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

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Numerical Simulation of the Heat Transfer from a Heated Solid Wall to an Impinging Swirling Jet 17

5 10 15 20 25 30

7E3

9E3

11E3

13E3

15E3

17E3

18.3E3

Re ↑

H/D

Fig. 14. Evolution of Nu for the constant values of Re indicated in the legend.

can be seen in Fig. 14, where the evolution of

Nu for each Re studied is shown versus the

nozzle-to-plate distance: for a given Reynolds number, the heat transfer always decreases

when the separation increases; the decreasing will be higher or lower depending on the

corresponding value of Re.

5. Conclusion

In this work, numerical simulations of the impingement of a turbulent swirling jet against a

solid hot plate at constant temperature have been presented. The jet has been modeled by

experimental measurements taken by means of a LDA equipment at the exit of a nozzle that

imparts the swirl to the jet by guided blades, and with the jet swirl intensity depending only

on the Reynolds number. Seven Reynolds numbers and three nozzle-to-plate distances have

been simulated, which gives a total of 21 numerical simulations. The analysis of the results

gives the following conclusions:

Firstly, taken into account that the main objective of this work was to see if the performance

of the heat transfer was higher for the highest nozzle-to-plate distance than for lower ones,

as in Ortega-Casanova et al. (2011) under seabed excavation tasks, it must be said that the

question is answered negatively: the heat transfer from the solid hot plate to the impinging

swirling jet always decreases when the separation increases, at least for the type of jets used

here. Therefore, the sand, that is, the granular media used in Ortega-Casanova et al. (2011)

on which the swirling jet impinged, plays an important role in reaching the scour its ﬁnal

shape, especially when H/D

= 30 and the highest Reynolds number jets are used. For that

combination, the impinging swirling jet creates the deepest and widest scours. When the

same swirling jet impinges against undeformable solid surfaces, the qualitative results, in term

of the heat transferred from the surface, are totally different from those obtained when the

impingement takes place against granular media and cannot be extrapolated from excavation

related problems to heat transfer ones, and vice versa.

189

Numerical Simulation of the Heat Transfer from

a Heated Solid Wall to an Impinging Swirling Jet

18 Will-be-set-by-IN-TECH

Secondly, the area-weighted average Nusselt number on the solid hot plate, always increases

with Reynolds number and for any value of the nozzle-to-plate distance, i.e. for any H/D.

Almost the same happens with the Nusselt evaluated at the stagnation point r

= 0: only for

≈ 9E3 and H/D = 5, this is not true due to the combination of the highest jet swirl intensity

and the smallest nozzle-to-plate distance, for which a deceleration of the swirling jet takes

place but without being high enough for the vortex breakdown to be observed. The effect of

the deceleration was not to increase the Nusselt number at the stagnation point but to create

amoreuniformNu region around the axis. Despite that decreasing in the stagnation point

Nusselt number, the mean heat transfer, i.e.

Nu, on the surface always increases with Re.On

the other hand, for high nozzle-to-plate distances, the beneﬁts of using high Reynolds number

jets instead of low ones, are higher than at small nozzle-to-plate distances. This fact has much

to do with the displacement of the azimuthal motion of the swirling jet to an annular region

off the axis that has more inﬂuence on the heat transferred from the solid hot plate when the

nozzle-to-plate is the highest studied. The above mentioned displacement of the azimuthal

motion takes place for Reynolds numbers greater than 13E3, approximately. It could have

been interesting to study the heat transfer when the solid hot wall is impinged with swirling

jets that have undergone breakdown and to compare the Nusselt distributions on the solid

hot wall due to the impingement of swirling jets with and without breakdown. Unfortunately,

vortex breakdown has not been observed experimentally with the nozzle conﬁguration and

ﬂow rates used in this work.

And ﬁnally, the area-weighted average Nusselt number always decreases with the increasing

of the nozzle-to-plate distance: for a given Reynolds number, the smaller the nozzle-to-plate

distance, the higher the heat transferred from the plate to the jet.

6. Acknowledgement

The author wants to thank Nicolás Campos Alonso, who was the responsible for taking

the LDA measurements at the laboratory of the Fluid Mechanics Group at the University of

Málaga.

All the numerical simulations were performed in the computer facility ``Taylor´´ at the

Computational Fluid Dynamic Laboratory of the Fluid Mechanic Group at the University of

Málaga.

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192

Two Phase Flow, Phase Change and Numerical Modeling

Recent Advances in Modeling Axisymmetric

Swirl and Applications for Enhanced Heat

Transfer and Flow Mixing

Sal B. Rodriguez

and Mohamed S. El-Genk

Sandia National Laboratories,

University of New Mexico

USA

1. Introduction

The concept of enhanced heat transfer and flow mixing using swirling jets has been

investigated for nearly seven decades (Burgers, 1948; Watson and Clarke, 1947). Many

practical applications of swirling jets include combustion, pharmaceuticals, tempering of

metals, electrochemical mass transfer, metallurgy, propulsion, cooling of high-power

electronics and computer chips, atomization, and the food industry, such as improved pizza

ovens. Recently, swirling jet models have been applied to investigate heat transfer and flow

mixing in nuclear reactors, including the usage of swirling jets in the lower plenum (LP) of

generation-IV very high temperature gas-cooled reactors (VHTRs) to enhance mixing of the

helium coolant and eliminate the formation of hot spots in the lower support plate, a safety

concern (Johnson, 2008; Kim, Lim, and Lee, 2007; Laurien, Lavante, and Wang, 2010;

Lavante and Laurien, 2007; Nematollahi and Nazifi, 2007; Rodriguez and El-Genk, 2008a, b,

c, and d; Rodriguez, Domino, and El-Genk, 2010; Rodriguez and El-Genk, 2010a and b;

Rodriguez and El-Genk, 2011).

There are many devices and processes for generating vortex fields to enhance flow mixing

and convective heat transfer. Figure 1 shows a static helicoid device that can be used to

generate vortex fields based on the swirling angle, θ. Recent advances in swirling jet

technology exploit common characteristics found in axisymmetric vortex flows, and these

traits can be employed to design the vortex flow field according to the desired applications;

among these are the degree of swirl (based on the swirl number, S) and the spatial

distributions of the radial, azimuthal, and axial velocities.

For a 3D helicoid, the vortex velocity in Cartesian coordinates can be approximated as:

oo o o

V(x,y,z) = usin(2π

)i - u sin(2πx)

+ w k





(1)

For a jet with small radius r, the above velocity distribution can be represented in cylindrical

coordinates as

v (r) = u sin(πr) (2)

Two Phase Flow, Phase Change and Numerical Modeling

194

and

w(r) = w . (3)

Whereas there are about a dozen or so definitions of the swirling number S, we define it as a

pure geometric entity, consistent with various investigators in the literature (Arzutug and

Yapici, 2009; Bilen

et al. 2002; Kerr and Fraser, 1965; Mathur and MacCallum, 1967):

()

S = tan θ

(4)

Fig. 1. Helicoid swirl device and associated velocity distribution

Fig. 2. Comparison of conventional vs. swirling jet velocity streamlines

Recent Advances in Modeling Axisymmetric Swirl

and Applications for Enhanced Heat Transfer and Flow Mixing

195

Increasing S has been shown to increase the entrainment of the surrounding fluid (K

)

linearly (Kerr and Fraser, 1965) as:

K = 0.35 + 1.4S . (5)

As a result of the increased entrainment, mixing, and turbulence, a swirling jet has a wider

core diameter than a conventional impinging jet, as shown in Figure 2. Further, the

azimuthal velocity of the helicoid vortex increases as S increases,

1 +







(6)

while the axial velocity decreases (Rodriguez and El-Genk, 2010b),

z,0

= .

V3S

1 +







(7)

is the velocity at the jet exit when no swirling occurs (conventional jet).

2. Vortex modeling

For this discussion, the broad field of turbulent vortex flows is narrowed down to swirling

Newtonian fluid jets that are axisymmetric and incompressible. These vortices have been

studied in many coordinates. A rather “unnatural” selection is Cartesian coordinates, as the

flow field rotates about the axisymmetric axis, which is generally selected as the z

coordinate. Therefore, the axisymmetric vortex flow is symmetric about the z axis as the

flow rotates in the azimuthal direction, and continues to expand in the z direction.

As noted in Figure 2, both impinging and swirling jets have the same source diameter and

expand into an open domain. While the conventional jet experiences no azimuthal rotation,

the swirling jet with θ = 45º has a strong rotational flow component. Note that the swirling

jet has a wider jet core diameter, a higher degree of entrainment of surrounding fluid, and

its azimuthal rotation causes more fluid mixing than the conventional impinging jet.

Because part of the axial momentum in the swirling jet is converted to azimuthal

momentum, the axial velocity component decays much faster than for the conventional jet.

Vortex research in Cartesian coordinates includes the Green-Taylor vortex (Taylor and

Green, 1937) and the helicoid vortex discussed herein (Rodriguez and El-Genk, 2008a, 2010b

and d). Various researchers selected spherical coordinates (Gol’Dshtik and Yavorskii, 1986;

Hwang and Chwang, 1992; Tsukker, 1955), but the vast majority of the research found in the

literature is in cylindrical coordinates (Aboelkassem, Vatistas, and Esmail, 2005; Batchelor,

1964; Burgers, 1948; Chepura, 1969; Gortler, 1954; Lamb, 1932; Loitsyanskiy, 1953;

Martynenko, 1989; Newman, 1959; Rankine, 1858; Rodriguez and El-Genk, 2008a, 2010b and

d; Sullivan, 1959; Squire, 1965). Cylindrical coordinates are chosen primarily due to its

geometric simplicity and excellent mapping of the vortex behavior onto a coordinate

system—in particular, as a 3D vortex spins azimuthally, stretching about the z axis, the

vortex velocity field engulfs a cylindrical geometry. Certainly, conical coordinates could be

Two Phase Flow, Phase Change and Numerical Modeling

196

used, but they are not as convenient to manipulate mathematically. Indeed, as the vortex

expands, forming a 3D cone, care needs to be taken such that the swirl field is not so large

that it generates a field with an angle that is larger than the conical coordinate system.

Consider a cylindrical coordinate system with r, θ, and z as the radial, azimuthal, and axial

components, as defined in Figure 1. For 3D, steady state, negligible gravitational effect,

incompressible, Newtonian fluids with symmetry about the z axis (axisymmetry), the

momentum and conservation of mass equations are given as:

222

v1 1

uu uuuu

rr z r rrrr z



∂

∂∂ ∂∂∂

−+ =− +ν + −+



∂∂ρ∂∂∂∂



(8)

222

vv v v1vv vu

rr z rrrr z



∂∂∂∂∂

++ =ν + −+



∂∂∂∂∂



(9)

ww www

rz zrrrz



∂

∂∂ ∂ ∂∂

+=−+ν++



∂∂ρ∂ ∂ ∂∂



(10)

uu w

rr z

∂∂

++ =

∂∂

(11)

An inspection of the literature over the past 150 years shows that the above equations are

generally the departure point for generating analytic solutions for axisymmetric swirling

flows (vortices) (Aboelkassem, Vatistas, and Esmail, 2005; Batchelor, 1964; Burgers, 1948;

Chepura, 1969; Gortler, 1954; Lamb, 1932; Loitsyanskiy, 1953; Martynenko, 1989; Newman,

1959; Rankine, 1858; Rodriguez and El-Genk, 2010b; Sullivan, 1959; Squire, 1965).

3. Helicoid swirl modeling

Helicoids for the generation of swirling-flow fields may be produced by various methods.

One method is via a geometrical specification of a static, swirl device that consists of

helicoidal surfaces, and another is a mathematical description of a swirl boundary condition

(BC) that reproduces the flow field. It has been reported in the literature and confirmed

herein that if a computational fluid mechanics (CFD) code does not have a swirl boundary

option, it would be customary to develop the geometry for a swirl device and then mesh it

(Duwig

et al. 2005; Fujimoto, Inokuchi, and Yamasaki, 2005; Garcia-Villalba, Frohlich, and

Rodi, 2005; Rodriguez and El-Genk, 2008a, b, c, and d; Rodriguez, Domino, and El-Genk,

2010; Rodriguez and El-Genk, 2010a and b; Rodriguez and El-Genk, 2011). Several static

swirl devices are shown in Figure 3 (Huang, 1996; Huang and El-Genk, 1998; Larocque,

2004; Rodriguez and El-Genk, 2010a and b; Rodriguez and El-Genk, 2011). The swirl device

considered in this chapter consists of a sharp cone that surrounds four helical surfaces off-

set by 90° and spiral symmetrically around the cylinder (Figure 4, LHS). Because the

swirling device is static, the fluid flows around the helicoid surfaces, producing a swirling

motion as it travels around the surfaces.

Deciding on a particular swirl angle

a priori for the swirling device, and then meshing its

geometry, is computationally-intensive and time consuming, not to mention that it is a

tedious, error-prone, and expensive process. Instead, as will be shown

a posteriori, having a

Recent Advances in Modeling Axisymmetric Swirl

and Applications for Enhanced Heat Transfer and Flow Mixing

197

closed form, mathematical formulation for the swirling field not only simplifies the

computation requirements but also yields significant insights concerning the behavior of the

velocity fields and their impact on heat transfer.

Fig. 3. Various Swirl Devices Found in the Literature (Huang, 1996; Huang and El-Genk,

1998; Larocque, 2004; Rodriguez and El-Genk, 2010a and b; Rodriguez and El-Genk, 2011)

Fig. 4. Comparison of Geometric and BC Swirl Models

Now, once a mathematical swirl formulation is derived, it is a straightforward matter to apportion the

jet velocity fields such that a given swirl angle is uniquely specified (Equations 4, 6, and 7).

As a

result, a mathematically-generated velocity field with no helicoid surfaces (i.e. just swirl BCs)

can very closely approximate the swirl fields of the geometric helicoid devices shown in Figure

1. To demonstrate this, two meshes are shown: one with a geometric swirl device and the other

with swirl BCs (see Figure 4). The CFD computation for both meshes has the same S.

A comparison of both models shows that the total velocity vector streamlines are essentially

indistinguishable, as shown in Figure 5. The azimuthal velocity compared in Figure 6, again

shows that the velocity streamlines are fairly identical. This methodology results in

Two Phase Flow, Phase Change and Numerical Modeling

198

significant savings in the meshing and computational effort, especially because the helicoid

geometry requires finite elements that are about four times smaller than the rest of the

model, and further, the time step is dominated by the smallest finite element.

Fig. 5. Comparison of the Geometric and BC Swirl Models: U + V + W Streamlines

Fig. 6. Comparison of the Geometric and BC Swirl Models: Azimuthal Velocity Streamlines