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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 means to compute the hydraulic properties of a cross section including the correction factors for curvilinearity are provided in the CHANNEL command.

Notes: In the commands FEQX, FEQXLST, and FEQXEXT, the sinuosity is assumed to be zero unless a sinuosity value is specified in a CHANNEL command. Consequently, the input for the CHANNEL command is divided into two parts: the sinuosity-definition block, which defines the sinuosities for a series of cross sections, and the cross-section function-table definition block, which consists of a series of FEQX, FEQXLST, or FEQXEXT commands, one for each cross section involved in the sinuosity-definition block. An example CHANNEL command for a curved reach in a prismatic channel is the following.

The sinuosity-definition block starts with the input item SINDEF and ends with the END item. Then after the sinuosity-definition block, only the first two lines of input for the FEQX commands are shown in the example. The input for FEQX, FEQXLST, or FEQXEXT is unchanged if these commands appear within a CHANNEL command. The stations for the cross-section function tables in the FEQX commands are the stations along the axis of the sinuosity-definition block. The CHANNEL command ends with the ENDCHAN Line.

Each flow line in the sinuosity-definition block is given a name on the HEAD Line. Points along each of the named flow lines are defined by subsequent lines of input. Each of these input lines starts with a four-character identifier giving the nature of the data items on that line. The line identifiers are: HEAD--defining the names for the flow lines; OFFS-defining the offsets of the flow lines; STAT-defining the station for each flow line for a cross section; SINU-defining the numeric value of the sinuosity for a flow line at a cross section or defining how the sinuosity is computed; LENG-gives the incremental length along each flow line from the previous cross section; and END-signals end of the sinuosity definition table. These identifiers must begin in column 1 of the line. The identifiers and their order in the input are discussed in the line-by-line input description that follows.

In the FEQX, FEQXLST, or FEQXEXT commands within a CHANNEL command, the output of a table type that tabulates the correction factors for curvilinearity must be specified if these factors are to appear in the cross-section function tables computed. These factors are tabulated in table types 23, 24, and 25. The stations given for these tables in the cross-section computation commands must match the stations given for the longitudinal axis in the sinuosity-definition block.

Most of the input for the CHANNEL command is format free and the requested values need not appear in any particular columns. The exceptions are the standard line names, the SINDEF Line, and the FEQX, FEQXLST, or FEQXEXT command specifications that follow the sinuosity definitions. However, the input on a line that is format free is still order dependent, and some restrictions are placed on the order of the lines. Items on a line must be separated by one or more spaces, and all items for a line must appear on a single line. The items listed below for a given line cannot be split between two or more lines of input. Even though the format of much of the data is free of column restrictions, orderly columns should be utilized as shown in the previous example. Orderly columns make reading and checking the input much easier.

LINE 1Variable: SINDEF

Format: 7X, A8

Example: SINDEF=LINEAR

Explanation:

SINDEF is the global value of the method to be used in computing the sinuosity of each nonaxis flow line at each cross section. This must be the first line of input after the CHANNEL command is specified. The sinuosity is the derivative of the flow-line length relative to the length along the axis. An example of a flow-line distance function is shown in figure 17. The variation in flow-line length relative to the length along the distance axis also is shown in figure 17. Typically, the flow-line distance function will oscillate about the line of constant sinuosity (sinuosity=1.0). Three different options for approximating the derivative of the flow-line distance function are available in the CHANNEL command. These options are LINEAR, PARABOLA, and CUBIC. |

The derivative at a point like point *M* in figure 17 is
computed as the arithmetic average of the slope of the
piecewise-linear approximation to the flow-line distance
function in the LINEAR option. The slope (sinuosity) of the
line segment to the left of point *M* is

and the slope of the line segment to the right of point *M
* is

where

The sinuosity at point *M* along flow line *i*
computed in the LINEAR option is

This approximation is reasonable only at interior points such
as *M* in figure 17. At the first cross section, the
slope of the line to the right of the point *,
*, must be applied, and at the last cross section, the
slope of the line to the left of the point,
, must be applied.

The derivative at a point such as *M* is computed as the
derivative of the parabola fitted to the three points *L
*, *M*, and *R* in the PARABOLA option.
Again, at the first and last cross section, the derivative must
be taken at the beginning or ending point of the parabolic
segment.

The derivative at a point is computed as the derivative of a cubic spline fitted to three or more consecutive points on the flow-line distance function in the CUBIC option. Computation of a cubic spline requires specified endpoint conditions. The default option for the end conditions is to force the second derivative to be zero. This implies that the cubic function should approach linear variation as the end points are approached. The default end condition may be changed with the SINU input line.

LINE 2

Variables: CRDNAM, VALUES (*)

Format: A4, free

Example: HEAD LFP AXIS RFP

Explanation: The names for the flow lines that will be defined
on subsequent lines of input are specified on this line.

CRDNAM must be "HEAD" in the first four characters of
the first line of input after the SINDEF value is given.
VALUES are the labels for flow lines. One of the flow lines must be labeled AXIS or axis to identify the flow line that will be used as the distance axis. The other flow lines must be given a valid name of eight or less alphanumeric characters with the first character required to be alphabetic. The number of flow lines is defined by the number of flow-line names that appear on the HEAD Line. |

LINE 3 (Repeated in combination with Lines 4 and 5 as a group until the curvilinear portion of the channel is input. The STAT or LENG Line must be the first line for each cross section or incremental distance for each flow line.)

Variables: CRDNAM, VALUES (*)Format: A4, free

Example 1: STAT 0.0 0.0 0.0

Example 2: LENG 125.0 115.0 110.0

Explanation:

CRDNAM defines whether the station of a cross section or the
incremental distance from the
previous cross section are specified on this line. The STAT
Line must be given for the first cross section in a sequence
because the first cross section has no previous cross section
from which to define the incremental distance values for the
LENG Line.
VALUES are the values of station or incremental distance for each flow line. The units of the station or incremental distance must be the same for all cross sections in the CHANNEL command. The same units must be used for the station values given in the FEQX, FEQXLST, or FEQXEXT commands that follow the definition of the sinuosity. The order of the values is the same as the order of the flow-line names. The STAT or LENG Line must be the first line for each cross section in the sinuosity-definition block of the CHANNEL command. |

LINE 4 (Repeated in combination with Lines 3 and 5 as a group until the curvilinear portion of the channel is input. This line must be specified for each cross section in the sinuosity-definition block.)

Variable: CRDNAM, VALUES (*)Format: A4, free

Example: OFFS 0.0 10.0 20.0

Explanation:

CRDNAM must be OFFS to indicate that the offsets of the flow
lines in a cross section are specified on this line. This line
must be specified for each cross section in the
sinuosity-definition block.
VALUES are the values of the offsets for each flow line. The order of the values is the same as the order on the HEAD Line. The units must be the same as the units for the offsets in the cross sections used in the FEQX, FEQXLST, or FEQXEXT commands that follow the sinuosity-definition block. The number of offsets relative to the number of flow-line names determines the sinuosity variation as piecewise constant (PWC) or piecewise linear (PWL). If the number of offsets matches the number of flow-line names, then the sinuosity variation is PWL. If the number of offsets is one less than the number of flow-line names, then the sinuosity variation is PWC. Any other number of offsets is an error. Also, the sinuosity variation cannot change once specified. |

LINE 5 (Repeated in combination with Lines 3 and 4 as a group until the curvilinear portion of the channel is input. This line of input is optional.)

Variable: CRDNAM, VALUESFormat: A4, free

Example 1: SINU linearu 1.0 1.05

Example 2: SINU * * linearu

Example 3: SINU cubic

Explanation:

CRDNAM must be SINU to indicate input of either a local option
for computing the sinuosity or a local value of sinuosity.
VALUES are the values that are sufficient to define the sinuosity for the flow line or lines of interest. The sinuosity for the axis flow line is always 1.0, and any other value given is ignored. In the first example, a local option for computing the sinuosity is specified. The second example illustrates how a local sinuosity-definition option may be specified for one of the flow lines while the global option is applied for the other flow lines. The asterisk denotes the default value for the sinuosity definition. In the third example, the sinuosity definition is assigned to the first flow line given in the HEAD Line. Subsequent flow lines will take the global value from SINDEF. If the linearu option is specified, the slope of the line segment to the left (upstream) of the current cross section is used to define the sinuosity at this cross section. This option may be specified as linearu, LINEARU, linu, and LINU. The slope of the line segment to the right (downstream) of the current cross section is selected by application of lineard, LINEARD, lind, or LIND as the local option. Local options for a parabolic definition of the derivative are parabola, PARABOLA, parab, PARAB, paraboli, and PARABOLI. Local options for a cubic-spline definition of the derivative are cubic, CUBIC, cspline, and CSPLINE. If no local sinuosity-definition option is specified and no numeric sinuosity value is specified, then the sinuosity definition given by SINDEF is applied. Any values on the SINU Line are assigned in left to right order to the flow lines named in the HEAD Line. |

Variable: CRDNAM

Format: A4

Example 1: END

Example 2: ENDCHAN

Explanation:

CRDNAM equals END terminates the sinuosity-definition block, whereas CRDNAM equals ENDCHAN terminates the entire CHANNEL command. |

The following example of a CHANNEL command is presented below to illustrate some of the options not previously discussed.

This example of the CHANNEL command is adapted from one created in the CHANNEL option in the HEC2X command (section 5.15). In the HEC2X command, a HEC-2 input file is processed, the cross sections are extracted, and a sinuosity-definition table is computed with the data in the HEC-2 input file on the flow distances in the left overbank (LOB), the main channel (AXIS), and the right overbank (ROB). In this case, only two offsets result even though there are three flow lines. Thus, PWC variation is selected for sinuosity across a section. The offset values in this case give the offsets at the banks of the stream. The computation option CUBIC has been selected in the global sinuosity-definition option, SINDEF. A SINU line has been added for the first and last cross section. These lines override the default end condition of the cubic spline fit computed in the CUBIC option. At the first section, the sinuosity has been defined as 1.0 for all flow lines. This is appropriate if the stream channel at that point becomes essentially linear. The default end condition of the second derivative equal to zero may not produce appropriate values in this case. At the last section, the sinuosity has been defined by the average sinuosity in the last flow-line segment.

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