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Full Equations Utilities (FEQUTL) Model for the Approximation of Hydraulic Characteristics of Open Channels and Control Structures During Unsteady Flow

U.S. GEOLOGICAL SURVEY WATER-RESOURCES INVESTIGATIONS REPORT 97-4037

2.2 Two-Dimensional Function Tables


One-dimensional tables are limited to one argument. Therefore, 1-D tables can only represent a limited range of functions. Two cases for representing more complex functions with 2-D function tables are provided in FEQUTL. Thus, two types of function tables are computed in FEQUTL. The concepts underlying the two cases are presented in detail in section 11 in the documentation report for the Full Equations model (Franz and Melching, 1997). Only a brief outline is given here.

In the first case, represented with tables of type 13, a decrease (or drop) in piezometric head in the direction of flow always results across the control structure from the upstream, approach section to the downstream, departure section. In tables of type 13, the flow through the structure is listed as a function of piezometric head upstream from the structure in the approach section and the drop in piezometric head across the structure. Thus, in tables of type 13, flow is a function of upstream and downstream piezometric head (equal to the upstream piezometric head minus the drop) referenced to the stream-system datum. When this drop in piezometric head across the structure becomes large enough for a fixed upstream piezometric head (HUP), a critical control will result at some point in the structure. Further increases in the drop for the given upstream piezometric head will not increase the flow through the structure. The value of the drop in piezometric head across the structure at the point where further increases in drop have no effect is called the drop to free flow or just the free drop (FDROP). Free flow is the flow rate when the tail-water (downstream) piezometric head no longer affects the flow through the structure. Given the free drop for a specified upstream piezometric head, the flow for a range of drops that are smaller than the free drop, called partial free drops, is computed in FEQUTL. These partial free drops are further normalized by dividing by the drop to free flow, and the resulting normalized, partial free drop is abbreviated as PFD. Thus, a PFD of 0.0 implies that the flow is zero, and a PFD of 1.0 implies that the flow is equal to the free flow (the flow when the drop is the free drop).

The following is an example of a cross-section table of type 13 as output from FEQUTL for input to FEQ.

Table

NHUP is the number of upstream piezometric heads and NPFD is the number of normalized, partial free drops. The values of the normalized, partial free drops considered are listed below the PFD in the example table. The values of the fixed upstream piezometric heads considered are listed in the row beginning with HUP. The values of the free drop for a fixed upstream piezometric head are listed in the row beginning with FDROP immediately below the corresponding upstream piezometric head. The drop is computed as the product of PFD and FDROP, and the downstream piezometric head is computed as HUP - (PFD x FDROP). The numbers in the table are listed in a compact notation with the integer after the plus or minus sign giving the power of 10 to apply to the number that precedes the sign. For example, the maximum upstream piezometric head, the last number in the row that starts with HUP, is 1.0. The free flow at an upstream head of 0.8 appears at the bottom of the next to last column of the function table and is 26.43. The drop to free flow from this upstream piezometric head is in the second to the last entry in the row that starts with FDROP and is 0.5293. Values not tabulated are interpolated linearly during FEQ simulation as a function of upstream piezometric head and proportion of free drop. If the drop between the upstream piezometric head and the downstream piezometric head exceeds the free drop at the given upstream piezometric head, then the control structure is in a free-flow state and the flow is interpolated in terms of upstream piezometric head and free flow. The example table has been shortened relative to the table size normally used in FEQ simulation and illustrated in the example data files that may be obtained by electronic retrieval as described in section 1.1. In an application to a stream system, more upstream piezometric heads and more partial free drops would be computed so that linear interpolation would yield a better approximation to the variation of the flows.

In the second case, represented with tables of type 14, the downstream piezometric head is fixed and the upstream piezometric head is computed for a range of flows. Thus, in tables of type 14, upstream piezometric head is a function of flow and downstream piezometric head. When the flow is zero, the two piezometric heads are equal. As the flow increases from zero, the piezometric heads will differ. Eventually a flow will be reached for the fixed downstream piezometric head (HDN) that results in a critical control at some point in the structure. The upstream piezometric head at that point is at the free-flow limit. The upstream piezometric head for a series of partial free flows is then computed in FEQUTL. The flows are normalized by dividing by the free flow (QFREE) so that a partial free flow (PFQ) of 0.0 results in zero flow and a PFQ of 1.0 results in the free flow for the specified value of downstream piezometric head.

The following is an example of a table type 14 as output from FEQUTL for input to FEQ.

Table

NHDN is the number of downstream piezometric heads, and NPFQ is the number of partial free flows listed in the table. The numbers in the line beginning with QFREE are the flow rate required to yield free flow at the downstream piezometric head listed immediately below in the line beginning with HDN. For a given downstream piezometric head, a flow greater than the QFREE value is a free flow, and in this case the relation between flow and upstream piezometric head must be applied in the interpolation. The upstream piezometric head for each of the values of QFREE appears in the last line of the table except at a PFQ of 1.00. For example, at a downstream piezometric head of 9.6, the upstream piezometric head at free flow is 11.72 and the free flow is 699.9. As an example, a downstream piezometric head of 9.6 and a flow of 840.4 are used. The flow would be computed as free in FEQ simulation because the value of the flow argument, 840.4, is larger than the critical flow at the free-flow limit for a downstream piezometric head of 9.6. The upstream piezometric head sequence at free flow and the free flow would be used to determine the upstream piezometric head for the given flow. As an example of table interpretation, if the downstream piezometric head is 9.60 and the flow is 237, a partial free flow of 0.3386 is obtained. For this flow and downstream piezometric head, the upstream piezometric head is 10.09.

A third type of 2-D table, applied only to supply parameters to FEQUTL for the CULVERT and UFGATE commands, is a table of type 10. This table represents an input parameter for the CULVERT or UFGATE command as a function of two arguments by tabulating the parameter values at the intersections of a rectangular grid defined by the two arguments. Linear interpolation is applied in both the row and column directions for computing values intermediate to the values listed in the table. The arguments and function values can be any value needed in FEQUTL calculations. The file, TYPE5.TAB, required as input to FEQUTL in the FTABIN command and available by electronic retrieval as described in section 1.1, contains 10 examples of this table type. One of these example tables of type 10 is given below.

Table

The line of alphanumeric strings in quotes following the TYPE designation is the definition of an optional series of formats for processing the table. The first string is the format for the input of the column heading in the table. The second string is the format for the output of the column headings in the output from FEQUTL for checking purposes. The third string is the format for processing any numeric value in the column heading line. The fourth string is the format for processing a row of the table. The fifth and last string gives the format for the output of a row of the table in the output from FEQUTL. If this format information is omitted, each column in the table has six characters available on the row. In either case, the number of columns for function values is given by the parameter MCTD10 in the file ARSIZE.PRM associated with FEQUTL (appendix 2). The current value for MCTD10 is 9, which sets a maximum of 10 columns for the table including the argument value for each row of the table. The example table of type 10 lists the maximum ratio between successive arguments to a simple power function, axb, such that linear interpolation for the function between arguments will have a relative error less than the value given to the right of the column heading POWER. For example, if the power in the simple power function is 0.5 and the desired maximum relative error is 0.02, then the arguments at the end of the interpolation interval must have a ratio less than or equal to 2.2406. Linear interpolation in both the row and column argument is utilized for intermediate values. The value of -1 in the last line indicates the end of the table. All input for tables of type 10 is predefined, and user-generated input of this table type is not required. This information is provided so that users may examine the tables of type 10 included in TYPES5.TAB available by electronic retrieval, as described in section 1.1, in case problems result during simulation.


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