1 introduction - wit press · 2014. 5. 12. · bry ant ©hyperbolic, math, ualberta. ca abstract in...

Front speeds for gravity currents on

an incline

P. J. Montgomery & T. B. Moodie

Applied Mathematics Institute, Department of

Mathematical Sciences, University of Alberta,

Edmonton, Alberta, Canada. TOG 2G1

bry ant ©hyperbolic, math, ualberta. ca

Abstract

In this paper we shall present some recent results concerning a two-layershallow-water formulation for gravity currents resulting from the initial re-lease of a block of dense fluid on an incline or sloping bottom. Model equa-tions are developed for the time dependent two-layer situation in two spatialdimensions under the assumptions of constant density layers which are in-compressible, immiscible, and inviscid. Effects such as rotation, surfacetension, or turbulent entrainment between the fluids are also neglected. Forlow aspect ratio flows, the subsequent shallow-water formulation is statedand includes a forcing term with a frictional drag focused at the head of thegravity current which is introduced to balance the momentum increase dueto the acceleration down the incline. The appropriateness of this Chezy-type frictional drag is discussed with reference to physical experiments. Anew finite-difference scheme is adapted to produce solutions to the initialrelease gravity current problem with and without a source of fluid and pro-vide an interpretation of the thin lower layer front speed, improving thepresent theoretical understanding.

Transactions on Engineering Sciences vol 18, © 1998 WIT Press, www.witpress.com, ISSN 1743-3533

328 Advances in Fluid Mechanics II

1 Introduction

A gravity current consists of one fluid flowing within another when

this flow is driven by relatively small density differences between the

fluids. These flows, also known as density or buoyancy currents, can

take the form of either top or bottom boundary currents, or mayoccur as intermediate intrusions. Gravity currents can be either nat-

ural or man-made, and a thorough categorization of these flows hasrecently been completed by Simpson.^ Typical natural examples are

thunderstorm outflows, avalanches, salt-water intrusions at a river

mouth, and sea-breeze fronts.

The gravity current problem considered in this paper is that of

the release of a wedge of heavy fluid at one end of a rectangular tank of

less dense fluid. This experimental setup has been used extensively

(Ellison & Turner,5 Rottman & Simpson,^ Huppert & Simpson^),

and the denser fluid spreads out along the bottom of the tank, be-

coming a typical gravity current. For a tank with a horizontal bottom

such experiments have been analyzed using shallow-water or depth-

averaged models.^ This modelling approach is not as successful whenthe tank is inclined from the horizontal, although experiments haveshown that for small slopes (less than about five degrees,) the struc-ture of the gravity currents are similar. The differences between the

currents in the two cases need clarification, a statement substantiatedby Simpson:^ "the transition zone between steady currents down a

slope and time-dependent currents along a horizontal surface has not

yet been thoroughly investigated."A shallow-water model for time-dependent gravity currents in

two dimensions is derived in Section 3 under the assumptions thatthe layers are inviscid, incompressible, and immiscible. A term is

introduced in the lower layer momentum equation to incorporate aspatially dependent viscous drag on the gravity current, a physical

effect discussed in Section 2. This new forcing term allows a compar-ison of some special limiting cases to underscore the importance ofthe viscous term, and therefore a steady-state and a travelling wave

solution to the model equations are discussed in Section 3. A numeri-cal method of solution is outlined in Section 4, with some results and

interpretations focussing on the front speed of the gravity currentresulting from initial release on an incline with and without a source

term. Some concluding remarks follow in Section 5.


Advances in Fluid Mechanics II 329

2 Viscous drag at the gravity current head

At the front of a gravity current, a region called the 'head' which is

deeper than the following flow is formed. At the head, many pro-

cesses occur such as mixing, turbulence, lobe-and-cleft structures orbillows, and contact with the bottom boundary.^ The effect of these

forces on the front shape and velocity of the gravity current has un-dergone considerable study, and it is generally understood that for

small slopes of not more than a few degrees, the frictional forces

are of primary importance.^ Entrainment has been found to play an

important role for greater slopes but is of negligible importance for

nearly horizontal low-velocity flow.^ Entrainment has also been shown

to have neglectable effect for gravity currents which result from low

aspect-ratio initial conditions/The initial release problem on a sloping boundary has been clas-

sified as either an inclined starting plume (a gravity current followed

by a continuous flow) or an inclined thermal which is created from the

release of a finite volume.^ For inclined plumes, Britter & Linden^

found that although the front speed was steady for small slopes, in the

horizontal or nearly horizontal case (less than a degree), the velocity

was not steady and exhibited deceleration after the gravity current

was established. This behaviour was attributed to the frictional drag

force which balances the component of the gravitational force down

the slope except in the nearly-horizontal case, where it deceleratesthe flow. For inclined thermals, various predictions of front speedbased on the intial release volume have also been examined, with

similar results.*The prevailing shallow-water methods used for modelling grav-

ity currents do not take into account frictional or entrainment drag.

Rottman and Simpson^ were able to achieve good results for hori-

zontal thermals by neglecting such effects, although their approachdoes not include the case of inclined thermals. Middleton^ suggested

that a Chezy-type frictional drag for a steady gravity current suchas that used for roll waves (see, for example, Whitham^) was not

a correct formulation, and that the velocity and height of the headwere important in determining the front speed. A gravity current is

in general not of uniform velocity, and the following flow within a

gravity current moves faster than the head, with as wide a variation

as the ratio of 1 : 0.7 reported by Middleton.^

We propose that a viscous drag be incorporated as a nonlinear



forcing term in the lower layer momentum equation in such a way that

it is present at the head of the gravity current, but zero away from the

front. In this way, neither uniform velocity gravity currents nor roll

waves are considered, despite the presence of a frictional Chezy-type

force. The form of truncation function is somewhat arbitrary, how-

ever it is felt that such a generalization is a logical step arising from

knowledge of the physical processes. The viscous term is therefore

introduced to be of the form

F = Cĵ T(x), (1)

where C/ is a coefficient of friction, u the velocity of the head, and d

the height. T(x) is a truncation function which is zero away from the

front of the gravity current, and as such serves to focus the force at the

front. For smoothness properties, we choose a Gaussian truncation

function, which will be stated precisely in the next section, instead

of a discontinuous step function.

3 The model equations

The physical system under consideration is shown in Figure 1. Avolume of heavy fluid of constant density p^ over a variable bottom

of height z — a(x) underlies a less dense fluid of constant densitypi. The system is simplified by symmetry and we consider only two-

dimensional motion with velocities in the x and z directions given byU{ and Wi, respectively, where the subscript i denotes the upper andlower layers (i = 1,2). The interface and surface heights are given

by h(x,t) and rj(x,t) where the surface height 77 is measured with

reference to an initial value of z = H.We state the equations of motion (see Kundu^) under the sim-

plifying assumptions that the fluid is incompressible and inviscid, and

neglect such physical effects such as rotation, entrainment, and sur-

face tension between the fluids. The dimensional equations consist

of: mass conservation,

vertical momentum balance,



and horizontal momentum balance,

(4)

(5)^ ^

Eqn (5) contains the frictional drag term (1) discussed in Section

2 with a Gaussian truncation function employed. Here, xp denotes

the position of the front, C/ is a small dimensionless coefficient of

friction, and K is a parameter with units of length that represents thewidth of the head at the front of the current.

z=H

x=x

Figure 1: The two-layer system variables.

Usual physical conditions at the boundary are imposed, such asno net flow across z — a(x) and both kinematic and dynamic bound-

ary conditions at z = h(x, t) and z = H+r)(x, t). These commonplace



boundary conditions can be found in most fluid dynamics books. *°

A nondimensionalization scheme is employed to focus on the

gravity wave processes. Nondimensional quantities, denoted with a

superscript tilde, are given by

x — Lx, xp — Lxp, z — Hz, a = Ha, h = H h,

%, = (75;, ^ = ~Y~̂ *' ^ ̂ — ̂ ^ ̂ [7^ ̂ ̂ ̂APn (6)

H ~ft = Z/ft, Cf = -j-Cf.

In the scaling (6), U* = igH, where 7 = (p? - pi) /pi- Eqns (2)-(5)

can now be reduced using standard shallow-water theory^ based on

the smallness of the aspect ratio S = H/L.

The resulting equations can be written with the tilde notation

suppressed, and are stated using the four variables %,-, h, and rj, with

the parameters 7, Cf, and K. They are:

*!i + |2 = o, (7)dx ox

+ [(1 - A + 777)̂ 1 + %2(A - 4] = 0, (8)

, , ,-7 ̂ +̂ ~ = -c/r~ — exp{-( - }, (9)^ r -

dt dx dx ox h - a

Eqns (7)-(10) can now be used to emphasize the effect of the viscousdamping term C/. We consider the simplification that the motion of

the upper layer is neglected (that is, when HI = 0 and 77 = 0), and

a constant bottom slope given by a(x) = -/3x, with /? > 0 a small

constant.The initial boundary value problem (IBVP) to be solved consists

of the instantaneous release of a fixed volume of heavy fluid at rest.

This IBVP has the form

(%, 0) = 0, %2(0, f) = 0, and (11)

^^ ^ 0 < X < 1,



where 0 < e < 1 is constant.

For the case without friction, Cj = 0, a solution exists in thelower layer which is given by

(13)

whenever

This solution (13), (14) represents a constant lower layer shape whichaccelerates linearly down the slope.

In contrast to this situation, when Cj > 0, there exists a steadysolution moving at a constant velocity u. In this case, there is a

travelling wave solution to eqn (10) of the form

Defining the variable f = x - ut allows eqn (9) to be reduced to anordinary differential equation,

(16)h - a

h-a

Figure 2: Steady-state solution to eqn (16)



Although an analytic expression for h — a is unavailable, a typical

numerical profile is plotted below in Figure 2 for the variable values

Cf — 0.25, 7 = 0.2, K = 0.1, u — 0.5, and the boundary value h — a =

0.3 at £ = 1. These two limiting cases show the importance of

the factor C/, and the need to include it in the shallow-water theory,

if the steady-state solutions which are observed experimentally are to

occur. In addition, the shape of the gravity current is only determined

when Cf ^ 0 as the initial conditions descibe the lower layer height

for C/ = 0.

4 The numerical scheme and results

To formulate eqns (7)-(10) into a form applicable to our numerical

solution scheme, we rewrite the equations of motion as the system

TIT ~> 7T~* \**"> ̂ ~~ *•0t 02

where the vector u is given by

/ 7/1 \

(17)

U = (18)

and the vector functions f, b are given by

-a)

b(u,ar) =

-a)

00

, (19)

0

(20)

\ " /

A finite-difference method was developed recently by Jin andto solve such systems of conservation laws (17) in the special

case with b = 0 and f = f (u). This method produces excellent shockresolution, and has the additional benefit that the eigenvalues of the

Jacobian matrix for f (u, x) (the matrix composed of the derivatives



of f with respect to 'the components of u need not be calculated as is

necessary when using such standard schemes as Godunov's method.

Jin and Xin's* method used to calculate solutions to the gravity

current problem on a horizontal bottom (Montgomery & Moodie^)

in which case b = 0 and f = f (u). For nonzero slope, this reduction

of the problem to a system of conservation laws does not occur, and

the scheme is generalized as follows. Associated with the system (17),

we define the relaxation system* to be solved for the vectors u(x,t),and v(x,t), to be given by the two vector equations:


size h and the parameter a are chosen, a uniform time step k is fixed

which will satisfy the CFL condition

\/a^


MUSCL scheme.^** For completeness, these are stated as

The slopes af in eqn (27) are given by

(28)

^ ^where the slope limiter function 0 used is

It should be noted that although there are some implicit steps in the

algorithm (25), these occur only when calculating u% and u™ and

since the variation in these values is small with changes in time, this

difficulty is avoided by using the values from the previous time stepto find f (u*, x) and f (u**, x).

The numerical scheme is applied to the problem of sudden release

of a block of fluid into an area of less dense fluid at rest which is

bounded by a solid wall at x = 0 and unbounded (semi-infinite) tothe right. The eqns of motion (17)-(20) are then solved numericallystarting from the initial value

%i (%, 0) = 0, %2(%, 0) = 0, 77(2, 0) = 0,

*> f ***> (3D0 for x > 1,

and subject to the boundary condition of no flow across the wallx = 0, that is,

%i(0,f) = 0, %2(0,f) = 0. (32)

The initial and boundary conditions are implemented numericallyfrom these physical values for the required vectors u, v as suggested

by Jin & Xin* via v(s,0) = f(u(x,0),a?) and v(0,f) = f(u(0,*),0).The components b(0, t) and 7?(0,f) are interpolated from the approx-imate solution to the right of the boundary using a second-orderpolynomial approximation.^



The boundary values given in eqns (26), (27) produce an inclined

thermal. To include inclined starting plumes in our calculation, we

need to implement a line source of lower layer fluid somewhere in the

lower layer region. This has been done experimentally by using a line

source at one end of the tank^ and we represent this line source by

introducing a constant addition of fluid volume per unit time over

a fixed region within the lower layer. Let Q denote the nondimen-

sionalized addition of mass considered as a volume per unit length

per unit time. If it is added in the region 0 < x < 0.1, then conser-

vation of mass implies that mass addition is given numerically by a

simple increase in the lower layer height over the region 0 < x < 0.1,and a corresponding increase in free surface deflection. The numer-

ical scheme (25)-(30) is changed such that after each time step, the

surfaces h and 77 are altered for the cells j = 0, 1, ...9 according to

The bottom topography considered here the same as that given

in Section 3, in particular with a constant bottom slope,

o(z) = -/)%, /3 > 0. (34)

Although this simplifies the lower boundary considerably, this is done

for illustrative purposes only; more general topographical forms can

be handled easily by the numerical scheme.Approximation of the front positions XB and XF was achieved

by truncating the approximate solution of h — a as its value fell below-J^Q-IO npĵ precise location of xp was assumed to occur to the left

of the truncation position, and the calculation of the upper layerwas continued to the right using only the two upper layer equations.

In this way, we avoided the numerical difficulties arising from eqn(9) when h - a is zero. For non-zero slopes, the phenomenon of

separation of the lower layer from the end wall at x = 0 required asimilar treatment at the point %#, namely truncation of the lower

layer solution at the point where the thickness fell below the same

value of 10-™.To investigate the effects of slope 0 and volume flux Q, several

calculations were performed for varying values of these parameters.

The physical parameters used throughout were 7 = 0.2, ho = 0.5,

K — 0.1, and Cj = 0.25, with the numerical scheme parameters fixed



at CFLno. = 0.75, a = 10, e =

k = 0.75/i/x/a from eqn (24).

, and grid width h = 0.01, and

40

30 -

£O

O04

Okl

20 —

10 -

0 . I ' I ' I '0 10 20 30 40

Time (nondimensional)

Figure 3: Front position vs. time for varying slope parameter.

Figure 3 displays the front position of the lower layer as a func-tion of time for Q = 0 and varying values of slope parameter. It can

be observed from the graph that for j3 = 0.05 a steady solution isattained quickly, while for ft — 1 this does not occur until later at

approximately t = 35. For the horizontal bottom /3 = 0 there is a

steady initial phase until about t = 13 after which time a gradual

deceleration occurs. This compares well with experimental evidence,

in particular Rottman & Simpson's*^ Figure 8.

In Figure 4, the volume flux Q is varied for a constant slope of

(3 = 0.1. There is not much difference in the behaviour of the front

speed in the two cases given, although there slightly faster flow results

in the case of nonzero volume flux. This behaviour is expected, andthe low variation in front speed coincides with the experimental ob-

servations that the front speed primarily determines the speed of the



current. Although not shown in Figure 4, non-zero slope will cause

separation of the lower layer gravity current from the left boundary

at x — 0 unless Q ̂ 0. This phenomenon of rear-wall separation and

its dependence on slope and volume flux will be explored in subse-

quent research.

40

co-H-P

ocu

30

20 -

10 -

:«:'0:

10 20 30Time (noridimensional)

40

Figure 4: Front position vs. time for varying volume flux Q.

5 Conclusion

A model for gravity currents resulting from the instantaneous release

of heavy fluid has been presented. The equations include both time-

evolution of two-layer flow and deflection of the free surface, aspectswhich are not generally discussed theoretically. In addition, variable

bottom topography was considered, and a forcing term in the lowerlayer horizontal momentum equation was added to balance the accel-

eration down the incline. Numerical solution of the equations gave



some

idence, an

preliminary results which compare well with experimental ev-

?, an achievement done without resorting to large computations.

References

[1] Beghin, P., Hopfinger, E.J. & Britter, R.E. Gravitational convec-

tion from instantaneous sources on inclined boundaries, J. Fluid

Mech., 107, pp. 407-422, 1981.

[2] Benjamin, T.B. Gravity currents and related phenomena, J. Fluid

MecA., 31, pp. 209-248, 1968.

[3] Britter, R.E. & Linden, P.P. The motion of the front of a gravity

current travelling down an incline, J. Fluid Mech., 99, pp. 531-

543,1980.

[4] D'Alessio, S.J.D., Moodie, T.B., Pascal, J.P. & Swaters, G.E.

Gravity currents produced by sudden release of a fixed volume of

heavy fluid, Stud. Appl. Math., 96, pp. 359-385, 1996.

[5] Ellison, T.H. & Turner, J.S. Turbulent entrainment in stratified

flows, J. Fluid Mech., 6, pp. 423-448, 1959.

[6] Huppert, H.E. & Simpson, J.E. The slumping of gravity currents,

J. Fluid Mech., 00, pp. 785-799, 1980.

[7] Huq, P. The role of aspect ratio on entrainment rates of instanta-

neous, axisymmetric finite volume releases of dense fluid, J. Haz.

Mat., 49, pp. 89-101, 1996.

[8] Jin, S. & Xin, Z. The relaxation schemes for systems of conserva-

tion laws in arbitrary space dimensions, Comm. Pure and Appl.

Math., 48 pp. 235-276, 1995.

[9] Von Karman, T. The engineer grapples with nonlinear problems,Bull Amer. Math. Soc., 46, pp. 615-683, 1940.

[10] Kundu, P.K. Fluid Mechanics, Academic Press, San Diego, Cal-

ifornia, 1990.

[11] Leveque, R.J. Numerical Methods for Conservation Laws,

Birkhauser-Verlag, Basel, 1992.



[12] Middleton, G.V. Experiments on density and turbidity currents,

I. Motion of the head, Canadian Journal of Earth Sciences, 3,

pp. 523-546, 1966.

[13] Montgomery, P.J. & Moodie, T.B. Analytical and numerical re-

sults for flow and shock formation in two-layer gravity currents,

J. AWr. MofA. 5oc. Senes B., 39, pp. 1-23, 1998.

[14] Pedlosky, J. Geophysical Fluid Dynamics, Springer- Verlag, New

York,1987.

[15] Rottman, J.W. fe Simpson, J.E. Gravity currents produced by

instantaneous release of a heavy fluid in a rectangular channel, J.

Fluid Mech., 135, pp. 95-110, 1983.

[16] Simpson, J.E. Gravity Currents in the Environment and the Lab-

oratory (2nd edition), Cambridge University Press, Cambridge,

1997.

[17] Whitham, G.B. Linear and Nonlinear Waves, J. Wiley and Sons,

New York, 1974.


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