Simulation of crossflow around a bluff body using URANS
method: paving the way for Fluid-Structure Interaction analyses
relatori: Prof. Walter Ambrosini Doctor Kevin Zwijsen
Anno accademico 2019-2020
Corso di laurea magistrale in ingegneria nucleare
Contents
1 Introduction 7
2 Problem description 11
2.1 The vortex shedding and the force induced vibrations . . . 11
2.2 The 2 cylinders cases . . . 14
3 Reference experimental data 17 4 Adopted models 23 4.1 Balance equations . . . 23
4.2 The problem of turbulence and RANS approach . . . 26
4.3 The two equations models . . . 32
4.3.1 The k-epsilon model . . . 33
4.3.2 The k-omega model . . . 35
4.3.3 The 2 layer k-epsilon model . . . 36
4.4 The non-linear constitutive equations . . . 38
5 Obtained results 41 5.1 The one-cylinder case . . . 41
5.1.1 Introduction . . . 41
5.1.2 Description of the numerical problem . . . 43
5.1.3 Results . . . 48
6 The Two-cylinder cases 55 6.0.1 The two tandem cylinders cases . . . 59
6.0.2 Results . . . 61
6.0.3 The two side-by-side cylinders cases . . . 68
6.0.4 Results . . . 71 3
Abstract
The primary purpose of this work is to model the cross-flow around a bluff body, using CFD methods. During the last decades, studying the effects of a turbulent fluid around a U-tube bundle raised great interest, particularly in conditions such as those of a steam generator in a nuclear power plant. Many experimental setups had been operated during the last decades to understand the displacements in the pipes caused by a turbulent fluid. Nowadays, thanks to the improvements reached in the hardware and software developments, we can use numerical methods to replicate the same conditions of these experimental setups. The prediction of the forces caused by a turbulent fluid using CFD methods could be challenging and expensive in terms of computational time; for this reason, it is not feasible using turbulence models such as DNS (Direct Numerical Simulations) or LES (Large-Eddy Simulations). The RANS (Reynolds Averaged Navier-Stokes) approach here is used for the turbulence modeling coupled with correlations that take account of the Reynolds stress tensor component anisotropy. In this work, cross-flow around a single-cylinder for a Reynolds number of 3900, is studied for understanding which are the main parameters to be taken account for having an accurate prediction of the forces acting on the cylinder. Then, the interaction between the wakes of 2 cylinders, is studied for different configurations and Reynolds numbers. The software used for this study is the version 11.06 of the Siemens-CFD code Starccm+.
Chapter 1
Introduction
In the fight against global warming, and in enstablishing the most reliable and clean energy pro-duction sources, nuclear energy is one of the most important candidates. According to researches performed by the European Union (see, for example [1]), for the following years nuclear energy will be, together with renewable sources energy, essential for electric energy production.
Fig.1 : Fuel consumption forecast [1]
8 CHAPTER 1. INTRODUCTION According to the latest projections for the energy market performed by IAEA [2] , under a high case scenario, we expect an increase of global nuclear electrical generating capacity by 82 per cent in the period between 2020 and 2050; for that reason, nuclear development technologies are essential for the future generations.
In order to better understand the motivations of this work, it is mandatory to focus on the most important nuclear technologies present in the market and, in particular, on the most used during this period: the technology of the Light Water Reactors or LWRs. In this type of reactors, light water is used to moderate the neutrons for fission energy production and, at the same time, it is used to cool the reactor core itself. We can make a further distinction between two types of LWRs:
• if steam is generated inside the reactor vessel without any intermediate loop, we are talking about Boiling Water Reactors (BWR): one of the significant advantages of this kind of design is the operating pressure being lower (about 70 MPa) than in the other designs where light water is required.
• if the thermal energy generated by the chain reaction is exchanged between liquid water inside a primary loop (at a pressure of about 150 MPa) and the water of a secondary loop, we are talking about Pressurized Water Reactors: this kind of design is the most widespread and used worldwide (300 operating reactors nowadays) thanks to the great experience obtained during the last decades.
Let us focus our attention on the PWR configuration: as already explained a primary and a secondary loop in the PWR configuration require such components that allow the heat transfer between them. This kind of element is the steam generator and is one of the most critical parts of a nuclear power plant.
9
Fig 2: Scheme of a Westinghouse’s Steam generator
As it can be noted from the Fig.2, we can distinguish two main zones inside the steam generator: the first one, consisting of the so-called primary loop, where hot water from the nuclear reactor flows inside a U-tube bundle exchanges heat with the secondary side fluid. In the secondary loop, feedwater coming from the condenser receives thermal energy from the primary loop, becoming steam to be routed to the inlet of the turbine, this occurs thanks to natural circulation phenomena in the Westinghouse design. Considering the significant amount of operational time for a nuclear power plant (60 years for the last generation plants), monitoring structural property changes is strictly mandatory for economic and safety reasons. In a steam generator the long-term effect of a turbulent fluid around the U-tube bundle can cause vibrations that can lead to fretting wear rupture.
10 CHAPTER 1. INTRODUCTION One of the most famous accidents related to this kind of break happened to the S.Onofre NPP (2012), where a U-tube crack caused a leak of radioactive material (and the following shut down and decommissioning of the plant).The collection of possibly the most interesting reference data related to the vibrations induced by a turbulent flow, was performed by EDF several decades ago, with the so-called VISCACHE experiment, where the displacement of a measurement tube caused by a turbulent flow was recorded for different flow parameters (see Fig. 3 for the sketch of the facility [11]).
Fig.3 : VISCACHE experimental setup [11]
The purpose of present work is to enstablish a solid background in the study of the phenomena involved in the interactions between a U-tube bundle and a turbulent cross-flow, using numerical methods. The first chapters will introduce the vortex shedding phenomenon and the balance equations solved inside the domain; then, we will consider the different kinds of models employed for simulating phenomena such as turbulence. Then, we study the cross-flow around a single-cylinder to understand the main parameter and conditions for having the right prediction of the forces acting on the body. Finally, to understand the possible interactions between the wakes of two cylinders, we decided to study the two-cylinders cases for two different configurations: tandem and side-by-side.
Chapter 2
Problem description
2.1
The vortex shedding and the force induced vibrations
The study of crossflow around a circular cylinder is one of the classical problems approached in the fluid-dynamics field. In dealing with similar problems, Prandt introduced in 1905 the concept of boundary layer. From a theoretical perspective, the boundary layer is a zone near the cylinder walls where we observe a drastic variation on the fluid’s velocity profile. So the viscous forces in this zone are no longer negligible. Before starting with the study, it is mandatory to understand the entity of the forces acting on a circular cylinder if subjected to a crossflow (also knows as Force Induced Vibrations or FIV). We can distinguish the FIV as following:
• Fluid elastic instability: they are generated from the interaction between the damping of the structure (the cylinder) and the forces exerted by the fluid: is a threshold phenomenon, so the instability occurs if a critical velocity is reached (the velocity which corresponds zero dampings);
• Turbulence induced excitation: there are random pressure fluctuations generated by a turbulent fluid; this kind of phenomena can be generated locally (near field excitation) and can cause long-term fretting wear damages;
• Periodic wake shedding : crossflow around bluff bodies such as cylinders, induces regular detachment and roll-up of the boundary layer, causing fluctuating forces on the cylinder itself: if the fluid density is high, the periodic vortex shedding can generate very large amplitude vibrations.
12 CHAPTER 2. PROBLEM DESCRIPTION Considering that a proper prediction in the flow parameters is mandatory before implement-ing any Fluid-Structure interaction model in codes, this study will focus only on the vortex shedding force prediction. As already explained, the vortex shedding induces the detachment of the boundary layer at the cylinder surface. Thanks to a pressure gradient on the cylinder surface, this detachment is observed where the shear stresses are equal to zero (considering a no-slip boundary condition at the walls):
(∂u
∂y)wall = 0 (2.1)
Where u is the component of the velocity vector in the x direction. The periodic (and sym-metric) separation of the boundary layer generates fluctuating lift forces on the cylinder. Let’s focus our attention on the map of regimes shown in Fig. 4:
Fig. 4: Vortex shedding regimes for the single cylinder case [20]
When the Reynolds number increases above 40, the cylinder wake starts to assume a particular pattern. The path begins to be unstable and generates the so-called Von Karmàn vortex street (from Theodore Von Karmàn, the first who observed this pattern in 1912).
2.1. THE VORTEX SHEDDING AND THE FORCE INDUCED VIBRATIONS 13 Before proceeding, we define the Reynolds number as:
Re = ρV D
µ (2.2)
Where ρ is the density of the fluid, V the free-stream velocity of the fluid, D a characteristic length and µ the dynamic viscosity. If the Reynolds number increases not beyond a value of 150, the phenomenon is strictly 2D. Above this value, two types of 3D instabilities are observed for different Reynolds number:
• A primary vortex core instability (or mode A instability) for Re ∼ 180/190 ; • A secondary shear layer instability (or mode B instability) for Re ∼ 230/260. For Reynolds numbers of 300 up to 3 · 105, the boundary layer is laminar; simultaneously, the
vortex street is fully turbulent (sub-critical vortex shedding regime) . Considering that the velocities range for the great part of the industrial problems is in this regime, this study will try to replicate the same conditions. The studies conducted on this phenomenon during the last years will be useful for the validation of our model; in particular, we will focus our attention on:
• DNS simulations (Dong et al. 2005 [10])
• comparison between DNS simulations and Particle Image Velocimetry studies (Dong et al. 2006 [9])
• comparison between DNS and LES simulations (Ma et al. 1998 [14])
• comparison between LES and RANS simulations (Rodi et al. 1997 [19] and Lubcke et al. 2001 [13])
14 CHAPTER 2. PROBLEM DESCRIPTION
2.2
The 2 cylinders cases
The study of the interactions between the wakes of 2 (or more) cylinders is the next step of this study. Thanks to the work of Zdravkovich (1986) [23], we can distinguish the different wake regimes function of the adimensional center to center distance T
D (where T is the gap distance
and D is the diameter of the cylinder) in both the configurations of the problem (tandem and side by side). Let us focus on the map of the regimes (as in the fig.5):
Fig.5 map of 2 cylinders’ regimes [23]
As it can be noted from Fig.5, for the side by side arrangment, there are the following regimes (P-SS or Proximity side by side arrangment) :
• If 1 < T
D < 1.2, the cylinders are so near that we observe only a single vortex street (PSS-A
regime in fig.5); • If 1.2 < T
D < 2.2, the situation is slightly different because the so-called bi-stable flow is
present in the gap; this particular path consists of the alternating from a narrow wake and a wide wake from one cylinder to another, at irregular time steps (PSS-B regime in fig.5); • If 2.7 < T
D < 5, the wake configuration for each cylinder is the same as for a singular
2.2. THE 2 CYLINDERS CASES 15 For the Tandem arrangment, the regimes are the following:
• If 1 < T
D < 1.8 For 1 < x < 1.8, the bodies are so close that a single vortex street is
observed (without any reattachment from the upstream cylinder to the downstream one, like the P-SSA case for the side by side arrangement, W-T1 regime in fig. 5);
• If 1.8 < T
D < 3.8, even in this case a single vortex street is observed. The difference with
the previous case is the boundary layer reattachment from the upstream cylinder to the downstream one. (W-T2 regime in fig.5);
• If T
D > 3.8, two vortex streets are observed (behind the wake of the bodies). There is a
binary behavior behind the downstream cylinder because two vortices are observed (one behind the downstream cylinder, the other one behind the upstream one, W-T (1+2) regime in fig.5).
As already explained in the previous section, to validate our results it is mandatory to compare them with other studies performed in the previous years. For the side-by-side configuration, the most crucial reference studies will be the experiments conducted by Sumner (1999) [22] and the LES simulations performed by Afgan [3] (2011, at Reynolds number of 3000) and Chen [8] (2003, at Reynolds number of 750). For the tandem arrangement, the references will be the studies performed by Alam [5] (2003, experiments performed at different pitches with Reynolds number of 6.5·104) and Carmo [7] (2006, a spectral element method study for a range of 160 < Re < 320).
Chapter 3
Reference experimental data
To compare our results with the experimental and computational studies previously mentioned, the choice of the main reference parameters will be an essential part of this study, even for a better physical knowledge of the phenomenon. The first property to be taken account is the frequency of the boundary layer detachment (and so the frequency of the fluctuating lift forces, acting on the cylinder) described by an adimensional parameter know as Strouhal number St:
St = f D
U (3.1)
where D is a characteristic lenght (in our case is the diameter of the cylinder), U is the free-stream velocity and f is the frequency of the vortex detachment. According to the experimental data (see, for example, the work of Shin, 1977 [20]), in the sub-critical vortex shedding regime (for Re < 2 · 105), the Strouhal number is almost constant and equal to about 0.2. It is also well
known the strong relationship between the Strouhal number and Reynolds number, supported by a considerable number of empirical correlations; according to Baracu (2011) [6], it is recommended to use, for Reynolds numbers of our interest, the so-called Roshko formula:
St = 0.212(1 − 21.2
Re ) (3.2)
The next parameters to be taken into account are strictly related to the forces acting on the cylinder; following for ease of reference the Starccm+ user’s guide [21], the following adimensional parameters can describe these forces:
18 CHAPTER 3. REFERENCE EXPERIMENTAL DATA • Drag force coefficient, defined as:
cd=
2Fd
ρ∞U02A (3.3)
where Fdis the total drag force acting on the cylinder, ρ is the reference fluid density, U0
is the free-stream velocity and A is a reference area of the bluff body. Even in this case, there is an empirical correlation that links the drag force coefficient with Reynolds number. According to the work of Baracu (2011) [6], the most useful one for Reynolds number in the range of 1 < Re < 2 · 105 will be the so-called Munson formula:
cd= 1.17 +
5.93 √
Re (3.4)
19 • pressure coefficient, defined as:
cpressure=
2(P − P∞)
ρ∞U02 (3.5)
where P is the static pressure in the point where the pressure coefficient is evaluated, P∞ is the static pressure in the freestream (far away from any kind of disturbance), P0
is the stagnation pressure in the freestream, ρ∞ is the density of the fluid and U0 is the
freestream velocity. The correct prediction of these parameters is essential for having an accurate description of the forces acting on the cylinder. Thanks to Ma (2000) [14] work, we can compare the results of this study with the following data obtained through DNS simulations (see Fig.7).
Fig. 7: Pressure coefficient distribution vs circumferential angle of the cylinder [14], circles denoted experimental data obtained by [15] at Re = 3000. The dash line are DNS data obtained
20 CHAPTER 3. REFERENCE EXPERIMENTAL DATA • Lift force coefficient, defined as:
cl=
2Fl
ρ∞U02A
(3.6)
where Fl is the total lift force acting on the cylinder, ρ is a reference density, U0 is the
free-stream velocity and A is a reference area. Considering the oscillatory trend of this parameter, it is appropriate to analyze its root mean square value and make some distinctions. According to Norberg (2003) [16], the sectional lift force coefficient c0
l(0)
is the root mean square value of the lift force coefficient for a very small cylinder segment (L
D −→ 0, where L is the lenght of the cylinder and D its diameter). So it is assumed that
this parameter is the lift force coefficient referred for a 2D simulation. As well as for the drag force coefficient, many empirical correlations link the sectional lift force coefficient with Reynolds number:
c0l(0) = 0.52 − 0.06x−2.6(f or5.4 · 103 < Re < 2.2 · 105) c0l(0) = 0.045 + 3x4.6(f or1.5 · 103 < Re < 5.4 · 103) (3.7) where x = log( Re 1.5·103)
Considering the 3D nature of the problem (remember the vortex shedding in the sub-critical regime, already mentioned), let us define the ratio between the root mean square lift force coefficient for a 3D simulation over the 2D equivalent (assuming homogeneity over the spanwise length) as:
γl = 1 lc [2 Z lc 0 (lc− s)Rll(s)ds]1/2 (3.8)
where Rll is the correlation coefficient, at zero time delay, between sectional lift forces
21 to [18], trying to obtain an empirical correlation for Rll(s) is pretty difficult, because
demand very complex instrumentation to measure it. However, a good approximation of this parameter can be calculated in other ways; according to [15] exist a strong relation between Rll(s) and the one-sided spanwise correlation length Λ :
Rll(s) = exp(
−s
Λ ) (3.9)
so it is possible to calculate the ratio γl as: γl= √
2
a [exp(−a) + a − 1]1/2(where a = lc
Λ). To
fin the Λ parameter is possible to use the following empirical correlations of Norberg [16] (2003): Λ = d · [7.6 · (1.72·10Re 3)0.6](f or 1.72 ·103< Re < 5.1 · 103) Λ = d · [2.6 · (2.4·Re5)−0.2](f or 8 ·103< Re < 2.4 · 105) (3.10)
The last properties to be taken into account are related to the flow pattern and to the separation point from the cylinder surface. Let us analyze the stream-wise velocity profile in the wake centerline of the bluff body:
Fig.8 Experimental data: centreline velocity normalized by the free-stream velocity at Re = 3900. Squares are data of Lourenco,Shih [20], circles of Ong,Wallace [17]
22 CHAPTER 3. REFERENCE EXPERIMENTAL DATA As it can be noted from fig.8, there is a specific adimensional distance x
D at which an inversion
of the sign in the stream-wise velocity distribution appears. From a physical point of view, this zone coincides with the so-called recirculation length or bubble length and is an excellent parameter to be considered in the calculations. Last but not least the correct prediction of the detachment angleis also relevant; according to the classical definition found in literature, the separation of the boundary layer around the cylinder surface is observed for a specific angle θ, if the condition:
(∂u
∂y)wall = 0 (3.11)
Chapter 4
Adopted models
4.1
Balance equations
The variation of a generic fluid property φ (velocity field, pressure, etc.) inside the reference frame in Fig.9 is calculated in the following format:
∂ρφ
∂t + div(ρφu) = div(Γgradφ) + Sφ (4.1)
where φ could be equal to 1 (for the conservation of mass), u v and w (for the momentum equations) or i(T0; h0) (for the energy equation). The other two terms are called Diffusion
term with diffusion coefficient(Γ) and source term (Sφ). Remember that int the problem
of our concern, we are not interested at this moment in the study of the energy balance equations, so that we will focus our attention only on the following:
24 CHAPTER 4. ADOPTED MODELS Fig.9: example of a control volume. The letter u,v and w are the component of the velocity
vector ~u ∂ρ ∂t + div(ρ~u) = 0 (4.2) ρDuDt = ∂(−P +τxx) ∂x + ∂τyx ∂y + ∂τzx ∂z + Smx ρDvDt = ∂τxy ∂x + ∂(−P +τyy) ∂y + ∂τzy ∂z + Smy ρDwDt = ∂τxz ∂x + ∂τyz ∂y + ∂(−P +τzz) ∂z + Smz (4.3)
The equations are also known as Navier-Stokes equations.Let us analyze the "degrees of free-dom" of the previous equations: as already explained, we are interested only in resolving mo-mentum and mass balance (4 equations). On the other hand, we have 9 unknowns to find (the three components of the velocity vector: u,v, w, and the elements of the so-called viscous stresses tensor τij). To obtain the problem closure it is mandatory to introduce some relation between
the velocity vector components and the viscous stresses. This relation is written as (assuming an isotropic and Newtonian fluid):
τij = λdiv(u) + µ( ∂ui ∂xj +∂uj ∂xi ) (4.4)
4.1. BALANCE EQUATIONS 25 where µ is the dynamic (or first) viscosity and λ is the second viscosity (which relates τij with
the volumetric deformations).The closure of the problem is guaranteed if the Stoke hypothesis:
λ = −2
3µ (4.5)
is taken. In conclusion, we need to specify that these equations are valid for a laminar fluid. If we also need to take account of the turbulence phenomena, further additional models are required. The next sections will introduce the physical description and the necessary models for describing the turbulence phenomena. As already explained in the previous chapter, we will focus our attention on the RANS (Reynolds avareged Navier-Stokes) models and the two equations (k-omega and k-epsilon) models.
26 CHAPTER 4. ADOPTED MODELS
4.2
The problem of turbulence and RANS approach
As already explained in the previous section, the study of a turbulent fluid is more complicated to model because of the other phenomena involved, which lead to modify the balance equations already shown in section 2.1. To be more specific, in a turbulent fluid, a massive number of vortices (also called eddies) with different sizes are observed, and the interactions between these vortices are not to easy to model. It is a transient and random phenomenon that increases friction and diffusivity phenomena, thanks to the so-called energy cascade, which transfers energy from the bigger vortices (or Large eddies) to the smaller ones (or Small eddies, where the turbulent energy is converted in to internal energy trough viscosity dissipation) .
Fig. 10: typical example of turbulent flow
It is possible to distinguish the Large eddies from the smaller ones? Referring to the similarity theories presented by Andrej Kolmogorov. Assuming the following hypothesis:
• For Re large enough, the motions of the small scales eddies are statically isotropic (Kolmogorov hypothesis of local isotropy): u0i2 = u0
j2 = u0k2 ;
• Through the energy cascade, all the geometric informations are lost. As a result, the small-scale eddies geometry is universal, independently from Reynolds number or boundary con-ditions;
4.2. THE PROBLEM OF TURBULENCE AND RANS APPROACH 27 Let us introduce the so-called Kolmogorov similarity hypotheses:
• First similarity hypothesis: For a sufficiently high Reynolds number, the statistic prop-erties of the fluid are uniquely determined by the kinematic viscosity ν (m2
s ) and the
turbulent kinetic energy dissipation rate (m2
s3 );
• Second similarity hypothesis: in the sub range L >> l >> η (where L is the lenght scale of the larger eddies and η the lenght scale of the small eddies), the statistical properties of the fluid are determined by the turbulent kinetic energy dissipation , independently on ν. From this hypotheses, it is possible to distinguish two different macro-regimes:
1. the inertial sub-range where the viscous forces are negligible, so the fluid undergoing in particular inertial forces;
2. the dissipation range where the viscous forces convert the turbulent kinetic energy into internal energy.
28 CHAPTER 4. ADOPTED MODELS Trough a dimensional analysis is possible to calculate the scales of motion (length, time and velocity) for the small eddies:
η = (ν3)1/4 uη = (ν)1/4 τη = (ν)1/2 (4.6)
Where η is the lenght scale, uηis the velocity scale and τη is the time scale. The relations between
the scales of motion for a small eddy and a large eddy are:
L η = ( ν U L) −3/4= Re3/4 (4.7) U uη = ( ν U L) −1/4 = Re1/4 (4.8) t τη = ( L U (UνL3)1/2 ) = Re1/2 (4.9)
Although this theory has a solid background for the study of turbulence, it presents some limits and drawbacks: for example, it works better for very high Reynolds numbers and does not consider the inversion of the backscatter phenomena (inversion of the energy cascade). We focus our attention on the effects of the turbulence in relation with Navier-Stokes equations. Considering the random and chaotic behavior of the phenomenon, the laminar fluid hypothesis will not be realistic anymore because it does not consider the apparent increase of properties such as diffusivity or viscosity in a turbulent fluid. It is possible to distinguish the different approaches for the turbulence, basing on the accuracy and the computational time required of the approach itself; generally they are classified as follows:
4.2. THE PROBLEM OF TURBULENCE AND RANS APPROACH 29 • Direct numerical simulations or DNS: with this method, the Navier-Stokes equations are solved directly inside the computational domain. It is the most accurate, but also the more time expensive method (the CPU time required is proportional to Re11/4) and
generally it is used for validation of experimental techniques;
• Large eddy simulations or LES: this method is less accurate than a DNS simulation but it is also more convenient in terms of the computational resources. These methods apply a filtering approach in order to separate the larger eddies from the smaller ones; after this filtering, only the larger eddies are directly solved, while the smaller ones are modeled; • Reynolds averaged Navier-Stokes or RANS: this method introduces a Reynolds
decomposition for the generic property φ:
φ = φ + φ0(t) (4.10)
where φ is the mean value of φ and φ0(t) is the fluctuating value. If we introduce an
averaging process, inside the generic Navier-Stokes equation:
∂ρφ
∂t + div(ρφu) = div(Γgradφ) + Sφ (4.11)
Considering the following properties for the averaging process:
cc0 = 0 c0= 0 c01c02 6= 0 (4.12)
30 CHAPTER 4. ADOPTED MODELS And introducing the Reynolds’ decomposition for the generic property φ, we obtain:
∂
∂t(ρφ) + div(ρφu) = div(Γgrad(φ)) − div(ρφ
0u0) + S (4.13)
From this last equation, we can notice that the averaging process introduces new terms (described by div(ρφ0u0)) , which can no longer guarantee the closure of the problem. This
term takes into account the momentum transfer between the eddies. From a physical point of view, in a turbulent fluid are present additional shear stresses, described by :
τijT = −ρu0iu0j (4.14)
this is also called Reynolds stress tensor components. To guarantee the problem clo-sure, additional models for the Reynolds stress tensor components are required. There are two different ways to obtain this condition:
1. Is it possible to model every component of the Reynolds stress tensor with a single equation (in this case, we will talk about Reynolds stress models or RSM). Con-sidering that six equations are required (one for each component of the tensor), we discard this approach because it will be too much time consuming;
2. Considering the analogy between the viscous stresses and Reynolds stresses, it is possible to assume the Boussinesq approximation:
Tlin = −ρvv0 = 2µTS −
2
3(µT∇ · v + ρk)I (4.15)
where S = 1
2(∇v + ∇v
4.2. THE PROBLEM OF TURBULENCE AND RANS APPROACH 31 where µT is called eddy viscosity.The great advantage of this approach is the need
to model only the eddy viscosity, so the number of equations required for the problem closure is remarkably lower than for the Reynolds stress models . This kind of model complexity is proportional to the number of partial derivative equations used to model the turbulent viscosity; the most used are the so-called: two equations models (the description of this models is postponed to the following sections).
The choice of URANS methods for our case is due to the lower computational time required for performing the simulation. On the contrary, these methods accuracy is lower in comparison with LES and DNS approaches. To prevent (at least in part) this lack, different additional models are required for our simulations to take into account problems such as the anisotropy of the Reynolds stress tensor components.
32 CHAPTER 4. ADOPTED MODELS
4.3
The two equations models
The assumption of isotropy of the eddy viscosity introduced by the Boussinesq approximation can guarantee, using the proper models, the closure of the Navier-Stokes problem for a turbulent fluid. Considering that the eddy viscosity is related to the mean flow, it is mandatory to understand the relationship between these two properties.The degree of complexity of the model is strictly linked to the number of partial derivative equations (PDE) employed in the model; we can distinguish between: :
• algebraic (or zero equations) models, if any PDE are employed; • one equation models;
• two equations models;
More PDE are employed, the more complicated will be the model itself. Let’s focus our attention on the two equations models (considering that are the most versatile and reliable): to taking into account the "memory" of the fluid, the eddy viscosity is modeled as a function of the turbulent kinetic energy k, defined as:
k = 1 2[u
02+ v02+ w02] (4.16)
where u0, v0 and w0 are the fluctuating velocities in the x, y and z direction. The second variable
is chosen arbitrarily to the method employed: • If the turbulent dissipation rate [m2
s3]is chosen, we will talk about k-epsilon models;
• If the specific dissipation rate ω[1
4.3. THE TWO EQUATIONS MODELS 33
4.3.1 The k-epsilon model
In the k-epsilon model, the eddy viscosity is a function of the turbulent kinetic energy k (already mentioned in previous sections) and the kinetic energy dissipation rate , defined as:
= ν(∂u 0 i ∂xk ∂u0j xk ) ∼k 3/2 l (4.17)
where l is the turbulent length scale. Then, the eddy viscosity is calculated as:
µT = ρ
cµk2
(4.18)
Where cµ = 0.09 from experiments performed with water and air. In addition, two transport
equations (one for the parameter k and the other one for ) are introduced and solved:
∂k ∂t + uj ∂ui ∂xj = τijT∂ui ∂xj − + ∂ ∂xj [(ν + νT σk ) ∂k ∂xj ] (4.19) ∂ ∂t + uj ∂ ∂xj = c1τijT ∂ui ∂xj − c2ρ 2 k + ∂ ∂xj [(ν + νT σ) ∂ ∂xj ] (4.20) σk= 1.00 σ= 1.3 c1 = 1.44 c2 = 1.92 (4.21)
34 CHAPTER 4. ADOPTED MODELS The k-epsilon model is one of the most used methods worldwide, thanks to its high reliability for a very high number of industrial cases. Considering the technique approximation, the k-epsilon model could present some unphysical results, particularly for problems that involve large pressure gradients or separation flows (such as our cases).
4.3. THE TWO EQUATIONS MODELS 35
4.3.2 The k-omega model
The k-omega model uses the same approach as the k-epsilon model.The eddy viscosity is a func-tion of the turbulent kinetic energy k and the specific dissipafunc-tion rate ω, which is the conversion rate of the turbulent energy in thermal energy (per unit of time and volume). So the eddy viscosity will be modeled as:
µT = ρ
k
ω (4.22)
Even for the k-omega model, two more transport equations are implemented and solved to guarantee the closure.
∂k ∂t + uj ∂k ∂xj = τijT ∂ui ∂xj − β∗kω + ∂ ∂xj [(ν + σ∗νT) ∂k ∂xj ] (4.23) ∂ω ∂t + uj ∂ω ∂xj = αω kτ T ij ∂ui ∂xj − βω2+ ∂ ∂xj [(ν + σT) ∂ω ∂xj ] (4.24) α = 59 β = 403 β∗= 1009 σ = 12 = β∗ωk (4.25)
This model has better performance for flows near the wall zones because it can switch from a vis-cous sub-layer formulation to using proper wall functions at different adimensional wall distances y+. In 1992 Menter formulated a "hybrid" approach: considering the better performances of the k-epsilon method far away from the walls and the standard k-omega version near the wall zones, the SST Menter k-omega method use some blending functions to switch from one mode to another one.
36 CHAPTER 4. ADOPTED MODELS
4.3.3 The 2 layer k-epsilon model
Let’s conclude this section with the last turbulence model implemented in this study. According to [21], the 2 layer k-epsilon model in a similar way to the SST Menter k-omega model, switches between a standard k-epsilon model to a model that solves the transport equation (PDE equation) for the turbulent kinetic energy and an algebraic equation for the epsilon parameter as a function of the adimensional distance from the wall y+. The dissipation rate , is calculated
ad:
= k
3/2
l (4.26)
Where the length scale function l is defined as:
l= f (y, Rey) (4.27)
and Rey = √
ky
ν is the wall distance Reynolds number. The Rey parameter, is used for the
definition of the following blending function α:
1 2[1 + tanh( Rey− Re∗y A )] (4.28) where: |∆Rey| atanh(0.98) (4.29)
4.3. THE TWO EQUATIONS MODELS 37 So, the turbulent viscosity µT is calculated as:
µT = (αµT)k−+ (1 − α)[µ(
µT
µ )2L] (4.30)
To model the length scale function, we refer to the Wolfenstein model:
l= cly[1 − exp(− Rey A )] (4.31) where : A= 2cl cl= kc3/4µ (4.32) and: cµ= 0.09 k = 0.42 (4.33)
Adopting this method, the "turbulent viscosity ratio" will be:
µT µ = Reyc 1/4 µ k[1 − exp(− Rey Aµ )] (4.34)
38 CHAPTER 4. ADOPTED MODELS
4.4
The non-linear constitutive equations
In the models described in the previous section, we assume the isotropy of the eddy viscosity through the so-called Boussinesq approximation:
Tlin= −ρvv0= 2µTS −
2
3(µT∇ · v + ρk)I (4.35)
where S = 1
2(∇v + ∇v
T)is the strain rate tensor
According to [13], this model cannot obtain good results if strong curvature streamlines are present, so they cannot realistically predict the fluid behaviour. To fix this problem without any help of higher methods such as LES or RSM simulations, a different way to model the eddy viscosity is required; in Starccm+ are implemented several models, also to consider the effects of the strain rate and vorticity tensors, trough quadratic (Tquad):
Tquad= Tlin−4c1µT k [S ·S − 1 3I(S : S)]−4c2µT k [W ·S +S ·W T]−4c 3µT k [W ·W T−1 3(W : W T)] (4.36) where W = 1 2(∇v − ∇v
T) is the vorticity tensor
and cubic (Tcubic) correlations:
Tcubic= Tquad− 8c4µT µ2 2[(S · S) · W + W T · (S · S)] − 8c 5µT k2 2(S : S − W : W T) (4.37)
4.4. THE NON-LINEAR CONSTITUTIVE EQUATIONS 39 where : c1 = (C CN l1 N l6+CN l7S3)cµ c2 = C CN l2 N l6+CN l7S3)cµ c3 = C CN l3 N l6+CN l7S3)cµ c4 = CN l4c2µ c5 = CN l5c2µ and: ca0= 0.667 ca1= 1.25 ca2= 1 ca3= 0.3 CN l1= 0.75 CN l2= 3.75 CN l3= 4.75 CN l4= −10 CN l5= −2
In the next chapter, we will see the advantages of using this kind of correlation; in particular, we will observe a significant improvement in predicting the forces (in the x and y direction) and pressure coefficients. Considering the need to use a RANS approach (due to the high number of volume cells of the discretized domain), these correlations (in particular, the cubic one) will be essential in this (and also for the future) study.
Chapter 5
Obtained results
5.1
The one-cylinder case
5.1.1 Introduction
Cross-flow around a bluff body (such as a cylinder) is a well-known problem since the beginning of the previous century (see, for example, the work of Prandt in 1905, as already explained in the previous chapters) and will be one of the leading problems for this study. In this chapter, we will start to face all the drawbacks to be taken into account for performing good simulations of the vortex shedding for a single-cylinder; as already explained in the previous sections (at least partially), the main problems are strictly related to:
• The need to perform 3-D simulations to have a realistic prediction of the forces acting on the cylinder. The choice is due to the shedding pattern in the sub-critical regime: for Reynolds numbers higher than 260, the so-called mode B instability (already shown in the previous chapter) is present. The differences between a 2-D simulation and a 3-D simulation will be more evident in the following sections.
• The need to have a suitable discretization of the problem. A fine enough mesh is mandatory to capture the smaller structures effects, but too much fine mesh could lead to a higher computational time for performing the simulation. For that reason, we have to conduct a mesh independency study (we will see the definition in the following sections).
42 CHAPTER 5. OBTAINED RESULTS • The need to have a good enough mesh refinement in the zone near the cylinder walls and
model the wall zone with an all y+ approach.
• The need to implement a suitable turbulence model. Considering that we should face a turbulent fluid (after detaching the boundary layer in the sub-critical shedding regime), it is mandatory to choose a proper model. The other issues to take into account for this kind of choice will be the possibility to implement additional models (such as quadratic or cubic correlations for the eddy viscosity) and near wall resolution (we will focus the attention only on those models where there is the possibility to use an all y+ approach).
• The need to choose a good enough time step and a proper time discretization scheme. If this condition is not satisfied, there will be a high probability of introducing numerical diffusion effects in the results, leading to poor outcomes.
As already mentioned, we decided to use version 11.06 of Starccm+, a commercial CFD code developed by CD-Adapco; this program most crucial feature is creating personalized field func-tions through a specific command window (this feature will be essential for the post-processing of this study).
5.1. THE ONE-CYLINDER CASE 43
5.1.2 Description of the numerical problem
This section introduces the geometry of the domain, the main settings and parameters used for performing the simulations, and the reason behind these choices.The reference domain is the same introduced in the Starccm+ user guide [21]:
Fig.12 : Domain geometry for the single cylinder case [21] (D is the diameter of the cylinder) The center of the cylinder will be the reference frame origin.For this (and future) studies, the cylinder diameter is taken equal to D = 0.01 m . The boundary conditions of the problem, are chosen as the following:
• At the inlet location (at a distance of 5D from the center of the cylinder), we decided to put a velocity inlet condition to control the fluid velocity.
• A pressure outlet condition is employed at the outlet location (at a distance of 20D from the center of the cylinder) to control the pressure easily inside the domain.
• In the wall regions ( for y = +5D , y = −5D and on the cylinder’s surface), a non-slip boundary conditionis used.
• In the bulk fluid zone, along the spanwise length, the symmetry condition is used. Before describing the simulation settings, it is mandatory to dedicate more attention to the domain discretization (or meshing) . This process is one of the most crucial parts of every CFD problem because it can heavily influence the results’ quality. The conditions to obtain
44 CHAPTER 5. OBTAINED RESULTS a "good mesh" require time and a good knowledge of the phenomenon. We need a mesh fine enough to bring to convergence the results with the lowerest numbers elements as possible (this condition is also called mesh independency). Considering that for our case there are different parameters to take into account (such as the turbulence model or the need to perform a 3D simulation), we will try to reach the mesh independency using the following conditions:
• Turbulence model: considering that we are interested only to obtain convergent results, it is not required to introduce some additional models (such as the cubic correlations for the eddy viscosity) that can lead to a higher computational time. For that reason, the k-omega SST modelwill be our choice because it is one of the most reliable turbulence models and, thanks to the blending functions, works well either in the bulk fluid either in the near-wall zones.
• 2D vs. 3D simulations: as already mentioned in the previous sections, the vortex shed-ding presents some 3D instabilities in the sub-critical regime. We can notice that, for a 2D simulation, there will be an overestimation of parameters such as the average drag coefficient cd and an overestimation of the recirculation length LR; on the contrary,
a 3D simulation predicts very well these parameters, according to the experimental data. The drawback of performing 3D simulations is the massive number of elements of volume generated during the meshing phase, which can lead to a higher computational time; con-sidering that up to now we are interested only to reach the convergency of the results (so an higher number of simulations, using less time as possible are required), 2D simulations will be performed for the mesh independency study.
• Reynolds number: the simulations will be perform for a Reynolds number of 3900; this choice is due to the many experiments and studies conducted at these conditions, so it will be easier to compare our results with the previous studies.
5.1. THE ONE-CYLINDER CASE 45 • Physical time of the simulations: the simulation should be running for enough time to calculate the time average properties such as the mean drag coefficient. A common rule for this kind of problem is running the simulations for a physical time equal to 250 times the convective unit time t = D
U0 (where U0 is the free-stream velocity and D the
cylinder diameter); for that reason, every simulation for the mesh independency study will be running for 7.2 seconds.
• Type of mesh: a polyhedral mesh will be used for the domain’s discretization. To have a suitable wall treatment, we will decide to use 15 prism layers (either in the cylinder zone either in the walls zones) with a stretching equal to 1.1.
Four types of meshes with different base sizes (uniform for the whole domain) will be tested (see Table 1); we decided to look at the convergency of the following properties: the average drag force coefficient cd, the adimensional recirculation length Lr/D (where D is the
cylinder’s diameter), the separation angle θsep, and the Strouhal’s number St (see the
previous sections for the definition of these properties).
type of mesh base size (m) numb. of elements cd Lr/D θsep (degree) St
1 0.003 7112 1.117 1.411 90° 0.193
2 0.0015 18116 1.392 1.392 90° 0.228
3 0.001 43138 1.665 0.858 84° 0.242
4 0.00075 46748 1.671 0.858 84° 0.242
46 CHAPTER 5. OBTAINED RESULTS Considering these results, we decided to use the "type 3" mesh (with a base size of 0.001 m) to perform our simulations; the necessity to perform 3D simulations with a certain spanwise length was already highlighted in the study of Lei (2001) [12], where it is demonstrated that it is mandatory to put a spanwise distance greater than two times the cylinder diameter, to have a right prediction of the force coefficients (see fig. 13).
Fig. 13 Drag and lift force coefficients trend for a spanwise lenght of Lz/D = 1 (up) and
5.1. THE ONE-CYLINDER CASE 47 As already mentioned, the need to perform 3D simulations (with a spanwise length equal to πD) leads to a considerable drawback: the number of volume cells inside the domain increases , so the computational time increases in consequencess. To fix (at least in part) this problem, we decided to introduce a so-called "refinement box" :a zone where the mesh is fine enough to catch the small structures (we chose to focus our attention in the wake of the cylinder, where we can observe the vortex shedding, see fig. 14).
Fig.14 : final mesh for the single cylinder case
We conclude this section by talking about the time step and the time discretization scheme used for our studies; to avoid problems of " numerical diffusion," a second-order temporal schemeis used. Regarding the time step used, we decide to apply the standard rule to take the Convective Courant number CFLbelow 1:
CF L = U ∆t∆x < 1
(where U is the free-stream velocity, ∆t is the time step applied and ∆x is the base size of the mesh) ; we applied a time step of 0.001 seconds for these reasons.
48 CHAPTER 5. OBTAINED RESULTS
5.1.3 Results
Considering the settings and the parameter shown in the previous section, we now analyze the results for the single-cylinder case; as explained in the same section, the need to perform 3D simulations to obtain the right force prediction is one of the main objectives of this study. To support this point, we can observe the results of Lei (2001) [12] in fig. 13 : the introduction of a spanwise length greater than two times the diameter (in this case, we can observe for a distance equal to 6 D) lead to a more "irregular" trend of the force coefficients (drag and lift). Fig.15 show the drag force coefficient trend using a k-epsilon two-layer turbulence model with the addition of a cubic correlation for the Reynolds stress tensor’s components (For a 3D simulation, we decided to use a spanwise length of π times the diameter of the cylinder)
Fig. 15 : Drag force coefficients trend for a 2D simulations vs. 3D simulations
As we can notice, for the 3D case the trend looks more similar to the data of the Fig.13 for a spanwise length of six times the cylinder diameter. Introducing a spanwise size, we can observe a further improvement in the quality results, even watching the averaged properties: for the 2D case, the average drag force coefficient cd∼ 1.529, while for the 3D case cd∼ 1.27(it is important
to remember that the main empirical data for calculating the average drag, are referred to the Munson formula: cd= 1.17 +√5.93Re ∼ 1.265for Re = 3900).
5.1. THE ONE-CYLINDER CASE 49 A similar approach can be used for the Lift coefficient trend:
Fig.16: lift force coefficients trend for a 2D simulations vs. 3D simulations
considering his oscillatory trend, we cannot rely on the classical definition of the averaging process. As already explained in section 1.4, a more useful way is the root mean square averaging concept. For a generic property x is definied as:
RootM eanSquare = (
PN i=1x2i
n )1/2
This definition is adopted for comparing our results with the ones obtained thanks to Norberg’s work (2002) [16]. The results obtained seem to be coherent with the study mentioned; however, there is an underestimation of the ratio γl defined in section 1.4 (in our research, γl∼ 0.315, and
according to Norberg [16], γl∼ 0.96); this is due to the model’s poor performance for Re < 8000
50 CHAPTER 5. OBTAINED RESULTS The next step of this study will be to show the results’ quality if a different turbulence model is employed. We will also use different ways to model the Reynolds stress tensor’s components (with the Boussinesq hypothesis, with a cubic correlation and with Reynolds stress model) to understand which one is the best choice, to obtain the right force prediction. To have a fair comparison with the data from [14], we will first analyze the Pressure coefficient distribution on the cylinder’s surface:
Fig. 17: Pressure coefficient distribution on the cylinder surface (3D simulations)
It is evident from Fig.17 that the introduction of a superior model than the linear one (with the Boussinesq hypothesis) gives a better prediction of the forces. It is unclear why the better results were associated with the k-epsilon 2 layer approach compared to the elliptic blending model (derived from the Reynolds stress model approach, so with a higher number of correlations to model the components of the Reynolds stress tensor); one of the most considered hypothesis is linked to a wrong implementation of the model in the Starccm+ code for our problem.
5.1. THE ONE-CYLINDER CASE 51 After these considerations, the force coefficients (drag and lift) analysis will be more clear. Fig. 18 shows the drag force trend with the different models already established in the pressure coefficient distribution study.
Fig.18 : drag force coefficient trend for different turbulence’s models (3D simulations) The trend looks like having a more regular pattern after a certain period without using the cubic correlation for the eddy viscosity. Analyzing the averaging properties, we can say that the average drag force coefficient cdhas a different value if a different turbulence model is employed
(cd ∼ 1.270 for the cubic version of the k-epsilon 2 layer model, cd ∼ 1.007 for the standard
version of the k-epsilon 2 layer model and cd ∼ 1.37 for the elliptic blending model). For the
same reasons already explained previously, these results confirm the better performances of the cubic version of the k-epsilon 2 layer turbulence model, with a good compromise from the point of view of the computational time required (see Tab.2).
52 CHAPTER 5. OBTAINED RESULTS Fig. 19 shows the lift force coefficient trend for the different turbulence models
Fig. 19 :lift force coefficient trend for different turbulence’s models (3D simulations) Even in this case, we can notice the different patterns if the cubic correlation for the eddy viscosity is employed. The differences are evident even in calculating the lift force coefficient root mean square (see tab.2). Considering that this study wants to put a solid background for future Fluid-structure interaction studies, it is mandatory to spend some efforts to optimize computational time.The main reason for using RANS instead of the LES approach is the lower time required for performing a simulation. The implementation of additional models for taking into account the Fluid-Structure interaction requires the fastest (and the most accurate) models possible; from tab.2, we can clearly say that the cubic version of the k-epsilon 2 layer model is the right compromise for our purposes.
5.1. THE ONE-CYLINDER CASE 53 In conclusion, although the RANS methods present some limitations, we can see that the presented results seem to agree with the experimental data and the empirical correlations already present in the previous sections. In future studies, we suggest testing other turbulence models with the proper correlations for the eddy viscosity, such as the k-omega SST (already used in this study for the mesh independency study) to demonstrate if is it possible to have a better force prediction.
Turbulence model cd c0l computational time (s)
"standard" k-epsilon 2L 1.007 0.0054 8789374.8 "cubic" k-epsilon 2L 1.270 0.0076 9422038.9
elliptic blending 1.378 0.0139 15586357.3
Reference data 1.265 * 0.009 **
Tab.2 : comparison of the different models employed (data for 3D simulations) * applying the Munson’s formula 3.4
Chapter 6
The Two-cylinder cases
The cross-flow study around a single bluff body put a solid background for future studies of the interactions between two (or more) bodies. According to Zdravkovich (1987) [23], if another cylinder is placed nearby a bluff body, the possible interactions will be more complicated to predict because there are many other phenomena to take into account. Fig.20 shows the map of the potential interferences between two wakes of two bluff bodies (as already shown in the previous sections):
Fig. 20: Definition of regions of flow interference for two pipes arrangements, the lines delimits the bi-stable flow regions [23]
56 CHAPTER 6. THE TWO-CYLINDER CASES As it can be noted, we can make some distinctions:
• If one of the body is not "submerged" in the wake of the adjacent one, we will be talking about proximity interference (the "P-S1, P-SS and PS2 regions" in Fig.20); the side-by-side configurations belong to this kind of interaction;
• If one of the cylinders is located in the wake of another cylinder, we will be talking about wake interference (the "W-SD and W-SG regions" in fig.20). This kind of interaction occurs only in the body located in the downstream position (if we are talking about tandem configuration)
• If the bodies are located at intermediate distances of the previous regimes, there is a possibility of a combination of wake interference and proximity interference (the "P+W" region in fig.20);
• If the bodies are located far away enough from each other, the interference is negligible, and every cylinder behaves as a single bluff body.
In the previous chapter, we described the different kinds of regimes for every wake interaction. We need to remember that our interest is focused only on actual components such as the U-tube bundle inside a steam generator. For that reason, we will focus our attention only on those distances involved in such systems. To study all the possible interactions as a function of the gap distance, we decided to perform the following simulations; for the "side-by-side" configuration:
• A simulation with a Reynolds number of 3000 and a gap distance of 0.1 D (where D = 0.01 m is the diameter of the cylinder);
57 and for the "tandem" configuration:
• A simulation with a Reynolds number of 6.5 · 104 and a gap distance of 0.1 D (where
D=0.01 m is the diameter of the cylinder);
• A simulation with a Reynolds number of 6.5 · 104 and a gap distance of 1.4 D
The reason behind the choice of these particular configurations is linked to our will to analyze the wake interactions for small gap distances, to understand which kind of phenomena could be involved inside a steam generator (where small pitches are present). Before starting with the description of the results, we need to remind the reference data to compare with them. For the tandem arrangement, Alam’s (2003) work [5] is the most important paper for this study: the experiments of that paper were performed with a Reynolds number of 6.5 · 104 and the main
parameter used in this study were recorded (such as pressure coefficient distribution for both cylinders, see fig.21-22). For the side-by-side configuration, we focused on the work performed by Afgan (2011) [3], where some LES simulations were performed with a Reynolds number of 3000 for different spacings. We need to specify that, for the side-by-side configurations, we will notice a so-called "bi-stable flow" where an alternance between narrow wake (NW) and wide wake (WW) is observed behind the tails of both the cylinders. It is mandatory to record the effects of this effect for both the bluff bodies, such as specified in fig.23.
58 CHAPTER 6. THE TWO-CYLINDER CASES
Fig.21: Pressure coefficient distribution (upstream cylinder) for different spacings [5]
Fig.22: Pressure coefficient distribution (downstream cylinder) for different spacings [5]
Fig.23: Comparisons for mean and fluctuating forces on upper and lower cylinders for various pitch to diameter ratios (The red line refers to the narrow wake NW, the black one to the wide wake WW) (T/D = 1 − 5). (a) cl, (b)cd , (c) c0l, and (d) c0d (- WW, - - NW). Case 1 Re = 3000
59
6.0.1 The two tandem cylinders cases
This section will start to analyze the wake interactions between two cylinders in a tandem configuration.The spacing between the cylinders is one of the most important parameters to consider for the two cylinders cases because it can affect the forces acting on the cylinders (see fig.24).
Fig.24: Effects on tandem spacing on the average drag force coefficient [5]
From this figure, we can better understand the choice of that particular spacing already men-tioned before: for adimensional gap distances L < 3.0 (the so-called critical spacing), the average drag for the upstream cylinder decreases if the spacing increase (because the afterlength of the body is higher). On the other hand, for the downstream cylinder, the average drag reaches a maximum for L = 1.4 and has negative values (there is a "forward thurst").
60 CHAPTER 6. THE TWO-CYLINDER CASES To understand the interaction between the wakes of the two cylinders (before to reaching the critical spacing), we decided to study two particular situations:
• When the cylinders are so close that may be considered as a single body (for L=0.1); • When the spacing is large enough to observe the wake interactions (for L= 1.4) .
Like the single-cylinder case, we decided to discretize our domain with a polyhedral mesh with a base size of 0.001 m (the so-called "type 3" mesh). To reduce the number of volume elements, we adopted a "refinement zone," where we have an interest in analyzing the shedding phenomenon. The mesh in the cylinder zone needs to be fine enough to predict the forces on the bodies; we decided to use the same settings for the single-cylinder case (see Fig.25).
Fig.25: near wall refinement mesh (2 cylinders in tandem configuration)
For the same reason as for the single-cylinder case, the simulations will be performed for a physical time of ∆t = 1.5 s and a time step of 0.001 seconds (the time discretization scheme will be of the second order). Considering the excellent results obtained for the single-cylinder case, we will use the k-epsilon 2 layer turbulence model (using a cubic correlation to model the eddy viscosity).We decided to focus our attention on the force coefficient comparison and on the topology of the vortices generated.
61
6.0.2 Results
Fig.26 shows the vorticity scene for a gap distance L/D = 0.1 D: we can notice that for this case, the observed shedding has the same topology as in the single-cylinder simulations, without any presence of boundary layer reattachment on the downstream cylinder.
62 CHAPTER 6. THE TWO-CYLINDER CASES
Fig.27: drag force distribution for 2 cylinders in tandem configuration (gap distance L/D = 0.1)
We can do the same consideration analyzing the force coefficient distribution. in Fig.27, it is appropriate to focus the attention on the drag coefficient trend for the downstream cylinder. From Alam’s experimet [5], it is clear that the downstream cylinder exerts a negative drag (already mentioned as "forward thurst" before), such as in our case. Fig.28 shows the lift force coefficienttrend for both the bodies; as it can be noted, the detachment of the boundary layer, which corresponds to a peak on the lift, have the same frequency, and so the same Strouhal’s number. We can confirm that this configuration behaves in the same way as a single-cylinder from these considerations. Finally, we will analyze the pressure coefficient distribution on the upstream cylinder surface, and we will compare it with the results obtained by Alam [5]. From this comparison (see fig. 30), we can notice that our results are in good agreement with that experimental data ;
63
Fig. 28: Lift force distribution for 2 cylinders in tandem configuration (gap distance L/D = 0.1)
64 CHAPTER 6. THE TWO-CYLINDER CASES If the cylinders are located at a certain distance, we will observe a boundary layer reat-tachmenton the downstream cylinder’s surface; this reattachment affects the pressure coefficient distribution itself. The main phenomena involved are much more complicated with respect to the previous case because, in the gap region, the so-called "forward shear layer" (the boundary layer separation from the downstream cylinder’s surface and the following shedding towards the upstream cylinder’s surface, see fig.31) is observed. For that reason, we will expect a distribution trend very different from the case of the upstream cylinder (see Fig.22). From the time average pressure coefficient distribution, we can predict the reattachment position and the influence of this reattachment in the pressure (and so in the forces) acting on the downstream cylinder sur-face. From Alam’s experience [5], it is clear that the maximum angle for the pressure coefficient distribution for the downstream cylinder coincides with the reattachment position of the bound-ary layer itself. The vorticity scene in fig.32 confirms the forward shear layer presence in the gap zone for our simulations, with a good approximation of the angle of reattachment, as we can see in the pressure distribution of the downstream cylinder’s surface (fig.33).
65
Fig. 32: vorticity scene for 2 cylinders in tandem (L/D = 1.4)
Fig.33 : Pressure coefficient distribution for the downstream cylinder (L/D = 1.4)
The results shown in this latter figure seem to agree with the results exposed in Alam’s work, although we can notice huge differences for an angle lower than 50 degrees.
66 CHAPTER 6. THE TWO-CYLINDER CASES
Fig. 34: drag force coefficients for 2 cylinders in tandem (L/D = 1.4)
Larger spacing effects also affect the force coefficient (drag and lift) trend differently from the previous case. From the drag trend in fig.34, we can notice a more oscillating trend for the down-stream cylinder, probably due to the continuing detachment and reattachment of the boundary layer in the cylinder’s surface. Analyzing the lift force coefficients trend in fig. 35, we can no-tice a larger amplitude for the downstream cylinder, associated with two different detachments occurring on the surface (backward and forward), leading to a higher lift force.
67
68 CHAPTER 6. THE TWO-CYLINDER CASES 6.0.3 The two side-by-side cylinders cases
The phenomena occuring with two cylinders in the so-called side-by-side configuration are much different compared to the previous cases. According to the Fig.20, we identify proximity inter-ference; considering a different kind of interaction and we need to focus our attention on other reference papers to understand if our study had been conducted correctly. The best candidate for our purposes is the work performed by Afgan (2011) [3], who focuses his attention on the influence of the gap spacing on the cylinders forces. Particular attention this configuration must be paid to is the phenomena involved for the intermediate gap spacings (1.1 < L/D < 2.5) where the so-called bi-stable flow occurs:
Fig.36: classical example of "bi-stable flow"
From fig.36, we can have a better idea of this phenomenon. Considering the wake of the upper cylinder as a reference, we can notice the alternating narrow wake (NW in the figure) and a wide wake (WW in the figure). The effects of this unstable behavior can be observed in fig.37:
Fig. 37: Lift coefficient trend for 2 cylinders in side-by-side configuration [3]
If compared with the single-cylinder case, the side-by-side configuration shows a higher ampli-tude. The other relevant aspect is the substantial variation of the amplitude observed when there
69 is a switch between "mode one" and "mode two" due to the different recovery pressure between the narrow wake and the wide wake. By the way, this phenomenon will be triggered after a very long period (as it can be noted from fig.37, mode one will be activated after 1000 seconds, and we will observe the mode switch after 2000 seconds. Considering the need to perform 3D simulations (already explained in the previous chapters) and so the massive number of volume elements in our domain, we cannot observe this phenomenon for our cases. In this work, we decided to study the proximity interaction effects without any presence of bi-stable flow to understand if the adopted models will be good enough for future studies. The adopted mesh has the same properties as the mesh employed for the tandem configuration (including the mesh in the near-wall zones).
Fig. 38: meshing domain for the 2 side-by-side cylinders case (L/D = 0.1)
The simulations are performed with a Reynolds number of 3000 (The same used in Alam’s work) and a time step of 0.001 seconds, with a second-order discretization for the time; for the same reasons mentioned before, the physical time of the simulation is 7.2 seconds.
70 CHAPTER 6. THE TWO-CYLINDER CASES The turbulence model employed is the same k-epsilon 2 layer using a cubic correlation for the turbulence intensity, already used for the previous cases. To highlight the influence of the gap spacing, we decided to perform the following simulations:
• With a gap distance of L/D = 0.1 (to understand the force trend without the gap interac-tions) ;
• With a gap distance of L/D = 1.5 (to understand the effects of greater spacings between the cylinders ).
Fig. 39: Flow visualization of two side-by-side circular cylinders in steady cross flow (L/D = 1 and Re = 1320) [22]
71
6.0.4 Results
The results for the case with a gap distance L/D = 0.1 is similar to ones of the the simulation with a tandem configuration and the same spacing; the vortex street generated has the same structure (although the average drag force coefficient is higher) as the single-cylinder case (see fig.40). These results agree with the experimental flow visualization performed by Sumner (1998) [22] through PIV methods.
72 CHAPTER 6. THE TWO-CYLINDER CASES Fig. 41 shows the drag force coefficient trend for a gap spacing L/D = 0.1. The trend seems to be equal for both the cylinders. However, the cylinder placed on the lower side of the domain shows a slightly lower average drag force coefficient (cd ∼ 1.67 vs. cd ∼ 1.92 than the
cylinder located on the domain superior side). It is interisting to notice that these values result relatively larger than ones for the single-cylinder case (cd∼ 1.27for a Reynolds number of 3900).
73 Fig.42 shows the lift force coefficient trend for gap spacing L/D = 0.1; we can see that both the cylinders, if placed at this distance, seems to behave as a single cylinder. The peaks shown in this plot, which correspond to the boundary layer detachment, demonstrate a frequency (and so a Strouhal number) almost equal, such as the single-cylinder case. Also in this case, it is interesting to note the root mean square values of the lift force coefficient for both the cylinders (c0
l ∼ 0.0148for the cylinder located in the superior part of the domain vs. c 0
l ∼ 0.0207for the
lower one); these results are essential for the future comparison for the next case.
