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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


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:

  1. 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.
  2. 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.
  3. 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:

  1. The free-flow limit, Qf, is found for the given tail-water head, Ht.
  2. 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.
  3. 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 (1997).

11.2.1 Type 13
11.2.2 Type 14

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