Y. Niño1, J. Helmbrecht2, F. Lopez2, G. Buscaglia3,
L. Reyes1 and S. Morillo2
1Department of Civil Engineering, University of Chile. Chile
2National Institute of Water and Environment,
National University of Cordoba. Argentina
3Atomic Center of Bariloche and Balseiro Institute. Argentina
Abstract: Results of both, an experimental and a numerical
study on the behavior of the density stratification in water bodies under the
action of a surface shear stress, such as that exerted by the wind, are
presented. The experimental study was conducted in a tank with a conveyor belt
that applies a shear stress on the surface of a two-layer salinity-stratified
volume of water. The experimental results were conducted under unsteady surface
shear stress conditions. Video images of the flow were analyzed to study the
shear-induced mixing and tilting of the density interface. The numerical study
consisted of a 3D simulation of the hydrodynamics of Lake San Roque, driven by
measured hydro-meteorological data. The experimental results show the existence
of a mode 2 response of the density stratification to surface shear. The tilting
of the density interface responds to the variable Richardson number induced by
the surface shear and results to be similar to that measured in similar
experiments with steady shear conditions. The rate of mixing induced by the
unsteady shear, once the flow conditions are established, is similar to that
observed in the final stage of mixing in the steady case, for corresponding
values of the Richardson number. The interface tilting obtained from the
simulations agrees better with the predictions of the classical theory than that
of the experimental results. The values of the entrainment velocity obtained
from the simulations are generally larger than expected according to different
entrainment relationships. A definite explanation for this result is currently
being investigated.
Keywords: stratified lakes, metalimnion, wind-induced mixing, interface tilting, seiches, laboratory experiments, 3D numerical simulations
In stratified water bodies, such as lakes and reservoirs, the wind plays an important role as a source of turbulent kinetic energy, which influences mixing processes in the density interface between the well-mixed surface layer and the bottom region. The wind is also responsible for the tilting of the density interface, and the generation of seiches and internal waves, all of which results in a complicated dynamical response of the density stratification to the surface shear stress induced by the wind. Previous work by the authors, based on numerical and experimental studies, has explored the behavior of both, the rate of wind-induced mixing and the slope of the density interface, relating them to the Richardson number of the flow and the length to depth ratio of the stratified water body (Niño et al., 2000; López et al., 2000a, b). The experimental study and most of the numerical work have been done considering steady conditions for the surface shear stress. This is true also, for most of other similar research (e.g., Kranenburg, 1984, 1985), however wind conditions prevailing in the field situation are intrinsically unsteady (e.g., López et al., 2000a). In this paper, experimental and numerical results regarding the effect of a variable surface shear stress, such as that associated with wind events occurring in the field, on the behavior of the density stratification in a water body are presented and discussed.
The experiments were conducted in a laboratory tank, 2 m long, 0.2 m wide, and 0.7 m deep, located at the University of Chile. A conveyor belt installed at half depth was used to exert a shear stress on the surface of a volume of water filling the bottom part of the tank. A two-layer, stable density stratification was created inside the tank by setting a layer of tap water over a solution of salt and water dyed with potassium permanganate. A previously developed experimental technique was used to determine instantaneous density fields, by analyzing the pixel intensity information of the video images and applying a calibrated relationship between pixel intensity and the salinity of dyed saline solutions.
The experiments were conducted under steady and unsteady surface shear stress conditions. In this paper results of a particular experiment conducted under unsteady conditions are reported and compared with previous results under steady conditions reported elsewhere (Niño et al., 2000; López et al., 2000b). A wind event measured in the field, in Lake San Roque, Córdoba, Argentina, was scaled down and reproduced in the laboratory by varying in time the conveyor belt speed. The forcing surface shear velocity and the resulting values of the associated Richardson number of the flow, Ri = Dr/r g D/u*2, (determined from measurements of Dr, r, and D in time) are plotted in Fig. 1. Here r is the density of the surface mixed layer, Dr is the density difference between the surface and bottom layer, D is the thickness of the surface layer, and u* is the surface shear velocity. The length to height ratio of the experimental volume was 9.
The numerical model used is an ad-hoc modification of the 1998 version of POM (Princeton Ocean Model). This is a 3D model that solves the system of conservation equations including continuity, momentum and thermal energy. Turbulent transport processes are estimated using the Mellor-Yamada model, which requires solving a conservation equation for the turbulent kinetic energy and an algebraic equation for the turbulence macro-scale. Horizontal diffusion is modeled using a Smagorinsky type of equation. The model works with a sigma coordinate in the vertical and an orthogonal curvilinear mesh in the horizontal. The numerical scheme is based on finite differences and is explicit in the horizontal and implicit in the vertical. The model has been previously applied, and validated, by the authors to simulate steady state flows in water bodies with simple shapes, the flow in Lakes Nahuel Huapi and San Roque, in Argentina, under steady wind conditions and, in the latter case, also under real forcing meteorological conditions measured through a network of sensors located around the lake (López et al., 2000a). In this paper, selected simulated data of the behavior of the temperature stratification of Lake San Roque under wind events that caused the tilting and deepening of the thermocline due to mixing effects, during the month of February 1999, are analyzed.
Shortly after starting the application of the first step of the variable surface shear of Fig. 1, the density interface responded by tilting in the downstream direction, reaching a relatively constant slope after about 6 min (Fig. 2). At the same time, a secondary interface became apparent below the main one, defining an intermediate zone of large density gradients. This zone, the metalimnion, has been termed wedge in the past ( Kranenburg, 1985) because its thickness increases in the upstream direction. The values of the buoyancy frequency, N, associated to the primary and secondary interfaces remain relatively constant during the first 10 min of the experiment (Fig. 2), although in the case of the primary interface N tends to decrease at the end of the 10 min period. This tendency, which is maintained until the end of the experiment, is caused by mixing induced by the surface shear, which tends to decrease the initially sharp gradient of the primary interface. The behavior of both interfaces is indicative of a mode 2 response of the density stratification (Monismith, 1986). The thickness of the metalimnion was measured in two sections of the tank: the middle section and an upper section located at a distance equal to 0.6 m from the upstream wall of the tank. In both sections there is a tendency for this thickness to oscillate in time after the slopes of the primary and secondary interfaces are fully developed. The period of the oscillation is of the order of one to two minutes. The oscillations appear to be caused by seiching of both, the primary and secondary interfaces.
The overall response of the slope of the primary density interface to the variable Richardson number of the flow during the whole experiment is shown in Fig. 3. In the same figure, experimental values of this slope obtained for different Richardson numbers under steady shear stress, for the same length to height ratio of the water body (López et al., 2000b), and the slope predicted by the classical theory, Si = Ri-1 (López et al., 2000b), are also plotted. The values of the slope measured under unsteady conditions do not differ significantly from the corresponding values measured under steady conditions, which is indicative of the short time scale of the response of the flow, in terms of the adjustment of the longitudinal pressure gradients, to the surface shear stress. Both sets of experimental data show that the classical theory tends to underestimate the value of the slope.
The experimental values of the dimensionless entrainment velocity, ue/u*, where ue is the rate of growth of D, are plotted as function of Ri in Fig. 4, together with corresponding values measured in the final stage of mixing under steady shear conditions (Niño et al., 2000). In the same figure, the predictions of an entrainment relationship proposed by Niño et al. (2000) evaluated for a value of the length to height ratio equal to 9 and the predictions of Kranenburg’s (1984) entrainment relationship valid for a water body of infinite length are also plotted. The entrainment velocity induced by the unsteady shear is similar to that observed in the final stage of mixing in the steady case, for corresponding values of the Richardson number of the flow. During the initial stage of mixing, however, the entrainment velocity observed in the unsteady case was, as in the steady case, larger than that measured in later stages of the mixing process. These results are indicative that the time scale of the response of the mixing rate to the changing surface shear is smaller than the time scale of change of such variable.
The results of the numerical simulations show a metalimnion dynamics that is similar to that observed in the experiments. The input of turbulent kinetic energy by the wind causes the deepening of the surface layer, which appears well mixed with a rather constant temperature in the vertical. By analyzing the numerical data corresponding to different wind events, the slope of the thermocline predicted by the model was estimated. The results are plotted as function of the corresponding values of the Richardson number in Fig. 5, together with the prediction of the classical theory. The theory tends to slightly overestimate the slopes resulting from the simulations, except for a couple of cases of perfect agreement. A much better prediction of the theory in the case of the simulations than in the case of the experimental results presented previously is evident. This may be indicative of relatively weaker shear stresses acting on the interface in the field situation than in the experiments.
Values of the dimensionless entrainment velocity, ue/u*, resulting from the simulations are plotted in Fig. 6 as function of Ri. In the same figure predictions of Kranenburg’s (1984) entrainment relationship valid for a water body of infinite length and Niño et al.’s (2000) entrainment relationship for a length to height ratio of 200, representative of Lake San Roque, are also plotted. The values of the entrainment velocity obtained from the simulations are generally larger than those predicted by both entrainment relationships. These results may be indicative of a number of things. For example, they may indicate that the interface deepening simulated by the model is not a consequence of the wind alone but it may also be a caused by heat transport effects. On the other hand, it may be indicative that the Mellor-Yamada closure is not suitable for modeling wind-induced mixing in lakes. A deeper analysis of the numerical results is currently been conducted. More definite conclusions regarding these aspects of the numerical simulations are expected soon.
The experimental results show the existence of a mode 2 response of the density stratification to surface shear. Quasi-periodic oscillations of the metalimnion thickness were observed in different sections along the tank. The tilting of the density interface responds to the variable Richardson number induced by the surface shear and results to be of the same order as that measured in similar experiments with steady shear conditions. The rate of mixing induced by the unsteady shear, once the flow conditions are established, is similar to that observed in the final stage of mixing in the steady case, for corresponding values of the Richardson number. The results of the numerical simulation show a metalimnion dynamics that is similar to that observed in the experiments. The interface tilting obtained from the simulations agrees better with the predictions of the classical theory than that of the experimental results. This may be indicative of relatively weaker shear stresses acting on the interface in the field situation than in the experiments. The values of the entrainment velocity obtained from the simulations are generally larger than expected according to different entrainment relationships. A definite explanation for this result is result is currently being investigated.
Acknowledgements
The authors gratefully acknowledge the support provided by the International Cooperation Project funded by CONICYT (Chile, 1999-3-02-183) and SECyT (Argentina, CH/10/99), and also by Projects FONDECYT 1981180 (Chile) and FONCyT-PICT97 1005 and 0982 (Argentina).
Kranenburg, C. (1984). “Wind-induced entrainment in a stably stratified fluid”. J. Fluid Mech., 145, pp. 253-273.
Kranenburg, C. (1985). “Mixed-layer deepening in lakes after wind set-up”. J. Hydraul. Div. ASCE, 111 (HY9), pp. 1279-1297.
López, F., Buscaglia, G. C., Arnica, D. L. (2000a). "3D numerical modeling of wind-induced circulation and transport processes in the San Roque reservoir (Cordoba, Argentina). Hydroinformatics 2000, IAHR. Iowa, USA.
López, F., Niño, Y., Caballero, R., Buscaglia, G., Helmbrecht, J., and Morillo, S. (2000b). “Metalimnion mechanics in stratified water bodies”. XIX IAHR Latin American Congress on Hydraulics. Córdoba, Argentina. In Spanish.
Monismith, S. (1986). “An experimental study of the upwelling response of stratified reservoirs to surface shear stress”. J. Fluid Mech., 171, pp. 407-439.
Niño, Y., Hillmer, I., and Caballero, R. (2000b) “Experiments on mixing in a stratified fluid due to surface shear stress”. Fifth International Symposium on Stratified Flows, IAHR. Vancouver, Canada.
Fig.1 Forcing conditions for the unsteady shear
experiment. Time evolution of the surface shear velocity and corresponding
Richardson number.
Fig.2 Evolution of the slope and buoyancy frequencies
of the primary and secondary interfaces.
Fig.3 Slope of the
primary interface as a function of the Richardson number. Comparison with
experimental results for steady shear and predictions of the classical theory.
Fig.4 Dimensionless entrainment velocity as a function
of the Richardson number. Comparison with experimental results for steady shear
and predictions of entrainment relationships.
Fig.5 Slope of the thermocline as a function of the
Richardson number resulting from the numerical simulations. Comparison with
predictions of the classical theory.
Fig.6 Dimensionless entrainment velocity as a function of the Richardson number resulting from the numerical simulations of Lake San Roque. Comparison with predictions of entrainment relationships by Kranenburg (1984) and Niño et al. (2000).