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

APPENDIX 1: LIST OF NOTATION

a	An averaging parameter used to compute the mean conveyance in a flow expansion or contraction
a i, b i	Limits of a tabulation interval in a cross-section table
A	Total cross-sectional area
A c	Flow area at critical depth
A if	Area for full flow in the culvert barrel at section i
B g	Width of a gate opening
B i	Channel width at section i at an underflow gate
	The ratio between the channel width at section i at an underflow gate and the gate-opening width
c (y 1)	Flood-wave celerity at water-surface height y 1
C c	Contraction coefficient of the full barrel area of a culvert giving the flow area at the vena contracta
Cco	Contraction coefficient for orifice flow at an underflow gate
Cd	Culvert discharge coefficient
C dc	Discharge coefficient for a critical contraction
C dfo	Discharge coefficient for free-orifice flow at an underflow gate
C dfw	Discharge coefficient for free-weir flow at an underflow gate
C dso	Discharge coefficient for submerged-orifice flow at an underflow gate
C dsw	Discharge coefficient for submerged-weir flow at an underflow gate
C i	A constant coefficient used when computing a nonuniform velocity distribution equal to a function of Manning's n and the cross-section boundary slope at line segment i
C u	Unit conversion for Manning's equation equal to 1.49 in English units and 1 in metric units
C wr	Weir coefficient for a roadway embankment
D	Maximum inside vertical dimension of a culvert barrel
D vc	Maximum inside vertical dimension of a culvert barrel at the vena contracta
E 1 f	Energy loss from the approach section (section 1) to the roadway crest when flow over the roadway is free of downstream effects
E 1 s	Energy loss from the approach section (section 1) to the roadway crest when flow over the roadway is submerged from downstream
E 143 i	Energy loss from the approach section (section 1) to section 43 for flow over a roadway crest at incipient flow submergence from downstream
E 143 s	Energy loss from the approach section (section 1) to section 43 for flow over a roadway crest when the flow is submerged from downstream
f c (C c)	Piezometric depth ratio to the maximum vertical dimension of a culvert as a function of the contraction coefficient
f CH ( )	Function that gives the weir discharge coefficient for high-head flow
f CL ( )	Function that gives the weir discharge coefficient for low-head flow
f DB (y)	Function that yields the flow through a dam breach for each water-surface height in the reservoir
f ec (X)	Function yielding the product of the energy-loss coefficient and the difference in the product of the velocity and the energy-flux correction coefficient from the downstream section to the upstream section of a flow expansion or contraction
f s ( )	Weir-submergence function
f t (y t)	Piecewise linear approximation in a function table of function f t at argument value y t
	The i th tabulated value in a function table
F	Froude number
F E	Total flux of kinetic energy through the cross section
	The flux of kinetic energy for line segment i of the cross section
Fg	Gate-opening Froude number for orifice flow at an underflow gate
F M	Total flux of momentum through the cross section
	The flux of momentum for line segment i of the cross section
g	Acceleration of gravity
h (s)	Local height of the water surface in a cross section at offset s
	Average local water-surface height between adjacent cross sections at offset s in a curvilinear channel
h g	Gate-opening height
	The ratio between the piezometric head at section i at an underflow gate and the gate-opening height
h I	Piezometric downstream head at incipient flow submergence measured from the roadway crest
	Piezometric head at section i of an underflow gate
h t	Downstream (tail water) head causing flow submergence at a weir
h w	Water-surface head on a roadway embankment, roadway crest, or weir
h 1 foll	Upstream piezometric head at section 1 (approach section) for an underflow gate at the lower limit of free-orifice flow
h 1 fwul	Upstream piezometric head at section 1 (approach section) for an underflow gate at the upper limit of free-weir flow
h 4 fwul	Downstream piezometric head at section 4 (departure section) for an underflow gate at the upper limit of free-weir flow
h 4 swso	The piezometric head at section 4 (departure section) for an underflow gate at the boundary between submerged-weir and submerged-orifice flow for a piezometric head at section 1 (approach section) midway between the piezometric head equal to the gate opening and the piezometric head equal to the upper limit of free-weir flow
H	Piezometric head on a weir or roadway embankment measured from the minimum point on the weir or roadway embankment
J	First moment of area with respect to the water surface
k A	Energy-loss coefficient for accelerating (contracting) flow
k D	Energy-loss coefficient for decelerating (expanding) flow
k ec	Energy-loss factor for flow expansion or contraction
k(s)	Local conveyance at offset s in a cross section of a curvilinear channel
K	Total channel conveyance
	Mean conveyance
K f	The mean full-flow conveyance in a culvert barrel
K 2 f	Conveyance at full flow at the culvert entrance (section 2)
K 23	The value of conveyance for a culvert that gives the correct barrel-friction loss computed in the steady-flow profile computations for flow types 2 and 3
L aj	The cumulative distance along the main-channel axis to the jth cross section
L ij	The cumulative distance along the ith flow line at the jth cross section
L(s)	Distance between two adjacent cross sections at offset s in a curvilinear channel
m	Number of subsections in a cross section
m s	Slope of a boundary line on the channel cross section
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 y 0
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 y 0
M r	Momentum flux for flow over the roadway at a culvert
n c	Number of cross sections considered in the computation of sinuosity
n i	Manning's roughness coefficient for subsection i of a cross section
n p	Number of surveyed points used to describe a cross section
n s	Number of line segments below the water surface in the cross section; used when computing a nonuniform velocity distribution
NDDABS	The allowable absolute deviation from normal depth in the computation of the water-surface profile in the CHANRAT command
NDDREL	The allowable relative deviation from normal depth in the computation of the water-surface profile in the CHANRAT command
q (s)	Flow per unit width in a cross section at offset s
q r	Flow over a roadway-embankment weir per unit width of roadway
q w (s)	Flow over a weir per unit width at offset s
Q	Total flow rate in the cross section
Q B	Flow rate in a culvert barrel
Q C	Critical flow rate determined for steady flow in a compact channel ( = = 1)
Q E	Critical flow rate determined from minimization of specific energy in a channel where 1
Q f	Free-weir or free-orifice flow at an underflow gate
Q i	Flow rate represented by conditions at line segment i in the cross section
Q M	Critical flow rate determined from minimization of specific force in a channel where 1
Q r	Flow rate over the roadway at a culvert
Q w (z w)	Total flow over a weir for a given upstream water-surface elevation
P e	Roadway embankment height
P i	The wetted perimeter for subsection i of the cross section
R (s)	Hydraulic radius at offset s in the cross section
R hg	Fraction of the gate-opening width over which the transition in contraction coefficient between weir flow and orifice flow is applied
s	Offset distance across a cross section
s B	Beginning offset for the top width in a cross section at a given water-surface height
s E	Ending offset for the top width in a cross section at a given water-surface height
S A	Appropriate slope to use when approximating a nonuniform velocity distribution for estimation of or ; should be the energy slope, S e, when estimating , and the friction (momentum) slope, S f, when estimating
S c	Critical slope in a culvert barrel
S e	The energy slope for flow in a channel
S f	The friction (momentum) slope for flow in a channel
S h (x 1, x 2 )	The correct volume of water between cross sections at locations x 1 and x 2 for a given water-surface height
S q (x 1, x 2 )	The correct momentum content of the flow between cross sections at locations x 1 and x 2 for a given water-surface height
S 0	Bottom slope of the channel, positive when the bottom elevation decreases in the downstream direction
T	Top width of the flow
T c	Top width of the water surface at critical depth
T i	Top width of the water surface for subsection i of the cross section
V (s)	Flow velocity at offset s in the cross section
V	Cross-sectional average velocity
V D	Velocity at a dam site after failure
V Di	Velocity at a dam site before failure
V	Velocity difference specified to define a continuous energy-loss function as flow reverses in the vicinity of a flow expansion or contraction
W (s)	Crest width of a weir represented at the local offset s
x	Distance along the distance axis for the channel (main-channel axis)
dx	Distance increment along the main-channel axis
dx (i)	Distance increment on flow line i in a curvilinear channel
x	Distance between cross sections
x 12	Distance between the approach section (section 1) and entrance section (section 2) of a culvert
x 23	Length of a culvert
X	The difference in the product of the velocity and the energy-flux correction coefficient from the downstream section to the upstream section of a flow expansion or contraction
X f	A general designation of parameters on which the first moment of area with respect to the water surface is functionally dependent
y (x)	The height of the water surface above the minimum point in the cross section at the location given by x
y c	Critical depth (water-surface height)
y n	Normal depth (water-surface height) for a given flow
y s	The stopping water-surface height for computation of the water-surface profile in the CHANRAT command
y t	A nontabulated argument value for which a function value is sought by interpolation from a function table
	The i th argument value tabulated in a function table
y 1 fw	The limiting water-surface height at section 1 (approach section) of an underflow gate for free-weir flow for a given gate opening
y 2 max	Maximum water-surface height at the culvert entrance (section 2) for culvert-flow type 1
y 4 f	The water-surface height at section 4 (departure section) of an underflow gate at the free-flow limit for a given gate opening
y 4 sw	The limiting water-surface height at section 4 (departure section) of an underflow gate for submerged-weir flow for a given gate opening
z (s)	Elevation of the cross section boundary at offset s
	Minimum bottom elevation at section i of a culvert
z ci	Elevation of the culvert invert at the downstream end
z e	Total-energy line elevation
z i	Boundary elevations of the surveyed points used to describe a cross section
z j	Bottom elevation at section j of an underflow gate relative to the datum for the stream system
z m	Elevation of the minimum point on the cross-section boundary
z w	Water-surface elevation
	Maximum water-surface elevation at section 1 for culvert-flow type 1
z 3 p	Elevation of the piezometric surface at the exit section (section 3) of a culvert
dz e	Differential increment in the total-energy line elevation
z	Difference in bottom elevation between sections 2 and 3 of an underflow gate
	Kinetic-energy-flux correction coefficient
	Momentum-flux correction coefficient
µ	The slope of the sinuosity for a submerged line segment on the cross-section boundary
i	The sinuosity in the cross section applicable to the i th subsection
ij	The sinuosity at the j th cross section of the i th flow line
(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)
(y)	Escoffier stage variable used in the generalized Ritter dam-break solution