Chapter 4.
Heat/Mass Transfer Analogy
- Turbulent Boundary Layers
Boundary layers on gas turbine airfoils, initially laminar, go through transition to
turbulent boundary layers. A large part of the airfoil is covered with a turbulent boundary
layer; therefore, it is of an interest of the present study to determine how heat/mass
transfer analogy varies in such condition.
Furthermore, today’s experiments in gas
turbine heat transfer are interested in determining the effects of secondary flows. The
triangular region of high heat transfer rates caused by the passage vortex has a highly
turbulent structure, different from an ordinary turbulent boundary layer. Furthermore, the
vortex is neither homogeneous nor isotropic.
This chapter first introduces how conventional relationships for the analogy
function for turbulent boundary layers have developed. Then, previously performed
experiments are combined to form an experimentally determined distribution of the
analogy function. This distribution is particularly interesting in the region influenced by
the passage vortex.
4.1
Conventional Relationship
The conventional heat/mass transfer analogy function for turbulent boundary layers is
given by the following expression.
47
. Re x −1/10 Sc −1/ 6 ⋅ ( Sc − 1)
Nu Pr 1 + 148
=
⋅
Sh Sc 1 + 148
. Re x −1/10 Pr −1/ 6 ⋅ (Pr − 1)
(4.1.1)
At the Reynolds number of 106, the analogy function between air (Pr = 0.707) and
naphthalene (Sc = 2.28) yields:
Nu
= 0 .5
Sh
(4.1.2)
The Reynolds number dependency in Eqn. (4.1.1) is weak; therefore, the above ratio of
0.5 may be used in the range of Reynolds numbers between 105 and 107 with less than 10
percent deviation from Eqn. (4.1.1) (see Fig. 10 in Goldstein and Cho, 1995).
This analogy function, Eqn. (4.1.1), is derived from an analytical expression of
Stanton number distribution for a turbulent boundary layer on a flat plate. The derivation
of the expression is discussed in detail in Eckert and Drake (1959) and has the following
form.
0.0296 Re x −1/ 6
St =
1 + 148
. Re x −1/10 Pr −1/ 6 ⋅ ( Pr − 1)
(4.1.3)
An identical equation can be written for mass transfer simply by replacing Pr with Sc in
the above expression.
St M =
0.0296 Re x −1 / 6
1 + 1.48 Re x −1 / 10 Sc −1 / 6 ⋅ (Sc − 1)
(4.1.4)
Dividing Eqn. (4.1.3) by Eqn. (4.1.4) yields the form shown as Eqn. (4.1.1).
The Stanton number correlation, Eqn. (4.1.3), was derived empirically from
boundary layer profiles.
However, the analogy function could be derived from the
governing partial differential equations for turbulent boundary layers, just as the analogy
function for laminar boundary layers was derived in the previous chapter.
To do this, one must face the so-called turbulence “closure” problem which
prevents one from solving the governing equations. This comes about as follows. To
engineers, Reynolds decompositions to separate mean quantities and fluctuating
quantities due to turbulence are useful since engineers are typically interested in
48
variations in the mean quantities. The x-momentum (in the direction of flow) equation
for two-dimensional boundary layer results in the following form.
u

∂u
∂u 1 dp ∂  ∂u
+v
=
+  ν
− u ′v′ 
∂x
∂y ρ dx ∂y  ∂y

(4.1.5)
The “closure” problem comes about in the above equation because of the presence of the
last term consisting of turbulence quantities. The term is sometimes called Reynolds
stress or turbulent shear stress. To overcome this problem, a frequently used technique is
to define the turbulent diffusivity of momentum, εM. This is considered to be equivalent
to the molecular diffusivity of momentum in a case of laminar flow which is usually
called kinematic viscosity, ν, and it is defined by the expression:
u ′v′ = −ε M
∂u
∂y
(4.1.6)
Applying this to Eqn. (4.1.5) yields the following form of the time-averaged, x-momentum, boundary layer equation.
u
∂u
∂u 1 d p ∂ 
=
+
(ν + ε M ) ∂u 
+v

∂x
∂y ρ dx ∂y 
∂y 
(4.1.7)
Similarly, the time-averaged energy equation for constant property, twodimensional, turbulent boundary layer has the form:
u

∂T
∂T
∂  ∂T
 α
+v
=
− u ′T ′ 
∂x
∂y ∂y  ∂y

(4.1.8)
The turbulent diffusivity for heat transfer is defined by:
u ′T ′ = −ε H
∂T
∂y
(4.1.9)
The form that is equivalent to Eqn. (4.1.7) yields:
u
∂T
∂T
∂ 
+v
=
(α + ε H ) ∂T 

∂x
∂y ∂y 
∂y 
(4.1.10)
Finally, the corresponding mass transfer equation can be modified in the same way by
introducing the turbulent diffusivity for mass transfer.
49
These three turbulent diffusivities, those for momentum, heat and mass transfer
processes, are to be determined empirically. Then, these values can be introduced into
the above time-averaged equations for the analogy function for turbulent boundary layers
to be determined.
4.2
Heat and Mass Transfer Data from Experiments
The last, and a major, issue of the present study is to address the behavior of heat/mass
analogy functions on the suction-side wall of a gas-turbine airfoil which has mixed
effects: adverse streamwise pressure gradient, flow separation, transition, turbulence, and
an intense vortex structure impinging on surface. To do this, two sets of data from two
different experiments, a heat and a mass transfer experiment, are merged to form a
distribution of experimentally determined, heat/mass transfer analogy functions over the
suction surface of the CF6 airfoil profile near the endwall. This is done and discussed in
the next section. For this reason, this section introduces the two experiments that are to
be merged and prepares the data sets for merging.
4.2.1
The heat transfer experiment
The heat transfer experiment was performed by Chung and Simon (1993). The detailed
experimental procedures are discussed in Chung’s dissertation (1992). The experimental
method used to capture variations of heat transfer rates at the suction surface was liquid
crystal thermography.
A photograph of the liquid crystal sheet is attached in his
dissertation. It is a task of the present study to digitally process the photograph to
generate a distribution of heat transfer rates.
Details of the photo processing are presented in Appendix C, and the main
concerns in the process are from hue-temperature correlation, true and apparent color, and
location of the photograph on the airfoil.
Using a color separation technique, the
photograph was first separated to hue, saturation and value. Hue is the determinant of
50
color, and hue-temperature correlation is a key to the analysis. Based on the calibration
colors and information from the literature, temperature was correlated with hue by a
linear function. These issues are dealt with in the Appendix, and a field of Stanton
numbers is generated.
Chung’s experiment was done in a two-airfoil simulator (Fig. 4.1). Table 4.1
summarizes relevant parameters of the experiment compared against those of the mass
transfer experiment. Figure 4.2 shows the resulting Stanton number distribution.
Figure 4.1 Chung’s facility for the heat transfer experiment (Chung 1992)
51
0.5
0.0036
0.45
0.003
0.0018
0.0016
0.4
0.003
0.0016
0.35
z/C
0.0022
0.0024
0.0016
0.3
0.0034
0.0032
0.002
0.0028
0.0028
0.0018
0.0028
0.0026
0.25
0.0050
0.2
0.003
0.85
0.0032
0.9
0.95
0.0032
0.0036
0.003
0.15
0.8
0.0032
0.0034
0.0036
0.004
1
1.05
x/C
1.1
1.15
1.2
Figure 4.2 Stanton number distribution calculated from
the liquid crystal photograph in Chung (1992)
52
1.25
4.2.2
The mass transfer experiment
The mass transfer experiment was done by Chen and Goldstein (1992). Details of the
experiment are recorded in Chen’s dissertation (1988). The experimental method used
here was naphthalene-sublimation mass transfer. Distributions of mass-transfer Stanton
number are available for the entire surface of the airfoil; therefore, the only task of the
present study is to digitize the data.
The wind tunnel and the airfoil cascade are shown schematically in Fig. 4.3. Chen
and Goldstein used a boundary-layer tripping wire to have two different cases of
boundary layer thickness at the inlet endwall. As shown in Table 4.1, the case that
matched the thickness of Chung’s experiment was processed. Figure 4.4 shows the
distribution of mass-transfer Stanton numbers. The dashed square represents the area that
the liquid crystal data (Fig. 4.2) covers; therefore, the analogy functions can be
determined only in this region.
Table 4.1: Comparison of the experiments
Mass Transfer
Heat Transfer
(Chen 1988 and
(Chung 1992 and
Chen and Goldstein 1992)
Chung and Simon 1993)
Uin [m/s]
10.45
20
TI [%]
1.31
0.6
Rein
104,000
293,000
Ree
171,000
482,000
CF6
CF6
BC*
Tw = const.
qw = const
P/C
0.77
0.77
δ1/C
0.013
0.013
Profile
*
BC: boundary condition
53
As shown in the Table, the chord-Reynolds numbers are different between the two
experiments. This means that the growth rates of the boundary layers are different, even
in the nondimensionalized coordinates of Figs. 4.2 and 4.4.
One thing to note that is not listed in the above table is the difference in time
scales between the two experiments. As typically done, the naphthalene-coated airfoil in
the Chen’s experiment was exposed to airflow for over an hour. Therefore, the resulting
mass transfer distribution consists of time-averaged values. On the other hand, since the
heat transfer Stanton number distribution from the Chung’s experiment is generated from
a photograph of liquid crystal, the results are more nearly instantaneous.
Figure 4.3 Chen’s facility for the mass transfer experiment (Chen 1988)
54
x/C
Figure 4.4 Mass-transfer Stanton number distribution from the mass transfer experiment,
the dashed line indicates the regions shown in Figs. 4.2 and 4.5 (Chen 1988)
55
4.3
Experimentally Measured Analogy Factors
To determine the analogy factors in the region indicated in Figs. 4.2 and 4.4, the Stanton
numbers were first converted to Nusselt and Sherwood numbers, respectively, using the
following relations.
Nu x = St ⋅ Re x ⋅ Pr
(4.3.1)
Sh x = St M ⋅ Re x ⋅ Pr
(4.3.2)
Then, a field of analogy factors was generated by dividing the Nusselt numbers by the
Sherwood numbers at the corresponding locations. Figure 4.5 presents the distribution of
the experimentally measured analogy factors.
0.5
1.7
1.6 1.5
0.70
0.45
1.5
1.6
1.5
1.3
0.4
0.4
1.2
1.4 1.3
1.4
1.4
0.5
1.3
1
1.1
0.35
0.6
0.3
1.4
1.4
1.3
z/C
0.2
1.5
0.5
0.9
0.5
0.9 0.8
1
0.7
0.3
0.6
0.5
0.25
0.2
0.4
0.5
0.5
0.5
1.10
0.6
0.7
0.15
0.5
0.5
0.8
0.85
0.9
0.95
1
1.05
x/C
0.5
0.6
0.7
1.1
1.15
0.7
1.2
Figure 4.5 Analogy factor, Nu/Sh, distribution near the corner
of endwall and trailing edge on the suction side of the CF6 airfoil.
56
1.25
0.5
At first glance, one sees that Fig. 4.5 has values from 0.3 to over 1.5 at some
points, which do not seem to agree with the value 0.5 in Eqn. (4.1.2) and 0.677 in Eqn.
(3.1.10). It seems that the values vary, based on the flow conditions. The upper left
portion of the field shows relatively high values and is considered to be a region of flow
separation and transition indicated by a steep gradient in Nu/Sh. The other portion of the
figure is turbulent everywhere although the structures of turbulence differ from location
to location.
The triangular region affected by the passage vortex is clear in the analogy factor
distribution. The diagonal band of low values (lower than 0.4) matches with the band of
high values of mass-transfer Stanton number (Fig. 4.4). This may be due to artificially
low values in the liquid crystal data, which might be caused by the data reduction process
from the liquid crystal image (Fig. 4.2). Another possibility is that analogy factors at
locations under a strong vortex, in fact, may become as low as those shown in Fig. 4.5.
However, the true reason remains unclear because of the uncertainty in the liquid crystal
data.
The liquid crystal photograph was not taken with the intention for further
processing, such as this.
Another region of interest is the flow transition to turbulence region in the twodimensional flow portion of the airfoil. This runs through about z/C of 0.45 from about
x/C of 1.05 to the trailing edge. First, near x/C of 1.05, it appears that transition is
underway. Here, there is a sudden decrease in the analogy factors. This transition is
thought not to be triggered by the passage vortex or any other secondary flow systems
since the transition seems to begin at about the same x/C locations and last for a range of
x/C values. In other words, there is no dependency in the spanwise direction (the zdirection). Instead, it is natural transition of a two-dimensional flow.
Besides the gradients being steep in this region, another reason that this region
might indicate flow transition is that the values of analogy factors decrease with
transition. A decrease in the analogy factor is expected in flow transition from laminar to
turbulence just by comparing Eqns. (3.1.10) and (4.1.2), at least from 0.677 to 0.5. What
is different is that, in Fig. 4.5, the decrease comes from about 1.2 to below 0.5. Such a
57
high value as 1.2 leads one to be somewhat suspicious of extrapolation of the liquid
crystal photograph processing to far beyond the calibration range. However, there is a
way to defend such high values. Figure 3.2 plots a few of the TEXSTAN calculations on
the CF6 velocity profile. The 3/1 case matches the set of boundary conditions in Fig. 4.5.
These two figures cannot be directly compared since the heat transfer experiment in Fig.
4.5 has a different chord-Reynolds number. Nevertheless, Fig. 3.2 indicates that the 3/1
case goes through flow separation before x/C of 1.0, and, at that point, the analogy factor
is over 1.0.
The question of why analogy factors drop below 0.3 after the transition remains
unanswered (upper right portion). One possibility is the difference in the boundary
conditions, which is considered less significant than it is in a case of laminar flow. In
addition, some very local variations, shown at low z/C locations and between x/C of 1.05
and 1.25, mainly come from the liquid crystal photograph. They are thought not to be
relevant to the present study, considering that there are no such variations in the mass
transfer data.
The values with the largest area in Fig. 4.5 is perhaps those between 0.4 and 0.5;
therefore, the average value of the analogy factor over the turbulent region might also be
in between 0.4 and 0.5. Considering that the liquid crystal photograph was not taken for
this purpose of digital processing, these experimentally derived analogy functions agree
quite well with the conventional relationships of Eqn. (4.1.2). The functions also agree
with what one might expect when extrapolating the values and the computed values to (1)
very intense turbulence and (2) into and through transition, separation and reattachment.
4.4
Conclusions
First, the conventional relationship of heat/mass transfer analogy function for turbulent
boundary layer flow on a flat plate was reviewed. It was derived from Stanton number
profiles as functions of Reynolds number and Prandtl or Schmidt number.
It was
determined that for an air-naphthalene combination, the ratio of Nusselt to Sherwood
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number is 0.5 at the Reynolds number of 105. It is then noted that the difference between
the analogy factors for laminar and turbulent boundary layers on the plane wall, 0.677 and
0.5 respectively, gives rise to a further interest in the analogy function under flow
separation and transition.
Secondly, two data sets were collected from two different experiments (one being
heat and the other being mass transfer experiment) done with the same airfoil geometry.
Because of the high three-dimensionality, the data taken on the suction side of the airfoil
are used in the present study. Utilizing a hue separation technique, a liquid crystal
photograph was digitally processed to yield a field of Nusselt numbers. The resulting
Nusselt number field exhibits the triangular region of high values indicating the presence
of a passage vortex. This phenomenon is also shown in the Sherwood number field from
the mass transfer experiment.
Lastly, an experimentally determined field of the heat/mass analogy function was
generated from the above Nusselt and Sherwood number fields. The analogy-function
field included the region which was affected by the passage vortex on the airfoil surface.
The majority of the region presented resulted in values that were close to that of the
conventional relation of 0.5 for turbulent boundary layers. Some general trends should be
addressed with regard to the difference in the conventional relations between laminar and
turbulent flows. Where the passage vortex may be most intense, the analogy factors are
on the order of 0.3-0.4. This is lower than the conventional factor of 0.5 for turbulent
boundary layers. In the region which appears to be separated and transitional, the values
are greater than 1.0. This is supported by the previous discussions in Chapter 3 where the
analogy factor for laminar flow increased to 1.1 just before separation, for this
mismatching-BC case (see line 3/1 in Fig. 3.2). Future investigations are encouraged in
order to precisely determine how the analogy factors result in extreme values in such
extreme conditions of flow separation, transition and intense turbulence.
59