An experimental study on plunging depth of density currents

Abstract

Mass density of the current flows is the one of the important problem in the hydraulics of the dam reservoir. Plunge point occurs when the mass density current penetrates in the stagnant fluid. Recognition the place of this point is very important because of clearing the boundary of the density current flow and ambient fluid. In this study the influences of bed slope and hydraulic parameters on plunging depth were experimentally investigated. The results show that the slope has a minor effect on the plunging depth. The height of plunging depth is increased by increasing the density of the current flow. Also increasing the densimetric Froude number caused of decreasing the plunging depth. Finally an equation was proposed to estimate the plunging depth using as function of flow characteristics.

Keywords

dense current / plunge point / Densimetric Froude Number / dam reservoirs

Introduction

The incoming density fluid will move into the ambient fluid because of the difference between the density of the ambient fluid and current flow. Due to the density of the current flow which may be lesser or more than the density of the ambient flow, it moves in as over flow or under flow and if the ambient fluid is stratified, a density current will find its appropriate density environment. Temperature, dissolved substances and suspended particles may cause density differences. The problem of sedimentation has been regarded the most crucial element in shortening the life time of dams, Many reservoirs have been discarded due to overloading of sediments. International Commission of Large Dams (LCOLD) by considering the existent volume of all world`s dams, has declared that one percent of reservoirs capacity will be reduced each year [[1]]. One of the most important phenomena that affect on sedimentation process is density current, so additional research is needed to recognize the role of this phenomena on sedimentation process. In reservoirs and lakes, density currents are important in managing siltation and water quality. Gravity currents are primarily horizontal, occurring as either top or bottom boundary currents or as plunging at some intermediate level. Turbidity currents which are the case in this study are gravity currents in which the excess density or unit weight providing the driving buoyancy forces is due to the presence of sediment being held in suspension by fluid turbulence. Thus whether or not a sediment-bearing gravity current is turbidity current depends to a great extent upon the ordering that exists between the density of the interstitial fluid and that of the ambient fluid. Turbidity currents play an important role in a vast array of geophysical and engineering applications and have been the subject of extensive theoretical, numerical, and experimental investigations over the past few decades. So, the plunging depth that bounds between density currents and ambient fluid, and investigation parameters hydraulic of density current that plunging in reservoir such as the velocity of head, body density current and water entrainment depended on this point, knowing the depth is necessary when a dense suspended flow enters a reservoir. Therefore researchers proposed different formulas to calculate the density current depth at plunging area [2–4]. Singh and Shah [5] could find an equation to calculate the plunge point depth as Eq. (1):

(1)h_p=\mathrm{1.85}+\mathrm{1.3}{\left({\displaystyle \frac{q^{\mathrm{2}}}{g\text{'}}}\right)}^{\frac{\mathrm{1}}{\mathrm{3}}},

where q is the dense current discharge in unit of width, g' is the Densimetric gravitational and h_p is the height of plunge point. Ghomeshi [6] proposed a formula for head of density currents using experimental data from sediment and solute density currents with fixed bed slope. He also considered the height of current head as a function of density current discharge and densimetric gravitational acceleration. Haghiabi [7] obtained a relation between buoyancy flux, velocity of current head and bed slope. Akiyama and Stefan [2] stated that the plunge point appears when the Densimetric Froude Number is 0.68. Furthermore Savage and Brimberg [8] studied the plunging phenomenon using motion equations terms for a two-layered gradual varied current. They assumed the pressure distribution is hydrostatically and didn’t consider rate of entrainment. Basson [9] found that for steep slopes (So>2%) and having the momentum equilibrium conditions at plunging point, the maximum of Densimetric Froude Number should be 0.67. Hong-Yuan and Wei-Sheng [10] investigated the hydraulic characteristics of dense current in a reservoir. They claimed that the plunge point initially is unstable however after moving forward it could be stable, they also defined the plunge area and declared that the stable plunge area will happen at the fixed rate of entrainment. In their studies the Densimetric Froude Numbers at plunge point in the first stage and the stable state were 1 and 0.6, respectively moreover the length of plunge area was 15 times of the water depth at the stable plunge point. Dallimore et al. [11] investigated model of bottom dense current. They also used a hybrid two-dimensional model of bottom dense current and three dimensional hydrodynamic models to simulate the plunging currents in a reservoir. Dai and Garcia [12] analyzed the plunging phenomenon in dense currents and identified that the ratio of a bottom dense current thickness just after plunging (h_d) to the current depth before plunging (h_p), and their related Densimetric Froude Numbers which can be assigned as functions of the initial entrainment. Huang et al. [13] developed a numerical model of turbidity current solving Reynolds-Averaged Navier-Stokes (RANS) equations for dilute suspension using a finite volume method. The bottom boundary in the grid system is allowed to adjust in response to sediment deposition and entrainment during the computations. The model was applied to simulate the evolution of a conservative saline density current and turbidity currents along an 11 m long flume. Sutherland et al. [14] investigated the motion of gravity current along the interface of two layered fluids. Their theory well predicted the motion of intrusive currents of density equal to the average density of the ambient but under predicted the case where the density of the intrusive current was different from the average density of the ambient. Singh et al. [15] developed a three-dimensional subgrid model solving Navier- Stokes equations for simulation of saline and turbid currents on a horizontal bottom boundary. The model predicted the gravity current characteristics such as head height, intrusion speed and sediment deposition to a reasonably good accuracy. Barahmand and Shamsai [16] by using of experimental and theoretical study, density jumps on smooth and rough beds were investigated .According to the above research, more focus has been on flow characteristics after plunge point and experimental investigations very little has been done on the plunge point. In paper has attemped the impact of slope and Densimetric Froude Number on creation of plunge point was surveyed and relationship between depth of plunging point of density currents with flow parameters were presented.

Laboratory model

The studies on plunging phenomenon show that the dense current depth in plunge point (h_p) can be affected by some variables like: the gravitational acceleration; bed slope of the dense current channel; dense current discharge and the density of ambient water [17]. In Fig. 1 the sketch of a density current has been shown:

In Fig. 1, q can be the density current discharge and hp is the plunge height.

Materials and methods

Analyzing the depth dimension of a dense current in a plunge point

The empirical tests about plunging phenomenon show that the dense current depth in the plunge point (h_p) is depended to the following:

(2)h_p=f(q,g,s,{\rho }_d,{\rho }_a),

Where h_p is the plunging depth, q is the discharge per unit width, g is the gravitational acceleration, s is the channel slope, r_d is the inflow fluid density and r_a is the ambient fluid density. Numerous equations have been proposed for predicting the plunge point depth, hence we can write a new general equation as following:

(3)h_p=\alpha {\left({\displaystyle \frac{q^{\mathrm{2}}}{g\mathrm{\Delta }}}\right)}^{\beta }f(s),

In above equation \mathrm{\Delta }={\displaystyle \frac{{\rho }_d-{\rho }_a}{{\rho }_a}}is the relative density difference between inflow and ambient water. The coefficient a and b can be determined using experiments.

Experiments

The experiments were conducted in a tilting flume 0.5-m wide, 0.8-m deep and 9.25-m long. The sides of flume were made from acrylic sheet (Plexiglas). The flume was equipped with several apparatus to prepare the dense fluid and to control the steady state of the turbidity current during the experiments. The flume and the associated equipment are shown schematically in Fig. 2. All of the measuring instruments were prepared properly and the flume was filled with clear water before starting the experiments. Adjacent to the flume a tank served to prepare and store the density current. The mixing tank equipped was equipped with a mixer. The dense fluid was prepared in the mixing tank by mixing salt in water. The produced dense fluid was pumped from mixing tank to head tank at a constant rate through the supply pipe. The dense fluid was delivered from the head tank to the flume with a system of hoses, valves and a flow meter. Inflow rate of density current was regulated using a valve. The upstream end of the flume was closed using a plate. The entire flume was divided into two sections of unequal length by sliding gate. The entrance gate created the initial depth of the current. The current was driven into the flume after pulling upward the gate and then it flowed downstream of the flume. A stilling basin with sills was provided at the end of flume to prevent turbulence of ambient water when clear water was supplied to the flume. When the dense current was driven into the ambient water, considering its more density would be plunged. The plunging depth was measured in stable condition. It was a measurement the depth of plunging, when density current move into the ambient water because of the strong inflow density current in ambient water was moved. When the power incoming density current was lowed, this moving was indicated such as plunging flow. After stabilization the incoming density current and fixity plunging point, plunging depth was measured with a ruler. In Table 1 the ranges of experiments has been shown:

Results and discussion

The effects of slope and discharge of density current on plunging depth

Figure 3 shows the plunging depth variations against the slopes for different discharges. According to Fig. 3 plunge point depth in a constant discharge for all the slopes would be almost fixed, so we can claim that the slope has no special effect on the plunging depth. The data are shown an increasing plunging depth as the inflow density current discharge gets higher. In a certain density, when the inflow of the current is increased, the height of plunge point also is increased. Also for a value constant of discharge of density current, with increasing concentration inflow of density current, the height of plunging point was decreased. Therefore the height of plunge point would not depend on the slope and the Eq. (3) can be written as following:

(4)h_p=\alpha {\left({\displaystyle \frac{q^{\mathrm{2}}}{g\mathrm{\Delta }}}\right)}^{\beta }.

Moreover the data also show an increasing plunging depth as the inflow density current discharge gets higher.

Densimetric Froude Number at plunge point

The Densimetric Froude Number at plunge point may write as following:

(5)F{r}_D={\displaystyle \frac{v}{\sqrt{(g{\displaystyle \frac{\mathrm{\Delta }\rho }{\rho }}}{h}_p)}}.

The Densimetric Froude Numbers at the plunge point were calculated for the three different slopes 8%, 12% and 16%. As it was mentioned in previous section the effect of slope on the plunging depth can be assumed negligible. Then variation of Densimetric Froude Number with plunging depth can be shown as Fig. 4.

According to Fig. 4, at the plunge point, the flow is subcritical, since the Densimetric Froude Number at plunge point for different slopes is less than one. Moreover when Densimetric Froude Number at the plunge point increases, the plunging depth will decrease. The minimum and maximum of Densimetric Froude Number at the plunge point for different slopes is shown Table 2.

The data show that the Densimetric Froude Number is almost the same for all mentioned slopes at the plunge point, i.e., the amount of slope has no effect on the plunging depth. Table 3 shows the range of Densimetric Froude Number at plunge point obtained by other researchers.

The relationship between the plunging depth and characteristics of density current

In Fig. 5 the formation of plunge point on a slope, is shown:

We can estimate the values of coefficients in Eq. (4) using the Fig. 6 after calculating \left({\displaystyle \frac{q^{\mathrm{2}}}{g\mathrm{\Delta }}}\right) for each experiment.

As we can see in Fig. 6. the amounts of a and b in Eq. (4) are 2.887 and 0.414, respectively and the related coefficient of correlation is 0.88 which shows the resulted equation from Fig. 6 has a good fit with the data. To gain the coefficients in Eq. (4) the 80% of the data was used, so it would result as following:

(6)h_p=\mathrm{2.887}{\left({\displaystyle \frac{q^{\mathrm{2}}}{g\mathrm{\Delta }}}\right)}^{\mathrm{0.414}},

where h_p is the plunging depth, q is the discharge per unit width, g the gravitational acceleration, s the slope, r_d the inflow fluid density and r_a is the ambient fluid density and \mathrm{\Delta }={\displaystyle \frac{{\rho }_d-{\rho }_a}{{\rho }_a}}is the relative density difference between inflow and ambient water. For verification Eq. (6), the 20% remaining of the experimental data was used and the result is shown in Fig. 7.

Figure 7 shows that the Eq. (6) can predicts the experimental data with a good accuracy. Also the total observed and calculated values of plunging depth are compared in Fig. 8.

Figure 8 shows the resulted data values from Eq. (6) have a suitable fit to the 45° line.

The results can be compared with that of other researcher. For example Wunderlich and Elder [4] and Singh and Shah [5] obtained the following equations:

(7)h_p={\left({\displaystyle \frac{\mathrm{1}}{\mathrm{0.5}}}\right)}^{\frac{\mathrm{2}}{\mathrm{3}}}{\left({\displaystyle \frac{q^{\mathrm{2}}}{g\mathrm{\Delta }}}\right)}^{\frac{\mathrm{1}}{\mathrm{3}}},

(8)h_p=\mathrm{0.0185}+\mathrm{1.3}{\left({\displaystyle \frac{q^{\mathrm{2}}}{g^{\prime }}}\right)}^{\frac{\mathrm{1}}{\mathrm{3}}}.

The parameters in Eqs. (6), (7) and (8) are similar. In Fig. 9 the resulted obtained from Eqs. (6), (7) and (8) are compared with the observed data.

Based on the statistical analysis the discrepancy between the data obtained from the Wunderlich and Elder’s equation [4] and measured data is 21.85%, i.e., Eq. (6) is more accurate as 20.94%. Considering Fig. 9 it’s clear that the resulted data convergence in Eq. (6) to the 45° line is more than that of Wunderlich and Elder’s equation. Moreover the discrepancy between the data obtained from Singh and Shah’s equation [5] and measured data is 37.77%, i.e., Eq. (6) is more accurate as 36.86%. According to Fig. 9 resulted data convergence from Eq. (6) to the 45° line is better than that of Singh and Shah [5]. The reason of more difference between the data obtained from Eqs. (7) and (8) with experimental data was the experimental conditions. Eqs. (7) and (8) on the basis of a series of laboratory work and under conditions almost different with this research is obtained and this subject was became the reason of more difference between the data obtained from Eqs. (7) and (8) with experimental data and Eq. (6). Since, in obtaining the Eq. (7) the effect of friction bed has been, the accuracy of the Eq. (7) relative to the Eq. (8) in the estimation of experimental data is higher.

Conclusions

The relationship between the plunging depth and characteristics of density currents was experimentally investigated. It was found that the slope has a minor effect on plunging depth. In a certain density, when the inflow of the current is increased, the plunging depth also increases. The Densimetric Froude Numbers at plunge point were less than one in all of the three mentioned slopes, i.e., the currents was subcritical. The results show that if the Densimetric Froude Number at plunge point increases, the plunging depth will decrease. The Densimetric Froude Numbers are the same in three different mentioned slopes which shows the amount of slope never affects the plunging depth. The relationship between the plunging depth and the characteristics of flow is as following:

h_p=\mathrm{2.887}{\left({\displaystyle \frac{q^{\mathrm{2}}}{g\mathrm{\Delta }}}\right)}^{\mathrm{0.414}}.

The results obtained from the above equation show a good fit to observed data. The data convergence in above equation to the measured data is more than that of Wunderlich and Elder [4] and Singh and Shah [5].

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