Ahsan A. Two Phase Flow, Phase Change and Numerical Modeling

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Modelling of Profile Evolution by Transport Transitions in Fusion Plasmas

169

integral form of the primary equation and change to an integral counterpart y of the variable

and (iii) approximation of the second order ODE for y in the vicinities of grid knots by

ODEs with constant coefficients whose exact analytical solutions are conjugated by

requiring the continuity of the solution and its first spatial derivative.

The approach elaborated is demonstrated by modelling of the development of the edge

transport barrier in a tokamak. The plasma heat conduction dependence on the

temperature gradient is assumed as a dropping abruptly at a critical value of the

temperature e-folding length. First a situation with a heating exceeding a critical level is

considered that corresponds to a L

magnitude at the LCMS smaller than L

. Calculations

have shown that, depending on the initial temperature profile, the whole range of the

radial position of the interface between ETB and the core region with strong transport,

predicted by an analytical model, can be realized in the final stationary state.

Computations also demonstrate that an ETB can be triggered even with a subcritical heating

if impurity radiation localized at the very plasma edge is enhanced to a certain critical level.

This may improve the understanding of the radiation improved mode observed in several

tokamaks.

Modelling of the plasma density profile involves additional non-linearity through the

coupling between the densities of charged and neutral particles. This can lead to numerical

instabilities in calculations with both too small and too large memory times. With a proper

chosen

such difficulties can be avoided. An approach is proposed to choose the memory

time. Itis based on the equilibration of different contributions to the free term in the equation

for y and relates

to the confinement time. The optimization of the

selection procedure

allows reducing noticeably the total number of iterations required to achieve the stationary

solution. It allows also modelling reliably the plasma density evolution by a spontaneous

formation of the ETB. The methods developed can be applied by simulations of other

physical phenomena in fusion plasmas having a bifurcation-like character, e.g., density

limits, and phase transitions in other media.

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Numerical Simulation of the Heat Transfer from a

Heated Solid Wall to an Impinging Swirling Jet

Joaquín Ortega-Casanova

Área de Mecánica de Fluidos, ETS de Ingeniería Industrial, C/ Dr. Ortiz Ramos s/n,

Universidad de Málaga, 29071 Málaga

Spain

1. Introduction

Swirling jets are frequently used in many industrial applications such as those related

with propulsion, cleaning, combustion, excavation and, of course, with heat transfer (e.g.

cooling/heating), among others. The azimuthal motion is usually given to the jet by different

mechanisms, being the most used by means of nozzles with guided-blades (e.g. Harvey, 1962);

by entering the ﬂuid radially to the device (e.g. Gallaire et al., 2004); by the rotation of some

solid parts of the device (e.g. Escudier et al., 1980); or by inserting helical pieces inside a

cylindrical tube (e.g. Lee et al., 2002), among other conﬁgurations. The way the swirl is given

to the ﬂow will ﬁnally depend on the particular application it will be used for.

Impinging swirling (or not swirling) jets against heated solid walls have been extensively

used as a tool to transfer heat from the wall to the jet. In the literature, one can ﬁnd many

works that study this kind of heat transfer related problem from a theoretical, experimental

or numerical point of view, being the last two techniques presented in many papers during

the last decade. In that sense, Sagot et al. (2008) study the non-swirling jet impingement

heat transfer problem from a ﬂat plate, when its temperature is constant, both numerically

and experimentally to obtain an average Nusselt number correlation as a function of 4

non-dimensional parameters. And, what is most important from a numerical point of view,

their numerical results, obtained with the commercial code Fluent© and the Shear Stress

Transport (SST) k

− ω turbulence model for values of Reynolds number (Re)rangingfrom

10E3 to 30E3, agree very well with previous experimental results obtained by Fenot et al.

(2005), Lee et al. (2002) and Baughn et al. (1991).

More experimental results are given by O’Donovan & Murray (2007), who studied the

impinging of non-swirling jets, and by Bakirci et al. (2007), about the impinging of a swirling

jet, against a solid wall. The last ones visualize the temperature distribution on the wall and

evaluate the heat transfer rate. In Bakirci et al. (2007), the swirl is given to the jet by means of

a helical solid insert with four narrow slots machined on its surface and located inside a tube.

The swirl angle of the slots can be varied in order to have jets with different swirl intensity

levels. This is a commonly extended way of giving swirl to impinging jets in heat transfer

applications, as can be seen in Huang & El-Genk (1998), Lee et al. (2002), Wen & Jang (2003)

or Ianiro et al. (2010). On the other hand, Angioletti et al. (2005), and for Reynolds numbers

ranging between 1E3 and 4E3, present turbulent numerical simulations of the impingement

of a non-swirling jet against a solid wall. Their results are later validated by Particle Image

2 Will-be-set-by-IN-TECH

Velocimetry (PIV) experimental data: when the Reynolds number is small, their numerical

results, obtained with the SST k

− ω turbulence model, ﬁt very well the experimental data,

while for high Reynolds number values, either the Re-Normalization Group (RNG) k

− 

model or the Reynolds Stress Model (RSM) works better. Others previous numerical studies,

as the ones by Akansu (2006) and by Olson et al. (2004), show that the SST k

− ω turbulence

model is able to predict very well the turbulence in the near-wall region in comparison with

other turbulence models. This fact is essential to obtain accurately the turbulent heat transfer

from the wall.

Another different turbulent model, presented by Durbin (1991), is used by Behnia et al.

(1998; 1999) to predict numerically the heat transfer from a ﬂat solid plate by means of

turbulent impinging jets, showing their results good agreement with experimental data. The

inconvenient of this last turbulent model is that it does not come originally with Fluent

package, so it is ruled out as an available turbulent model.

The work presented here in this chapter deals with the numerical study about the heat transfer

from a ﬂat uniform solid surface at a constant temperature to a turbulent swirling jet that

impinges against it. To that end, the commercial code Fluent© is used with the corresponding

turbulent model and boundary conditions. As any turbulent numerical study where jets are

involved, it needs as boundary condition the velocity and turbulence intensity proﬁles of the

jet, and the ones measured experimentally, by means of a Laser Doppler Anemometry (LDA)

technique, at the exit of a swirl generator nozzle will be used. The nozzle, experimental

measurements and some ﬁtting of the experimental data will be shown in Section 2. Different

information, about the computational tasks and decisions taken, will be presented in Section

3, such as those related with the computational domain, its discretization, the numerical

methods and boundary conditions used and the grid convergence study. After that, in

Section 4 the different results obtained from the numerical simulations will be presented and

discussed. They will be divided into two subsections: one to see the effect of varying the

Reynolds number; and another to see the effect of increasing or decreasing the nozzle-to-plate

distance. Finally, the document will conclude with Section 5, where a summary of the main

conclusions will be presented together with some recommendations one should take into

account to enhance the heat transfer from a ﬂat plate when a turbulent swirling jet impinges

against it.

2. Experimental considerations

Regarding the experimental swirling jet generation, it is created by a nozzle where the swirl

is given to the ﬂow by means of swirl blades with adjustable angles located at the bottom of

the nozzle (see Fig. 1). After the ﬂuid moves through the blades, it ﬁnally exits the nozzle as

a swirling jet. Due to the fact that blades can be mounted with ﬁve different angles, swirling

jets with different swirl intensities can be generated. Thus, for a given ﬂow rate, or Reynolds

number (deﬁned below), through the nozzle, ﬁve different swirling jets with ﬁve different

swirl intensities, or swirl numbers (deﬁned below), can be obtained. When the blades are

mounted radially, no swirl is imparted to the jet and the swirl number will be practically

zero. This blade conﬁguration will be referred in what follows as R. However, with the

blades rotated the maximum possible angle, the jet will have the highest swirl levels (and

then the highest swirl numbers). This conﬁguration will be referred as S2. Between R and S2

conﬁgurations there are other 3 possible blade orientations, but only the one with the most

tangential orientation, S2, will be considered in this work. Fig. 2 shows a 2-D view of the

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Two Phase Flow, Phase Change and Numerical Modeling

Numerical Simulation of the Heat Transfer from a Heated Solid Wall to an Impinging Swirling Jet 3

hub

blades

Fig. 1. 2D view of the nozzle. The dimensions are in mm.

swirl blades mounted radially and with the most tangential angle, R and S2 conﬁguration,

respectively.

Similar devices to the one used here to generate the swirling jet are commonly used in

several industrial applications, but its use in this work is motivated by two reasons: ﬁrstly,

by seabed excavation devices that usually use a swirl component to enhance their excavation

performance (see Redding, 2002), instead of using a totally axial jet; and secondly, to compare

the heat transfer performance of the impinging swirling jet with that obtained experimentally

by the same kind of impinging swirling jets but under seabed excavation tasks and reported

in Ortega-Casanova et al. (2011). They show that better results (in terms of the size of the

scour created) are obtained when the swirl blades are rotated the maximum possible angle, S2

conﬁguration, and for the highest nozzle-to-plate distance studied. Thus, the objective of this

numerical study is to be able to answer the question about whether or not the S2 conﬁguration

and the largest nozzle-to-plate distance, also give the highest heat transfer from the plate to

the jet.

To model the swirling turbulent jet created by the nozzle is necessary to know both the average

velocity ﬁeld and its turbulent structure at the exit nozzle. In a cylindrical coordinate system

175

Numerical Simulation of the Heat Transfer from

a Heated Solid Wall to an Impinging Swirling Jet

4 Will-be-set-by-IN-TECH

(a) (b)

Fig. 2. 2D view of the guided blades: mounted radially, R conﬁguration, (a); and rotated the

maximum angle, S2 conﬁguration, (b).

(r, θ, z), the mean velocity components of the velocity vector will be indicated by



V(r, θ, z)=

(

U, V, W), while the jet turbulence will be take into account by the velocity ﬂuctuations,

v



(r, θ, z) =(u



, v



, w



). Both vectors have been previously measured experimentally by

means of a LDA system and, due to the shape of the exit tube of the nozzle (see Fig. 1),

the radial component of both



V and v



has been considered small enough to be neglected:

= 0 = u



. Typical non-dimensional mean velocity proﬁles at the nozzle exit, together with

its ﬂuctuations, are shown in Fig. 3 for two ﬂow rates, the smallest and the highest used,

≈ 100 l/h and Q ≈ 270 l/h, respectively. In the same ﬁgure is also included, with a solid

line, the ﬁtting of the experimental data (see Ortega-Casanova et al., 2011, for more details

about the ﬁtting models used). In Fig. 3, the velocity has been made dimensionless using the

mean velocity W

based on the ﬂow rate through the nozzle, W

= 4Q/(πD

), and the radial

coordinate with the radius of the nozzle exit D/2.

In addition, Fig. 3 shows that, for a given blade orientation, S2 in our case, the swirl intensity

of the jet will depend on the ﬂow rate Q through the nozzle, since the azimuthal velocity

proﬁle is different depending on Q, too. Due to this, the one and only non-dimensional

parameter governing the kind of jet at the nozzle exit is the Reynolds number:

ρW

4ρQ

μπD

,(1)

where ρ and μ are the density and viscosity of the ﬂuid, respectively: in Ortega-Casanova et al.

(2011) the ﬂow rate ranges from 100 l/h to 270 l/h, so the Reynolds number ranges from 7E3 to

18.3E3, approximately. On the other hand, once the blade orientation is given, S2[shownFig.

2(b)], the swirl intensity of the jet will depend only on the Reynolds number, and following

Chigier et al. (1967), an integral swirl number S

can be deﬁned to quantify the swirl intensity

of the jet as



∞

WV dr

(D/2)



∞



−



.(2)

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Two Phase Flow, Phase Change and Numerical Modeling

Numerical Simulation of the Heat Transfer from a Heated Solid Wall to an Impinging Swirling Jet 5

0 0.5 1 1.5

−0.4

−0.2

0.2

0.4

0.6

0.8

0 0.5 1 1.5

−0.4

−0.2

0.2

0.4

0.6

0.8

(a) (b)

0 0.5 1 1.5

−0.2

0.2

0.4

0.6

0.8

1.2

1.4

1.6

0 0.5 1 1.5

−0.2

0.2

0.4

0.6

0.8

1.2

1.4

1.6

Fig. 3. Dimensionless azimuthal, (a) and (b), and axial, (c) and (d), velocity proﬁles for S2

conﬁguration. Q

≈ 100 l/h for (a) and (c); and Q ≈ 270 l/h for (b) and (d). The circles

indicate mean velocity values and the error bars its ﬂuctuations.

The evolution of S

versus the Reynolds number for the blade orientation under study is

shown in Fig. 4.

As it has been pointed out previously, the swirl intensity of the jet S

will depend on the

blade orientation and the ﬂow rate. As can be seen in Fig. 4, S2 conﬁguration produces jets

with variable levels of swirl, with its maximum around Re

≈ 9E3. This Reynolds number

divides the curve in two parts: the left one, Re

 9E3, in which S

increases with Re;and

the right one, Re

 9E3, in which S

decreases with Re. S

has been calculated using (2) and

the non-dimensional mean axial and azimuthal velocity proﬁles measured just downstream

of the nozzle exit. Both components of the velocity are depicted in Fig. 5 for all Reynolds

numbers experimentally studied. From this ﬁgure can easily be understood the behavior of

for S2 conﬁguration. These proﬁles are also shown in Ortega-Casanova et al. (2011), but

are reproduced here again in order to have a complete and general idea of the swirling jets

generated by the nozzle conﬁguration under study. When the swirl increases with the rotation

of the blades, not only the dimensional azimuthal velocity increases, as it was expected, but

also the maximum axial velocity at the axis, appearing a well deﬁned overshoot around it (see

the axial and azimuthal velocity proﬁles for other blade orientations in Ortega-Casanova et al.,

2011). In addition to this, another effect associated with the increasing of the blade rotation

177

Numerical Simulation of the Heat Transfer from

a Heated Solid Wall to an Impinging Swirling Jet

6 Will-be-set-by-IN-TECH

0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Fig. 4. Integral swirl number S

as a function of the Reynolds number. S2 conﬁguration.

is the appearance of a swirless region near the axis and a shift of all the azimuthal motion to

a region off the axis when the Reynolds number is above a certain value, as can be seen in

Fig. 5(a) for Re

> 11E3. This swirless region has nothing to do with vortex breakdown since

the axial velocity [Fig. 5(b)] does not have any characteristic of this phenomena, like a reverse

ﬂow at the axis with a stagnation point at a certain radius of the proﬁle. This phenomena has

been recently observed experimentally by Alekseenko et al. (2007), where vortex breakdown

occurs for jet swirl intensities above a critical value (see, e.g., Lucca-Negro & O’Doherty, 2001,

for a recent review about that phenomena).

Also, in Ortega-Casanova et al. (2011) is shown that the best combination for excavation

purposes in order to produce deeper and wider scours on sand beach is the axial overshoot

together with the shift of the azimuthal motion to an annular region. They also discuss and

give the mathematical models that better ﬁt the experimental data, shown also in Fig. 3 with

solid lines. Obviously, when S2 conﬁguration is used, as it is here, the azimuthal velocity

models depend on the Reynolds number considered, being different the one used for low

Reynolds numbers (Re

≤ 11E3) than for high ones (Re ≥ 13E3).

Those models will be used now as a boundary condition to specify the velocity components

of the swirling jet in the numerical simulations. However, not only the model of the velocity

proﬁles are needed to model the turbulent jet, but also is necessary to model its turbulence.

Once the velocity ﬂuctuations

v



have been measured, the turbulent intensity I of the jet can

be estimated as

√

2

+ v

2

+ w

2



√

2

+ w

2

.(3)

In order to have an analytical function of the turbulent intensity proﬁle to be used as boundary

condition, all turbulent intensity I proﬁles must be ﬁtted and it is found that the best ﬁtting is

achieved with the Gaussian model

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Two Phase Flow, Phase Change and Numerical Modeling