OPEN CHANNEL HYDRAULICHydrology and Water Resources RG

REVIEW OF FLUID MECHANICS

Fluid mechanics Weight Mass Density Specific weight Specific gravity Hydrostatics Continuity equation Types of flow Energy and Energy Head Bernoulli’s Equation Flow through open channel

Properties of a Fluid Weight

W = mg (kN, lb) m = mass of fluid (kg, slugs) g = acceleration due to

gravity 9.81 m2/sec, 32.2 ft2/sec Mass Density

mass of the fluid per unit volume at a standard temperature and pressure

r = m/V (kg/m3, slugs/ft3) V = volume of fluid (m3, ft3) In the case of water, neglect the variation in mass

density and consider it at a temperature of 4oC and at atmospheric pressure; then r = 1,000 kg/m3

Properties of a Fluid Specific Weight

gravitational force per unit volume Units: kN/m3, lb/ft3 In SI units, the specific weight of water at a standard reference temperature of 4oC

and atmospheric pressure is 9.81 kN/m3

g = W/V Specific Gravity

ratio of the specific weight of a given liquid to the specific weight of pure water at a standard reference temperature

Units????Sg (fluid) = g fluid/ g water

Specific Gravity of water = ?

Problem? A reservoir of glycerin has a mass of

1,200 kg and a volume of 0.925 m3. Calculate

1. Weight of the glycerin2. Mass density of glycerin3. Specific weight of glycerin4. Specific gravity of glycerin

g = 9.81 ft/sec2, g w = 9800 N/m3.

OPEN CHANNEL FLOW

Terminology Open channel flow – any flow path with a free

surface (open to atmosphere) Can be classified as

Prismatic channel With constant x-section and a constant bed slope

Non-prismatic Varies in both the x-sectional shape and bed slope

between any two selected points along the channel length Atmospheric pressure acts continuously,

constantly and at every location on water surface therefore is neglected

X-section: natural channel & floodplain

Prismatic & Non-prismatic Channels

X-section for open channel flow

Open Channel Hydraulics Variables of open channel flow analysis Open channel flow classification based

on various criteria Time Depth Space Regime (subcritical or supercritical)

Depth of Flow Elevation difference between water

surface and deepest part of the channel

Channel top width & wetted perimeter

Channel Slope Difference in the channel invert

elevation between two locations divided by the distance between them

In prismatic channel the slope is often constant over a significant channel distance

Hydraulic depth & hydraulic radius Hydraulic depth: average depth across

the channel

Discharge & Velocity Discharge or flow rate: amount of water moving in a

channel or stream system Velocity: speed at which water moves in an open

channelV = Q/A

V= average channel velocity, Q= discharge, A = x-sec area Water movement adds kinetic energy to the system Channel velocity is not constant at any location Varies both horizontally and vertically for any given

channel cross-section Velocity near the channel banks is less than the

velocity in the center of the channel

Velocity Profile in channel x-sections

Flow Classification

Uniform vs. non-uniform Steady vs. unsteady flow One-dimensional vs. multidimensional

flows Gradually varied vs. rapidly varied Subcritical vs. supercritical

Types of Flow

Uniform Flow in which the flow velocity and depth do not change from point to

point along any of the streamlines otherwise it is called non-uniform or varied flow

Laminar Flow in which each liquid particle has a definite path and the paths of

individual particles do not cross each other Turbulent Flow

if each particle does not have a definite path and the paths of individual particles also cross each other, the flow is called turbulent

Types of Flow Steady Flow

in which the depth and velocity at a point are constant with respect to time

Unsteady Flow if Q is not constant

One-dimensional Flow flow, whose streamlines may be

represented by straight lines as opposed to curved lines

Subcritical & Supercritical Flow Classification is based on ratio of inertial to

gravitational forces at a stream location – Froude number

If Fr > 1 – flow is ‘supercritical’ and inertial forces dominate, associated with steeper slopes (high velocity and shallow depth)

If Fr < 1 – flow is ‘subcritical’ – gravitational forces dominate usually calm and tranquil –small slope usually in natural channels - (low velocity and high depth)

For Fr = 1 both depth and flow are call ‘critical’

HYDROSTATICS

Energy What is energy?

Ability to do work? Moving fluids possess energy by virtue

of its Velocity Position Pressure

Energy and Head 3 kinds of energies that can be stored in a

waterbody1. Potential: due to elevation/position ‘Z’ (elevation above

a fixed datum)PE = WZ= mgZ

2. Kinetic: due to velocity/motionKE = mv2 = (W/g) v2

3. Pressure: amount of work done in moving the fluid element a distance equals to the segment’s length ‘d’

Force F = PAWork done (Pressure energy) = Fxd = PAd = P(Ad) = P(Volume) = PW/ g

Total Energy Total Energy = Potential + Kinetic +

Pressure

TE =WZ + (W/g)v2 + PW/ g

Energy may be expressed as ‘Head’ divide by ‘W’ throughout Represents total energy per unit weight of

the fluid

Energy Head Total Head

H = Z + v2/g + P/ g

Z = Elevation Head (units of length)

v2/g = Velocity Head (units of length)

P/ g = Pressure Head (units of length)

Velocity head at a cross-section

Example? Given:

Water in a 6 in diameter pipe with a velocity of 8 ft/s

Fluid pressure is 4 lb/in2

Elevation of the center of the pipe above datum is 10 ft

Required? What is total energy head?

Bernoulli’s Equation

Bernoulli’s Equation – conservation of energy

During a steady flow of a frictionless incompressible fluid, the total energy (total head) remains constant along the flow path

Z + v2/g + P/ g = constant

Z1 + v12/g + P1/ g = Z2 + v2

2/g + P2/ g

Continuity equation Based on the conservation of mass Assumption: flowing fluids have constant mass density

(incompressible liquid) States that the quantity of liquid passing per time unit is

the same at all sectionsQ1 = Q2 = Q3= ….

OR A1V1 = A2V2 = A3V3 = ….

Q = flow discharge [m3/s]; V = average velocity of the liquid [m/s]; A = area of the cross-section [m2]; and 1, 2, 3 = the number of sections 1-3

THIS IS ALL ABOUT RG744 FALL SEMESTER 2013

GOOD LUCK ;-)