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Y. Kumar et al.
q l Lateral inflow
Q Flow discharge
S Vector comprising of A, g, q l , S 0, S f and V x
S 0 Slope of the channel
S f Slope of the energy line
U Vector comprising of A and V
V Velocity of flow
V x Component of the velocity of lateral inflow
y Distance from the water surface to the centroid of the area
1 Introduction
Flow in real world open channels is generally unsteady, with flow conditions varying
with respect to time. Unsteady flows are also called transients. From an engineering
perspective, a transient is referred as any pressure wave that is short-lived (i.e., not
static pressure or pressure differential due to friction or minor inflow loss) [15]. The
unsteadiness may be due to natural causes or due to human action. The unsteady
flow conditions are a function of time and space. [4].
The partial differential equations delineate the unsteady flows, and except in very
simplified cases, closed-form solutions are not available for these equations. Finite
difference methods are numerical methods and used for obtaining solutions of partial
differential equations using approximations for derivatives. The MacCormack finite
difference scheme is an explicit numerical technique for analyzing one-dimension
and two-dimensional unsteady flow problems [4].
The rapidly varied transient phenomenon in an open channel, generally known
as "surge," occurs when there is a sudden change in depth or discharge or both.
Such type of transients’ situations occurs during the sudden closure or opening of a
gate in flowing water. A surge with increasing depth is called positive surge while
one with decreasing depth is known as negative surge [15]. Over the past century,
hydraulicians and applied mathematicians studied positive surges. Mathematician
Barré de Saint-Venant [1], Boussinesq [3], Swiss professor Favre [6] and several
other researchers have discussed surge development [2, 6, 9, 10, 13, 14, 16, 18].
Garcia-Navarro and Saviron [7] uses MacCormack finite difference method as a
predictive numerical tool for simulation of unsteady open channel flow. Gualtieri
and Chanson [8] performed several experiments with six different gate opening using
acoustic droppler velocimetry (ADV) and non-intrusive devices for observation of
surge and experimental results matched with various theories. Leng and Chanson
[11] experimentally studied upstream propagation of surges and bores in a large
sized rectangular open channel with a smooth bed having Froude number ranging
between 1.1 and 2.3. Viero et al. [17] analyze experimentally positive surge propagation in slopping open channels and compare experiment results with 0D model
Y. Kumar et al.
q l Lateral inflow
Q Flow discharge
S Vector comprising of A, g, q l , S 0, S f and V x
S 0 Slope of the channel
S f Slope of the energy line
U Vector comprising of A and V
V Velocity of flow
V x Component of the velocity of lateral inflow
y Distance from the water surface to the centroid of the area
1 Introduction
Flow in real world open channels is generally unsteady, with flow conditions varying
with respect to time. Unsteady flows are also called transients. From an engineering
perspective, a transient is referred as any pressure wave that is short-lived (i.e., not
static pressure or pressure differential due to friction or minor inflow loss) [15]. The
unsteadiness may be due to natural causes or due to human action. The unsteady
flow conditions are a function of time and space. [4].
The partial differential equations delineate the unsteady flows, and except in very
simplified cases, closed-form solutions are not available for these equations. Finite
difference methods are numerical methods and used for obtaining solutions of partial
differential equations using approximations for derivatives. The MacCormack finite
difference scheme is an explicit numerical technique for analyzing one-dimension
and two-dimensional unsteady flow problems [4].
The rapidly varied transient phenomenon in an open channel, generally known
as "surge," occurs when there is a sudden change in depth or discharge or both.
Such type of transients’ situations occurs during the sudden closure or opening of a
gate in flowing water. A surge with increasing depth is called positive surge while
one with decreasing depth is known as negative surge [15]. Over the past century,
hydraulicians and applied mathematicians studied positive surges. Mathematician
Barré de Saint-Venant [1], Boussinesq [3], Swiss professor Favre [6] and several
other researchers have discussed surge development [2, 6, 9, 10, 13, 14, 16, 18].
Garcia-Navarro and Saviron [7] uses MacCormack finite difference method as a
predictive numerical tool for simulation of unsteady open channel flow. Gualtieri
and Chanson [8] performed several experiments with six different gate opening using
acoustic droppler velocimetry (ADV) and non-intrusive devices for observation of
surge and experimental results matched with various theories. Leng and Chanson
[11] experimentally studied upstream propagation of surges and bores in a large
sized rectangular open channel with a smooth bed having Froude number ranging
between 1.1 and 2.3. Viero et al. [17] analyze experimentally positive surge propagation in slopping open channels and compare experiment results with 0D model