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

APPENDIX 1:LIST OF NOTATION

a	Lower bound of the function region integrated in the weighted four-point scheme of numerical analysis
A	Total cross-sectional area
A [ x, y ( x )]	Total cross-sectional area at the location along the distance axis given by x
AC	Total cross-sectional area at critical flow
AO	Exit area of the outlet conduit for a variable-speed pump
Ap	The area of a submerged object projected on a cross section orthogonal to the approach velocity
As	Constant surface area for the reservoir assumed to be present at a dummy branch
Asi	Effective area of a storm-sewer inlet
	Derivative of the cross-sectional area with respect to distance x if water-surface height y is held constant
A	A small incremental area in the channel cross section
dA	The differential area taken as a vector normal to the control surface of the control volume
b	Upper bound of the function region integrated in the weighted four-point scheme of numerical analysis
b	Vector of residuals in the solution of a system of nonlinear equations with Newton's method
c	Wave celerity
cai	Coefficient for the power equation used to interpolate tabulated critical flows in tabulation interval i
cbi	Power for the power equation used to interpolate tabulated critical flows in tabulation interval i
C+	Upstream wave trajectory
C-	Downstream wave trajectory
CD	Dimensionless drag coefficient representing the drag resulting from a submerged object or obstruction in an open channel
CD (w)	Dimensionless drag coefficient for wind-shear stress
Cdu (pG)	The weir coefficient when flow is from the downstream node to the upstream node at a variable-height weir
CSW (l)	Side-weir coefficient
Cud (pG)	The weir coefficient when flow is from the upstream node to the downstream node at a variable-height weir
CL	Level of discharge or water-surface elevation monitored at a control point for gate operation
	Rate of movement of water-surface elevation or discharge (depending on which is monitored) at the control point
CLcp	Level of discharge or water-surface elevation monitored at a control point for pump operation
CLL	The lower limit of the null zone for simulation of gate or pump operation
	The minimum rate of movement of water-surface elevation or discharge (depending on which is monitored) at the control point toward the null zone for which the gate setting remains the same
CLU	The upper limit of the null zone for simulation of gate or pump operation
CS	Control surface of the control volume
CV	Control volume in open-channel flow
d	The piezometric head of the tail water on the crest of a weir
da	The actual head drop across a hydraulic structure represented by a table of type 13
df	The free drop at headwater head, Hh , for a hydraulic structure represented by a table of type 13
dL	Nominal upstream piezometric head on a variable-height weir
dR	Nominal downstream piezometric head on a variable-height weir
DC	The control structure sign
DD	Direction of flow specified by the user
DF	The sign the flow at the discharge node of a control structure must have
Dp	Pumping direction
Dq	The discharge node sign
DS	The system sign for a two-node control structure
DT	The transition sign for a flow expansion
E (Q, Zw)	Elevation of the total energy line for water-surface elevation, Zw , and flow, Q.
Es (Q, y)	Specific energy for flow rate, Q, and water-surface height, y.
fC (zw)	Function specifying critical flow at the upstream node of an abrupt expansion with a diversion channel
fcp (.)	Control function for a pump
ff (Qq)	The head losses resulting from separation at the entrance to the pump intake, flow resistance in the intake conduit, and flow resistance in the outlet conduit
fgc (pG)	Gate-crest-elevation function for a variable-height weir
fi(.)	Function specifying flow through structure i that conveys flow between the upstream and downstream nodes at a special feature in the stream system
	Function defining inflow to a channel over a side weir
fK (zw)	The conveyance function for the flow path between two nodes in the simple-conveyance option
fm ( zw)	Maximum-capacity function for a variable geometry, one-node control structure
	Flow function for a one-node control structure
fqb (t)	Boundary condition of flow as a function of time
fqs (hL, hR, wg)	Function defining the flow through the sluice-gate opening
fQ ( h1)	The flow rate delivered by a variable-speed pump operating at the standard operating speed against a head difference of Dh1
fQdu (hw)	Weir-flow function for flow from downstream node to upstream node
fQud (hw)	Weir-flow function for flow from upstream node to downstream node
fs (rh)	Submergence factor for flow over a variable-height weir
fSdu(.)	Function denoting the table of weir-flow submergence factors for flow from downstream node to upstream node
fSud(.)	Function denoting the table of weir-flow submergence factors for flow from upstream node to downstream node
ftw (t)	The tail-water condition for a simple pump at any time, t
ftwc (xtc)	Tail-water conversion function for a simple pump, where xtc equals the tail-water condition obtained from ftw (t)
fT (y)	Tabulated function value for argument value y
	Interpolated function value for argument value y
fT´(y)	Tabulated derivative of tabulated function value with respect to argument y
	Interpolated derivative of tabulated function value with respect to argument y
	Function defining the flow over a weir
fz (t)	Boundary condition of water-surface elevation as a function of time
f14(.)	A function that yields the head at the upstream node given the piezometric head at the downstream node and the flow at the flow node
	The drag rescaled by the upstream hydrostatic-pressure force at a submerged object or obstruction in the channel
Fa	Froude number for the flow approaching a submerged object or obstruction in the channel
FW	Froude number of the channel flow relative to the head on the side weir
FD	Drag from a submerged object or obstruction determined from measured drag coefficients
	Term included in the equation of motion representing drag from a submerged object or obstruction determined from measured drag coefficients
FDE	Drag from a submerged object or obstruction approximated from an equivalent energy slope determined from head-loss relations
FDEC	The equivalent drag combining drag resulting from submerged objects of obstructions and eddy losses resulting from channel expansions or contractions
Fp	Hydrostatic pressure force on the cross section
Fx	Forces acting on a control volume in the x direction
F(.)	The residual function
F(u)	Vector of values of the residual function for the equations of motion describing the stream system
Fij´(.)	Partial derivative of the residual function of equation of motion i with respect to variable j
FN (u)	Nonlinear function solved with Newton's method
FN´(u)	Derivative with respect to u of nonlinear function solved with Newton's method
	The value on the line tangent to FN, the point of tangency being û0
g	Acceleration of gravity
G	First point on the x-t plane affected by disturbances at XL and XR at time t0; lower boundary in time of the region of influence for XL and XR
	The average piezometric head in the source channel for a side weir
h (s)	Local flow depth at offset s in a cross section
	Upstream piezometric head relative to a head reference point for structure i at a special feature in the stream system represented with an explicit two-dimensional table
hM	Head at the middle node in the representation of flow over a side weir
	Downstream piezometric head relative to a head reference point for structure i at a special feature in the stream system represented with an explicit two-dimensional table
hSW (l)	Head on the side weir
hw	The piezometric head on a weir
h0	Base stage for the numerical experiments of Xia and Yen (1994) of the effect of on the momentum equation
h	The head difference between water ponded at the entrance to a storm sewer and the pressure head in the storm sewer
h*	A small value of the head difference used to linearize the computation of the maximum current rate of inflow to a storm sewer to avoid computational failure when h < h*
had	Head loss for accelerating or decelerating flow in channel contractions or expansions, respectively
hp	Head loss in a control volume in open-channel flow resulting from a submerged object or obstruction
HSW (l)	Height of the side-weir crest above the channel bottom
Hh	Headwater head for a hydraulic structure represented by a table of type 13
Ht	Tail-water head for a hydraulic structure represented by a table of types 13 and 14
I (t)	Inflow of water that enters a control volume over or through the sides of the channel
J [x, y (x, t)]	First moment of area with respect to the water surface at location x for water-surface height y at time, t
J(u)	Matrix of partial derivatives (Jacobian matrix) of the residual functions for the equations describing the stream system
	Derivative of the first moment of area with respect to the water surface with distance x if water- surface height y is held constant
k(s)	The conveyance per unit width in a cross section at offset s
k, k´	Correction coefficient for a nonhydrostatic pressure distribution in the nearly exact momentum equations
ka	Head-loss coefficient for accelerating flow in a channel contraction
kd	Head-loss coefficient for decelerating flow in a channel expansion
kp	Head-loss coefficient for the submerged object or obstruction at location xp
K	Total channel conveyance
KE	Head-loss coefficient for flow through an expansion corrected for flow direction
KF	Friction-loss coefficient applied at a dummy branch
KM	Mean conveyance in a computational element
Ko	Exit-loss coefficient for a variable-speed pump
Kvh	A correction factor applied to the approach-channel velocity head to account for the increase in energy head because of converging flow in the immediate vicinity of a variable-height weir
K+	The head-loss coefficient when the flow is from the upstream node to the downstream node in a flow expansion
K-	The head-loss coefficient when the flow is from the downstream node to the upstream node in a flow expansion
l	Distance measured along a side weir
L	Length of the side weir
L	Lower triangular matrix of the Jacobian matrix, J
LG	Length of an overflow gate
m	Bandwidth of a Jacobian matrix
mc	The number of control structures conveying water between the upstream and downstream nodes at a special feature represented with an explicit two-dimensional table
mD	The number of submerged objects or obstructions in a computational element for which drag may be estimated from measured drag coefficients
mp	The number of submerged objects or obstructions in a computational element for which drag must be estimated from an equivalent energy slope determined from a head-loss relation
M (Q, y)	Specific force for flow rate, Q, and water-surface height, y
MA (x, y0)	The weight coefficient that will result in a valid volume per unit length when multiplied with the cross-sectional area at location x for water-surface height y0
Mf	The weight coefficient to correct for the effects of channel curvilinearity on the friction term in the momentum equation
MF	Momentum flux through the cross section.
ML	Rate of change of the fraction of maximum discharge capacity for a simulated gate with the monitored level at the control point below the null zone
MQ (x, y0)	The weight coefficient that will result in a valid momentum content per unit length when multiplied with the total flow rate through the cross section at location x for water-surface height y0
MT	The weight coefficient that will result in a valid water-surface area per unit length when multiplied by the top width
MU	Rate of change of the fraction of maximum discharge capacity for a simulated gate with the monitored level at the control point above the null zone
n	Manning's roughness coefficient
ne	Number of nonlinear equations to be solved simultaneously with Newton's method
nj	Number of flow-path end nodes at a junction
nr	Relative operating speed of a variable-speed pump
ns	Pump operation speed
q (x, t)	Lateral inflow per unit length along the channel
qSW (l)	Rate of outflow per unit length of side weir
qw (s)	The flow per unit width in a cross section at offset s
Q (x, t)	The total flow rate through the cross section at location x at time, t
Q (x, y0)	The total flow rate through the cross section at location x for water-surface height, y0
QC	Critical flow rate determined for steady flow in a compact channel
Qcs	The flow through a variable-geometry, one-node control structure
QE	Critical flow rate determined from minimization of specific energy in a channel where 1
QEXi	The flow at the ith flow-path end node at a junction
Qf	Free-flow limit for a given tail-water head for a hydraulic structure represented by a table of type 14
QM	Critical flow rate determined from minimization of specific force in a channel where 1
QMAX	The maximum current rate of inflow to a storm sewer
QP	Pumping rate
Qq	Discharge at the discharge node for a control structure
Qqb	Discharge at an external boundary of the stream system
QSW	Flow over a side weir
QSWM	Flow over a side weir at the midpoint of a computational interval of the side weir length
Qw	Flow over a weir
p (t)	The opening-fraction function for a variable-geometry control structure
	The rate of change of the gate-opening fraction
pD	Partial free drop for a hydraulic structure represented by a table of type 13
pF	Partial free flow for a hydraulic structure represented by a table of type 14
	Absolute value of the maximum rate of change in the gate-opening fraction
pG (t)	Gate-position fraction for a variable-height weir
pI	The proportion of the function-table interval represented by the point of interpolation
P [x, y (x, t)]	The wetted perimeter of the channel at location x for water-surface height y at time t
PGF	The numerical approximation to the integral of the pressure, gravity, and friction terms in the momentum equation
PRH	Priority assigned for gate-setting changes when the level at the control point is above the null zone
PRL	Priority assigned for gate-setting changes when the level at the control point is below the null zone
PRN	Priority assigned for gate-setting changes when the level at the control point is in the null zone
r	The ratio of downstream water-surface height to upstream water-surface height at a submerged object or obstruction in the channel
rh	The ratio upstream piezometric head to downstream piezometric head for a variable-height weir
R (x, y)	Hydraulic radius for location x and water-surface height y
s	Offset distance across a cross section
sB	offset at the beginning of the wetted top width for the cross section
sE	Offset at the end of the wetted top width for the cross section
S	The volume of water between adjacent cross sections in the stream channel
Sh (x1, x2)	The correct volume of water between cross sections at locations x1 and x2 for a given water-surface height
Sq (x1, x2)	The correct momentum content of the flow between cross sections at locations x1 and x 2 for a given water-surface height
SR	Storage volume of a level-pool reservoir
S0	Bottom slope of the channel, positive when the bottom elevation decreases in the downstream direction
Sad	Equivalent slope corresponding to the eddy loss distributed over the length of the control volume for accelerating or decelerating flow in channel contractions or expansions, respectively
Sf	Friction (momentum) slope for flow in a channel
Sp	Slope introduced to represent the loss of head (mechanical energy) in a control volume in open-channel flow resulting from a submerged object or obstruction
Sw	Water-surface slope between two nodes applied in the simple-conveyance option, positive when the water-surface elevation decreases in the downstream direction
ST (z)	Storage capacity of a level-pool reservoir at water-surface elevation z
t	Time
tD	The time point that is down relative to the center of the rectangular box in the x-t plane used to define the four-point numerical solution scheme
tU	The time point that is up relative to the center of the rectangular box in the x-t plane used to define the four-point numerical solution scheme
t0	Initial time for unsteady-flow computations
ts	Time elapsed since the pump speed was changed
t	Time step in FEQ computations
T	Top width of the channel
TC	Top width at critical flow
	Derivative of the top width with respect to distance x if water-surface height y is held constant
u	Independent variable for a nonlinear function solved with Newton's method
u	Vector of unknown discharge and water-surface elevation values for the stream system
û0	Point near a root in the solution of a nonlinear equation solved with Newton's method
ûi+1	The root of the tangent line, which becomes the next approximation to a root of the nonlinear equation solved with Newton's method
U	Wind velocity
U	Upper triangular matrix of the Jacobian matrix J
v	Local velocity for a small incremental area, A, in a cross section
vx	Local velocity in the x direction
V	Cross-sectional average velocity
V	The velocity vector
Va	Cross-sectional average velocity in the approach reach upstream from a submerged object
v	The deviation of the point velocity from the average velocity
d	The volume differential
w	Vector product of the upper triangular matrix of the Jacobian matrix J, and the vector of unknown discharges and water-surface elevations for the stream system, u
wg	Sluice-gate opening distance
W	Weight applied in the weighted four-point scheme of numerical analysis
WA	Weight applied to integrals of area with respect to distance in the weighted four-point scheme of numerical analysis
WE	Weight applied to average water-surface elevation if an unmodeled side channel joins the channel
WSM	Time-averaged wind-stress term in the numerical approximation of the momentum equation
WT	Weight applied to integrals with respect to time in the weighted four-point scheme of numerical analysis
WX	Weight applied to integrals of conveyance with respect to distance in the weighted four-point scheme of numerical analysis
x	Distance along the distance axis for the channel
xD	Location of a drag-producing submerged object or obstruction in an open channel
xL	Location of upstream face of a control volume in open-channel flow
xR	Location of downstream face of a control volume in open-channel flow
xp	Location in an open channel of a drag-producing submerged object or obstruction for which drag is determined from an equivalent energy slope
x	Distance between cross sections in FEQ computations; also, length of computational element in steady- and unsteady-flow analysis
xc	Distance between the nodes in the simple-conveyance option
xCV	Length of a control volume in open-channel flow
xs	Small increment along the distance axis of the channel, applied to determine the weight coefficients that correct for the effects of channel curvilinearity
XL	Upstream point along a channel disturbed at time t0
XR	Downstream point along a channel disturbed at time t0
y (x)	The height of the water surface above the minimum point in the cross section at the location given by x
yG	The height of the ground surface above the storm-sewer inlet
yM	Mean value for the water-surface height in the control volume for a computational time step
yR	Maximum depth of water in a level-pool reservoir
y0	Constant water-surface height applied to determine the weight coefficients that correct for the effects of channel curvilinearity
y	The tabular interval of a function table
z	Height above the thalweg
	The average water-surface elevation in the source channel for a side weir
zhSW	The elevation of the datum for defining heads for flow over a side weir
	Head-reference point for special feature i represented with an explicit two-dimensional table
zlhs	The total energy head at the upstream node of a flow expansion
zm	The minimum elevation of a weir
zrhs	The total energy head at the downstream node of a flow expansion
zw	Water-surface elevation
	Average water-surface elevation between two nodes in the simple-conveyance option
	Water-surface elevation at a boundary node
	Water-surface elevation at a critical control
	Water-surface elevation at the head node for a control structure
z	Incremental height
zw	The change in water-surface elevation across a computational element
	Kinetic-energy-flux correction coefficient
	Momentum-flux correction coefficient
	Series of solutions used in empirical tests of method convergence
	A constant of proportionality applied in the Richardson extrapolation
	A measure of the size of the time and distance steps in empirical tests of method convergence
	The angle between the direction of the water velocity and the direction of a line normal to the small incremental area, A
	Density of water
a	Density of air
(s)	The rate of change of distance along the flow line at offset s to the rate of change of distance along the main-channel axis (sinuosity at offset s)
	The average shear stress exerted on the water by the channel boundary
	The angle between the direction of the velocity and the direction of the distance axis at the location of the cross section
	Angle between the downstream flow direction in the channel and the velocity of the wind
	A correction factor applied to the normal weir coefficient to approximate the discharge coefficient for each location along a side weir
	A correction factor applied to the normal weir coefficient to approximate the discharge coefficient for each location along a side weir computed at the average values at the upstream and downstream nodes