Open Channel Hydraulic Reference book: Floodplain Modelong using HEC-RAS - Haestad Methods Water Solutions Hydrology 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 Mass Density 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

Sg (fluid) = g fluid/ g water 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 Weight of the glycerin Mass density of glycerin Specific weight of glycerin Specific gravity of glycerin g = 9.81 ft/sec2, g w = 9800 N/m3.

Open channel flow

Terminology Free surface Prismatic Channel Non-prismatic Channel Open to atmosphere Prismatic Channel With constant x-section and bed slope (usually manmade channels are like that) Non-prismatic Channel Varies both in x-sectional shape and bed slope between 2 selected points along the channel length (usually natural channels are like that)

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 Prismatic: manmade channels such as sewers, drainage ditches, etc.) Non-prismatic: natural streams and rivers

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 Note: all three values of depth are essentially equal until the slope of the channel becomes quite steep. A slope is considered steep when there’s at least a one-percent difference between y & h (10 percent slope or 5.7 degree theta).

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

V= average channel velocity, Q= discharge, A = x-sec area 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 channel V = 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 Laminar Flow Turbulent 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 Unsteady Flow One-dimensional 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? Moving fluids possess energy by virtue of its Ability to do work? Moving fluids possess energy by virtue of its Velocity Position Pressure

Work done (Pressure energy) = Fxd = PAd = P(Ad) = P(Volume) = PW/ g Energy and Head 3 kinds of energies that can be stored in a waterbody Potential: due to elevation/position ‘Z’ (elevation above a fixed datum) PE = WZ= mgZ Kinetic: due to velocity/motion KE = mv2 = (W/g) v2 Pressure: amount of work done in moving the fluid element a distance equals to the segment’s length ‘d’ Force F = PA Work done (Pressure energy) = Fxd = PAd = P(Ad) = P(Volume) = PW/ g PE = WZ v = velocity of fluid P= pressure per unit area A = area Z= elevation above a fixed datum

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: Required? 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? = 10 + 0.5x82/32.2 +4lb/in2 /64.4 lb/ft3 x 144 in2/ft2 = 1

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 + v22/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 sections Q1 = 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 ;-)