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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,

Spain

1 Introduction

Swirling jets are frequently used in many industrial applications such as those relatedwith 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 differentmechanisms, 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 somesolid parts of the device (e.g Escudier et al., 1980); or by inserting helical pieces inside acylindrical 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 extensivelyused as a tool to transfer heat from the wall to the jet In the literature, one can find manyworks 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 duringthe last decade In that sense, Sagot et al (2008) study the non-swirling jet impingementheat transfer problem from a flat plate, when its temperature is constant, both numericallyand experimentally to obtain an average Nusselt number correlation as a function of 4non-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) ranging from

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 theimpinging of non-swirling jets, and by Bakirci et al (2007), about the impinging of a swirlingjet, against a solid wall The last ones visualize the temperature distribution on the wall andevaluate 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 intensitylevels This is a commonly extended way of giving swirl to impinging jets in heat transferapplications, 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 numbersranging 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

8

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 withother turbulence models This fact is essential to obtain accurately the turbulent heat transferfrom 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 ofturbulent impinging jets, showing their results good agreement with experimental data Theinconvenient of this last turbulent model is that it does not come originally with Fluentpackage, 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 transferfrom a flat uniform solid surface at a constant temperature to a turbulent swirling jet thatimpinges against it To that end, the commercial code Fluent© is used with the correspondingturbulent model and boundary conditions As any turbulent numerical study where jets areinvolved, it needs as boundary condition the velocity and turbulence intensity profiles of thejet, 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, experimentalmeasurements and some fitting of the experimental data will be shown in Section 2 Differentinformation, about the computational tasks and decisions taken, will be presented in Section

3, such as those related with the computational domain, its discretization, the numericalmethods and boundary conditions used and the grid convergence study After that, inSection 4 the different results obtained from the numerical simulations will be presented anddiscussed They will be divided into two subsections: one to see the effect of varying theReynolds number; and another to see the effect of increasing or decreasing the nozzle-to-platedistance Finally, the document will conclude with Section 5, where a summary of the mainconclusions will be presented together with some recommendations one should take intoaccount to enhance the heat transfer from a flat plate when a turbulent swirling jet impingesagainst 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 ofthe 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, swirlingjets with different swirl intensities can be generated Thus, for a given flow rate, or Reynoldsnumber (defined below), through the nozzle, five different swirling jets with five differentswirl intensities, or swirl numbers (defined below), can be obtained When the blades aremounted 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

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,

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 thescourcreated) 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 tothe jet

To model the swirling turbulent jet created by the nozzle is necessary to know both the averagevelocity field and its turbulent structure at the exit nozzle In a cylindrical coordinate system

(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 byV (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 bymeans of a LDA system and, due to the shape of the exit tube of the nozzle (see Fig 1),the radial component of both Vand v has been considered small enough to be neglected:

U=0=u  Typical non-dimensional mean velocity profiles at the nozzle exit, together withits fluctuations, are shown in Fig 3 for two flow rates, the smallest and the highest used,

Q ≈ 100 l/h and Q ≈270 l/h, respectively In the same figure is also included, with a solidline, the fitting of the experimental data (see Ortega-Casanova et al., 2011, for more detailsabout the fitting models used) In Fig 3, the velocity has been made dimensionless using the

mean velocity W c based on the flow rate through the nozzle, W c=4Q/(πD2), 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:

whereρ 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 [shown Fig.

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 ican be defined to quantify the swirl intensity

The evolution of S i versus the Reynolds number for the blade orientation under study isshown in Fig 4.

As it has been pointed out previously, the swirl intensity of the jet S i 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 i increases with Re; and the right one, Re9E3, in which S i decreases with Re S ihas been calculated using (2) andthe 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 Reynoldsnumbers experimentally studied From this figure can easily be understood the behavior of

S i 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 jetsgenerated by the nozzleconfigurationunder study When the swirl increases with the rotation

of the blades, not only the dimensional azimuthal velocity increases, as it was expected, butalso the maximum axial velocity at the axis, appearing a well defined overshoot around it (seethe 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

0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10 4

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

S i

Re

Fig 4 Integral swirl number S i 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 sincethe axial velocity [Fig 5(b)] does not have any characteristic of this phenomena, like a reverseflow at the axis with a stagnation point at a certain radius of the profile This phenomena hasbeen recently observed experimentally by Alekseenko et al (2007), where vortex breakdownoccurs 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 excavationpurposes in order to produce deeper and widerscourson sand beach is the axial overshoottogether with the shift of the azimuthal motion to an annular region They also discuss andgive 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 velocityprofiles 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

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

0 0.5 1 1.5 0

0.1 0.2 0.3 0.4 0.5 0.6

0.7

7E3 11E3 15E3 18.3E3

1.5

7E3 11E3 15E3 18.3E3

r

(b)Fig 5 Mean dimensionless azimuthal (a) and axial (b) velocities measured just downstreamthe nozzle exit

I=∑n

i=1a ie



−r −bi ci

 2 

where r is the dimensionless radial coordinate and a i , b i and c i are fitting parameters

depending on the Reynolds number It has been checked that n=3 is enough to fit quite well,

and with the simplest model, the radial I profile for any value of Re Fig 6 shows the profile

of I for two values of the Reynolds number For low Re and almost all radial positions, the swirling jet is more turbulent than for high Re, with the highest levels of turbulence around the axis, while for high Re, the turbulence is more uniform The profiles shown in Fig 6

0 0.5 1 1.5 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35

in the turbulent model used (see next section)

The turbulent swirling jets measured experimentally at the nozzle exit by means of a LDAsystem are ready to be used as boundary condition in the numerical simulations thank themodels of both azimuthal and axial velocity componentsas well as the one of theturbulentintensity

3 Computational considerations

The numerical simulations have been carried out by means of the commercial codeFluent© (version 6.2.16) As for any numerical turbulent simulations, some previous thingsmust be chosen, such as the turbulent model, the optimum grid, the computational geometryand boundary conditions, etc

Firstly, the computational geometry together with the corresponding boundary conditions

used will be presented The problem is considered to be axisymmetric, so only a 2D r − z

section of the three-dimensional geometry will be solved Fig 7 shows a sketch of theheat transfer problem solved in this work together with the different boundary conditions

used: the swirl generator nozzle is located at a distance H above the solid hot plate (whose

radius isR) where the swirling jet will impinge once it leaves the nozzle as a swirling jet

(the non-dimensional nozzle-to-plate distance will be indicated by the ratio H/D); once the

impinging takes place, the fluid leaves the computational domain through either the side ortop surface The velocity and turbulent intensity profiles shown in the previous section will beintroduced into the simulations by a ``velocity inlet´´ boundary condition at the left-top of thedomain by means of a User Defined Function (abbreviated as UDF in what follows) in order

to model the nozzle above the plate As can be seen in Fig 7, the nozzle exit is surrounded by

an annular solid part of the nozzle It will be modeled giving to the velocity components inthat region an almost zero value through the velocity profile at the ``velocity inlet´´ boundary

z r

R

D/2

Axis Velocity inlet Pressure outlet Wall

Fig 7 Sketch of the computation domain The nozzle and type of boundary condition usedare also included

condition However, Fluent does not allow to specify a turbulent intensity distribution

or profile but a constant value Due to this, in order to indicate the turbulent structure

of the swirling jet when it leaves the swirl generator nozzle by the measurements takenexperimentally, the turbulent intensity I must be turned into other turbulent magnitudes that

will depend on the turbulent model used, asitwill be shown later The surfaces where thefluid is allowed to leave the computational domain (the right and top side) will be indicated

as ``pressure-outlet´´ boundary conditions The bottom of the geometry represents the solidhot plate where the fluid will impinge and is considered as a no-slip surface with a prescribedtemperature and modeled as a ``wall´´ boundary condition (Sagot et al., 2008, showed thatalmost similar results can be obtained when the boundary condition on the solid plate is either

a prescribed temperature or heat flux) Finally, the left line from the nozzle exit to the plate atthe bottom will be indicated as an ``axis´´ boundary condition, since it represents the axis ofsymmetry of the problem

Regarding the turbulent model, the k − ω one will be used, in particular, its version SST This

decision is based on the previous works review presented in the Introduction because is theone used by Sagot et al (2008) (where good agreement between numerical and experimentalsolutions are shown) and because the Reynolds number used here, in this work, rangesbetween 7E3 and 18.3E3, close to those employed in Sagot et al (2008)

The flow we are interested in solving with this problem is considered turbulent, steady andaxisymmetric with the fluid (water) having its density constant (incompressible fluid) as inOrtega-Casanova et al (2011) Thus, the steady Reynolds Averaged Navier-Stokes (RANS)

equations are solved numerically to obtain any fluid magnitude They can be written inCartesian tensor notation as:

the continuity equation:

e=h − ρ p+V  ·  V

where ν is the kinematic viscosity, h is the enthalpy, K is the thermal conductivity and

K e f f = K+K t is the effective thermal conductivity that takes into account the turbulent

thermal conductivity K t : K t = c p μ t /Pr t c p is the specific heat,μ t is the turbulent dynamic

viscosity and Pr tis the turbulent Prandtl number Also, two closure equations are needed: one

to know the turbulent kinetic energy k and another one for the specific turbulent dissipation

where: ΓkandΓω are the effective diffusivity of k and ω, respectively; G k and G ω are the

generation of k and ω, respectively, due to mean velocity gradients; and Y k and Y ωare the

dissipation of k and ω, respectively To know more about their definition and implementation

in Fluent, the reader is remitted to Fluent 6.2 User’s Guide (2005)

Regarding the boundary conditions shown in Fig 7, their implementation in Fluent was asfollows:

pressure, so the boundary condition ``pressure-outlet´´ was chosen;

temperature Tw, so the boundary condition ``wall´´ was chosen, imposing its temperature

at the known constant value

Velocity inlet: in this surface, the corresponding radial dependence axial and azimuthalvelocity profile associated with the corresponding Reynolds number under study wasimposed trough an UDF file through a ``velocity-inlet´´ boundary condition The modelsused to fit the velocity profiles shown in Fig 3 are given in Ortega-Casanova et al (2011),and the reader is remitted there to know more about them On the other hand, regardingthe specification of the swirling jet turbulence levels, the turbulence intensity can beestimated from the LDA measurements, eq (3), and fitting to a radial profile, eq (4),but Fluent does not allow to specify as boundary condition a radial dependence profile forthe turbulence intensity but a constant value For that reason, and in order to specify theradial turbulence distribution of the jet, the turbulence intensity is turned into the variables

k and ω for which are possible to indicate a radial profile as boundary condition Once the

mean axial and azimuthal velocities are measured, W and V, respectively, together with its fluctuations, w  and v  , respectively, and with the turbulent intensity I given by (3), k and

different values to U, I and D Hon a velocity inlet boundary condition, and relating theω

value giving by Fluent on that boundary with them [(11) has been also confirmed by thesame methodology] On the other hand, the jet leaves the nozzle at a constant temperature

grid point to the solid hot plate must be as close to the surface as possible to have an y+

of unity order To achieve this, rectangular stretched meshes with different node densitieshave been generated with the total nodes ranging from 13 000 to 60 000 All meshes have incommon that the mesh nodes density is higher near the solid hot plate, the axis, the mixinglayer and the nozzle exit The grid independence study were done with five grids in order to

choose from them the optimum one The number of nodes, with the maximum value of y+along the solid hot plate indicated in parenthesis, used were: 13 041 (8.0); 22 321 (4.0); 30 000

(0.4); 37 901 (0.4) and 60 551 (0.4) The y+values previously indicated were obtained from thenumerical simulation of the most unfavorable case studied (see next section): the one with

the highest Reynolds number (Re ≈ 18.3E3), and the shortest nozzle-to-plate distance, i.e

H/D = 5 The grid density near the solid hot plate selected as the optimum for this H/D will be reproduced, in that zone, for other nozzle-to-plate distances, or H/D values, that is,

the radial node distribution and the one next to the plate along axial direction: meshes for

different values of H/D will differ only on the axial node distribution and the number of

nodes along that direction

The minimum y+obtained in the grid independence process was 0.4, but in 3 different grids,

so the optimum will be selected in terms of the area-weighted average Nusselt number along