front speeds for axisymmetric gravity currents · the usual approach to solving for the...

CANADIAN APPLIED

MATHEMATICS QUARTERLY

Volume 12, Number 3, Fall 2004

FRONT SPEEDS FOR AXISYMMETRIC

GRAVITY CURRENTS

P. J. MONTGOMERY AND T. B. MOODIE

ABSTRACT. The initial phase of gravity current slumpingcaused by the instantaneous release of a cylinder of dense fluid isconsidered theoretically and numerically. The resulting axisym-metric flow is examined in the framework of the solution to asystem of nonlinear hyperbolic partial differential equations forwhich the advancing front of the gravity current is consideredas a jump discontinuity in the solution. The Rankine-Hugoniot

jump conditions are employed to determine the possible frontconditions that such a jump may take. In the nonrotating case,the standard Froude number relationship is obtained, while inthe rotating case, an approximation for constant rotation isderived. These relationships are tested through numerical so-lutions of the equations of motion, and the new front conditionis shown to be an improvement on the standard one.

1 Introduction The primarily horizontal motion of two fluids ofslightly different densities often will result in a flow called a density, orgravity current. There are various types of gravity currents, and exam-ples may be found in environmental, industrial, and natural settings [6].Much has been learned about gravity current structure over the past fewdecades, especially when the geometry of the situation produces flow inmainly one direction. However, the situation is not as well understoodfor problems in which the flow is truly three dimensional, and researchin this area is continuing. For example, one situation recently studied isthat of a gravity current resulting from the sudden release of a cylinderof dense fluid within a rotating volume of less dense fluid [7]. Moreover,additional experiments and analysis are still being conducted to create

Both of the authors are grateful for financial support in the form of Individ-ual Research Grants from the Natural Sciences and Engineering Research Council(NSERC) of Canada. The author P.J.M. also appreciates the use of the computingresources (HPC) at UNBC.

Keywords: scientific computation, fluid dynamics, shallow-water flow.Copyright c©Applied Mathematics Institute, University of Alberta.

371

372 P. J. MONTGOMERY AND T. B. MOODIE

accurate predictive expressions for the spread of oil on water [1]. Thistype of activity motivated the present study of axisymmetric gravitycurrents through analytical and numerical methods.

A gravity current created from the instantaneous release of a finitevolume of dense fluid within a quiescent layer of less dense fluid is usu-ally characterized as having three distinct stages of spreading [1]. Inthe first stage, the gravitational acceleration is balanced with changes inthe inertia of the flow, while the later stages are described by a balancebetween gravity and either viscosity or surface tension. In this paper,the first stage of the flow is considered, and the dynamics of the floware further simplified by assuming that the gravity currents are axisym-metric. This latter assumption simplifies the equations of motion alongwith the removal of terms due to viscous, turbulent or other dissipativeforces. One important factor which is not neglected is the effect of theCoriolis force which is included in the equations of motion as a constantparameter.

The model equations are first considered for a single layer of homoge-neous inviscid fluid, expressed in cylindrical polar co-ordinates [3]. Fora single fluid layer of height h, radial velocity u and azimuthal velocityv the shallow-water model equations simplify to: conservation of mass

(1.1)∂h

∂t+

∂

∂r(hu) +

∂

∂θ(hv) +

hu

r= 0,

conservation of radial momentum

(1.2)∂u

∂t+ u

∂u

∂r+

∂h

∂r+

u

r

∂u

∂θ−

v2

r= εv,

and conservation of azimuthal momentum

(1.3)∂v

∂t+ u

∂v

∂r+

1

r

∂h

∂θ+

v

r

∂v

∂θ+

uv

r= −εu.

These equations are expressed with nondimensional quantities, and havebeen simplified to contain only one parameter as a result of the scalings.This parameter, labelled ε, represents the ratio of the rotational to iner-tial velocity scalings, and may be observed as the inverse of the RossbyNumber,

(1.4) ε =fL

U

where f is the rotation frequency, L is the characteristic length scale ofthe motion, and U is the characteristic velocity scale of the motion. A

FRONT SPEEDS 373

more thorough and detailed derivation of the equations of motion maybe found in standard textbooks in fluid dynamics [3], although specificcalculations relevant to the results herein have been completed elsewhere[4].

The equations (1.1)–(1.3) are simplified in the case of axisymmetricflow where it is assumed that the flow is independent of the azimuthalangle θ. The flow variables, u, v, and h are then simply functions of thenondimensional radial position (r) and time (t) only, with the subsequentequations of motion for axisymmetric flow given by

∂h

∂t+

∂

∂r(hu) +

hu

r= 0,(1.5)

∂u

∂t+ u

∂u

∂r+

∂h

∂r−

v2

r= εv,(1.6)

and

(1.7)∂v

∂t+ u

∂v

∂r+

uv

r= −εu.

Some simple manipulations with the product rule permit equation (1.5)to be rewritten in the conservation form

(1.8)∂

∂t(rh) +

∂

∂r(rhu) = 0.

Hence, the equations of motion to be subsequently discussed consist ofequations (1.6)–(1.8).

The usual approach to solving for the time-dependent axisymmetricflow of gravity currents in the inertial spreading stage involves applyingthe method of characteristics to a suitable two-point boundary valueproblem created by scaling the equations of motion such that the radiusis always in a fixed domain [7]. This approach is limited by the factthat the boundary conditions at r = 0 and at the front of the advancinggravity current must be specified both theoretically and numerically fora properly-posed problem. Such a specification of the front conditionsbefore the flow is actually known is often problematic, and a discussionof the difficulties involved in the simpler planar flow problem has beencompleted by the authors [5], where the results show that an alternatemethod of solution is possible. The new method proposed by the authorsis to treat the boundary value problem as a purely initial value problemwhere the advancing gravity current front is interpreted as a shock in


the solution. A discussion of the mathematical concepts involved maybe found in various textbooks (see [8] or [9], for example, with specificsreviewed in [4]) with the relevant points mentioned below.

The idea of a shock is sensible when considering weak solutions tosystems of conservation laws in their integral form. In general, for ap-propriate scalar functions ϕ (x, t) , f(ϕ, x) and g(ϕ, x), the differentialform of a conservation law

(1.9)∂

∂tg (ϕ, x) +

∂

∂xf (ϕ, x) = 0

permits solutions which may be piecewise continuous, given that anydiscontinuity x = s(t) satisfies the celebrated Rankine-Hugoniot relation

(1.10)ds

dt[g] = [f ] .

The bracket notation in equation (1.10) is standard [8], and representsa jump in value across the discontinuity at position s, i.e.,

[g] = limx→s+

g (ϕ(x, t), x) − limx→s−

g (ϕ(x, t), x)

with a similar definition for [f ].

The plan of the remainder of this paper is to apply the condition (1.10)to equations (1.6)–(1.8) in an attempt to find a suitable relation whichcaptures the essence of the front conditions employed elsewhere [7]. InSection 2, two new front conditions are derived, and it is shown thatalthough relation (1.10) is not directly applicable to equations (1.6)–(1.8), approximate methods may be employed to obtain new relationsbetween the gravity current speed and height at the advancing front.These new front conditions are subsequently examined through the useof numerical solutions to the equations obtained from a numerical tech-nique introduced for systems of nonlinear conservation laws which arenot necessarily hyperbolic [2]. The numerical results suggest that one ofthe new front conditions in particular is a qualitative and quantitativeimprovement over the one which is currently in use.

2 Approximate front conditions An initial value problem forequations (1.6)–(1.8) is considered which contains an initial discontinuityin the lower layer height. The equations of motion are written again in

FRONT SPEEDS 375

conservation form as

∂

∂t(rh) +

∂

∂r(rhu) = 0,(2.1)

∂u

∂t+

∂

∂r

(

1

2u2 + h

)

=v2

r+ εv,(2.2)

∂v

∂t+ u

∂v

∂r= −

uv

r− εu.(2.3)

The initial value problem then consists of solving equations (2.1)–(2.3)for the instantaneous release of a cylinder with initial height profile

(2.4) h (r, 0) =

{

0.5, 0 ≤ r ≤ 1

0, r > 1,

and initial rest velocities

(2.5) u(r, 0) = v(r, 0) = 0.

The initial value problem (2.1)–(2.5) has a starting discontinuity im-posed upon the height h imposed at r = 1. To obtain informationregarding how this initial discontinuity develops in time, we consider adiscontinuous solution to (2.1)–(2.5) with a single discontinuity r = s(t),with initial position s(0) = 1.

First, it should be noted that equation (2.1) is of the form (1.9), andcondition (1.10) may be used. The resulting jump condition is thengiven by the equation

(2.6)ds

dt[rh] = [rhu] .

By introducing a different notation,

(2.7) h+ = limr→s+

h(s, t), h− = limr→s−

h(s, t)

and similarly for v± or u±, equation (2.6) may be written in the form

(2.8)ds

dt= u−.

The result (2.8) is in accordance with the physical notion that the shockmoves outward with the radial velocity of the fluid, and that the discon-tinuity represents a material interface or gravity current front.


In general for a system of scalar conservation laws, a similar analysisfor each equation is completed to obtain a relation between the values ofthe components near the shock. However, in this case the method doesnot directly lead to a straightforward result (see [4] for more details), andan alternate approach is necessary. One of the simplest cases to consideroccurs when the flow is entirely radial and the azimuthal componentvanishes (v = 0). In this case, equation (2.3) is not applicable, andequation (2.2) simplifies to the form

(2.9)∂u

∂t+

∂

∂r

(

1

2u2 + h

)

= 0.

In this special case, relation (1.10) applied to equation (2.9) gives thesimplified jump condition

(2.10)ds

dtu− =

1

2u2− + h−,

which, when combined with the result (2.8) produces the classical resultused previously [7],

(2.11) u− =√

2h−.

Equation (2.11) is similar to the Froude number front condition for pla-nar flow [5].

In the general case when the flow is not entirely radial (v 6= 0), thevalidity of the result (2.11) is questionable. It is presumptive to as-sume that it holds for equations (2.1)–(2.3) when only a special case ofthe possible flow has been examined. However, due to its form, it isnot possible to incorporate equation (2.3) directly into the jump con-dition (1.10) to obtain a front condition. In the following subsections,two approximations are therefore derived by using (2.3) to simplify theright hand side of equation (2.2) to a point where an approximate jumpcondition may be calculated.

2.1 Constant azimuthal velocity approximation The first methodof approximation is to assume that the azimuthal velocity, v, is a con-stant either throughout the flow or in a neighbourhood of the front. Al-though not entirely physically realistic, this assumption is an improve-ment to the requirement that v = 0 identically. Using the notationv = v0, equation (2.2) for the radial flow becomes

(2.12)∂u

∂t+

∂

∂r

(

1

2u2 + h

)

=v20

r+ εv0,

FRONT SPEEDS 377

which, after an integration can be stated as

(2.13)∂u

∂t+

∂

∂r

(

1

2u2 + h − v2

0 ln r − εv0r

)

= 0.

Equation (2.13) is now in the form (1.9) and therefore, and the jumpcondition (1.10) is subsequently applied.

Weak solutions with a discontinuity at r = s(t) which are zero forr > s then satisfy the relation (1.10) for (2.13),

(2.14)ds

dt[u] =

[

1

2u2 + h − v2

0 ln r − εv0r

]

,

which simplifies using the notation (2.7) to

(2.15)ds

dtu− =

1

2u2− + h− − v2

0 ln s − εv0s.

Using the result (2.8) to remove the derivative then allows equation(2.15) to then be rewritten as the result

(2.16) u− =√

2h− − 2v0 (v0 ln s + εs).

Equation (2.16) is similar in form to the front condition (2.11), andin fact reduces to this result in the special case v0 = 0. The frontcondition (2.16) is also seen to be physically sensible by consideringthe case when v0 < 0, which occurs in the case of positive ε (counter-clockwise rotation). For such an assumption, the front speed u− givenby (2.16) is in fact reduced from the predicted value for the non-rotatingstate (2.11) for any height h−.

Of additional interest is the steady-state case, with the special limitu− = 0 which then allows (2.16) to be rewritten as

(2.17) h− = v0 (v0 ln s + εs) .

Although relation (2.17) is a nonlinear equation for s, in the specialcase that the front is quite small or nonexistent it can be assumed thath− = 0. This simplification produces a result which gives the radiusof the steady-state gravity current, ln s = −εs/v0. It is an interestingexercise to find solutions to this equation for s ≥ 1, and ε/v0 < 0.One can show that there are no solutions for ε/v0 < −e−1, exactly onesolution for ε/v0 = −e−1, and two solutions for 0 > ε/v0 > −e−1.


For our situation, it is noted that when ε/v0 = −e−1, the solution to(2.17) with h− = 0 is s =e1, and as ε/v0 increases to zero, one solutiondecreases to 1 while the other increases without bound. Such an analysisis purely academic, however, since in such late stages of the flow theadvancing gravity current is stopped by viscous or other forces and adifferent governing equation must be considered.

2.2 Local plane wave analysis In the case when the azimuthal ve-locity is not a constant, it is not clear how to solve the problem dueto the difficulties posed by the nonlinearities. However, by adaptingthe method of plane wave analysis (see, for example, [8]) an asymp-totic result similar to equation (2.16) may be obtained. The standardplane wave solution is obtained by assuming a solution of the formv(r, t) = g(r)f(τ) where τ is the wave variable, to be determined througha technique dependent on the equations themselves. In this case, anasymptotic solution to equation (2.3) based on the plane wave form isattempted, and implemented in equation (2.2) to determine a front con-dition.

The azimuthal velocity is assumed to be varying from a constant nearthe front, and is given in the form

(2.18) v(r, t) = v0 + v1 (t) δ + v2(t)δ2 + O

(

δ3)

,

where the term δ = r − r0 represents a small deviation from a pointof constant radius r0 and azimuthal velocity v0. The expansion (2.18)is more general than the solution in subsection (2.1), and it is assumedthat |δ| � 1 so that the analysis is a local one. Substitution of theexpansion (2.18) into equation (2.3) then provides a decoupling of theresulting equations according to the order of δ in the standard way. Anintermediate step gives

(2.19) uv1 + δ

(

dv1

dt+ 2uv2

)

= −u

r0 + δ(v0 + δv1) − εu + O

(

δ2)

,

which then may be written after expansion of (r0 + δ)−1 as the equation

(2.20) uv1+δ

(

dv1

dt+ 2uv2

)

= −u

r0

(v0 + δv1)

(

1 −δ

r0

)

−εu+O(

δ2)

.

Equation (2.20) may be decomposed through orders of δ to give theO(1) equation,

(2.21) v1 = −v0

r0

− ε,

FRONT SPEEDS 379

and the O (δ) equation

(2.22)dv1

dt+ 2uv2 = u

(

−v1

2r0

+v0

r20

)

.

Substitution of (2.21) into (2.22) then provides an expression for v2,

(2.23) v2 =3v0

2r20

+ε

2r0

.

The expressions (2.21) and (2.23) for v1 and v2 respectively may besubstituted back into the expansion (2.18). Formally, this yields anapproximation to the azimuthal velocity with the first few terms writtenexplicitly as

(2.24) v = v0 −

(

v0

r0

+ ε

)

(r − r0) +

(

3v0

2r20

+ε

2r0

)

(r − r0)2

+ O(

δ3)

.

The result (2.24) may then be reorganized to the form

(2.25) v = q0 (r) v0 + εq1 (r) + O(

δ3)

,

where q0 and q1 are quadratic functions in r given by

(2.26) q0 =7

2− 4

r

r0

+3

2

r2

r20

, and q1 =3

2r0 − 2r +

1

2

r2

r0

.

Now, the expansion (2.25) may be used in equation (2.2) to give, aftersome simplification,

(2.27)∂u

∂t+

∂

∂r

(

1

2u2 + h

)

=q20v

20

r+ 2

q0q1v0

r+ εq0v0 + O(δ2, ε2).

Equation (2.27) is further modified by an integration of the terms onthe right hand side so that the result may be stated up to O(δ2) in theconservative form

∂u

∂t+

∂

∂r

{

1

2u2 + h − v2

0

∫

q20

r

− 2v0

∫

q0q1

r− εv0

∫

q0

}

= O(δ2, ε2).

(2.28)


Since this equation is (asymptotically) in the form (1.9), the relation(1.10) is assumed to also hold in the same sense. The jump relationmay be specified by calculating the integrated terms. This is done bydefining some notation to obtain equation (2.28) as

(2.29)∂u

∂t+

∂

∂r

{

1

2u2 + h − v2

0p0 − 2v0p1 − εv0p2

}

= O(δ2, ε2),

where in (2.29) the new terms are given by

p0 =49

4ln r − 28

r

r0

+53

4

r2

r20

− 4r3

r30

+9

16

r4

r40

,(2.30)

p1 =21

4r0 ln r − 13r + 6

r2

r0

−5

3

r3

r20

+3

16

r4

r30

,(2.31)

and

p2 =7

2r − 2

r2

r0

+1

2

r3

r20

.(2.32)

That is, the jump condition (1.10) for equation (2.29), using the short-hand notation (2.30)–(2.32) is then given by

(2.33)ds

dtu− =

1

2u2− + h− − v2

0p0− − 2v0p1− − εv0p2−,

which further simplifies using r = r0 = s (t) in the left hand limit as

ds

dtu− =

1

2u2− + h− − v2

0

(

49

4ln s −

291

16

)

− 2v0s

(

21

4ln s −

407

48

)

− 2εv0s.

(2.34)

Upon employing equation (2.8) in the result (2.34), it follows that thefront speed u− is related to the front height and position through theapproximation

(2.35) u2− = 2h−−2v0

[(

49

4v0 +

21

2s

)

ln s +

(

407

24+ 2ε

)

s −291

16v0

]

.

It should be noted that the front condition (2.35) reduces to (2.11) inthe case v0 = 0, and is similar in character to the constant azimuthalvelocity approximation (2.16).

FRONT SPEEDS 381

3 Numerical results In order to investigate the usefulness of thefront conditions (2.16) and (2.35) obtained in Section 2, it is helpful toresort to numerical solutions of the equations of motion (1.6)–(1.8). Anumerical scheme has been developed elsewhere [2] and modified by theauthors for application to systems of nonlinear hyperbolic conservationlaws with forcing terms [5]. This scheme is a finite difference relaxationscheme which has the advantage that it is stable, conservative, and cap-tures discontinuous solution in a nonoscillatory manner. Details of thenumerical scheme are not discussed herein, and may be found elsewhere(see [2] or [4]), but it is useful to understand the basic idea underlyingthe numerical method.

The relaxation method is applied to solve systems equations in the(vector) form

(3.1)∂

∂tu (x, y, t) +

∂

∂xf (u, x, y, t) +

∂

∂yg (u, x, y, t) = b(u, x, y)

which are hyperbolic. The system (3.1) is not solved directly, rathera relaxation system is solved which is thrice as large as (3.1), and iswritten as the separate systems:

(3.2)∂

∂tu+

∂

∂xv1 +

∂

∂yv2 = b,

(3.3)∂

∂tv1+α1

∂

∂xu =

1

ε(f − v1) ,

and

(3.4)∂

∂tv2+α2

∂

∂yu =

1

ε(g − v2) .

The advantage of solving the three linear systems (3.2)–(3.4) simulta-neously is that any implicit difficulties due to nonlinearities in f or g

are avoided, and in the limit that the parameter ε → 0, the originalsystem (3.1) is recovered. The constant ε is the relaxation constant (notto be confused with the Coriolis parameter ε), and the constants α1,2

are determined through consideration of the CFL condition for the dif-ference scheme in conjunction with straightforward conditions on theJacobian matrices of the flux functions. These parameters are discussedelsewhere [4] for the specific system considered in this paper.


As a simple attempt at solving the initial value problem (2.1)–(2.5),the relaxation method is not cast into polar co-ordinates, and the carte-sian form of equations (2.1)–(2.3) are used. The equations of motion aregiven therefore as

∂

∂t

uhvhh

+∂

∂x

hu2 + 1

2h2

huvhu

+∂

∂y

huvhv2 + 1

2h2

hv

=

εhv−εhu

0

,

(3.5)

where u represents the velocity in the x-direction, and v represents thevelocity in the y-direction. The Cauchy Problem consists of solvingequation (3.5) with the Cauchy data at t = 0 of zero velocity (u, v), anda finite volume. The conditions (2.4) and (2.5) were used as the initialcondition for equation (3.5), and a typical resulting solution profile ofthe lower layer surface height is shown in Figure 3.1. For this calcu-lation, the rotational parameter was set to zero so that the right handside of (3.5) vanished. The first-order spatial discretization of the nu-merical scheme resulted in such typically smooth profiles, however theparameters introduced in equations (3.3) and (3.4) are important in theaccuracy of the numerical method [2].

Since the finite difference scheme is only first-order accurate in space,it is helpful to view a cross-section of the radially expanding volumesince we are interested in studying the motion of the front. Such adiagram of the height profile shows the differences obtained in usingvarious values of the parameter α = α1 = α2 introduced in equations(3.3) and (3.4). In general, one finds that for smaller values of α thefront is more clearly defined, with the associated trade-off being that thescheme becomes unstable for values of α which are too small. A cross-section of the surface in Figure 3.1 is plotted in Figure 3.2 for the valuesof α = 10 (used in the calculation for Figure 3.1) and the larger value ofα = 100. The profiles are seen to be quite similar, although the positionof the front near h = 0 varies quite substantially as compared with theintersecting value at approximately h = 0.03. It is clear from Figure 3.2that the position of the front is not precisely determined by the numericcalculations, and this observation is important when when interpretingthe front condition relations which are evaluated at the front.

Another limitation in determining front position was imposed by themaximum size of the computational array used in the finite difference

FRONT SPEEDS 383

−2

−1

0

1

2

−2

−1

0

1

20

0.02

0.04

0.06

xy

FIGURE 3.1: Lower layer height h at t = 1 for the initial release of acylinder of diameter 1 and height 1.

calculations. At a certain radius (usually at multiples of about 6) thenumerical solution had to be rescaled and the calculation continued.This rescaling resulted in small jumps of the front position, as seen inFigure 3.3. These jumps could be easily removed through a more ac-curate interpolation, but the rough nature of the numerical calculationsis in keeping with the use of the Cartesian co-ordinate system and isreasonable for this investigation. The front position is measured as thepoint at which the layer thickness is less than 10% of the peak value, butincreasing as one approaches the middle such that the next grid space isabove this 10% value. Figure 3.3 also shows the effects of rotation on theradial speed of the advancing gravity current. As expected, when therotation rate is increased through the rotational parameter ε, the grav-ity current’s spread is delayed as more momentum is transferred intothe axial component. For the value ε = 2, it is seen that the advancinggravity current reaches a finite radius, at which point the current makesthe transition to steady state flow.

To examine the utility of the front conditions (2.11), (2.16), and(2.35), numerical simulations were completed and the relevant variables


−3 −2 −1 0 1 2 30

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

x

ζ

α=10 α=100

FIGURE 3.2: Cross-section of the surface h(x, y) at y = 0 in Figure3.1 for two values of α. Note the determination of front position isdependent on the choice of α.

recorded at the front position. With this data, the correlation functionwas plotted such that exact agreement would be seen through a calcu-lated correlation value of 1. For example, for the condition (2.11), thefunction plotted on the vertical axis would be u−/

√

2h− versus time onthe horizontal axis, with similar ratios for the other two front conditionexpressions (2.16) and (2.35). To illustrate the importance of the pointof measurement, the ratio (2.11) is plotted in Figure 3.4 at three frontpositions as the gravity current advances in time.

In interpreting Figure 3.4, several comments about the graph shouldbe made. The minor oscillations are due to the rough calculation ofthe front ratio at the chosen grid point for which the layer height isclosest to, but not above the maximum height. This method of fronttracking results in having the front position move along in time in smallincrements rather than in a smooth manner. The oscillations could easilybe removed by an interpolation between grid points to track the positionmore precisely; however, it was felt that for this initial investigationsuch fine detail was not necessary to determine the general featuresdesired. The larger jumps just after t = 2 are due to the grid sizereadjustment already discussed, and are imposed by memory allocationsof the computational platform. The most important observation from

FRONT SPEEDS 385

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

45

Time

Fron

t Pos

ition

ε=1/1000ε=1/10 ε=2

FIGURE 3.3: Front position versus time for various levels of rotation asper the parameter ε.

Figure 3.4 is that the correlation ratio for the front condition (2.11) isvery poor, and increases in what seems to be a linear manner for allthree of the front position points. It is thus inferred that the qualitativenature of the front condition does not depend on the point at which thefront is measured, but rather on the development of the variables of theflow.

Before investigating the effects of rotation on the front position, recallthat all three of the front condition relations become the same if v0 = 0.In this case, Figure 3.4 thus shows the poor correlation for all three ofthe front conditions. Once the rotation parameter is included, the frontratios differ, and may be calculated at any of the three positions used totrack the front. This is done in the remaining figures labelled 3.5 to 3.7.In these figures, the front ratio for expression (2.35) is not used due tothe fact that the calculation often resulted in imaginary numbers, andwas thus (2.35) not considered to be physically relevant. In Figure 3.5rotation is included in the ratio ε = 1/10. In Figure 3.6 the rotation isincreased to ε = 1/5, and in Figure 3.7 the rotation parameter is set ata value of ε = 2.0. In all figures, the ratios are calculated at the threepositions as in Figure 3.4, and it is observed that the position is mostimportant for the front relation (2.11).

In Figure 3.5, the correlation of relation (2.16) is observed to be much


FIGURE 3.4: The ratio u/√

2h evaluated at three positions. Position 1is tracked at 10% of the maximum height, while positions 2 and 3 arerespectively one and two grid steps behind the advancing front. Therotation parameter is ε = 0

better than (2.11). Although not a constant, it is much closer to unitythan the increasing values of the front condition (2.11). This phenomenais better seen in Figure 3.6 where the front is tracked over a longer periodof time, and it is observed that the front ratio for expression (2.11) peaksmidway through the domain. After the peak, the radial expansion slows,and it is thought that the second stage of gravity current spreading ismore applicable than the earlier stage of inertial spreading.

Surprisingly, the simpler calculation resulting in (2.16) produced abetter correlation than (2.11), and the physically irrelevant expression(2.35). However, expression (2.16) is still somewhat unsatisfactory inthat a constant correlation does not exist. Overall, it can be seen thatequation (2.16) represents an improvement over (2.11) in the sense that

FRONT SPEEDS 387

FIGURE 3.5: Correlation front ratios for relation (2.11) and (2.16) mea-sured at three front positions. The rotation parameter is ε = 0.1

variations in the slope of the plotted front ratios are less drastic, althoughsimilar behaviour exists for both (2.11) and (2.16), lending the supportthat neither expression is useful after the initial inertial phase.

The final figure is Figure 3.7 which shows similar results as per Figures3.4–3.6 for which the rotation parameter is set at the high value ofε = 2. In this case, the front is tracked until just prior to the onsetof steady-state rotation when the front ratio necessarily vanishes withthe spreading velocity. It is interesting to note that in Figure 3.7, theslope of the ratio for expression (2.16) is still less volatile than expression(2.11), even throughout the later stages of the flow


FIGURE 3.6: Correlation ratios for relation (2.11) and (2.16) measuredat position 1. The rotation parameter is ε = 0.2

4 Conclusions The initial inertial slumping phase of a gravity cur-rent created by the instantaneous release of a cylinder of fluid has beenexamined theoretically and numerically. To obtain a theoretical ex-pression of a Froude number at the front of a gravity current, weaksolutions and the Rankine-Hugoniot conditions were used. For a non-rotating gravity current, the classical (nondimensional) front conditionu2 = 2h was obtained, and in the rotating case, this was generalized tou2 = 2h − 2v (v ln s + εs) where the variables u, h, and v are measuredat the front position r = s. An additional front condition (2.35) wasderived which, although similar to the improved front condition, did notseem to produce as sensible a result.

FRONT SPEEDS 389

FIGURE 3.7: Correlation ratios for relation (2.11) and (2.16) measuredat position 1. The rotation parameter is ε = 2.0

For an axisymmetric gravity current, it was observed by using a roughnumerical scheme that the new front condition gave an improved quali-tative and quantitative agreement over the standard one. This approxi-mation is most vaild for the first stage of inertial spreading, and is betterfor small rotation parameters. However, the result still represents an im-proved qualitative agreement for fast rotation until steady-state flow isreached.


REFERENCES

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2. S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws inarbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), 235–276.

3. P. K. Kundu, Fluid Mechanics, Academic Press, California, 1990.4. P. J. Montgomery, Shallow-Water Models for Gravity Currents, Ph.D. thesis,

University of Alberta, Edmonton, 1999.5. P. J. Montgomery and T. B. Moodie, Generalization of a relaxation scheme for

systems of forced nonlinear hyperbolic conservation laws with spatially depen-dent flux functions, Stud. Appl. Math. 110 (2003), 1–19.

6. J. E. Simpson, Gravity Currents in the Environment and the Laboratory (secondedition), Cambridge University Press, Cambridge, 1997.

7. M. Ungarish and H. E. Huppert, The effects of rotation on axisymmetric gravitycurrents, J. Fluid Mech. 362 (1998), 17–51.

8. G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.9. Y. Zheng, Systems of Conservation Laws: Two-Dimensional Reiemann Prob-

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Mathematics Program, University of Northern British Columbia, 3333 Uni-

versity Way, Prince George, B.C., Canada, V2N 4Z9.

E-mail address: [email protected]

Applied Mathematics institute, Department of Mathematical and Statis-

tical Sciences, University of Alberta, Edmonton, AB, Canada, T6G 2G1.

E-mail address: [email protected]

front speeds for axisymmetric gravity currents · the usual approach to solving for the...

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