Research Journal of Environmental and Earth Sciences 3(3): 187-192, 2011
ISSN: 2041-0492
© Maxwell Scientific Organization, 2011
Received: July 17, 2010
Accepted: September 17, 2010
Published: April 05, 2011
An Analytical Model for Predicting Drag Resistance Caused by Internal Waves
in a Stably Stratified Open Channel Flow
1
E.C. Obinabo and 2O.L. Ndubuisi
Department of Electrical and Electronic Engineering,
2
Department of Civil Engineering, Ambrose Alli University, P.M.B., 14, Ekpoma,
Edo State, Nigeria
1
Abstract: This study explores optimal control via modelling of an open-channel flow, primarily pertaining to
the general problem of drag resistance associated with gravity waves in a stably stratified fluid, and the
influence of gravity waves on offshore structures. The study applies the results reported in the existing literature
to gravity gradient waves using essentially both analytical and numerical techniques, and establishes, among
others, detailed characterization of drag resistance effects for application to free and forced conditions in open
channel flow.
Key words: Drag resistance, internal gravity waves, open channel flow, stratified fluid
An earlier study by Scorer (1949) proposed a lee
wave term of the type studied by Obinabo (1978) and
although his basic equations were based on perturbation
theory, he approached his problem by considering a
model of two layers instead of one, and in which GO was
constant in each. He showed that the length of the waves
is such that their speed in the upstream direction relative
to the water is equal to the speed of the water in the
downstream direction. He also showed that the only
quantity which could influence the lee wavelength is the
wind speed, and inferred that the wavelength must
therefore be determined solely by the terms and Uo and
their variations with height. Large amplitude wave theory
was developed by Long (1953), and he produced an exact
steady state equation of motion and continuity of a perfect
liquid moving in two-dimensions with an arbitrary
vertical distribution of density and velocity. He integrated
the equation once to yield a second order differential
equation which he examined with regard to uniqueness
and stability of the motion. He developed a criterion
giving a sufficient condition for the motion to be uniquely
determined by the configuration of the topography over
which the fluid moves. He also approached the problem
on non-linear solution of the equation of motion of the
fluid by considering specific cases in which the non-linear
terms were identically zero before he produced analytical
results. Groves (2004) regarded the non-linear terms of
the equation as "perturbations" on the linearized case and
produced a solution which he argued would provide wave
solutions not entirely dissimilar to those obtained from the
linearized equation. His work also includes the
INTRODUCTION
The first contributions to the subject of wave forces
included valuable results of theoretical and field studies
(Baddour and Abbink, 1983) and were similar to those
studied by Lukomsky and Gandzha, (2003). the theory
assumes that the waves are periodic in the horizontal
direction, and independent of time (Jamaloddin et al.,
2005,). Mathematical relations were subsequently
formulated for computing the forces exerted by breaking
waves (Eyo, 2007a; Drennan, 1992).
Motion of stratified fluids with laminar flow has
been studied extensively in the past several decades.
Long (1953) put forward a theory for waves in an
airstream which had a uniform stability and velocity up to
infinity, and demonstrated that there could be waves
characterized by more than one wave crest on each
streamline although there is one on the ground. He also
showed that these waves have amplitude which increased
upwards, and decreased downwards. similar results were
computed in which slight modifications were made for the
effect of the earth's rotation (Sen-Gupta, 1973;
Phillips, 1977). It was shown (Ndubuisi and
Obinabo, 2010) that for ordinary sized mountains which
had no steep sides the second the subsequent waves had
a very small amplitude. The theory of both of these
authors is equivalent to the assumption that the
wavelengths of the perturbations are small compared to
the depth of the atmosphere. The theory, however, was
considered inadequate as it was observed that in the
troposphere there was only one harmonic wave of large
amplitudes known as a lee wave.
Corresponding Author: E.C. Obinabo, Department of Electrical and Electronic Engineering, Ambrose Alli University, P.M.B. 14,
Ekpoma, Edo State, Nigeria
187
Res. J. Environ. Earth Sci., 3(2): 187-192, 2011
application of the Fourier series in the solution of the
wave equation (Oyetunde, 2007; Oyetunde and
Okeke, 2007). In fact, although a lot of work has been
done on the subject of the internal gravity waves, in
stratified fluid (Polton et al., 2004; Naser et al., 1980),
none of the published works was devoted to the
investigation of the drag resistance due to the internal
gravity waves. Consequently this project is, to a great
extent, a pioneering venture in this field.
The purpose of the study was to develop a twodimensional model of a train of free internal gravity
waves in a stably stratified fluid bounded by the four sides
of an open tank and free surface. Computation of the
model was enhanced by the assumption that the flow in
the open channel is linear and stabilizable. Our results are
presented for values of the amplitude which was varied
from small to large to demonstrate the development of
rotor regions, the effect of which was shown to have some
considerable influence on the values of the drag (Obinabo,
1978). The measured values of the drag were quite small
but they are within the expected magnitude for the
laboratory model considered.
∂P ∂
.
∂ x ∂Z
⎛ q2 ⎞ ∂P ∂ ⎛ q2 ⎞
⎜⎜ ⎟⎟ +
. ⎜⎜ ⎟⎟
⎝ 2 ⎠ ∂z ∂ x ⎝ 2 ⎠
u∂ P
∂2P
∂P
+w
=−
−g
∂x
∂ x∂ z
∂x
Subtract (3) from (4) we obtain:
∂P ∂
.
∂ x ∂Z
⎛ q2 ⎞ ∂P ∂ ⎛ q2 ⎞
⎜⎜ ⎟⎟ −
. ⎜⎜ ⎟⎟
⎝ 2 ⎠ ∂z ∂ x ⎝ 2 ⎠
⎛ u∂ P w∂ P ⎞
∂P
+ w⎜
+
⎟ = −g
∂z ⎠
∂x
⎝ ∂x
Dividing through by P and re-arranging, we obtain:
∂w 1 ∂ P d ⎛ q 2 ⎞ 1 ∂ P d ⎛ q 2 ⎞
⎜ ⎟−
+
. ⎜ ⎟
∂t p ∂ x ∂ z ⎜⎝ 2 ⎟⎠ p ∂ z ∂ x ⎜⎝ 2 ⎟⎠
g ∂P
+ .
=0
p ∂x
MATERIALS AND METHODS
The theoretical study reported in the literature
(Ostrowski and szmytkiemicz, 2006; Eyo, 2007b)
considers a streamline wave motion in which an
instantaneous displacement of a fluid particle from the
undisturbed level applied to Lagrange’s equation to the
fluid particle to obtain:
∂ ⎛ q2 ⎞
∂
⎜ ⎟ − wPw = −
P
∂x⎝ 2 ⎠
∂x
P
where, q2 = u2 + w2 by multiplying (1) by
by
d
d
d
=u
+w
dt
∂x
∂z
Since the fluid is incompressible, there is a streamline
functions * (x,z) such that:
U =−
(2)
−w
∂ P
w∂ P
=−
∂z
∂ x∂ z
2
∂δ
∂δ
, w=
∂z
∂x
Also, the density P is conserved so that P = P (*)
Equation (5) can be written as:
∂
and (2)
∂Z
∂w 1 ∂ P ⎛ d ⎛ q 2 ⎞ ⎞
d ⎛ q2 ⎞
⎜
⎟
u ⎜ ⎟ +w ⎜ ⎟
+
∂t p ∂δ ⎜⎝ ∂ x ⎝ 2 ⎠ ⎟⎠
∂z ⎝ 2 ⎠
∂
we obtain:
∂x
∂P ∂ ⎛ q2 ⎞ ∂P ∂ ⎛ q2 ⎞
. ⎜ ⎟+
. ⎜ ⎟
∂ Z ∂ x ⎜⎝ 2 ⎟⎠ ∂ x ∂ z ⎜⎝ 2 ⎟⎠
(5)
where,
(1)
∂ ⎛ q2 ⎞
∂P
⎜ ⎟ + wPw = −
− Pg
∂Z ⎝ 2 ⎠
∂Z
(4)
(6)
g d P dz
+ .
=
=0
p dδ dt
(3)
p and
dp
are not functions of time, hence (6) may be
dδ
written as:
188
Res. J. Environ. Earth Sci., 3(2): 187-192, 2011
d
dt
⎧
⎞⎫
1 ∂p ⎛ q2
⎪
⎪
⎜⎜
+ gz⎟⎟ ⎬ = 0
⎨w +
p ∂δ ⎝ 2
⎠⎪
⎪
⎩
⎭
Clearly this equation is nonlinear and cannot be solved
easily by a mathematical analysis. There is, however, one
obvious model for which the equation can be linearised,
(7)
2
and this is one in which the term U ο p remains constant
with depth and the density is linear in Zo. Eq. (12) then
reduces to:
Integrating (7) and noting that:
∂ w ∂ u ∂ 2δ ∂ 2δ
w=
−
=
+
∂ x ∂z ∂ x2 ∂z2
∇ 2δ + Gοδ = 0
we obtain:
g
⎫
∂ p ⎧⎪ ( ∇δ )
⎪
+
gz
⎨
⎬ = H (δ )
dδ ⎪ 2
⎪
2
∇ 2δ +
⎩
where, Gο =
(8)
⎭
⎫
⎪
+ g Z0 − Z ⎬
⎪⎭
(
δ ( x , z ) = δ ( 0, z ) Coskx
(9)
)
where, k =
This result can be written in a different and useful form by
substituting Zo for the dependable variable *. If Uo does
not vanish anywhere, the relation:
d
1 d
=−
dδ
U ο dzο
(10)
or − k
{( )2 − 1} dzd ( LnU ο2 P)
1
∇ zο
2
(14)
(15)
U ο2 P
(
d
Z −Z
dzο ο
λ
δ+
2
then (15) reduces to:
∂ 2δ
Coskx + GοδCoskx = 0
∂z2
(17)
∂ 2δ
+ Gοδ = 0
∂z2
δ ( 0, z ) = ASinν z + BCosν z
(11)
)
(18)
where, A and B are constants. Applying the boundary
conditions:
With further substitution * = Zo – Z:
* = 0, when z = 0 then B = 0
dδ ⎫ d
1⎧
2
LnU ο2 P
⎨ (∇ δ ) − 2 ⎬
∂ z ⎭ dzο
2⎩
(12)
g ∂d
= 2
.δ
U ο P dzο
(
(16)
The solution of this equation is of the form:
ο
g
2π
k 2 Coskx +
leads to:
∇ 2δ +
gβ
= constant
U ο2
Assuming a periodic motion of the fluid particle, that is:
( )2
=
=
∂ 2δ ∂ 2δ
+
+ Gοδ = 0
∂ x2 ∂z2
1 ∂p
=
p dδ
⎧
1 ∂ p ⎪ Uο
w(δ ) +
⎨
p dδ ⎪ 2
⎩
∇ 2 zο +
U ο2
Solution of the wave equation: We generalized the
model (13) as follows:
where, H(*) is a function of * to be determined by the
conditions upstream. Thus for the height zo(*) of the
streamline * = cons tant and T(*) is the vorticity
upstream, the final result becomes:
∇ 2δ +
1 dρ
ρ dzο
(13)
)
For z = H, Eq. (18) becomes:
A Sin< H = 0
Since A cannot be zero, then:
Sin< H = 0
189
Res. J. Environ. Earth Sci., 3(2): 187-192, 2011
that is < H = n B (n = 1, 2, 3,...)
we then obtain:
(19)
υ 2 H 2 = n 2π 2 = Gο − k 2
and
(20)
i.e.,
2π
λ=
Therefore, Yx = Cos
(21)
⎛
n 2π 2 ⎞
⎜ Gο −
⎟
H2 ⎠
⎝
2
gβ
⎧
⎛ δ
⎞⎫ 1
− υ = ⎨Cos −1 ⎜
⎟⎬
⎝ A sin υz ⎠ ⎭ x
⎩
2
U ο2
2
⎫
gβ 1 ⎧
δ ⎞⎤
⎪⎡
2 ⎪
−1 ⎛
+
Cos
υ
x
=
⎟
⎜
⎨⎢
⎬
⎝ A sin υz ⎠ ⎥⎦
U ο2 x ⎪⎣
⎪
⎩
⎭
Also zο = z − δ = z − δ ( 0, z ) Coskx . Therefore,
zο = z − ASinυz Cos kx
Therefore,
(22)
U ο2 =
The constant A is an arbitrary amplitude.
Significance of n: The expression (----) give the
distribution of * at x = 0, From (19)< H = n B, and for n =
1 we obtain:
D=
<H=B
For which the distribution may be obtained to represent
the resulting waveform.
Drag =
The displacement of a fluid particle from an undisturbed
layer can be written as * = z ! zo,
But zo = z ! A SinLz Cos kx
K is a function of the wavelength and of the streamline
velocity Uo so that in (23) K was written as:
gβ
2
⎝ Uο
1
CD AS ρU ο2
2
(25)
(26)
CD AS ρ
gβx
(27)
2
2
⎧
δ ⎞⎫
−1 ⎛
2
⎟⎬ +υ x
⎨Cos ⎜⎝
A sin υz ⎠ ⎭
⎩
Solution by Fourier's method: In this solution, a linear
model for internal waves in a stratified fluid is considered.
The equations are generalized by Fourier's analysis in
order to consider motions past an obstacle on the surface.
In his approach to this problem Scorer (1949) considered
the infinite atmosphere. Sen-Gupta (1973) considered the
same problem but applied the theory to a linearized model
of stratified water for a laboratory scale experiment. We
obtained:
(24)
Therefore, from (23)
⎛
2
⎧
δ ⎞⎫
−1 ⎛
2
⎟⎬ +υ x
⎨Cos ⎜⎝
⎠
A
z
sin
υ
⎩
⎭
where, A = wave amplitude, As = projected area of the
body in the direction of the motion.
The values of * can be obtained by applying Fourier's
analysis to the linearized wave equation. This solution
takes into account the geometry of the mountain model
used in generating the waves in the fluid.
Therefore, * = A SinLz Cos kx = A sin L z cosk x (23)
δ = ASinυzCos⎜⎜
gβx
Substituting (25) in (26) we obtain:
RESULTS AND DISCUSSION
⎛ gβ
⎞
k = Gο − υ 2 = ⎜⎜ 2 − υ 2 ⎟⎟
⎝ Uο
⎠
δ ⎞
⎜
⎟
⎝ A sin υz ⎠
−1 ⎛
⎞
− υ 2 ⎟⎟ x
⎠
∇ 2δ + Gοδ = 0
Writing the transform;
⎛ gβ
⎞
1 ⎞
−υ2⎟
⎜
⎟ = CosYx , Y = ⎜
A ⎝ Sinυz ⎠
⎝ Uο
⎠
δ ⎛
δ for δ = δ ( x , z )
190
(28)
Res. J. Environ. Earth Sci., 3(2): 187-192, 2011
δ=
∫
∞
e −ikxδdx
The integral (36) was evaluated by the method of
contour integration (Scorer, 1949). From Cauchy’s
theorem, the integral round the closed contour C was as
follows:
(29)
0
hence (28) become transformed to:
d 2δ
− k δ + 2 + Gο δ = 0
dz
2
∫ f ( k )dk = 2πi × sum of the residues at the poles
(30)
e
inside the contour
The residue at the pole was calculated using the relation:
This has the solution:
δ = ASinυz + BCosυz
f (k) =
We now apply the boundary conditions
C
C
At z = 0,
At z = H
δ = 0 therefore B = 0
and we show that the pole exists where g2(k2). With the
assumption that k = k* at the pole, the residue was
computed from the relation:
M
∫
δ ( H ,k ) = ASinυH = e −ikxδ( N ,k ) dx = F ( k ) (31)
R=
0
Therefore, we obtain:
δ ( H ,k ) =
Hence
δ( z , k ) =
∫
∞
0
g1 ( K1 )
g2 ( K2 )
F ( K ) sin υz
Sinυz
(32)
F ( K ) sin υz ikz
e dk
SinυH
(33)
g1 ( k *)
g2′ ( k *)
It is assumed that the pole is of the first order, and.
that the formula is not general. A more rigorous
mathematical argument would not necessarily serve a
better purpose as the theory of this problem is at best an
approximation to the physical features encountered in
practice. From Cauchy's theorem:
∞
∫
δ( z , k ) = − α 2 e − kb +ibx
This integral was solved by choosing a suitable value for
F(k). Scorer has used a ridge profile of shape:
0
∞
δ( H ,k ) =
−a b
b2 + x 2
(37)
∫
δ( z ,k ) = − α 2 e − kb +ikx
2
(34)
+
This shape was mountain model used in the
experiment
(Obinabo,
1978;
Ndubuisi and
Obinabo, 2010). From above:
0
2 − k *b +ik * x
πia e
Sinυz
dk
SinυH
Sinυz
dk
SinυH
Sinυ * z
⎛ ∂υ *⎞
HCosυ * H ⎜
⎟
⎝ ∂k ⎠
(38)
*
,
( x > 0)
(39)
∞
− a 2b
δ( H ,k ) = 2
= − a 2 eikx − kh dk
2
b +x
∫
CONCLUSION
(35)
0
The analysis of the result by Fourier's method gave
rise to the values of the wave amplitudes which was
compared with the experimental results (Ndubuisi and
Obinabo, 2010). The results also show that the formation
of the rotors is a function of the wave amplitude, and they
tend to present some complications in the flow of the fluid
which may affect the values of the drag. the drag bears a
linear relationship with the wave amplitude.
This reduces the problem to the solution of the integral:
∞
δ( Z ,k ) = − a
2
∫
0
eikx − kh
Sinυz
SinυH
(36)
191
Res. J. Environ. Earth Sci., 3(2): 187-192, 2011
The study relates the dynamics of the fluid flow to
the drag and lift components of the resultant dynamic
force exerted on the body immersed in the fluid. Drag, the
resistance to motion, is the component of the resultant
force in the direction of the relative flow ahead of the
body, and the lift is the component normal to this
direction. The former, however, is the component we
were involved with in this study, and the steady state
solution for the displacement of the fluid particle for each
velocity variation was obtained by Fourier's analysis. This
took into account the geometry of the mountain model
developed by Obinabo (1978) to generate the waves in the
stratified fluid. The analysis of the drag coefficient of the
body was based strictly on the geometry of the mountain
model.
REFERENCES
Baddour, R.E. and H. Abbink, 1983. Turbulent
Underflow in a Short Channel of Limited Depth, J.
Hydr. Div. ASCE, 109(HY6): 722-740.
Drennan, W.M., 1992. Accurate calculations of stokes
water waves of large amplitude. Z. Angew. Math.
Phys., 43: 367-384.
Eyo, A.E., 2007a. Mathematical modelling of uniform
flow in three open channels. J. Nig. Ass. Math. Phys.,
11: 587-596.
Eyo, A.E., 2007b. Mathematical modelling and its impact
on dredging a rectangular open channel with
hydraulic jump. J. Sci. Eng. Tech., 14(2): 7375-7387.
Groves, M.D., 2004. steady water waves. J. Nonlinear
Math. Phys., 11(4): 435-460.
Jamaloddin, N., I.B. Samsul, S.J. Mohammad,
A.M. Waleed and M. Shahrim, 2005. Simulation of
waves and current forces on template offshore
structure. Suranaree J. Sci. Tech., 12(3): 193-210.
Long, R.R., 1953. Some aspects of flow of stratified
fluids. 1. a theoretical investigation. Tellus, 5: 42-58.
Lukomsky, V.P. and I.S. Gandzha, 2003. fractional
fourier approximations for potential gravity waves in
deep water. Nonlinear Proc. Geophy., 10: 599-614.
Naser, M.S., P. Venkataraman and A.S. Ramamurthy,
1980. Flow in a channel with a slot in the bed. J.
Hydr. Res., 18(4): 359-367.
Ndubuisi, O.L. and E.C. Obinabo, 2010. Experimental
evaluation of gravity wave effects in a stably
stratified open channel flow. Res. J. Environ. Earth
Sci., 3(3): 193-196.
Obinabo, E.C., 1978. Drag resistance caused by gravity
waves in a stratified fluid. B.Sc. Thesis, University of
Portsmouth, England, United Kingdom.
Ostrowski, R. and M. szmytkiemicz, 2006. Modelling
longshore sediment transport under asymmetric
waves. Oceanologia, 48(3): 395-412.
Oyetunde, B.S., 2007. On the effects of wave steepness
on higher order stokes waves. J. Nig. Ass. Math.
Phys., 11: 369-374.
Oyetunde, B.S. and E.O. Okeke, 2007. on the impact of
wave-current on stokes waves. J. Nig. Ass. Math.
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Phillips, D.M., 1977. On the generation of surface waves
by turbulent wind. J. Fluid Mech., 2: 417-452.
Polton, J.A., D.M. Lewis and S.E. Belcher, 2004. The
role of wave-induced coriolis-stokes forcing on the
wind driven mixed layer. J. Phys. Oceanogr., 35:
444-457.
Scorer, R.S., 1949. Theory of waves in the lee of
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ACKNOWLEDGEMENT
The authors gratefully acknowledge the cooperation
received from their colleagues in the Faculty of
Engineering and Technology, Ambrose Alli University,
Ekpoma, Edo State, Nigeria, for the success of this
research.
NOMENCLATURE
x, z
u, w
a, b
A
AS
*
*1
Z
Uo
Po
g
Space co-ordinates in two-dimensional system
Velocities in and directions respectively
Dimensions of the mountain model
Wave amplitude
Projected area of the mountain model in the
direction of motion.
Vertical displacement from undisturbed position
Boundary layer thickness
Height of streamline when undisturbed
Undisturbed stream velocity
Density in undisturbed stream
Gravitational constant
$o
Static stability =
G
Stability parameter =
K
Horizontal wave number =
H
H
8
w
)2
CD
D
Horizontal wave number
Depth of fluid model
Wavelength
Vorticity vector
Laplacian operator
Coefficient of drag
Drag force
1 ∂p
p0 ∂ Z0
gβ0
U 02
2π
wavelength
192