Hydrology Exam 2
rating curve
a graph of discharge versus stage for a given point on a stream, usually at gauging stations, where the stream discharge is measured across the stream channel with a flow meter The rating curve is usually plotted as discharge on x-axis versus stage (surface elevation) on y-axis
Streamlie
a line that is everywhere tangent to the velocity field streamline and pathline are identical for steady flows.
The velocity profile in channels most often is approximated by:
a logarithmic curve.
Pressure downstream
p2 = p1 + (dp/ds)(ds) s = s direction > positive pressure force in the upstream s direction > negative pressure force in the downstream s direction
Partial derivative
partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).
Viscosity is:
temperature dependent. a measure of how easily a fluid deforms or flows under an applied shear stress. given by the ratio of shear stress to velocity gradient at a point.
Specific energy (E [L])
the energy per unit weight of the flowing water relative to the stream bottom. If the elevation of the channel bottom remains constant, then because H is constant, E must also be constant.
Specific energy is:
the energy per unit weight relative to the channel bottom. E+zb=H equal to the sum of velocity head and water depth. E=U2/2g+h
Measurement of the height of the pool behind a broad- crested weir (relative to the crest of the weir) can be used to estimate the discharge in a channel because:
the flow is forced to be critical over the weir.
6) If the local acceleration of a flow is zero, then:
the flow is steady
The velocity of water (temperature, 20°C) through a pipe with a circular cross-section and a diameter of 0.02 m is 0.4 m s-1. Based on this:
the flow is turbulent. R=ρUL/μ=8000 the friction factor is 0.033. f=0.316R-1/4
Local acceleration
the rate of change in temperature with time at a fixed location dT/dx Steady flow: Velocity at a point does not change with time
Convective acceleration
the rate of change of temperature with distance at a fixed time DT/dt Uniform flow: Velocity does not change from point to point along the flow path
water over a step
the water surface drops and velocity increases as water flows over the step.
Water is flowing at a steady rate through a horizontal pipe of constant cross-section. Manometers indicate that pressure in the pipe drops in the downflow direction. Based on this we can say that:
there is a loss of head along the pipe.
When a steady flow of water through a horizontal pipe passes through an expansion (increase in pipe diameter),
velocity decreases. pressure increases.
12) If we know that flow through a pipe is laminar, then:
velocity varies with the square of pipe diameter (D ). U=-dp/dx(D /32μ)
The unit weight (γ)
γ of fluid is the weight per unit volume and is equal to the product of density and the acceleration of gravity, that is γ = ρg.
Specific discharge is:
he discharge per unit width (Q/w), equal to Uh in a rectangular channel.
Mean-Section Method - average velocity
(1) Divide the stream into a number of rectangular elements (2) Use current meter to measure speed of the flow in each rectangular. The velocity is approximately the average velocity for that rectangular. (3) Multiply the average velocity by the area of the rectangular. (4) Sum across the stream. (5) Divide the sum by total area of the cross-section.
weight of water
(F˚g) =(pdv)g (F˚g)=body force or weight [M L T−2] dV = volume = Ads [L3]. Friction (neglected here)
Barnes (1977) reports measurements for Esopus Creek at Coldbrook, New York. The cross-sectional area of flow is 1470 ft2, the hydraulic radius is 8.28 ft, the fall is 1.15 ft over 258 ft, and the discharge is 13,900 cfs (ft3 s−1). What is the value of Manning's n for this reach?
0.043
Flow occurs over a broad-crested weir of length 1.3 m. The pool height, hweir, is measured to be 10 cm. What is the discharge over the weir?
0.07 m3 s−1
If the water depth d in Figure 3.1 (textbook or note) is 1 cm, the water temperature is 20°C, the applied force F is 0.1 N, and the area of the floating plate A is 10 m2, what is the speed of the plate?
10 cm s-1 F/A=μ uplate/d μ=10-3Pa s
Hydrograph
A graph of river stage or discharge versus time at a time or stage (depth) at a time The discharge hydrograph is not measured directly, but is inferred from the stage hydrograph. Hydrograph peaks downstreamof a reservoir are smaller in magnitudeand delayed in timecompared to those on the same stream upstreamof the reservoir.
VIscosity.
A measure of a fluid's ability to resist deformation (motion under shear stress). The rate at which fluid deformation occurs, depends not only the shear stress, but also the fluid viscosity. decreases with increasing temp independent of pressure
Bernoulli's equation assumptions
Assume the fluid is both homogeneous and incompressible Assumptions satisfied? 1. Incompressible fluid 2. Homogeneous fluid 3. Flow steady with time 4. No friction 5. Apply along a streamline Streamline: from 1 to 2
Mean velocity:
Averaged velocity over all streamlines U=Q/A, Q: discharge rate at a cross-section and A: area of the cross-section
Density
Depends on Temp + Pressure
discharge / normalized discharge
Discharge generally increases downstream as drainage area increases when normalized by drainage area, peak flood discharge generally does decrease in the downstream direction owing to channel friction and storage
Froude Number
F = U/ (sqrt (gh)) F=1: critical flow F<1: subcritical flow F>1: supercritical flow
subcritical vs supercritical limb
Flow in Reservoir: Subcritical, slow and deep Flow after spillway: Supercritical, fast and shallow Subcritical flow: structure downstream Supercritical flow: structure upstream
Lagrangian method
Follow individual fluid particles as they move about and determine how the fluid properties associated with these particles change as a function of time u(t). The fluid particles are "tagged" or identified, and their properties determined as they move.
continuity equation
Q1 = Q2
Bernoulli's equation (relationships)
Relationship between pressure, velocity, and height (u^2 / 2g) + z + (P/pg) Pressure -upstream pressure: p1 force in the downstream direction: p1A -downstream pressure: p2 force in the upstream direction, p2A
continuum assumption:
The fluid is idealized macroscopically as being continuous throughout its entirety.
Eulerian method
The fluid motion is given by completely prescribing the necessary properties (pressure, density, velocity, etc.) as functions of space and time, u(x,y,z,t). From this method we obtain information about the flow in terms of what happens at fixed point in space as the fluid flows past those points.
hydrostatic equation
The pressure in a liquid at a given depth is called the hydrostatic pressure. This can be calculated using the hydrostatic equation: P = rho * g * d, where P is the pressure, rho is the density of the liquid, g is gravity (9.8 m/s^2) and d is the depth (or height) of the liquid.
Total acceleration
The total rate of change of temperature with time dT/dt = DT/dx + u(DT/Dt)
Shear stresses are zero in a fluid at rest.
True
4) A homogeneous fluid is one with a constant density.
True
Velocity in rivers
Velocity and discharge in rivers usually are related by a power function, which plots as a straight line on logarithmic paper. The equation of the straight line shown in the graph is U = 0.28Q0.62.
Reynolds number
a dimensionless number R which describes the flow properties: > R = (ULp)/viscositiy U = a characteristic velocity [L T−1]; (mean velocity for a hose) L = a characteristic length [L]; (diameter) ρ= fluid density [M L−3]; μ = viscosity [M L−1 T−1]. The break in the measurements at Reynolds numbers between 2000 and 4000 marks the transition from laminar to turbulent flow. in between is the transition region, The relationship between friction factor f and Reynolds number R for smooth pipes is measured in laboratory experiments over a range of Reynolds numbers.
7) The Bernoulli equation [equation (3.20)]:
assumes the flow is frictionless.
The equation that defines specific energy,
can be used to calculate the two allowable values for water depth for given values of E and qw.
10) The friction factor for a stream channel is essentially similar to that for pipe flow. On the basis of this identity, we expect that the friction factor for an open channel will:
decrease as a power of the Reynolds number. F=0.316R-1/4
Head Loss
depends on (1) viscosity of the fluid, (2) velocity of the flow, (3) hose diameter and (4) hose length. Because head loss also is a function of the roughness of the wall material itself, we must include a friction factor in its definition
flood routing
determination of river discharge at a point based on knowledge of the discharge at some upstream location(inflow) and the characteristics of the intervening river channel or reservoir.
Finding velocity of a fluid
difficult to do so for fluid, because different fluid particles move at different velocity at different time. > At a single time, two particles at two locations move with different velocity. > At a single point, a particle moves at different velocities at different times.
At critical flow mean water velocity is:
equal to the square root of g times h.
H, the total energy per unit weight of water flowing in a channel is:
equal to the sum of specific energy and elevation of the channel bottom. equal to the sum of the pressure head, elevation head, and velocity head. constant in channels in the absence of friction.
friction factor for smooth pipes
f = 0.316R^(-1/4)
Manning coefficient
function of bed roughness n=0.025 for sand bed n=0.075 for coarse, weedy bed U sand = 3U weedy
Consider steady, frictionless flow in a rectangular channel under subcritical conditions. A smooth vertical step decreases the bottom elevation over a short reach of the channel. There is no change in channel width. Which of the following statements are true?
he specific energy at the downstream station is greater than that at the upstream station. The specific discharge at the downstream station is the same as that at the upstream station. The water depth increases over the step. The flow at the downstream station is subcritical.
Consider again steady, frictionless flow in a rectangular channel under subcritical conditions. A smooth change in channel width occurs, narrowing the channel. There is no change in the elevation of the channel bottom. Which of the following statements are true?
he water depth decreases through the contraction. E =U2/2g+h
friction factor/head loss
hl = f (L/D)(U^2 / 2g) f = friction factor [dimensionless] L = length [L] D = diameter [L] hl = head loss
5) The pressure below the surface of a liquid:
increases with increasing depth. depends on the liquid density.
The friction factor:
is a function of the Reynolds number. depends on the roughness of the surfaces bounding the flow.
Pathline
is the line tracing out by a given particles as it flows from one point to another. The path line is a Lagrangian concept that can be produced in the laboratory by making a fluid particle and taking a time exposure photograph of its motion. streamline and pathline are identical for steady flows.
