Analysis of the near-wake past an axisymmetric blunt-based body

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UNIVERSITA’ DI PISA

Dipartimento di Ingegneria Civile ed Industriale

Corso di Laurea in Ingegneria Aerospaziale

Tesi di Laurea Magistrale

Analysis of the near-wake past an

axisymmetric blunt-based body

Relatori Allievo

Chia.ma Prof.ssa M. V. Salvetti Stefano D’Andrea

Dott. S. Camarri Ing. A. Mariotti Ing. L. Siconolfi

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...Alla mia Famiglia e a Catia

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Abstract

In the present work the main findings of a numerical research activity aimed at characterizing and reducing the base drag of bluff bodies are presented. The flow around an axisymmetric body with a sharp-edged base perpendicular to its axis is considered. Variational MultiScale Large Eddy Simulations carried out at Reynolds Re = 5.5 × 105_{, based on the body length and the freestream}

velocity, corresponding to Red = 9.6 × 104, based on the body diameter are

analyzed. Direct Numerical Simulations are performed at a Reynolds numbers roughly two orders of magnitude lower, i.e. Red = 1500 are also considered.

VMS-LES and DNS simulations show that a decrease of the base suctions is directly proportional to an increase of the length of the mean recirculation region behind the body. Indeed although the different set-ups and Reynolds numbers in the numerical simulations, in all cases the data collapse on a single straight line when the base pressure is plotted against the length of the mean recirculation region behind the body. The lengthening of near-weak region, in turn, can be obtained by increasing the boundary layer thickness before sepa-ration. Moreover, it is fundamental that the length of the mean recirculation region is connected with the location of the incipient instability of the detach-ing shear layers. It is shown that the location of this instability can be moved downstream, and thus base drag can be reduced, by increasing the thickness of the separating boundary layer. The analysis of the instantaneous dynamic of the wake has been carried out for all numerical simulations. The vortex indicator λ2 is used. Hairpin vortical structures are found for on simulations.

The DNS simulations at Re = 800 and Re = 1500 show a different wake dy-namics. In particular, at the lower Re the flow is characterized by a planar symmetry and the plane of symmetry remains costant over time, while at the higher Re the flow is characterized by vortical structures forming and evolving on planes rotating with time. These two different behaviors of the flow are consistent with the studies present in literature. The VMS-LES simulations show a wake dynamics similar to the are found at Re = 1500 in DNS, but the hairpin vortices form chose to the body base and shed at higher frequency. An analysis of local stability is studied on cross-plane after the body. Four unstable modes are found that characterizing the dynamics of wake.

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Acknowledgments

Desidero ringraziare la Prof.ssa Maria Vittoria Salvetti ed il Dott. Simone Camarri per avermi dato la possibilit`a di svolgere questo lavoro di tesi, per la loro attenzione e disponibilit`a nel guidarmi durante questo percorso e per i loro preziosi insegnamenti.

Ringrazio l’Ing. Alessandro Mariotti e l’Ing. Lorenzo Siconolfi per il loro importantissimo aiuto, per la passione e per la costanza con cui mi hanno seguito. Il loro entusiamo `e stato uno stimolo fondametale nei momenti pi`u difficili.

Voglio ringraziare la mia Famiglia ed in particolare mio Padre e mia Madre, capi saldi della mia vita, porto sicuro sui cui poter sempre contare.

Dedico questa tesi a Catia, il mio più grande amore, compagna di viaggio e avventure. Ti ringrazio per il sostegno e la pazienza che hai avuto nei miei confronti. Mi hai dato la forza quando pensavo di averla persa, il sorriso an-che nei momenti più bui. Ho sceso milioni di scale dandoti il braccio non già perchè con quattr’occhi forse si vede di più. Con te le ho scese perchè sapevo che di noi due le sole vere pupille, sebbene tanto offuscate, sono le tue.

Ringrazio tutti i miei Amici, mi avete dato tutti un pezzetto di quello che siete e se sono arrivato fin qui `e anche merito vostro.

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Chapter 1 Introduction

This introductory chapter is intended to introduce the thesis work done, then a report on the state of the art on the near-wake flow behind a bluff body is made. The present work contributes to a wider research project that aims at giving a better understanding of the flow around an axisymmetric blunt-based body. In particular, this work gives a contribution to analysis and the synthesis at the research activities carried out through both numerical simulations [19] and [26] and experimental investigations [31] and [20].

Axisymmetric blunt-based body

Blunt-based bodies are characterized by a premature separation of the bound-ary layer from their surface, due to an abrupt geometry change which causes the wake having a significant lateral dimension and, generally, an unsteady velocity field. A schematic representation of the typical flow behind an ax-isymmetric bluff body is shown in Fig. 1.1 (see e.g. [10]). The boundary layer separation generates a shear layer, a stagnation point and a recirculation zone, usually denoted as the near-wake.

The portion of a bluff body surface lying within the separated wake is usually indicated as the base of the body and is subjected to low pressure val-ues, which give a significant contribution to the total drag force acting on the body. Approaching the centreline, the shear layer velocity decreases and the streamline curvature changes, with a consequently reduction of the suctions. The shear layer with enough kinetic energy to overcome this pressure gradient

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1 Introduction 2

Figure 1.1: Schematic representation of the wake flow behind a bluff body

realigns with the centreline downstream of the wake. The part of shear layer without enough energy creates a recirculation zone immediately downstream of the body. These studies, in primis focused on the analysis of the connection between separating boundary layer thickness, near wake flow features and base pressure, which contributes to a significant part of the total drag force acting on a bluff body. The base pressure drag of bluff bodies is closely connected with the flow conditions in the near wake and in the separating boundary lay-ers. In particular, if we consider a body moving in an otherwise still fluid, the work done on the fluid by the opposite of the drag force acting on the body corresponds to an increase of the total energy of the fluid (sum of the kinetic and internal energies of the fluid). This perturbation energy is mainly found in the wake of the body, and is a function of the amount and organization of the vorticity continuously generated over the body surface and introduced in the wake. The blunt-based axisymmetric body can be understood as a simplifica-tion of a road vehicle such as a truck or sports vehicle like the bob (Olympic sport). The chosen axisymmetric blunt-based body guarantees near-wake flow features analogous to those of a road vehicle, as for instance the presence of a recirculation region. This simplification allows us to show fundamental geo-metrical and fluid dynamical parameters that characterize the flow behaviour, otherwise difficult to assess because the parameters of the specific problem are many.

In the literature [6] and [7] are present works describing the reduction of the base pressure with increasing boundary layer thickness for bluff bodies with

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1 Introduction 3

sharp-edged flat base, but a complete physical explanation for this result is still not present. Therefore, further investigations are still necessary to achieve a deeper understanding of the physical origin of this influence.

Furthermore this thesis is focused on the analysis of the wake structure. In particular has been studied the presence of vortical structures. A better de-scription of the flow field is indeed important because it is known that the base pressure drag of bluff bodies is closely connected with the flow dynam-ics in the near wake and in the separating boundary layers. The alternate vortex shedding behind two-dimensional bluff bodies is due to a strong global instability present in their wakes (see e.g. [23]) and causes large fluctuating velocities. When the characteristic Reynolds numbers are sufficiently high, velocity fluctuations are usually present also in the wakes of axisymmetric bluff bodies. However, these fluctuations are definitely smaller than those that characterize two-dimensional wakes and are connected with the appearance of a region of absolute instability in the near wake (see e.g.[25]); In particular, the most unstable mode in the wake of disks and spheres is found to be an azimuthal m = 1 helical mode. A characteristic feature of the resulting un-steady field is the shedding of hairpin-shaped vortices, each composed of two stream-wise legs and one final cross-flow portion. These vortical structures originate from the shear layers bounding the mean recirculation region present immediately downstream of the body and produce an instantaneously non-axisymmetric wake configuration having a symmetry plane; however, in most cases this plane rotates randomly, thus producing an average axisymmetric wake (see, e.g. [29], [18]). Nonetheless, one particular plane may be fixed by proper perturbations introduced in the wake, by means of control bodies such as spheres [33] or cylinders [12]. Also the identifier of vortices λ2 is considered

among the most studied parameters. It is a most popular methods for vortex identification are based on the analysis of the velocity gradient tensor.

It is useful analyse the mean recirculation region when alternate vortex shed-ding takes place. A measure of the length of this recirculation region is given by the vortex-formation length, defined in [3] as the distance downstream of the body base, along the wake centreline, where the rms value of the velocity fluctuations reaches a maximum. The close connection between the vortex for-mation distance and the end of the mean recirculation region, already pointed

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1 Introduction 4

out in [34], is clearly demonstrated by the results described by [1]. Further-more, considering data for two-dimensional blunt-based bodies in which the regular vortex shedding was modified or inhibited using either splitter plates with different lengths or various levels of base bleed, it was shown in[4] that the base pressure of a two-dimensional body decreases linearly with the decrease of the vortex formation distance. More specifically, Bearman pointed out that “base pressure is dependent on the vortex formation distance but apparently independent of the method adopted to interfere with vortex formation”. This result is consistent with the findings of [32] and [24], who showed that the drag reduction provided by a small circular cylinder placed in the near wake of a two-dimensional blunt body was connected with an increase of the length of the recirculation region, caused by a downstream movement of the maximum global instability region. Analogous results for axisymmetric wakes are less numerous, but also for this type of flow high fluctuations are normally encoun-tered immediately downstream of the stagnation point present at the end of a mean recirculation region (see, e.g. [14]). Furthermore, a decrease of drag when the recirculation region of a blunt body was lengthened by perturbing the wake through cylinders and rings was reported in [12], who also highlighted the importance of delaying the separated shear layer instability.

In previous works ([26] , [20]) Variational MultiScale (VMS) Large-Eddy Sim-ulations (LES) were carried out. The axisymmetric model configuration com-prises a forebody with a 3:1 elliptical contour, and a cylindrical main body whit a sharp-edged base perpendicular to the axis. The Reynolds number is Red = u∞d/ν = 96000. The shape of the model and the value of Reynolds

number are the same one used in the experiments (see e.g. [20]). Differences are that the upstream flow is laminar in the simulations, while in the experiments the freestream turbulence intensity was 0.9%, and the numerical simulations are carried out without the model support that was present in the wind tunnel tests. Direct Numerical Simulations (DNS) were also carried out at Reynolds numbers roughly two orders of magnitude lower than in the VMS-LES simula-tions (see e.g. [19]) , i.e. Red= u∞d/ν = 1500, in order to investigate on the

sensitivity of the relevant phenomena to the Reynolds number. In the present work the results obtained in VMS-LES, DNS simulations and in experimental investigations are compared, in order to see whether general conclusions on

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1 Introduction 5

the relationship between boundary layer thickness, base drag and near-wake feature may be drawer.

The stability of the wake of an axisymmetric blunt-based body is also on interesting. The global dynamics of shear flows is known to closely depend on the local instability characteristics, either convective or absolute (see e.g. [13]). Convectively unstable systems are sensitive to inflow perturbations and they behave as amplifiers of external noise. In contrast, absolutely unstable systems display non-trivial dynamics without external input, often leading to self-sustained oscillations. A link between the naturally selected global flow and the dynamics prevailing locally (see e.g. [25]), may be established by analysing the wake structure and its local stability characteristics. As also shown in this work, the flow behind an axisymmetric bluff body is sensitive to varying the Reynolds number. In this work the local stability analysis is carried out at Red = 1500, where the flow in the wake is characterized by

detachment of hairpin vortexes at appropriate Strouhal number. The aim of this analysis is to derive local stability features of the flow past the blunt-base body, by identifying unstable modes in the wake.

The present thesis is organized as follows:

• In chapter 2 the VMS-LES simulations and in chapter 3 the DNS simu-lations are described: the model geometry, the computational ste-up and numerical methodology are presented. The instantaneous and the mean field characteristics are presented for the reference configuration.

• In Chapter 4 the results of VMS-LES are comparated with those of DNS. Similarities and differences between the different simulations are considered.

• In Chapter 5 local stability is described. The analysis is applied directly to DNS flow fields at Red= 1500.

• In Chapter 6 conclusions and possible future developments of this re-search are summarized.

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Chapter 2 VMS-LES simulations

2.1 Geometry definition

The simulation model configuration comprises a forebody with 3:1 elliptical contour, and a cylindrical main body with a sharp-edged base perpendicular to the axis. This model is equal to that used in the wind tunnel experiments. Figure 5.11 shows the geometry of the body together with its dimensions, the direction of the free-stream velocity and the reference system.

Figure 2.1: Sketch of the geometry and main parameters.

The ratio between the diameter, d, and the overall length, l, is d/l = 0.175. The Reynolds number is Re = u∞l/ν = 5.5 × 105 and then some Red =

u∞d/ν = 9.5 × 104. In the following work all quantities are non dimensional,

therefore the lengths are divided by the diameter, d, and the velocity by the free-stream velocity, u∞.

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2.2 Simulations set-up 7

2.2 Simulations set-up

Computational domain and grid analysis

The computational domain is cylindrical. The diameter is 15d and the lenght is 50d. The body is collocated so that his axis of symmetry coincides with that of domain and 30d is the distance from the body base to the outflow. The domain is discretized by an unstructured grid having approximately 2.4 ∗ 106 geometrical nodes. The grid is particularly refined near the body surface and in the near wake (the wall y+ is lower than 1). Figure 2.2 shows the entire grid in the x/d - z/d plane and the figure 2.3 shows a detail of the grid to nose of the body.

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Figure 2.3: Frontal part of the body.

Figure 2.4 shows the rear part of the body in the plane x/d - z/d plane.

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Figure 2.5(a) shows Cross-flow plane at x/d = −0.1 and Figure 2.5(b) shows Cross-flow plane at x/d = 0.

(a) Cross-flow plane at x/d = −0.1d.

(b) Cross-flow plane at x/d = 0d.

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Description of boundary conditions

Figure 2.6: Configurations with modified boundary conditions, Case 0.75l.

Figure 2.7: Configurations with modified boundary conditions, Case 0.375l.

The method to change the boundary layer thickness in correspondence of the base was obtained by using a free-slip buondary condition over different ini-tial portions of the body surface. In particular, three simulations are presen-tated,which correspond to no-slip over the entire body, over the last 75% and over the last 37.5% of the body lateral surface, Figure 2.6 and Figure 2.7. The three simulations will be referred as case l, case 0.75l, and 0.375l by indicating the portion of the body along which the no-slip condition is imposed. The method to change the thickness of the boundary layer is different from that used in the tests in the wind tunnel. In the wind tunnel tests strips of emery cloth were wrapped around the body circumference in various positions in or-der to cause an earlier transition of the boundary layer and, thus, to increase its thickness (see e.g. [27]). Also a substantial difference between the two con-figurations in that in the wind-tunnel tests the model is located above a flat plate by means of a faired strut (see Figure 2.8), and its diameter and overall length are, respectively, d = 70mm and l = 400mm. The chosen support

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implies that the flow is not symmetrical in the plane, while it remains sym-metrical in the horizontal plane. It was preferred to rear sting support, which directly interferes with near-wake flow development and effectively changes the body base geometry. In the numerical simulations conversely the support is not considered.

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2.3 Numerical methodology 12

2.3 Numerical methodology

A variational multi-scale large-eddy simulation (VMS-LESS) approach to tur-bolence is adopted here. The Smagorinsky model is used as subgrid scale model (SGS) in order to close the VMS-LES equations. The code AERO has been used for the simulations,it is a Navier-Stokes solver for Newtonian, compress-ible and tridimensional flows. ( For more detail see e.g. [26]). The Navier-Stokes equations are discretized in space using a mixed finite-volume/finite-element method applied to unstructured tetrahedrizations. A finite-volume formulation is used for the convective terms and a finite-element formulation for the diffusive term. Finally, an implicit linearized time-marching algorithm is used for the discretization of the time derivative. The numerical method is second-order accurate in space and time.

General features of large-eddy simulation

The large eddy simulations, LES, is a mathematical model used in computa-tional fluid dynamics to study turbulent flows. In classical large-eddy simula-tion, LES, a spatial filter is applied to the Navier-Stokes equations in order to reduce the number of unknowns and to get rid of the scales smaller than the filter width. Thus only the large scales are computed while the small scales are modeled. In the LES approach, the flow variables are decomposed as follows:

W = W |{z}

LRS

+WSGS (2.1)

Where W are the largest resolved scales and WSGS _{are the smallest}

unre-solved scales.

The Navier-Stokes equations for a compressible Newtonian fluid are consid-ered. To avoid to model the sub-filter scales of the mass the Favre filtering is introduced denoted by a ˜, and it is defined for an arbitrary quantity φ as:

˜

φ = ( ¯ρφ)/( ¯ρ),the over-line denoting the grid filter. ∂ ¯ρ

∂t +

∂( ¯ρ˜uj)

∂xj

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2.3 Numerical methodology 13 ∂( ¯ρ˜ui) ∂t + ∂( ¯ρ˜uiu˜j) ∂xj = −∂ ¯p ∂xi + ∂(µσfij) ∂xj − ∂M (1) ij ∂xj + ∂M (2) ij ∂xj (2.3) ∂( ¯ρ ˜E) ∂t + ∂[( ¯ρE + ¯p)˜uj] ∂xj = −∂(˜ujσ˜ij) ∂xi − ∂ ˜qj ∂xj + ∂ ∂xj (Q(1)_j + Q(2)_j + Q(3)_j ) (2.4)

where µ is the viscosity, p is the pressure, E is the total energy, ui is the

velocity component in the i direction, ˜qj is the resolved heat flux. The tensor

f σij is defined as: f σij = − 2 3Sfkkδij + 2 fSij, (2.5) f

Sij being the resolved strain tensor:

f Sij = 1 2( ∂ ˜ui ∂xj +∂ ˜uj ∂xi ), (2.6)

In modeling the SGS terms resulting from filtering the Navier-Stokes equa-tions, it is assumed that low compressibility effects are present in the SGS fluctuations. We also assume that the heat transfer and temperature gradi-ents are moderate. The SGS term in the momentum equation is thus given by the classical stress tensor:

M_ij(1) = ρuiuj − ¯ρueiuej (2.7) and by the SGS term M_ij(2) that takes into account the transport of viscous terms due to the small scale fluctuations. M_ij(2) can be neglected because we are interested in high Reynold number flows. The isotropic part of Mij can also

be neglected under the assumption of low compressibility effects in the SGS fluctuations. The deviatoric part, Tij can be expressed by an eddy-viscosity

term in accordance with the Smagorinsky model extended to a compressible flow (see [27]):

Tij = −2µsgs( fSij −

1

3Sfkk), (2.8)

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In the total energy equation, the effect of the SGS fluctuations, i.e. all the SGS terms Qj(1), Qj(2) and Qj(3), are modeled by introducing a constant

SGS Prandtl number to be fixed a priori:

P rsgs = Cp

µsgs

Ksgs

, (2.9)

Ksgs is the SGS conductivity coefficient. It takes into account the diffusion

of total energy due to SGS fluctuations. In the filtered and normalized equation of energy, this term is added to the molecular conductivity coefficient.

In a LES model, large part of turbulent energy is accounted in simulated eddies; two sources of dissipation can affect this model: the numerical dissi-pation can affect eddies just larger than grid scale, model which may damp simulated eddies, even in laminar case.

Variational MultiScale LES approach

In the VMS-LES approach, the flow variables are decomposed in three parts as follows: W = W |{z} LRS + W0 |{z} SRS +WSGS (2.10)

where W are the largest resolved scales, W0 are the smallest resolved scales and WSGS _{are the unresolved scales. The scales of turbulences are separated a}

priori and the effects of the unresolved structures modeled only in the equations governing the small resolved structures. This method does not compute WSGS,

but models its effect by damping the scales which have a dimension comparable with the discretization size, while preserving the Navier-Stokes model for the largest resolved scales. Given an approximation space Vh, the unmodified Navier-Stokes system for W is discretized on a coarser subspace of Vh by means of a Galerkin formulation. In the complementary space, a LES damping model which applies only on the SRS components is introduced. (see e.g. [5])

Smagorinsky subgrid scale model definition

The SGS eddy-viscosity Smagorinsky model is the best known closure model (see [27]). It is well known that this model has some drawbacks but, thanks to

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the simplicity of implementation and the low computational costs, it is very attractive for complex industrial applications. For the classical Smagorinsky model the eddy-viscosity µsgs is defined as follows:

µsgs = ¯ρ(Cs∆)2| eS|, (2.11)

| eS| = q

2 fSijSfij, (2.12)

where ∆ is the filter width and Cs is a specific constant that must be a

priori assigned. The value typically used for shear flows is Cs = 0.1. This value

is adopted for the numerical simulation. The grid width filter corresponding to the numerical discretization must be computed. The filter for each grid element is defined herein as follows:

∆(l) = V ol(Tl) 1/3

(2.13) in which V ol(Tl) is the volume of the lth tetrahedron of the mesh. Many

studies have shown that the value assigned to the constant Cs plays an

im-portant role in the quality of the simulation. This value is generally flow de-pendent. Moreover, the Smagorinsky model is characterized by the following drawbacks:

1. a wrong behaviour of the flow is predicted in the near wall region, Tij

not vanishing with the correct trend;

2. only a dissipative effect of the small scales is obtained and so it becomes impossible to properly take into account the backscatter of energy from the small scales to large scales;

3. it is not able to properly handle transition, since the SGS viscosity does not vanish for laminar flows with shear.

These problems were partially overcame with the appearance of the dy-namic version of the Smagorinsky model. In this method, the constant Cs

is calculated during the simulation and takes alocal value, obtained from the smallest resolved scales, which can be negative. This local value is calculated using an algebric equation and applying a coarser filter than that used for the

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filtering of the Navier-Stokes equations (see e.g. [11]). This allows the above mentioned problems of the Smagorinsky model to be overcome. However, if too high fluctuations in the value of Cs appear during a simulation, this can

lead to instabilities.

In the variational multiscale approach adopted herein, the Smagorinsky closure terms is computed as a function of the smallest resolved scales and it is added to smallest resolved scales only. This approach is obtained a good compromise between accuracy and computational requirements.

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2.4 Mean flow and instantaneous flow features 17

2.4 Mean flow and instantaneous flow features

2.4.1 Mean flow features

An overall impression of the main flow features is provided by the streamlines of the velocity and pressure coefficient fields averaged in time and in the az-imuthal direction for the case in which the no-slip condition is applied over the whole body surface (case l), which are shown in Fig. 2.9(a) and 2.9(b). In the considered operating conditions, the boundary layer remains completely attached over the lateral surface of the model up to the separation at the sharp-edged base contour. The flow separation at the base leads to the devel-opment of a free shear layer, to the creation of a trailing stagnation point and to a flow recirculation behind the base. The mean-velocity streamlines bend inwards after the end of the body, aside the previous mentioned recirculation zone. The shape and length of the mean recirculation region are expected to be connected with the base drag. Indeed, the base pressure is connected with the velocity outside the separating boundary layer, which, in turn, increases with the convex curvature of the outer streamlines bounding the recirculation region in the near wake.

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2.4.1 Mean flow features 18

(a) x-velocity field

(b) Pressure coefficient field

Figure 2.9: Streamlines of the velocity field averaged in time and in the azimuthal direction.

Three different boundary layers are obtained in the simulations by changing the extent of the surface portion where a no-slip condition is applied. The non-dimensional profiles of the x-velocity, of the rms of x-velocity fluctuations and of the azimuthal vorticity, obtained at x/d = −0.1 and averaged in time and in azimuthal direction, are shown in Figs. 2.10(a), 2.10(b) and 2.10(c), respectively. Note that all the Chapter the quantities averaged in time and in azimuthal direction are indicated as “mean quantities”, while the ones averaged only in time are referred to as “time-averaged quantities”. The mean values of the boundary layer thickness, δ/d, at x/d = −0.1 are reported in Table 2.1, where the corresponding mean values of the displacement thickness δ∗/d, of the momentum thickness θ/d and of the shape factor H are also given. As the considered position lies within a region where a negative pressure gradient along the surface is present, with a consequent accelerating velocity at the edge of the boundary layer, the values of δ/d were obtained by identifying the distance normal to the surface at which the mean velocity was equal to the 99% of the maximum one in the considered section. Within numerical uncertainty,

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this position coincided with the one corresponding to 99% of the integral of the vorticity in the normal direction. The boundary layer thickness, δ/d, the displacement thickness δ∗/d and the momentum thickness θ/d are defined as follows: • Displacement thickness δ?_: δ? = δ Z 0 1 − u uext dy (2.14) • Momentum thickness θ: θ = δ Z 0 u uext 1 − u uext dy (2.15) • Shape parameter H: H = δ ? θ (2.16)

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2.4.1 Mean flow features 20 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 u/u_∞ (r−d/2)/d l 0.75l 0.375l

(a) Mean x-velocity

−150 −100 −50 0 0 0.05 0.1 ω_θd/u_∞ (r−d/2)/d l 0.75l 0.375l

(b) Mean azimuthal vorticity

0 0.01 0.02 0 0.05 0.1 rms(u‘/u_∞) (r−d/2)/d l 0.75l 0.375l (c) x-velocity fluctuations

Figure 2.10: Non-dimensional boundary layer profiles evaluated at x/d = −0.1 for the three simulations.

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Case δ/d δ∗/d θ/d H lr/d lrrms/d Cpbase Cpr≤0.4base

0.375l 0.0306 0.0078 0.0036 2.15 1.45 1.55 -0.135 -0.136

0.75l 0.0411 0.0107 0.0050 2.14 1.48 1.59 -0.130 -0.131

l 0.0663 0.0146 0.0072 2.03 1.54 1.64 -0.122 -0.124

Table 2.1: Summary of the mean boundary layer characteristics, the length of the mean recirculation region, and the values of the mean local pressure coefficient averaged over the body base.

As for the effect of the variation of the boundary layer thickness on base pressure, the time-averaged values of the local pressure coefficients on the base of the three considered cases are shown in Figs. 2.11(a), 2.11(b) and 2.11(c). As can be seen, the pressure coefficient is almost uniformly distributed on the base of the model and the base suctions decrease with increasing boundary layer thickness. This is confirmed also by analysing the mean values of the pressure coefficients for the three considered simulations (see Fig. 2.12). The corresponding values of the mean pressure coefficient averaged also over the whole base surface (Cpbase) are reported in Table 2.1.

Moreover, considering that the experimental pressure measurements (see e.g. [31]) are available only for r ≤ 0.4, the mean pressure coefficient averaged only over this limited portion of the base is calculated, and the corresponding values are denoted as Cpr≤0.4_base in Table 2.1. These measures in this way are not subject to a numerical uncertainty due to the presence of the edge at the end of the body.

The base-pressure variation is found to be connected with an increase of the length of the mean recirculation region that is present behind the body. This length lr/d is evaluated as the distance from the body base of the point

at which the mean streamwise velocity on the x-axis is equal to zero (see Fig. 2.13(a)). The values of lr/d are reported in Table 2.1.

The length of the mean recirculation region was also evaluated another procedure, i.e. by identifying the distance from the base of the point on the x-axis at which the maximum of velocity fluctuations occurs (see Fig. 2.13(b)); the obtained values, denoted as lrms

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(a) Case 0.375l (b) Case 0.75l

(c) Case l

Figure 2.11: Time-averaged pressure coefficient distribution on the base of the body.

0 0.25 0.5 −0.16 −0.14 −0.12 −0.1 r/d Cp l 0.75l 0.375l

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rms of the velocity fluctuation is defined as: rms(u0/u∞) = q (u02 x/u2∞) + (u 0₂ y/u2∞) + (u 0₂ z/u2∞) (2.17)

The maximum values of the fluctuations are found slightly downstream of the end of the recirculation region lr/d. The fluctuations of the three velocity

components along the x-axis are shown in Fig. 2.14. Higher values of fluctu-ations are found for the velocity components in the cross-flow plane and no preferential directions of the fluctuations are found within this plane. Nonethe-less, significant fluctuations are present also in streamwise direction, probably related to a pulsation of the recirculation region.

0 1 2 3 4 5 0 0.5 1 x/d u/u ∞ l 0.75l 0.375l

(a) Mean x-velocity

0 1 2 3 4 5 0 0.2 0.4 0.6 x/d rms(u‘/u ∞ ) l 0.75l 0.375l (b) Velocity fluctuations

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2.4.1 Mean flow features 24 0 1 2 3 4 5 0 0.02 0.04 0.06 0.08 0.1 x/d u‘ x 2 /u ∞ 2 l 0.75l 0.375l

(a) x-velocity fluctuations

0 1 2 3 4 5 0 0.02 0.04 0.06 0.08 0.1 x/d u‘ y 2 /u ∞ 2 l 0.75l 0.375l (b) y-velocity fluctuations 0 1 2 3 4 5 0 0.02 0.04 0.06 0.08 0.1 x/d u‘ z 2 /u ∞ 2 l 0.75l 0.375l (c) z-velocity fluctuations

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The mean flow streamlines bounding the recirculation region, i.e. averaged in time and in the azimuthal direction, are shown in Fig. 2.15, together with the wake edges, evaluated as the position corresponding to 99% of the integral of the vorticity in the normal direction. It is evident that the reduction of the length of the mean recirculation region is connected with an increase of the inward curvature of the wake edges. This leads to an increase of the velocity and a decrease of the pressure outside the separating boundary layer and over the body base, as can be appreciated from Fig. 2.16, where the velocities along the edge of the separating boundary layer and of the first part of the near-wake are shown for the various flow conditions. As can be appreciated, the velocities are practically equal for the three cases up to slightly upstream of the base. Conversely, more downstream a decreasing acceleration of the flow is found for increasing boundary layer thickness, which confirms the link between the velocity along the near-wake boundary and the pressure acting over the body base. In other words, shorter lengths of the recirculation region lead to higher curvatures and higher velocities along its boundary and to lower pressures over the base surface.

−0.50 0 0.5 1 1.5 2 2.5 0.5 1 x/d r/d l 0.75l 0.375l

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2.4.1 Mean flow features 26 −0.5 0 0.5 1 1.05 1.1 x/d || u ||/u ∞ l 0.75l 0.375l

Figure 2.16: Velocity magnitude along the edge of the boundary layer and of the near wake.

Moreover, the rms of x-velocity fluctuations along the edge of the boundary layer and of the near wake for the three simulations are shown in Fig. 2.17. As will be further explained by considering the instantaneous flow, the onset of the shear-layer instability along the separating shear layer roughly corresponds to the increase in the rms curves. It is clear from Fig. 2.17 that the effect of increasing the boundary layer thickness is to move downstream the increase in the fluctuations, and, therefore, the shear-layer instability; this corresponds to the already observed increase of the length of the recirculation region.

0 0.2 0.4 0.6 0.8 1 1.2 0 0.05 0.1 0.15 0.2 x/d std/u ∞ l 0.75l 0.375l

Figure 2.17: Values of the rms of x-velocity fluctuations along the edge of the boundary layer and of the near wake.

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2.4.2 Instantaneous flow features 27

2.4.2 Instantaneous flow features

Vortices identifier

The λ2 criterion has been used to investigate on the vortical structure present

in the threedimensional wake. The λ2 criterion is formulated based on the

observation that the concept of a local pressure minimum in a plane fails to identify vortices under strong unsteady and viscous effects (Hussian 1995). Indeed the non stazionarity of the flow can introduce depressions that do not necessarily imply a motion of vorticity and that the viscosity can eliminate area of swirling motion where vortex would be present. Taking the gradient of the Navier-Stokes equations is determined the Equation 2.18, the infromation on local pressure extrema is contained in the Hessian of pressure , p,ij.

ai,j = −

1

ρp,ij+ (ν) ui,jkk (2.18)

Where ai,j is the acceleration gradient, and p,ij is symmetric. Then, ai,j

can be decomposed into symmetric and antisymmetric parts as follow:

ai,j = DSij Dt + ΩikΩkj + SikSkj + DΩij Dt + ΩikSkj+ SikΩkj (2.19) Where: Sij = 1 2(ui,j+ uj,i) (2.20) Ωij = 1 2(ui,j − uj,i) (2.21)

The antisymmetric part of Equation 2.18 is the vorticity transport equa-tion. The symmetric part of Equation 2.18 is:

DSij

Dt − (ν) Sij,kk+ ΩikΩkj+ SikSkj = − 1

ρp,ij (2.22)

The contributions of the first two terms in the left-hand side of Equation 2.22 will not be considered, since the first term represents unsteady irrotational straining and the second term represents viscous effects. Only ΩikΩkj+ SikSkj

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is considered to determine the existence of a local pressure minimum due to vortical motion and define a vortex core. Therefore, the equation considered is the following:

ΩikΩkj + SikSkj = −

1

ρp,ij (2.23)

The occurrence of a local pressure minimum in a plane requires two posi-tive eigenvalues of the tensor p,ij, this corresponds to region with two negative

eigenvalues of ΩikΩkj+ SikSkj. Note that since ΩikΩkj+ SikSkj is symmetric, it

has real eigenvalues only. If λ1, λ2and λ3are the eigenvalues and λ1 ≥ λ2 ≥ λ3,

the new definition is equivalent to the requirement that λ2 < 0 within the

vor-tex core.

Instantaneous flow features

For the instantaneous flow dynamics, the near wake is characterized by the formation and evolution of hairpin vortices, in agreement with the results obtained on similar bodies at lower Reynolds numbers (see e.g. [30, 2, 6, 7, 8, 16] for axisymmetrical bodies and [21, 17, 22, 18] for sphere and disks). Indeed, in the VMS-LESs the hairpin vortical structures form and evolve on planes rotating with time, without any preferential direction. This can be qualitatively seen for instance in Fig. 2.18, where the time behaviour of the isosurfaces of the vortex-indicator λ2 (see e.g. [15, 9]) for the simulation with

no-slip condition over the whole body is shown.

The present analysis suggests that the length of the mean recirculation re-gion, and hence the base drag, is directly connected with the location where the instability of the detaching shear layers occurs. Indeed, as can be seen in Fig. 2.19, where the dynamic of the vorticity component normal to the considered plane is shown, the onset of the shear-layer instability roughly corresponds to the increase in the value of the rms of x-velocity fluctuations in Fig. 2.17.

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(a) Time t ∗ u∞/d = 16.34 (b) Time t ∗ u∞/d = 17.08

(c) Time t ∗ u∞/d = 17.71 (d) Time t ∗ u∞/d = 18.45

(e) Time t ∗ u∞/d = 19.02 (f) Time t ∗ u∞/d = 20.17

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(a) Time t ∗ u∞/d = 16.97

(b) Time t ∗ u∞/d = 17.14

(c) Time t ∗ u∞/d = 17.25

(d) Time t ∗ u∞/d = 17.42

Figure 2.19: Time behaviour of the z-vorticity; the dashed line represents the length of the mean recirculation region, while the black symbols show the positions of the onset of the shear-layer instability along the separating shear layer.

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Chapter 3 DNS simulations

3.1 Geometry and numerical set-up

Problem definition and numerical methodology

The Direct Numerical Simulations have been carried out only for the final portion of the previously described axisymmetric body. Thus, the considered geometry comprises a cylindrical main body of length l = 2d with a sharp-edged base perpendicular to the axis, where d is a diameter (see Fig. 3.1). The physical quantities are dimensionless, then d = 1 and u∞ = 1. The

computational domain is cylindrical, with a diameter of 10d and a length of 30d (28d being the distance from the body base to the outflow); it was discretized through a hexahedral structured grid having approximately 3.5×106 _{nodes. In}

particular, this grid is obtained by using a planar grid having 7.2 × 104 _nodes

and 48 nodes in azimuthal direction ([19]).

The method to change the boundary layer thickness is again different, com-pared to the one of the VMS-LESs. An axisymmetric autosimilar boundary-layer velocity profile is assumed at the inlet of the computational domain, with four different values of the thickness δin/d, viz. δin/d = 0, δin/d = 0.1,

δin/d = 0.2, and δin/d = 0.3. (see e.g. [19]).

The chosen boundary conditions are: no-slip is imposed on the body (lateral surface and the base), free slip was used on the domain lateral surface and together with pressure outflow conditions.

The Reynolds numeber of the simulations are Red = u∞d/ν = 1500, Red=

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3.1 Geometry and numerical set-up 32

Figure 3.1: Sketch of the geometry and main parameters.

800 and Red = 300.

Direct numerical simulation (DNS) of the considered configuration were carried out by using the open-source code OpenFOAM, based on the finite-volume discretization method. The accuracy of the numerical method was second order both in space and time. In particular the solution scheme setting are: for the pressure discretization the Guassian integration and for momen-tum discretization the Gaussian discretizaztion, limitedLinearV total variation diminishing (TVD) interpolation scheme. The discretization in time is such that the CFL number is maintained approximately equal to 20. The CFL number is defined as:

CF L = δt|U |

δt (3.1)

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3.2 Numerical validation 33

3.2 Numerical validation

The simulations are carried out for a cylindrical main body with a sharp-edged base perpendicular to the axis (see Fig. 3.1), having an inlet boundary layer thickness equal to δin/d = 0.1.

The DNS set-up is validated through comparison against results available in the literature for the same type of bodies at low Reynolds numbers (see e.g. [6, 7, 8, 16]).

Three different Reynolds numbers are chosen, viz. Red = 300, 800 and

1500, which are representative of the three characteristic flow regimes that can be found downstream of this body at low Reynolds numbers. These flow regimes has been recently studied by using linear stability analysis techniques in [6, 7], in which are defined two critical values of Red for all the considered

Mach numbers. As for the incompressible flows, i.e. for M = 0, the flow downstream of the body is stable for Red ≤ 460 and it is characterized by

a stationary axisymmetric recirculation region. On the other hand, in the interval 460 ≤ Re ≤ 910 the flow is characterized by a planar-symmetric solution. Indeed, with increasing Reynolds number, the initially stable and axisymmetric base flow undergoes a first stationary bifurcation which breaks the axisymmetry and develops two parallel steady counter-rotating vortices. As the Reynolds number increases above a second threshold, i.e. Red ≥ 910,

a second instability is found that is defined as a three-dimensional peristaltic oscillation which modulates the vortices, similar to the sphere wake, which leads to a shedding of hairpin vortices.

The three Direct Numerical Simulations have been able to well capture the three previously described flow patterns. In particular, the flow at Red = 300

is characterized by a stationary axisymmetric recirculation, as can be seen in Fig. 3.2, where the instantaneous streamlines are reported together with the x-velocity. Note that in Fig. 3.2 the results for the plane z/d = 0 are considered, but, since the solution is axisymmetric, they can be considered representative of every generic streamwise plane in the near wake.

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Figure 3.2: Streamlines downstream of the body at Red= 300.

The instantaneous streamlines downstream of the body at Red = 800 in

two generic planes are shown in Fig. 3.3. The recirculation region is no more axisymmetric and a planar symmetry is found, with two couples of principal planes in the near and in the far wake, called planes y0/d = 0 and z0/d = 0 and planes y00/d = 0 and z00/d = 0, respectively. A rotation of the two couples of principal plane of about 45◦ is found as in [6]. As for the near wake, from the instantaneous base pressure distribution (see Fig. 3.4) and from the visualization of the near-wake streamlines in the principal planes y0/d = 0 and z0/d = 0 (see Figs. 3.3(a) and 3.3(b)) it is evident that two counter-rotating steady recirculations are found in the plane z0/d = 0, while an oscillatory dynamics is present in the plane y0/d = 0.

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(a) Plane z/d = 0

(b) Plane y/d = 0

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Figure 3.4: Pressure coefficient distribution on the base of the body and sketch of the principal planes at Red= 800.

(a) Plane z0/d = 0

(b) Plane y0/d = 0

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In the far-wake there is again a planar symmetry in the plane z00/d = 0, while an oscillatory dynamic is found in the plane y00/d = 0. This is evident in Fig. 3.6, where the isosurfaces of the vortex-indicator λ2 (see e.g. [15, 9])

are reported.

Figure 3.6: Isosurfaces of the vortex-indicator λ2 at Red= 800.

The flow in the DNS at Red = 1500 is unsteady and completely

three-dimensional, as can be seen from the instantaneous velocity streamlines in Fig. 3.7. The instantaneous pressure coefficient distribution on the base of the body is almost uniformly distributed (see Fig. 3.8).

At this Reynolds number, the instantaneous flow dynamics is characterized by the formation and evolution of hairpin vortices (see e.g. [30, 2, 6, 7, 8, 16]), as can be qualitatively seen for instance from the isosurfaces of the vortex-indicator λ2 (see e.g. [15, 9]) in Fig. 3.9.

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(a) Plane z/d = 0

(b) Plane y/d = 0

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Figure 3.8: Instantaneous pressure coefficient distribution on the base of the body at Red= 1500.

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3.3 Mean flow and instantaneous flow features

After a numerical transient of ∆T = ∆t ∗ u∞/d = 110, a time length of the

simulations equal to ∆T = 60 is considered for all the mean flow statistics presented in this section (see e.g. [19]) for the convergence analysis of the mean statistics).

Four different boundary layers are obtained by changing the thickness of the autosimilar velocity profile imposed at the inlet section (x/d = −2). The non-dimensional profiles of the mean x-velocity, of the rms of x-velocity fluctuations and of the mean azimuthal vorticity, obtained at x/d = −0.1, are shown in Figs. 3.10(a), 3.10(b) and 3.10(c), respectively.

0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 u/u_∞ (r−d/2)/d δ/d=0.131 δ/d=0.171 δ/d=0.234 δ/d=0.318

(a) Mean x-velocity

0 0.005 0.01 0 0.1 0.2 0.3 rms (u‘/u_∞) (r−d/2)/d δ/d=0.131 δ/d=0.171 δ/d=0.234 δ/d=0.318 (b) x-velocity fluctuations −15 −10 −5 0 0 0.1 0.2 0.3 ω_θd/u_∞ (r−d/2)/d δ/d=0.131 δ/d=0.171 δ/d=0.234 δ/d=0.318 (c) Azimuthal vorticity

Figure 3.10: Profiles of boundary layer quantities at x/d = −0.1.

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position, are reported in Table 3.1, together with the correspondent values of the displacement thickness δ∗/d, of the momentum thickness θ/d and of the shape factor H.

The values of δ/d were obtained by identifying the distance normal to the surface at which the mean velocity was equal to the 99% of the maximum one in the considered section. Within numerical uncertainty, this position coincided with the one corresponding to 99% of the integral of the vorticity in the normal direction.

δ/d δ∗/d θ/d H lr/d Cpr≤0.4base

0.131 0.0433 0.0184 2.35 1.59 -0.118

0.171 0.0539 0.0232 2.32 1.70 -0.103

0.234 0.0739 0.0321 2.30 1.86 -0.087

0.318 0.1024 0.0441 2.32 2.07 -0.064

Table 3.1: Summary of the boundary layer characteristics, the length of the mean recircu-lation region, and the integral of the base suctions on the body base.

An overall impression of the effect of the variation of the boundary layer thickness on the main flow features is provided by the streamlines of the ve-locity and pressure coefficient fields averaged in time and in the azimuthal direction for the four considered cases, which are shown in Figs. 3.11 and 3.12, respectively. The time average has been done considering only the instants corresponding to the simulation convergence. The boundary layer remains completely attached over the lateral surface of the model up to the separation at the sharp-edged base contour and the flow separation at the base leads to the development of a free shear layer and to a flow recirculation downstream of the base (compare Figs. 3.11 and 3.12 with Figs. 2.9(a) and 2.9(b)).

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(a) δ/d = 0.131

(b) δ/d = 0.171

(c) δ/d = 0.234

(d) δ/d = 0.318

Figure 3.11: Streamlines of the velocity field averaged in time and in the azimuthal direc-tion and x-velocity field.

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(a) δ/d = 0.131

(b) δ/d = 0.171

(c) δ/d = 0.234

(d) δ/d = 0.318

Figure 3.12: Streamlines of the velocity field averaged in time and in the azimuthal direc-tion and pressure coefficient field.

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It is immediately evident in Figs. 3.11 and 3.12 that the increase of the boundary layer thickness produces also in DNS simulations a lengthening of the mean recirculation region behind the body and a reduction of the base suctions.

This is confirmed also by analysing the mean values of the pressure coeffi-cients for the four considered simulations (see Fig. 3.13).

The time-averaged pressure coefficient is almost uniformly distributed on the base of the model and the base suctions decrease with increasing boundary layer thickness.

The corresponding values averaged also over the base surface for r/d ≤ 0.4 (Cpr≤0.4_base ) are reported in Table 3.1.

0 0.25 0.5 −0.16 −0.12 −0.08 −0.04 r/d Cp δ_/d=0.131 δ_/d=0.171 δ_/d=0.234 δ_/d=0.318

Figure 3.13: Variation of the averaged Cp over the base.

This base-pressure variation is found to be connected with an increase of the length of the mean recirculation region lr/d, evaluated as the distance

from the body base of the point at which the mean streamwise velocity on the centreline is equal to zero (see Fig. 3.14(a)).

The correspondent values of lr/d are reported in Table 3.1. As for the

velocity fluctuations along the near-wake centreline (see Fig. 3.14(b)), a well localized maximum of the fluctuation is clearly present only for the simulations with the thinner boundary layers.

Thus, the quantity lrms

r /d can not be accurately evaluated in DNS

simula-tions. However, the fluctuations of the velocity components along the centre-line, shown in Fig. 3.15, confirm that higher values are found in the cross-flow plane and no preferential directions of the fluctuations are found within this

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plane. On the other hand, lower fluctuations are present in streamwise direc-tion. 0 1 2 3 4 5 0 0.5 1 x/d u/u ∞ δ/d=0.131 δ/d=0.171 δ/d=0.234 δ/d=0.318

(a) Mean x-velocity

0 1 2 3 4 5 0 0.2 0.4 0.6 x/d rms(u‘/u ∞ ) δ/d=0.131 δ/d=0.171 δ/d=0.234 δ/d=0.318 (b) Velocity fluctuations

Figure 3.14: Results along the centreline.

The mean flow streamlines bounding the mean recirculation region, aver-aged in time and in the azimuthal direction, are shown in Fig. 3.16, together with the wake edges, evaluated as the position corresponding to 99% of the integral of the vorticity in the normal direction. Again, the reduction of the length of the mean recirculation region is connected with an increase of the inward curvature of the wake edges and to decrease of the pressure outside the separating boundary layer and over the body base (see Fig. 3.12).

The rms of x-velocity fluctuations along the edge of the boundary layer and of the near wake for the three simulations are shown in Fig. 3.17.

The effect of increasing the boundary layer thickness is to move downstream the onset of significant velocity fluctuations along the separating shear layer,

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3.3 Mean flow and instantaneous flow features 46 0 1 2 3 4 5 0 0.02 0.04 0.06 0.08 0.1 x/d u‘ x 2 /u ∞ 2 δ/d=0.131 δ/d=0.171 δ/d=0.234 δ/d=0.318

(a) x-velocity fluctuations

0 1 2 3 4 5 0 0.02 0.04 0.06 0.08 0.1 x/d u‘ y 2 /u ∞ 2 δ/d=0.131 δ/d=0.171 δ/d=0.234 δ/d=0.318 (b) y-velocity fluctuations 0 1 2 3 4 5 0 0.02 0.04 0.06 0.08 0.1 x/d u‘ z 2 /u ∞ 2 δ/d=0.131 δ/d=0.171 δ/d=0.234 δ/d=0.318 (c) z-velocity fluctuations

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which are connected with the onset of the shear-layer instability. This, in turn, produces an increase of the length of the recirculation region.

−0.50 0 0.5 1 1.5 2 2.5 0.5 1 x/d r/d δ/d=0.131 δ/d=0.171 δ/d=0.234 δ/d=0.318

Figure 3.16: Mean flow streamlines bounding the recirculation zone and wake edges.

0 0.2 0.4 0.6 0.8 1 1.2 0 0.02 0.04 x/d std/u ∞ δ/d=0.131 δ/d=0.171 δ/d=0.234 δ/d=0.318

Figure 3.17: Values of the rms of x-velocity fluctuations along the edge of the boundary layer and of the near wake.

The previously described mean flow features in the near wake are the result of the instantaneous flow dynamics. Also at Red = 1500 the near wake is

found to be characterized by the formation and evolution of hairpin vortices, in agreement with the results obtained on similar bodies at low Reynolds numbers (see e.g. [30, 2, 6, 7, 8, 16] and in the VMS-LES al higher Reynolds number (see section 2.4.2).

These vortical structures form and evolve on planes rotating with time, as can be qualitatively seen from the time behaviour of the isosurfaces of the vortex-indicator λ2 (see e.g. [15, 9]) in Fig. 3.18 for the simulation with

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(a) Time t ∗ u∞/d = 55 (b) Time t ∗ u∞/d = 56

(c) Time t ∗ u∞/d = 57 (d) Time t ∗ u∞/d = 58

(e) Time t ∗ u∞/d = 59 (f) Time t ∗ u∞/d = 60

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A tentative sketch of the hairpin vortices downstream of an axisymmetric body is shown in Fig. 3.19. A complete cycle ∆T is described.

(a) Time t = T∗

(b) Time t = T∗+1₄∆T

(c) Time t = T∗+1₂∆T

(d) Time t = T∗+3₄∆T

Figure 3.19: Sketch of the hairpin vortices downstream of the axisymmetric body.

Due to the lower Reynolds number, the formation these vortical structures is more downstream and their evolution is slower in the DNS simulations rather than in VMS-LES ones. The onset of the shear-layer instability roughly cor-responds to the increase in the values of the rms of x-velocity fluctuations and this instability moves downstream by increasing the boundary layer thickness.

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(a) Time t ∗ u∞/d = 56

(b) Time t ∗ u∞/d = 57

(c) Time t ∗ u∞/d = 58

(d) Time t ∗ u∞/d = 59

Figure 3.20: Time behaviour of the z-vorticity; the dashed line represents the length of the mean recirculation region, while the black symbols show the positions of the onset of the shear-layer instability along the separating shear layer.

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The streamlines in cross-flow planes downstream of the body for the sim-ulation with δin/d = 0.1 at the time t ∗ u∞/d = 59 are presented in Fig. 3.21.

The two stremwise legs of the hairpin are counter-rotating and move from the center of the wake to the external part. The formation of the new hairpin vortex is usually closer to the wake centreline compared to the position of the already-formed hairpin vortex, and, characterized by opposite directions of ro-tation. Thus, once it is formed, it moves in the opposite direction. Moreover, these vortical structures form and evolve on planes rotating with time.

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(a) Plane x/d = 4 (b) Plane x/d = 4.5

(c) Plane x/d = 5 (d) Plane x/d = 5.5

(e) Plane x/d = 6 (f) Plane x/d = 6.5

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Chapter 4 Comparison between DNS and

VMS LES simulations

DNS and VMS-LES simulations

The increase of the boundary layer thickness before separation produces a reduction of the pressure drag of the body and a lengthening of the mean recirculation region behind the body in both VMS-LES and DNS. In spite of the differences in the Reynolds number, the correspondent relationship are found to be quantitatively in good accordance for the two computational set-ups (see Figs. 4.1 and 4.2). Moreover, we again found that the base pressure reduction is directly proportional to an increase of the length of the mean recirculation region behind the body, because the DNS and VMS-LES results almost collapse on a single straight line.

In order to deeply investigate on these results, the DNS and VMS-LES characteristic of the separating boundary layers are compared in terms of the integral of the vorticity and of the vorticity flux. The integral of vorticity is evaluated as:

2π Z δ

0

ωθr dr (4.1)

The vorticity flux Φωθ is calculated as:

Φωθ = 2π

Z δ

0

ωθuxr dr (4.2)

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4 Comparison between DNS and VMS LES simulations 54

The results for VMS-LES and DNS are compared in Figs. 4.3(a) and 4.1. As can be seen both the considered quantities scale linearly with the boundary layer thickness δ/d, indicating that the status of the two separating boundary layer is similar in spite of the different Reynolds numbers of the two simula-tion set-ups. This is consistent with the similar shape factor values previously found. The similar status of the boundary layer thickness (laminar in DNS and laminar or almost transitional in VMS-LES) seems to be the reason of the good accordance of DNS and VMS-LES simulations not only in the re-lationship between base pressure and mean recirculation length, but also for the ones between the boundary layer thickness and the other two considered parameters. Note that, if we neglected the result for the DNS with δin/d = 0

(for which slightly differences in the vorticity integral and flux are found), for the other simulations the variations in base pressure and mean recirculation length are almost directly proportional to the boundary layer thickness mod-ification. This, plausibly indicate a negligible effect of the Reynolds number until this does not produce a significant modification of the state of the bound-ary layer. The Reynolds number is the only significant difference between the DNS and VMS-LES, since the effect of the model support and of the turbulent free-stream oncoming flow in wind tunnel tests are not taken into account in both numerical set-ups.

0 0.1 0.2 0.3 0.4 1.2 1.4 1.6 1.8 2 2.2 δ/d l r /d VMS−LES DNS

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4 Comparison between DNS and VMS LES simulations 55 0 0.1 0.2 0.3 0.4 −0.16 −0.12 −0.08 −0.04 δ/d Cp base r ≤ 0.4 VMS−LES DNS (a) Cpr≤0.4_base vs. δ/d 1.2 1.4 1.6 1.8 2 2.2 −0.16 −0.12 −0.08 −0.04 l r/d Cp base r ≤ 0.4 VMS−LES DNS (b) Cpr≤0.4_base vs. lr/d

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4 Comparison between DNS and VMS LES simulations 56 3.2 3.4 3.6 3.8 0 0.1 0.2 0.3 0.4 2 π∫ 0 δ_ω θ r dr δ /d VMS−LES DNS

(a) Integral of vorticity

1.7 1.8 1.9 2 2.1 0 0.1 0.2 0.3 0.4 δ /d Φ_ω θ VMS−LES DNS (b) Flux of vorticity

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Numerical simulations and experimental investigations

The results obtained with numerical simulations are compared whit the experi-mental ones ( see e.g. [31]).The result of the present analysis is that, in spite of the underlined differences, the numerical and the experimental data show the same linear relationship between base pressure and mean recirculation length, as can be seen in Fig. 4.4.

1.2 1.4 1.6 1.8 2 2.2 −0.16 −0.12 −0.08 −0.04 l r/d Cp base r ≤ 0.4 Experiments VMS−LES DNS

Figure 4.4: Comparison between the experimental, the VMS-LES and the DNS results: lr/d vs. δ/d.

This suggest that the parameter directly controlling the base pressure is the recirculation length, independently of the status of the boundary layer before separation, of the turbulence level of the oncoming flow and of the Reynolds number.

The effect of the separating boundary layer on the mean recirculation length is graphically summarized in Fig. 4.5(b). It appears that in all cases the mean recirculation length can be increased by increasing the thickness of the separat-ing boundary layer. However, the relationship between the mean recirculation length and the boundary-layer thickness is quantitatively different for the ex-periments and the two sets of numerical simulations. This is probably due to the different status of the boundary layer, which is fully-developed turbulent in the experiments, while in the simulations is laminar or almost transitional. Moreover, thanks to the linear relationship between base pressure and mean recirculation length, the similarity of the effect of the boundary layer thickness on the mean recirculation length and on the base pressure can be explained ( Fig. 4.5(b) and Fig. 4.5(a)).

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4 Comparison between DNS and VMS LES simulations 58 0 0.1 0.2 0.3 0.4 −0.16 −0.12 −0.08 −0.04 δ/d Cp base r ≤ 0.4 Experiments VMS−LES DNS (a) Cpr≤0.4_base vs. δ/d 0 0.1 0.2 0.3 0.4 1.2 1.4 1.6 1.8 2 2.2 δ/d l r /d Experiments VMS−LES DNS (b) Cpr≤0.4_base vs. lr/d

Figure 4.5: Comparison between the experimental, the VMS-LES and the DNS results.

The mean flow features in the near wake are the result of the instantaneous flow dynamics. The near wake is found to be characterized by the formation and evolution of hairpin vortices, in agreement with the results obtained on similar bodies and on spheres at low Reynolds numbers. In this general frame-work, differences are again observed between the results of the experiments and of the numerical simulations. In numerical simulations, these vortical struc-tures form and evolve on planes rotating with time, but, due to the higher Reynolds number, their formation is more upstream and their evolution more rapid in the VMS-LES simulations than in DNS ones. In the experiments, the hairpin vortices form only on a preferential fixed plane and do no rotate due to the presence of the fairing. In spite of these differences, the present analysis suggests that the length of the mean recirculation region is directly connected with the location where the instability of the detaching shear layers occurs. The onset of the shear-layer instability roughly corresponds to the increase in the rms curves, and the effect of increasing the boundary layer thickness is to

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move downstream the increase in the fluctuations, and, therefore, the shear-layer instability, which, in turn, corresponds to the already observed increase of the length of the recirculation region.

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Chapter 5 Local stability analysis

5.1 Stability problem formulation

Mathematical model

Hydrodynamic stability theory allows to study the behaviour possible pertur-bations inside a flow, from now on called base flow. Perturpertur-bations will be damped if the flow is stable and, on the contrary, they will be amplified if the flow is unstable. In order to carry a linear stability analysis, perturbations are infinitesimal and it is possible to neglect the second order terms. The ana-lyzed base flow is the mean flow obtained from previous incompressible DNS simulations at Red= 1500.

Denoting with (−→Vo, Po) the base flow, the perturbed fields (

− →

V , P ) are obtained adding the disturbances (−→v , p):

− →

V =−→Vo+ −→v (5.1)

P = Po+ p (5.2)

The base flow and the perturbed flow must respect the continuity equations. The dynamics of the disturbances is obtained subtracting the continuity and momentum equations of the base flow from those of the whole flow:

div(−→v ) = 0 (5.3)

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5.1 Stability problem formulation 61 ∂(−→v ) ∂t + − → Vograd(−→v )+−→v grad( − → Vo)+−→v grad(−→v ) = −grad(p)+ 1 Red ∇2₍₋→_{v ) (5.4)}

A stationary base flow is assumed in order to make the components of the vector −→Vo time independent,but still variable in space. Also, the perturbation

terms are small enough to neglect the second order terms −

→_{v grad(−}→_{v ) = 0} _(5.5)

Modal form of the disturbances

In the BiGlobal model the base flow is assumed to be almost parallel and slowly variable in the streamwise direction x. The analysis concerns the spanwise y,z sections of the base flow, each of which is considered independent of x. Consequently, the disturbance is assumed to be generic in the y,z plane and totally periodic in the streamwise direction:

− →_{q =}

b

q(X, y, z)ei(kx−ωt) (5.6)

Where q = (_b _bu,_bv,w,_b _bp) is the modal coefficient.

Substituing the 5.6 in the disturbance equations and neglecting the derivates in x, a classic genarilezed eigenvalue problem is obtained:

[A0+ kA1 + k2A2]q = ωBb qb (5.7)

Completed by the no-slip conditions _bu,_bv,w = 0 at solid boundaries._b Two kinds of analysis can now be introduced:

• In the temporal analysis the wavelength k ∈ R is assigned; the equations are an eigenvalue problem with ω ∈ C as an eigenvalue.

• In the spatial analysis the frequency ω ∈ R is assigned; the equations are an eigenvalue problem with k ∈ C as an eigenvalue.

In both cases the system (5.7) takes the form of a generalized eigenvalue problem, where k or ω represent the complex eigenvalue andq is the associated_b

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5.1 Stability problem formulation 62

eigenvector. Note that in the spatial case, the eigenvalue appears up to the second power and specific solution methods must be employed.

When k ∈ R, therefore in temporal analysis, the disturbance is a wave with wavelength k and whose amplitude and frequency can vary in time. In particular, ωi is temporal growth rate, if it is positive the disturbance grows in

time, and the mode is temporally unstable, while it is negative the disturbance is dumped, and the mode is temporally stable. The disturbance in temporal analysis takes the following form:

− →_{q =}

b

q(X, y, z)ei(kx−ωrt)_eωit _(5.8)

When ω ∈ R, therefore in spatial analysis, the disturbance in temporal analysis takes the following form:

− →_{q =}

b

q(X, y, z)ei(krx−ωt)_e−kit _(5.9)

The disturbance is a wave with assigned frequency ω/2π, if kiis negative the

amplitude grows in x-direction, while if the term ki is positive the amplitude

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5.2 Local temporal analysis 63

5.2 Local temporal analysis

The local stability analysis is applied on some representative cross sections in streamwise direction after the recirculation zone. Sections at at 2.5d and 4d to base have been considered. The equations have been discretized using the finite-element method. The numerical fields are interpolated in the mesh (fig. 5.11). The domain has been discretized on triangular elements, the number of elements is 4000, P 2 and P 1 Taylor-Hood elements for the velocity fields and for the pressure field (see e.g. [28]).

Figure 5.1: Geometry of the mesh used for the interpolation of DNS simulation fields

Figure 5.2: Basic functions

The eigenvalue value problem has been solved by the Arnoldi method using the parallel implementation available in the SPLECc library. In order to ac-celerate the convergence, a shift-invert strategy has been used, which implies the resolution of a large sparse eigenvalue problem through MUMPS library. Figs. 5.3 and 5.4 shows the results of the temporal stability analysis. On both blanes four unstable modes are found.

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5.2 Local temporal analysis 64 0 1 2 3 4 5 6 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 k ω i mode 1 mode 2 mode 3 mode 4

Figure 5.3: Local temporal curve for x = 2.5d

0 1 2 3 4 5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 k ω i mode 1 mode 2 mode 3 mode 4