A.W. Bruens1, J. Pietrzak1 and J. C. Winterwerp1,2
1 Delft University of Technology, Faculty of Civil Engineering and Geosciences, P. O. Box 5048, 2600 GA Delft, The Netherlands, tel: +31-15-2784070, fax: +31-15-2785975,
E-mail: a.bruens@ct.tudelft.nl
2 WL|delft hydraulics P.O Box 117, 2600 MH Delft, The Netherlands.
Abstract: This paper deals with experiments on the decay of non-locally produced turbulence over a pool filled with a dense fluid. Upstream of the depression, turbulence is produced mainly as a result of bed friction. In the depression friction and the production of turbulence is for the larger part suppressed and the non-locally produced turbulence is advected over the dense fluid and gradually decays in downstream direction. The scaling laws which have to be taken into account when interpreting the laboratory experiment with respect to field conditions are discussed in this paper. During the experiments, mean velocities as well as turbulent fluctuations are measured at several positions over the depression. The obtained data indicate a decrease of turbulence in downstream direction over the pool. While decaying though, this non-locally produced turbulence might entrain material from the dense fluid. The entrainment rates are also measured. The results indicate a decrease in downstream direction, as the local Richardson number decreases in this direction.
Keywords: turbulence decay, density stratification, entrainment, cohesive sediments, concentrated benthic suspension
High deposition rates of cohesive sediments lead to many coastal management problems. One of the largest problems is the siltation of harbors and navigation channels. To keep these areas navigable, it is often necessary to dredge tons of material on a yearly basis. This is both a time and money consuming occupation, especially if the sediment is polluted. To deal with this type of problem, it is essential to understand the behavior of cohesive sediment. In particular improved understanding of the interaction between the turbulent flow-field and sediment-induced density stratification is needed. This paper deals with laboratory experiments carried out to enhance the understanding of this interaction.
In the deeper part of an estuary, for example within a navigation channel, the deposition rate of cohesive sediments may be high, especially during slack water. In that case a concentrated benthic suspension (CBS) may be formed, which behaves as a dense fluid for several hours. In the flow upstream of this depression, turbulence is produced mainly as a result of bed friction. In the depression, friction and the production of turbulence is for the larger part suppressed due to the presence of the sediment-induced density stratification. The turbulence produced over the rigid bed upstream of the deeper part is advected over the CBS and gradually decays in the downstream direction, as no or only little turbulence is produced locally. While decaying though, this non-locally produced turbulence might entrain material from the CBS.
At Delft University of Technology, experiments were carried out to study the importance of non-locally produced turbulence on CBS-dynamics (i.e. entrainment). The objectives of the experiments were 1) to obtain a relationship between hydrodynamic parameters, decay of turbulence and entrainment rates, 2) to generate data to validate and improve computer models.
In order to scale the field conditions with the laboratory experiments, several scaling laws have to be taken into account. These laws are discussed in the next section of this paper. The experimental set-up and program are described in the third section, followed by the experimental results in the fourth section. Finally, conclusions are drawn in the last section of this paper. The validation of a 3D numerical simulation model, using the data, will be presented in a future paper.
The first scaling law is concerned with the turbulence structure. It is required that the Reynolds numbers (Re = uh/ν, where u is mean velocity, h is depth of the flow and ν is viscosity) in the physical model is sufficiently high and that upstream of the depression hydraulically rough conditions prevail.
The second law is concerned with the scaling of turbulence decay. A lower bound of the length scale L for decay can be obtained from a balance of the advection and dissipation terms in the transport equation for turbulent kinetic energy. The length scale L scales with the water depth (L ≈ 9h) and for significant decay to be observable, the length of the depression should be at least several times L.
The third scaling law concerns the consolidation time. The consolidation time (Tc) is proportional to the inverse of the thickness (d ) squared of the CBS layer (Tc µ1/d 2). This implies that in the physical model it is nearly impossible to keep the CBS in a fluidised state. A fluid mud layer, possessing strength, will be formed and the turbulence production at the water/fluid mud interface then would not be essentially different from that over the upstream rigid bed.
The last scaling law deals with entrainment. The entrainment process is governed by the overall Richardson number
where Δρ is the excess density of the CBS, ρ the density of the overlying water, g is the acceleration of gravity, h the height of the water layer and u* a friction velocity. Simulation of the entrainment process in a hydraulic model requires scaling at constant Ri*, which yields for constant g and ρ:
where subscript f refers to the field and subscript m to the physical model. Substituting realistic values for the field parameters, very low bed shear stresses in the physical model are required, which is not likely to be achieved with a concentrated layer.
In order to obey the last two scaling laws, the CBS in the physical model is replaced by water in which sodium chloride is dissolved. In earlier work it was already shown that, in terms of initial entrainment rate, CBS behaves in a similar manner to saline water.
The physical experiments were conducted in a straight flume with an overall length of 30 m, 1 m wide and 0.3 m deep. Two series of experiments were carried out. The first series were concerned with decay of turbulence, the second series with entrainment. A longitudinal cross-section of the experimental set-up for the first series is shown in Figure 1. The inlet structure at the beginning of the flume (consisting of a weir and flow straighteners) was designed to eliminate secondary flow patterns. The upstream part of the flume was 6 m long and 0.15 m deep. Here, hydraulically rough conditions prevailed and created fully turbulent shear flow conditions. After this upstream part, the flume deepened to 0.3 m. A ramp linked the two sections of the flume. To prevent flow separation the angle of this ramp was small (8°). The deeper part of the flume was filled with saline water up to 0.1 m, so some internal wave motion was possible without saline water flowing in the upstream part. The saline water level was kept constant at 0.1 m by an internal weir at the end of the deeper part, and feeding the saline layer with water from a reservoir. The rate at which saline water was added was small and just compensated the amount of saline water entrained. To avoid accumulation of saline water behind the internal weir, it could be withdrawn from the bottom of the flume. Fresh water from the upper layer was caught in an outlet structure behind a second, larger weir. From here fresh water was recirculated to the inlet structure by a pump. In case the water in the upper layer became too saline (as a consequence of entrainment of the lower layer), it was replaced by fresh water in the outlet structure.
The geometry of the flume had to meet two
requirements. First, it had to be wide enough, so that wall-influences can be
neglected in the center of the flume (where the measurements were taken).
Secondly, the upstream part had to be long enough for the flow to be fully
developed before reaching the ramp. Preliminary experiments to test the
applicability of the flume indicated that both requirements were fulfilled (Wissmann
& Bruens, 2000).
Four experiments on the decay of turbulence (D1-D4) were carried out. The experimental program was set-up by varying the flow velocity (Table 1). A mean velocity of 0.154 m/s was the maximum attainable velocity in the present set-up. The flow velocity for experiment D4 was similar to the velocity of experiment D1. The density difference between experiment D1 and D4 was varied in order to assess the influence of the density difference on the rate of overall decay. Vertical velocity profiles (including mean velocities and turbulent fluctuations) and (mean) density profiles were measured at eight positions (figure 1). The first position was situated 0.5 m upstream of the ramp. The mutual distance between the successive positions was 2 m.
The entrainment rate decreases with decreasing turbulence level in the downstream direction over the depression. To measure the integral entrainment rate at different positions, the experimental set-up was slightly changed from the one shown in Figure 1. The internal weir was replaced by an internal barrier, which could be placed at various positions (X2 to X8 in Figure 1). For each position the exact amount of saline water that had to be fed into the flume to keep the saline water layer constant at 0.1 m (balancing the loss by entrainment) was measured. Two experiments have been carried out (E1 and E2 in Table 1).
The results for the four different experiments
on the decay of turbulence do not deviate from each other in their qualitative
behavior. Therefore only the results from experiment D3 are presented in this
paper. For the results of the remaining three experiments the reader is referred
to Wissmann & Bruens, 2000.
Conductivity probes were used to measure densities. At each measurement point fluid was pumped through a small tube towards a sensor outside the flume. As the measurements were not taken in-situ, high frequency measurements and measurements at small vertical intervals could not be taken. The density profiles for all positions are plotted in Figure 2. Also the time, at which the profiles were taken, is given. The profiles indicate that the interface is more or less uniform and stationary.
Velocities were measured using an immersible
laser-Doppler anemometer (lda) at a frequency of 100 Hz. Mean velocity profiles
and profiles of Reynold stresses (
) are plotted in Figure 3 and 4 respectively. To retain a better overview of
the data, profiles at position X1 to X4 are plotted in Figures a, while profiles
at position X4 to X8 are plotted in Figures b (note the change of scales). To
reduce the amount of data in the figures and increase the overview, results at
position X3 are not plotted. The conditions in the upstream part of the flume at
position X1 are shown in Figure 3a and Figure 4a. The lowest measuring point was
located 1.4 cm above the bed; it was the lowest position within reach of the lda.
In the region z/h < 0.6 of the velocity profile, the log-law distribution is
valid. Closer to the water surface, the velocity decreases. This decrease was
also observed by Nakagawa and Nezu (1995), who based the occurrence of this ‘velocity-dip
phenomenon’ on secondary currents induced by turbulence in channels or rivers
with an aspect ratio of approximately 5-6. Velocity gradients in the upstream
part are highest near the bed. Since production of turbulence is directly linked
to the velocity gradient, highest turbulence intensities must also occur closer
to the bed. Figure 4a shows that Reynold stresses at position x1 are indeed
highest in this region.
Due to the deepening of the flow, mean flow velocities decrease between position X1 and X2 (Figure 3a). Downstream of position X2, the velocity profile adapts to the new boundary conditions (Figures 3a and 3b). These figures show that besides a decrease in velocity, also a decrease in the vertical velocity gradient takes place in the downstream direction. This decrease leads to little local production of turbulence over the depression, compared to the production rate in the upstream part. The effect of this small local production of turbulence is visible in Figure 4a and 4b; the turbulent stresses decrease in the downstream direction. Towards the end of the flume (between position X7 and X8) a new equilibrium between the small amount of turbulence generated near the interface and the decay of this locally produced turbulence, is nearly reached.
From Figure 4a also another phenomenon can be observed. Between position X2 and X4, the Reynold stresses (and therefore the turbulent intensities) higher in the water column increase, before they start to decrease in downstream direction of position X4. To explain this phenomenon, the situation in the upstream part has to be considered. In this part of the flume vortices with a large amount of energy are produced near the rigid bed. These vortices are diffused to regions higher in the water column, and eventually they reach the water surface. In the upstream part, these vortices diffuse in a flow regime where high velocity gradients prevail, resulting in a high dissipation rate of their energy. Vortices generated near the end of the upstream part diffuse in the deeper part of the flume, where vertical velocity gradients, and therefore dissipation rates, are smaller. These vortices will now possess more energy when they enter regions closer to the water surface. This phenomenon will be quantitatively elaborated upon in a future paper.
The turbulent kinetic energy, k = (u’2+w’2)/2, is determined for
all measurement points. The total turbulent kinetic energy (K) at position X1 to X8 is obtained by integrating the vertical
energy profiles k(z). The
non-dimensional turbulent kinetic energy (=K/(hu2), where h and u are depth of flow and mean velocity of the upstream flow
respectively) is plotted versus longitudinal distance in Figure 5a. As it has
not been possible to measure close to the rigid bed in the upstream part, the
accuracy of the integration at position X1 is low. Probably because of the
deceleration of the flow between position X1 and X2, the amount of turbulence
increases. From position X2 onwards, the turbulent kinetic energy decreases.
To obtain the scaling law for turbulent decay, a
balance of the advection and dissipation terms in the transport equation for
turbulent kinetic energy was used:
where x is the horizontal coordinate positive in downstream direction, l represents the length scales of the large eddies (proportional to the water depth) and cd is a coefficient, cd»0.15. This balance indicates that k µ 1/x2. In Figure 5b the non-dimensional K is plotted versus x on a log-log-scale indicating that this relationship is valid for the present experiments.
During the second series of experiments, the integral entrainment over the pool has been measured. The size of the pool has been varied by shifting the internal barrier from position X1 to position X8. From the increase of the integral entrainment in the downstream direction, the entrainment rate (we) as a function of x could be determined. In Figure 6a these entrainment rates are plotted versus longitudinal distance. As expected, entrainment rates decrease in the downstream direction. The overall Richardson number for experiment 2 is smaller, leading to larger entrainment rates during this experiment.
The entrainment rates are expected to be a function of a local Richardson number. In the present study, this Richardson number is based on the mean turbulent kinetic energy (km=K/hl, where hl is the local depth of the upper layer):
As km decreases in the downstream direction, the Richardson number increases and the entrainment decreases. In Figure 6b the non-dimensional entrainment rate (E=we/Ökm) is plotted versus Rik on a log-log-scale. These preliminary results do not show a general relation in the form EµRi-n, as found in previous studies on the entrainment of a dense fluid. In the present analyses, the turbulent kinetic energy from experiment D1 and D2 are used to determine Rik and to scale we for experiment E1 and E2 respectively. It should be noted though, that the density difference for experiment E2 (40 g/l) is substantially smaller than for experiment D2 (100 g/l). In the near future, the influence of the density difference on the decay of turbulence has to be assessed. If it is found that the density difference has a significant effect on the total amount of turbulent kinetic energy, the discrepancy between E1 and E2 in Figure 6b may be caused by an inproper scaling, and a another scaling for E2 has to be used.
In this paper experiments on the decay of non-locally produced turbulence over a pool filled with a dense fluid are presented. The obtained data indicate a decrease of turbulence in downstream direction over the pool. This decrease is proportional to the inverse square of the downstream distance. Entrainment rates are also measured. The results also indicate a decrease in the downstream direction, as the local Richardson number decreases in this direction. From the data, no uniform relation between entrainment rates and this local Richardson number has been obtained at present. The experiments have only recently been carried out. Therefore a more quantitative analyses of the relation between hydrodynamic parameters, decay of turbulence (including the effect of the density difference on the total decay) and the subsequent decrease in entrainment will be carried out in the future.
Acknowledgement
This work was partially funded by the European Commission, Directorate General XII for Science, Research & Development through the COSINUS-project within the framework of the MAST-3 programme, contract MASC3-CT97-0082. Dr C. Kranenburg is gratefully acknowledged for his preparatory work on the design of the flume and his contribution to the scaling laws. Jochen Wissmann, of the technical University of Karlsruhe, is gratefully acknowledged for his contribution to the experiments and data processing.
References
Nakagawa, H. & Nezu, I. (1995). Turbulent structures and the related environment in various water flows. 1994 Scientific Research Activities Vol. 8, Department of Civil & Global Environment Engineering, Kyoto University, Kyote 606 Japan.
Nezu, I. & Nakagawa, H. (1993). Turbulence in open-channel flow. International Association for Hydraulic Research, Monograph Series, A. A. Balkema, Rotterdam.
Wissmann, J. & Bruens, A. W. 2000. Experiments on the decay of turbulence due to density stratification. Tech. Rept. 6-00. Delft University.
Fig. 1 Schematic cross-section of the experimental set-up for the first series of experiments (not to scale).
Fig. 2 Density profiles during experiment D3.
Fig. 3a Mean horizontal velocities at Fig. 3b Mean horizontal velocities at
positions X1, X2 and X4. positions X4, X5 and X7.
.
Fig. 4a
Reynold stresses at positions
Fig. 4b Reynold stresses at positions X4
X1, X2 and X4.
to X8.
Fig. 5a Non-dimensional turbulent kinetic energy versus downstream direction.
Fig. 5b Log of non-dimensional turbulentkinetic energy versus log of thedownstream direction.
Fig. 6a Entrainment rate versus downstream distance.
Fig. 6b Log of non-dimensional entrainment rate versus log of Richardson number.