George C. Christodoulou
Department of Civil Engineering
National Technical University of Athens
5 Iroon Polytechniou St., Athens 15780, Greece
Tel: (+30)-1-7722813, Fax: (+30)-1-7722814
E-mail: christod@hydro.civil.ntua.gr
Abstract: The rate of lateral growth of 3-D bottom-attached density currents is examined. After a brief review of previous relevant work, the results of an extensive experimental study are presented. Non-dimensional plots of the current width b vs distance x from the source are given for five different slopes (2° to 15°), normalized with respect to the buoyancy length scale l = (Q3/B)1/5. It is found that the rate of growth is different for x/l>10 and for x/l < 10. At x/l » 10 the non-dimensional width is b/l » 20 irrespective of slope, within the accuracy of the observed experimental scatter. For x/l > 10, the width grows approximately as b ~ x1/3, with no apparent dependence on slope. For x/l < 10 the rate of growth is faster, and there is an influence of the bottom slope.
The behaviour of bottom-attached, laterally unbounded density currents has been the subject of several experimental or theoretical investigations in recent years, notably by Fietz & Wood (1967), Hauenstein & Dracos (1984), Alavian (1986), Tsihrintzis (1988), Christodoulou & Tzachou (1994), Tsihrintzis & Alavian (1996), Horsch et al (1998), Choi (1999 a,b). Such heavier-than-ambient currents arise either naturally, e.g. when sediment-laden rivers enter into lakes or reservoirs, or due to human activities, e.g. in the disposal of waste from some industrial, mining, or dredging operations.
The lateral growth of a 3-D density current flowing on an open slope is generally much larger than its vertical growth; moreover, the lateral growth is observed to be quite rapid near the source and gradually decreases downstream until a normal state is reached, characterised by a constant Richardson number and marginal further growth due solely to entrainment (Tsihrintzis, 1988). The effect of geometrical parameters, such as the bottom slope, or flow parameters, such as the buoyancy flux, on the lateral spreading rate has been assessed mostly qualitatively so far. The objective of this paper is to examine the lateral spreading rate based on experiments in a large laboratory basin and present the relevant laws in a general non-dimensional form.
The first experiments on 3-D density were conducted by Fietz & Wood (1967). They suggest that, not too close to the origin, the width of the current grows linearly, at an angle which is larger for turbulent currents as compared to laminar ones. It is to be noted, however, that their experiments were conducted in a tank with a wall boundary along the current’s symmetry plane. Hauenstein & Dracos (1984) based on similarity analysis and neglecting bottom friction derived for steady currents also a linear relation between the width b and the distance from a virtual origin xv.
Alavian (1986) observed that the width of the current tends to become nearly constant at some distance from the origin of the order of 30 times the initial width at the source, and that the final width was larger for larger buoyancy fluxes and for smaller bottom slopes. Tsihrintzis (1988), based on a large number of experiments, arrived at similar conclusions concerning the growth of the current width. Tsihrintzis (1988) and Tsihrintzis and Alavian (1986) also developed an integral model to simulate the evolution of the current and showed fair agreement with experimental data.
Christodoulou and Tzachou (1994) suggested a nearly linear dependence of the final width of the current on the length scale (B2/g’3)1/5, where the coefficient of proportionality increased with decreasing slope. The above length scale can be written equivalently as l = (Q3/B)1/5, where Q is the flowrate and B is the buoyancy flux. The same scale was used recently by Choi (1999.a) for expressing his experimental results for the maximum width of unstready currents as a logarithmic function of time.
Tsihrintzis and
Alavian (1996) presented a systematic analysis of flow regimes depending on the
governing forces (gravity, pressure, inertia, viscosity), the bottom slope and
the initial conditions (supercritical or subcritical), and they derived laws for
the evolution of 2-D and 3-D laminar density currents vs time, which compare
favorably with experimental data.
Eliminating time, their expressions for supercritical slopes and initially supercritical flow imply:
l in the inertia-viscous regime b ~ x7/11 (1)
l in the gravity-viscous regime b ~ x1/2 (2)
Similarly, for initially subcritical flow:
l in the buoyancy-inertia regime b ~ x (3)
l in the gravity-inertia and gravity-viscous regimes b ~ x1/2 (4)
Numerical models for the simulation of 3-D currents were presented recently by Horsch et al (1999) and Choi (1999.b). In the former, a 3-D parabolic simulation of laminar currents is attempted, which produces a diminishing rate of lateral growth with distance from the source. In the latter, a 2-D finite element model originally developed for turbidity currents is applied and tested against experimental results obtained for saline density currents. The experimental data show a nearly linear increase of the lateral boundaries up to a distance of about 0.50 m from the source, beyond which the sidewalls of the experimental tank are reached.
The variety of laws proposed in previous studies, ranging from b~x to b » const, and the need to better define the range of applicability of possibly different laws governing the lateral growth of the current depending on distance from the inlet or source, motivated the re-examination of experimental data as shown below.
A total of 74 experiments were conducted in a laboratory basin 7 m long, 5 m wide and 0.7 m deep. The bottom slope ranged between 2° and 15°, the flowrate Q between 25×10-6 m3/sec and 200×10-6 m3/sec, and the relative density difference Δρ/ρ between 0.005 and 0.038. The spreading of the coloured dense layer on a 0.10 x 0.10 m mesh drawn on the bottom slab was observed visually through photographic records. Details of the experimental setup and procedure are given by Christodoulou and Tzachou (1994). Sample photos showing a turbulent and a laminar current are shown in Figures 8 and 9, respectively.
The experimental results were analysed in several ways and are presented herein in non-dimensional plots of b/l vs x/l, where l=(Q3/B)1/5 is the buoyancy length scale, b is the visual width and x is the respective distance from the inlet. Figures 1 to 5 show the results for each slope in log-log scales.
Figure 1 shows the results of all runs with a 15° bottom slope. It is noticed that a change of trend occurs near x/l » 10. For smaller values of x/l the width grows faster, at a rate of about x3/5, whereas for larger values of x/l the rate of growth is about x1/3. Somewhat similar results are found for the 10° and the 5° slopes, although with slightly smaller exponents: As seen in Figures 2 and 4, the rate of width increase is about x1/2 for x/l<10 and x1/4 for x/l > 10. Data for the 7.5° and the 2° slope, presented in Figures 3 and 5 respectively, suggest a law of the form x1/3 for x/l > 10. In Figure 6, results obtained from Alavian (1986) are presented; they are seen to also support a x1/3 power law for x/l ³ 10.
Figure 7 shows all the results of the present experiments. It is clear that for x/l > 10 all data can be satisfactorily expressed through a 1/3 power law, with minor dependence on slope, indistinguistable from the experimental scatter. However, for x/l < 10 the slope does have an influence, such that for steeper slopes the exponent of the power law is higher compared to smaller slopes.
It is to be noted that, within the observed experimental scatter, the width of the density current reaches at x/l » 10 a value of b/l » 20, which is essentially independent of the bottom slope. Downstream from this point the lateral growth of the current seems to follow closely a 1/3 law, irrespective of slope, at least up to x/l » 200, where is the limit of experimental observations. The slower observed growth rate in this region as compared to the theoretical predictions discussed previously suggest that the balance of forces may be more complicated and there might be other factors, such as frontal drag, which restrict the lateral expansion and deserve further investigation.
The faster growth rate for larger slopes in the range x/l < 10, down to the limit of experimental observation x/l » 2 may be explained as follows: According to previous investigations, the initial lateral spread very close to the source, i.e. x/l £ 2, is larger for smaller slopes; therefore, to reach the same value of width b/l » 20 at x/l » 10, the plume should grow faster for larger slopes in the region 2 < x/l < 10. The rate of growth in this region is about x1/2 for ¶ £ 10° and somewhat larger for larger slopes. Finally, although there are no experimental data in the region x/l < 2, it may be indirectly deduced that the rate of growth is even larger, of the order of b ~ x. The width in this region obviously depends also on the initial width at the inlet; in the present experiments this was bo = 0.05 m or 0.028 m.
Based on experimental results from a large laboratory tank, the width b of a 3-D density current can be expressed in non-dimensional form in terms of distance x from the source as:
b/l ~ (x/l)n (5)
where l=(Q3/B)1/5 is the buoyancy length scale. Three regimes can be identified:
(1) Very close to the source, x/l < 2, the lateral growth is quite large, increasing for smaller slopes and apparently depends on the initial width bo. No quantitative data were available in this region.
(2) In the range 2<x/l < 10 the plume grows faster for larger slopes, so that eventually at x/l » 10 the non-dimensional width reaches a value of b/l » 20, nearly the same for all slopes. The exponent of the power law (eq.5) in this range is n ³ ½.
(3) In the range 10 < x/l < 200 the rate of lateral growth is essentially independent of the bottom slope, with n » 1/3.
Theoretical justification of the observed rates of growth is the subject of further investigation.
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[2] Choi, S.U. (1999.a). Bidirectional spreading of gravity underflows on an incline. Korea Water Resources Assoc., 32(2), 83-94.
[3] Choi, S.U. (1999.b). Layer-average modeling of two-dimensional turbidity currents with a dissipative-Galerkin finite element method, Part II: Sensitivity analysis and experimental verification, J. Hydraulic Res. 37(2), 257-271.
[4] Christodoulou, G.C. and Tzachou, F.E. (1994), Experiments on 3-D density currents, Preprints 4th Intern. Symposium on Stratified Flows, Vol.3, Grenoble, France.
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Fig.1 Experimental results for 15°slope
Fig.2 Experimental results for 10°slope
Fig.3 Experimental results for 7.5°slope
Fig.4 Experimental results for 5°slope
Fig.5 Experimental results for 2°slope
Fig.6 Results of Alavian (1986) for 10°slope
Fig.7 Proposed power laws compared to experimental results for all slopes
Fig. 8 Photograph of a turbulent density current
Fig.9 Photograph of a laminar density current
(¶=15°，Δρ/ρ=0.03, Q=100.10-6m3/sec). (¶=5°, Δρ/ρ=0.03, Q=100.10-6m3/sec).