CONFINEMENT AND BED-FRICTION EFFECTS IN SHALLOW TURBULENT MIXING LAYERS
By Vincent H. Chu,1 Member, ASCE, and Sofia Babarutsi2
ABSTRACT: The transverse development of the turbulent mixing layers in an open channel flow of shallow water depth was investigated experimentally to study the confinement and bed-friction effects. Mean and r.m.s. velocity profiles were obtained, using a hot-film anemometer, at a number of cross sections downstream of a splitter plate between two streams of different velocities. In the confinement between the free surface and the channel bed, the transverse spreading rate of the shallow mixing layer was initially twice as large as the nominal rate for the free mixing layer. The spreading rate reduces with distance from the splitter plate under the stabilizing influence of bed-friction, and diminishes to zero in the far field region when the bed-friction number exceeds a critical value of about 0.09.
INTRODUCTION
Turbulent shear flows in nature often exist in shallow environments, where the horizontal length scale is large compared with the depth of the flow field. The aerial photograph in Fig. 1 shows the development of such a shallow shear flow at the confluence of two rivers. The mixing layer between the two rivers spreads out in the horizontal direction and then approaches a width of about 140 m as the water depth in the region varies from 0.8-1.7 m. The water surface and the riverbed in the shallow flow impose a restriction on the vertical length scale of the turbulent motions. Two distinctive length scales exist: the large one is associated with the transverse shear and the small one with the bed friction. The turbulence generated by the bed friction is not visible since it is small compared with the resolution of the aerial photograph. The highly anisotropic nature of the shallow mixing layer in the river confluence is a characteristic quite different from that of a free mixing layer created in a deep apparatus, where the length scale of vortices is permitted to grow in the spanwise direction in proportion to the width of the mixing layer (Browand and Troutt 1980).
Beside the restriction of vertical length scale mentioned, the turbulent motion in the shallow mixing layer is under the stabilizing influence of the bed friction. If averaging is performed over the water depth, the small-scale motion would be filtered out (as it is invisible in the aerial photographs), and bed-friction terms would be introduced into the depth-averaged equations of motion. Chu et al. (1983) and Alavian and Chu (1985) have examined this bed-friction influence and have introduced a
•Assoc. Prof., Dept. of Civ. Engrg. and Appl. Mech., McGill Univ., Montreal H3A 2K6, Canada.
2Grad. Asst., Dept. of Civ. Engrg. and Appl. Mech., McGill Univ., Montreal H3A 2K6, Canada.
FIG. 1. Shallow Mixing Layer at Confluence of Nottaway River and Broadback River, Rupert Bay, Quebec; Large-Scale Instabilities Are Visible between Fast Flow and Turbid Water of Nottaway (Shown Here in Upper Part of Aerial Photograph) and Slow Flow and Clear Water of Broadback; Flow Directions Are from Left to Right in Photograph
bed-friction number, which is related to the bed-friction coefficient cf, the water depth h, the width of the transverse shear flow 8, the velocity difference across the shear layer A, and the average velocity across the shear layer U, as follows:
Using an inviscid theory and considering a parallel shear flow with a hyperbolic-tangent velocity profile, Chu et al. (1983) found a critical bed-friction number of 0.12 for the stability of the transverse shear flow. Alavian and Chu (1985) included the effect of turbulent motions in a viscous theory and found the value of the critical bed-friction number to be 0.06. The analyses of Chu et al. (1983) and Alavian and Chu (1985) were for parallel flows subjected to small disturbances. However, a similar stabilizing influence is anticipated in the turbulent mixing layers being investigated in this paper.
EXPERIMENT
The experiment of the shallow turbulent mixing layer was conducted in a 61-cm wide, 13-cm deep and 7-m long open channel as shown in Fig. 2. A splitter plate located at the midplane of the stilling basin divided the flow into two streams of different velocities. The flows passed through both a lateral and a vertical contraction before entering the test section from the
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two sides of the splitter plate. The velocity difference across the splitter plate was controlled by a set of screens in the stilling basin. The splitter plate was blunt-ended and was 2.5 mm in thickness.
Velocity measurements were made across the mixing layer using a cylindrical hot-film probe (TSI model 1210-W10) in conjunction with a TSI 1053B constant temperature anemometer. The bridge voltage was linearized by a TSI 1052 polynomial linearizer. The output voltage from the anemometer was digitized and analyzed by a PDPll/03 instrumentation computer. The turbulent statistics were obtained from 4,096 samples; the sampling rate was 64 samples/sec.
Velocity profiles were obtained at the entrance (x = 1 cm) to the test section and at five other sections (x = 50 cm, 100 cm, 150 cm, 200 cm, and 300 cm) downstream of the splitter plate. Typical lateral spacing between measurements was 1 cm inside the mixing layer and 2 cm outside in the ambient streams. The calibration of the hot-film probe was made against the average velocities of a large number of confetti floating on the water surface. During the experiment and the calibration, the hot-film probe was placed at a depth of 0.5 cm below the water surface. While the measurements were made at a depth of 0.5 cm below the water surface, the velocity data were the same as the velocity on the water surface. The performance of the hot-film probe was extremely sensitive to small changes in water temperature and the presence of suspended matter and air bubbles. Calibration of the probe was made before and after each traverse across the mixing layer. The probe was cleaned with a fine brush regularly during the experiment. The overheat ratio of the probe was kept at about 4%.
The bed-friction coefficient cy was evaluated using the following formula recommended by the A.S.C.E. Task Force on Friction Factor in Open Channel (Carter et al. 1963):
1 / 1.25 \
^r-4 l 0 8(^vi) ® in which the Reynolds number Re = AUhlv. The friction coefficient in the first 20-40 cm from the entrance is affected by the bottom boundary layer development. Since the development length is small compared with the length of the test section, which is 300 cm, the effect is ignored, and the friction coefficient is calculated by Eq. 2. The effect of boundary layer development has been considered in Babarutsi and Chu (1985).
Since the friction coefficient is weakly dependent on the local velocity, it varies slightly across and along the mixing layer. The friction coefficients in the ambient streams are cn and c^ . A local average across the mixing layer is cf = (cn + c^)l2. Longitudinal averages of the ambient friction coefficient are cn and c^ • The overall average is cf = (df, + Cp)l2. Since the turbulent eddy moves downstream with an average velocity U = (U1 + U2)/2, which stays practically constant along the mixing layer, the local average cy is approximately equal to the overall average cf.
A series of five tests was conducted. The water depth varied in a range from 2-5 cm, and the velocity difference across the splitter plate varied from 11.6-26.4 cm/s. The bed-friction length scales h/cr for the five tests Tl, T2, T3, T4, and T5, were 867 cm, 289 cm, 372 cm, 459 cm, and 400 cm,
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TABLE 1. Test Conditions
Test
(1)
Tl T2 T3 T4 T5
Depth (cm)
(2)
5.00 2.00 2.50 2.96 2.64
f l o (cm/s)
(3)
36.0 28.4 24.8 26.4 23.0
Ulo (cm/s)
(4)
9.6 16.8 12.7 11.1 9.8
sfl (5)
0.0048 0.0066 0.0063 0.0059 0.0061
92 (6)
0.0067 0.0072 0.0071 0.0070 0.0071
hlcf
(cm)
(7)
867 289 372 459 400
respectively. The test conditions are summarized in Table 1. The details of the experimental investigation are given in a report by Babarutsi and Chu (1985).
RESULTS
Typical mean and r.m.s. (root-mean-square) velocity profiles for two tests, Tl and T3, are shown in Figs. 3 and 4. A top-view photograph of the mixing layer is shown in Fig. 5. The turbulent motion is made visible by the injection of dye along both sides of the splitter plate. Despite the influence of the bed friction, a rather organized and coherent structure can be observed. The presence of the mixing layer is evident by the steep velocity gradient in the mean profiles and the peak in the r.m.s. profiles. A maximum-slope width of the mixing layer is defined as 8 = (U} -U2)l(dUldy)m3L% , in which (dU/dy)max = the maximum of the mean velocity gradient; and Ul and U2 = the mean velocities in the fast and slow ambient streams, respectively. (dU/dy)max , Ux, and U2 were determined as marked in Figs. 3(a) and 4(a).
The flow in test Tl was least affected by bed friction. In this test, the bed-friction length scale hlcf was 867 cm, which was large compared with the length of the test section. The width of the mixing layer in this test increased almost linearly with distance from the splitter plate (8 = 7.4 cm, 16.5 cm, and 25.1 cm at x = 50 cm, 100 cm, and 150 cm, respectively; see Fig. 3). At x = 200 cm, the mixing layer attached to the side wall, and the width could not be determined here accurately.
The flow in test T3, with a bed-friction length scale hlcf = 372 cm, was subjected to a stronger bed-friction influence than test Tl . The width of the mixing layer in this case no longer increased linearly with distance, but tended to approach an asymptoptic width of about 16 cm [8 = 3.8 cm, 9.0 cm, 11.8 cm, 14.5 cm, and 15.2 cm at x = 50 cm, 100 cm, 150 cm, 200 cm, and 300 cm, respectively; see Fig. 4(a)].
Important length scales and velocity scales of the shallow mixing layer are: h = the water depth; hlcf = the bed-friction length scale; A = (U1 -U2) = the velocity difference across the mixing layer; and U = (U1 + U2)/2 = the average velocity across the mixing layer.
A dimensionless presentation showing the longitudinal development of the maximum slope width is shown in Fig. 6. The data of the five tests collapse along a single curve when the following dimensionless variables are used in the presentation:
8* = 4" IT (3fl) h A0
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FIG. 3. (a) Mean Velocity Profiles for Test T1: hlcf - 867 cm, h = 5.0cm;(/b)r.m.s. Velocity Profiles for Test T1: hlcf = 867 cm, h = 5.0 cm
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2 0
20 •
2 0
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20 -
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0
(a)
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y(om)
FIG. 4. (a) Mean Velocity Profiles for Test T3:h/cf = 372 cm, h = 2.5 cm; (b) r.m.s. Velocity Profiles for Test T3: hlcf = 372 cm, h = 2.5 cm
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1 -
x=300 cm
x = 200 cm
i
X=150 cm
x=100 cm
2 -x=50 cm
x=1 cm
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(b) FIG. 4. Continued
10
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20 30
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200 250 300
FIG. 5. Mixing Layer in Shallow Open Channel Made Visible by Injection of Dye along Two Sides of Splitter Plate; Test T5: hlcf = 400 cm, h = 2.64 cm: (a) x = 0-150 cm; (b) x = 150-300 cm (Flow Directions Are from Left to Right in Photographs)
cfx Ob)
in which the subscript 0 = the condition at the entrance to the test section where x = 1 cm.
Doubling of Initial Spreading Rate The initial slope of the curve, for the near field region where the distance
from the splitter plate is small compared with the bed-friction length scale,
dh* dx1* 0.18 (4)
or
d8 Ul0 - U2Q A0
-r- = 0.36 '° f = 0.18-,-dx Ui0 + U20 u, o
(5)
This initial spreading rate of the shallow mixing layer is twice as large as the nominal rate proposed by Brown and Roshko (1974) for the free mixing layer.
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0.12
0.08
0.04
1
—
-
1
^ O o - 0 1 8
d5 0£. = 0O9 ax A0
a=0.18(1-^g)
a = 0.18
/ /*
' / > / / ' V""^
.AS
/A n'
- // .
/V I
/
I
/
m A _ ^ -
^—""*
Test
T1 T2 T3 T4 T5
I
h/c ,
867 289 372 459 4 0 0
I
Sy
T
— a
mbols
d B
*
1
-
-
-
-
0.4 0.8 1.2 .
FIG. 6. Maximum Slope Width of Shallow Mixing Layer
It should be pointed out that there are two possible limits when xcjh approaches zero. The first one is the free mixing layer when x is small compared with h. The second limits is when x is small compared with hlcf, but still large compared with h. The initial development of the mixing layer in the present experiment belongs to the latter situation when the bed-friction is negligible, but the turbulent length scale is still restricted in the vertical direction.
Entrainment Hypothesis The transverse spreading rate of the mixing layer reduces as the
bed-friction influence becomes more important in the far field region. The reduction of the spreading rate is associated with a reduction of an entrainment coefficient a, which is defined here as the ratio of an entrainment velocity dhldt and the velocity difference across the mixing layer A. Since the large turbulent eddy moves along the mixing layer with a velocity approximately equal to the average velocity U, then
]_db A dt
1 dx dh A dtTx
Udb A dx
(6)
The spreading rate db/dx was determined from curve-fitting through the data in a plot of 8 against x for each test. Fig. 7 shows the entrainment coefficient obtained from this direct method and its dependency on the bed-friction number S, defined in Eq. 1. The entrainment coefficient decreases with the increase in the value of the bed-friction number. The following relation fits the experimental data:
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0.16
0.08
1
*
8 » \ • X
e >v
^\A
\ • \
A
_
1
1
•
• A \ H
V
Alavian & ChuN (1985)
1
Test
T1 T2 T3 T4 T5
h/o,
867 2 8 9 372 459 4 0 0
B
I
1
Symbo ls
e m
* A
Y
Chu et al (1983)
• 0.04 0.08 0.12
FIG. 7. Variation of Entrainment Coefficient with Bed-Friction Number
= ° - 1 8 U ~ iTno » for 5 < 0.09 {la)
a = 0,
0.09
for S > 0.09 {lb)
The entrainment coefficient becomes negligible when the bed-friction number exceeds the critical value of 0.09. This critical value is in the range of values obtained from stability analysis of parallel flows by Chu et al. (1983) and Alavian and Chu (1985). The expression in Eq. 7 is similar to a relation obtained for jets and plumes between confined walls by Chu and Baines (1988).
Integral Analysis The curve-fitting of the experimental data to find db/dx and then the
entrainment coefficient previously described is subjective. Another method was used here to check the accuracy of Eq. 7. This was done by carrying out an integral analysis of the mixing layer using the expressions
a = a0( 1 - y for S < S„
a = 0, for S > Sr
(8fl)
(8*0
for the entrainment coefficient. The integral analysis gives a relation between 8* and x* as described in Appendix I. The 8*-x* relation depends on the value of a0 and Sc in Eqs. 8. The best fit of this relation with the data in Fig. 6 leads to a0 = 0.18 and Sc = 0.09; these values for a0 and Sc are in agreement with the earlier result obtained from the direct method. Another 8*-x* relation was obtained using a constant entrainment coefficient (a = a0 = 0.18), but this relation does not fit the experimental data
FIG. 8. Background Component of Turbulence; r.m.s. Velocities, u[ and u'2, Normalized by Bed-Friction Velocity, ;<*i and «*2
(see Fig. 6). It is clear that the entrainment coefficient is not constant but dependent on the bed-friction number.
Turbulent Motions Turbulence in the shallow mixing layer may be considered to have two
components with two distinct length scales. One is the small-scale turbulence, which is generated by the bed friction and limited in length scale by the water depth. The other is the large-scale turbulence, which is generated by the transverse shear and characterized by a length scale comparable with the width of the mixing layer. The bed friction generates not only the small-scale turbulence but, at the same time, exerts a stabilizing influence on the large-scale turbulence. To see the stabilizing influence on the large scale, a simple procedure is used here to separate the r.m.s. velocity
profiles into a transverse component and a background component. The background component, which is assumed here to be generated by bed friction alone, should be proportional to the bed-friction velocity, M* = Vc/2 U. The transverse component, on the other hand, should be related to the velocity difference across the mixing layer, A = C/j - U2 •
The two components of the turbulence are evident from the r.m.s. velocity profiles as shown in Figs. 3(b) and 4(b). Three characteristic r.m.s. velocities are obtained from these profiles as marked in the figures. These are the peak value u'm , obtained inside the mixing layer, and the ambient values u[ and u2, obtained in the ambient streams. If the ambient values were created by bed friction alone, they would scale with the local bed-friction velocities u^l and M#2 (W*I = VCfjl Ux and w*2
= vc^/2 U2). Fig. 8 shows the correlation of the ambient values with the bed-friction velocities. The ratios u'Ju^ and u'2lu*i, being generally in the order of unity, are consistent with the turbulent measurements in wide open channel flows (Nakagawa et al. 1975). Higher values are observed in some cases. For example, in the low velocity side of the ambient stream in test Tl, u2lu*2 = 1.85, 2.65, and 3.70 at x = 50 cm, 100 cm, and 150 cm, respectively. The high ambient turbulent intensity in test Tl is due to the interaction of the mixing layer with the boundary layer along the side wall.
If the high values of test Tl are ignored, the data in Fig. 8 suggest that the following formulas may be used to calculate the background components: u"= 1.1 w*! and u2= 1.1 z/̂ ,2 • In the middle of the mixing layer, the background component is (u'{ + u2)/2. An excess value, u'em, can be defined as the difference between the peak and the background:
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Item Mm j . . . . . . . \y)
This excess value represents the transverse component of the turbulent motion. Fig. 9 shows the variation of the excess value with bed-friction number. The data of all five tests collapse along a single curve, indicating that the method used to extract the transverse component was valid and that the stabilizing influence on the transverse component can be related with the bed-friction number. The curve leads to a critical bed-friction number of about 0.09, which is in agreement with the critical value obtained earlier from the entrainment coefficient relation. In the limit of small bed-friction number, the trend of the data in Fig. 9 suggests a slightly higher value of u'eJL than the value of about 0.18 obtained for the free mixing layer by Wygnanski and Fiedler (1970) and Champagne et al. (1976).
CONCLUSIONS
Two important characteristics of the shallow mixing layer are obtained from the present experimental investigation:
1. The initial spreading rate, dhldx, is equal to 0.18 Ao/C70, which is twice as large as the nominal spreading rate for a free mixing layer.
2. Both the entrainment coefficient and the transverse component of the turbulent motion diminish to zero as the bed-friction number exceeds a critical value of about 0.09; the critical value lies in the range between the theoretical values of 0.12 and 0.06 obtained by Chu et al. (1983) and by Alavian and Chu (1985) for parallel flows.
The following approximate relations for the entrainment coefficient are suggested (Eq. 8):
a = o J l - — j , fovS<Sc
a = 0, fox S>SC
in which a0 = 0.18 and Sc = 0.09. A number of processes that are unique in the development of the shallow
mixing layer can be identified. There is, of course, the friction stresses on the channel bed which generate the small-scale turbulence and, at the same time, exert a stabilizing influence on the transverse motion. There is also the restriction of the vertical length scale by the water surface and channel bed, which causes the doubling of the initial spreading rate. The production and dissipation of turbulent energy in this highly anisotropic flow field should be very different from that in the free mixing layer.
The free mixing layer has been a subject of intensive investigation. Large coherent turbulent structures have been observed by Brown and Roshko (1974) and Winant and Browand (1974). They attributed the growth of the mixing layer to pairing and coalescence processes. These processes are essentially two-dimensional and do not account for the development of turbulent motion in the third dimension. Konrad (1976) suggested the growth of spanwise vortices as a possible mechanism for
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three-dimensional turbulent motion. Browand and Troutt (1980) found the length scale of the spanwise vortices grows in proportion to the width of the mixing layer.
In a shallow mixing layer, the growth of spanwise vortices, if they exists at all, would be limited by the water depth. The transfer of turbulent energy from a horizontal motion toward a vertical motion is reduced in a shallow mixing layer, and this may be the reason for the larger initial spreading rate. It is possible that a small fraction of the turbulent energy generated from the bed friction may feed back toward the large-scale horizontal motion.
It is not the first time that a mixing layer has been observed to have a spreading rate, db/dx, significantly higher than the nominal rate of 0.09 A(/C/0 • Dependency of the spreading rate on the initial conditions at the splitter plate has been reported by Bradshaw (1966), Batt et al. (1971), and Champagne et al. (1976). Oster et al. (1977) performed experiments to control the initial shedding of vorticity and the triggering of the natural instability of the vortex sheet. Periodic forcing was introduced to a mixing layer in a frequency range of 100 Hz to 20 Hz. They found that doubling of the initial spreading rate is not only possible, but can be easily attained. Oziomba and Fiedler (1985) showed that even very weak periodic perturbation may cause significant nonlinear spreading of the mixing layers. The tuning of the mixing layer by its preferred mode of disturbance enhances the two-dimensional turbulent motion and possibly delays the transfer of energy to form spanwise vortices. This anisotropic production of turbulent motion through excitation of a preferred mode in a free mixing layer, as described in the experiment of Oster et al. (1977), is similar to the production of transverse motions in a shallow mixing layer.
ACKNOWLEDGMENTS
We would like to acknowledge the helpful discussions with Prof. R. G. Ingram of the Institute of Oceanography at McGill, who also supplied us with the photographs and the data in Fig. 1. The photograph (roll No. A37339-129, copyright 1976) in this article were reproduced from the collection of the National Air Photo Library with permission of Energy, Mines and Resources, Canada.
APPENDIX I. INTEGRAL ANALYSIS
The integral analysis of the shallow mixing layer is based on the depth-averaged formulation. In the ambient stream, the transverse exchange of momentum is negligible and the governing equations are
dU] cn . dPx
dU\ cn . dP2
-±r+fu^2-di-° • < • < » > The upstream boundary conditions are U1 = Ul0 and U2 = U20 , at x = 0.
In these expressions, subscript 0 = the condition at entrance to the test section, and subscripts 1 and 2 = the condition at the ambient streams. The 1272
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longitudinal pressure gradients, dPJdx and dP2ldx, are the same for both sides of the ambient stream. The friction coefficients, cn and cn, are replaced by the local average value cf. Integration of these equations leads to the expression as follows:
U\-U\ AU ( xc\
U20-U
220 &0U0 \ h)
Introducing this expression into the entrainment hypothesis, Eqs. 6 and 8, leads to the following equations for the relation between 8* and x*
dh* Ul — = a — exp (-*•) (13) dx* U in which the entrainment coefficient (Eq. 8)
a = ao( 1 - j | , for S < Sc
a = 0, for S > Sc
and
1 A0t7 S = - 8* - V (14)
2 AC/0
The integration of these equations was made numerically using the Runge-Kutta method. The relation between 8* and x* , for U/U0 = 1, a0 = 0.18, and Sc = 0.09, is shown as a solid line in Fig. 6. The virtual origin, where 8* = 0, was at x* = 0.025.
The b*-x* relation obtained for a constant entrainment coefficient (a = a0 = 0.18) is also shown in Fig. 6.
APPENDIX II. REFERENCES
Alavian, V., and Chu, V. H. (1985). "Turbulent exchange flow in a shallow compound channel." Proc, 21th Congress of Int. Assoc, of Hydraulic Research, v.3, 446-451.
Babarutsi, S., and Chu, V. H. (1985). "Experimental study of turbulent mixing layers in shallow open channel flows." Tech. Rept. No. 85-1 (FML), Fluid Mechanics Lab., Dept. of Civil Engrg. and Applied Mechanics, McGill University, Montreal, Canada.
Batt, R. G., Kubota, T., and Laufer, J. (1970). Proc, A.I.A.A. Reacting Turbulent Flow Conference, San Diego, Calif.
Bradshaw, P. (1966). "The effect of initial conditions on the development of a free shear layer." / . Fluid Mech., 26, 225-236.
Browand, F. K., and Troutt, T. R. (1980). "A note on spanwise structure in two-dimensional mixing layer." J. Fluid Mech., 97, 771-781.
Brown, G. L., and Roshko, A. (1974). "On density effect and large structure in turbulent mixing layer." J. Fluid Mech., 64, 775-816.
Carter, R. W., et al. (1963). "A.S.C.E. Task Force on Friction Factors in Open Channels." J. ofHydr. Div., ASCE, 89(HY2), 97-143.
Champagne, F. H., Pao, Y. H., and Wygnanski, I. J. (1976). "On the two-dimensional mixing region." / . Fluid Mech., 74, 209-250.
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Chu, V. H., and Baines, W. D. (1989). "Entrainment by a buoyant jet between confined walls." / . Hydr. Engrg. (in press).
Chu, V. H., Wu, J.-H., and Khayat, R. E. (1983). "Stability of turbulent shear flows in shallow channel." Proc, 20th Congress of Int. Assoc. Hydraulic Research, v.3, 128-133.
Dziomba, B., and Fiedler, H. E. (1985). "Effect of initial conditions on two-dimensional free shear layer." / . Fluid Mech., 152, 419-442.
Konrad, J. H. (1976). "An experimental investigation of mixing in two-dimensional turbulent shear flows with applications to diffusion-limited chemical reaction." Project SQUID, Tech. Report CIT-8-PU, Purdure University, West Lafayette, Ind.
Nakagawa, H., Nezu, I., and Ueda, H. (1975). "Turbulence in open channel flow over smooth and rough beds." Proc. Japan Soc. Civ. Eng., 241, 155-168.
Oster, D., et al. (1977). "On the effect of initial conditions on the two-dimensional turbulent mixing layer." Lecture note in physics {Proc, of the Symposium on Turbulence, I), H. Fiedler, ed., v.75, Springer-Verlag, Berlin, 48-64.
Winant, D. D., and Browand, F. K. (1974). "Vortex pairing: The mechanism of turbulent mixing layer growth at moderate Reynolds number." / . Fluid Mech., 63, 237-255.
Wygnanski, I., and Fiedler, H. E. (1970). "The two-dimensional mixing region." J. Fluid Mech., 41, 327-361.
APPENDIX III. NOTATION
The following symbols are used in this paper.
cf h P
Re S
sc U
u u' u"
u' " em "In
w* X
y a A 8
= = = = = = = = = = = = = = = = = =
bed-friction coefficient; water depth; pressure under a rigid lid; Reynolds number; bed-friction number; critical bed-friction number; mean velocity; average velocity across mixing layer; r.m.s. velocity; background component of r.m.s. velocity; transverse component of r.m.s. velocity (excess); peak value of r.m.s. velocity; bed shear velocity = Vc/72C/; longitudinal coordinate; transverse coordinate; entrainment coefficient; velocity difference across the mixing layers; and maximum-slope width.
Subscripts m = maximum at peak of r .m.s. velocity profile; 0 = conditions at entrance to test section (x = 1 cm); 1 = high-velocity side of ambient stream; 2 = low-velocity side of ambient stream; and * = shear velocity.
