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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

13.15 Backwater Analysis Block--Backwater Table


Purpose: The initial flows and the starting depth at control points are specified with this block to estimate the initial conditions at all nodes in steady-flow analysis.

Heading: One line of user-selected information. The suggested string is BACKWATER ANALYSIS.

LINE 1 (one for each branch; repeated in combination with Line 2)

Variable: NBR

Format: 14X, I5

Example: BRANCH NUMBER=00001

Explanation:NBR is the branch-number identification for the flow-specification table for a branch. Branch numbers must be given in ascending order.

If NBR < 0 on Line 1, then

LINE 2 (one for each branch; repeated in combination with Line 1)

Variable: DISCH

Format: 10X, F10.0

Example: DISCHARGE=250.0

Explanation:This is the flow in the branch if the flow is the same for all nodes on the branch. If the flows change along the branch, then a more detailed input is required.

If NBR > 0 on Line 1, then

LINE 2

Variable: HEAD

Format: A80

Example: NOD1 NOD2 DISCHARGE

Explanation:This is a user-selected heading for subsequent information.

LINE 3 (repeated as needed to define a flow for each node on the branch)

Variables: ND1, ND2, DISCH

Format: 2I5, F10.0

Explanation:

ND1 is the first node in range of node numbers.

ND2 is the second node in range of node numbers.

DISCH is the discharge to use for ND1 through ND2.

The first and last node in the range must be given even if they are the same number.

LINE 4

Variable: HEAD

Format: A80

Example: BRAN CODE ELEVATION EXN

Explanation:These are user-selected headings for subsequent information.

LINE 5 (one for each branch)

Variables: BRA, CODE, ELEV, EXN

Format: 2I5, F10.0, I5

Explanation:

BRA is the number of the simulated branch. If BRA=0 and if CODE > 0, then the flow-path end node given in CODE is assigned the elevation of the flow-path end node given in EXN.

CODE is the code defining the operation to be done:

CODE=0 indicates that the starting elevation for the branch given by BRA is the sum of the elevation of the water surface at the flow-path end node specified in EXN and the value specified for ELEV. The value specified for ELEV in this case may not be an elevation. It will likely be an increment in water-surface elevation.

CODE not equal to 0 indicates that the starting elevation is taken from ELEV. The starting elevation may be a first guess to be refined in FEQ simulation. The following options are available:

CODE=1 indicates that the starting elevation is used as the final elevation. This is the method for setting a known fixed elevation.

CODE=2 indicates that the critical depth is computed at the downstream end of the branch by use of the elevation given in ELEV as a first estimate. The resulting elevation at critical depth is then used as the starting elevation for the branch.

CODE=3 indicates that the water-surface height is computed from a stage-discharge relation as specified in the Network-Matrix Control Block (section 13.6) for CODE=4, TYPE=1, 2, or 3, by use of the elevation in ELEV as a first estimate. The resulting elevation is then used as the starting elevation for the branch.

CODE= - 3 indicates that the water-surface height is computed from a stage-discharge relation specified at the free node given in EXN in the same manner as for CODE=3, BRA=0 in this case. With this value, the initial condition for this process is taken from the Free Node Initial Conditions Block (section 13.14).

CODE=5 indicates that critical depth is computed at the downstream end of the branch by use of the elevation in ELEV as a first estimate. The resulting water-surface elevation is compared with the elevation at the node specified in EXN. The elevation is then used at critical depth if it is greater than the elevation specified in EXN. Otherwise, the elevation at the node specified in EXN is used.

CODE=6 indicates that the upstream water-surface elevation is computed from the flow and the downstream water-surface elevation for a two-node control structure is defined by CODE=5, TYPE=6 in the Network Matrix Control Block (section 13.6). The source for the downstream water-surface elevation is given in EXN. This must be the downstream node for head for the two-node control structure. The ELEV value gives an estimate of the initial water-surface elevation at the upstream node of the control structure. If this upstream node is on a branch, the branch number is given in BRA and the water-surface profile for the branch is computed starting at the elevation found for the upstream node of the control structure. If the upstream node is a new input-style free node, the node label is given in BRA and the elevation found is assigned to the free node. If the upstream node is an old input-style free node number, the node number prefixed by a minus sign is given in BRA, and the elevation found is assigned to the free node. In this option, it is assumed that the flow at the flow node of the control structure is non-zero. If the flow is zero, the user will know the elevation and should supply it directly.

Line 5 is repeated for each branch in the stream system but not necessarily in branch number order. The order is defined by the requirement that the water-surface elevation be known at the downstream end of the branch. Thus, the first branch appearing in the input will be the most downstream branch, where a boundary condition will define the water-surface elevation directly or indirectly. The branches are then given in an order such that the result of the analysis of one branch will provide an elevation for one or more upstream branches. A value of - 1 in the BRA column terminates input for the backwater computations and indicates that all flow depths have been defined.


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