead of a couple of vortices with periodic conditions on the lateral surfaces. The outflow condition, which is a stress-free condition, is imposed at the end and at the upper surface of the computational domain. Finally, a no-slip and no-penetration condition is applied to the bottom surface to simulate the flat plate. We are here interested in studying only the steady base flow, while the stability of the flow will be analysed in future works using the biglobal-stability analysis methods.

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3.4 Details on numerical tools

Gambit R _{is a Fluent meshing tool, that allows to define the geometries we have}

investigated with a parametric description. In some cases CATIA R _{has been}

use-ful to generate the .igs files used as staring point of the meshing process. Gambit has also been used to mesh models for computational fluid dynamic analysis. The commercial software Fluent R_{, fluid dynamic finite-volume solver, has been}

employed for a preliminary analysis of different configurations, before the far-wake simulations have been carried out.

The numerical code used to perform the DNS of the considered flow is NEK5000, which is an open-source massively parallel spectral element solver, employing hex-ahedral elements based on tensorial product of Gauss-Lobatto-Legendre nodes. Details on this code can be found directly on the reference website:

http://nek5000.mcs.anl.gov/index.php/Main_Page.

For the post-processing part of this work two different tools have been applied. Tecplot R _{has been used to analyse Fluent data files and the Nek5000 database}

after the conversion into Tecplot format. It allows to visualize the velocity fields e to interpolate such fields into structured grids, starting point for further analysis. Matlab R _{made possible to calculate all quantities we need after importing the}

fluid fields. For example the vorticity field and the circulation values has been evaluated by means finite differences method with a 2-nd order centred scheme.

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Chapter 4

Near-wake simulation results

for vortex generators

The main part of the near-wake simulations is aimed at approximating the first reference case, E1. However, during this work it has been highlighted, from other studies, that such configuration continues to present an instability region, even if the onset of this zone is moved forward to higher Rex numbers with respect to the uncontrolled case. This type of control allows anyway a remarkable delay in transition to turbulence. It has been demonstrated that the instability region is due to a too high streaks amplitude, which triggers the instability of the streaks and, in turn, the by-pass transition of the boundary layer. The case E2, with a lower streak amplitude has been therefore analysed. However, since this has been done only recently, we have studied here only one configuration representing this case. This chapter is organized into three main parts, each one focused on the results obtained for a particular geometry of the devices. The first one, §4.1, is centred on the analysis of delta-shaped winglets, the second one, §4.2 is focused on the investigation over the flow generated by the MVGs studied by Shihanfar et al. which have been shifted outside the boundary layer. Finally, in the last section §4.3, winglets with a rectangular shape have been studied.

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Near-wake simulation results for vortex generators

4.1 Delta-shaped winglets

Since the E1 and E2 configurations are based on couples of counter-rotating vor-tices, the flow produced by delta-shaped winglets, which are the most powerful devices to generate such type of vortices, has been investigated as a first attempt. The characteristic dimensions of a delta winglet are shown in Fig.4.1.

Figure 4.1: Characteristic dimensions of a delta winglet (only half winglet is shown)

4.1.1 Computational domain reduction

Taking advantage of the symmetrical configuration of the problem, we tried to reduce the computational costs by limiting the calculation domain. A first set of simulations, whose rationale has been reported in Fig.4.2, has been executed to validate the adopted procedure. With reference to Fig.4.2, three computational domains have been used:

• 3 delta winglets with a spanwise spacing of 10href and a total spanwise width equal to 30href (Case A)

• 1 delta winglet with a spanwise box extension of 10href (Case B) • 1

2 delta winglet with a spanwise box extension of 5href (Case C)

where we recall that href = 1.3mm is the reference length, equal to the experi-mental MVGs height (see Shihanfar et al.). The size of the discretization on each winglet for cases A, B and C (see Tab.4.1)is always the same in order to compare the obtained results: The results of these simulations, which are not reported here

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(a) Case A (b) Case B (c) Case C

Figure 4.2: Schemes of computational domain reduction

Element size 0.075 Wing Growth rate 1.2

Max Size 0.15 Fluid (Vol.1) Growth rate 1.2

Max Size 0.3 Fluid (Vol.2) Growth rate 1.2

Max Size 2.5

Table 4.1: Mesh elements size for "domain reduction" simulations

for sake of brevity, demonstrate that the force coefficients obtained for each winglet are the same for all three cases since the simulated flow is steady and stable and respects the geometric symmetries of the problem. For this reason the computa-tional domain of case C has been chosen as reference computation domain for the other simulations, so as to save computational costs. Its dimensions are reported in Tab.4.2.

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l1 w1 h1 l2 w2 h2

65href 5href 8href 80href 5href 40href

Table 4.2: Geometrical dimensions of the computational domain

First of all a delta winglet with a thickness t = 0.3mm, a base width b = 2href, a chord c = 3.85href and an incidence α = 15◦ has been chosen for our preliminary proofs. The distance of the device from the flat plate has been fixed equal to hv = 5href. The thickness t = 0.3mm has been chosen equal to that of the MVGs in the experiments by Shaninfar et al.. The other quantities have been selected so as to approach the reference case E1.

4.1.2 Mesh convergence

Four different grids, with a progressively higher spatial resolution, have been tested to check the grid convergence of the results. The features of each grid has been summarized in Tab.4.3:

C1 C2 C3 C4

Element number 2.25·106 _(Vol.1) _4.1·106_(Vol.1) _7.9·106 _(Vol.1) _19.2·106_(Vol.1)

1.06·106 _(Vol.2) _1.8·106_(Vol.2) _3.05·106_(Vol.2) _4.2·106 _(Vol.2)

Element size 0.075 0.05 0.025 0.02

Wing Growth rate 1.2 Growth rate 1.1 Growth rate 1.05 Growth rate 1.05 Max Size 0.15 Max Size 0.1 Max Size 0.1 Max Size 0.08 Fluid (Vol.1) Growth rate.1.2 Growth rate 1.1 Growth rate 1.05 Growth rate 1.05

Max Size 0.3 Max Size 0.25 Max Size 0.2 Max Size 0.15 Fluid (Vol.2) Growth rate 1.2 Growth rate 1.2 Growth rate 1.2 Growth rate 1.2

Max Size 2.5 Max Size 2 Max Size 2 Max Size 2

Table 4.3: Mesh features of all analysed grids

Force coefficients Cl and Cd have been evaluated together with the circulation Γ0 for all four cases and reported in Tab.4.4. It has been assumed that grid independence is reached when two different refined grids show a difference of less than 2% in the three previous quantities. The main results of the grid sensitivity analysis are summarized in Tab.4.4:

Since it can be observed that the results do not experience significant changes for computational grid more refined than C3, the dimensions of the elements of grid C3 have been taken as reference for the successive simulations.

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C1 C2 C3 C4

Cl 0.61 0.58 0.57 0.57 Cd 0.31 0.3 0.29 0.29 Γ0 0.95 0.94 0.93 0.93

Table 4.4: Forces and circulation evaluation

4.1.3 Delta winglet C3

The velocity fields for case C3 have been extracted at a Xsection= 10href past the device. In Fig.4.3 the velocity fields normalized with the reference velocity Uref = U∞ are presented, in particular Fig.4.3(a) shows the wake velocity deficit for the tip vortex core and Fig.4.3(b) and Fig.4.3(c)represent the induced veloc-ities of the vortex, respectively in y- and z- direction, with maximum picks of 0.15U∞. In Fig.4.4(a) the axial vorticity, normalized with the freestream velocity U∞ and reference length href, is presented. Fig.4.4 also shows that the radius of the core is about equal to 1.2href. The x-vorticity field has been calculated ap-plying a finite differences method with a 2-nd order centred scheme. In Fig.4.4(b) the no-dimensional circulation trend is presented for C3 respect with the core ra-dius, normalized with the reference length href. This trend has been calculated evaluating the circulation of the velocity vector on iso-vorticity contours.

Z/h ref Y/h ref X Velocity/Uref 0 2 4 −3 −2 −1 0 1 2 3 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 (a) X Velocity Z/h_ref Y/h ref Y_Velocity/U_ref 0 2 4 −3 −2 −1 0 1 2 3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 (b) Y Velocity Z/h_ref Y/h ref Z_Velocity/U_ref 0 2 4 −3 −2 −1 0 1 2 3 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 (c) Z Velocity Figure 4.3: Velocity fields generated by winglet C3

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Z/h_ref

Y/h

ref

X_vorticity⋅ (h_ref/U_ref)

0 2 4 −3 −2 −1 0 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 (a) X-Vorticity 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r 0/href Γ0 /( U ref ⋅ href ) Circulation (b) Circulation trend

Figure 4.4: X-vorticity field and circulation trend for winglet C3

4.1.4 Incidence reduction

Reminding that the reference no-dimensional circulation value in E1 case is Γ0 = 0.09, it can be observed in Fig.4.4(b) that the circulation obtained has an order of magnitude larger value with respect to the E1 case. In order to obtain a lower circulation value, the incidence α of this device has been reduced. The analysed cases for different angles of attack are listed in Tab.4.5:

Case α (deg)

C3 15

C5 12

C6 10

Table 4.5: Analysed cases for different values of α

The values of the circulation obtained with the same procedure applied before are shown in Fig.4.5. It can be observed that the value of the circulation has decreased for both cases according to the reduction of the angle of attack, however it continues to have a too large value with respect to the reference case.

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Near-wake simulation results for vortex generators 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r 0/href Γ0 /( U ref ⋅ href ) Circulation (a) Case C5 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r 0/href Γ0 /( U ref ⋅ href ) Circulation (b) Case C6 Figure 4.5: Circulation curve for C5 and C6

4.1.5 Chord dimension reduction

The results of the previous simulations (see Fig.4.5 ) showed again too large values of the circulation Γ0. Starting from the C6 case, it has been tried to reduce the intensity of the vortices, reducing the chord length in order to reduce the length of the lateral edge where the vortices detach. It has been considered a 65% and a 35% chord length respect with C6 case (see Tab.4.6) . During the analysis of these

Case c/href

C7 2.5

C8 1.25

Table 4.6: Chord dimensions for the analysed cases

cases a new problem arose. In fact vortex shedding starts to be present on the device invalidating our results. This is probably due to an excessive thickness of the device respect with the streamwise extension of winglet section in the proximity of the tip. In the next section we present in fact the results for a lower thickness t of the device.

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4.1.6 Thickness reduction

The previous thickness of the delta winglets was based on the dimensions of the MVGs experimentally tested at KHT Royal Institute of Technology. A new thick-ness of the device t = 0.1mm has been investigated. The case C8, with α = 10◦ and c = 1.25href, named here C9, has been again investigated with the new values of the thickness together with two other cases with a reduced incidence C10 with α = 8◦ and C11 with α = 6◦. For these cases the vortex shedding is no more present and the circulation has been calculated and reported in Fig.4.6. The val-ues reached with these new configurations are extremely lower with respect to the results obtained for the previous investigated devices. It can be observed that the value of the thickness of the device has an important role in allowing to reduce the circulation value. In fact for small chord lengths a small thickness is necessary to overcome the vortex-shedding occurrence. Observing the results it can be briefly conclude that all these alterations of the initial dimensions of the delta-shaped devices allowed to reduce drastically the circulation value, even if the value of the circulation of the case E1 has not been reached yet.

In Tab.4.7 all the delta winglets studied in this section are summarized report-ing the Γ0 = _U_refΓ·h0ref reached and the parameters of the devices.

Case c/href b/href t (mm) α (deg) Γ0

C3 3.85 2 0.3 15 0.93 C5 3.85 2 0.3 12 0.81 C6 3.85 2 0.3 10 0.7 C7 2.5 2 0.3 10 (-) C8 1.25 2 0.3 10 (-) C9 1.25 2 0.1 10 0.31 C10 1.25 2 0.1 8 0.25 C11 1.25 2 0.1 6 0.2

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Near-wake simulation results for vortex generators 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r 0/href Γ 0 /( U ref ⋅ h ref ) Circulation (a) α = 10◦ 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r 0/href Γ 0 /( U ref ⋅ h ref ) Circulation (b) α = 8◦ 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r 0/href Γ 0 /( U ref ⋅ h ref ) Circulation (c) α = 6◦

Figure 4.6: Circulation trends for thickness t = 0.1mm, cases C9(a), C10(b) and C11(c)

4.1.7 "Gapped" delta winglets

In this section a new configuration, named G1, will be proposed. This new delta winglet is characterised by the same dimensions of experimental MVGs. The counter-rotating vortices are generated by two winglets with a gap of 2g = 1.2href (see Fig.4.7). The characteristic dimensions of this device are: t = 0.3mm, c = 2.5href and b = href. The boundary conditions of the flow simulations are the same of all the cases analysed before.

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Figure 4.7: Sketch of half gapped winglets

Fig.4.8 and in Fig.4.9.

Z/h_ref Y/h ref X_Velocity/U_ref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 (a) X-Velocity Z/h_ref Y/h ref Y_Velocity/U_ref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 (b) Y-Velocity Z/h_ref Y/h ref Z_Velocity/U_ref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 (c) Z-Velocity Figure 4.8: Velocity fields generated by the gapped delta winglet G1

In Fig.4.8(a) the streamwise velocity component, normalised with the reference velocity Uref, is shown. The velocity fields observed in Fig.4.8(b) and in Fig.4.8(c) represent the induced velocity of a dipole with a tilted axis, whose presence is also confirmed by Fig.4.9(a). The vorticity field shown in Fig.4.9(a) highlights in fact the presence of a clockwise vortex (coloured with blue), counter-rotating with respect to the tip vortex. It can be also observed, from Fig.4.9, that a too high circulation is still present. This field configuration could be even better than the reference one for transition control, but it should have to be tested. In this work our purpose is to approximate the E1 case, so this particular case will be studied in future works. This configuration will be so disregarded for this moment, and

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Z/h_ref

Y/h

ref

0 1 2 3 4 5 −3 −2 −1 0 1 2 3 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 (a) X-Vorticity 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r₀/h_ref Γ0 /( U ref ⋅ href ) Circulation (b) Circulation trend

Figure 4.9: X-vorticity field and circulation trend for the counter-clockwise vortex

we will continue investigating other configurations.

4.1.8 Trapezoidal device

The G1 device geometry has been again modified to avoid the presence of the clockwise-vortex detected in §4.1.7. The new configuration is a trapezoidal winglet, whose geometrical parameters t, c, h and b are presented in Fig.4.10. As usual, a

Figure 4.10: Characteristic dimensions of the trapezoidal winglet

sensitivity analysis with respect to the parameters c and α has been carried out in order to reach the expected circulation value. The characteristics of the three simulations carried out for this geometry are summarized in Tab.4.8:

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Case b/href c/href h/href t α

(–) (–) (–) (mm) (deg)

T1 1.6 2.5 1 0.3 15

T2 1.6 1.875 1 0.3 10

T3 1.6 1.25 1 0.3 10

Table 4.8: Trapezoidal winglet dimensions for analysed cases

The flow fields and the circulation for case T1 are presented in Fig.4.11 and in Fig.4.12. In Fig.4.11(c) and in Fig.4.11(b) the velocity components induced by

Z/h_ref Y/h ref X_Velocity/U_ref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 0.75 0.8 0.85 0.9 0.95 1 (a) X-Velocity Z/href Y/h ref YVelocity/Uref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 (b) Y-Velocity Z/href Y/h ref ZVelocity/Uref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 (c) Z-Velocity Figure 4.11: Velocity fields generated by T1

the vortex in y- and z-directions are presented, in Fig.4.11(a) instead the effect of the vortex on the x-velocity component is shown. Observing Fig.4.12 it can be noticed that the vortex has a core radius r = 1.2href but again a high circulation value is obtained. For the case T2, the velocity fields have not been reported in this work, because they are qualitatively similar to Fig.4.11. Comparing the circulation of T2 shown in Fig.4.13 with Fig.4.12(b) it can been seen that T2 device produces a lower value of circulation, even if the E1 reference value has not been reached yet. Finally, since case T3 showed again vortex shedding and, thus, its results have been disregarded.

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Z/h_ref

Y/h

ref

0 1 2 3 4 5 −3 −2 −1 0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(a) X-Vorticity (b) Circulation trend

Figure 4.12: X-vorticity field and circulation trend for case T1

Figure 4.13: Circulation trend for T2 case

4.2 External Miniature Vortex Generators (EMVGs)

In this section, the flow generated by the same MVGs of the experimental reference configuration placed outside the boundary layer over a flat plate, has been inves-tigated. A sketch of this configuration is shown in Fig.4.14 and the geometrical dimensions are listed in Tab.4.9:

Case c/href h/href t d/href tl α

(–) (–) (mm) (–) (mm) (deg)

EMVG1 2.5 1 0.3 2.5 0.15 15

Table 4.9: Characteristic dimensions of EMVG1

The velocity fields, the axial vorticity and the circulation trend generated by EMVG1 are presented in Fig.4.15 and Fig.4.16. In Fig.4.15(a) a largely extended

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Figure 4.14: Scheme of EMVG1 configuration

wake velocity deficit caused by the EMVG1 is shown. The presence of a second counter-rotating vortex in the field can be deduced by the maps of y- and z-velocity showed in Fig.4.15(b) and in Fig.4.15(c) and confirmed by the Fig.4.16(a) where the x-vorticity field is presented. This vortex is however weak and its presence is not a priori negative for the control on the boundary layer transition.

Z/href Y/h ref XVelocity/Uref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 (a) X-Velocity Z/h_ref Y/h ref Y_Velocity/U_ref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 (b) Y-Velocity Z/h_ref Y/h ref Z_Velocity/U_ref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 (c) Z-Velocity Figure 4.15: Velocity fields generated by EMVG1

As explained before in §4.1.7 our purpose is to approximate E1 configuration so we tried to avoid the generation of this vortex. Assuming that the origin of this vortex was due to the interference between the support and the EMVG1, it has been tried to eliminate this vortex using a chamfer between these two surfaces as represented in Fig.4.17. The value of the radius of the chamfer has been chosen equal to the radius of the weaker vortex, however this problem still persists.

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Z/h_ref

Y/h

ref

0 1 2 3 4 5 −3 −2 −1 0 1 2 3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Figure 4.16: X-vorticity field and circulation trend for counter-clockwise vortex gener-ated by EMVG1

Figure 4.17: Characteristic dimensions of EMVG2

4.3 Rectangular winglets

In this last section, the numerical investigation is simply based on a rectangular winglet, whose characteristic dimensions are shown in Fig.4.18.

For this particular case the analytic solution for rectangular wings present in lit-erature (Lifting-line theory) has been used to define the winglet dimensions. Since the values of the circulation (Γ0 = 0.09 for E1 and Γ0 = 0.06 for E2) and the distance between the tip vortices (b0 = lv = 2.4href for E1 and E2) are a priori known, the geometry of the wings can be easily defined. An aspect ratio ofA=6 has been chosen. The wings set-up for reaching expected values of Γ0 = 0.09 and Γ0 = 0.06, respectively R1 and R2, are reported in Tab.4.10. From Tab.4.10, it can be also noticed that a lower value of the circulation calculated in the

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simula-Figure 4.18: Characteristic dimensions of rectangular winglet

Case A Γ0ref Γ0sim b0/href c/href b/href t α

(–) (–) (–) (–) (–) (–) (mm) (deg)

R1 6 0.09 0.069 2.4 0.5 3 0.05 4.5

R2 6 0.06 0.051 2.4 0.5 3 0.05 3.5

Table 4.10: Rectangular winglets R1 and R2 set-up

tion (Γ0sim) respect to the expected analytic results (Γ0ref) for both configurations (R1 and R2) has been obtained. For both cases the velocity fields are extracted at a Xsection = 10href past the device. The results of the numerical analysis are presented from Fig.4.19 to Fig.4.22. In Fig.4.19(a) it can be observed that the wake velocity deficit is not only in the proximity of the tip vortex, but it has a wide spanwise extension. The same behaviour will be also noticed in Fig.4.21(a). The y- and z-velocities induced by the vortex are shown in Fig.4.19(b) and in Fig.4.19(c) for case R1, the same quantities are instead reported in Fig.4.21(b) and in Fig.4.21(c) for the case R2. The core radius of the vortices for both cases, R1 and R2, is about r0 = 0.7href, which results smaller than the E1 and E2 core radius r0 = href. The half spanwise distance between the counter-rotating vortices of the same couple results equal to lv/2 = b0/2 = 1.2href (see Fig.4.20(a) and Fig.4.22(a)). It can be concluded that the characteristics of the vortices generated by R1 and R2 are similar to the reference vortices of the cases E1 and E2.

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Z/h_ref Y/h ref X_Velocity/U_ref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 0.88 0.9 0.92 0.94 0.96 0.98 1 (a) X-Velocity Z/h ref Y/h ref Y Velocity/Uref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 −0.02 −0.01 0 0.01 (b) Y-Velocity Z/h ref Y/h ref Z Velocity/Uref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 −0.02 −0.01 0 0.01 (c) Z-Velocity Figure 4.19: Velocity fields generated by R1

Z/h_ref

Y/h

ref

0 1 2 3 4 5 −3 −2 −1 0 1 2 3 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

(a) X-vorticity (b) Circulation trend

Figure 4.20: X-vorticity field and circulation trend of R1

Z/h_ref Y/h ref X_Velocity/U_ref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 0.88 0.9 0.92 0.94 0.96 0.98 1 (a) X-Velocity Z/h ref Y/h ref Y Velocity/Uref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 −0.015 −0.01 −0.005 0 0.005 0.01 (b) Y-Velocity Z/h ref Y/h ref Z Velocity/Uref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 (c) Z-Velocity Figure 4.21: Velocity fields generated by R2

The last configuration analysed R3 is presented in Tab.4.11. In this case the distance between the cores of the vortices has been modified in order to obtain a

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Z/h_ref

Y/h

ref

0 1 2 3 4 5 −3 −2 −1 0 1 2 3 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

uniform spanwise spacing (b0 = 5href). In this configuration the velocity induced Case A Γ0ref Γ0sim b0/href c/href b/href t α

(–) (–) (–) (–) (–) (–) (mm) (deg)

R3 6 0.09 0.086 5 1.05 6.4 0.05 2.5

Table 4.11: Rectangular winglet R3 set-up

on each centre of the vortices by the other counter-rotating vortices in the field is negligible and the vortices remain, in theory, at the same height respect to the flat plate. The numerical results for this configuration will be shown in Fig.4.23 and in Fig.4.24. As observed before in Fig.4.19(a) and in Fig.4.21(a), even in Fig.4.23(a) the wake velocity deficit has a wide extension in spanwise direction. In Fig.4.23(b) and in Fig.4.23(c), the induced y- and z-velocities are shown and it can be observed that they reach picks of 0.015U∞.

In Tab.4.12 and in Tab.4.13 all the cases analysed in this chapter will be re-sumed and the most interesting cases will be highlighted. In particular, in Tab.4.12 the circulation values for the delta-shaped winglets are reported and the circulation values for the other geometries will be presented in Tab.4.13.

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Z/h_ref Y/h ref X_Velocity/U_ref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 0.9 0.92 0.94 0.96 0.98 1 (a) X-Velocity Z/h ref Y/h ref Y Velocity/Uref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 −0.01 −0.005 0 0.005 0.01 0.015 (b) Y-Velocity Z/h ref Y/h ref Z Velocity/Uref 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 (c) Z-Velocity Figure 4.23: Velocity fields generated by R3

Z/href

Y/h

ref

Xvorticity⋅ (href/Uref)

0 1 2 3 4 5 −3 −2 −1 0 1 2 3 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

(a) X-Vorticicty (b) Circulation trend

Case C3 C5 C6 C7 C8 C9 C10 C11

Γ0 0.93 0.81 0.7 (–) (–) 0.31 0.25 0.2

Table 4.12: Circulation values for delta-shaped winglets

Case G1 T1 T2 T3 EMVG1 R1 R2 R3

Γ0 0.28 0.85 0.61 (–) 0.28 0.069 0.051 0.086

Table 4.13: Circulation values for the analysed configurations

Looking at Tab.4.13 it can be observed that only rectangular-shaped winglets allow to reach circulation value not too much different from the reference configu-rations. In particular the R3 case allows to reach a circulation value similar to the

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reference case E1. The R2 configuration instead reaches a lower circulation value with respect to the reference case E2. Since the vortices generated in R3 have similar dimensions and almost the same circulation of the E1 case (see Tab.3.1), they will be used in NEK5000 to study the evolution of the generated streaks.

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Chapter 5

Analysis of the far-wake

simulations

In this chapter the post-processing analysis of the results of the far-wake simulation is presented. The maximum amplitude of the streaks generated by R3 has been calculated and compared with the reference case E1. Velocity fields for different x-sections will be presented to highlight the evolution of the streamwise velocity modulation.

5.1 Nek5000 simulation results

One of the most interesting characteristics of a streak is represented by its local maximum amplitude. This quantity is introduced in order to make a direct com-parison with the reference case, where the same definition has been used. The definition of the local maximum amplitude is reported in the equation 5.1.

AST_max(X) = 1 2U∞ max y {∆U (X, y)} (5.1) where: ∆U (X, y) = max

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Analysis of the far-wake simulations

It can be noticed that this amplitude measure maximizes over both z and y, mean-ing that the spanwise distance between the points used to calculate ∆U (X, y) will differ depending on the downstream location and the considered wall-normal po-sition. The evolution of the maximum amplitude of the streaks generated by the vortices of R3 device is shown in Fig.5.1, together with the amplitude of the streaks generated by the reference case E1 simulated with both Rankine and Batchelor vortices. It can be observed that the maximum of the streak amplitude for R3 is ∼ 27% of U∞, reached for a x-section of 800 mm. The evolution of the streaks amplitude for the present case R3 is in agreement with the evolution of the streaks amplitude evaluated for the idealized vortices. In particular the maximum of the amplitude reached for the present case is exactly the same obtained for the case E1 with the Batchelor vortex, while the maximum amplitude obtained for E1 Rankine vortex is slightly higher and it is ∼ 30% of U∞.

0 100 200 300 400 500 600 700 800 0 0.05 0.1 0.15 0.2 0.25 A

max with Gamma = 0.09

X(mm) Am a x R3 E1 Batchelor E1 Rankine

Figure 5.1: Streamwise streak amplitude evolution of the case R3 and the reference case E1 calculated for Batchelor and Rankine vortices

It is also clear from Fig.5.1 that the streak for case R3 evolves faster than the cases with the ideal vortices. This behaviour can be explained by observing the velocity modulation inside the boundary layer generated by the real vortices of R3 case in Fig.5.2, where the contours of the streamwise velocity field U, normalized

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Analysis of the far-wake simulations Z (mm) Y (mm) X−Velocity 0 2 4 6 0 2 4 6 8 10 12 (a) X=330 mm Z (mm) Y (mm) X−Velocity 0 2 4 6 0 2 4 6 8 10 12 (b) X=500mm Z (mm) Y (mm) X−Velocity 0 2 4 6 0 2 4 6 8 10 12 (c) X=750mm

Figure 5.2: Cross-sections of the streamwise velocity component, normalized with U∞,

calculated for case R3

with the reference velocity U∞ are presented.

The velocity induced by the couple of vortices in the y-z plane pushes down the external flow with a higher momentum in the region between the vortices, while the zones with a lower momentum are moved away from the flat plate. It is possible to compare these velocity fields shown in Fig.5.2 with the same quantities for the E1 case with the Batchelor vortex presented in Fig.5.3. We recall in fact that in E1 case the distance between the vortices of the same couple is lv = 2.4href, while for R3 case this distance should be equal to lv = 5href. From these figures it is possible to observe that the different spanwise position of the vortices in the two cases leads to a different spanwise modulation of the velocity and this is the reason why the slope of R3 amplitude curve is higher if compared to the other two cases presented before in Fig.5.1. The different behaviour of the velocity field for cases R3 and E1 can be also confirmed by the comparison of the x-velocity perturbation fields for the same sections as it is shown in Fig.5.4 and in Fig.5.5. From these figures it can be observed in fact that the maximum difference of the x-velocity component for each y-z plane, quantity defined in eq.(5.2), is higher for R3 with respect to the Batchelor vortex and this explains why even the local maximum

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Analysis of the far-wake simulations Z (mm) Y (mm) X−Velocity 0 2 4 6 0 2 4 6 8 10 12 (a) X=330 mm Z (mm) Y (mm) X−Velocity 0 2 4 6 0 2 4 6 8 10 12 (b) X=500mm Z (mm) Y (mm) X−Velocity 0 2 4 6 0 2 4 6 8 10 12 (c) X=750mm

Figure 5.3: Cross-sections of the streamwise velocity component, normalized with U∞,

calculated for case E1

amplitude of the streak results always higher for case R3 than the case E1 with a Batchelor vortex.

(a) X=330 mm (b) X=500 mm (c) X=750 mm

Figure 5.4: X-velocity perturbation fields for case R3 at three subsequent cross-sections

Moreover the higher distance between the vortices of each couple leads to an higher lift-up velocity in the low-speed streak region for case R3 with respect to the case E1 and it is the reason why the thickness of the boundary layer in

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Analysis of the far-wake simulations

(a) X=330 mm (b) X=500 mm (c) X=750 mm

Figure 5.5: X-velocity perturbation fields for case E1 at three subsequent cross-sections

the present case results higher than the case E1. In the same time the low speed streaks are narrower and this makes the velocity isocontour steeper in the low speed region in the case R3 with respect to the case E1. It is possible to suppose that this behaviour moves forward the boundary layer instability, which could be here inflectional. Therefore, the effect of a different spanwise spacing of the vortices is twofold. From one side there is an advantage in generating vortices with a uniform spanwise spacing, because they moves slowly in the wall-normal direction since low induced velocities act on each vortex. However, the different spanwise position of the vortices for the case R3 creates a velocity modulation whose shape is less appropriate with respect to the case E1 to allow the control of the flow and moreover the generated streaks result unstable. In fact even if a complete stability analysis has not been carried out, a preliminary temporal analysis has highlighted that there is not a net gain in comparison with the Blasius boundary layer. However this problem has to be studied in deep and a complete biglobal stability analysis has to be performed. Since the result obtained from the analysis of this case is not satisfactory, two possible solutions can be used. The intensity of the vortices can be reduced in order to obtain the reference case E2 and the reference spacing of the vortices can be restored, or the streak intensity can be reduced modifying the incidence of the device or increasing the distance between the device and the flat plate.

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Chapter 6

Conclusions

In this thesis different devices configurations have been analysed with the purpose of recreating a reference configuration of idealised counter-rotating vortices ob-tained as results of previous studies. From the analysis of all the configurations it has been observed that reproducing a configuration which gives exactly the same characteristics of the reference case it is not easy to be obtained. A delta-shaped device has been chosen as first attempt in order to recreate the reference counter-rotating vortices, but the values of the circulation obtained for vortices generated with such devices result too large. The changes of the chord length and the re-duction of the angle of attack did not allow a sufficient rere-duction of the vortices intensity so as to reach the value of the reference configuration. Other configu-rations, such as trapezoidal devices, previously presented in this work have been studied, but no remarkable results have been obtained. One problem occurred during this work has been the presence of the vortex shedding caused by a too large value of the thickness of the analysed devices, whose value has been initially taken equal to the experimental MVGs thickness. To overcome this problem, con-figurations with a new lower value of the thickness have been numerically studied. In this way the vortex shedding is no more present but the circulation obtained continues to be too high. The depth reduction can be possible considering that this new thickness can be technologically obtained. In fact this work has the pur-pose of individuating configurations that can be experimentally tested. Finally, the design of rectangular winglets configurations have been investigated. The

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geo-Conclusions

metrical dimensions of such devices have been determined applying the lifting-line theory. For this geometry interesting results have been highlighted and a configu-ration with the desired vortices has been detected. For such configuconfigu-ration a DNS simulation to evaluate the effect of the vortices generated by the rectangular de-vice on the Blasius boundary layer has been carried out. The amplitude evolution of the streaks generated in this work has been compared with the reference case, highlighting a good agreement between the two results, even if some differences between the two cases have occurred caused by the different spanwise distribution of the vortices. In conclusion the rectangular winglets are the most appropriated devices, among the configurations here analysed, to recreate the reference vortices configuration. As subsequence future developments, a numerical boundary layer solver is under development in order to avoid the DNS simulations in base flow study, allowing a drastic decrease of the computational costs. In this way the calculation times will be reduced only at the computational costs of the near-wake simulations. Otherwise DNS simulations still remain necessary in transition delay study. Moreover other configurations could be investigated and an optimization procedure to define the best device configuration can be carried out. Finally, stud-ies on the interference effects created by the supports used to physically hold in position these devices could be executed.

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References

[1] P. Andersson et al. “On the breakdown of boundary layers streaks”. In: J. Fluid. Mech. 482 (2001), pp. 29–60.

[2] A.A. Bakchinov et al. “Transition expereriments in a boundary layer with embedded streamwise vortices”. In: Phys. Fluids 7 (1995), pp. 820–832. [3] S. Camarri, J.H.M. Fransson, and A. Talamelli. “Numerical Investigation of

the AFRODITE Transition Control Strategy”. In: Progress in Turbulence V. Springer International Publishing, 2014.

[4] C. Cossu and L. Brandt. “On Tollmien–Schlichting-like waves in streaky boundary layers”. In: European Journal of Mechanics - B/Fluids (2004). [5] C. Cossu and L. Brandt. “Stabilization of Tollmien-Schlichting waves by

finite amplitude optimal streaks in the Blasius boundary layer”. In: Physics of Fluids (2002).

[6] J. H. M. Fransson and A. Talamelli. “On the generation of steady streamwise streaks in flat-plate boundary”. In: J. Fluid Mech (2012).

[7] J. H. M. Fransson et al. “Delaying transition to turbulence by a passive mechanism”. In: Phys. Rev. Lett. (2006).

[8] J. H. M. Fransson et al. “Experimental study of the stabilization of Tollmien– Schlichting waves by finite amplitude streaks”. In: Physics of Fluids (2005). [9] J.H.M Fransson et al. “Experimental and theoretical investigation of the nonmodal growth of steady streaks in a flat plate boundary layer”. In: Physics of Fluids 16.10 (2004), pp. 3627–3638.

Numerical investigation of the design of transition delay devices for a Blasius boundary layer

UNIVERSITÁ DI PISA

Dipartimento di Ingegneria Civile e Industriale

Sezione Ingegneria Aerospaziale

Tesi di Laurea Magistrale

NUMERICAL INVESTIGATION OF THE DESIGN OF

TRANSITION DELAY DEVICES FOR A BLASIUS

BOUNDARY LAYER

Abstract

Contents

List of Figures

List of Tables

Chapter 1

Introduction

Chapter 2

Passive methods for transition

delay

2.1

Streaks generation previous studies

2.2

Streaks generation by means of external

vor-tices

Chapter 3

Flow configuration and

numerical set-up

3.1

Experimental reference configuration

3.2

External vortices reference configurations

3.3

DNS simulations

3.3.1

Near-wake simulations

3.3.2

Far-wake simulations

3.4

Details on numerical tools

Chapter 4

Near-wake simulation results

for vortex generators

4.1

Delta-shaped winglets

4.1.1

Computational domain reduction

4.1.2

Mesh convergence

4.1.3

Delta winglet C3

4.1.4

Incidence reduction

4.1.5

Chord dimension reduction

4.1.6

Thickness reduction

4.1.7

"Gapped" delta winglets

4.1.8

Trapezoidal device

4.2

External Miniature Vortex Generators (EMVGs)

4.3

Rectangular winglets

Chapter 5

Analysis of the far-wake

simulations

5.1

Nek5000 simulation results

Chapter 6

Conclusions

References