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Full Equations (FEQ) Model for the Solution of the Full, Dynamic Equations of Motion for One-Dimensional Unsteady Flow in Open Channels and Through
Control Structures
U.S. GEOLOGICAL SURVEY WATER-RESOURCES INVESTIGATIONS REPORT 96-4240
11.2 Two-Dimensional Function Tables
One-dimensional tables are limited to one argument. Therefore,
only a limited range of functions can be
represented in 1-D tables. Two approaches to representing more
complex functions by use of 2-D function tables are provided in
FEQ. The concepts underlying the two tables are illustrated by a
hypothetical case of a hydraulic structure that is subject to
tail-water effects.
11.2.1 Type 13
The culvert shown in figure 37 is subject to
tail-water effects.
Tables of type 13 are developed as follows: Initially, the
headwater and tail-water surface elevations are identical, and no
flow results; then, the tail water is lowered at the same time
that flow rate is increased from the initial zero flow so that the
headwater level is maintained. The possible variation in flow for
this case is shown in figure 38. As the tail
water continues to
lower, the flow increases to maintain the headwater level. There
is a limit, however, to the continued increase of flow as the tail
water is lowered. At some point, called the free-flow limit, a
critical-flow or equivalent condition will be reached so that
further reductions in tail water do not result in increases in
flow rate to maintain a constant headwater level. The difference
between the headwater level and the tail-water level at the limit
of the effect of the tail water is called the drop to free flow or
the free drop. Free flow denotes flow free of tail-water effects.
If the relation between flow and tail water is assessed for
several initial water levels, a family of lines of constant
headwater level is obtained, as shown in
figure 39. The figure is
divided into two regions, one-node control and two-node control.
In the one-node control region, the water level at only one
flow-path end node is needed to define the flow. In the two-node
control region, the water levels at two flow-path end nodes are
needed to define the flow.
In the two-node region, a partial free drop is used to
facilitate interpolation. The dashed line that defines the
boundary between the one-node and two-node regions is the line
of complete or total free drop (the free-flow limit). The
horizontal axis where the flow is zero is the line of zero free
drop (that is, no difference between headwater and tail water).
Other lines of partial free drop are shown in
figure 39. For
tables of type 13, headwater head and partial free drop are
linearly interpolated in the two-node flow region. The flow as
a function of headwater head and the partial free drop is
tabulated in tables of type 13, the partial free drop varying
from 0.0 to 1.0. The free drop for each headwater level also
is tabulated in table type 13.
Given a headwater head, Hh > 0 and a tail
water head, Ht, the current flow rate is
calculated as follows:
-
The free drop for the headwater head is computed by linear
interpolation from the tabulated values of free drop and
headwater head by use of the principles and concepts
previously outlined. The free drop at Hh is
defined as df.
-
The actual drop, da = Hh - H
t, is computed. This drop must be
greater than or equal to zero in tables of type 13. If
da = 0, then the flow is zero. If d
a > df, the drop is
greater than the drop to free flow and the flow is under one-node
control. The free flow for the given headwater head is
determined. If da < df, then
the flow is under two-node control.
-
To determine the flow, the partial free drop, pD =
d a / d f,
is computed. Using the headwater head and the partial free
drop, the flow is computed by bivariate linear interpolation
in the tabulated values.
The requirement of a nonnegative drop from upstream to downstream
for table type 13 limits its application somewhat. At some
structures, notably expansions in flow, a tail-water head that is
higher than the headwater head may result because velocity head
recovers somewhat (kinetic energy converts to potential energy).
11.2.2 Type 14
A structure similar to that described in type 13 tables is shown
in figure 40, but in this example,
the downstream water level is
held fixed and the upstream water level is estimated. This is the
defining basis for tables of type 14. Potential variations of the
headwater head as the flow is varied are shown in figure 41.
Again, at some point as the flow is increased, critical or some
equivalent flow results. At that point, the curve for a given
downstream head must end because the relation among the three
variables does not apply. For each downstream head, a maximum
flow is assumed in type 14 tables above which the relation is not
valid. The flow is again called the free flow, and the headwater
head for this condition is the head at free flow. The relation
defined by the free flow and the headwater head at free flow
determines the one-node control relation for the structure. These
results are shown graphically in figure 42.
In the two-node
control region, heads are interpolated in the stream with the
downstream head and partial free flow.
Table lookup for type 14 is more complex than for type 13. The
direction of flow for type 13 was always from the higher
piezometric head to the lower piezometric head. This does not
apply for type 14 tables. The current flow rate determines the
direction of flow. However, if the flow rate is zero, as is
possible for some water levels, the direction of flow is
undefined. This situation could result in problems when flow
is initiated through or over a structure. In this case, the
flow is zero because the water has not yet overtopped the
minimum point of the structure. However, if the flow was zero
in the previous time step and one or both of the heads are
greater than zero in the current time step, free flow is
assumed and is computed in the direction of decreasing
piezometric head. This computed direction may be in error, but
the error does not matter: The flow has been estimated as
nonzero, and subsequent iterations in the solution for the
current time step will determine the correct flow direction and
magnitude.
The following steps illustrate the procedure for table lookup
in a table of type 14 when the special cases involving zero
flow have been excluded. If Q is the current flow
with Q > 0 and Ht is the
tail-water head, then the following steps define the headwater
head:
-
The free-flow limit, Qf, is found for the
given tail-water head, Ht.
-
If Q > Qf, then the flow is
under one-node control. The headwater head at the free-flow
limit is determined for Q by linear interpolation
in the tabulated values of limiting free flow and the
headwater head at the limiting free flow. Otherwise, step 3
is done.
-
The partial free flow, pF = Q / Q
f, is computed. The headwater head is interpolated
by use of bivariate linear interpolation on partial free
flow and tail-water head.
Tables of type 13 and 14 for some control structures are computed
in the utility program, FEQUTL (Franz and Melching, 1997). An
outdated 2-D table, type 6, equivalent to table type 13 with a
different format, is
supported in FEQ simulation. Table type 13 is preferred because
it uses less space and is easier to read.
If structures described by tables of type 13 are in parallel
alignment in the stream system, they may be
combined into a single table of type 13 because flow rates can
be added. Tables of type 14 cannot be combined to describe
parallel structures because values of head cannot be added.
Therefore, multiple flow paths between two nodes must be
represented explicitly if tables of type 14 are used.
Two-dimensional tables of type 10 are used for the CULVERT and
UFGATE command parameters in FEQUTL (Franz and Melching, 1997),
but tables of this type are not used in FEQ simulation.
Bivariate linear interpolation for a function of two variables
is applied in tables of type 10. Tables of type 10 are
mentioned here for completeness. Detailed discussion and
examples of tables of type 10 are given in
Franz and Melching
(written commun., 1997).
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11.2.1 Type 13
-
11.2.2 Type 14
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Last modified: Fri Nov 28 08:56:31 CST 1997