• Non ci sono risultati.

CFD modelling of stratified two-phase flow for oil and gas transportation

N/A
N/A
Protected

Academic year: 2021

Condividi "CFD modelling of stratified two-phase flow for oil and gas transportation"

Copied!
69
0
0

Testo completo

(1)

Universit`

a degli Studi di Pisa

Laurea Magistrale in Ingegneria Energetica

CFD modelling of stratified two phase flow for

oil and gas transportation

Autore:

Gabriele Chinello

Supervisori: Prof. Ing. R.A.W.M. Henkes Ing. Marco Antonelli Prof. Ing. Luigi Martorano

(2)

Contents

1 Introduction 1 2 Background 3 2.1 Literature review . . . 3 2.2 Fluent results . . . 10 2.3 Objective . . . 14 3 Turbulence modelling 20 3.1 Two phase flow in channel . . . 20

3.2 Eddy viscosity assumption . . . 22

3.3 Turbulent kinetic energy equation . . . 23

3.4 Turbulent length scale . . . 25

3.5 Wall boundary conditions . . . 26

3.5.1 High Reynolds model . . . 28

3.5.2 Low Reynolds models . . . 29

4 Numerical method 31 4.1 Grid . . . 31

4.2 Discretization . . . 34

4.3 Numerical solution procedure . . . 36

4.3.1 Single phase flow . . . 36

4.3.2 Two phase flow . . . 38

5 Single phase flow results 40 6 Two phase flow results 46 6.1 Laminar case . . . 46

6.2 Turbulent case . . . 48

6.2.1 Validation for the classical k− ω model . . . . 48

6.2.2 Modified k− ω model . . . 51 7 Conclusions 58 References 59 Appendices 63 A Lockhart-Martinelli parameter 63 B 1-D model 65

(3)

1

Introduction

Stratified two-phase flow is a flow regime typically encountered in gas transportation pipelines, where a small amount of hydrocarbon condensate, and in multiphase oil/gas pipelines. Due to gravity, the heavier fluid settles in the lower region of the pipe and the lighter fluid settles in the upper region. They are separated by a continuous interface that can be smooth or wavy, depending on the mutual velocities of the two phases and the pipe inclination (fig. 1, SS and SW). Among all the different gas-liquid flow patterns, stratified flow is perhaps the simplest and can be considered as a starting point to evaluate transition to the other flow regimes. The key parameters of gas-liquid flow are the superficial velocities of the phases, the pressure gradient, and the liquid or gas holdup. For safe and optimal operation of the pipeline, it is crucial to carry out accurate predictions of the pressure gradient and holdup as the flow rates are generally input quantities. This is the main objective of the modelling. The early studies on stratified two-phase flows focused on the determination of semi-empirical correlations, e.g. Lockhart and Martinelli (1949), Chisholm (1967). In these studies, the fric-tional pressure gradient is obtained by means of two-phase multipliers that relate the single phase condition with the two-phase condition pressure drop, see appendix A. The accuracy of the semi-empirical correlations is not always satisfactory, and it depends largely on the operating conditions for which the correlation was developed.

A more detailed approach is to average over the pipe cross section area the local instanta-neous momentum balances of the two phases, see appendix B. In this way, a one-dimensional model is derived at the expense of losing the information in the direction normal to the main flow. Therefore, in 1-D models the main issues are related to the formulation of closure relations for the shear stress, especially at the interface. Many studies were focused on the derivation of empirical relations for inter facial and wall shear stress. A review of them is provided by Espedal (1998). The purpose of the contemporary studies is to develop closure relations based on turbulence modelling, in order to reduce the inaccuracies and loss of gener-ality due to the adoption of empirical closure relations, e.g. Biberg (2007). One-dimensional models are largely used in oil-gas industries, and they are employed in commercial codes to simulate pipeline systems. It is clear that nowadays this is the only practical solution for simulating long pipelines (in the order of several km in length) due to the extremely high cost associated with CFD multi-dimensional simulations. Although one-dimensional models generally achieve fast and reliable predictions, they present limitations where a detailed flow field computation is necessary (3-D geometry effects, V-section, complex geometry, thermal effects etc.). Another issue is related to low liquid loading in slightly inclined pipe as multi-ple solutions of the hold up equation are found and shear stress relations are unsatisfactory, Grolman and Fortuin (1997), Birvalski et al. (2012), Birvalski et al. (2014a).

In these cases, CFD becomes an interesting tool since it can help to improve the existing models, or it can be directly employed in combination with 1-D codes. This last case is quite attractive. In fact, detailed flow field computation would be achieved only where needed, thus maintaining reasonable computational time. Nevertheless it is not well established if contemporary CFD codes can lead to a satisfactory prediction of stratified two-phase flow quantities. The keystone of the modelling is to correctly represent the turbulent behaviour of the interface, how it deforms and affects the momentum transfer between the phases. The objective of this thesis is to validate the results of computational fluid dynamics in the mod-elling of stratified two-phase flows.

The outline of the present thesis is as follows. In section 2.1, a review of the CFD studies on stratified two-phase flow is performed. Then, in order to investigate the actual capability of commercial packages, two-phase flow in a channel is simulated with RANS model through Fluent 13.0. The results of the simulations are reported in section 2.2 and compared with the experimental results of Fabre et al. (1987). The following sections are all dedicated to

(4)

Figure 1: Comparison of Taitel and Dukler (1976a) theory, solid lines, and Mandhane et al. (1974) experimental results, dashed line. Air-water flow in 0.025 m diameter pipe at 25◦C and 1 atm. US

L and UGS are respectively the liquid and gas superficial velocity.

the development of a RANS code (with Matlab) able to simulate steady state fully developed two-phase flow in a channel. The goal of the code is to overcome the issues experienced with the classical turbulence models when simulating stratified-two phase flow. From section 3.1 to section 4.3 the steps followed to develop the code are described. In section 5, a validation for single phase flow is performed against the DNS data by Hoyas and Jim´enez (2006). Section 6 reports the code results for two-phase flow and their comparison with the experimental results by Akai et al. (1980) and Fabre et al. (1987). Section 7 presents the final conclusions and raccomandations.

(5)

2

Background

2.1 Literature review

A great number of studies were conducted to investigate stratified multiphase flow behaviour, and it is difficult to make a precise classification. However, as suggested by Mouza et al. (2001), three categories can be identified:

Experimental studies: Experiments are carried out in pipe or channel. Initially only pressure drop and liquid holdup were measured, leading to empirical correlation. Later more attention was given to the mechanism and details of the flow.The velocity profile was measured, shear stress and wave characteristic were investigated (e.g. Fabre et al. 1987, Lorencez et al. 1997, Strand 1993, Espedal 1998, Birvalski et al. 2014b).

Phenomenological models: Physically realistic models are developed relying on the dominant mechanisms, e.g. Taitel and Dukler (1976a), Hart et al. (1989). The phases are considered as one-dimensional bulk flows, so a cross sectional velocity description is neglected. Empirical relations link the bulk velocities with the shear stress at the walls and interface.

Detailed flow field computations: The aim of these studies is to compute details of the flow field such as velocity profiles, shear stress distribution, turbulent quantities. Turbulence is modelled and the effects of waves are taken into account by using interface conditions.

In this thesis, attention is given to the studies belonging to the third category, detailed flow field computations. Between these, one of the first and most relevant work is that by Akai et al. (1981). In this, RANS equations were solved for a two-phase fully developed steady state flow between two parallel flat plates. The results were then compared to experimental data of air-mercury channel flow by the same researchers (Akai et al. 1980). The turbulence was modelled with a modified version of the k−  low Reynolds model, Jones and Launder (1972) (1973). Volumetric flow rate of gas and liquid was chosen as the input value in the numerical procedure; thus holdup and pressure gradient were the output quantities. The equations (momentum, turbulent kinetic energy, turbulent kinetic energy dissipation rate) were solved separately for each phase, continuity of the shear stress and velocity were enforced at the interface. Two different sets of boundary conditions for k and  were employed at the interface, depending on its nature, smooth or wavy. Considering the smooth interface case, the same boundary conditions as for a smooth wall were adopted

kk= 0 k = 2νk

 ∂√kk ∂y

2

where ν is the kinematic viscosity, and the subscript k indicate the phase (liquid k = l or gas k = g). Considering the wavy interface case, the kinetic energy value at the interface was assumed dependent on the interface displacement velocity in the vertical direction.

kk = β  2∆h T 2 k= 2νk  ∂√kk ∂y 2

Here β is a numerical constant, ∆h and T are the wave height and the wave period, respectively. ∆h and T were evaluated as empirical functions of the gas and liquid Reynolds number. In addition (for the the wavy interface) a ’ wall function’ for the gas velocity in proximity of the interface was adopted. The wall function was the same for turbulent flow over a rough wall

(6)

u+= 1 κln

y− yi ks

+ C

where κ = 0.4, ks = ∆h/2 and C was chosen equal to 5.2 in order to fit the experimental logarithmic velocity profiles. The meaning of these boundary conditions is that the waves act as a source of kinetic energy for the liquid phase, whereas for the gas phase they induced flow separation that results in an increment of the shear stress. For the wavy cases, up to Reg = 2× 103 (fig. 2), holdup and pressure gradient were found in good agreement with experimental data. On the other hand, for the smooth cases, the pressure gradient was well predicted only for low gas flow rates, therefore when the interface is completely smooth and far from transition. For high flow rates, a deviation between computed and measured pressure gradient was observed.

Figure 2: Experimental flow map for horizontal air-mercury flow in channel, Akai et al. (1980). − − −,proposed line by Mandhane et al. (1974). Reg= Ub,g(H− hl)/νg , Rel= Ub,lhl/νl.

Issa (1988) extended the work of Akai to flow in inclined pipe and 2-D channel, re-writing the equations in a bi-polar coordinate system. For Re > 104 he used a high Reynolds model, Jones and Launder (1972); in other cases a low Reynolds model was employed. The low Reynolds model was the same as the one adopted by Akai with the only difference of  boundary condition in the wavy case, for which a symmetry condition was assumed

∂ ∂η = 0

where η is the direction normal to the interface in the bipolar coordinate system. This alternative was justified by the author with the unrealistically high level of turbulent viscosity, which yields the former boundary condition:  = 2ν∂√k

∂y 2

. In fact, the gradient of√k may vanish at the interface thus leading to very small values of the dissipation , and consequently high values of the turbulent viscosity, µt∼ ρk

2

 . Calculations were made for both channel and pipe flow. For the channel case results were compared with the experimental and numerical results of Akai, but in contrast with Akai, smooth boundary conditions were also employed for 2000 < Reg < 5000, see fig. 2. It was found that the effects of 2-D geometry are negligible in the channel case, thus demonstrating the validity of the one-dimensional approach operated

(7)

by Akai. A reasonable match of the pressure gradient and hold up was obtained, except for low Reynolds gas number where pressure drops were over predicted. This suggests that the choice of Akai to use wavy boundary conditions for 2000 < Reg< 5000 was correct and that the use of smooth boundary conditions should be restricted to a completely smooth interface. In 1997, Lorencez et al. performed both experimental and numerical computations of air-kerosene flow in a channel. It was shown in the experiment that the mechanism of momentum transport from gas to liquid is mainly due to the generation of high-momentum carrying lumps of liquid, which move from the interface to the bulk of the liquid increasing the turbulent fluctuations. It was seen that the wavy motion increases the velocity fluctuation in the vertical direction v0. The same was for the velocity fluctuation in the stream-wise direction u0. However in this latter case the increment was believed to be mainly due to the non-uniform distribution of the shear stress on the windward and leeward of the waves, rather than to the wave motion itself. Concerning the numerical procedure, 2-D equations for steady state fully developed channel flow were solved. Turbulence was accounted by the same models in the work by Issa (1988), and only the wavy case was taken into account. Both boundary conditions for k and  in the liquid phase were changed respect to Akai and Issa. Another term was introduced in the kinetic energy inter facial boundary condition, due to the contribution of the stream wise velocity fluctuations. Moreover the dissipation was imposed equal to the boundary condition for an open channel by Celik and Rodi (1984)

kl= β  2∆h T 2 + ρG ρl  u2 τ,ig l= k3/2l ahl

where a is a constant. The equations were solved iteratively by a SIMPLER algorithm. Velocity and kinetic energy profile were found in good agreement with experimental data. It is reported that the holdup and pressure drop prediction matched the experimental results for less than 15 %, however the comparison is not shown.

In 2000, Newton and Behnia modelled stratified smooth gas-liquid flow in a pipe and in 2001 they extended to stratified wavy flow. Their work can be seen as a continuum of the work by Issa (1988). The governing equations were transformed into bi-polar coordinate system. The turbulent viscosity was calculated with a k− low Reynolds model, and damping functions by Lam and Bremhorst (1981). Some minor modifications to the damping functions were made to account for the non-circular flow geometry in each phase. For the smooth case, the following boundary conditions at the interface were applied

kk= 0

∂k

∂η = 0

For the wavy case the interface turbulent kinetic energy was chosen in accordance with the study by Rodi (1984) on wind induced shear stress at the free surface

kk(ξ) = τi(ξ) ρkpCµ

∂k

∂η = 0,

where Cµ= 0.09 is a k−  model closure coefficient, τi(ξ) is the shear stress distribution at the interface, and it was imposed empirically a priori. Numerical results were compared with experimental data of an air-water flow in a horizontal pipe, 50 mm diameter and 16 m long (Newton and Behnia 1996). The pressure gradients were over predicted with an absolute average error of 20 %. Liquid holdup scored slightly under the prediction of the model, and the absolute average error was 19 %. Three mechanistic models were also employed (Taitel and Dukler 1976a, Andritsos and Hanratty 1987, Spedding and Hand 1997). The last model shows the best results. However, it was an overall agreement of mechanistic models and the CFD calculations.

(8)

A work that differs from all the previous is the one by Berthelsen and Ytrehus (2005), and Berthelsen and Ytrehus (2007). They studied smooth and wavy stratified flow in pipes, introducing two main differences regarding the turbulence model and the numerical treatment of inter facial boundary conditions. The turbulence model was a revised version of the Chen and Patel two-layer model (1988) . This does not require boundary conditions for the dissipation  since its limiting behaviour is already defined by the one-equation k− l model that is turned on in the proximity of the interface and at the walls (inner layer). Moreover, the model was modified in order to treat the wavy interface as a rough surface, without the need of ’wall function’. With these two features, it is necessary to only specify the surface roughness and the turbulent kinetic boundary conditions at the interface. The surface roughness was linked to the friction velocity with the expression proposed by Charnock (1955)

Rs= β u2

τ,i g

where β is the Charnock parameter, which was tuned according to the experimental results by Espedal (1998). The boundary conditions for the turbulent kinetic energy at the interface were taken equal to Newton and Behnia (2001), but this time the shear stress was estimated from the flow field, and not imposed a priori. For what concerns the numerical treatment of the interface boundary conditions, the immersed interface method developed by Berthelsen (2004) was implemented. This solve arbitrary shape interface, unlike the body fitted grid usually adopted. The numerical predictions were in good agreement with the experimental results by Espedal (1998). It was found that the choice of the Charnock parameter affected the calculated pressure gradient and holdup, but the deviation was not dramatic. As reported previously by Newton and Behnia (2001) also in this case the level of turbulent kinetic en-ergy on the gas side was founded too high. This suggests that different interface boundary conditions for the turbulent kinetic energy could be implemented.

The studies presented until now followed the ”Two-fluid approach”. One set of equations is solved in each phase, continuity of velocity and shear stress is imposed at the interface, boundary conditions for the turbulence quantities are applied at the interface. A list of this type of studies is reported in Table 1. Between these only de Sampaio et al. (2008) used the k− ω model. Meknassi et al. (2000) extended the work of Lin´e et al. (1996) to circular pipes. Both account also for secondary flow in the gas phase, with an algebraic stress model. Mouza et al. (2001) used the commercial software CFX instead of an in-house solver. Despite the differences that can be found between these studies, it is clear that the main issue is related to the correct specification of the boundary conditions for the turbulent quantities at the interface. In this sense, there aren’t widely accepted values and an ad-hoc tuning is necessary to fit the experimental results.

A different approach is the ”One fluid”, where a single set of equations is solved for both phases, and the interface is captured by a marker function. Holm˚as et al. (2005) used the commercial CFD code COMET to simulate stratified two-phase flow in a channel with the VOF method. Both smooth and wavy case were investigated, and the results were compared with the experiment by Akai et al. (1980) and Lorencez et al. (1997). Periodic boundary conditions were adopted, imposing a fixed pressure gradient. In the smooth case, gas and liquid flow rates were largely underestimated by both the turbulence models employed, k− ω MSST and k−  RNG. It was found an overestimation of the turbulent kinetic energy close to the interface, and an upward shift of the maximum in the gas velocity profile. In the wavy case, the gas flow rate was too low, whereas the liquid flow rate was too high. As in the smooth case the turbulent kinetic energy was overestimated at the interface and too high upward tilt in the gas velocity profile was observed. In the liquid phase, the kinetic energy was underestimated everywhere except close to the interface, where it was overestimated. This

(9)

explains why the liquid velocity profile was not flat, but similar to the Coutte flow. In fact, a flat velocity profile is the result of an enhanced turbulent kinetic energy level due to wave motion. Moreover, it is supposed that the turbulent kinetic energy overestimation results in too high shear stress at the interface, instead the under prediction of the turbulent kinetic energy in the liquid phase causes a too low shear stress at the wall. This may explain the mismatch found between predicted and experimental flow rates. The conclusion of Holm˚as et al. was that the turbulence model has to be modified in order to correctly predict the turbulence at the interface.

The same conclusion was drawn by Terzuoli et al. (2008). They used the commercial codes CFX 10.0 and Fluent 6.1 to simulate the experiment by Fabre et al. (1987). Only the smooth case experimental results were taken into account. The VOF method was employed, the entire channel was simulated and constant velocities of the phases were applied at the inlet. Constant gas pressure and liquid hydrostatic pressure were imposed on the outlet section. The predicted liquid level was about 24% lower than the experimental value, thus the gas velocity was under predicted and the liquid velocity was over predicted. An upward shift of the maximum gas velocity was observed. Due to these results, it was suggested that the frictional drag between the two phases was overestimated. A single phase analysis was also carried out. The channel was divided according to the experimental value of the liquid level and the interface was simulated with a moving wall. The velocity of the moving wall was imposed according to the experimental value. Calculations were made in 2D and 3D geometry. It was found that 2D calculations underestimated the velocity values of about 10% of the 3D ones. The velocity profile was well simulated by the single phase calculation. However, the pressure gradient predictions were not reported.

Lo and Tomasello (2010) performed simulations of gas-liquid flow in pipes with the com-mercial code Star-CD using the VOF method. Three turbulence models were tested, k−  standard, k− ω, k − ω SST in all cases wall functions were adopted. The effects of surface tension were also accounted for. Simulated pressure drop and liquid level were compared with the experimental results of Espedal (1998). The same trend for pressure drop and liquid level was found. However, the pressure drop was overestimated and the liquid level was underesti-mated in all cases. To solve this problem an artificial damping of the turbulent kinetic energy in the region close to the interface was adopted. As proposed by Egorov (2004), an additional source term was introduced in the standard ω equation formulated by Wilcox (1998)

∂ω ∂t + uj ∂ω ∂xj = αω k τij,T ρ ∂ui ∂xj − βω 2+ ∂ ∂xj  (ν + σνT) ∂ω ∂xj  + S (1)

where α = 5/9, β = 0.072, σ = 0.5 are the k− ω closure coefficients, τij,T is the Reynolds stress tensor. The idea is to balance the destruction term−βω2 with a positive similar term, S, in order to increase the value of ω at the interface thus dampening the turbulence νT = ωk. The source term assumed the following form

S = A∆nβ(Bωwall)2 A = 2r|∇r|

here r is the volume fraction, A is the interface area density and works as the activation term of S (A = 0 outside the interface), ωwall is a function that describes the asymptotic behaviour of ω near a smooth wall (N.B. It is valid only when y+ < 2.5). The ω

wallformulation has been derived by Wilcox (1998) with a perturbation analysis of the boundary layer

ωwall= 6µk βρk∆n2

(10)

where ∆n is the grid size in the direction normal to the interface, B is a tunable parame-ter which deparame-termines the damping strength, ρk and µk are the liquid (k = L) or gas (k = G) density and viscosity. As formulated above, the source term performs a symmetric damping of the turbulence, enforcing the same value of ω would be near a wall. The main issue of such a formulation of the source term is that it is mesh dependent because of ∆n, and because the zone where A6= 0 is determined by the mesh. Moreover, the choice of the tunable parameter B is almost a tentative affair and its value changes with the mesh. Egorov (2004) suggested B=100. In their work Lo and Tomasello chose B=2500 to match the experimental results. They found a sensible improvement in the predictions of holdup and pressure gradient in respect to the non-damped cases. As specified by the authors, the additional term corrects the interface turbulence only when a strong velocity gradient is detected. For this reason, and for the limitations exposed earlier further improvement seems to be necessary for the proposed damping.

Some of the major CFD works on stratified two-phase flow have been briefly described above.A list of the studies available in the literature is reported in table 1.It can be con-cluded that a robust method to simulate stratified two-phase flow has not been developed yet, and further investigation is needed.

(11)

Approach Pipe Channel – Issa (1988), k− 

– Meknassi et al.

(2000),k − 

modi-fied for turbulence

anisotropy

– Newton and Behnia

(2000)(2001), k−  – Mouza et al. (2001),

CFX, k− 

– Berthelsen and Ytrehus (2005)(2007), k− 

– de Sampaio et al.

(2008), k− ω

– Duan et al. (2014), k− 

– Akai et al. (1981), k−  Low Reynolds modified – Issa (1988), k−  – Lin´e et al. (1996), k− 

modified for turbulence anisotropy – Lorencez et al. (1997), k−  – Mouza et al. (2001), CFX, k−  – Yao et al. (2005), CHATARE, k −  modified

– Gao et al. (2003), Two

layer model k−  RNG

+ k−  Low Reynolds

– Ghorai and Nigam

(2006), Fluent 6.0, Two layer model k− RNG+ k−  Low Reynolds – Lo and Tomasello (2010), Star-CD, k−  standard, k − ω, k− ω SST – Egorov (2004), CFX, k− ω – Holm˚as et al. (2005), Comet, k−  RNG k − ω M SST

– Banerjee and Isaac

(2006), Fluent 6.1, k−  RNG – Terzuoli et al. (2008), Fluent 6.1 CFX 10.0, k− ω SST – Sawko (2012), Open-Foam, k− ω

– H¨ohne and Mehlhoop

(2014), AIAD k− ω

Table 1: List of some major works available in the literature where stratified two-phase flow is simulated with RANS.

(12)

2.2 Fluent results

In order to assess the performance of the current commercial available CFD software in the study of two-phase flow, simulations with Fluent 13.0 were carried out. Stratified two-phase flow in a channel was simulated since for channel 2-D calculations can be conducted as first approximation. The simulations results have been compared with the experimental data by Fabre et al. (1987). The experiments were performed in a channel of length 12.60 m (L), height 0.1 m (H), width 0.2 m (W) and thickness 0.2 m. The channel had a downward inclination of 0,1 %, which correspond to an angle of θ = 0.0573◦. The working fluid were water and air at ambient pressure and temperature. Three different experiments with a constant water flow rate and an increasing air flow rate were performed. The operating conditions for each experiment are reported in table 2.

Figure 3: Wavy stratified two-phase flow in rectangular cross-section channel. Adapted form Lin´e et al. (1996). Run Ql(m 3 s ) Qg( m3 s ) hl(m) dp/dx( P a m) Ul(m/s) Ug(m/s) Rel Reg 250 0.003 0.0454 0.0380 -2.1 0.395 3.66 43100 22900 400 0.003 0.0754 0.0315 -6.7 0.476 5.50 45300 37100 600 0.003 0.1187 0.0215 -14.8 0.698 7.56 49000 56400

Table 2: Experimental results for stratified air-water flow in a channel. Fabre et al. (1987). In table 2 the liquid and gas velocities have been calculated dividing the volumetric flow rates by the cross section area occupied by each phase, Ul = Ql/(hlW ) and Ug = Qg/[(H− hl) W ]. The Reynolds number is based on the liquid and gas hydraulic diameter, which are defined as

DHl= 4Al Pl = 4W hl (2hl+ W ) DHg = 4Ag Pg+ Pi = 4W (H− hl) [2(H− hl) + W ] + W

The density and dynamic viscosity of water and air have been taken as: ρl = 997 (kg/m3), µl = 1.005 10−3(P a s) ; ρg = 1.21 (kg/m3), µg = 1.83 10−5(P a s). The average liquid level hland the instantaneous velocity profile of both phases, were measured at a distance of 9.1m (91 H) from the channel inlet, where the flow has been found fully developed. The velocity measurement of the water was performed along a vertical line in the center of the channel, adopting Laser Doppler Anemometry. For air, Hot Wire Anemometry was employed. Both longitudinal (stream-wise) and vertical velocity were measured. The pressure gradient was obtained by measuring the pressure difference in the gas phase with a micro manometer. The experimental horizontal and vertical velocity profile are shown in figure 4. Turbulent kinetic energy profile and Reynolds shear stress profile are plotted in figure 5. It can be seen that for a smooth interface (Run 250) the flow is a superimposition of two single-phase

(13)

flow layer: Couette flow (liquid) and Poiseuille flow (gas). The gas velocity profile is almost symmetric in run 250, instead for run 400 and 600 it becomes asymmetric with an upward shift of the maximum. This is explained by the occurrence of inter facial waves that increase the momentum transfer from the gas to the liquid. As a consequence the shear stress at the interface becomes higher than the gas shear stress at the wall. Also the turbulent kinetic energy profile loses its symmetry in run 400 and 600, because of the enhanced turbulence level associated with the appearances of waves. Secondary flow was observed in both phases for runs 400 and 600, as it can be seen from the vertical velocity profile and the non linearity of the Reynolds stress. The secondary flow in the gas is due to the anisotropy of the turbulence generated by the crosswise variation of the wave amplitude. In the liquid phase, the secondary flow is due to the interaction between mean flow and waves (see Lin´e et al. 1996 for more details). In fact in run 250, where the interface is smooth, the vertical velocity is almost zero and the Reynolds stress exhibits a linear profile in both phases. In the gas phase (run 250) the Reynolds stress is positive for y/H < 0.6 and negative for y/H > 0.6, which means that the mean momentum is transferred to the interface and the wall, respectively. In the liquid phase the Reynolds stress is positive, therefore the mean momentum is transferred to the wall only. As pointed out by Fabre et al., the velocity logarithmic law has been observed near the wall and above the interface. This is an important conclusion for turbulence modelling, since confirmed the validity of using ’wall function’ at the wall and at the gas interface. As already reported by Terzuoli et al. 2008, 3-D geometry effects are important. Values of the bulk velocity, different than those reported in table 2, are found when integrating over the cross section height the gas and water velocity profile given by Fabre at al. (see table 3). It is worthy to highlight that in run 600 the bulk gas velocity obtained by integration of the velocity profile is lower in value than the bulk velocity obtained dividing the volumetric flow rate by the gas cross section. This means that the transversal distribution of the stream-wise velocity do not have its maximum at the centreline as proved by Lin´e et al. (1996).

Run Ubl(m/s) Ubg(m/s)

250 0.415 4.262

400 0.523 5.561

600 0.722 7.245

Table 3: Bulk water and air velocities obtained integrating the experimental velocity profile by Fabre et al. (1987).

It is clear that simulating the experimental results by Fabre et al. (1987) is not a simple task due to the many effects that are present and the limitations imposed by the turbulence model. However it was decided to perform simplified simulations as preliminary test cases. Initially, only run 250 was simulated, since in this case secondary motion and 3-D effects are less important. In the simulations with Fluent, the VOF method was adopted and a constant pressure gradient was imposed as periodic boundary condition. Periodic boundary conditions represent an easy and cheap option to simulate fully developed flow rather than considering the entire channel domain. The mesh adopted was two-dimensional with two cells in the stream-wise direction and a refinement at the walls in order to collocate the first grid point near the wall at y+ < 2.5. The interface location was imposed a priori according to the experimental value (table 2). Mesh with and without refinement at the interface were investigated. Unsteady calculation were carried out with a time step of 1 second and 20 iterations per time step, since our interest is for the fully developed solution. The equations were discretized in space with a second order upwind scheme and in time with a first order implicit method. The SIMPLE method was used to calculate the pressure field and to couple it with the velocity field. The equations were solved iteratively with a segregated algorithm. The effects of the gravity were considered negligible, therefore gravity was not accounted

(14)

for. The k− w model with Low Reynolds correction was employed. The simulations were considered converged when a constant value of the shear stress at both the walls was reached and the residuals were at least below 10−5 for all quantities. It was found that in order to get a solution independent of the mesh, a refinement at the interface is needed. In table 4 liquid and gas bulk velocity are reported for different meshes without refinement at the interface. The velocity deviation (emesh) between two consecutive meshes is also shown. In table 4 the same results are shown for mesh with refinement at the interface. From these tables, it can be seen that a mesh dependent solution is reached for a mesh of 100 nodes, and with refinement at the interface. Instead, the same mesh without interface refinement exhibits solution dependent on the mesh. It is interesting to note that what is influenced by the mesh is mainly the gas velocity. This seems to be due to the necessity of correctly representing the steep gradient in velocity and kinetic energy present on the gas side of the interface. In fact the liquid acts on the gas as a moving wall, so it is correct to expect the same mesh refinement as a normal wall. Comparing the simulations results, table 5, with the experimental results, table 3, it is clear that the simulated gas and liquid mean velocity are largely underestimated. This is in agreement with the CFD results of Holm˚as et al. (2005). They found an under prediction of the mean velocity in both phases when smooth stratified two phase-flow was simulated with the k− ω SST and k −  RNG models (see section 2.1).

Nodes liquid Nodes gas Ubl(m/s) Ubg(m/s) emesh,l(%) emesh,g(%)

20 30 0.2150 0.5789 -

-40 60 0.2167 0.6067 +0.76 +4.72

80 120 0.2129 0.6332 -1.78 +4,04

160 240 0.2102 0.6528 -1.28 +3.00

Table 4: Simulated bulk liquid and gas velocity for different meshes without refinement at the interface. Run 250, k− w model with Low Reynolds correction. Gravity is not accounted for.

Nodes liquid Nodes gas Ubl(m/s) Ubg(m/s) emesh,l(%) emesh,g(%)

20 30 0.2042 0.6544 -

-40 60 0.2102 0.6724 +2.85 +2.67

80 120 0.2097 0.6775 -0.24 +0.75

Table 5: Simulated bulk liquid and gas velocity for different meshes with refinement at the interface. Run 250, k− w model with Low Reynolds correction. Gravity is not accounted for. In order to test also the other turbulence models available in Fluent, simulations with k− ωSST , k −  RNG, k −  EW T were carried out for a mesh of 100 nodes with refinement at the wall and at the interface. The simulations with k− ω SST and k −  RNG, did not converge and are not presented. A comparison of the experimental velocity profile for run 250 and the simulated velocity profile with k−ω and k − EW T is shown in figure 6. In table 6 the deviations between the computed and experimental values are reported (e). It can be seen that both k− ω and k −  EW T under predict the experimental value and an upward shift of the maximum gas velocity is observed. To overtake this limitation, we introduce a damping of the turbulence at the interface. Simulations with the damping source term proposed by Egorov (see section 2.1) were carried out in Fluent. The value of the constant B was chosen equal to 10. Results for the simulation with damping are also shown in figure 6 and table 6. It can be seen that the damping of the turbulence at the interface improve sensible the performance of the k− ω model.

However, the under prediction in the water bulk velocity is still present. This can be attributed to the gravity effects which have more influence on the water than to the air due

(15)

Ubl(m/s) Ubg(m/s) el(%) eg(%)

exp. 0.415 4.262 -

-k− ω 0.210 0.673 -49 -84

k−  0.236 1.523 -43 -64

k− ω damping 0.219 4.290 -47 +0.65

Table 6: Simulated and experimental bulk liquid and gas velocity. Run 250. Gravity is not accounted for.

to the different densities. For this reason, simulations with gravity were also carried out. It was found that in order to get convergence when including gravity, it is necessary to use a mesh with more than 2 cells in the stream-wise direction. Thus a mesh with 100 nodes in the vertical direction and 25 nodes in the stream-wise direction has been employed. Some changes in the simulations settings were made. The implicit body force treatment was enabled, which makes the solution more robust when body force exists in multiphase flow (see user guide Fluent). Then, the volume fraction equation was discretized in space using the high-resolution capturing scheme (HRIC), instead of the second order upwind scheme previously employed. Simulations were conducted with k−  EWT, k − ω Low Reynolds correction model with and without damping. In table 7, the simulations results are reported. In figure 7, the velocity profile, the kinetic energy profile, and the Reynolds shear stress profile are shown.

Ubl(m/s) Ubg(m/s) el(%) eg(%) τwg(P a) τwl(P a)

exp. 0.415 4.262 - - 0.424 0.0541

k− ω 0.415 0.932 -0.05 -78 0.569 0.0138

k−  0.464 2.555 +11 -40 0.543 0.0379

k− ω damping 0.444 4.169 +6 -2.2 0.514 0.0673

Table 7: Simulated and experimental bulk liquid and gas velocity. Run 250. Gravity is accounted for.

It can be seen that it is fundamental to take into account the inclination of the channel. In fact also if it is small, the inclination results in a not negligible force due to the high density of the water and the relatively small pressure gradient. According to equation (22), the gravity contribution is 3.70 Pa/m for run 250. Again a damping of the turbulence is needed at the interface in order to get values of the gas velocity that compare well with the experimental results. This time the constant B was chosen equal to 1. Simulation for the wavy case (Run 400) were also carried out employed the k− ω Low Reynolds correction with and without the damping source term. Results for run 400 are reported in table 8 and in figure 8. The constant B was 0.0025. It turns out that also for wavy case the damping of turbulence at the interface is necessary.

Ubl(m/s) Ubg(m/s) l(%) g(%) τwg(P a) τwl(P a)

exp. 0.523 5.561 - - 0.6167 0.1227

k− ω 0.516 1.498 -1.33 -73 0.9442 0.0360

k− ω damping 0.585 5.749 +11.9 +3.38 0.8363 0.1418

Table 8: Simulated and experimental bulk liquid and gas velocity. Run 400. Gravity is accounted for.

(16)

2.3 Objective

From the literature review and the simulations conducted with Fluent it can be inferred that when stratified two-phase flow is simulated with RANS and VOF, the turbulence model needs to be modified in order to correctly predict the turbulence behaviour of the interface. Actually the only remedy to the turbulence model lack is to damp the turbulence at the interface thus reproducing the effects of the near wall turbulence. The simulations carried out in the former section shows that the damping source term suggested by Egorov (2004) leads to a considerable improvement of the k− ω model performance. This was also found by Lo and Tomasello (2010). However, both in our simulations and in the simulations performed by Lo and Tomasello, the damping source term was found to be dependent on the mesh. Moreover the constant B that determine the damping strength has to be tuned, and its value changes with the mesh and with the flow parameters. It is clear that this is a point of weakness for a turbulence model. The objective of this thesis is to developed a code to simulate stratified two-phase flow between two parallel flat plates. The code is intended to be a fast, reliable and easy-to-modify tool to test different turbulence models and to investigate and develop more robust damping methods. The next sections of this thesis report the steps followed in the development of such a code with Matlab. It is clear that the improvement of the existing turbulence models depends on the knowledge of the turbulence behaviour of the interface which is nowadays still a debated issue. At the moment the turbulence damping at the interface is only a remedy for the limitations show by the classical turbulence models and it does not solve the fundamental base of the problem.

(17)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 u/Ul(-) y/ H (-) Run 250 Run 400 Run 600 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u/Ug(-) y/ H (-) Run 250 Run 400 Run 600 −0.070 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 v/Ul(-) y/ H (-) Run 250 Run 400 Run 600 0 0.02 0.04 0.06 0.08 0.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 v/Ug(-) y/ H (-) Run 250 Run 400 Run 600

Figure 4: Horizontal and vertical mean velocity profile. Water (left), air (right). Experimental results by Fabre et al. 1987.

(18)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 k/U2 l (-) y/ H (-) Run 250 Run 400 Run 600 0 0.005 0.01 0.015 0.02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 k/U2 g(-) y/ H (-) Run 250 Run 400 Run 600 −0.5 0 0.5 1 1.5 2 2.5 x 10−3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 −u0v0/U2 l (-) y/ H (-) Run 250 Run 400 Run 600 −4 −3 −2 −1 0 1 2 3 4 x 10−3 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −u0v0/U2 g(-) y/ H (-) Run 250 Run 400 Run 600

Figure 5: Turbulent kinetic energy profile and Reynolds shear stress profile. Water (left), air (right). Experimental results by Fabre et al. 1987.

(19)

0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 u(m/s) y/ H (-) exp. k − ω k − 0 k − ω damp 0 1 2 3 4 5 0.4 0.5 0.6 0.7 0.8 0.9 1 u(m/s) y/ H (-) exp. k − ω k − 0 k − ω damp 0 0.01 0.02 0.03 0.04 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 k(m2/s2) y/ H (-) exp. k − ω k − 0 k − ω damp 0 0.1 0.2 0.3 0.4 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1 k(m2/s2) y/ H (-) exp. k − ω k − 0 k − ω damp 0 2 4 6 8 x 10−3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 −u0v0(m2/s2) y/ H (-) exp. k − ω k − 0 k − ω damp −0.05 0 0.05 0.1 0.15 0.4 0.5 0.6 0.7 0.8 0.9 1 −u0v0(m2/s2) y/ H (-) exp. k − ω k − 0 k − ω damp

Figure 6: Comparison between experimental results by Fabre et al. 1987 (Run 250) and simulations with Fluent 13.0. Water (left), Air (right). Gravity is not accounted for.

(20)

0 0.2 0.4 0.6 0.8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 u(m/s) y/ H (-) exp. k − ω k − 0 k − ω damp 0 1 2 3 4 5 0.4 0.5 0.6 0.7 0.8 0.9 1 u(m/s) y/ H (-) exp. k − ω k − 0 k − ω damp 0 0.005 0.01 0.015 0.02 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 k(m2/s2) y/ H (-) exp. k − ω k − 0 k − ω damp 0 0.1 0.2 0.3 0.4 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1 k(m2/s2) y/ H (-) exp. k − ω k − 0 k − ω damp 0 1 2 3 4 5 6 x 10−3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 −u0v0(m2/s2) y/ H (-) exp. k − ω k − 0 k − ω damp −0.05 0 0.05 0.1 0.15 0.4 0.5 0.6 0.7 0.8 0.9 1 −u0v0(m2/s2) y/ H (-) exp. k − ω k − 0 k − ω damp

Figure 7: Comparison between experimental results by Fabre et al. 1987 (Run 250) and simulations with Fluent 13.0. Water (left), Air (right). Gravity is accounted for.

(21)

0 0.5 1 1.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 u(m/s) y/ H (-) exp. k − ω k − ω damp 0 2 4 6 8 0.4 0.5 0.6 0.7 0.8 0.9 1 u(m/s) y/ H (-) exp. k − ω k − ω damp 0 0.2 0.4 0.6 0.8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 k(m2/s2) y/ H (-) exp. k − ω k − ω damp 0 0.2 0.4 0.6 0.8 1 0.4 0.5 0.6 0.7 0.8 0.9 1 k(m2/s2) y/ H (-) exp. k − ω k − ω damp −0.020 0 0.02 0.04 0.06 0.05 0.1 0.15 0.2 0.25 0.3 0.35 −u0v0(m2/s2) y/ H (-) exp. k − ω k − ω damp −0.2 0 0.2 0.4 0.6 0.4 0.5 0.6 0.7 0.8 0.9 1 −u0v0(m2/s2) y/ H (-) exp. k − ω k − ω damp

Figure 8: Comparison between experimental results by Fabre et al. (1987) and simulations with Fluent 13.0. Water (left), Air (right). Gravity is accounted for.

(22)

3

Turbulence modelling

3.1 Two phase flow in channel

Stratified smooth two-phase flow in a channel is considered. As shown in figure 9, the heavier phase (l) is flowing on the bottom, and the lighter (g) on the top. The two fluids are considered Newtonian, immiscible and incompressible.

Figure 9: Schematic representation of stratified flow.

Continuity and momentum equations in Cartesian coordinate system, adopting the Ein-stein summation notation are

∂Ui ∂xi = 0 (2) ρ∂Ui ∂t + ρUj ∂Ui ∂xj =−∂P ∂xi +∂Tij ∂xj + ρgi (3) Tij = 2µSij = µ  ∂Ui ∂xj +∂Uj ∂xi  (4) Substituting (4) in (3), assuming constant property fluid give the Navier-Stokes equations

ρ∂Ui ∂t + ρUj ∂Ui ∂xj =∂P ∂xi + µ ∂ 2U i ∂xj∂xj + ρgi (5)

In turbulent flow, the flow variables are decomposed into mean φ and fluctuating φ0 parts. For the generic variable Φ it holds

Φ = φ + φ0, φ = 1 ∆t Z t+∆t/2 t−∆t/2 Φ dt, Z t+∆t/2 t−∆t/2 φ0 dt = 0 (6)

Replacing all the flow variables in equation (2) and (5) by their mean and fluctuating parts and performing time averaging yields to

∂ui ∂xi = 0 (7) ρ∂ui ∂t + ρuj ∂ui ∂xj =∂p ∂xi + ∂ ∂xj  µ∂ui ∂xj − ρu 0 iu0j  + ρgi (8)

Eqs. (8) are the so-called RANS. Turbulent fully developed steady state is assumed and the flow is considered independent from z coordinate (channel width>>channel height). Under these assumptions, for the generic fluid k (k=g or k=l), equations (7) and (8) reduce to

(23)

dvk dy = 0 (9) 0 =−∂pk ∂x + d dy  µk duk dy − ρu 0 kv 0 k  − ρkg sinθ (10) 0 =−∂pk ∂y + d dy  −ρv0 kv 0 k  − ρkg cosθ (11)

here x1 = x, x2 = y, x3 = z, u1 = u, u2 = v, u3 = w. Impermeability and no slip boundary conditions are imposed at the walls

vk= vk0 = 0 at y = 0, H (12)

uk= u0k= 0 at y = 0, H (13)

thus vk0v0k = u0kv0k = 0 at the walls. For equation (9) and (12), vk = 0 everywhere. Surface tension effects are neglected (W e >> 1), and no slip condition is imposed between the phases

τg,xy = τl,xy = τi at y = hl (14) pg = pl= pi at y = hl (15) ug= ul= ui at y = hl (16) where τk,xy = µk duk dy − ρu 0 kv 0

k. Integrating equation (11) and combining it with equation (15) leads to

pk+ ρkvk0v0k= pi+ ρk(vk0v0k)i− ρkg cosθ(y− hl) (17) differentiating this respect to x yields

∂pk ∂x = dpi dx + ρkgcosθ ∂hl ∂x → ∂pk ∂x = dpi dx (18)

due to ∂hl/∂x = 0. From (18) it can be deduced that the axial pressure gradient is indepen-dent of y. Integrating eq. (10), and considering equations (14) and (18) we obtain the shear stress distribution τk,xy(y) =  ∂pk ∂x + ρkg sinθ  (y− hl) + τi (19)

evaluating this at the walls yields  ∂pg ∂x + ρgg sinθ  hg+ τwg+ τi = 0 (20)  ∂pl ∂x + ρlg sinθ  hl+ τwl− τi = 0 (21)

(24)

where τwg =−τg,xy(y = H) and τwl = +τl,xy(y = 0). These are the integral x-momentum balances for the two phases. Adding eq. (20) and (21) give the total integral x-momentum balance ∂p ∂x+ (ρgαg+ ρlαl) g sinθ + τwg H + τwl H = 0 (22)

where αg = hg/H and αl= hl/H are respectively the gas and liquid hold-up, αG = (1− αl). The hold up equation is obtained combining (20) and (21)

αlτwg− αgτwl+ τi− αgαlH(ρl− ρg)gsinθ = 0 (23) In order to compute the axial pressure gradient and the hold up with eqs. (22) and (23), wall and inter-facial shear stress have to be known. Assuming that the flow rates of the two phases are given, it is necessary to introduce some relations of the type τ = f (Ub). The bulk velocities are defined as

Ub,g= 1 hg hg Z hl ugdy Ub,l= 1 hl hl Z 0 uldy (24)

If such relations for the shear stress are not available, the velocity distribution u(y) has to be derived from eq. (10). In order to do that, a formulation for u0

kvk0 is required.

3.2 Eddy viscosity assumption

According to the turbulent viscosity assumption (Boussinesq 1877), the Reynolds stress can be defined as

−u0v0 = ν t

du

dy (25)

where νtis the turbulent or eddy viscosity. It has the same measurement unit as the molecular viscosity ν, (m2/s), but conversely is not a fluid property since it depends also on the flow field. For simplicity the subscript k, indicating the phase (g or l), will be omitted from now on. Introducing eq. (25) in the axial momentum balance ( eq. (10)), leads to

d dy  (ν + νt) du dy  = 1 ρ ∂p ∂x+ g sinθ (26)

The eddy viscosity can be written as the product of a velocity scale u∗ and a length scale l m

νt∼ u∗lm (27)

It can be seen that both eq. (25) and (27) are in analogy with the molecular momentum transport process(see Wilcox 2006, pg.54). Kolmogorov (1942) and Prandtl (1945) suggested to scale u∗ with the turbulent kinetic energy, k = 12(u02 + v02+ w02). From dimensional consideration

u∗ = c k1/2 (28)

thus the turbulent viscosity become

(25)

where c is a constant to define. In order to close the problem, a definition of k and lm is needed. The derivation of k is achieved by solving a differential transport equation for k. In one-equation model, lm is directly specified. In two-equation models, lm is defined by solving an additional transport equation for an appropriate quantity. The introduction of k allows to incorporate flow history and non local effects in the eddy viscosity.

3.3 Turbulent kinetic energy equation

Subtracting RANS equations (8) to the Navier-Stokes equations (5) leads to ∂u0i ∂t + uj ∂u0i ∂xj + u0j∂ui ∂xj + u0j∂u 0 i ∂xj =−1 ρ ∂p0 ∂xi + ν ∂ 2u0 i ∂xj∂xj + ∂ ∂xj u0 iu0j (30)

multiplying the resulting equation for u0i and performing time average yields to the trans-port equation for the turbulent kinetic energy k

∂k ∂t + uj ∂k ∂xj = ∂ ∂xj  ν ∂k ∂xj − 1 2u 0 iu0iu0j − 1 ρp 0u0 j  +P −  (31)

The k transport equation can also be derived from the transport equation of the Reynolds stress tensor (see Wilcox 2006). Eq. (31) consists of seven terms:

Unsteady term: local temporal variation of k. ∂k ∂t

Convective term (or advection): transport of the turbulent kinetic energy per unit mass due to the mean fluid motion.

uj ∂k ∂xj

Viscous diffusion: diffusive transport of k at the molecular level. ν ∂k

∂xj

Turbulent convection: transport of the turbulent kinetic energy through the fluid, by the turbulent velocities fluctuations (u0iu0i= 2k).

1 2u

0 iu0iu0j

Pressure transport: turbulent transport due to the correlation between velocity and pressure fluctuation (analogous term in the energy equation).

1 ρp

0u0 j

Production term: production of k by the work of the Reynolds stress on the mean flow. Turbulent kinetic energy is removed from the mean flow and transferred to the turbulence.

P = −u0 iu0j

∂ui ∂xj

(26)

Dissipation term: rate at which turbulent kinetic energy is converted into thermal energy (Kg sJ = ms32).  in (31) is the pseudo-dissipation, defined as

 = ν∂u 0 i ∂xj ∂u0 i ∂xj

For fully developed steady-state channel flow, the turbulent kinetic energy equation (31) reduces to 0 = d dy  νdk dy − 1 2u 0 iu0iv0− 1 ρp 0v0  − u0v0du dy − ν ∂u0 i ∂y ∂u0 i ∂y (32)

Figure 10 shows the behaviour of each term in the above equation for the viscous wall region, y+ < 50. It can be seen that, at the wall, the dissipation balance the viscous diffusion and the other terms are zero.

 = νd 2k

dy2 at the wall (33)

In the log-law region, y+ > 30, the production is balanced by the dissipation. The peak in production is located in the buffer layer y+ ∼ 12. Here the Reynolds shear stress has its maximum (fig. 11). The production exceeds the dissipation and the energy in excess is transported away by the transport terms. In equation (32), the production is in closed form

P = −u0v0du dy = νt  du dy 2 (34) according to the eddy viscosity assumption, eq. (25). The viscous diffusion is in closed form too. The other terms are not and they have to be modelled. Turbulent convection and pressure transport are generally modelled with the gradient-diffusion hypothesis, for which in turbulent flow the transport of a scalar quantity is given by the mean scalar gradient multiplied by a diffusivity 1 2u 0 iu0iv0+ 1 ρp 0v0= 1 2k 0v0+1 ρp 0v0 =νt σk dk dy (35)

where σk is the turbulent Prandtl number for kinetic energy and it is generally taken equal to unity. The gradient-diffusion hypothesis is an analogous of molecular transport process, e.g. Fourier’s law, Fick’s law. Looking at (35) it can be concluded that the pressure transport has been neglected, since in the last term only the gradient of k appears. As fig. 10 shows this is an acceptable simplification in our case. The specification of  is based on the unknown parameters k and lm. From dimensional considerations

 = CDk3/2/lm (36)

Subsituting (34) and (35) in (32), the kinetic energy transport equation for fully developed turbulent flow in a channel is obtained

d dy  ν + νt σk  dk dy  =−νt  du dy 2 +  (37)

(27)

3.4 Turbulent length scale

Two widely used turbulent models are the k−  and the k − ω. In the k −  the turbulent length scale is specified by solving a transport equation for the dissipation rate  (m2/s3), thus the turbulent viscosity is given substituting (36) in (29)

νt= c CD k2

 = Cµ k2

 (38)

where Cµ is a closure coefficient. It can be derived in different ways that Cµ = 0.09 (see Pope 2000). This has also been verified far from the walls (P/ ∼ 1) by DNS data (fig. 12). Contrary to the k-equation (31), which is an exact equation, the  transport equation is empirical. Generally it has this form:

∂ ∂t + uj ∂ ∂xj = ∂ ∂xj  ν + νt σ  ∂ ∂xj  + C1P k − C2 2 k (39)

which for fully-developed steady state channel flow reduces to d dy  ν + νt σ  d dy  =−C1  kνt  du dy 2 + C2 2 k (40)

The Launder and Spalding (1972) model, also referred as the ’standard’ k−  model, adopts the following closure coefficients

Cµ= 0.09 C1= 1.44 C2 = 1.92 σk= 1 σ= 1.3 (41)

The values of the closure coefficients are obtained studying the behaviour of the model re-spect to simple flows (decaying homogeneous isotropic turbulence, homogeneous shear flow, logarithmic layer) or by comparison with experimental data. The closure coefficients can be tuned in order to give better results according to a particular flow type. The k−  model as presented by equations (63),(38), (40) and (41) is valid for the fully turbulent region ν/νt<< 1. Therefore the molecular diffusion term is generally neglected. The wall boundary layer is then simulated with ’wall functions’, (High Reynolds number models). In fact, as it can be seen from fig. 12, in order to solve the equations at the wall, it is necessary to adopt different closure coefficients in order to account for the viscous effects, (Low Reynolds number models). A detailed description about this will be given later.

In the k−ω model the turbulent length scale is defined by solving a transport equation for the specific dissipation rate ω (1/s). ω can be considered as the rate of dissipation of energy per unit time and volume or as a characteristic frequency of the turbulence decay process. From dimensional argument

ω = CSD k1/2

lm

(42) Substituting (42) into (29) gives the turbulent viscosity

νt= c CSD k

ω (43)

(28)

postulated by Kolmogorv (1942), later was modified by Wilcox (1998) with the introduction of a viscous diffusion term and a production term

∂ω ∂t + uj ∂ω ∂xj = ∂ ∂xj  (ν + σνt) ∂ω ∂xj  + αPω k − β ω 2 (44)

which, for steady state fully developed turbulent channel flow reduces to d dy  (ν + σνt) dω dy  =−αω kνt  du dy 2 + β ω2 (45)

In the k-equation (63), the dissipation rate  has to be rewritten in term of ω. The relation between  and ω is obtained equating (38) and (43)

 = Cµkω (46)

then substituting this in eq. (63) leads to the kinetic energy equation for the k− ω model d dy  (ν + σ∗νt) dk dy  =−νt  du dy 2 + β∗kω (47)

According to Wilcox (1998),the closure coefficients are Cµ= β∗= 0.09 α = 5 9 β = 3 40 σ ∗ = 0.5 σ = 0.5 (48)

The main advantage of the k− ω model respect to the k −  model is that the first can be directly integrated in the viscous sub-layer without modification. The k−  model is more stable than the k− ω in treating the free stream edge. Looking at the  and ω equations, a difference lies in the cross diffusion term. As described in Pope (2000), deriving the ω equation implied by the k−  model result in

Dω Dt = ∂ ∂xj  ν + νt σω  ∂ω ∂xj  + (C1− 1)Pω k − (C2− 1)ω 2+2νt σω ∂ω ∂xj ∂k ∂xj (49) This equation differs from (44) for the last term. In the last version of his k−ω model Wilcox (2006) add this term in the ω equation calling it ’cross diffusion’.

3.5 Wall boundary conditions

Concerning the wall treatment, there are two categories of turbulence model. The first are the High Reynolds number models (HR), which make use of wall functions to simulate the near wall region. The second are the Low Reynolds number models (LR), where the equations are integrated through the wall boundary layer in order to capture the physics of the wall in detail. In the latter case a refined mesh is needed close to the wall, which results in a higher computational cost respect to the HR models. Higher the Reynolds number higher is the computational cost since the vicious sub-layer is thinner for high Reynolds number flows. In this section the HR and LR models of both k−  and k − ω are reported.

(29)

0 5 10 15 20 25 30 35 40 45 50 –0.20 –0.10 0.00 0.10 0.20 y+ Gain Loss Production Dissipation Viscous diffusion Turbulent convection Pressure transport

Figure 10: The turbulent kientic energy budget in the viscous wall region of channel flow: terms normalized by viscous scale. From DNS data of Kim et al. (1987). Re=13750. Taken from Pope (2000). 0 5 10 15 20 25 30 35 40 45 50 -1 0 1 2 3 4 5 6 7 8 〈uiuj〉 uτ2 y+ 〈u2 〈w2 〈uv〉 〈v 2 k

Figure 11: Profiles of Reynolds stresses and kinetic energy normalized by friction velocity in the viscous wall region of turbulent channel flow: DNS of Kim et al. (1987). Re=13750. Taken from Pope (2000).

Figure 12: Profile of Cµ = νt/k2 from DNS data of channel flow at Re=13750 (Kim et al. 1987). Taken from Pope (2000).

(30)

3.5.1 High Reynolds model

Simulate the near-wall region introduce complication in the turbulence model, and compu-tational cost. In fact, the wall causes effects which require a modifications in the turbulence model coefficients. Approaching the wall, the turbulence become anisotropic and viscous effects become important. In the wall-region are present the highest gradients of the flow quantities, which can show singular behaviour (ωy→yw = ∞). Particular attention has to

be paid in order to accurately represented such phenomenon. To overcome these issues, the boundary conditions for k and  (or ω) are not specify at the wall (yw) but at a distance away from it (yp). Thus, the turbulence-model equations are not solved between yw and yp. By locating yp in the log-law region (e.g. yp+> 30) it is possible to use the log-law to relate the velocity and the wall shear stress

u uτ = 1 κln uτy ν + C (50)

where uτ =pτw/ρ is the friction velocity, κ = 0.41 is the Von Karman constant and C = 5.2 (see Pope 2000 for log-law derivation and wall regions description). The value of κ and C varies in the literature within the 5%. The friction velocity is computed iteratively from (50). Then the wall shear stress is obtained (τw= ρu2τ) and imposed as boundary condition for the momentum equation. The boundary conditions for k,  and ω have to be obtained. Deriving (50) respect to y leads to du dy = uτ κyp (51) According to eq. (34) the turbulent viscosity can be rewritten has

νt=P du dy

−2

(52)

Substituting this in (38) and applying (51) yields P = Cµk

2u2 τ (κyp)2

(53) In the log-low regionP ∼  thus

p=

pCµkpuτ κyp

(54) which is the value of  in the log-law region. Substituting (46) in (54) gives the value of ω in the log-law region

ωp = uτ pCµκyp

(55) These boundary conditions for  and ω have been derived from the turbulent viscosity hy-pothesis, the log law and the assumption of P ∼ . Generally it is assumed that the shear stress in the logarithmic region is constant, and that the laminar part can be neglected

τxy = ρν du

dy − ρu

0v0 ∼ −ρu0v0= ρu2

(31)

see for example fig. 11. Combining this assumption with the experimental evidence that −u0v0 ∼ 0.3k in the regions where P ∼  ,leads to

kp = u2

τ pCµ

(57) Which can be used as a boundary condition for k. Similar boundary conditions have been derived, with a different procedure, by Wilcox (2006) pg.157. These are

kp = u2 τ √ β∗ p= uτ3 κyp ωp= uτ √ β∗κy p (58) Note that Cµ = β∗ and that (54) reduce to the above p if (57) is substituted in (54). The conditions (58), have been obtained by Wilcox under the assumption (56). Assumption (56) is valid when the pressure gradient in the momentum equation (10) is zero and it is a reasonable assumption for high Reynolds number flow where an overlap region is present. It can been expected that the accuracy of the solution deteriorates when a constant shear stress is not present (56) or when the local equilibrium is not satisfy (P 6= ), since these two are the main assumptions made to derive the above wall functions. Other wall functions are present in the literature, which lead to more robust solutions also in case when (58) orP ∼  are not satisfied (e.g. separated flow). However, due to the simplicity of our case, it was decided to used (58).

3.5.2 Low Reynolds models

The k−  HR model reported in section 3.4, if integrated near the wall, does not lead to satisfactory results. The turbulence has to be damped in order to account for the viscous effects. This is achieve with damping functions. The k−  LR model is as follows

d dy  ν + νt σk  dk dy  =−P +  (59) d dy  ν + ν σ  d˜ dy  =−C1f1 ˜ P k + C2f2 ˜ 2 k − E (60) νt= Cµfµ k2 ˜  (61)

where f1, f2 and fµ are the damping functions. Equation (60) is solved for ˜ instead for . The relation between the two quantities is

˜

 = − 0 (62)

It has been reported in section 3.4 that the dissipation rate , at the wall, balance the viscous diffusion term (33). 0 represent the value that the dissipation rate  assumes at the wall. Subtracting 0from  make possible to specify a zero boundary condition for the new quantity ˜

, which is a simplification from the computational point of view. The quantity ˜ can be seen as the isotropic part of the dissipation. In equation (60), E is a term introduced empirically. A variety of formulation for f1, f2, fµ, E and closure coefficients is present in the literature (see for example Wilcox (2006) pg.196). Here the model of Launder and Sharma (1974) is

(32)

considered fµ= e−3.4/(1+Ret/50) 2 f1= 1 f2= 1− 0.3e−Re 2 t 0 = 2ν ∂√k ∂y !2 E = 2ννt ∂ 2u ∂y2 2 Cµ= 0.09 C1= 1.44 C2 = 1.92 σk= 1 σ= 1.3

The damping functions depend on the local Reynolds number Ret=

k2 ˜ ν

For Ret → ∞, fµ = f1 = f2 → 1 and the High Reynolds model is recovered. Boundary condition for a fixed smooth wall are: u = 0, k = 0, ˜ = 0.

When integrated in the near wall region, the k− ω model successfully predict the most important flow properties. However it require damping functions in order to accurately pre-dict the near wall effects (e.g. the kinetic energy peak is underestimate). The k− ω model with near wall correction functions (Wilcox 1998), is as follows

d dy  (ν + σ∗νt) dk dy  =−P + β∗kω (63) d dy  (ν + σνt) dω dy  =−αω kP + β ω 2 (64) νt= α∗ k ω (65)

where the closure coefficients are defined as

α∗ = α ∗ 0+ Ret/Rek 1 + Ret/Rek α = 5 9 α0+ Ret/Reω 1 + Ret/Reω 1 α∗ β∗ = 0.095/18 + (Ret/Reβ) 4 1 + (Ret/Reβ)4

(33)

β = 3/40 σ∗ = σ = 0.5 α∗0 = β/3 α0 = 1/10

Reβ = 8 Rek = 6 Reω= 27/10

The local Reynolds number is

Ret= k ων

The HR model is obtained taking the limit Ret→ ∞ which gives α∗ = 1, α = 5/9, β∗ = 0.09. The boundary conditions for a fixed smooth wall are

u = 0 k = 0 ω = limy→yw(6ν/βy

2)

Because ω becomes unbounded at the wall, the boundary condition was not prescribe at the wall but in the first 5 grid points above the wall. According to Wilcox (2006) pg. 385, enforcing the ω condition in more points is a remedy to eliminate the numerical error gives when ω is only specified in the first grid point above the wall. Note that ω = 6ν/βy2 is valid only for y+= 2.5, so the mesh has to be very refined close to the wall.

4

Numerical method

4.1 Grid

The equations for turbulent, fully developed, steady-state flow between two flat plates have been derived in the previous sections. The flow is only dependent on the y-coordinate (fig. 9). The flow equations are composed by a diffusion term and by source (or sink) terms, no advection or unsteady terms are present. This simplifies a lot the solution of such equations. In order to solve the equations, it is necessary to operate a spatial discretization of the channel height. Three different types of grid have been used. The first one, is a bi-exponential grid suitable for Low Reynolds models. The second, is a uniform grid where the height of the cell near the wall can be adjusted (High Reynolds models). The third grid is a bi-exponential grid with an additional refinement at a prescribed channel height. This third grid is suitable for two-phase flow where a refinement is needed at the interface. The three different grids are show in figure 13.

A biexponential grid is nothing more than a composition of two exponential grid. In order to create an exponential grid (fig.14), the ratio between lengths of succeeding interval should be set equal to

R = hi+1 hi

= e(10/ny)(x−12) (66)

where ny is the number of cells (or intervals) and x is a parameter determining the grade of refinement, see fig.15. The length of each cell can be related to the length of the first cell with

hi = h1Ri−1 (67)

The height of the channel is given by

H = ny

X

i=1

(34)

Figure 13: Different grids adopted. Form left to right: bi-exponential grid, uniform grid, bi-exponential grid with refinement at the interface. (The grid nodes are plotted).

Combining (67) and (68) yields H = h1 ny X i=1 Ri−1= h1 1− Rny 1− R The steps to create an exponential grid are given as follows

– Define H, ny, x and derive R from (66). – Derive the length of the first interval: h1= H

1− R 1− Rny.

– Derive the length of each interval: hi= h1Ri−1with i = 1 : ny. – Calculate the coordinate of each node: ji+1= ji+ hi with i = 1 : ny.

– Calculate the coordinate of each cell center: yi= (ji− ji−1)/2 + ji−1 with i = 2 : ny+ 1 With the procedure described above an exponential grid will be obtained. In order to con-struct a bi-exponential grid, it is just necessary to divide the channel height H in two halves and treat each half as an exponential grid. The same concept is for a bi-exponential grid with refinement at the interface, but, in this case the channel height has to be divided in 4 parts. An exponential grid exhibits particular advantages. As can be seen from figure 15, by changing the value of the parameter x, the grid refines at the top wall (x < 0.5) or at the bottom wall (x > 0.5). For x∼ 0.5 reduce to an uniform grid. Another property of an exponential grid is that if the number of nodes is doubles, the original nodes are unchanged. This property can be derive analytically, and it is shown graphically in figure 16. The steps to create an uniform grid with adjustment of the first grid point height are not reported here, since the procedure is straightforward.

(35)

Figure 14: Exponent grid.

Figure 15: Exponent grid for increasing value of the parameter x. From left to right: 0.1, 0.25, 0.49, 0.75, 1. (NB: are plotted the grid nodes).

Figure 16: Exponent grid for doubling the number of nodes. From left to right: 5, 10, 20, 40. (NB: are plotted the grid nodes).

(36)

4.2 Discretization

The equations for u, k and  (or ω), are now discretized with the finite difference method over a non-uniform grid, fig. 17. The first derivative, and the diffusion term are discretized with a second order central difference approximation

 ∂φ ∂y  i ≈ φi+1− φi−1 yi+1− yi−1 (69)  ∂ ∂y  Γ∂φ ∂y  i ≈  Γ∂φ∂y i+1 2 −Γ∂φ∂y i−1 2  yi+1 2 − yi− 1 2  (70)

where φ is a general variable, Γ is a general diffusive coefficient, i is the cell center, i−1/2 and i + 1/2 are the cell nodes (fig.17). The inner derivative in the diffusion term is approximated with (69), which leads to

 ∂ ∂y  Γ∂φ ∂y  i ≈ Γ1+1 2 φi+1− φi yi+1− yi − Γ1− 1 2 φi− φi−1 yi− yi−1  yi+1 2 − yi− 1 2  = aiφi−1+ biφi+ ciφi+1 (71) with ai = Γi−1/2

(yi− yi−1) yi+1/2− yi−1/2

 ci=

Γi+1/2

(yi+1− yi) yi+1/2− yi−1/2

 bi =−ai− ci

Figure 17: Non-uniform grid.

Performing the discretization of u, k,  (or ω) equations, with (69) and (71), leads to a common formulation

aiφi−1+ b∗iφi+ ciφi+1= RHSi (72)

which represents a system of equations where the coefficient matrix is tridiagonal. The coefficients ai, ci and bihave been already defined above, and they are determined in relation to Γ. The right hand side (RHSi) represents all the constant terms. b∗i is the coefficient bi

Riferimenti

Documenti correlati

The movie shows the flow strealmlines coloured with temperature scalars for a steady state RANS simulation. Highest temperatures

The modernisation of the regasification facilities located in Spain concerned, above all, building a new tank in the terminal in Bilbao in 2014 with the storage capacity of

With reference to figure 4 during systole the LV contracts, the mitral valve is closed, its volume decreases and flow is ejected (S-wave) at systolic pressure through the Aorta,

Keywords: Sexual Rights, Sexual Health, Sexual Politics Source of Funding: Social Science and Humanities Research Council of Canada and the Ford Foundation Conflict of Interest

Il territorio di Lunigiana (Val di Vara compresa) sotto gli imperatori franchi venne inserito nella Marca della Liguria orientale, affidata alla famiglia degli Obertenghi, dalla

The results of this study confirm the high specificity of US find- ings indicative of crystal deposits at hyaline cartilage level and indicate that the supraspinatus tendon and

Item statuerunt et ordinaverunt quod singulo anno eligantur in dicto communi Hortinovi duo sindici dicti communis pro uno anno continuo, qui sindici teneantur et

The experimentation described above was divided into two phases: one, in static conditions, where different compounds (acetylcholine, L-NAME, resveratrol, NS 1619,