Front Velocity and Front Location of Lock-exchange Gravity Currents Descending a Slope in a Linearly Stratified Environment

Transcription

1 Front Velocity and Front Location o Lock-exchange Gravity Currents Descending a Slope in a Linearly Stratiied Environment Liang Zhao 1, Zhiguo He, M.ASCE, Yaei Lv 3, Ying-Tien Lin, Peng Hu 5, Thomas Pähtz 6 1 Ph.D. Candidate, Ocean College, Zhejiang University, Zhoushan 3161, China. liangz@zju.edu.cn Proessor, Ocean College, Zhejiang University, Zhoushan 3161, China; and, State Key Laboratory o Satellite Ocean Environment Dynamics, The Second Institute o Oceanography, State Oceanic Administration, Hangzhou 311, China (Corresponding author). hezhiguo@zju.edu.cn 3 Graduate Student, Ocean College, Zhejiang University, Zhoushan 3161, China. lvyaei@zju.edu.cn Associate Proessor, Ocean College, Zhejiang University, Zhoushan 3161, China. kevinlin@zju.edu.cn 5 Associate Proessor, Ocean College, Zhejiang University, Zhoushan 3161, China. pengphu@zju.edu.cn 6 Research Proessor, Ocean College, Zhejiang University, Zhoushan 3161, China. tpaehtz@gmail.com Abstract: Gravity currents descending a slope in a linearly stratiied environment can be requently encountered in nature. However, ew studies have quantitatively investigated the evolution process o lock-exchange gravity currents in such environments. A new set o analytical ormulae is proposed by integrating both mass conservation and linear momentum equations to determine the ront velocity and the ront location o a downslope current. Based on the thermal theory, the ormula considers the inluence o ambient stratiication by introducing a newly deined stratiication coeicient in the acceleration stage. As or the deceleration stage, the ormula is derived by adding a parameter that takes into account the density distribution o the ambient water. The transition point between the acceleration and deceleration stages and the maximum ront velocity are also determined by the proposed ormulae. Lock-exchange gravity current experiments are conducted in the lume with linear stratiications to provide data or the validation o the ormulae. The comparisons between the calculated and measured data in terms o ront location and ront velocity show satisactory agreements, which reveal that ront velocity presents a rapid acceleration stage and then a deceleration stage in a linearly stratiied ambient. Keywords: Gravity current; Front velocity; Front location; Slope; Stratiication 1

2 Introduction Gravity currents are lows driven by the horizontal density contrast between the current and the ambient water (Simpson 198). They are an important phenomenon in both nature and engineering practices. Examples o gravity currents include thunderstorm outlows, salt water intrusions in estuaries, sediment-laden river discharges in lakes, and heavier pollutant discharge in water bodies, etc. (Simpson 198; Ottolenghi et al. 16; Steenhauer et al. 17). Gravity currents have been extensively studied by a continuous-inlow or by a lock-exchange technique (Ho and Lin 15; Ottolenghi et al. 16; He et al. 17) in a laboratory lume. A lock-exchange gravity current is usually generated by a sudden release o a locked volume o dense luid into ambient lighter water (Ottolenghi et al. 16) in the lume, leading to an exchange between current and ambient water. The typical structure o a lock-exchange gravity current includes a dense and semi-elliptic ront head propagating in the lume, ollowed by a thin tail. The ront location and velocity o the dense head determine the distance that the current can reach and the time the current arrives at a certain point. Thereore, quantitatively understanding the ront location and ront velocity is o key signiicance in investigating the dynamic process o lock-exchange gravity currents. Previous studies (Huppert and Simpson 198; Meiburg et al. 15) have demonstrated that the propagation o a lock-exchange gravity current on a lat bed can be divided into three stages, i.e., a relatively short acceleration stage, ollowed by a slumping stage with a nearly constant ront velocity, and then a sel-similar deceleration stage. However, in reality, gravity currents oten occur in varying topographic beds (Steenhauer et al. 17), which means that the above three stages may not always exist. Several experimental studies have already shown that the

3 current irst accelerates and then decelerates when the lock-exchange gravity current propagates down a slope in a homogenous environment (Beghin et al. 1981; Maxworthy and Nokes 7). A thermal theory (Beghin et al. 1981), assuming that the current develops rom a 'virtual origin' and propagates with a constant ratio o head height to head length, was proposed to calculate ront velocity and location, and this was validated by lock-exchange experiments o gravity currents moving down slopes between 5 and 9. Furthermore, various models based on the thermal theory were developed to investigate dierent kinds o gravity currents, such as Non-Boussinesq gravity currents (Dai 1; 15), powder-snow avalanches (Rastello and Hopinger ), and particle-laden currents (Dade et al. 199). However, those models are only applicable to gravity currents in homogeneous environments. In real geophysical environments, gravity currents also propagate in non-homogeneous/stratiied environments (Baines 1; Wells and Nadarajah 9; Cortés et al. 1; Snow and Sutherland 1) since the vertical density variation o the ambient oten exists, such as thermoclines in lakes and haloclines in estuaries and oceans (Fernandez and Imberger 8; Meiburg et al. 15). In recent years, experiments o lock-exchange gravity currents in stratiied environments have also been carried out to investigate the ront velocity (Snow and Sutherland 1; He et al. 17), the velocity ield (He et al. 17), the mixing and entrainment (Samothrakis and Cotel 6), and the separation depth (Snow and Sutherland 1; He et al. 17). By itting with experimental data, Maxworthy et al. () proposed empirical ormulae to determine the ront velocity o gravity currents using Froude numbers in a lat and linearly stratiied environment, which were also validated by comparing with the results rom direct numerical simulations. Based on the steady-state theory 3

4 proposed by Benjamin (1968), Ungarish (6) discussed the Froude number o gravity currents on a lat bed in weak and strong linearly stratiied environments, which were urther tested and discussed by Birman et al. (7) using numerical simulations. Most recently, Longo et al. (16) conducted experiments o gravity currents in a lat and linearly stratiied environment in both rectangular and semi-circular channels to validate a simpliied theoretical model to describe the ront velocity at the slumping stage. Moreover, He et al. (17) experimentally studied the hydrodynamics o lock-exchange gravity currents down a slope in a linearly stratiied environment and extended the thermal theory to examine current velocity in the deceleration stage, however, the velocity in the acceleration stage was neglected. Thereore, although some studies have documented the hydrodynamics o a lock-exchange gravity current, an entirely quantitative study o its propagation has not yet been ully investigated, especially in an inclined and stratiied environment. The objective o this study is to develop a set o ormulae to determine the ront velocity and ront location o lock-exchange gravity currents down a slope in linearly stratiied environments by considering both the ambient density variation and water entrainment. The proposed ormulae are urther validated by comparing with the experimental data in both relatively weak and strong ambient stratiications. This paper is organized as ollows. First, we describe the experimental set-up and results. Then, we derive the ormulae used to determine the development o the gravity current and validate these ormulae with the experimental data. Finally, some discussions and conclusions are presented.

5 Experiments and Parameters Experimental set-up and procedure Table 1. Experimental runs and the relevant parameters ρ c ρ B ρ s m g ' Run θ S Re (kg/m³) (kg/m³) (kg/m³) (1/m) (m/s ) A series o lock-exchange gravity current experiments were conducted in a rectangular plexiglass lume with linearly stratiied salt water, as shown in Fig. 1. A brie introduction o the experimental set-up and procedure is summarized here. One can reer to He et al. (17) or details. A locked head tank used to store dense salt water was placed at one side o the lume, connected by inclined perspex boards o dierent lengths to create dierent slopes. Beore each experiment, linearly stratiied water was gradually illed in the lume to a water depth o 3 cm using a two-tank system (Ghajar and Bang 1993). During this process, the opening height o lock 1 was kept as cm and lock was closed. In all the experiments, dense saline water dyed with permanganate was gently injected into the head tank to a height o 9 cm. By 5

6 suddenly liting up lock, the dense luid could intrude into the ambient linearly stratiied water and propagate along the slope. Lock 1 was set in ront o lock so it could lessen the water level luctuation arising rom the liting o lock. A digital camcorder at a rame rate o 5 ps with a resolution o 98 pixel 36 pixel was employed to obtain an overall view o the developmental process. A total o 16 experimental runs were conducted under dierent conditions as given in Table 1. Plan view Head tank Preset interstice 15 Side view 8 Camera Connecting to the two-tank system lock1 lock 6 3 Linearly stratiied water θ Gravity current 5 Parameters and Front Velocity Fig. 1. Plan and side views o the experimental set-up (Unit: cm). The ambient stratiication in the lume is described by the density gradient, deined as ( B s)sin m, (1) H where ρs is the density o the ambient luid at the start point o the slope; ρb is the density o the ambient luid at the bottom o the lume; Ha is the vertical distance between the gate and the bottom o the lume; and θ is the slope angle. For a gravity current descending down a slope with limited length, the relative stratiication S can be used to determine whether the gravity current can separate rom the slope, which is deined as (He et al. 17) s a 6

7 B s S, () c s where ρc is the density o the initial dense luid. For S > 1, the gravity current can separate rom the slope at the neutral density level where the density contrast between the current and the ambient water vanishes and then intrudes into the environment horizontally (Cortés et al. 1; He et al. 17). The bulk Reynolds number Re is deined as (Dai 13) Re = g h h l l, (3) where g' = g(c s) / c is the initially reduced gravity; hl is the opening height o lock 1 and ν is the kinematic viscosity o water. In all the runs in this study, the Reynolds number is larger than 11 (as listed in Table 1, ranging rom 118 to 3515), which means that the low is turbulent, the viscous eect can be ignored (Dai 13) and the motion o the current is dominated by the gravity body orce. The ront location X is deined as the distance rom the start point o the slope to the oreront o the current. The corresponding ront velocity U can be ound by U = dx / dt. Fig. shows the typical time evolution o the ront velocity o a gravity current in an inclined and linearly stratiied environment. Once the lock is lited, the gravity current is released and starts to accelerate along the slope. A velocity shear is then produced between the current and the ambient luid, which generates Kelvin Helmholtz instabilities at the upper interace (Baines 1). The ambient lighter water is entrained into the downslope current. Meanwhile, the increase o the ambient water density quickly reduces the density contrast between the current and the ambient. These two eects lead the gravity current to accelerate in a shorter period than that in the uniorm ambient. With the reduction o the density contrast, the 7

8 downslope current then begins to decelerate. In addition, the ront velocity also presents a luctuation (e.g., < t < 3, 35 < t < 7 in Fig. ), which has been proved to be a result o the changing shape o the dense ront with time and the three-dimensional action o the cross-stream water entrainment (Ieong et al. 6). This phenomenon can also be seen in previous studies (Dai 13; Snow and Sutherland 1)..5 Acceleration stage. Deceleration stage U (cm/s) t (s) Fig.. Typical development o the ront velocity (U ) o a gravity current down a slope in a linearly stratiied environment (Run ). When the density contrast vanishes, the current stops descending along the slope (Baines 1; Guo et al. 1). It comes to a separation stage in which the gravity current separates rom the slope and then intrudes into the ambient environment horizontally at a quite low speed with a thin and sharp orward motion. For the predication o the vertical separation depth, one can reer to the work o previous researchers (Wells and Nadarajah 9; Snow and Sutherland 1; He et al. 17) or more details. Although the downslope current inevitably excites internal waves, prior research has demonstrated that these internal waves have little eect on the ront velocity until the current separates rom the slope (Snow and Sutherland 1). Since the current has let the slope in the separation stage, we only ocus on the ront velocity in the acceleration and deceleration stages and do not consider the inluence 8

9 o internal waves in the present study. Formulae o ront velocity and ront location The thermal theory (Fig. 3) was irst proposed by Beghin et al. (1981) to determine the development o gravity currents down a slope in a uniorm ambient. In this section, we revisit this theory and extend it to describe the evolution process o lock-exchange gravity currents down a slope in linearly stratiied environments. Lock gate Virtual origin T L S H X H a B Fig. 3. Sketch o a gravity current in an inclined and linearly stratiied environment. ρ T is the density o the ambient luid at the top. H and L are the height and length o the semi-elliptical head, respectively. X is the distance rom the gate to the virtual origin. α is the growth angle o the head. There are two main assumptions in the thermal theory. Firstly, the gravity current is assumed to be developed rom a 'virtual origin' located behind the gate, which is determined by the slope and the growth o the current, as shown in Fig. 3. Secondly, the theory assumes that the head o a gravity current keeps a semi-elliptical shape, with a constant ratio o head depth H to head length L. This study adopts these two assumptions. Thereore, the linear momentum equation o the gravity current, ignoring the bottom riction, is (Beghin et al. 1981) d( k ) S HLU dt a v a 1 m B sin, () c where t is time, Um is the mass-center velocity o the current head, ρa is the density o the ambient luid at the position o the mass-center o the current head, kv = k = H/L is the added mass coeicient (Batchelor ), and S1 = π/ is the shape actor with which the sectional area o the head is calculated by S1HL (Dai 13). Bc is the 9

10 buoyancy contained in the head o the current, which is expressed by (Dai 13): B ( ) ga, (5) c c a where is a raction actor and A is the volume o the initial dense luid which can low down the slope. Note that, in a linearly stratiied environment in this study, ρa should be determined by a s 1 ( X X ) m, (6) where X is the distance rom the virtual origin to the mass-center o the current head. By introducing the entrainment ratio E, the entrainment velocity Ue can be deined by (Ellison and Turner 1959) U e EU m. (7) 1981) The mass conservation equation o the gravity current has the orm (Beghin et al. d.5 ( S 1 HL ) S ( HL ) U e, (8) dt where S = (π/ 1.5 )(k +1).5 /k.5 ; this is another shape actor, by which the circumerence o the semi-elliptical head is determined by S(HL).5 (Dai 13). The entrainment ratio E is related with α by E = αs1/(k.5 S) (Dai 13). Substituting Eq. (7) into Eq. (8) and then integrating it leads to (Beghin et al. 1981) 1 S = S.5 H k EX 1 and 1 S. (9) S.5 L k EX 1 written as Substituting Eqs. (5), (6) and (9) into Eq. (), the momentum equation can be 1

11 3 Umd( UmX ) Umd( UmX ) (1 X m) m R( c s X m) RmX, (1) dx dx S1sin ga where R. (1 k ) E S v s By integrating Eq.(1), one can then obtain the mass-center velocity o gravity current in the ollowing orm: U c S SmX X 1mX mx X U X R 1 mx mx 3 c S S m X 1 mx mx 3X 1 mx mx R 1 mx mx mr mx 3X 1 mx mx 1 mx mx X mr c S S c S S, (11) where U is the initial mass-center velocity o the current. As the ront velocity U is much easier to be measured than the mass-center velocity Um, by substituting U = (1 + α / k)um (Dai 13) into Eq. (11), the relationship between the ront velocity U and ront location o lock-exchange gravity currents down a slope in linearly stratiied environments can be easily obtained: U 1 / k Um = 1 / k [ mx mx X U 1 ] X mx mx X R 1 mx mx R 1 mx mx 3X 1 mx mx 3X 1 mx mx 3 c S S c S S. (1) mr c S SmX X mr c S SmX 1 X 1mX mx Note that, when the ambient environment is uniorm, i.e. the ambient stratiication m =, Eq. (11) can be simpliied as U X 8S sin B X U ( ) [1 ( ) ]. (13) 1 c 3 m X 3 X (1 kv) E S s X Eq. (13) is the same as the result rom Beghin et al. (1981). I the gravity current starts rom a quiescent state, the initial mass-center velocity U can be assumed to be zero 11

12 (Dai 13). One can get the ront velocity o lock-exchange gravity currents down a slope in unstratiied environments as: X 3 1 8sinS1( c a) ga U (1 ) [1 ( ) ]. (1) k X 3X ( 1 kv) E S a In Eq. (1), there are several parameters that need to be determined to predict the relationship between the ront velocity and ront location o lock-exchange gravity currents down a slope in linearly stratiied environments. The parameters m, ρc, ρs, S1, and, are related to the initial experimental conditions and can be directly measured. The parameters α, k = H / L, X, U can be measured during the motion o the gravity current. The parameters kv = k and E = αs1/(k.5 S) are calculated using the values o the above parameters. However, the parameter, deined as the raction o the heavy luid contained in the head o the gravity current, is diicult to be determined. This raction is important because it is used to calculate the buoyancy (Bc) contained in the head o the current. Its value varies during the propagation o a gravity current in stratiied water. First, this raction changes due to entrainment and mixing with ambient water. Second, the raction in the deceleration stage should be much smaller than that in the acceleration stage since the dense luids in the downslope current at dierent depths attempt to ind their own neutral density levels and detrain into the environment during the deceleration stage (He et al. 17). Because is dierent in the accelerating and the decelerating stage, we simpliy Eq. (1) in the ollowing in order to obtain easy-to-use equations or each regime that may ind application in uture studies. Front velocity o gravity current in a linearly stratiied ambient in the acceleration stage As the mechanisms and the dynamic eatures o the gravity current in the two 1

13 stages are greatly dierent, ollowing the previous researchers (Beghin et al. 1981; Dai 13), we apply dierent methods in the acceleration and deceleration stages to urther simpliy the ormula. In the acceleration stage, the lock-exchange gravity current in an inclined and linearly stratiied environment only moves a short distance along the depth. Meanwhile, the gravity current is not ully developed so the length o its head is relatively small. Thereore, or the acceleration stage, the ollowing simpliications can be urther applied: (1) The gravity current starts rom a quiescent state, so U (Dai 13); () The order o magnitude o mx, i.e., ( ) sin B s X, is much smaller than H a s that o c s, so c s mx c s ; (3) During the acceleration stage, the length o head o the current is relatively small, consequently, X X X ; () Compared to the total vertical distance Ha, the vertical movement distance o the current X sin in the acceleration stage is relatively small, so X sin 1. H a By applying the above simpliications (1-) and the relationship o ( ) sin a X B s 1 mx mx 1 in Eq.(1), one can get the ront velocity o H s s a a gravity current in linearly stratiied ambience as: 1 / k { 1 U R mx mr c S X X X [1 ( ) ] c S ( X X ) } X. (15) In this step, the expression associated with the location o the current 3 1 X / ( X X ) / 3X is approximated by 13 X / ( X X ) according to the

14 suggestion o Beghin et al. (1981). Similarly, we can approximate X X X 1 / ( ) / with X / ( X X ) to urther simpliy Eq. (15) as: U X (1 ) R c S s, (16) k ( X X ) where the new parameter, i.e., the stratiication coeicient αs is expressed by: 1 mx s S g c S g, (17) where g' = g(c a) / c is the reduced gravitational acceleration at the head o the gravity current. Note that αs considers the inluence o the ambient stratiication on the movement o the gravity current at the acceleration stage. The raction actor in Eq. (16) was diicult to determine, and was assumed to be unity by Beghin et al. (1981), or estimated by itting with the experimental data by Dai (13). In all previous experiments related to the thermal theory (Beghin et al. 1981; Rastello and Hopinger ; Maxworthy and Nokes 7; Maxworthy 1; Dai 13), the experimental tanks were almost the same, in that locks were set vertically to the slope (see Fig. 3). However, as the present experiments were conducted in a linearly stratiied environment, a similar set-up to that in Baines (1) was adopted, in which the head tank was set horizontally (see Fig. 1). Under this set-up, the experimental results show that only a small part o the initial dense luids in the horizontal tank could low down the slope. These dierent experimental conditions and inluential mechanisms make the values o and A dierent rom those in the previous studies. It is hard to determine these two values at the same time. To avoid this problem, a new parameter, the geometric coniguration coeicient ca, is introduced. We can rewrite Eq. (16): 1

15 U dx P X (1 WX ) dt X X, (18) where P 8sin S ( ) g( c A ) 1 c s a ( 1 ) and k ( 1 kv) E Ss W m. (19) = s c s In this way, and A can also be assumed to be unity and the total volume o the initial dense luid respectively, as done in the previous study (Beghin et al. 1981). As is set as unity, we do not put it into the equations anymore. The geometric coniguration coeicient ca then can be treated as the combined eect o the experimental coniguration and dierent mechanisms on and A. Eqs. (18) and (19) and are the ormulae or the ront velocity and ront location o lock-exchange gravity currents in an inclined and linearly stratiied environment in the acceleration stage. As the dominant driving orce controlling the motion and the dynamic eatures o the current are dierent in the acceleration and deceleration stages, it is necessary to determine where the transition point between the two stages is. Theoretically, assuming that the current turns to the deceleration stage right ater reaching the maximum velocity in the end o the acceleration stage, the transition point (i.e., X,p) is deined as the linkage between the acceleration and deceleration stage. X,p can be determined by setting the derivative with respect to distance o Eq. (18) equal to be zero. Thus, the transition point X,p is calculated as X, p X 1 XW. () The maximum ront velocity then can be determined by taking Eq. () into Eq. (18): 15

16 U,max P X (1 WX ). (1) X X (1 WX ) Front velocity o gravity current in a linearly stratiied ambient in the deceleration stage When the gravity current is suiciently ar into the deceleration stage, its propagation distance becomes longer. In addition to the simpliications (1) and (), the ollowing simpliication is adopted (Dai 13): X X 1. () Meanwhile, according to the deinition, ront location X and X has the relationship as (Dai 13) X X (1 ) X. (3) k By substituting Eqs. () and (3) into Eq. (1), He et al. (17) has derived the ront velocity in the deceleration stage. To keep consistency, this paper summarizes it as U dx I ( X X ) J dt M G( X X ), () where I 8S sin g( c A ) (1 3(1 ) 1 d kv E SS k )( c S mx), S m G, (5) 1 k and J ms sin g( c A ) ( c S mx ), (1 ) 1 d kv E SS M 1. (6) 1 k Similarly, the geometric coniguration coeicient cd in the deceleration stage has been considered in I and J. 16

17 Eqs. (), (5) and (6) are the ormulae to determine the ront velocity and ront location o the lock-exchange gravity currents down a slope in linearly stratiied environments in the deceleration stage. Thereore, Eqs. () - (6) together with Eqs. (18) and (19) orm the complete ormulae to describe the whole propagation o lock-exchange gravity currents along a slope in a linear stratiication. Eqs. () and (1) are used to determine the transition point between the acceleration and deceleration stages and the corresponding maximum ront velocity. Validation o the proposed ormulae The measured data o X and U rom the present experiment are employed to validate the above ormulae. For each experiment, we irst determine ca and cd by setting the best-itting lines through plots o the measured ront velocity versus ront location in the respective acceleration and deceleration stages. Then, the equations o the relationship between the ront location and time are solved and the results are compared with the experimental data. The consistency check towards Eq. (1) is also perormed by comparing the calculated ront velocity, in which the itted ca and cd rom the irst step are used, with the experimental data. 17

18 U (cm/s) X (cm) U (cm/s) U (cm/s) X (cm) 8 6 Experimental data (Run 5) Fitted line in A.S. Fitted line in D.S. Calculated data Eq. (1) X (cm) U (cm/s) Experimental data (Run 1) Fitted line in A.S. Fitted line in D.S. Calculated data Eq. (1) X (cm) Experimental data (Run 1) Calculated data in A.S. Calculated data in D.S X (cm) 1 3 t (s) Fig.. Validation o the simpliied ormulae and consistency check o the uniied ormula. In weak stratiication. A.S. and D.S. mean the acceleration stage and the deceleration stage, respectively. indicates the maximum ront velocity and the corresponding turning point. 8 6 Experimental data (Run ) Fitted line in A.S. Fitted line in D.S. Calculated data Eq. (1) Experimental data (Run ) Calculated data in A.S. Calculated data in D.S X (cm) Experimental data (Run 7) Fitted line in A.S. Fitted line in D.S. Calculated data Eq. (1) t (s) Experimental data (Run 7) Calculated data in A.S. Calculated data in D.S X (cm) t (s) Fig. 5. Validation o the simpliied ormulae and consistency check o the uniied ormula. In weak-medium stratiication. 18

19 U (cm/s) X (cm) U (cm/s) U (cm/s) X (cm) U (cm/s) 6 Experimental data (Run 8) Fitted line in A.S. Fitted line in D.S. Calculated data Eq. (1) 6 X (cm) 6 Experimental data (Run 15) Fitted line in A.S. Fitted line in D.S. Calculated data Eq. (1) 3 Experimental data (Run 15) Calculated data in A.S. Calculated data in D.S X (cm) t (s) Fig. 6. Validation o the simpliied ormulae and consistency check o the uniied ormula. In medium-strong stratiication. 6 Experimental data (Run ) Fitted line in A.S. Fitted line in D.S. Calculated data Eq. (1) X (cm) Experimental data (Run 16) Fitted line in A.S. Fitted line in D.S. Calculated data Eq. (1) 3 Experimental data (Run 16) Calculated data in A.S. Calculated data in D.S X (cm) t (s) Fig. 7. Validation o the simpliied ormulae and consistency check o the uniied ormula. In strong stratiication. The parameters in all 16 experimental runs are listed in Table. Eight comparisons between the experimental data and calculated data are shown in Figs

20 or convenience, in which, Fig. represents the cases with weak stratiication, Fig. 5 shows the cases with medium-weak stratiication, Fig. 6 presents the cases with medium-strong stratiication, and Fig. 7 shows the cases with strong stratiication. It can be seen that the acceleration and deceleration propagations and the maximum ront velocity with the transition point can be well described by the proposed ormulae. The uniied equation, i.e., Eq. (1), is also validated. From the calculation, we also notice the value o J is much smaller than the other term I ( X X ), as shown in Table, so Eq. () can be urther simpliied into U dx I ( X X ) dt M G( X X ) (7) Run a k E Table. The relevant parameters in the proposed ormulae X (m) cd ca I G J P W M (m 3 /s ) (1/m) (m /s ) (m 1.5 /s) (1/m) Discussion The propagation o lock-exchange gravity currents down a slope in linearly stratiied environments is very complicated due to entrainment and mixing between

21 the current and the stratiied ambient. During the initial time (acceleration stage), the density dierence between the heavy current and the light ambient water is the main actor driving its movement. As the current moves urther down the slope, the density dierence gradually decreases due to the entrainment eect and the density increase o the ambient water. In the present ormulae, these two actors are well relected by the entrainment ratio E and the density gradient m, respectively. The motion o the current is generally controlled by the buoyancy orce and then viscous orce. In dierent stages, the distance that the current propagates is greatly determined by the density contrast. Although the whole propagating process o gravity current can be described by the uniied expression o Eq. (1), this ormula involves several parameters that are diicult to be determined. For instance, the mixing and entrainment between the current and ambient water changes the raction o the heavy luid contained in the head in the acceleration and deceleration stages. Thereore, several simpliications are applied in two stages to simpliy the ormula to easily calculate the ront velocity and ront location o the current. Fig. 8. Gravity current in an inclined and linearly stratiied environment (Run ). (a) acceleration stage, t = 3 s, more dense luids are contained in the head; (b) deceleration stage, t = 3 s, the tail o the gravity current gets thick and more dense luids stay in the tail. In act, the value o parameter (i.e. the raction o the heavy luid contained in the head) has varied greatly in previous studies o gravity currents in unstratiied environments. It was assumed to be unity by Beghin et al. (1981), while it was itted to be about. by Maxworthy (1) and about.8 by Dai (13). In a stratiied 1

22 environment, the situation is much more complicated as the evolution process is signiicantly inluenced not only by the entrainment but also by the vertical stratiication. By introducing geometric coniguration coeicients ca and cd in the present ormulae, we do not directly determine the value o but assume it to be unity as done by Beghin et al. (1981). The data in Table show that ca is larger than cd in all the experimental cases, which implies that the actual raction o the heavy luid contained in the head in the acceleration stage is also larger than that in the deceleration stage. This is because the dominant driving orce o gravity currents in the acceleration and deceleration stages is dierent. In the acceleration stage, the density contrast between the gravity current and the ambient water is suiciently large to drive the current down the slope so the head contains a larger raction o the buoyancy (see Fig. 8a). Subsequently, the density o the downslope current decreases due to entrainment and density increase o the ambient water. Furthermore, the dense luids in the downslope current at dierent depths attempt to ind their own neutral density levels and detrain into the environment (He et al. 17). Thus, a large raction o the dense luids stays within the tail (see Fig. 8b). Consequently, the head o the gravity current contains a smaller raction o dense luid in the deceleration stage. One o the main assumptions in the thermal theory is that the gravity current was developed rom a 'virtual origin', which is determined by the slope and the growth o the head height. When the gravity current propagates on a horizontal plane, the height o the head does not increase with distance so the assumption o the virtual origin might not be applied (Dai 13). The previous researchers (Beghin et al. 1981; Dai 13) have indicated that the thermal theory is not applicable or gravity currents on a horizontal boundary. Similarly, the present ormulas have the same limitation. For the theory in this situation, the reader can reer to the work o Maxworthy et al. ()

23 and Ungarish (6). However, the assumptions and derivations are not limited to larger slope angles so the results in the present study are essentially suitable or steep slopes, though the speciic parameters may have to be re-calibrated. The equations developed in this study have been validated using experimental data with a slope varied rom 6 to 18. It is suggested that urther experiments on more steep slopes should be conducted to investigate the lock-exchange gravity currents in linearly stratiied environments. Conclusion This study presents a complete set o analytical ormulae to determine the ront velocity and ront location o lock-exchange gravity currents down a slope in linearly stratiied environments. The ormulae are developed rom mass conversation and momentum equations based on the thermal theory, by urther considering the vertical linear stratiication o ambient water, i.e. parameter m. The lock-exchange experiments show the evolution o the gravity current can be distinguished as a short acceleration stage and then a deceleration stage based on its ront velocity beore it leaves the slope. In the acceleration stage, the ormula or ront velocity takes into account the inluence o the ambient stratiication by the stratiication coeicient αs. As or the deceleration stage, the U - X relationship is derived by adding a parameter which describes the density distribution o ambient water. Two geometric coniguration coeicients are introduced in the ormulae in the respective acceleration and deceleration stages to consider the inluences o the experimental coniguration and mechanisms on the raction o the buoyancy contained in the head and the volume o the downslope current. The transition point between the acceleration and deceleration stages and the corresponding maximum ront velocity can be also determined by the proposed ormulae. The good agreements between the data rom 3

24 the experiments and the ormulae validate the capacity o the proposed ormulae to describe the evolution process o lock-exchange gravity currents down a slope in linearly stratiied environments. Furthermore, the ormulae could also be applied to describe the development o lock-exchange gravity currents down a slope in other kinds o stratiied environments by modiying the stratiication parameter m. The present study mainly ocuses on the development o particle-ree gravity current down a slope in linearly stratiied environments. The applicability o the present theory to particulate gravity currents needs urther experimental and theoretical work in the uture. Acknowledgement This work was partially supported by the National Key Research and Development Program o China (17YFC55), National Natural Science Foundation o China (116767), Natural Science Foundation o Zhejiang Province (LR16E91), and Research Funding o Shenzhen City (JCYJ ). Reerences Baines, P. G. (1). "Mixing in lows down gentle slopes into stratiied environments." J. Fluid Mech., 3, Batchelor, G. K. (). An introduction to luid dynamics, Cambridge university press, Cambridge. Beghin, P., Hopinger, E. J., and Britter, R. E. (1981). "Gravitational convection rom instantaneous sources on inclined boundaries." J. Fluid Mech., 17, 7-.

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28 o planar gravity currents propagating down an inclined surace." Phys. Fluids, 9(3), 366. Ungarish, M. (6). "On gravity currents in a linearly stratiied ambient: a generalization o Benjamin's steady-state propagation results." J. Fluid Mech., 58, Wells, M., and Nadarajah, P. (9). "The Intrusion Depth o Density Currents Flowing into Stratiied Water Bodies." J Phys. Oceanogr., 39(8),