ISTP-16, 2005, PRAGUE
16TH INTERNATIONAL SYMPOSIUM ON TRANSPORT PHENOMENA
CFD ANALYSIS OF TURBULENT THERMAL MIXING
OF HOT AND COLD AIR IN AUTOMOBILE HVAC UNIT
Hideo Asano(1)(2), Kazuhiko Suga(3), Masafumi Hirota(2), Hiroshi Nakayama(2),
Shunsaku Hirayama(1) and Yasuhiro Mizuno(1)
(1) DENSO CORPORATION, Kariya 448-8661, Japan.
(2) Nagoya University, Nagoya 464-8603, Japan.
(3) Toyota Central R & D Labs., Inc., Nagakute, Aichi 480-1192, Japan.
Corresponding author: [email protected]
Phone: +81-52-789-2702 Fax: +81-52-789-2703
Keywords: Automobile air-conditioning, Turbulent thermal mixing, Planar shear layer,
Second moment closure, GGDH heat flux model,
Abstract
The results of numerical simulations on
turbulent flow and thermal mixing of hot and
cold airflows in the HVAC unit used in
automobile
air-conditioning
system
are
presented, and their reliability is examined by
comparing them with experimental data. To
simplify the problems, the thermal mixing in the
HVAC unit has been modeled by the twodimensional planar turbulent mixing layer.
Three turbulence models have been tested;
namely, standard linear k-ε model, low-Re cubic
nonlinear k-ε model, and two-component-limit
second moment closure (TCL SMC). For
turbulent heat fluxes, prescribed turbulent
Prandtl numbers are applied along with the k-ε
models.
TCL SMC is coupled with a
generalized gradient diffusion hypothesis
(GGDH) or its higher order version (HOGGDH)
model. The results suggest that TCL SMC +
(HO)GGDH model can predict the distributions
of turbulent heat fluxes and mean temperature
of the mixed flow successfully.
1 Introduction
Turbulent mixing of two flows with different
velocities, temperatures and/or concentrations is
encountered in many engineering applications,
such as chemical reactor, piping system in
power plant, combustion chamber, etc. One of
the typical examples can be also found in the
HVAC (Heating, Ventilating and AirConditioning) unit used in the automobile airconditioning system [1]. Figure 1 shows a
schematic of the HVAC unit, in which a fan, an
evaporator and a heater-core are packaged. In
this unit, all air taken by the fan is once cooled
down by the evaporator to reduce humidity, and
a part of this cold air is heated by the heatercore. Then, hot and cold air is mixed at
appropriate flow-rate ratios to control the air
temperature blown into the compartment of an
automobile. The temperature of the mixed
airflow is determined by the flow-rate ratio of
the hot and cold air, and it is controlled by the
opening of the air-mix door that is settled
between the evaporator and the heater-core.
Thus it follows that, in the HVAC unit, the hot
and cold airflows meet at various angles and
various velocity ratios depending on the air
temperature required in the automobile
compartment.
Nowadays it is strongly desired that the
HVAC unit be designed virtually by making the
most use of CAE to reduce the developing time.
This digital engineering requires the numerical
simulations of turbulent mixing of hot and cold
airflows. At the present stage, however, the
reliability of the calculated velocity and
temperature distributions in the mixed airflow is
not high enough to be directly applicable to the
Defroster Duct
Air-Mix Door
Cold Air
Cold Air
Blower
Ho
Evaporator
r
t Ai
Face Duct
Heater Core
Foot Duct
Fig. 1. Conceptual illustration of HVAC unit
design of the HVAC unit. This is because the
velocity field of the mixed flow usually shows
complex features, and the present turbulence
models are not designed to predict it with
sufficient accuracy. In addition to the problems
of turbulence models, the modeling of the
turbulent heat fluxes is also a key issue for
improving the performance of numerical
simulations of the turbulent thermal mixing
encountered in the HVAC unit [1]. Detailed
data on the flow and temperature fields in the
HVAC unit are, however, quite scarce, and thus
the suitability of turbulence and turbulent heat
flux models for the simulation of thermal
mixing process encountered in the HVAC unit
has not been examined in detail yet.
With these points as background, in this study,
we have made numerical simulations on
turbulent thermal mixing of hot and cold
airflows in the HVAC unit, and examined their
reliability by comparing the results with
experimental data.
The turbulent thermal
mixing in HVAC unit has been modeled by the
mixing of simple two parallel flows with
different velocities and temperatures, i.e., twodimensional planar turbulent mixing layer, as
shown in Fig. 2.
We have tested three turbulence models for
velocity field and they have been coupled with
three turbulent heat flux models; the first
turbulence model is the standard linear k-ε
model and the second one is the low-Reynoldsnumber cubic nonlinear k-ε model [2], both of
which have been coupled with prescribed
turbulent Prandtl numbers for turbulent heat
fluxes. The third turbulence model is the twocomponent-limit second moment closure (TCL
SMC) [3], which is coupled with a generalized
gradient diffusion hypothesis (GGDH) [4] or its
higher order version (HOGGDH) [5] for
turbulent heat fluxes. TCL SMC is the latest
version of the Reynolds stress transport model,
and its usefulness has been confirmed for a few
complex flows. The tested cases are, however,
wall turbulence and its performance in the
application to the mixing layer is not clarified
yet. In this study, we compare the simulated
results with measured ones [6], and evaluate the
performance and suitability of these models in
the application to the turbulent thermal mixing
in the HVAC unit.
2 Flow geometry
As described above, in this study, the
turbulent thermal mixing in the HVAC unit has
been modeled by the two-dimensional planar
turbulent mixing layer. Figure 2 shows the
schematic of the flow geometry. Cold airflow at
Tc = 30°C and hot airflow at Th = 80°C are
mixed in the test section after flowing through
the settling chambers, flow nozzles and
developing regions. Each developing region has
a cross section of 200 mm × 97.5 mm, and the
splitter plate that divides the hot and cold flows
is 5 mm thick. The mixing section has a cross
section of 200 mm × 200 mm.
These
dimensions of the test channel were determined
referring to the practical HVAC unit [1].
The velocity of the cold flow Uc was kept at 4
m/s, and that of the hot flow Uh was set at 2 m/s
and 4 m/s (velocity ratio r = Uh/Uc = 0.5 and 1,
respectively). Under both velocity ratios, cold
air flows in the upper half of the test channel
and hot air flows in the lower half. It was
confirmed that in the cold flow side a turbulent
boundary layer about 15 mm thick was formed
at the end of the splitter plate. The Reynolds
number based on the momentum thickness of
this boundary layer is about 380. Under these
experimental conditions, the Richardson number
is as small as 1.84×10-4, thus the influence of
the buoyancy force is negligibly small. We
confirmed that the velocity distribution
measured under this non-isothermal condition
agreed well with that obtained under the
2
CFD ANALYSIS OF TURBULENT THERMAL MIXING OF HOT AND
COLD AIR IN AUTOMOBILE HVAC UNIT
500
3 Turbulence models
700
Cold air
Uc, Tc
200
X2
X1
Hot air
Uh, Th
enlarged view
5
8
Nozzle
35
Splitter plate
Fig. 2. Test channel used in the experiment
(2-D mixing layer)
Fig. 3 Grid system used around the tip of the
splitter plate
isothermal condition. The coordinate system is
also shown in Fig. 2. The mean and fluctuating
velocity components in each direction are
denoted as U1, U2 and u1, u2, respectively. T
and t denote the mean and fluctuating
temperatures. Details of the experiments are
described in a reference [6].
In the numerical simulation, two-dimensional
calculation has been made in this study. The
exit of the calculation region is set at 1000 mm
downstream from the tip of the splitter plate to
avoid the influence of the exit condition on the
simulated flow field. In order to calculate the
flow field just after the flow merging accurately,
quite a fine grid system is formed around the tip
of the splitter plate. Figure 3 shows the grid
system near the splitter-plate tip used in the
present numerical simulation. The number of
the grid points has been changed from 51,000 to
125,000 depending on the turbulence models.
The flow parameters measured at 35 mm
upstream from the tip of the splitter plate have
been used as the initial values of the numerical
simulations.
3.1 Flow field
Considering the computation time allowed in
the practical design of HVAC unit, three kinds
of RANS turbulence models for flow field have
been tested in this study. The first model is the
standard linear k-ε model that has been widely
used in various engineering applications. The
second model is the low-Reynolds-number
cubic nonlinear k-ε model [2]. This model can
reproduce the anisotropy of turbulent stresses
accurately and thus it is expected that the
reliability of predictions of complex turbulent
flows be improved.
The third model is the two-component-limit
second moment closure (TCL SMC) [3]. This is
the latest Reynolds stress transport model, and
can predict the turbulence anisotropy more
successfully than other models. Although its
usefulness has been confirmed for a few
complex flows, such as 3-D curved duct flow
and turbulent obstacle flow, the tested cases are
still limited to wall turbulence and its
performance in the application to the free
turbulence is not fully clarified yet. Recently,
Suga applied this model to the turbulent mixing
layer and obtained satisfactory results. Since
these turbulence models need lengthy
expressions, details of each turbulence model
are not described in this paper: see the
references for details of these models.
3.2 Turbulent heat fluxes
The standard k-ε models and the lowReynolds-number cubic nonlinear k-ε model
adopted in this study have been combined with
prescribed turbulent Prandtl numbers Prt. This
is the simplest way to express the turbulent heat
fluxes. Prt is assumed to be 0.9 for the low-Re
nonlinear k-ε model. Three values of Prt, 0.1,
0.5 and 0.9 are tested with the standard k-ε
model to examine the influence of Prt-values on
the mean temperature distributions. As for TCL
SMC, two turbulent heat flux models have been
tested. One is a generalized gradient diffusion
3
hypothesis (GGDH) model [4], and the other is
its high-order version (HOGGDH). These
models are expressed as follows [5].
Prt model:
u iθ = −
ν t ∂Θ
(1)
Prt ∂xi
GGDH model:
u iθ = −cθ τ u i u j
k
∂Θ
, τ=
∂x j
ε
(2)
HOGGDH model:
u iθ = −cθ kτ (σ ij + α ij )
σ il = cσ 1
ui u j
k
⎛
ul u j
⎜
⎝
k
α ij = cα 1τ ⎜ Ω il
Ω ij =
+ cσ 2
∂U i ∂U j
−
∂x j
∂xi
∂Θ
∂x j
ui ul ul u j
k2
+ Ω li
u j ul ⎞
⎟
k ⎟⎠
(3)
(4)
(5)
(6)
See the reference for details of the model
coefficients [5].
In general, Prt model cannot predict the
turbulent heat fluxes reasonably in complex
turbulent flows that have mean temperature and
velocity gradients in multiple directions. In
other words, this model cannot reproduce the
streamwise turbulent heat flux under the fully
developed thermal condition. This is because
Prt model assumes that turbulent heat flux in the
Xi-direction uit is generated only by the
contribution of the mean temperature gradient in
the Xi-direction, although uit is generated by the
mean temperature gradients not only in the Xidirection but also in the Xj-directions (i ≠ j).
GGDH heat flux model is generally
successful in the computation of complex
thermal field. It is, however, known that GGDH
cannot predict the streamwise heat flux
component reasonably well [5] though it is
much better than Prt. The high order version of
GGDH mode, i.e., HOGGDH model, was
developed by expanding GGDH with extra
terms including a quadratic product of the
Reynolds stress tensor to improve the
performance of the GGDH model. It is more
effective to predict the streamwise turbulent
heat flux. In general, the streamwise turbulent
heat flux does not exert important influences on
the mean temperature distribution in a fully
developed flow. The flow in a practical HVAC
unit is, however, so complex accompanied by
frequent changes of its direction that the
streamwise turbulent heat flux can become as
important as transverse one to predict the mean
temperature distribution. Since the performance
of the GGDH-type heat flux models relies on
the accuracy of the predicted turbulence
anisotropy, GGDH and HOGGDH have been
coupled with TCL SMC in this study.
4 Results and discussion
In this paper, the results obtained with k-ε
models coupled with Prt are presented at first.
Then the results of TCL SMC + (HO)GGDH
models are shown and their performance on the
prediction of thermal field in the plane turbulent
mixing layer is examined in detail.
4.1 Flow field with k-ε models
Figure 4 shows the distributions of the
streamwise mean velocity U1/Uc (left) and the
turbulent shear stress u1u2 (right) obtained at the
velocity ratio r = Uh/Uc = 1. The solid line
shows the results calculated by the standard
(STD) k-ε model, and the broken line shows
those by low-Reynolds-number nonlinear
(LRN) k-ε model. The experimental results are
shown by open symbols. In this paper, the
results obtained at three streamwise locations,
X1/D = 0.5, 1.0 and 2.0, are presented, where D
denotes a half of the side length of the mixing
cross section (= 100 mm). These locations have
been determined considering the size of
practical HVAC units, at which the flow does
not attain the self-similar state [7].
As shown in Fig. 4, the distributions of U1
obtained by STD k-ε model agree well with
those by LRN k-ε model at all X1/D, and these
simulated results are in good agreement with
experimental ones at all X1/D. The distributions
of u1u2 are also successfully reproduced by
these k-ε models, although their peak values are
4
CFD ANALYSIS OF TURBULENT THERMAL MIXING OF HOT AND
COLD AIR IN AUTOMOBILE HVAC UNIT
X1/D=0.5
0.5
r =1
X1/D=1.0
X1/D=2.0
X1/D=0.5
4.2 Thermal field with k-ε + Prt models
X1/D=2.0
X1/D=1.0
r =1
X2/D
X2/D
STD k-ε
LRN k-ε
EXP
0.0
0.0
-0.5
0.5
STD k-ε
LRN k-ε
EXP
0
0.5
1.0 0
0.5 1.0 0
U1/Uc
0.5
1.0
-0.5
-1
0
-1
0
-1
0
1
2
u1u2/Uc *100
Fig. 4. Mean velocity and turbulent shear stress
distributions predicted by k-ε models (r = 1)
X1/D=0.5
0.5
X1/D=1.0
X1/D=2.0
r =0.5
0.5
X1/D=0.5
STD k-ε
LRN k-ε
EXP
X2/D
X2/D
STD k-ε
LRN k-ε
EXP
0.0
0.0
-0.5
0
0.5
1.0 0
0.5 1.0 0
U1/Uc
X1/D=2.0
X1/D=1.0
r =0.5
0.5
1.0
-0.5
-1
0
-1
0
-1
0
1
2
u1u2/Uc *100
Fig. 5. Mean velocity and turbulent shear stress
distributions predicted by k-ε models (r = 0.5)
somewhat underpredicted at X1/D = 0.5. These
results suggest that, at r = 1, the STD k-ε model
as well as the LRN k-ε model is good enough
for predicting the flow field in the planar
turbulent mixing layer.
At the velocity ratio of r = 0.5 shown in Fig. 5,
however, the reliability of the simulated results
becomes much lower than that for r = 1. From
the U1 distributions shown in Fig. 5 (left), it is
found that the thickness of the mixing layer is
overpredicted with LRN k-ε model and that the
recovery of the velocity deficit region at X1/D =
2.0 is delayed in both k-ε models. As for u1u2
distributions, LRN k-ε model generally
overpredicts the experimental results, and the
locations of the peak values are not well
predicted by both models. As a whole, the
reliability of the results obtained by LRN k-ε
model is lower than that of STD k-ε model
under the velocity ratio of r = 0.5.
At first, we show the results of the thermal
fields calculated with a prescribed turbulent
Prandtl number of 0.9; the flow field is
calculated with two k-ε models. The results at r
= 1 are presented in Fig. 6; distributions of
mean temperature (T-Tc)/(Th-Tc), streamwise
turbulent heat flux u1t/Uc(Th-Tc) and transverse
turbulent heat flux u2t/Uc(Th-Tc) are shown from
the top of this figure. The transverse component
of turbulent heat flux u2t, which dominates the
heat transport in the mixing layer, is
underpredicted over all X1/D. Moreover, the
streamwise component u1t, which is in almost
the same level as u2t in the experiment, is nearly
zero in the simulations. As a result of such
underpredictions of the turbulent heat fluxes, the
simulation with Prt = 0.9 tends to underpredict
the development of the thermal mixing layer.
The difference between the measured mean
temperature and calculated one becomes larger
in the region further downstream from the origin
of the mixing layer. Quite similar results are
obtained at r = 0.5 shown in Fig. 7 and the
difference between the experimental results and
numerical ones is increased in comparison with
the case of r = 1.
It is thought that such an underprediction of
u2t as observed above can be improved by
assuming smaller turbulent Prandtl number.
Hence, in this study, we made the calculations
with three turbulent Prantdl numbers of Prt =
0.1, 0.5 and 0.9. In these calculations, the
standard k-ε model has been coupled with Prt,
because the reliability of the flow field predicted
by STD k-ε model is higher than that of LRN kε model. Figures 8 and 9 show the results for r
= 1 and 0.5, respectively. At r = 1, the
reliability of u2t prediction is much improved
with the turbulent Prandtl number of 0.5. In
free turbulence, Prt is often assumed to be in the
range of 0.55 - 0.7: the result of r = 1 supports
this assumption about Prt-value. On the other
hand, at r = 0.5, the reliability of u2t prediction
is not improved so much by decreasing Prt.
This suggests that Prt changes depending on the
velocity field even in the relatively simple
planar thermal mixing layer.
5
X1/D=2.0
X1/D=1.0
X1/D=0.5
0.5
r =1
X1/D=2.0
X1/D=0.5
0.5
r =0.5
X2/D
0.0
0.5
0.5 1.0 0
1.0 0
(T-Tc)/(Th-Tc)
X1/D=0.5
0.5
X1/D=1.0
0.5
0.0
-0.5
0
1.0
X1/D=2.0
0.5
r =1
0.5
0.5 1.0 0
1.0 0
(T-Tc)/(Th-Tc)
X1/D=0.5
0.5
-0.5
1.0
X1/D=2.0
X1/D=1.0
STD k-ε
LRN k-ε
EXP
0
0.5
r =0.5
0.5
1.0 0
0.5
1.0 0
(T-Tc)/(Th-Tc)
X1/D=0.5
X1/D=1.0
0.5
-0.5
1.0
X1/D=2.0
0
0.5
r =1.0
0.5
1.0 0
0.5
1.0 0
(T-Tc)/(Th-Tc)
X1/D=0.5
X1/D=1.0
0.5
1.0
X1/D=2.0
r =0.5
STD k-ε
LRN k-ε
EXP
0.0
0.0
X2/D
X2/D
0.0
0.0
Prt=0.9
Prt=0.5
Prt=0.1
EXP
-0.5
-0.5
-2 -1 0 1 -2 -1 0 1 -2 -1 0 1 2
-2 -1 0 1 -2 -1 0 1 -2 -1 0 1 2
X1/D=1.0
u1t/Uc/(Th-Tc)*100
X1/D=2.0
0.5
X2/D
X2/D
r =1
0.0
0.0
X1/D=0.5
0
1
-1
X1/D=2.0
u1t/Uc/(Th-Tc)*100
0.5
r =0.5
0
1
-1
0
1
Fig. 6. Thermal field
by Prt model (r = 1)
2
-0.5
-1
0
X1/D=1.0
u1t/Uc/(Th-Tc)*100
X1/D=2.0
0.5
0.0
0.0
STD k-ε
LRN k-ε
EXP
u2t/Uc/(Th-Tc)*100
X1/D=0.5
r =1.0
STD k-ε
LRN k-ε
EXP
-0.5
-1
X1/D=1.0
-0.5
-2 -1 0 1 -2 -1 0 1 -2 -1 0 1 2 -0.5
-2 -1 0 1 -2 -1 0 1 -2 -1 0 1 2
X2/D
X1/D=0.5
Prt=0.9
Prt=0.5
Prt=0.1
EXP
X2/D
u1t/Uc/(Th-Tc)*100
0.5
X1/D=2.0
0.0
X2/D
0
X1/D=1.0
Prt=0.9
Prt=0.5
Prt=0.1
EXP
X2/D
-0.5
X1/D=0.5
0.5
Prt=0.9
Prt=0.5
Prt=0.1
EXP
X2/D
X2/D
0.0
X1/D=2.0
r =0.5
STD k-ε
LRN k-ε
EXP
STD k-ε
LRN k-ε
EXP
X1/D=1.0
r =1
X2/D
X1/D=1.0
X1/D=0.5
0.5
1
-1
0
1
-1
0
1
2
u2t/Uc/(Th-Tc)*100
Fig. 7. Thermal field
by Prt model (r = 0.5)
From the results described above, it is thought
that the reliability of u2t prediction may be
improved by giving the Prt distributions with
some appropriate functions.
As for the
streamwise component u1t, however, Prt given
by functions cannot improve the reliability of
the simulated result because the streamwise
temperature gradient ∂T/∂X1, to which u1t is
assumed to be proportional, is almost zero in the
present thermal mixing layer. In order to
examine the reason for such complex u1t
distributions as obtained in the measurement,
-0.5
-1
Prt=0.9
Prt=0.5
Prt=0.1
EXP
0
1
-1
0
1
-1
0
1
u2t/Uc/(Th-Tc)*100
Fig. 8. Thermal field
with different Prt
(r = 1)
2
-0.5
-1
X1/D=0.5
X1/D=1.0
X1/D=2.0
r =0.5
Prt=0.9
Prt=0.5
Prt=0.1
EXP
0
1
-1
0
1
-1
0
1
2
u2t/Uc/(Th-Tc)*100
Fig. 9. Thermal field
with different Prt
(r = 0.5)
the production terms in the transport equation of
u1t have been evaluated based on the
experimental data.
Figure 10 shows the
distributions of each term of u1t production,
which is expressed as follows, measured at X1/D
= 0.5 [6].
P1 A = −u1u2
P1C = −u2t
D
∂U
D
∂T
, P1B = −u1t 1 2
2
∂X 2 U c ∆T
∂X 1 U c ∆T
∂U1 D
∂X 2 U c2 ∆T
(7)
6
CFD ANALYSIS OF TURBULENT THERMAL MIXING OF HOT AND
COLD AIR IN AUTOMOBILE HVAC UNIT
X1/D=0.5
0.5
X1/D=1.0
X1/D=2.0
0.5
r =1
X1/D=0.5
TCL
EXP
X2/D
X2/D
TCL
EXP
0.0
0.0
-0.5
(a) r = 1
0
0.5
X1/D=2.0
X1/D=1.0
r =1
1.0 0
0.5 1.0 0
U1/Uc
0.5
1.0
-0.5
-1
0
-1
0
-1
0
1
2
u1u2/Uc *100
Fig. 11. Mean velocity and turbulent shear stress
distributions predicted by TCL SMC (r = 1)
X1/D=0.5
0.5
X1/D=1.0
X1/D=2.0
r =0.5
0.5
X1/D=0.5
TCL
EXP
It is found that the distributions of the sum of
these production terms are qualitatively similar
to those of u1t shown above. In particular, the
contributions of P1A and P1C, both of which
include the gradient of mean quantity in the
transverse (X2) direction, are much larger than
that of P1B. The term including ∂T/∂X1 was
smaller than P1B and its contribution to u1t
production was negligible. From these results,
it follows that the influences of ∂U1/∂X2 and
∂T/∂X2 must be included to reproduce
reasonably the complex distributions of u1t.
This is a reason for the selection of GGDH-type
turbulent heat flux models to calculate the
thermal field of the present mixing layer.
4.3 Results of calculation with TCL SMC +
(HO)GGDH models
In this section, the results of the velocity and
temperature fields calculated by TCL SMC +
(HO)GGDH models are compared with the
experimental data.
Figure 11 shows the
distributions of U1/Uc and u1u2 calculated for r =
1 with TCL SMC. The calculated results agree
X2/D
X2/D
TCL
EXP
(b) r = 0.5
Fig. 10. Production of u1t
0.0
0.0
-0.5
0
0.5
X1/D=2.0
X1/D=1.0
r =1
1.0 0
0.5 1.0 0
U1/Uc
0.5
1.0
-0.5
-1
0
-1
0
-1
0
1
2
u1u2/Uc *100
Fig. 12. Mean velocity and turbulent shear stress
distributions predicted by TCL SMC (r = 0.5)
well with the experimental ones, although a
slight delay of the recovery of the velocity
deficit region is observed in U1 distributions.
Figure 12 shows the results for r = 0.5. The
difference between the calculated distributions
of u1u2 and measured ones is increased in
comparison with that for r = 1; in particular,
u1u2 is underpredicted in a downstream region.
From a comparison of Fig. 12 with Fig. 5,
however, it is understood that the reliability of
the flow field calculated by TCL SMC is
improved in comparison with the k-ε models.
Thus, it follows that the difference between the
numerical results of the thermal field shown
below and those shown in Figs. 6 - 9 reflects the
performance of both the turbulence model and
turbulent heat flux model.
Next, the thermal field is examined. Similar
to Figs. 6 - 9, the distributions of the mean
temperature, turbulent heat fluxes u1t and u2t are
compared with the experimental results. Figure
13 shows the results at r = 1. The values of u1t
7
X1/D=0.5
0.5
X1/D=1.0
X1/D=2.0
X2/D
X2/D
HOGGDH
GGDH
EXP
0.0
0.0
0
0.5
1.0 0
0.5
1.0 0
(T-Tc)/(Th-Tc)
X1/D=1.0
X1/D=0.5
0.5
X1/D=2.0
r =0.5
HOGGDH
GGDH
EXP
-0.5
X1/D=1.0
X1/D=0.5
0.5
r =1
0.5
-0.5
0
1.0
X1/D=2.0
0.5
r =1
0.5
0.5 1.0 0
1.0 0
(T-Tc)/(Th-Tc)
X1/D=0.5
0.5
1.0
X1/D=2.0
X1/D=1.0
r =0.5
X2/D
X2/D
HOGGDH
GGDH
EXP
0.0
0.0
HOGGDH
GGDH
EXP
-0.5
-0.5
-2 -1 0 1 -2 -1 0 1 -2 -1 0 1 2
-2 -1 0 1 -2 -1 0 1 -2 -1 0 1 2
X1/D=0.5
u1t/Uc/(Th-Tc)*100
X1/D=1.0
u1t/Uc/(Th-Tc)*100
X1/D=2.0
0.5
X2/D
r =1
X2/D
0.5
0.0
0.0
X1/D=0.5
HOGGDH
GGDH
EXP
-0.5
-1
0
1
-1
X1/D=1.0
X1/D=2.0
r =0.5
HOGGDH
GGDH
EXP
0
1
-1
0
1
u2t/Uc/(Th-Tc)*100
Fig. 13. Thermal field
by (HO)GGDH model
(r = 1)
2
-0.5
-1
0
1
-1
0
1
-1
0
1
2
u2t/Uc/(Th-Tc)*100
Fig. 14. Thermal field
by (HO)GGDH model
(r = 0.5)
calculated with GGDH are slightly smaller than
the experimental results at X1/D = 0.5, but they
agree well with the measured values in a further
downstream region. The higher-order version
of GGDH, i.e., HOGGDH, generally
These results show that
overpredicts u1t.
GGDH model can predict the u1t distribution far
more successfully than Prt model. This is
because GGDH-type turbulent heat flux model
can incorporate the contribution of ∂T/∂X2 into
the formulation of u1t. As to u2t, the difference
between GGDH and HOGGDH is quite smaller
than that for u1t. Both models underpredict the
values of u2t near the origin of the mixing layer,
but the difference between the measured and
calculated values becomes smaller in the
downstream region. The reliability of u2t
calculated by GGDH and HOGGDH models is
higher than that obtained with the prescribed Prt
model. As a result of such improvement in the
calculation of turbulent heat fluxes, the
distributions of the mean temperature at r = 1,
shown at the top of Fig. 13, are more
successfully reproduced by (HO)GGDH models
than the Prt model.
Figure 14 shows the results at r = 0.5.
Although the qualitative features of u1t
distributions are well captured by both GGDHtype models, HOGGDH generally overpredicts
u1t as is the case with r = 1 while GGDH
underpredicts it. On the other hand, u2t is
underpredicted by both GGDH-type models in
the region of X1/D > 1; the disagreement
between the experimental and numerical results
is increased than the case of r = 1. This
tendency is similar to that calculated with k-ε +
Prt models, but as a whole (HO)GGDH models
can reproduce the distributions of u2t with
higher accuracy than the prescribed Prt model.
Similar to the case of r = 1, the reliability of the
mean temperature distributions predicted by
(HO)GGDH models is improved in comparison
with the Prt model, although the difference
between measured and calculated values is
somewhat increased in comparison with that for
r = 1.
Here it should be noted that the TCL SMC
and (HO)GGDH models tested in this study
were originally developed to the prediction of
turbulent heat transfer in complex wall shear
flows such as 3-D curved duct [5]. Thus, the
model coefficients were optimized referring to
the experimental data or DNS results in those
flows. In the present study, the original models
are used without any modifications. Since the
characteristics of the present flow geometry are
quite different from those of the wall turbulence,
the performance of the predictions may be
improved by adjusting the model coefficients to
the mixing layer. In the numerical simulations
8
CFD ANALYSIS OF TURBULENT THERMAL MIXING OF HOT AND
COLD AIR IN AUTOMOBILE HVAC UNIT
of turbulent thermal mixing in the HVAC unit,
another important point is the computation time.
It should be noted that the CPU time needed to
attain the final result by TCL SMC + HOGGDH
model was about three times as long as that for
standard k-ε + Prt model; this is within the
scope of practical use.
4 Conclusions
The numerical simulations of turbulent
thermal mixing in a planar shear layer have
been conducted, which simulates the mixing of
hot and cold airflows in the HVAC unit used for
automobile air-conditioning system. We have
tested three turbulence models: standard k-ε
model, low-Re nonlinear k-ε model and TCL
SMC. The first two k-ε models have been
coupled with prescribed Prt, and TCL SMC has
been combined with GGDH model and its
higher-order version for turbulent heat fluxes.
By optimizing the Prt-values, the k-ε + Prt
models can predict the distributions of u2t with a
sufficient accuracy at r = 1, but the reliability of
u2t prediction becomes lower at r = 0.5. The Prt
model cannot reproduce the u1t distributions in
principle, and the reliability of the predicted
mean temperature distributions is insufficient.
TCL SMC + (HO)GGDH models can
reasonably reproduce the turbulent heat fluxes
and successfully predict the mean temperature
distributions in the mixing layer. These results
suggest that TCL SMC + (HO)GGDH models
have high potential to the prediction of turbulent
thermal mixing encountered in the HVAC unit
for automobiles.
[5] Suga, K. Predicting Turbulence and Heat Transfer in
3-D Curved Ducts by Near-Wall Second Moment
Closures, Int. J. Heat Mass Transfer, Vol. 46, pp. 161–
173, 2003.
[6] Asano, H., Hirota, M., Nakayama, H., Mizuno, Y. and
Hirayama, S. Turbulent Thermal Mixing of Hot and Cold
Air in Planar Shear Layer (Thermal Mixing in
Automobile HVAC Unit), Proc. 6th World Conf. Exp.
Heat Transfer, Fluid Mech., Thermodynamics, 3-a-13,
Matsushima, Japan, 2005 (in CD-ROM).
[7] Abdul Asim, M. and Sadrul Islam, A. K. M. Plane
Mixing Layers from Parallel and Non-Parallel Merging of
Two Streams, Exp. in Fluids, Vol. 34, pp. 220-226, 2003.
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9