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International Conference “Passive and Low Energy Cooling 409
for the Built Environment”, May 2005, Santorini, Greece
Local dynamic similarity concept as applied to evaluation of discharge
coefficients of ventilated buildings. Part 1: Basic idea and underlying wind
tunnel tests
T. Kurabuchi and T. Endo
Tokyo University of Science, Japan
M. Ohba and T. Goto
Tokyo Polytechnic University, Japan
Y. Akamine
University of Tokyo, Japan
ABSTRACT
A model has been proposed for evaluating the
discharge coefficient and flow angle at an inflow
opening for cross-ventilation. This model is
based on the fact that the cross-ventilation flow
structure in the vicinity of an inflow opening
creates dynamic similarity under the condition
that the ratio of cross-ventilation driving
pressure to dynamic pressure of cross flow at the
opening is consistent. It was confirmed from a
wind tunnel experiment that the proposed model
can be applied almost regardless of wind
direction and opening position. Change of
pressure along the stream tube of a
cross-ventilated flow was estimated from the
results of Large Eddy Simulation, and was set as
the basis of model preparation.
1. INTRODUCTION
In recent years, there has been a lot of interest
and concern about the utilization of air flow for
improving indoor thermal conditions in hot and
humid rooms, which is important for
energy-saving in buildings. To expand the use of
natural ventilation and to establish a reliable and
effective utilization method, a much more
profound understanding of the mechanism of
natural ventilation is required. Wind tunnel
experiments have demonstrated that the
discharge coefficient relating to wind pressure
with ventilation flow rate varies with wind
direction and opening position (Vickery and
Karakatsanis, 1987, Kiyota and Sekine, 1989).
However, no model has yet been presented that
adequately explain how the discharge coefficient
is changed. Under such circumstances, we tried
in the present study to accurately identify
ventilation phenomena through use of both
experiments and CFD. In the process, we tried to
make it clear that total pressure can be
considered as a parameter specific to an opening
in a manner similar to wind pressure. We then
proposed a dynamic similarity model using the
total pressure at the opening in addition to wind
pressure and room pressure, and tried to explain
how the discharge coefficient is changed.
2. LARGE EDDY SIMULATION APPLIED
TO CROSS-VENTILATION
2.1 Outline of CFD and wind tunnel model
experiment
Cross-Ventilation air flow is characterized by
rapid acceleration and rapid deceleration.
Because it is considered difficult to apply an
eddy viscosity model such as the k-ε model
(Kurabuchi et al., 2000), alternatively we chose
Large Eddy Simulation (LES) where the
Smagorinsky coefficient is regarded as constant
(Cs = 0.13). As shown in Figure 1, the study was
performed on a ventilation air flow in a building
where the boundary layer flow is regarded as an
approach flow. The building under study was in
the form of a rectangular parallelepiped of 2:2:1.
The direction of the approach flow was varied in
the range of 0 degree to 67.5 degree.
2.2 Determination of stream tube shape
We tried to elucidate the structure of ventilation
410 International Conference “Passive and Low Energy Cooling
for the Built Environment”, May 2005, Santorini, Greece
air flow from the analysis of calculation results.
First, passive markers were set out from the
opening’s section. By tracing these trajectories
upstream and downstream, stream tube shapes
before and after the passing at the opening were
determined. When the wind direction is other
than 0 degree, the stream tube contacts the wall
surface before it reaches the opening, as shown
by the result of the case in Figure 2, where the
wind direction is set to 45 degree. It is turned to a
flow along the wall surface and reaches the
opening. This means that in most cases the
ventilation air flow may be approximated by a
wall jet or boundary layer flow before it flows
into the opening.
2.3 Pressure change along stream tube
The flow rate weighted average values of total
pressure, static pressure and dynamic pressure in
each cross-section of the stream tube were
calculated. The changes of these values together
with the shape of the stream tube are shown in
Figure 3. From this figure, it is apparent that
static pressure and dynamic pressure show
extreme changes before reaching the opening
when the wind direction is 45 degree or less,
while the total pressure, i.e. the sum of the two, is
almost constant, and pressure loss is low in the
process where the wind flows along the
windward wall surface. However, in the shape of
the stream tube when the wind direction is 60
degree, the flow is separated at the windward
corner and the flow reattaches again to the wall,
and total pressure is decreased in this process.
Figure 4 shows the changes of total pressure,
wind pressure and room pressure at the opening
where the approaching flow angle is changed.
Until the wind direction reaches 45 degree, total
pressure at the opening is constant. When the
wind direction exceeds 45 degree, air flow is
separated at the windward corner, and the total
pressure is greatly decreased.
3. LOCAL DYNAMIC SIMILARITY
3.1 Modeling of the flow around the opening
Based on the results of LES, a useful model is
presented, which characterizes the flow around
an opening. First, for a building with
cross-ventilation, total pressure at an opening for
ventilation air flow is split into three
components, i.e. dynamic pressure normal to the
50
0
67.5
0
degree
1
2
2
Size of opening
0.4×0.2
XY
22.5 45 60
45
degree 67.5
degree
degree
Figure 1: Building model and wind direction.
Wind direction : 45 degree
Figure 2: Shape of stream tube in the vicinity of opening.
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
-1.5 -1 -0 .5 0 0.5
Pres sur e
-1.0 0
w
ind angle
45°
Location o
f
opening
X direction
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
-1.5 -1 -0.5 0 0.5
Pres su re
0
-1.0
w
ind angle
60°
Location o
f
opening
Y dir ection X direction
total pressure
dynamic pressure
static pressure
Wind
direction
45 degree
Wind
direction
60 degree
Figure 3: Identified shape of stream tube and streamwise
change of pressure.
International Conference “Passive and Low Energy Cooling 411
for the Built Environment”, May 2005, Santorini, Greece
opening Pn, tangential dynamic pressure Pt, and
static pressure PS (Fig. 5). Next, room pressure
PR is picked up as an essential parameter on
room side. Because there is no meaning in
absolute pressure, static pressure loss “PS–PR”,
which is the difference between the static
pressure and the room pressure, is considered.
Further, by taking the conveniences into account,
special notice is given to “Pn+PS–PR”, i.e. the
static pressure loss plus Pn. Under the condition
where cross-ventilation flow rate is turned to 0,
the room pressure is equal to wind pressure PW.
In this case, Pn+PS=PW. If it is supposed that the
same condition exists even when there is air
flow, the value of Pn+PS–PR can be approximated
as ventilation driving force PW–PR. Therefore,
Pn, Pt and PW–PR are picked up as important
parameters to characterize flow around the
opening.
Suppose the building remains the same and
only the velocity of the approaching flow is
doubled. Then, all of these pressure values
should be quadrupled. This is because dynamic
similarity of total flow field is established. Under
the condition where dynamic similarity of total
flow field is established, the dimensionless
values calculated by combining the three
pressure values extracted above are turned to be
constant. As the combination of the
dimensionless values, the following values can
be used, for instance: Pn/Pt corresponding to
inflow angle β, and pressure loss coefficient
ζn=Pn/(PW–PS) of ventilation air related to
discharge coefficient Cd. With experiments
performed by changing the approaching flow
angle, the corresponding relation between the
inflow angle and the discharge coefficient is
determined.
In case air flow conditions acting around the
opening can be represented only by the three
pressure values extracted above, it is expected
that similarity of the flow can be established
without depending on the shape of the building,
position of the opening, approaching flow angle,
etc. This concept may be called as local dynamic
similarity in the meaning that dynamic similarity
exists -not in total flow field- but only in the
vicinity of the opening. In this case, a pair of two
dimensionless values prepared from three
pressure values represents a specific air flow
condition. If these values correspond to each
other at one by one, it may be deduced that, when
one of them is determined, the other is
automatically determined.
In order that local dynamic similarity of the
flow is established, the following conditions may
be required:
(a) The shape of the opening has geometrical
similarity.
(b) The direction of tangential flow of the
approaching flow with respect to the
opening is constant.
(c) The opening is positioned on a wall surface,
which is sufficiently large with respect to
the size of the opening.
(d) There is no wall to hinder the diffusion of
incoming air flow near the opening on
room side.
It must be confirmed by the experiments how
far these conditions must be satisfied in order
that local similarity is established.
The inflow angle β cannot be determined
unless ventilation flow rate is determined. Thus,
0
Wind direction [degree]
Pressure coefficient
-0.2
-0.4
-0.6
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70 80
total pressure at the opening
wind pressure averaged room pressure
2
2
A
Q
ρ
Figure 4: Evaluated pressure at different wind direction
angles.
PT
Pt
Pn
PS
PS-PR
PR
Figure 5: Characteristic pressures at opening
412 International Conference “Passive and Low Energy Cooling
for the Built Environment”, May 2005, Santorini, Greece
dimensionless room pressure PR* is defined
according to the equation (1), and this is used
instead of β.
*
R
W
R
t
PP
PP
−
= (1)
It is defined in such manner that it
corresponds to the inflow when PR* is negative,
and it corresponds to the outflow when it is
positive.
3.2 Validity of suction experiment
For the purpose of quickly conducting
experimental evaluation by assuming various
experimental conditions for the approaching
flow angle, the position of the opening, and the
ventilation flow rate, an experiment was
performed: A building model was connected
with a suction fan on leeward side and this model
was exposed to the approaching flow as shown in
Figure 6. To evaluate whether actual ventilation
condition can be correctly reproduced by this
experimental setup, suction air flow rate was
adjusted to achieve the same room pressure as
the room pressure at each approaching flow
angle of a ventilation model shown in Figure 1,
where both room pressure and cross-ventilation
flow rate had already been measured, and it was
evaluated whether this was consistent with
measured cross-ventilation flow rate. The results
are as shown in Figure 7. Because flow rate
showed good matching, it is possible to
determine ventilation performance at the
opening during ventilation by this method.
3.3 Validity of local dynamic similarity concept
To evaluate the validity of the proposed model,
wind tunnel experiment was conducted as shown
in Figure 6 by setting the position of the opening
and the wind direction as variable, and the
corresponding relation between PR* and
discharge coefficient was assessed. It was
assumed that the openings were located at 3
positions at the central height. The end of the
opening closest to the side wall concurred with
the side wall, and this may have conflicts with
the precondition (d) as given above.
First, it was evaluated whether the discharge
coefficient (Cd’s) always concurs in case
PR*→-∞. Two extreme cases were assumed: a
case where stagnant surrounding condition exists
around the opening and Pt=0 and suction is
performed by using a fan, and a case where
approaching flow angle is considered and the
suction flow rate is assumed large enough to
achieve Pn >> Pt. The results of the comparison
for each opening are summarized in Figure 8.
From this figure, it has been confirmed that
discharge coefficients concur well on all of the
openings.
Next, measurement was performed under the
condition where the value of Pt cannot be
neglected. It was difficult to obtain the value of
Total pressure tube
22.5
degree
200
200
100
Suction flow
Opening
position
A B
C
45
degree
67.5
degree
Figure 6: Experimental setup to evaluate validity of local
dynamic similarity concept (unit:mm).
0.006
Flow rate [m3/s]
Wind direction [degree]
cross-ventilation model
suction flow model
0.004
0.002
0
0 20 40 60 80
Figure 7: Comparison of suction flow rate and actual
cross-ventilation flow rate.
Discharge coefficient
0.1
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
A B C
Opening position
0.9
0 22.5 45 67.5
Stagnant surrounding
Figure 8: Comparison of Cd’s in case of PR*→-∞.
International Conference “Passive and Low Energy Cooling 413
for the Built Environment”, May 2005, Santorini, Greece
Pt at the opening with ventilation from the
measurement of wind velocity. In this respect,
we adopted the following method, which did not
depend on direct measurement. Total pressure PT
at the opening is Pn+Pt+Ps as already described.
If it is assumed that the value of Pw approximates
the value of Pn + Ps, the value of Pt can be
evaluated from the equation (2).
tTW
PPP=− (2)
By assuming that the value of Pw can be
substituted by the room pressure when the
ventilation flow rate is 0, the value of PT was
determined by directly measuring the value at
the center of the opening using total pressure
tube.
The relation between PR* and Cd is shown in
Figure 9. In the figure, the value of PT as
measured for each ventilation flow rate was
used. As a result, it was confirmed as shown in
Figure 9 that the relation between these two
values can be represented by the same curve
except some cases. It was extensively deviated
from the curve in the case where the approaching
flow angle was 67.5 degree at the windward end.
In this case, there are two possibilities: the flow
was separated near the opening and total
pressure could not be measured accurately, and
there were conflicts with the precondition of (b).
Also, in case the approaching flow angle was
22.5 degree at the windward end, the values of
total pressure PT and the wind pressure PW were
very close to each other. It was difficult to
evaluate the value of Pt in this experiment, and
this was exempted from the study. Except these
cases, it was confirmed that local dynamic
similarity could be established under extensive
conditions regardless of the position of the
opening and the approaching flow angle.
3.4 Simplification to assume total pressure
If it is necessary to have the value of PT
corresponding to the ventilation flow rate prior
to the prediction of discharge coefficient,
practical predicting method cannot be
established. However, the value of PT measured
in the above experiment takes nearly constant
value without depending on the ventilation flow
rate as shown in Figure 10. In this respect, we
attempted to simplify the measurement by using
the value of PT without depending on the
ventilation flow rate when suction flow rate was
increased as much as possible so that an inflow
angle of about 90 degree could be postulated. To
evaluate the validity of this method, a wind
tunnel experiment was conducted by using a
building model similar to the model used in the
previous experiment as shown in Figure 11.
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
PR*
Discharge coefficient
A-22.5 A-45 A-67.5
B-22.5 B-45 B-67.5
C-45 C-67.5
-2 -4 -6 -8 -10
Figure 9: Discharge coefficient curves as a function of PR*
for different wind direction an
g
les and o
p
enin
g
locations.
0 0.005 0.01 0.015
0
0.2
0.4
0.6
0.8
1.0
Q [m3/s]
A-45 B-45 C-45
Total pressure coefficient [-]
Figure 10: Relation of total pressure coefficient and
suction flow rate.
Split film probe
22.5
degree
300
300
150
Suction flow
Opening
position
ABC
45
degree
67.5
degree
DE
Figure 11: Experimental setup to evaluate relation of
discharge coefficient and inflow angles with PR*
(
unit:mm
)
.
414 International Conference “Passive and Low Energy Cooling
for the Built Environment”, May 2005, Santorini, Greece
Here, for the purpose of assessing unique
matching between PR* and the inflow angle β,
the inflow angle at the center of the opening was
measured by using a split film probe.
3.5 Verification of model validity
In order to verify the validity of the proposed
model, the relation between Cd and PR* is shown
in Figure 12 above, which summarizes a case
where the wind direction was changed and the
position of the opening was fixed at the front
center of the building (opening position C). In
this figure below, the relation where the wind
direction was fixed at 45 degree and the position
of the opening was changed is shown. According
to this figure, when PR* is less than -5, Cd is
almost constant. When it is -2 or more, Cd tends
to decrease rapidly. This relation remains almost
constant regardless of the position of the opening
and the wind direction. Similarly, Figure 13
shows the relation with the inflow angle at the
center of the opening when wind direction and
the position of the opening are changed. When
PR* increases, β approaches 90 degree. In this
way, by applying this local dynamic similarity
model, it is experimentally demonstrated that the
changes of Cd and β can be explained by a single
parameter PR*.
4. CONCLUSIONS
- Before the ventilation air flow reaches the
opening, the total pressure of the approach
flow is preserved almost completely
regardless of the approaching flow angle if
flow separation does not occur before it
reaches the opening.
- The dynamic structure of the ventilation air
flow becomes similar where dimensionless
indoor pressure PR* is consistent, which is
expressed by the ratio of the ventilation
driving force to the difference between total
pressure and wind pressure at the opening.
- When PR* is constant, the inflow discharge
coefficient is consistent with the inflow angle
even when the approach flow angle and the
position of the opening are changed.
REFERENCES
Kiyota, N. and T. Sekine, 1989. Experiment study on
pressure loss at the opening of wall surface (Part2), J.
Archt. Plann. Environ. Eng., AIJ, 398: pp47-57.
Kurabuchi, T., M. Ohba, A. Arashiguchi and T. Iwabuchi,
2000. Numerical study of airflow structure of a
cross-ventilated model building, The 7th International
Conference on Air Distribution in Rooms
‘ROOMVENT 2000’: pp313-318.
Vickery, B.J. and Karakatsanis, 1987. Experimental wind
pressure distribution and induced internal ventilation
flow in low-rise industrial and domestic structures.
ASHRAE Transactions, 93, Part2: pp565-568.
PR*
E-45 deg. D-45 deg.
C-45 deg. B-45 deg.
A-45 deg.
Discharge coefficient
0.1
0.3
0.4
0.5
0.6
0.7
0.8
0.2
0
0 -2 -4 -6 -8 -10
0.2
0
Discharge coefficient
PR*
C-0 deg. C-30 deg.
C-45 deg. C-67.5 deg. 0.1
0
0.3
0.4
0.5
0.6
0.7
0.8
-2 -4 -6 -8 -10
Figure 12: Discharge coefficient curves for different wind
direction angles and opening locations.
0
Inflow angle [degree]
C-22.5 deg. C-45 deg. C-
67.5 deg.
30
60
90
0
PR* -2 -4 -6 -8 -10
C-45 deg. B-45 deg. A-
45 deg.
0
Inflow angle [degree]
30
60
90
0
PR* -2 -4 -6 -8 -10
Figure 13: Inflow angle curves for different wind directio
n
angles and opening locations.