Benchmark on the Aerodynamics of a Rectangular Cylinder: sensitivity to inflow conditions

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UNIVERSITÀ DI PISA

Dipartimento di Ingegneria Civile e Industriale

Sezione Ingegneria Aerospaziale

Tesi di Laurea Magistrale

BENCHMARK ON THE

AERODYNAMICS OF A

RECTANGULAR CYLINDER:

SENSITIVITY TO INFLOW

CONDITIONS

Relatori Allievo

Prof. Maria Vittoria Salvetti Francesco Ascenso

Dott.Ing. Alessandro Mariotti

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Ai miei nonni..

che da lassù riescono comunque ad essere vicino a me

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Abstract

The aim of this thesis is to calculate the eects of the wind on a sharp-edged rectangular cylinder section with a chord-to-depth ratio equal to 5 and the sensi-tivity to inow conditions. In fact, past research activities show a big dispersion of data especially in terms of vertical forces. Even though the bare geometry, the knowledge of the ow characteristics can shed new light in the fundamentals of the aerodynamic behavior of a wide range of actual blu bodies such as high-rise buildings in the civil engineering elds. Unsteady RANS simulations of the rectan-gular cylinder have been carried out by using two dierent turbulence models and the commercial code Fluent. The SST k-ω and the Reynolds stress turbulence model are considered. First a grid sensitivity analysis is investigated. Then 25 simulations are performed varying three parameters: incidence, turbulence length and turbulence scale and the results are compared between a convergence and no-convergence grid. The Reynolds number is 40000. The same procedure is repeated for two models. At last, results are compared between the two models and with previous data in the literature and a deterministic analysis to inow condition is carried out. There is a good agreement in the outputs between the two models and the previous experimental data for all the bulk parameters. The main dif-ference between the two models, which however falls within the range described in literature concerns the root mean square value of vertical coecient and the presence of a modulation in time of the force coecients at a low frequency in the simulations performed with Reynolds Stress Model. Regarding the sensitivity to inow conditions, we can say that the main parameter that aects the results

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5 in terms of aerodynamic forces (mean and rms) and of the pressure eld is the incidence while turbulence length and scale have a minor eect. The sensitivity to inow conditions will be of great interest also for future experimental simulation where settings cannot be perfect. In fact, having a knowledge of the inuence of the input parameters, we can better forecast the eects on bulk parameters and on pressure eld of the unavoidable setting errors on experimental simulations.

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List of Figures

1.1 BARC model and domain geometry for the computational study . . 18

2.1 Sketch of the simulated geometry and reference frame . . . 22

2.2 Structure of the boxes used for the computational grid . . . 24

3.1 Sketch of the computational grid . . . 30

3.2 Time behaviour of the force coecients . . . 31

3.3 Side-averaged and time-averaged pressure coecient distribution for all the grids: comparison between dierent time ranges . . . 33

3.4 Side-averaged distribution of the standard deviation in time of the pressure coecient for all the grids: comparison between dierent time ranges . . . 34

3.5 Side-averaged distribution of the time averaged pressure coecient and of the standard deviation in time of the pressure coecient: comparison between the coarse and the ne grid . . . 35

3.6 Mean ow streamlines . . . 36

3.7 Time behaviour of the force coecients for the case 1 . . . 38

3.9 Case 1: comparison between the upper and the lower side . . . 39

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LIST OF FIGURES 10 3.12 Eect of the angle of incidence (turbulence intensity = 1.55% and

turbulence length scale = 2.55D) . . . 43 3.13 Eect of the turbulence intensity (angle of incidence = 0 deg and

turbulence length scale = 2.55D) . . . 44 3.14 Eect of the turbulence length scale (angle of incidence = 0 deg and

turbulence intensity = 1.55%) . . . 45

3.15 Eect of the angle of incidence on the reattachment points xr and

center points (xc, yc) of the mean recirculation aside of the model (turbulence intensity = 1.55% and turbulence length scale = 2.55D) 48

3.16 Eect of the turbulence intensity on the reattachment points xr and

center points (xc, yc) of the mean recirculation aside of the model (angle of incidence = 0 deg and turbulence length scale = 2.55D) . 48 3.17 Eect of the turbulence length scale on the reattachment points

xr and center points (xc, yc) of the mean recirculation aside of the

model (angle of incidence = 0 deg and turbulence intensity = 1.55%) 49

3.18 Eect of angle α on the upper side for the grid having 5×104 _nodes

(turbulence intensity = 1.55% and turbulence length scale = 2.55D) 50

3.19 Eect of angle α on the upper side for the grid having 1×105 _nodes

3.20 Eect of angle α on the lower side for the grid having 5 × 104 nodes

3.21 Eect of angle α on the lower side for the grid having 1 × 105 _nodes

3.22 Eect of the turbulence intensity for the grid having 5 × 104 _nodes

(angle of incidence = 0 deg and turbulence length scale = 2.55D) . 52

3.24 Eect of the turbulence scale for the grid having 5×104 nodes (angle

of incidence = 0 deg and turbulence intensity = 1.55%) . . . 54

3.25 Eect of the turbulence scale for the grid having 1×105 nodes (angle

of incidence = 0 deg and turbulence intensity = 1.55%) . . . 54 4.1 Sketch of the computational grid . . . 56

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LIST OF FIGURES 11

4.2 Time behaviour of the force coecients . . . 57

4.3 Side-averaged and time-averaged pressure coecient distribution for all the grids: comparison between dierent time ranges . . . 59

4.4 Side-averaged distribution of the standard deviation in time of the pressure coecient for all the grids: comparison between dierent time ranges . . . 60

4.5 Side-averaged distribution of the time averaged pressure coecient and of the standard deviation in time of the pressure coecient: comparison between the coarse and the ne grid . . . 61

4.12 Eect of the angle of incidence (turbulence intensity = 1.55% and turbulence length scale = 2.55D) . . . 69

4.13 Eect of the turbulence intensity (angle of incidence = 0 deg and turbulence length scale = 2.55D) . . . 70

4.14 Eect of the turbulence length scale (angle of incidence = 0 deg and turbulence intensity = 1.55%) . . . 71

4.15 Eect of the angle of incidence on the reattachment points xr and center points (xc, yc) of the mean recirculation aside of the model (turbulence intensity = 1.55% and turbulence length scale = 2.55D) 75 4.16 Eect of the turbulence intensity on the reattachment points xr and center points (xc, yc) of the mean recirculation aside of the model (angle of incidence = 0 deg and turbulence length scale = 2.55D) . 75 4.17 Eect of the turbulence length scale on the reattachment points xr and center points (xc, yc) of the mean recirculation aside of the model (angle of incidence = 0 deg and turbulence intensity = 1.55%) 76 4.18 Eect of angle α on the upper side for the grid having 1.2 × 104 nodes (turbulence intensity = 1.55% and turbulence length scale = 2.55D) . . . 76

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LIST OF FIGURES 12

4.19 Eect of angle α on the upper side for the grid having 5×104 nodes

4.20 Eect of angle α on the lower side for the grid having 1.2×104 _nodes

4.21 Eect of angle α on the lower side for the grid having 5 × 104 _nodes

4.22 Eect of the turbulence intensity for the grid having 1.2×104 _nodes

4.24 Eect of the turbulence scale for the grid having 1.2 × 104 nodes

(angle of incidence = 0 deg and turbulence intensity = 1.55%) . . . 80

4.25 Eect of the turbulence scale for the grid having 5×104 _{nodes (angle}

of incidence = 0 deg and turbulence intensity = 1.55%) . . . 81 5.1 Eect of the angle of attack for the two models and comparison

with range with the 95 % of condence level,maximum,minimum and averaged value of available data (turbulence intensity = 1.55% and turbulence length scale = 2.55D) . . . 84 5.2 Eect of the turbulence intensity for the two models and comparison

with range with the 95 % of condence level,maximum,minimum and averaged value of available data (angle of incidence = 0 deg and turbulence length scale = 2.55D) . . . 85 5.3 Eect of the turbulence length scale for the two models and with

range with the 95 % of condence level,maximum,minimum and averaged value of available data (angle of incidence = 0 deg and turbulence intensity = 1.55%) . . . 86 5.4 Comparison between the SST k − ω and the available data of the

eect of angle α on the upper side (turbulence intensity = 1.55% and turbulence length scale = 2.55D) . . . 89 5.5 Comparison between the RSM and the available data of the eect

of angle α on the upper side (turbulence intensity = 1.55% and turbulence length scale = 2.55D) . . . 90

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LIST OF FIGURES 13 5.6 Comparison between the SST k − ω and the available data of the

eect of angle α on the lower side (turbulence intensity = 1.55% and turbulence length scale = 2.55D) . . . 91 5.7 Comparison between the RSM and the available data of the eect

of angle α on the lower side (turbulence intensity = 1.55% and turbulence length scale = 2.55D) . . . 92 5.8 Comparison between the SST k − ω and the available data of the

eect of the turbulence intensity (angle of incidence= 0deg and turbulence length scale = 2.55D) . . . 95 5.9 Comparison between the RSM and the available data of the eect of

the turbulence intensity (angle of incidence= 0deg and turbulence length scale = 2.55D) . . . 96 5.10 Comparison between the SST k − ω and the available data of the

eect of the turbulence scale (angle of incidence = 0deg and turbu-lence intensity = 1.55%) . . . 98 5.11 Comparison between the RSM and the available data of the eect

of the turbulence scale (angle of incidence = 0deg and turbulence intensity = 1.55%) . . . 99

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List of Tables

2.1 Grid parameters for the SST k-ω model . . . 24 2.2 Grid parameters for the RSM . . . 25 3.1 Convergence of statistics for the dierent grids . . . 32 3.2 Reattachment points of the mean recirculation aside of the model

for the four dierent grids . . . 36 3.3 Bulk parameters for two dierent ow conguration . . . 37 3.4 Bulk parameters for the SST k − ω turbulence model (grid having

5 × 104 nodes) . . . 41 3.5 Bulk parameters for the SST k − ω turbulence model (grid having

1 × 105 nodes) . . . 42 3.6 Reattachment points xrand center points (xc, yc)of the mean

recir-culation aside of the model for the grid having 5 × 104 _{nodes (time}

range 150−300) . . . 46 3.7 Reattachment points xrand center points (xc, yc)of the mean

recir-culation aside of the model for the grid having 1 × 105 _{nodes (time}

range 150−300) . . . 47 4.1 Convergence of statistics for the dierent grids . . . 58 4.2 Reattachment points of the mean recirculation aside of the model

for the ve dierent grids . . . 63

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LIST OF TABLES 15 4.3 Bulk parameters for two dierent ow congurations and two

dif-ferent grids . . . 63

4.4 Bulk parameters for the Reynolds Stress Model (grid having 1.2×104

nodes) . . . 67

4.5 Bulk parameters for the Reynolds Stress Model (grid having 5×104

nodes) . . . 68

4.6 Reattachment points xr and center points (xc, yc) of the mean

re-circulation aside of the model for the grid having 1.2 × 104 _nodes

(time range 300−450) . . . 73 4.7 Reattachment points xrand center points (xc, yc)of the mean

recir-culation aside of the model for the grid having 5 × 104 nodes (time

range 300−450) . . . 74 5.1 Comparison of bulk parameters of the two employed models with

available data . . . 83 5.2 Comparison of the mean reattachment point and of the centre of

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Chapter 1

Introduction

The study of blu-body aerodynamics has always been a fundamental issue in aerodynamics problems. In particular every kind of body needs a study in itself because of the fact that geometry strongly inuences the motion eld and the forces on the body. The ow over an elongated rectangular cylinder, see Figure 1.1 at high Reynolds numbers is highly complex, being three dimensional, turbu-lent and characterized by unsteady ow separation and reattachment (see Bruno et al. [2014]). On the other hand, thanks to the simple geometry, a detailed anal-ysis of the ow dynamics can be carried out,and dierent patterns, which can also be found when dealing with more complex geometries,can be identied. It is well known that two-dimensional(2D)rectangular cylinders are characterized by one sin-gle geometric parameter,i.e.the ratio of the along wind dimension(Breadth)to the cross-wind dimension (Depth), B/D, which governs their aerodynamic behaviour (see e.g. Nakaguchi et al. [1968] and Stokes and Welsh [1986]). For B/D ratios greater than 3.5, reattachment is permanent and vortex shedding occurs from both the leading and the trailing edges. Then for B/D ratios between 3.5 and 6 there is a completely reattached ow and simultaneously one recirculation zone on each face of the cylinder.

In this thesis we study the eects of the wind on a sharp-edged rectangular cylin-der section with a chord-to-depth ratio equal to 5 and the sensitivity to inow conditions. In particular, we consider an international benchmark launched in 2008 (see Bartoli et al.) Despite the relatively simple geometry, the BARC case

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18 contains most diculties also found in realistic blu body ows of interest in wind engineering. The ow around the stationary elongated cylinder at the considered

high Reynolds number ReD = 40000 is turbulent with unsteady ow separation

from the upstream corners and reattachment on the cylinder side.

Figure 1.1: BARC model and domain geometry for the computational study The objective of the BARC benchmark is to collect dierent data sets for as-sessing the reliability and the dispersion of computational and experimental stud-ies (see Bruno et al. [2014]). The comparison of these results revealed that both the numerical simulations as well as the wind tunnel tests are impacted by various sources of uncertainty. In particular, besides modeling uncertainties and numerical errors, in numerical simulations it is dicult to exactly reproduce the experimental conditions due to uncertainties in the set up parameters,which sometimes cannot be exactly controlled or characterized.

The following uncertain set-up parameters are investigated here in the subsequent uniform ranges: the angle of incidence α (−1◦₋₁◦_{), the longitudinal turbulence}

in-tensity Ix (0.001 − 0.03), and the turbulence length scale L (0.1D − 5D). All these

parameters have signicant eects on the ow features and on the aerodynamic loads and they are often not characterized or dicult to be exactly controlled in experiments. Sensitivity analysis to small variations of the angle of incidence is also interesting to highlight to which extent this aects the symmetry of the

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19 ow. The analysis excludes the geometrical uncertainties to avoid re-meshing and post-processing visualization issues. Variations in the Reynolds number are not considered, because it has been found that Reynolds number eects on the ow are not among the predominant ones (see Bruno et al. [2014]). Moreover, varying

ReD would also imply to change the grid to maintain the same near wall resolution

in wall units.

The computations are performed using 2D Unsteady Reynolds Averaged Navier-Stokes (URANS) simulations in order to make the computational eort feasible. Simulations were carried out with two dierent turbulence models:

- SST k-ω turbulence model; - Reynolds Stress Model.

The topology of the grid for the two models is dierent for numerical reasons. We use a hybrid grid for the SST k-ω turbulence model with a structured part near the body and an unstructured part (triangular cells) far from it while the grid for the RSM is unstructured with triangular cells everywhere.

The outcomes will be compared to those of other RANS models and existing LES results to quantify the bias error due to turbulence modeling. The numerical error is estimated by comparing the results from computations on dierent mesh sizes. Among the output quantities of interest are the bulk parameters: time-average (t−avg) and root mean square (t−rms) of the vertical and horizontal force

coecients, cy and cx, and the nondimensionalized frequency of time oscillations

of cy, in terms of the Strouhal number St. In addition, the time-averaged ow

elds are considered as well as the surface pressure coecient Cp in terms of the

mean and standard deviation.

The simulations for each model are described in 3 and in 4. In these chapters, rst, grid sensitivity is investigated. This consists in performing the same simu-lation on dierent grids and results are then compared. If they are dierent in terms of mean and rms forces and in terms of eld parameters, new simulations are carried out on a grid with more cells. When the dierence between the results of two grid are enough similar, grid independence is reached. A time sensitivity analysis is performed together with the previous. The aim of this analysis is to determine the minimum time interval in which we should average the physical

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20 quantities. Then 25 simulations, obtained varying incidence α, turbulence inten-sity I and turbulence scale D, are performed with two grids. The rst one is the grid with a number of cells as minus as possible that is in convergence. The results of this grid will be compared later with the results obtained with RSM and with the previous data. The other is a no-convergence grid that is used to evaluate deterministically the dierence in the terms of eld parameters between the two grids and so the eect of the grid resolution on results.

In chapter 5.1 the two models are compared with previous data and also the eect of parameters α, I and D is investigated.

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Chapter 2

Methodology

In this chapter the methodology employed to perform all the simulations is pre-sented. First, the considered ow conguration is described. Then the grids used in the simulations are briey presented. Unsteady RANS simulation have been carried out by using SST k-ω turbulence model and RSM with the commercial code FLUENT. The second part of this chapter analyzes the features of these two turbulence.

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2.1 Geometry denition and computational grids 22

2.1 Geometry denition and computational grids

The simulated geometry is shown in Fig. 2.1. It is a rectangle with dimensions that start from x/D = −2.5 to x/D = 2.5 and from y/D = −0.5 to y/D = 0.5 where D is the length of the transverse section of the body. The computational domain extends from x/D = −75 to x/D = 125 and from y/D = −75 to y/D = 75 as recommended in the guidelines of BARC. The computational domain is divided in many blocks to obtain the desired mesh. No-slip conditions have been imposed at the cylinder surface (no wall laws are used) while pressure outlet is set at the outow and velocity inlet on the other three domain sides(the above, the below and the front wall).

Figure 2.1: Sketch of the simulated geometry and reference frame

The simulations are carried out at ReD = U∞D/ν = 40000 where U∞ is the

velocity of the incoming ow and ν is the kinematic viscosity.

Two dierent grid topologies are used for the two dierent turbulence models. A hybrid grid is used for the SST k-ω turbulence model, with structured cells near the body with a growth rate. The size of the rst cell is such that y+_{=1. y}+ _{is a} non-dimensional wall distance for a wall-bounded ow that can be dened in the following way:

y+ ≡ u∗y

ν (2.1)

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2.1 Geometry denition and computational grids 23 nearest wall and ν is the local kinematic viscosity of the uid. The friction velocity, u∗, can be dened in the following way:

u∗ ≡r τw

ρ (2.2)

where τw is the wall shear stress and ρ is the uid density at the wall. The wall shear stress, τw, is given by:

τw = µ ∂u ∂y y=0 (2.3) where µ is the dynamic viscosity, u is the ow velocity parallel to the wall and

y is the distance to the wall. y+ is commonly used in boundary layer theory and in

dening the law of the wall. An unstructured triangular grid is used far from the body to save cells. The growth rate of the grid is not uniform in the space because the ow eld behind the body has the major eects on the ow parameters and on the acting forces. A completely triangular grid is used for the RSM for convergence problems. Similar considerations as previously can be made. A grid sensitivity analysis is carried out for the two turbulence model. Grids with 5 × 104_{, 1 × 10}5_, 2 × 105 _{and 3 × 10}5 _{nodes for the SST k-ω and with 1.2 × 10}4_{, 2.5 × 10}4_{, 3.8 × 10}4_,

5 × 104 _{and 7.5 × 10}4 _{nodes for the RSM are investigated. The grids are generated}

by using GAMBIT. Some boxes are used (see Fig. 2.2), where dierent number of nodes and growth rates are used. They are summarized in Table 2.1 for SST k-ω and in Table 2.2 for the RSM.

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2.1 Geometry denition and computational grids 24 X Y -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -3 -2 -1 0 1 2 3 2 3 4 5 6 7 8 1 9 ₁₀ 11 12 13 14 15 16 17

Figure 2.2: Structure of the boxes used for the computational grid

Grid nodes 5 × 104 _{1 × 10}5 _{2 × 10}5 _{3 × 10}5 GR face 1-7 1.04 1.05 1.05 1.05 GR face 2-6 1.04 1.05 1.05 1.05 GR face 3-5 1.04 1.05 1.05 1.05 GR face 4 1.04 1.05 1.05 1.05 GR face 8 1.06 1.05 1.05 1.05 GR face 9-13 1.06 1.035 1.02 1.01 GR face 10-12 1.06 1.035 1.02 1.01 GR face 14-16 1.06−1.08 1.035 1.02 1.01 GR face 11 1.06 1.035 1.02 1.01 GR face 15 1.08 1.035 1.02 1.01 GR face 17 1.3 1.1 1.08 1.08 points edges y = ±0.5 96 193 379 483 points edges x = ±2.5 29 54 115 141

GR near the corners 1.27 1.117 1.04307 1.0444

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2.1 Geometry denition and computational grids 25 Grid nodes 1.2 × 104 2.5 × 104 3.8 × 104 5 × 104 7.5 × 104 GR face 1-7 1.22 1.2 1.2 1.19 1.19 GR face 2-6 1.22 1.2 1.2 1.19 1.19 GR face 3-5 1.22 1.2 1.2 1.19 1.19 GR face 4 1.22 1.2 1.2 1.19 1.19 GR face 8 1.22 1.2 1.2 1.19 1.19 GR face 9-13 1.48 1.42 1.42 1.38 1.38 GR face 10-12 1.34 1.31 1.31 1.29 1.29 GR face 14-16 1.84 1.8 1.8 1.7 1.7 GR face 11 1.28 1.24 1.24 1.21 1.21 GR face 15 1.72 1.64 1.64 1.56 1.56 GR face 17 1.1 1.1 1.1 1.1 1.1 points edges y = ±0.5 520 1042 1561 2082 3125 points edges x = ±2.5 104 208 314 418 625

GR near the corners 1 1 1 1 1

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2.2 Turbulence modeling and numerical

methodology 26

2.2 Turbulence modeling and numerical

methodology

Unsteady Reynolds−averaged incompressible Navier−Stokes equations are consid-ered as governing equations. Two dierent turbulence models have been used. The rst one is the shear-stress transport SST k-ω model (see Menter), so named because the denition of the turbulent viscosity is modied to account for the transport of the principal turbulent shear stress. It was indeed developed to blend the robust and accurate formulation of the k-ω model in the near-wall region with the free-stream independence of the k- model in the far eld. To achieve this, the k- model is converted into a k-ω formulation. The SST k-ω model is similar to the standard k-ω model, but it includes the following renements:

- The standard k-ω model and the transformed k- model are both multiplied by a blending function and both models are added together. The blending function is designed to be one in the near-wall region, which activates the standard k-ω model, and zero far from the surface, which activates the transformed k- model. - The SST model incorporates a damped cross-diusion derivative term in the ω

equation.

- The denition of the turbulent viscosity is modied to account for the transport of the turbulent shear stress.

- The modeling constants are dierent. These features make the SST k-ω model more accurate and reliable for a wider class of ows than the standard k-ω model. The Reynolds stress model (RSM) (see Launder and Rodi) is the most

elabo-rate turbulence model that FLUENT (see e.g. ANSYS R Fluent [2006]) provides.

Abandoning the isotropic eddy-viscosity hypothesis, the RSM closes the Reynolds-averaged Navier-Stokes equations by solving transport equations for the Reynolds stresses, together with an equation for the dissipation rate. This means that ve additional transport equations are required in 2D ows and seven additional trans-port equations must be solved in 3D. Since the RSM accounts for the eects of streamline curvature, swirl, rotation, and rapid changes in strain rate in a more

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methodology 27

rigorous manner than one-equation and two-equation models, it has greater poten-tial to give accurate predictions for complex ows. However, the delity of RSM predictions is still limited by the closure assumptions employed to model various terms in the exact transport equations for the Reynolds stresses. The modeling of the pressure-strain and dissipation-rate terms is particularly challenging, and often considered to be responsible for compromising the accuracy of RSM predic-tions. The RSM might not always yield results that are clearly superior to those of simpler models in all classes of ows to warrant the additional computational expense. However, use of the RSM is a must when the ow features of interest are the result of anisotropy in the Reynolds stresses.

As for the numerical methods FLUENT allows the use of two approaches: - Segregated;

- Coupled.

The Coupled method was used for the current simulations. It solves the equations simultaneously.

2.2.1 Set-up of the SST k-ω turbulence model

FLUENT is based on the nite-volume discretization method, and two-dimensional incompressible simulations are carried out for the current geometry. Unsteady time advancing is chosen together with a second-order implicit scheme. The implicit method allows to calculate the unknown value using both the existing and the unknown values of the cells. A second-order upwind scheme is used for the space discretization. The segregated PISO algorithm (Pressure-Implicit with Splitting of Operators) is chosen to couple the pressure and momentum equations (see e.g. Issa [1986]). PISO algorithm performs additional corrections: neighbor and skewness correction. The rst one moves the repeated calculations inside the solution stage of the p−correction equation. In this way after some PISO loops, the corrected velocities satisfy the continuity and momentum equations more closely (neighbor correction). It needs more CPU but there is a big decrease of the number of iterations. The second one recalculates the p−correction gradient that is used to update the mass uxes corrections. It reduces convergence diculties associated with highly distorted grides. For grids with some degrees of skewness the previous

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methodology 28

simultaneous corrections may cause divergence or lack of robustness. A method for handling the neighbor and skewness corrections is to apply one or more iterations of skewness correction for each separate iteration of neighbor correction.

The under−relaxation factors are set in this way: - pressure 0.3;

- density 1; - body forces 1; - momentum 0.7.

We use them to control the update of computed variables at each iteration. If residuals increase, we have to reduce the under−relaxation factors.

2.2.2 Set-up of the RSM

Also for the Reynolds Stress Model unsteady time advancing is chosen together with a second-order implicit scheme. A second-order upwind scheme is used for the space discretization. The PISO algorithm is chosen to couple the pressure and momentum equations. The under−relaxation factors are set as in the SST k-ω. The pressure strain term in the transport equation for the transport of the Reynolds stresses is modeled with a linear equation.

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Chapter 3

Simulations carried out with

the SST k-

ω

turbulence model

3.1 Convergence of statistics and grid sensitivity

analysis

The computational domain is discretized by using a hybrid grid, having structured quadrilateral cells in the neighbourhood of the body and unstructured triangular cells far from the body. An example of the grid topology is shown in Figs. 3.1.

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3.1 Convergence of statistics and grid sensitivity analysis 30

(a) (b)

Figure 3.1: Sketch of the computational grid

In this Section the grid sensitivity studies are presented. The time behaviour of the force coecients is shown in Fig. 3.2 for four dierent grids having 5 × 104, 1×105, 2×105and 3×105 nodes, respectively. As can be seen from Fig. 3.2, all the considered grids present a numerical transient equal approximately to t ∗ u∞/D =

150. This time interval is much lower than the time interval required to obtain

convergence in LES. This, however, is not surprising since 2D URANS give a ow dynamics closer to a periodic behaviour than in LES.

Dierent time intervals between t ∗ u∞/D = 150 and t ∗ u∞/D = 400 have

been considered for the analysis of the convergence of statistics on all the grids. The results of the convergence analysis, reported in Table 3.1, show that the time

interval between t∗u∞/D = 150and t∗u∞/D = 300is suitable for the convergence

of statistics in all the considered grids.

The Strouhal number, based on the body diameter and the free-stream velocity,

is equal to St = f ∗ D/u∞ = 0.116 ± 0.004for all the grids. This value is

indepen-dent of the time interval used for its computation, since the time oscillations have a practically constant period. Previous and BARC experiments give values of St in the range 0.105 − 0.12.

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3.1 Convergence of statistics and grid sensitivity analysis 31 0 50 100 150 200 250 300 350 400 −3 −2 −1 0 1 2 3 t*d/u_∞ cy

50k nodes 100k nodes 200k nodes 300k nodes

(a) 0 50 100 150 200 250 300 350 400 0.9 1 1.1 1.2 1.3 t*d/u_∞ cx

50k nodes 100k nodes 200k nodes 300k nodes

(b)

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Grid nodes Time range mean(cy) mean(cx) rms(cy) rms (cx)

150−200 -0.0485 1.0813 0.8959 0.0115 5 × 104 _150−300 _-0.0028 _1.0311 _0.9055 _0.0115 150−400 0.0083 1.0311 0.9073 0.0115 150−200 -0.0454 1.0980 0.9552 0.0173 1 × 105 _150−300 _-0.0024 _1.0983 _0.9639 _0.0173 150−400 -0.0035 1.0983 0.9664 0.0172 150−200 -0.0687 1.1019 0.9708 0.0154 2 × 105 _150−300 _-0.0010 _1.1019 _0.9852 _0.0155 150−400 -0.0075 1.1020 0.9837 0.0155 150−200 0.0205 1.1079 1.0296 0.0154 3 × 105 _150−300 _0.0024 _1.1079 _1.0297 _0.0154 150−400 -0.0012 1.1079 1.0297 0.0154 Experiments 1.029 0.4 Numerical contributions Ensemble average -0.04 1.002 0.54 Standard deviation 0.10 0.047 0.25

Table 3.1: Convergence of statistics for the dierent grids

Figures 3.3 and 3.4 show the distributions of the side-averaged and time-averaged pressure coecient and of the standard deviation in time of the pressure coecient for all the grids, which are evaluated in dierent time ranges. Again, a time interval from 150 to 200 is enough to assure the statistical convergence of the quantities of interest.

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3.1 Convergence of statistics and grid sensitivity analysis 33 0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D side−avg(t−avg(C p )) 150−200 150−300 150−400 (a) 5 × 104_nodes 0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D side−avg(t−avg(C p )) 150−200 150−300 150−400 (b) 1 × 105 _nodes 0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D side−avg(t−avg(C p )) 150−200 150−300 150−400 (c) 2 × 105 _nodes 0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D side−avg(t−avg(C p )) 150−200 150−300 150−400 (d) 3 × 105 _nodes

Figure 3.3: Side-averaged and time-averaged pressure coecient distribution for all the grids: comparison between dierent time ranges

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3.1 Convergence of statistics and grid sensitivity analysis 34 0 1 2 3 4 5 6 0 0.05 0.1 0.15 s/D side−avg(t−std(C p )) 150−200 150−300 150−400 (a) 5 × 104_nodes 0 1 2 3 4 5 6 0 0.05 0.1 0.15 s/D side−avg(t−std(C p )) 150−200 150−300 150−400 (b) 1 × 105 _nodes 0 1 2 3 4 5 6 0 0.05 0.1 0.15 s/D side−avg(t−std(C p )) 150−200 150−300 150−400 (c) 2 × 105 _nodes 0 1 2 3 4 5 6 0 0.05 0.1 0.15 s/D side−avg(t−std(C p )) 150−200 150−300 150−400 (d) 3 × 105 _nodes

Figure 3.4: Side-averaged distribution of the standard deviation in time of the pressure coecient for all the grids: comparison between dierent time ranges

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3.1 Convergence of statistics and grid sensitivity analysis 35 Once assured the statistical converge of the solutions, we focused on the grid in-dependence analysis. The results reported in Table 3.1 for the time range 150−300 and Fig. 3.5 show that grid independence is reached for the grid having about 1 × 105 nodes. 0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D side−avg(t−avg(C p )) 5.104 nodes 1.105 nodes 2.105 nodes 3.105 nodes

(a) Pressure coecient

0 1 2 3 4 5 6 0 0.05 0.1 0.15 s/D side−avg(t−avg(C p )) 5.104 nodes 1.105 nodes 2.105 nodes 3.105 nodes

(b) Standard deviation in time of the pres-sure coecient

Figure 3.5: Side-averaged distribution of the time averaged pressure coecient and of the standard deviation in time of the pressure coecient: comparison between the coarse and the ne grid

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3.1 Convergence of statistics and grid sensitivity analysis 36 The mean ow streamlines, reported in Fig. 3.6, and the reattachment points of the mean recirculation aside of the model also conrm that the suitable grid is

the one having 1 × 105 _nodes.

(a) 5 × 104_-nodes _{(b) 1 × 10}5_-nodes

(c) 2 × 105_-nodes _{(d) 3 × 10}5_-nodes

Figure 3.6: Mean ow streamlines Grid nodes reattachment point

5 × 104 _1.922

1 × 105 _1.926

2 × 105 _1.938

3 × 105 1.927

Table 3.2: Reattachment points of the mean recirculation aside of the model for the four dierent grids

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3.2 Analysis of the ow pattern around the cylinder at zero and non-zero

angle of incidence 37

3.2 Analysis of the ow pattern around the

cylin-der at zero and non-zero angle of incidence

In this Section we analysed in more details the results of the cases 1 and 2 in Table 3.3 to describe the eect of a non-zero angle of incidence of the ow. In particular, the case 1 is the same previously described in Sec. 3.1, while the case 2 has an angle of incidence of the oncoming ow equal to -1 deg. As for the bulk parameters, it can be seen in Table 3.3 that the angle of incidence α obviously

aects the mean value of cy, while the mean value of cx slightly increases with

increasing α. The same is for the standard deviation of cx. The eect on the amplitude of the cy time oscillations is very low.

Case Angle of incidence [deg] Turbulence intensity [%] Turbulence length scale [D] mean(cy) mean(cx) rms(cy) rms (cx) St 1 0 0.0155 2.5500 -0.0024 1.0983 0.9639 0.0173 0.116 ± 0.004 2 -1.0000 0.0155 2.5500 -0.9160 1.1149 0.9794 0.0856 0.116 ± 0.004

Table 3.3: Bulk parameters for two dierent ow conguration

The time behaviour of the force coecients are shown in Figs. 3.7 and 3.8 for the cases 1 and 2, respectively.

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3.2 Analysis of the ow pattern around the cylinder at zero and non-zero angle of incidence 38 0 50 100 150 200 250 300 0.9 1 1.1 1.2 1.3 t*d/u_∞ cx (a) 0 50 100 150 200 250 −3 −2 −1 0 1 2 3 t*d/u_∞ cy (b)

Figure 3.7: Time behaviour of the force coecients for the case 1

0 50 100 150 200 250 0.9 1 1.1 1.2 1.3 t*d/u_∞ cx (a) 0 50 100 150 200 250 −3 −2 −1 0 1 2 3 t*d/u ∞ cy (b)

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surfaces of the pressure coecient averaged in time and of its standard deviation for the cases 1 and 2 respectively.

0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D t−avg(C p )

upper side lower side

(a) Distribution of the time-averaged pres-sure coecient 0 1 2 3 4 5 6 0 0.05 0.1 0.15 s/D t−std(C p )

(b) Distribution of the standard deviation in time of the pressure coecient

Figure 3.9: Case 1: comparison between the upper and the lower side

0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D t−avg(C p )

(a) Distribution of the time-averaged pres-sure coecient 0 1 2 3 4 5 6 0 0.05 0.1 0.15 s/D t−std(C p )

As for the mean Cp and its uctuations, as expected, dierences are present

on the lateral side of the cylinder for the case 2, due to the incidence angle of the incoming ow. These are related to dierences in the topology of the mean ow. The main streamlines are shown in Fig. 3.11. The main recirculation regions are symmetric for the case 1, while they have dierent extents for the case 2. The

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smaller mean recirculation region on the upper side is consistent with an upstream

increase of the mean Cp on the this side, compared with that obtained on the lower

side.

(a) Case 1

(b) Case 2

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3.3 Summary of the bulk parameters 41

3.3 Summary of the bulk parameters

The summary of the operating conditions and of the resulting bulk parameters for

the 25 simulations for the grids having 5 × 104 _{and 1 × 10}5 _{nodes are summarized}

in Tables 3.4 and 3.5. Moreover the bulk results are compared in Figures 3.12,

3.13 and 3.14. The numerical transient is equal to t ∗ u∞/D = 150, and the time

interval between t ∗ u∞/D = 150 and t ∗ u∞/D = 300 has been considered to

evaluate the mean quantities of interest.

Case Angle of incidence [deg] Turbulence intensity [%] Turbulence length scale [D] mean(cy) mean(cx) rms(cy) rms (cx) St 1 0 0.0155 2.5500 -0.0187 1.0809 0.9117 0.0115 0.116 ± 0.004 2 -1.0000 0.0155 2.5500 -0.7840 1.0975 0.9172 0.0768 0.116 ± 0.004 3 1.0000 0.0155 2.5500 0.7863 1.0972 0.9141 0.0769 0.116 ± 0.004 4 -0.7071 0.0155 2.5500 -0.5470 1.0935 0.9250 0.0569 0.116 ± 0.004 5 0.7071 0.0155 2.5500 0.5502 1.0930 0.9220 0.0572 0.116 ± 0.004 6 0 0.0010 2.5500 -0.0186 1.0837 0.9292 0.0121 0.116 ± 0.004 7 0 0.0300 2.5500 -0.0029 1.0638 0.8056 0.0086 0.116 ± 0.004 8 -1.0000 0.0010 2.5500 -0.8132 1.1013 0.9412 0.0791 0.116 ± 0.004 9 1.0000 0.0010 2.5500 0.8180 1.1007 0.9372 0.0793 0.116 ± 0.004 10 -1.0000 0.0300 2.5500 -0.6877 1.0784 0.7971 0.0661 0.116 ± 0.004 11 1.0000 0.0300 2.5500 0.6841 1.0787 0.7974 0.0664 0.116 ± 0.004 12 0 0.0052 2.5500 -0.0198 1.0866 0.9431 0.0127 0.116 ± 0.004 13 0 0.0258 2.5500 -0.0103 1.0697 0.8440 0.0094 0.116 ± 0.004 14 0 0.0155 0.1000 -0.0201 1.0847 0.9339 0.0123 0.116 ± 0.004 15 0 0.0155 5.0000 -0.0182 1.0809 0.9107 0.0114 0.116 ± 0.004 16 -1.0000 0.0155 0.1000 -0.8138 1.1015 0.9411 0.0791 0.116 ± 0.004 17 1.0000 0.0155 0.1000 0.8174 1.1010 0.9380 0.0794 0.116 ± 0.004 18 -1.0000 0.0155 5.0000 -0.7830 1.0973 0.9160 0.0766 0.116 ± 0.004 19 1.0000 0.0155 5.0000 0.7856 1.0972 0.9129 0.0769 0.116 ± 0.004 20 0 0.0010 0.1000 -0.0209 1.0836 0.9305 0.0121 0.116 ± 0.004 21 0 0.0300 0.1000 -0.0202 1.0819 0.9206 0.0118 0.116 ± 0.004 22 0 0.0010 5.0000 -0.0116 1.0836 0.9267 0.0121 0.116 ± 0.004 23 0 0.0300 5.0000 -0.0004 1.0629 0.7971 0.0084 0.116 ± 0.004 24 0 0.0155 0.8176 -0.0190 1.0813 0.9163 0.0116 0.116 ± 0.004 25 0 0.0155 4.2824 -0.0180 1.0809 0.9105 0.0114 0.116 ± 0.004

Table 3.4: Bulk parameters for the SST k−ω turbulence model (grid having 5×104

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3.3 Summary of the bulk parameters 42 Case Angle of incidence [deg] Turbulence intensity [%] Turbulence length scale [D] mean(cy) mean(cx) rms(cy) rms (cx) St 1 0 0.0155 2.5500 -0.0024 1.0983 0.9639 0.0173 0.116 ± 0.004 2 -1.0000 0.0155 2.5500 -0.9160 1.1149 0.9794 0.0856 0.116 ± 0.004 3 1.0000 0.0155 2.5500 0.9087 1.1159 0.9807 0.0857 0.116 ± 0.004 4 -0.7071 0.0155 2.5500 -0.6334 1.1084 0.9766 0.0623 0.116 ± 0.004 5 0.7071 0.0155 2.5500 0.6241 1.1091 0.9773 0.0623 0.116 ± 0.004 6 0 0.0010 2.5500 -0.0136 1.1039 0.9997 0.0199 0.116 ± 0.004 7 0 0.0300 2.5500 0.0159 1.0859 0.8940 0.0130 0.116 ± 0.004 8 -1.0000 0.0010 2.5500 -0.9789 1.1216 1.0216 0.0897 0.116 ± 0.004 9 1.0000 0.0010 2.5500 0.9769 1.1221 1.0225 0.0897 0.116 ± 0.004 10 -1.0000 0.0300 2.5500 -0.8124 1.1028 0.8996 0.0782 0.116 ± 0.004 11 1.0000 0.0300 2.5500 0.8082 1.1039 0.9033 0.0787 0.116 ± 0.004 12 0 0.0052 2.5500 0.0027 1.1035 0.9971 0.0198 0.116 ± 0.004 13 0 0.0258 2.5500 0.0136 1.0900 0.9170 0.0143 0.116 ± 0.004 14 0 0.0155 0.1000 0.0023 1.1014 0.9837 0.0186 0.116 ± 0.004 15 0 0.0155 5.0000 0.0030 1.0983 0.9633 0.0173 0.116 ± 0.004 16 -1.0000 0.0155 0.1000 -0.9467 1.1184 1.0025 0.0878 0.116 ± 0.004 17 1.0000 0.0155 0.1000 0.9419 1.1192 1.0033 0.0878 0.116 ± 0.004 18 -1.0000 0.0155 5.0000 -0.9155 1.1149 0.9786 0.0856 0.116 ± 0.004 19 1.0000 0.0155 5.0000 0.9081 1.1160 0.9800 0.0856 0.116 ± 0.004 20 0 0.0010 0.1000 -0.0210 1.1035 1.0010 0.0199 0.116 ± 0.004 21 0 0.0300 0.1000 0.0044 1.0990 0.9704 0.0178 0.116 ± 0.004 22 0 0.0010 5.0000 -0.0118 1.1040 0.9996 0.0199 0.116 ± 0.004 23 0 0.0300 5.0000 0.0155 1.0854 0.8879 0.0127 0.116 ± 0.004 24 0 0.0155 0.8176 -0.0047 1.0988 0.9672 0.0176 0.116 ± 0.004 25 0 0.0155 4.2824 -0.0059 1.0984 0.9631 0.0173 0.116 ± 0.004

Table 3.5: Bulk parameters for the SST k−ω turbulence model (grid having 1×105

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3.3 Summary of the bulk parameters 43 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1

Angle of incidence [deg]

cy 5.104 nodes 1.105 nodes (a) mean(cy) −1 −0.5 0 0.5 1 1.06 1.08 1.1 1.12 1.14

cx 5.104 nodes 1.105 nodes (b) mean(cx) −1 −0.5 0 0.5 1 0.8 0.85 0.9 0.95 1 1.05

rms(c y ) 5.104 nodes 1.105 nodes (c) rms(cy) −1 −0.5 0 0.5 1 0 0.02 0.04 0.06 0.08 0.1

rms(c

x

)

5.104 nodes 1.105 nodes

(d) rms(cx)

Figure 3.12: Eect of the angle of incidence (turbulence intensity = 1.55% and turbulence length scale = 2.55D)

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3.3 Summary of the bulk parameters 44 0 0.5 1 1.5 2 2.5 3 −1 −0.5 0 0.5 1 Turbulence intensity [%] cy 5.104 nodes 1.105 nodes (a) mean(cy) 0 0.5 1 1.5 2 2.5 3 1.06 1.08 1.1 1.12 1.14 Turbulence intensity [%] cx 5.104 nodes 1.105 nodes (b) mean(cx) 0 0.5 1 1.5 2 2.5 3 0.8 0.85 0.9 0.95 1 1.05 Turbulence intensity [%] rms(c y ) 5.104 nodes 1.105 nodes (c) rms(cy) 0 0.5 1 1.5 2 2.5 3 0 0.02 0.04 0.06 0.08 0.1 Turbulence intensity [%] rms(c x ) 5.104 nodes 1.105 nodes (d) rms(cx)

Figure 3.13: Eect of the turbulence intensity (angle of incidence = 0 deg and turbulence length scale = 2.55D)

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3.3 Summary of the bulk parameters 45 0 1 2 3 4 5 −1 −0.5 0 0.5 1

Turbulence length scale [D]

cy 5.104 nodes 1.105 nodes (a) mean(cy) 0 1 2 3 4 5 1.06 1.08 1.1 1.12 1.14

cx 5.104 nodes 1.105 nodes (b) mean(cx) 0 1 2 3 4 5 0.8 0.85 0.9 0.95 1 1.05

rms(c y ) 5.104 nodes 1.105 nodes (c) rms(cy) 0 1 2 3 4 5 0 0.02 0.04 0.06 0.08 0.1

rms(c

x

)

5.104 nodes 1.105 nodes

(d) rms(cx)

Figure 3.14: Eect of the turbulence length scale (angle of incidence = 0 deg and turbulence intensity = 1.55%)

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3.4 Analysis of the ow behaviour on the cylinder lateral surfaces 46

3.4 Analysis of the ow behaviour on the cylinder

lateral surfaces

3.4.1 Reattachment points and centers of the bubbles

The reattachment points xr and center points (xc, yc) of the mean recirculation

aside of the model are evaluated for all the simulations in Tables 3.6 and 3.7. The results are also compared in Figures 3.15, 3.16 and 3.17.

Case Angle of incidence [deg] Turbulence intensity [%] Turbulence length scale [D] xupr /D xdownr /D x up c /D xdownc /D y up c /D ydownc /D 1 0 0.0155 2.5500 1.92 1.92 -0.48 -0.48 0.81 -0.81 2 -1.0000 0.0155 2.5500 1.42 2.30 -0.80 -0.11 0.77 -0.86 3 1.0000 0.0155 2.5500 2.29 1.33 -0.10 -0.80 0.86 -0.77 4 -0.7071 0.0155 2.5500 1.56 2.22 -0.71 -0.22 0.78 -0.85 5 0.7071 0.0155 2.5500 2.26 1.56 -0.21 -0.71 0.85 -0.78 6 0 0.0010 2.5500 1.92 1.93 -0.45 -0.48 0.81 -0.81 7 0 0.0300 2.5500 1.87 1.87 -0.60 -0.60 0.81 -0.81 8 -1.0000 0.0010 2.5500 1.42 2.30 -0.78 -0.08 0.77 -0.86 9 1.0000 0.0010 2.5500 2.32 1.42 -0.07 -0.78 0.86 -0.77 10 -1.0000 0.0300 2.5500 1.37 2.29 -0.88 -0.26 0.77 -0.85 11 1.0000 0.0300 2.5500 2.31 1.37 -0.26 -0.77 0.85 -0.88 12 0 0.0052 2.5500 1.93 1.94 -0.43 -0.45 0.81 -0.81 13 0 0.0258 2.5500 1.89 1.89 -0.56 -0.56 0.81 -0.81 14 0 0.0155 0.1000 1.93 1.94 -0.44 -0.47 0.81 -0.81 15 0 0.0155 5.0000 1.92 1.93 -0.47 -0.50 0.81 -0.81 16 -1.0000 0.0155 0.1000 1.43 2.30 -0.77 -0.06 0.77 -0.86 17 1.0000 0.0155 0.1000 2.32 1.43 -0.08 -0.77 0.86 -0.77 18 -1.0000 0.0155 5.0000 1.42 2.30 -0.80 -0.11 0.77 -0.86 19 1.0000 0.0155 5.0000 2.32 1.41 -0.10 -0.80 0.86 -0.77 20 0 0.0010 0.1000 1.92 1.93 -0.45 -0.48 0.81 -0.81 21 0 0.0300 0.1000 1.92 1.93 -0.46 -0.48 0.81 -0.81 22 0 0.0010 5.0000 1.92 1.93 -0.45 -0.48 0.81 -0.81 23 0 0.0300 5.0000 1.87 1.87 -0.61 -0.60 0.81 -0.81 24 0 0.0155 0.8176 1.92 1.93 -0.47 -0.49 0.81 -0.81 25 0 0.0155 4.2824 1.92 1.93 -0.47 -0.50 0.81 -0.81

Table 3.6: Reattachment points xr and center points (xc, yc) of the mean

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3.4 Analysis of the ow behaviour on the cylinder lateral surfaces 47 Case Angle of incidence [deg] Turbulence intensity [%] Turbulence length scale [D] xupr /D xdownr /D x up c /D xdownc /D y up c /D ydownc /D 1 0 0.0155 2.5500 1.94 1.94 -0.36 -0.36 0.82 -0.82 2 -1.0000 0.0155 2.5500 1.46 2.30 -0.67 0.04 0.78 -0.87 3 1.0000 0.0155 2.5500 2.31 1.46 0.04 -0.67 0.87 -0.78 4 -0.7071 0.0155 2.5500 1.61 2.24 -0.59 -0.08 0.79 -0.85 5 0.7071 0.0155 2.5500 2.28 1.61 -0.08 -0.59 0.85 -0.79 6 0 0.0010 2.5500 1.98 1.98 -0.31 -0.28 0.83 -0.83 7 0 0.0300 2.5500 1.92 1.92 -0.47 -0.45 0.82 -0.82 8 -1.0000 0.0010 2.5500 1.48 2.32 -0.62 0.11 0.78 -0.87 9 1.0000 0.0010 2.5500 2.32 1.48 0.11 0.62 0.87 -0.78 10 -1.0000 0.0300 2.5500 1.42 2.29 -0.76 -0.10 0.78 -0.87 11 1.0000 0.0300 2.5500 2.31 1.42 -0.10 -0.76 0.87 -0.78 12 0 0.0052 2.5500 1.99 1.98 -0.32 -0.28 0.83 -0.83 13 0 0.0258 2.5500 1.94 1.93 -0.44 -0.42 0.82 -0.82 14 0 0.0155 0.1000 1.98 1.97 -0.34 -0.31 0.82 -0.82 15 0 0.0155 5.0000 1.97 1.97 -0.37 -0.34 0.82 -0.82 16 -1.0000 0.0155 0.1000 1.48 2.31 -0.64 0.08 0.78 -0.87 17 1.0000 0.0155 0.1000 2.32 1.48 0.08 -0.64 0.87 -0.78 18 -1.0000 0.0155 5.0000 1.46 2.30 -0.67 0.04 0.78 -0.87 19 1.0000 0.0155 5.0000 2.31 1.46 0.04 -0.67 0.87 -0.78 20 0 0.0010 0.1000 1.98 1.99 -0.28 -0.31 0.83 -0.83 21 0 0.0300 0.1000 1.97 1.96 -0.36 -0.33 0.82 -0.82 22 0 0.0010 5.0000 1.99 1.98 -0.31 -0.28 0.83 -0.82 23 0 0.0300 5.0000 1.92 1.91 -0.48 -0.46 0.82 -0.82 24 0 0.0155 0.8176 1.97 1.96 -0.37 -0.34 0.82 -0.82 25 0 0.0155 4.2824 1.97 1.96 -0.37 -0.34 0.82 -0.82

Table 3.7: Reattachment points xr and center points (xc, yc) of the mean

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3.4 Analysis of the ow behaviour on the cylinder lateral surfaces 48 −3 −2 −1 0 1 2 3 −2 −1 0 1 2 x/D y/D α=0 deg α=−1 deg α=1 deg α=−0.707 deg α=0.707 deg

(a) Grid 5 × 104 _nodes

−3 −2 −1 0 1 2 3 −2 −1 0 1 2 x/D y/D α=0 deg α=−1 deg α=1 deg α=−0.707 deg α=0.707 deg (b) Grid 1 × 105 _nodes

Figure 3.15: Eect of the angle of incidence on the reattachment points xr and

center points (xc, yc) of the mean recirculation aside of the model (turbulence

intensity = 1.55% and turbulence length scale = 2.55D)

−3 −2 −1 0 1 2 3 −2 −1 0 1 2 x/D y/D 0.10 % 0.52 % 1.55 % 2.58 % 3 %

−3 −2 −1 0 1 2 3 −2 −1 0 1 2 x/D y/D 0.10 % 0.52 % 1.55 % 2.58 % 3 % (b) Grid 1 × 105 _nodes

Figure 3.16: Eect of the turbulence intensity on the reattachment points xr and center points (xc, yc) of the mean recirculation aside of the model (angle of incidence = 0 deg and turbulence length scale = 2.55D)

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3.4 Analysis of the ow behaviour on the cylinder lateral surfaces 49 −3 −2 −1 0 1 2 3 −2 −1 0 1 2 x/D y/D 0.1 D 0.82 D 2.55 D 4.28 D 5 D

−3 −2 −1 0 1 2 3 −2 −1 0 1 2 x/D y/D 0.1 D 0.82 D 2.55 D 4.28 D 5 D (b) Grid 1 × 105 _nodes

Figure 3.17: Eect of the turbulence length scale on the reattachment points xr and center points (xc, yc) of the mean recirculation aside of the model (angle of incidence = 0 deg and turbulence intensity = 1.55%)

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3.4.2 Eect of the angle of incidence on pressure eld

Figures. 3.18 and 3.21 compare the distributions of the mean value of Cp and

of its standard deviation for dierent values of the incidence angle α for the two considered grids. Similar considerations as previously can be made.

0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D t−avg(C p ) α=0 deg α=−1 deg α=1 deg α=−0.707 deg α=0.707 deg

(a) Distribution of the time-averaged pres-sure coecient 0 1 2 3 4 5 6 0 0.05 0.1 0.15 0.2 0.25 0.3 s/D t−std(C p ) α=0 deg α=−1 deg α=1 deg α=−0.707 deg α=0.707 deg

Figure 3.18: Eect of angle α on the upper side for the grid having 5 × 104 nodes

(turbulence intensity = 1.55% and turbulence length scale = 2.55D)

Figure 3.19: Eect of angle α on the upper side for the grid having 1 × 105 _nodes

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3.4 Analysis of the ow behaviour on the cylinder lateral surfaces 51 0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D t−avg(C p ) α=0 deg α=−1 deg α=1 deg α=−0.707 deg α=0.707 deg

Figure 3.20: Eect of angle α on the lower side for the grid having 5 × 104 _nodes

(turbulence intensity = 1.55% and turbulence length scale = 2.55D)

Figure 3.21: Eect of angle α on the lower side for the grid having 1 × 105 nodes

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3.4.3 Eect of the turbulence intensity on pressure eld

The eect of the turbulence intensity is highlighted in Figs. 3.22 and 3.23, where the side-averaged quantities are shown for the two considered grids. The turbulence intensity has a signicant eect only on the intensity of pressure uctuations on

the cylinder side. It aects the standard deviation of cy and the mean value of cx

(variation of base pressure not appreciable in Figs. 3.22(a) and 3.23(a)).

0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D side−avg(t−avg(C p )) 0.10% 0.52% 1.55% 2.58% 3.00%

(a) Distribution of the side-averaged and time-averaged pressure coecient

0 1 2 3 4 5 6 0 0.05 0.1 0.15 0.2 0.25 0.3 s/D side−avg(t−std(C p )) 0.10% 0.52% 1.55% 2.58% 3.00%

(b) Distribution of the side-averaged stan-dard deviation in time of the pressure co-ecient

Figure 3.22: Eect of the turbulence intensity for the grid having 5 × 104 nodes

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3.4 Analysis of the ow behaviour on the cylinder lateral surfaces 53 0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D side−avg(t−avg(C p )) 0.10% 0.52% 1.55% 2.58% 3.00%

0 1 2 3 4 5 6 0 0.05 0.1 0.15 0.2 0.25 0.3 s/D side−avg(t−std(C p )) 0.10% 0.52% 1.55% 2.58% 3.00%

Figure 3.23: Eect of the turbulence intensity for the grid having 1 × 105 _nodes

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3.4.4 Eect of the turbulence scale on pressure eld

The eect of the turbulence scale is highlighted in Figs. 3.24 and 3.25 , where the side-averaged quantities are shown for the two considered grids. The turbulence scale does not seem to have noticeable eects.

0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D side−avg(t−avg(C p )) 0.10 D 0.82 D 2.55 D 4.28 D 5.00 D

0 1 2 3 4 5 6 0 0.05 0.1 0.15 0.2 0.25 0.3 s/D side−avg(t−std(C p )) 0.10 D 0.82 D 2.55 D 4.28 D 5.00 D

Figure 3.24: Eect of the turbulence scale for the grid having 5 × 104 _{nodes (angle} of incidence = 0 deg and turbulence intensity = 1.55%)

0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D side−avg(t−avg(C p )) 0.10 D 0.82 D 2.55 D 4.28 D 5.00 D

0 1 2 3 4 5 6 0 0.05 0.1 0.15 0.2 0.25 0.3 s/D side−avg(t−std(C p )) 0.10 D 0.82 D 2.55 D 4.28 D 5.00 D

Figure 3.25: Eect of the turbulence scale for the grid having 1 × 105 _{nodes (angle} of incidence = 0 deg and turbulence intensity = 1.55%)

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Chapter 4

Simulations carried out with

the Reynolds Stress Model

4.1 Convergence of statistics and grid sensitivity

analysis

Unsteady RANS simulations have been carried out by using the Reynolds Stress Model and the commercial code FLUENT. The computational domain is dis-cretized by using an unstructured triangular grid. An example of the grid topology is shown in Figs. 4.1.

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4.1 Convergence of statistics and grid sensitivity analysis 56 X Y -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5

Figure 4.1: Sketch of the computational grid

In this Section the grid sensitivity studies are presented. The time behaviour of the force coecients is shown in Fig. 4.2 for ve dierent grids having 1.2×104_, 2.5 × 104, 3.8 × 104, 5.0 × 104 and 7.5 × 104 nodes, respectively. The numerical

transient, equal to t ∗ u∞/D = 300 is not considered for all the simulations.

Dierent time intervals between t ∗ u∞/D = 300 and t ∗ u∞/D = 550 have

been considered for the analysis of the convergence of statistics on all the grids. The results of the convergence analysis, reported in Table 4.1, show that the time

interval between t∗u∞/D = 300and t∗u∞/D = 450is suitable for the convergence

of statistics in all the considered grids.

The Strouhal number, based on the body diameter and the free-stream velocity,

is equal to St = f ∗ D/u∞ = 0.112 for all the grids. Time oscillations have a

practically constant period, while small modulations in amplitude are found. Figures 4.3 and 4.4 show the distributions of the side-averaged and time-averaged pressure coecient and of the standard deviation in time of the pressure coecient for all the grids, which are evaluated in dierent time ranges. Again, a time interval from 300 to 450 is enough to assure the statistical convergence of the quantities of interest.

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4.1 Convergence of statistics and grid sensitivity analysis 57 300 350 400 450 500 550 −2 −1 0 1 2 t*d/u_∞ cy

1.2x104 nodes 2.5x104 nodes 3.8x104 nodes 5.0x104 nodes 7.5x104 nodes

(a) 300 350 400 450 500 550 1.05 1.1 1.15 t*d/u_∞ cx

1.2x104 nodes 2.5x104 nodes 3.8x104 nodes 5.0x104 nodes 7.5x104 nodes

(b)

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Grid nodes Time range mean(cy) mean(cx) rms(cy) rms (cx)

300−350 -0.0450 1.0813 0.5133 0.0203 1.2 × 104 _300−450 _-0.0432 _1.0814 _0.5116 _0.0202 300−550 -0.0409 1.0816 0.5111 0.0202 300−350 -0.0054 1.1020 0.6130 0.0266 2.5 × 104 _300−450 _-0.0051 _1.1019 _0.6128 _0.0267 300−550 -0.0048 1.1019 0.6124 0.0267 300−350 0.0451 1.0875 0.5870 0.0265 3.8 × 104 _300−450 _0.0463 _1.0874 _0.5859 _0.0265 300−550 0.0477 1.0874 0.5847 0.0265 300−350 0.0355 1.0926 0.6082 0.0265 5.0 × 104 _300−450 _0.0347 _1.0921 _0.6074 _0.0266 300−550 0.0299 1.0923 0.6041 0.0266 300−350 0.0438 1.0875 0.6015 0.0259 7.5 × 104 _300−450 _0.0466 _1.0868 _0.6041 _0.0257 300−550 0.0404 1.0869 0.6020 0.0257 Experiments 1.029 0.4 Numerical contributions Ensemble average -0.04 1.002 0.54 Standard deviation 0.10 0.047 0.25

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4.1 Convergence of statistics and grid sensitivity analysis 59 0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D side−avg(t−avg(C p )) 300−350 300−450 300−550 (a) 1.2 × 104_nodes 0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D side−avg(t−avg(C p )) 300−350 300−450 300−550 (b) 2.5 × 104 _nodes 0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D side−avg(t−avg(C p )) 300−350 300−450 300−550 (c) 3.8 × 104 _nodes 0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D side−avg(t−avg(C p )) 300−350 300−450 300−550 (d) 5.0 × 104 _nodes 0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D side−avg(t−avg(C p )) 300−350 300−450 300−550 (e) 7.5 × 104 _nodes

Figure 4.3: Side-averaged and time-averaged pressure coecient distribution for all the grids: comparison between dierent time ranges

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4.1 Convergence of statistics and grid sensitivity analysis 60 0 1 2 3 4 5 6 0 0.05 0.1 0.15 s/D side−avg(t−std(C p )) 300−350 300−450 300−550 (a) 1.2 × 104_nodes 0 1 2 3 4 5 6 0 0.05 0.1 0.15 s/D side−avg(t−std(C p )) 300−350 300−450 300−550 (b) 2.5 × 104 _nodes 0 1 2 3 4 5 6 0 0.05 0.1 0.15 s/D side−avg(t−std(C p )) 300−350 300−450 300−550 (c) 3.8 × 104 _nodes 0 1 2 3 4 5 6 0 0.05 0.1 0.15 s/D side−avg(t−std(C p )) 300−350 300−450 300−550 (d) 5.0 × 104 _nodes 0 1 2 3 4 5 6 0 0.05 0.1 0.15 s/D side−avg(t−std(C p )) 300−350 300−450 300−550 (e) 7.5 × 104 _nodes

Figure 4.4: Side-averaged distribution of the standard deviation in time of the pressure coecient for all the grids: comparison between dierent time ranges

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4.1 Convergence of statistics and grid sensitivity analysis 61 Once assured the statistical converge of the solutions, we focused on the grid in-dependence analysis. The results reported in Table 4.1 for the time range 300−450 and Fig. 4.5 show that grid independence is reached for the grid having about 5.0 × 104 nodes. 0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D side−avg(t−avg(C p )) 1.2x104 nodes 2.5x104 nodes 3.8x104 nodes 5.0x104 nodes 7.5x104 nodes

(a) Pressure coecient

0 1 2 3 4 5 6 0 0.05 0.1 0.15 s/D side−avg(t−avg(C p )) 1.2x104 nodes 2.5x104 nodes 3.8x104 nodes 5.0x104 nodes 7.5x104 nodes

(b) Standard deviation in time of the pres-sure coecient

Figure 4.5: Side-averaged distribution of the time averaged pressure coecient and of the standard deviation in time of the pressure coecient: comparison between the coarse and the ne grid

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4.1 Convergence of statistics and grid sensitivity analysis 62 The mean ow streamlines, reported in Fig. 4.6, and the reattachment points of the mean recirculation aside of the model also conrm that the suitable grid is

the one having 5.0 × 104 _nodes.

x/D y/ D -2 0 2 -2 -1 0 1 2 (a) 1.2 × 104 _nodes x/D y/ D -2 0 2 -2 -1 0 1 2 (b) 2.5 × 104_nodes x/D y/ D -2 0 2 -2 -1 0 1 2 (c) 3.8 × 104 _nodes x/D y/ D -2 0 2 -2 -1 0 1 2 (d) 5.0 × 104_nodes x/D y/ D -2 0 2 -2 -1 0 1 2 (e) 7.5 × 104 _nodes

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angle of incidence for the Reynolds Stress Model 63

Grid nodes xupr /D xdownr /D x up c /D xdownc /D y up c /D ydownc /D 1.2 × 104 _1.141 _1.095 _-0.835 _-0.782 _0.814 _-0.814 2.5 × 104 _1.134 _1.119 _-1.049 _-0.865 _0.809 _-0.815 3.8 × 104 _1.220 _1.136 _-1.078 _-0.794 _0.812 _-0.817 5.0 × 104 _1.197 _1.161 _-1.003 _-0.772 _0.812 _-0.813 7.5 × 104 _1.198 _1.109 _-1.012 _-0.754 _0.812 _-0.811

Table 4.2: Reattachment points of the mean recirculation aside of the model for the ve dierent grids

4.2 Analysis of the ow pattern around the

cylin-der at zero and non-zero angle of incidence

for the Reynolds Stress Model

In this Section we analysed in more details the results of the cases 1 and 2 in

Table 4.3 for the grids having 1.2 × 104 _{and 5 × 10}4 _{nodes to describe the eect}

of a non-zero angle of incidence of the ow. In particular, the case 1 has an angle of incidence of the oncoming ow equal to 0 deg, while the case 2 has an angle of incidence of the oncoming ow equal to -1 deg. As for the bulk parameters, it can be seen in Table 4.3 that the angle of incidence α obviously aects the mean

value of cy, while the mean value of cx slightly increases with increasing α. The

same is for the standard deviation of cx. The eect on the amplitude of the cy time oscillations is very low.

Case Grid nodes Angle of

incidence [deg] Turbulence intensity [%] Turbulence length scale [D] mean(cy) mean(cx) rms(cy) rms (cx) St 1 1.2 × 104 ₀ _0.0155 _2.5500 _-0.0450 _1.0813 _0.5133 _0.0203 _0.1200 1 5 × 104 0 0.0155 2.5500 0.0347 1.0921 0.6074 0.0266 0.1200 2 1.2 × 104 _-1.0000 _0.0155 _2.5500 _-0.8419 _1.1069 _0.6358 _0.0556 _0.1200 2 5 × 104 _-1.0000 _0.0155 _2.5500 _-0.7940 _1.1118 _0.6758 _0.0580 _0.1200

Table 4.3: Bulk parameters for two dierent ow congurations and two dierent grids

The time behaviour of the force coecients are shown in Figs. 4.7 and 4.8 for the cases 1 and 2, respectively.

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300 350 400 450 −2 −1 0 1 2 t*d/u ∞ cy 1.2.₁₀4_nodes ₅.₁₀4_nodes (a) 300 350 400 450 1.05 1.1 1.15 t*d/u ∞ cx 1.2.₁₀4_nodes ₅.₁₀4_nodes (b)

300 350 400 450 −2 −1 0 1 2 t*d/u_∞ cy 1.2.104 nodes 5.104 nodes (a) 300 350 400 450 1 1.1 1.2 1.3 1.4 t*d/u_∞ cx 1.2.104 nodes 5.104 nodes (b)

Figures 4.9 and 4.10 show the distribution over the upper and lower cylinder surfaces of the pressure coecient averaged in time and of its standard deviation for the cases 1 and 2 respectively.

0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D t−avg(C p ) upper side upper side lower side lower side 5 .104 : 1.2 .₁₀4_:

(a) Distribution of the time-averaged pres-sure coecient 0 1 2 3 4 5 6 0 0.05 0.1 0.15 0.2 0.25 0.3 s/D t−std(C p ) upper side upper side lower side lower side 5 .104 : 1.2 .₁₀4_:

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0 1 2 3 4 5 6 −1 −0.5 0 0.5 1 s/D t−avg(C p ) upper side upper side lower side lower side 5 .₁₀4_: 1.2 .104 :

(a) Distribution of the time-averaged pres-sure coecient 0 1 2 3 4 5 6 0 0.05 0.1 0.15 0.2 0.25 0.3 s/D t−std(C p ) upper side upper side lower side lower side 5 .₁₀4_: 1.2 .104 :

As for the mean Cp and its uctuations, as expected, dierences are present

on the lateral side of the cylinder for the case 2, due to the incidence angle of the incoming ow. These are related to dierences in the topology of the mean ow. The main streamlines are shown in Fig. 4.11. The main recirculation regions are symmetric for the case 1, while they have dierent extents for the case 2. The smaller mean recirculation region on the upper side is consistent with an upstream

increase of the mean Cp on the this side, compared with that obtained on the lower

Benchmark on the Aerodynamics of a Rectangular Cylinder: sensitivity to inflow conditions

UNIVERSITÀ DI PISA

Dipartimento di Ingegneria Civile e Industriale

Sezione Ingegneria Aerospaziale

Tesi di Laurea Magistrale

BENCHMARK ON THE

AERODYNAMICS OF A

RECTANGULAR CYLINDER:

SENSITIVITY TO INFLOW

CONDITIONS

Ai miei nonni..

che da lassù riescono comunque ad essere vicino a me

Abstract

Contents

List of Figures

List of Tables

Chapter 1

Introduction

Chapter 2

Methodology

2.1 Geometry denition and computational grids

2.2 Turbulence modeling and numerical

methodology

2.2.1 Set-up of the SST k-ω turbulence model

2.2.2 Set-up of the RSM

Chapter 3

Simulations carried out with

the SST k-

ω

turbulence model

3.1 Convergence of statistics and grid sensitivity

analysis

3.2 Analysis of the ow pattern around the

cylin-der at zero and non-zero angle of incidence

3.3 Summary of the bulk parameters

3.4 Analysis of the ow behaviour on the cylinder

lateral surfaces

3.4.1 Reattachment points and centers of the bubbles

3.4.2 Eect of the angle of incidence on pressure eld

3.4.3 Eect of the turbulence intensity on pressure eld

3.4.4 Eect of the turbulence scale on pressure eld

Chapter 4

Simulations carried out with

the Reynolds Stress Model

4.1 Convergence of statistics and grid sensitivity

analysis

4.2 Analysis of the ow pattern around the

cylin-der at zero and non-zero angle of incidence

for the Reynolds Stress Model

2.1 Geometry denition and computational grids

3.2 Analysis of the ow pattern around the

3.4 Analysis of the ow behaviour on the cylinder

3.4.2 Eect of the angle of incidence on pressure eld

3.4.3 Eect of the turbulence intensity on pressure eld

3.4.4 Eect of the turbulence scale on pressure eld

4.2 Analysis of the ow pattern around the