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

Purpose: The hydraulic characteristics of an underflow gate (sluice gate or tainter gate) are computed in the UFGATE command and are placed in a series of 2-D tables each for a given gate setting referenced in a table of type 15. A detailed discussion of the computational procedures is given in section 4.8.

LINE 1

Variable: TAB

Format: 7X, I5

Example: TABLE *#=* 527

Explanation:

TAB is the table number of the table of type 15 to be computed. The gate opening in distance units is the argument, and the table number of the 2-D table of type 13 for that gate opening is the function value in this table. Key values for defining the boundaries between weir flow and orifice flow also are included in this table. |

Variable: LABEL

Format: 6X, A50

Example: LABEL

Explanation:

LABEL is a descriptive label for the table of type 15. Each of the 2-D tables computed for the table of type 15 contains this label, or as much of the label as will fit, given with the prefix of the gate opening. |

Variable: APPTAB

Format: 7X, I5

Example: APPTAB

Explanation:

APPTAB is the table number of the cross-section function table for the approach section of the gate. Table types 22 and 25 are suitable. The invert elevation for this cross section is determined from the elevation given in the table. |

Variable: DEPTAB

Format: 7X, I5

Example: DEPTAB

Explanation:

DEPTAB is the table number of the cross-section function table for the departure section of the gate. Table types 22 and 25 are suitable. The invert elevation for this cross section is determined from the elevation given in the table. |

Variable: SILLZ

Format: 9X, F10.0

Example: SILLELEV

Explanation:

SILLZ is the elevation of the gate sill. This elevation must not be less than the elevation of the departure section nor of the approach section. This elevation defines the datum for measuring heads. |

Variable: GATEW

Format: 8X, F10.0

Example: GATEWID

Explanation:

GATEW is the width of the gate. This value must be |

LINE 7

Variable: CD

Format: 3X, F10.0

Example: CD *=* 0.98

Explanation:

CD is the discharge coefficient to approximate losses in the approach reach. A value of 1.0 implies no losses. Losses are generally small, and values much smaller than 1.0 should be carefully reviewed. |

Variable: CCTAB

Format: 6X, I5

Example: CCTAB

Explanation:

CCTAB is the table number of the function table providing the coefficient of contraction for the gate. For sluice gates, the argument is the ratio of the gate opening to approach head. For tainter gates, the argument is the acute angle, measured in degrees, that the gate lip makes with the horizontal. This table is optional. If this table omitted, the user provides a contraction coefficient in the table below. If contraction coefficients are not given below, then CCTAB must be given. |

Variable: FWFOTR

Format: 9X, F10.0

Example: FWFOTRAN

Explanation:

FWFOTR is the proportion of the current gate opening over which a linear variation is applied to the contraction coefficient from 1.0 at the upper limit of free-weir flow to the value at the lower limit of standard free-orifice flow. Free-orifice and submerged-orifice flows in this interval are called nonstandard because they have a nonstandard value of contraction coefficient. |

Variable: MAXHU

Format: 8X, F10.0

Example: MAXHEAD

Explanation:

MAXHU is the maximum upstream head (at section 1) for flow through the gate. This is utilized in FEQUTL to determine the range of heads appearing in the 2-D tables. This head is measured from the sill of the gate. |

Variable: MINHU

Format: 8X, F10.0

Example: MINHEAD

Explanation:

MINHU is the minimum nonzero upstream head (at section 1) for flow through the gate. This is utilized in FEQUTL to determine the range of heads in the 2-D tables. |

Variable: LIPREC

Format: 10X, F10.0

Example: PRECISION

Explanation:

LIPREC is the linear interpolation precision required in FEQUTL computations defining the series of upstream heads. The precision is expressed in terms of the absolute value of the relative error. Thus, a value of 0.02 indicates that the absolute value of the interpolation error for the free flow in the 2-D tables computed in the UFGATE command will be less than or equal to 0.02 times the free-flow value for heads between the minimum and maximum head given in Lines 10 and 11. This is an approximate criterion; the errors may occasionally be larger than the precision requested. Precision values less than 0.001 and greater than 0.1 are not supported. |

This value also is used to define the sequence of partial free drops for the 2-D tables. The selected precision will only be met approximately because the variation of submerged flows is more complex than the variation of free flows.

The spacing of the upstream heads and the spacing of the
partial free drops is based on the accuracy of linear
interpolation applied to a simple power function. A simple
power function is *ax ^{b}* where

(122)

where *r* = *x* _{1} / *x
* and 1 / *R* _{1} <= * r *<= 1. If
*R* _{1} and *b* are fixed, the relative
error is defined and the maximum relative error in the interval
is a function of *R* _{1} and *b*. The
special file required in FEQUTL computations, TYPE5.TAB,
contains two 2-D tables of type 10 (table numbers 10001 and
10002) that define the value of *R* _{1} as a
function of *b* and the maximum absolute value of the
relative error. TYPE5.TAB may be retrieved electronically as
described in section 5.1. Thus, for simple power-function
interpolation, the relative accuracy of linear interpolation is
defined by the ratio of the function argument at the end points
of the interval of integration.

This method is applied in the UFGATE command for assignment of the upstream heads and the partial free drops to the 2-D tables of type 13 computed. For example, free-weir flow varies approximately at the 1.5 power of the head on the sill of the gate opening. Free-orifice flow varies approximately at the 0.5 power of the difference between the head at section 1 and the head at the vena contracta of the jet emerging from the gate opening. However, the drop to free flow for free-orifice flow appears to vary more like the head difference to the 1.5 power. The spacing ratio for the 1.5 power is smaller than for the 0.5 power. Consequently, the 1.5 power is applied to define the spacing. Submerged flows, at higher levels of submergence, vary at the 0.5 power of the partial free drop. However, as the degree of submergence becomes smaller, the power changes and can become very large at small levels of submergence for submerged-orifice flow. Key values of upstream head are always present and override the simple power-function spacing.

The effect that changing the value of PRECISION has on an example underflow gate is listed in table 7. The gates described in table 7 are the tainter gates at Lock and Dam 21 on the upper Mississippi River near Quincy, Ill. Each of these 10 gates are 20 by 64 ft. The table gives the size of the 2D table computed for a gate opening of 4 ft. The root-mean-square (RMS) error is computed for submerged flows from the values midway between the partial free drops used in the computation of the table. Thus, the information listed in table 7 is not a measure of the average error of interpolation; rather, it is a measure of the average error of interpolation at the points of interpolation where the error tends to be the largest.

LINE 13

Variable: HEAD

Format: A80

Example: Opening 2-D Table Cc Value Lip Angle

Explanation:

HEAD is a descriptive heading for the gate-opening table. |

LINE 14 (Repeated as needed for the range of computed gate
openings.)

Variables: HG, TAB2D, CCVAL, ANGLE

Format: F10.0, I10, 2F10.0

Example: 0.5 560

Explanation:

HG is the gate opening in feet or meters. TAB2D is the table number for the 2-D function table that is computed for that gate opening. CCVAL is an optional value. If CCVAL is given, the value in CCTAB (Line 8) is overridden. ANGLE indicates that a tainter gate is represented if a nonzero value is entered. For tainter gates, the contraction coefficient is primarily a function of the lip angle and only slightly dependent on the gate-opening ratio. |

The computed gate openings must be presented in increasing
order. The input is terminated by input of a gate opening that
is less than the preceding opening.

Interpolation between 2-D tables for tables of type 15 is based on heads relative to the gate opening. Each table must be computed to head values that permit interpolation in the interval between the two tables. This can result in large head values if care is not taken in the selection of the sequence of gate openings. For example, if the initial gate opening is 0.1 and the next gate opening in the sequence is 0.5, then the 2-D table for the gate opening of 0.5 must be computed to a maximum head five times larger than the maximum head for the 2-D table for a gate opening of 0.1. If the value of MAXHEAD (Line 10) is 30 for a gate opening of 0.1, the table for the 0.5 gate opening must be computed to a head of 150. This will be higher than the cross sections for the approach and departure section. These cross sections must be extended, most simply with vertical walls, so that the maximum head required can be computed. Thus, the ratio between adjacent gate openings in the gate-opening sequence should be two or less. This must be balanced by the number of tables that will be computed. One 2-D table of type 13 is computed for each gate opening in the sequence.

LINE 15

Variable: HEAD

Format: A80

Example: Partial free drop parameters

Explanation:

HEAD is the description for the next set of input. |

LINE 16

Variable: MINPFD

Format: 7X, F10.0

Example: MINPFD *=* 0.005

Explanation:

MINPFD is the minimum value of partial free drop to apply in computation of the submerged flows for the gate. |

If the minimum value is too small, excessive numbers of the partial free drop will be generated. Convergence problems may result in FEQ simulations of the flow through the gate if the flow through the gate is nearly completely submerged. When the flow is deeply submerged, the flow through the gate varies with the square root of the head difference across the gate. As this difference becomes small, the rate of change of flow with the change in head difference increases without bound. Thus, the first derivative of the flow with respect to each of the heads becomes very large. Also, the derivative changes rapidly because, in the limit as the head difference approaches zero, the rate of change of flow with change in head difference approaches infinity. Large derivatives such as these can sometimes prevent the iterative computations in FEQ from converging. This usually can be remedied by making the minimum partial free drop larger. The linear-interpolation approximation then reduces the computed first derivative at high levels of gate submergence relative to the computations with a smaller partial free drop. If the MINPFD is too large, the approximation to the flow at high submergence levels may be inaccurate. However, accurate measurement and computation of highly submerged flows is difficult. Therefore, high accuracy at high levels of submergence cannot be expected.

LINE 17

Variable: BRKPFD

Format: 7X, F10.0

Example: BRKPFD *=* 0.5

Explanation:

BRKPFD is the value of partial free drop at which variation of the submerged flows changes. The initial spacing of the partial free drops from MINPFD (Line 16) to BRKPFD is computed by application of a power of 0.5. Reasonable results up to a partial free drop of about 0.5 are usually obtained with this spacing. The power values for orifice flow will then begin to get larger and can become quite high as the partial free drop approaches 1.0. A value of 0.5 is a good starting value for BRKPFD and can be refined if needed on the basis of the results from the UFGATE command. |

LINE 18

Variable: LIMPFD

Format: 7X, F10.0

Example: LIMPFD *=* 0.99

Explanation:

LIMPFD is the limiting value of partial free drop less than 1.0. Orifice flow can decrease rapidly as the submergence of the emerging jet of water begins. It is helpful to have a value of partial free drop close to 1.0, so that this initial drop is isolated. The initial drop in flow tends to be large when the sluice gate width is close to the approach- and departure-channel widths. A starting value of 0.99 is suggested. If convergence problems at the gate result in FEQ simulations and these problems can be traced to the rapid change of the flow near the initiation of submergence, then it may be necessary to make LIMPFD smaller so that the drop in flow is spread over a larger interval. This makes the values in the table less precise in the reproduction of the computed flows for the underflow gate. However, the theory on which the submergence computations are based is less defined when the submergence levels are small. Therefore, additional computational smoothing at small submergence levels is acceptable if it facilitates convergence in FEQ simulation. |

LINE 19

Variable: FINPOW

Format: 7X, F10.0

Example: FINPOW *=* 2.0

Explanation:

FINPOW is the power to apply in computation of the spacing for the partial free drops from BRKPFD to LIMPFD. The value must be between 1.0 and 3.0. The display of the power in the UFGATE output provides a guide of the value to apply. The power will be changing continuously over this region of partial free drops. A power value commensurate with the local power near LIMPFD is a guide for the selection of FINPOW. The larger the power, the closer the spacing of the partial free drop values. |

User Output from UFGATE

A summary for each upstream head and for each gate opening is
output from the UFGATE command, in addition to the look up
tables placed in the function-table file. Each summary table
lists the partial free drop, the drop in water-surface level
from sections 1 to 4, the head at section 3 and section 4, the
flow type, the value of the contraction coefficient used (for
free-orifice and submerged-orifice flow), the discharge, and
the power of a simple power function fit to the variation of
discharge with partial free drop. The power can be used to
adjust BRKPFD and FINPOW to better approximate the flow under
the gate for different vertical gate-opening heights. An
estimate of the maximum relative error and the square root of
the average squared relative error (root-mean-squared, or RMS,
error) also are computed in UFGATE. The error is computed at
each point midway between the partial free drops defined by
PRECISION, MINPFD, BRKPFD, LIMPFD, and FINPOW with the
exception of the intervals between a partial free drop of 0.0
and MINPFD and between LIMPFD and 1.0. The error tends to be
near the largest value over an interval at the midpoint of the
interval. The maximum error can be larger, and sometimes
considerably larger, than indicated in the parameter PRECISION.
The maximum error seems to result for interpolation across the
boundary between submerged-weir and submerged-orifice flow
between the upstream head of the gate opening, *h _{g}*
, and the upstream head at the upper limit of free-weir flow,

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