Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
214 = 300 K. The effects of radiation are taken into account,
too. The Discrete Transfer Radiation Model DTRM is employed, assuming all surfaces to be diffuse. A grid
independence analysis has been lead in order to individuate the best compromise between the accuracy
and computational time.
Four different non-structured meshes have been developed considering the model characterized by l =
0.250 m, h = 0.110 m, H = 0.120 m and L = 0.850 m without thermal shields. They have a number of nodes
equal to 10676, 17194, 25206 and 41802, respectively. The third mesh has been adopted because the comparison
of the results in terms of tunnel and side wall temperatures with a coarsen mesh has revealed a little
difference, at most equal to 0.15 as shown in Figure 2.
Furthermore, a sensitivity analysis of the solution on the time step have been accomplished, choosing a value
equal to 0.2 s, and several simulations have been carried out to set properly the radiation model parameters.
Results are presented in terms of dimensionless parameters, defined by the next relations:
. d
f
qD Nu
T T
k =
−
1
d
T T
T T
θ
− =
−
2
2
t D
ν τ =
3
t [s] T[
K ]
10 20
30 40
50 340
360 380
400 420
440 460
480 500
520
Model 1 Model 2
Model 3 Model 4
Fig. 2. Temperature profiles of the tunnel walls depending on time for the four considered mesh grids
III. Results and Discussion
Results for the transient analysis are given in order to study the influence about the main geometric parameters
on the heat transfer interaction for a car under-body with the exhaust system and road surface. Several
characteristic ratios for lh have been studied such as 1.00, 1.45, 2.00, 2.27, 3.26 and 4.00 and for the base
configuration the behaviour depending on H and L has been investigated. The results are presented in terms of
dimensionless variables. Figure 3 depicts the Nusselt number profiles depending on dimensionless time for the
configurations without thermal shields and for different lh ratios. It is observed that Nu decreases as this
geometric ratio increases. In fact, for lh = 1.00 Nu is equal to 48 when the steady state condition is reached
while for lh = 4.00 Nu augments about 20 because the distance between the exhaust system and the under-body
is reduced. Moreover, the steady state condition is observed for growing times as lh decreases.
τ
Nu
0.5 1
1.5 50
55 60
65 70
lh = 1.00 lh = 1.46
lh = 2.00 lh = 2.27
lh = 3.26 lh = 4.00
Without Thermal Shield
Fig. 3. Nusselt number profiles depending on dimensionless time for different lh ratios without thermal shields
Figures 4 present the results in terms of average dimensionless temperature of the tunnel surface
depending on time for different lh ratios. Figure 4a is referred to the configurations without thermal shields and
show that the lower is the lh the higher are the average temperatures. The introduction of the thermal shields
reduces the temperatures and this behaviour is more evident as lh augments, as depicted in Fig. 4b. At lh =
2.27 a decrease equal to 6 is detected.
Because the vehicle is stationary the heat transfer mechanism is due to natural convection together with the
radiation. The exhaust system warms up the surrounding air and
the fluid moves up towards the tunnel only because density changes.
In this way, a thermal plume develops and rises from the duct creating two different branches.
Two convective cells are detected and the temperature and velocity fields are influenced by the introduction of
the thermal shields and by the geometric ratios Figs. 5- 6.
In fact, when the lh ratio changes the considered cavity stretches and the distance between the heated duct
and the tunnel reduces.
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
215
τ θ
0.5 1
1.5 0.2
0.3 0.4
0.5 0.6
0.7
lh = 1.00 lh = 1.46
lh = 2.00 lh = 2.27
lh = 3.26 lh = 4.00
Without Thermal Shield
a
τ θ
0.5 1
1.5 0.1
0.2 0.3
0.4 0.5
0.6 0.7
lh = 1.00 lh = 1.46
lh = 2.00 lh = 2.27
lh = 3.26 lh = 4.00
With Thermal Shield
b Figs. 4. Average dimensionless temperature profiles of tunnel surface
depending on time for different lh ratios: a without thermal shields; b with thermal shields s = 0.007 m
The fluid flow may be choked for the effect of the mutual positions of duct and surrounding surfaces and
the heat transfer becomes less efficient. Moreover, the radiation works parallel to the natural
convection and the heat transfer is more efficient when the duct reaches for the tunnel surfaces.
Fig. 5. Temperature fields at the steady state condition: a lh = 2.27, H = 0.12 m and L = 0.85 m
Fig. 6. Stream functions contours at the steady state condition: a lh = 2.27, H = 0.12 m and L = 0.85 m
Figures 7 show a similar behaviour regarding the average temperatures on the tunnel side walls as the lh
ratio changes. In fact, for lh = 2.27 a reduction of 10 is evaluated by introducing the thermal shields.
Temperatures are generally lower than ones calculated for the tunnel roof because the side walls are not directly
exposed to the thermal plume rising from the exhaust system.
τ θ
0.5 1
1.5 0.1
0.2 0.3
0.4 0.5
0.6 0.7
lh = 1.00 lh = 1.46
lh = 2.00 lh = 2.27
lh = 3.26 lh = 4.00
With Thermal Shield
a
τ θ
0.5 1
1.5 0.2
0.3 0.4
0.5 0.6
0.7
lh = 1.00 lh = 1.46
lh = 2.00 lh = 2.27
lh = 3.26 lh = 4.00
Without Thermal Shield
b Figs. 7. Average dimensionless temperature transient profiles of tunnel
side surface for different lh ratios: a without thermal shields; b with thermal shields s = 0.007 m
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
216 Figure 8 depicts the temperature distribution along the
tunnel roof wall when steady state condition is reached. For small lh ratios the temperature distribution are more
homogeneous, the radiation effect are less evident and the detected temperatures are the highest. When lh raises
the distance between surfaces and duct reduces and the central part of the roof walls is heated by radiation,
significantly. The introduction of the thermal shield decreases the temperature values and distribution are
more homogeneous.
x l
θ
0.25 0.5
0.75 1
1.4 1.45
1.5 1.55
1.6 1.65
lh = 1.00 lh = 1.46
lh = 2.00 lh = 2.27
lh = 3.26 lh = 4.00
With Thermal Shield
Fig. 8. Temperature profiles of tunnel roof surface for different lh at steady state condition with thermal shields
In Figure 9 results are presented in terms of average dimensionless temperature of the tunnel side walls for
different lh ratios and investigate the influence of the thermal shields.
As described, the introduction of the thermal shield tends to decrease the temperatures evaluated on the
tunnel side walls and this behavior is more evident as lh augments.
y h
θ
0.25 0.5
0.75 1
1.1 1.2
1.3 1.4
1.5 1.6
lh = 1.00 lh = 1.46
lh = 2.00 lh = 2.27
lh = 3.26 lh = 4.00
With Thermal Shield
Fig. 9. Dimensionless temperature profiles of tunnel side surface for different lh ratios at steady state condition for the configuration with
thermal shields s = 0.007 m
Figure 10 shows the Nusselt number profile depending on time for the base configuration with
thermal shields for different values of the underbody total width, L. Nusselt number raises as L decreases
although differences are not so significant.
In this way, it is possible to focus the attention on simpler models characterized by small L and reduce
computational times.
τ
Nu
1 2
55 60
65 70
L
tot
= 0.60 m L
tot
= 0.70 m L
tot
= 0.85 m L
tot
= 0.90 m L
tot
= 1.00 m
With Thermal Shield
Fig. 10. Nusselt number profile depending on time for the base configuration with thermal shields for different values of L
Figures 11 are referred to the base configuration characterized by lh = 2.27 with thermal shields and
describes the behaviour of the system by changing the distance of the under-body from the road surface. Figure
10a depicts the Nusselt number profiles for different values of H. It is observed that for the largest value of H
the steady state condition is not reached.
Nu rises as H grows although differences are not much evident.
The steady state condition, moreover, is detected for larger dimensionless times as H augments.
Figure 11b presents the results in terms of temperature distribution along the tunnel roof wall.
From the figure it is easy to observe two relative maximum. This behaviour is related to presence of the
convective cells rising from the heated duct. The maximum values are attained at xl = 0.3 and 0.7,
respectively and profiles are perfectly symmetric.
The highest maximum values of dimensionless temperature are detected for H = 0.009 m because the
fluid flows difficultly in the tunnel and in correspondence of the inlet and outlet sections.
The minimum value of the dimensionless temperature is detected at xl = 0.5.
IV. Conclusion
