Advanced Fluid Dynamics of The Environment- Spreading of a Shallow Mass on an Incline

7 Pages·0.08 MB·English

by  Chiang C. Mei

1
Lecture Notes on Fluid Dynamics
(1.63J/2.21J)
by Chiang C. Mei, MIT
2-4spreadmud.tex
2.4 Spreading of a shallow mass on an incline
References:
C. C. Mei, (1966), Nonlinear gravity waves in a thin sheet of viscous (cid:13)uid, J. Math. & Phys.
45, 482-496.
K. F. Liu & C. C. Mei, 1989, Slow spreading of a sheet of a Bingham (cid:13)uid on an inclined
plane, J. Fluid Mech. 207, 505-529.
X. Huang &M. H. Garcia,1997, Asymnptotic solution forBingham debris (cid:13)ows, Debris -(cid:13)ow
Hazards and Mitigation, ASCE Proc. pp.561-575.
The spreading of a (cid:12)nite mass of paint, paper pulp, mud or lava on an inclined plane is of
interest to a variety of industrial and geological problems. For mud modeled as a Bingham-
plastic non-Newtonian (cid:13)uid, Liu & Mei (1989) solved the equation similar to (2.3.22) numer-
ically. An analytical solution for Herschel-Bulkley (cid:13)uid was given later by Huang & Garcia
(1997). We modify their theories for the simpler case of Newtonion (cid:13)uid here.
The free surface of a thin layer is expected to (cid:13)atten in time over most of the pro(cid:12)le,
but it should steepen near the downstream front where the spatial derivative is much more
important than elsewhere. Let us study the problem by dividing the total (cid:13)uid extent into
two: the far (cid:12)eld not too close to the steep front and the near (cid:12)eld around the front.
2.4.1 Far (cid:12)eld away from the front
Let the total length of the thin layer be L and the maximum depth be H, with H=L 1.
(cid:28)
We introduce the normalized variables suitable for the far (cid:12)eld.
0 0 0
h = Hh; x = Lx; t = Tt (2.4.1)
where T will be chosen to simplify the appearance of the (cid:12)nal equation. Thus,
H @h0 (cid:26)gsin(cid:18)H3 @ H @h0
+ 1 h03 = 0
T @t0 3(cid:22) L @x0 " (cid:0) Ltan(cid:18)@x0! #
Let us choose the time time scale to be
L
T = (2.4.2)
(cid:26)gH2sin(cid:18)=3(cid:22)
2
where the denominator is the average (cid:13)uid speed across the depth. The dimensionless
equation for the far(cid:12)eld reads,
0 0
@h @ H @h
+ 1 h03 = 0 (2.4.3)
@t0 @x0 " (cid:0) Ltan(cid:18)@x0! #
Assume
H
(cid:15) 1 (2.4.4)
Ltan(cid:18) (cid:17) (cid:28)
(2.4.3) can then be approximated by the hyperbolic equation,
@h0 @h03 @h0 @h0
+ = +3h02 = 0 (2.4.5)
@t0 @x0 @t0 @x0
It is of the class called kinematic wave equation in (cid:13)ood hydrology. Solution can be obtained
by the theory of characteristics.
In particular (2.4.5) can be rewritten in the form
@h @h
dt+ dx = dh = 0 (2.4.6)
@t @x
if
0
dx
= 3h02 (2.4.7)
dt0
0 0
The last two equations imply that h remains constant for all t along the characteristic
0 0
curve x(t) de(cid:12)ned by the di(cid:11)erential equation (2.4.7). Moreover, if the initial pro(cid:12)le is
presecribed,
0 0 0 0
h(x;0) = h (x); (2.4.8)
o
then allcharacteristics arestraight but ofdi(cid:11)erent slopes. The characteristic originated from
0 0 0
the initial point x = (cid:24) at t = 0 is the straight line
x(t;(cid:24)) = 3h2((cid:24))t+(cid:24); (2.4.9)
o
along which
0 0 0 0 0
h(x;t) = h ((cid:24) ) (2.4.10)
o
0 0 0
In principle we can solve for (cid:24) from (2.4.9) in term of x;t and substitute the result
0 0 0
into (2.4.10) to get h(x;t). For any initial hump, the characteristics at the front must
intersect one another, impling multivaluedness of solution, which is phycially unacceptable.
The solution can still be proceeded if a discontinuity, shock, is allowed at x0 = x0(t0), as long
s
as mass is conserved.
As an example, consider a triangular initial pro(cid:12)le:
0 0
x 0 < x < 1
0 0
h (x) = (2.4.11)
o 0 0
0; x < 0; & x > 1:
(
3
The pro(cid:12)le has a shock front to begin with, which is a mathematical idealization of course.
From (2.4.9) we get
3(cid:24)02t0 +(cid:24)0; 0 < (cid:24)0 < 1;
0
x = (2.4.12)
0 0 0
(cid:24) ; (cid:24) < 0;(cid:24) > 1
(
0 0 0 0
Within 0 < (cid:24) < 1, (cid:24) can be solved in terms of x;t,
1
(cid:24)0 = 1+p1+12x0t0 (2.4.13)
6t0 (cid:0)
(cid:16) (cid:17)
0
For any t > 0,
0 0 0 0 0
h = 0; for x < 0; & x > x (t); (2.4.14)
s
and
1
h0 = (cid:24)0 = 1+p1+12x0t0 ; 0 < x0 < x0(t): (2.4.15)
6t0 (cid:0) s
(cid:16) (cid:17)
0 0
The shock front x (t) is unknown.
s
0 0
To locate the shock front x (t), we invoke mass conservation,
s
1 x0s(t) 0 0 h0s 0 dx0 0
initial volume = = h dx = h dh (2.4.16)
2 Z0 Z0 dh0(cid:12)(cid:12)t0
(cid:12)
(cid:12)
From (2.4.12), (cid:12)
x0 = 3h02t0 +h0 (2.4.17)
so that
0
dx
0 0
= 6ht +1
dh0(cid:12)(cid:12)t0
(cid:12)
Thus (cid:12)
(cid:12)
1 h0s 0 0 0 0 h0s2 0
= h(6ht +1)dh = (4h t+1)
s
2 2
Z0
or
1 h02
0 s
t = (cid:0) (2.4.18)
4h03
s
0 0 0 0
This gives the shock height h implicitly as a function of t. To get the shock postion x (t)
s s
we put this result into (2.4.17)
3+h02
0 s
x = (2.4.19)
s 4h0
s
0 0
Together (2.4.18) and (2.4.19) give x (t), as plotted in (cid:12)gure (2.4.1). To be needed later we
s
note that the shock speed is given by
dx0 dx0 dt0 1 3h0(cid:0)2 4h02
s s s s 02
= = = (cid:0) = h : (2.4.20)
dt0 dh0 dh0 4 1 3h0(cid:0)2 s
s
(cid:0)
4
3.5
3
x’ s (t’)
2.5
2
1.5
1
0.5
h’s (t’)
0
0 2 4 6 8 10 12 14 16 18 20
t’
0 0 0 0
Figure 2.4.1: Time variation of the location x (t) and depth h (t) of shock front.
s s
2.4.2 Near (cid:12)eld of downstream front
Clearly near the shock the local free surface slope cannot satisfy (2.4.4). We de(cid:12)ne the
longitudinal length scale for the near (cid:12)eld to be Ls so that
H Ls
= O(1); i.e., = O((cid:15)) (2.4.21)
Lstan(cid:18) L
and renormalize the near (cid:12)eld,
0 0 x xs(t) 0 0 0 0
h = HH ; t = Tt; (cid:0) = X ; or x = x +(cid:15)X (2.4.22)
s
Ls
0 0 0 0 0
Thus the near (cid:12)eld solution H is a function of X (x;t) and t. The derivatives are now
related by the chain rule,
0 0 0 0 0 0 0 0 0
@H (x;t) @H @H @X @H 1dx @H
s
= + = (2.4.23)
@t0 @t0 @X0 @t0 @t0 (cid:0) (cid:15) dt0 @X0
0 0 0 0 0 0
@H (x;t) @H @X 1@H
= = (2.4.24)
@x0 @X0 @x0 (cid:15) @X0
Eq. (2.4.3) becomes
0 0 0 0
@H 1dx @H 1 @ @H
s + H03 1 = 0 (2.4.25)
@t0 (cid:0) (cid:15) dt0 @X0 (cid:15) @X " (cid:0) @X0!#
To the leading order the near (cid:12)eld is approximately governed by
0 0 0
dx @H @ @H
s + H03 1 = 0 (2.4.26)
(cid:0) dt0 @X0 @X0 " (cid:0) @X0!#
5
H’
hs’
H’(0)
X’
*
X’
x’=x’s(t)
Figure 2.4.2: Matching the near-(cid:12)eld and far-(cid:12)eld at the front by preserving mass.
which is the ordinary di(cid:11)erential equation for a stationary wave, moving at the known shock
speed dx0=dt0 = h02. Integrating with respect to X0 once, we get
s s
0
dH
h02H0 +H03 1 = 0:
(cid:0) s (cid:0) dX0!
The constant of integration is taken to be zero so that the surface touches the dry bed where
0
H = 0. We may rewrite (2.4.26) as
H02dH0 h0 1 1
0 0 s
dX = = dH 1 +
(cid:0)h0s2 H02 " (cid:0) 2 h0s H0 h0s +H0!#
(cid:0) (cid:0)
It follows upon integration that
0 0 0
h h H
0 s s 0 0
H + log (cid:0) = X X (2.4.27)
2 h0s +H0! (cid:0) (cid:3)
0 0 0 0
This is a smooth surface decreasing monotonically from H = h at X = to H = 0
s
0 0 (cid:0)1
(dry bed) at X = X(cid:3). To determine the value of X(cid:3), we require that mass under the smooth
pro(cid:12)le be the same as that under the pro(cid:12)le with a shock, as sketched in Figure 2.4.2.
0 0 0 0 X(cid:3)0 0 0 0
(h H )dX = H (X )dX (2.4.28)
s
Z(cid:0)1 (cid:0) Z0
Referring to Figure ??, the integrals can be equivalently carried out as
h0s 0 0 0 H0(0) 0 0 0
X (H )dH = X (H )dH (2.4.29)
(cid:0)ZH0(0) Z0
0 0
where H (0) is the height at X = 0, given by
0 0 0
h h H (0)
0 s s 0
H (0)+ log (cid:0) = X (2.4.30)
2 h0s +H0(0)! (cid:0) (cid:3)
6
Substituting (2.4.27) into (2.4.29), we get, for the left-hand-side,
h0s 0 0 0 0 0 0 1 0 2 02 h0s h0s h0s H0 0
X (H )dH = X (H (0) h )+ H (0) h log (cid:0) dH
(cid:0)ZH0(0) (cid:3) (cid:0) s 2 (cid:16) (cid:0) s (cid:17)(cid:0) 2 ZH0(0) h0s +H0!
and for the right-hand side,
H0(0) 1 h0 H0(0) h0 H0
0 0 0 0 0 0 2 s s 0
X (H )dH = X (H (0) 0)+ H (0) 0 log (cid:0) dH
(cid:0)Z0 (cid:3) (cid:0) 2 (cid:16) (cid:0) (cid:17)(cid:0) 2 Z0 h0s +H0!
Therefore,
0 1 h0s h0s H0 0 0 h0s 1 1 z
X = log (cid:0) dH +h = log (cid:0) dz +1
s (cid:0)2 "Z0 h0s +H0! s# (cid:0) 2 (cid:20)Z0 (cid:18)1+z(cid:19) (cid:21)
Using the facts
1 1
log(1 z)dz = 1; log(1+z)dz = 2log2 1
(cid:0) (cid:0) (cid:0)
Z0 Z0
we get
1
0 0 0
X = log2 h = 0:19897h (2.4.31)
(cid:3) s s
2 (cid:0)
(cid:18) (cid:19)
The near (cid:12)eld solution is now complete.
Recall that the near and far (cid:12)elds coordinates are related by
0 0 0
x = x +(cid:15)X (2.4.32)
s
in general and
1
0 0 0 0 0 0
x (t) = x (t)+(cid:15) log2 h (t) (2.4.33)
(cid:3) s s
2 (cid:0)
(cid:18) (cid:19)
in particular.
We can get the uniformly valid solution (hU:V:) by adding the near and far (cid:12)eld solutions
and subtracting the common part. Thus, in terms of the far-(cid:12)eld variables,
0 0 0 0
h = h +H h (2.4.34)
U:V: s
(cid:0)
0 0
where h is given by (2.4.15) and H is given by (2.4.27), while the coordinates are related
by (2.4.32).
Sample pro(cid:12)les of the free surface are shown for (cid:15) = 0:1 in Figure 2.4.3 for various
instants.
7
h’
1.2
1
t’=0
0.8
0.6 t’=1.2
t’=2.4
t’=3.6
t’=4.8
0.4 t’=6 t’=7.2
0.2
x’
0
0 0.5 1 1.5 2 2.5
Figure 2.4.3: Slow evolution of the free surface of a vicous (cid:13)uid down a dry incline. The
initial pro(cid:12)le is triangular with (cid:15) = 0:1.