Steady Lateral Growth of Three-Dimensional Particle Laden Density Currents

In this paper the steady lateral growth of three-dimensional turbulent inclined turbidity current is investigated. To simulate the current, an experimental setup is developed to analyze the turbidity current for different regimes in the particle laden density currents environment. The Buckingham’s π theorem together with a dimensional analysis is implemented to derive the appropriate non-dimensional variables. The experimental results were normalized and plotted in the form of non-dimensional graphs from which a theoretical model is developed and analyzed. Based on the results obtained for the steady lateral growth, three different regimes, namely, inertia-viscous one as the first regime, buoyancy-viscous and gravity-viscous as the second and third regimes are distinguished within the current. In these regimes, the force balance is between the driving and resisting forces. Namely, in the first regime, the force balance is between the inertia and viscous forces, in the second regime, the buoyancy and viscous forces, and in the third regime, gravity and viscous forces are balanced. The experimental results indicate that the lateral growth rate in the first regime is smaller than that in the second and third regimes due to the magnitude and type of the forces involved in those regimes. According to the graphical results, the three different lateral growth rates appear when the normalized current length is smaller than about 3, between about 3 and 10, and larger than about 10. In those regions, the slopes of the data are different with respect to one another.


Introduction
Many geophysical flows can be classified as the gravity or density currents as they occur due to a density difference with the surrounding environment. In atmospheric gravity currents, such as sea-breeze fronts or thunderstorm outflows, the density difference is typically caused by a temperature difference between a dense spreading cold front and a relatively warm less dense ambient air (Simpson, 1987;Härtel et al., 2000). Gravity or density currents are flows driven by the density differences. Turbidity currents are the types of gravity currents where the driving force gained from the suspended sediment sand turbid water is greater than that of the clear water above it. Geological observations show turbidity currents in some common forms of sediment transport in many sedimentary basins (lakes, reservoirs, seas, oceans, etc.) (Hay, 1987).
Turbidity current is also defined as particle-laden underflows that occur in lakes and oceans bottoms. These currents play an important role in transporting fluvial littoral and shelf sediment into deep ocean environments (Salaheld-in et al., 2000). Unfortunately, natural turbidity currents are hard to be observed and studied owing to their large scale and often destructive nature (Salaheldin et al., 2000). Therefore, one of the best means for understanding the hydrodynamics of gravity currents is experimental study. A large number of experimental studies on density currents can be found in the literatures (Ellison and Turner, 1959;Middleton, 1966;Britter and Linden, 1980;Garcia andParker, 1991, 1993;Morris, 1992a, 1992b;Garcia, 1994;Altinakar et al., 1996;Kneller et al., 1997Kneller et al., , 1999Kneller and Buckee, 2000;Buckee et al., 2001;Sequeiros et al., 2010;Ungarish et al., 2014;Staguaro and Pittaluga, 2014;Tilston et al., 2015;Varjavand et al., 2015;Chamoun et al., 2016aChamoun et al., , 2016bSteel et al., 2016;Asghari Pari et al., 2016;Ho et al., 2016). Turbidity currents in the ocean are known to be a maker of submarine canyons (Salaheldin et al., 2000). Sediment deposition in reservoirs reduces the floodcontrol benefits downstream and the capacity of water storage increase flooding upstream due to the streambed aggregation in a deltaic region which causes impairment of navigability, sediment entrainment in hydropower equipment, blockage of gates and intakes, etc. Morris, 1992a, 1992b). Staguaro and Pittaluga (2014) presented a series of detailed experimental observations of saline and turbidity currents flowing in a straight channel. Experiments were performed by continuously feeding the channel with a dense mixture until a quasi-steady configuration was attained. Their longitudinal velocity profiles were measured using an ultrasound Doppler velocity profiler. They also measured the density of the mixture using a rake of siphons sampling at different heights from the bottom in order to obtain the vertical density distribution in a cross section where the flow had already attained a quasi-uniform configuration. The distributions turned out to be influenced by the Reynolds number of the flow, the relative bed roughness, and the presence of sediment in suspension. Unexpectedly, the densimetric Froude number of the current turned out to have no influence on the dimensionless velocity profiles.
The first set of experiments on a three-dimensional density current was conducted by Fietz and Wood (1967). They suggested that, not too close to its source, the width of the current grows linearly, at an angle which is larger for a turbulent current than that for an equivalent laminar one. Alavian (1986) investigated three-dimensional density currents of salt solution for the slopes of 5°, 10°, and 15°. He showed that the lateral spreading of the density current is dependent on the primary buoyancy flux and the slope of the bottom. He observed that the width of the current tends to become nearly constant at a certain distance from the origin, about 30 times larger than the initial width at the source. The final width was found to be influenced by larger buoyancy fluxes and smaller bottom slopes. Tsihrintzis (1988), based on a large number of experiments, arrived at similar conclusions as those of Alavian's concerning the growth of the current width. Christodoulou and Tzachou (1994), suggested a nearly linear dependence of the final width of the current on the length scale (B 2 /g′ 3 ) 1/5 , where B is the buoyancy flux and g is the gravity acceleration. The above length scale can be written equivalently as l = (Q 3 /B) 1/5 , where Q is the flow rate. The same scale has been used, recently, by Choi (1999) to express his experimental results for the maximum width of unsteady currents as a logarithmic function of time. Tsihrintzis and Alavian (1996) investigated the geometric and kinematic behavior of two-and three-dimensional negative buoyancy gravity plume spreading on a steep sloping surface in a laboratory tank. They presented simple analytical expressions based on the balance of driving and resisting forces, i.e., gravity, buoyancy, inertia, and friction, in both the longitudinal and latitudinal flow directions. They concluded that the analytical expressions for the spreading depend on the bottom slope, the initial buoyancy flux, the Richardson number, and the geometry of the plume at the source. Also, it was shown that the mathematical solution was in agreement with the measured experimental data. Based on their results, for different slopes, Table 1 for different spreading regimes is constructed. Christodoulou (2001) examined the rate of the lateral growth of three-dimensional bottom-attached density currents for the bottom slopes ranging between 2° and 15°, for the volumetric flow rate Q between 25×10 -6 and 200×10 -6 m 3 /s, and the relative density difference Δρ/ρ between 0.005 and 0.038. Based on his experimental results, he found that the width b of the three-dimensional density current can be expressed in the non-dimensional form in terms of the distance x from the source as b/l~(x/l) n , where l=(Q 3 /B) 1/5 is called the buoyancy length scale and b is the width that is normalized with respect to the buoyancy length scale (Christodoulou, 2001). Choi and Garcia (2001) investigated the spreading law for sediment-laden gravity currents. They employed saline density currents as surrogates for fine-grained turbidity flows. They used the dimensional analysis to develop a simple expression for the lateral spreading rates of two-dimensional flows on sloping beds. The characteristic length and time scales were determined by the volume flux and buoyancy flux at the inlet. By knowing the initial width of the flow, the spreading law was used to estimate the maximum width of the current at different times and at different longitudinal spreading rates. A regression analysis was performed to obtain a logarithmic relationship between the maximum half width and time (Choi and Garcia, 2001).
Based on a thorough literature search conducted, there is quite a dearth of experimental as well as theoretical work on the steady three-dimensional turbidity current arena, in particular, under the steady (no change of the turbidity current width with respect to time at any distance from the entrance) and turbulent conditions. In this study, efforts are under- Table 1 Spreading regimes for two-and three-dimensional gravity plumes (Tsihrintzis and Alavian, 1996)  taken to investigate the lateral growth of a turbulent steady supercritical inclined turbidity current. The study employs the force balance using the Buckingham's π theroem to derive some dimensionless relations for a better understanding of a turbulent turbid current propagating within another medium. The present study involves both experimental and theoretical work, and considers only the lateral growth portion of the turbid current under the steady condition.
2 Experimental setup and procedure

Experiments
The experimental setup used for the work of this project is shown in Fig. 1 and consists of a main channel (filled with tap water) with the length of 12 m, width of 1.5 m, and height of 0.6 m. Inside the main channel at its upper end, a housing with the dimension of 0.15 m long, 1.5 m wide, and 0.6 m high is located as shown in the figure.
A mixing tank (reservoir) made of a cylindrical rustproof stainless-steel with a volume of 2 m 3 is located at the upper end above the main channel. This reservoir is used to make different blends. Caloen with a density of 2650 kg/m 3 is blended with tap water to make the mixture within the reservoir. The average size of the caloen particles used is about 20 microns. To furnish a fixed head with a constant volumetric flow rate for the blend, a small stainless-steel over flow tank (shown in Fig. 1) with the dimension of 0.3 m high, 0.5 m long, and 0.5 m wide is designed and placed on the top of the mixing tank as shown in the figure. The mixture of caloen and water is pumped by a small submersible pump within the mixing tank into the over flow tank to produce the fixed head. The mixture then enters the upper portion of the main channel through a connecting pipe and a flow meter as shown in the figure. The remaining blend within the over flow tank goes back into the mixing tank. Another submersible pump is also set inside the mixing tank to keep the above circulation going. A small stirrer is also placed within the mixing tank to prevent the sedimentation of the particles inside the tank.
The mixture stream (turbidity current) containing the caloen particles passes through an underpass trapdoor (0.1 m long and 0.125 m high) of the housing and enters the main channel and flows underneath the tap water. The stream, then, travels all the way underneath the clean water to the end of the channel where it drains out (Fig. 1). The slope of the main channel can be changed by a hydraulic jack to the maximum of 3.5%. The experimental runs are performed for three slopes of 1%, 2%, and 3%. In order to measure the volumetric flow rate of the flow, an ultrasonic volume flow meter with an accuracy of 0.01 liter/min (1.66×10 -7 m 3 /s) is used (Fig. 1). The runs are performed for three volumetric flow rates of 10, 15, and 20 liter/min (1.66×10 -4 , 2.50×10 -4 , and 3.33×10 -4 m 3 /s). A digital camcorder is used to measure the latitudinal (b) and longitudinal (x) spreading of the turbidity current within the main channel, and to observe the stream head. The channel framework is made of glassy walls which enables to record films from the sides. Along one of the sides of the main channel, a metal rail is set up to let the filming carriage move on it, filming the longitudinal spreading, the turbidity current, the depth of the stream, and the flow front, all with an accuracy of 0.01 s. To measure the latitudinal and longitudinal spreading of the flow, the main channel floor is gridded. Another camera is located over the channel to film the lateral spreading of the density current. Also, an acoustic Doppler velocimeter is used to more accurately measure the steady lateral growth. In this experiment, three different concentrations, volumetric flow rates, and main channel slopes are used to observe their effects on the lateral spreading of the density current and its velocity, and on the thickness of the developing sediment. Fig. 2 shows the top and side schematics of the main channel of Fig. 1 in which the width of the turbidity current "b" appears at the distance "x" measured from the source (the channel inlet). The figure also indicates the method to measure the steady latitudinal growth of the turbidity cur-  China Ocean Eng., 2018, Vol. 32, No. 4, P. 467-475 469 rent (b) and its longitudinal growth (x) for each experimental run.

Measurements
C 0 Table 2 shows the conditions at which each experimental run is taken place. In this table, S represents the main channel slope, denotes the initial (inlet) mixed weight concentration defined by, where w c and w w are the weights of the water and caloen used, respectively. In this table is the initial volumetric flow rate, is the initial flow Reynolds number defined as: where is the initial (inlet) turbidity current thickness or the height of the inlet sluice gate, is the initial velocity of ν 0 Ri 0 the turbidity current in the flow direction, and is the initial flow kinematic viscosity. Also, in Table 2 is the initial flow Richardson number defined as: where is the initial reduced gravitational acceleration, θ is the main channel angle with the direction of the horizontal axis, and also in Table 2, T denotes the temperature of the clear water. With this equation when Ri 0 >1, the subcritical condition is reached and when Ri 0 <1, the flow becomes supercritical.
As the blend concentrations applied do not affect the water-caloen mixture viscosity much, the viscosity of the turbidity current is taken to be equal to that of water. It is also noted that, in Eq. (2), the range of the initial flow Reynolds number for the conducted runs is found to be between 1461.62 and 3027.34. The range of the initial Richardson number (in Eq. (3)) for the runs is between 0.0065 and 0.0406 as shown in Table 2.

Dimensional analysis
By using the Buckingham's π theorem and dimensional analysis, the dimensionless variables for the problem at hand can be derived as follows.
If the relationship between the effective independent variables for the steady lateral growth of the turbidity current in each force balance regime (the force balance regimes proposed by Tsihrintzis and Alavian (1996) for spreading the density current) is considered, the following relation can be written: where the subscript "0" denotes the initial values, Q 0 and B 0 denote the initial volumetric flow rate and buoyancy flux, respectively. In Eq. (4), θ is the slope of the channel bottom, b 0 and h 0 are the width and height of the inlet sluice gate (or the width and height at the source), respectively. Now, with the Buckingham's π theorem, the following dimensionless variables can be obtained: therefore, In Eq. (5), the quantity ( ) is the dimensionless current width, ( ) is the dimensionless current length, ( ) is the dimensionless initial current width, ( ) is the dimensionless initial current thickness, and θ is the channel slope. In all the above relations, is the initial buoyancy length scale. It is noted here that, b 0 and h 0 are different constants, then, Since θ (the channel slope) has very little effect on the dimensionless lateral growth ( ), then, in each of the force balance regime the following equations can be written: Eq. (9) above, in fact, relates the dimensionless current width to the m-th power of the dimensionless length (under the steady state condition) with "C" being a constant coefficient.

Results and discussion
In this study the lateral growth of a turbulent steady supercritical inclined turbidity current is to be examined empirically on the basis of the force balance. The study is conducted for different channel slopes of 1%, 2%, and 3% and three initial concentrations of 0.5%, 1%, and 1.5%, and three initial volumetric flow rates of 10, 15, and 20 liter/min (1.66×10 -4 , 2.50×10 -4 , and 3.33×10 -4 m 3 /s). The results are then normalized and plotted in the form of dimensionless graphs through which a theoretical model is obtained. It is to be noticed here that using the initial buoyancy length scale , both x and b are normalized in this study. Figs. 3-6 illustrate the plots of the dimensionless current width (hereinafter called the current width, ) vs. the dimensionless current length (hereinafter called the current length, ). The graphs, in fact, link the steady lateral growth of the turbidity current to its position. According to Figs. 3-6, the rate of the lateral growth is different for (the regime (zone) referred to as R1), compared with (the regime referred to as R2), and (the regime referred to as R3). In Fig. 3, the influence of variations of the initial volumetric flow rate on the dimensionless lateral growth (hereinafter called the lateral growth) is illustrated for two constant slopes of 1% and 3%, for two different initial concentrations of C 0 =0.5% and C 0 =1.5%. In Figs. 3a-3d, due to the slope change, the three regimes R1, R2 and R3 are recognized. By comparing Fig. 3a with Fig. 3b, the influence of the concentration increase (from 0.5% to 1.5%) on the later-  al growth at a constant slope (S=1%), but different volumetric flow rates can be observed. As it can be seen in Fig. 3b, the slope (lateral growth rate) for each run, in the area near the entrance (Regime R1), is different, while in Regimes R2 and R3, the data coincide and the slope can be considered as constant. Therefore, the lateral growth rate for Regime R1 at higher concentrations (C 0 =1.5%) and for different initial volumetric flow rates is different and not fixed. It is because, the concentration increase causes an increase of the driving force that includes the inertia force and the component of the apparent weight in the flow direction. Thus, the turbidity current speeds up and does not have sufficient time to spread. However, the concentration increase has no effect on Regimes R2 and R3. The same discussions can also be applied to Figs. 3c and 3d. As it can be seen from Figs. 3a and 3b, for higher concentration, the volumetric flow rate decrease causes the lateral growth rate to get steeper in Regime R1. In the lower concentrations, due to the lower driving force, the obtained data fall on the top of each other, and it seems that in the lower concentrations and lower slopes, the lateral growth rate in Regime R1 is less affected by the flow rate changes (Fig. 3a). Fig. 4 shows the influence of the initial concentrations on the lateral growth for two constant slopes of 1% and 3%, for two different initial volumetric flow rates of Q 0 =1.66× 10 -4 and Q 0 =3.33×10 -4 m 3 /s. As the figure indicates, for all the cases shown, the three regimes of R1, R2, and R3 can be recognized due to the slope changes in the three regions. Fig. 5 depicts the influence of the slopes of the main channel on the lateral growth for three different initial concentrations of C 0 =0.5%, C 0 =1%, and C 0 =1.5%, for three different initial volumetric flow rates of Q 0 =1.66×10 -4 , Q 0 =2.50×10 -4 , and Q 0 =3.33×10 -4 m 3 /s. Again, based on the cases shown in the figure, the change of the slope shows the recognition of the three regimes (R1, R2, and R3). However, due to the coincidence of the data for all the main channel slopes, one may conclude that in the range of the initial volumetric flow rates and initial concentrations of the experimental runs conducted, with good approximation, the growth rate seems to be independent of the channel slopes.
In Fig. 6, according to the plotted data, the following equation can be fitted in each force balance regime: In the above equation, m is the slope of the line in each  force balance regime (Fig. 6) and C is a proportionality constant which are both unknown and determined from the obtained experimental data. Fig. 6, actually, shows the results of all the experimental runs together with the theoretical ones obtained by the Buckingham's π theorem followed by the conducted dimensional analysis. The two results are shown to be in agreement.
x/l 0 ≈ 3, 10 b/l 0 ≈ 3, 13.4 x/l 0 ≈ 3 Based on the figure, when the current length gets to be around 3 and 10 ( ), the current width takes a value of about 3, and 13.4 ( ), correspondingly. These results show that the lateral growth in these regions is independent of the slope, initial concentration, and initial volumetric flow rate variation. At and 10 the slope changes (Fig. 6), therefore, the three regimes of R1 ( ), R2 ( ), and R3 ( ) can be identified. As it was mentioned before, the rate of the lateral growth in Regimes R1, R2, and R3 is different from one another. This is due to the different slopes of the fitted lines going through the plotted data in those regimes. This, in turn, shows that the types of the forces involved and their magnitudes in those regimes are different. It should be noted here that Tsihrintzis and Alavian (1996) in their study for the unsteady lateral growth of saline density current also identified different flow regimes according to the balance of the driving and resisting forces in the latitudinal and longitudinal directions. Under the supercritical initial flow and mild slope conditions, they identified three regimes of the inertia-viscous, buoyancy-viscous, and gravity-viscous in their experiment. In the present work, under the supercritical initial flow condition with mild slopes, the same conclusion as that of the Tsihrintzis and Alavian (1996) is also reached with this difference that the present work has been conducted for the steady lateral growth of particle laden density currents. It is to be noticed here that for the steady lateral growth the balance of the governing forces does not change with time, but for the unsteady lateral growth it does.
It can be seen in Fig. 6 that there are three regimes within the latitudinal turbidity spreading current. As mentioned above, for the mild slopes under the steady supercritical initial flow condition, the inertia is the driving force and buoyancy is the resisting one. In Regime R1, the force balance takes the inertia and viscous forces into account. In Regime R2, an internal hydraulic jump occurs and the flow becomes subcritical, hence, the type of the force balance changes to a balance of the buoyancy and viscous forces for a small latitudinal spreading current. In Regime R3, however, the type of the balance turns into a gravity-viscous force balance.
Also, in Fig. 6, based on the slope of the fitted line in the three regimes, the lateral growth rate in Regime R1 is smaller than those in Regimes R2 and R3 since in Regime R1 the driving force is large and the resisting force is small. Thus, the turbidity current speeds up and does not have suf-ficient time to spread. It should be noted that, here, for all the regimes mentioned above, the resisting force is the drag force and the driving force consists of the inertia force and the component of the apparent weight in the flow direction. Based on Fig. 6 with the changes of the concentration and volumetric flow rate in Regimes R2 and R3, the slope remains constant, on the other hand, in Regime R1, the slope changes due to the type of the flow which is supercritical and turbulent. This kind of the slope change is more pronounced for greater concentrations, higher volumetric flow rates, and steeper channel slopes. With the fact that in Regime R1 the channel slope changes and the concentration variations do not considerably influence the obtained results, the slope of the line passing through the data can be regarded as constant.
According to Eq. (10), the measured width b with respect to x (Fig. 2) follows the following relation for all the three regimes R1, R2, and R3: for Regime R2: and for Regime R3: Hence, the lateral growth in Regimes R1, R2, and R3 are proportional to x 0.37 , x 1.35 , and x 0.83 , respectively. Based on the relations above, the lateral growth rate in Regime R2 is larger than those in Regimes R1 and R3. However, the values for m under similar initial conditions in the case of the unsteady state flow in the work of Tsihrintzis and Alavian (1996) for the density current without particles are x 0.64 , x 0.20 , and x 0.50 for Regimes R1, R2, and R3, respectively. These results show that in comparison with the case of the steady state flow, the lateral growth rates in Regimes R2 and R3 for the unsteady flow are smaller than those in the same regimes for the steady flow. This growth rate, however, is reversed in Regime R1 when comparing the steady and unsteady cases.

Conclusions
In this study the lateral growth of the steady turbulent inclined turbidity current has been studied. The following concluding remarks are deducible from the obtained results.
According to the experimental results, three different regimes of the lateral growth rates namely, R1, R2, and R3 with different slopes with respect to one another are distin-guished. In these regimes, the force balance is between the driving and resisting forces. In Regime R1 the force balance is between the inertia and viscous forces. In Regime R2 the buoyancy and viscous forces are balanced, and in Regime R3 the force balance takes place between the gravity and viscous forces.
x/l 0 < 3 3 < x/l 0 < 10 x/l 0 > 10 Based on the graphical results, the three different rates of the lateral growth occur for the zones of , and , respectively. This indicates that the magnitude and types of the forces involved in those regimes differ from one another.
(1) The lateral growth rate for Regime R1 (only) at higher concentrations and for different initial volumetric flow rates is different.
(2) In Regime R1, for high concentrations, as the volumetric flow rate increases, the lateral growth decreases. This trend, however, reverses for low concentrations.
(3) For all the three Regimes R1, R2, and R3 the lateral growth shows to be independent of the implemented channel slopes.
(4) The rate of the lateral growth in Regime R1 is smaller than those in Regimes R2 and R3 due to the existence of higher driving and lower resisting forces in Regime R1 compared with the other two regimes.
x/l 0 ≈ 3 x/l 0 ≈ 10 b/l 0 ≈ b/l 0 ≈ 13.4 (5) When the current length gets to around 3, ( ), and 10, ( ), the current width takes a value of 3 and , respectively. These results show that the lateral growth in these regions is independent of the channel slope, initial concentration, and initial volumetric flow rate variation.
(6) For the steady state, the lateral growth rate in Regime R2 is larger than those both in Regimes R1 and R3.
(7) The results obtained from the experiments indicate that there is a non-linear single coefficient relation between the lateral growth of the turbidity current and the longitudinal position at which the growth appears.
(8) In the case of the supercritical inlet flow with the mild slope for the steady turbidity current, the lateral growth in Regimes R1, R2, and R3 is proportional to x 0.37 , x 1.35 , and x 0.83 , respectively.