Watershed Sciences 6900 - Amazon Web...
Transcript of Watershed Sciences 6900 - Amazon Web...
Watershed Sciences 6900FLUVIAL HYDRAULICS & ECOHYDRAULICS
FORCES & FLOW CLASSIFICATION + ENERGY & MOMENTUM PRINCIPLES
Joe Wheaton
Slides by Wheaton et al. (2009-2014) are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License
TODAYโS PLAN A
I. Force Classification
II. Overall Force Balance
III. Take-Homes from Summary of Force Magnitudes
IV. Reynolds Number as a Force Ratio
V. Froude Number as a Force Ratio
I. Froude Number & Hydraulic Jumps
FORCES & FLOW CLASSIFCATION
From Dingman (2009)
WEโVE ALREADY DEALT WITH FORCESโฆ BUT:
โข Understanding the relative magnitudes of all forces in different situations gives us a basis for ignoring themโฆ (i.e. simplifying assumptions)
โข To keep comparison simple, consider:
Wide, rectangular channel (Y=R) with constant width (W1=W2=W) but
spatially variable depth
๐น๐ท = ๐น๐ ๐น = ๐ โ ๐
From Dingman (2008), Chapter 7
TODAYโS PLAN A
I. Force Classification
II. Overall Force Balance
III. Take-Homes from Summary of Force Magnitudes
IV. Reynolds Number as a Force Ratio
V. Froude Number as a Force Ratio
I. Froude Number & Hydraulic Jumps
FORCES & FLOW CLASSIFCATION
From Dingman (2008)
A SIMPLE FORCE CLASSIFICATION
๐น = ๐ โ ๐ a= ๐น
๐
Forces per unit massโฆ
From Dingman (2008), Chapter 7
TODAYโS PLAN A
I. Force Classification
II. Overall Force Balance
III. Take-Homes from Summary of Force Magnitudes
IV. Reynolds Number as a Force Ratio
V. Froude Number as a Force Ratio
I. Froude Number & Hydraulic Jumps
FORCES & FLOW CLASSIFCATION
What did Dingman Mean by Force Balance?
From Dingman (2008), Chapter 7
RECALLโฆ 1D FORM OF NAVIER-STOKES
โข Conservation of Momentumโฆ in 1D:
๐ โ ๐ = ๐น
๐๐๐๐ฅ๐๐ก+ ๐๐ฅ๐๐๐ฅ๐๐ฅ= โ๐๐
๐๐ง
๐๐ฅโ๐๐
๐๐ฅ+ ๐น๐ฃ๐๐ ๐๐ฅ
๐ = ๐ โ ๐1 โ ๐1 = ๐ โ ๐2 โ ๐2
Wide, rectangular channel (Y=R) with
constant width (W1=W2=W) but
spatially variable depth
SO IN TERMS OF a= ๐น
๐CLASSIFICATION
๐๐๐๐ฅ๐๐ก+ ๐๐ฅ๐๐๐ฅ๐๐ฅ= โ๐๐
๐๐ง
๐๐ฅโ๐๐
๐๐ฅ+ ๐น๐ฃ๐๐ ๐๐ฅ
๐๐๐ฅ๐๐ก+ ๐๐ฅ๐๐๐ฅ๐๐ฅ= โ๐๐๐ง
๐๐ฅโ1
๐โ๐๐
๐๐ฅ+ ๐น๐ฃ๐๐ ๐๐ฅ๐
๐๐ก + ๐๐ = โ๐๐บ โ ๐๐ + ๐๐
1D Momentum Equationโฆ
Rearranged in terms of our
classificationโฆ
Matched to Table 7.1
From Dingman (2008), Chapter 7
RECALL: DEFINITIONS OF UNIFORM & STEADY FLOW
โข Uniform Flow: if the element velocity at any instant is constant along a streamline the flow is uniform (i.e. convective acceleration ๐๐ข ๐๐ฅ = 0),.
โข Steady Flow: if the element velocity ๐ข at any
given point on a streamline does not change with time, the flow is steady (i.e. local acceleration ๐๐ข ๐๐ก = 0) .
From Dingman (2008) โ Chapter 4& 6
THREE MAIN OPEN CHANNEL FLOW TYPES
1. Steady Uniform Flow
2. Steady Nonuniform Flow
3. Unsteady Nonuniform Flow
๐๐ก + ๐๐ = โ๐๐บ โ ๐๐ + ๐๐๐๐บ + ๐๐ โ ๐๐ + ๐๐ + ๐๐ + ๐๐ก = 0
๐น๐ท = ๐น๐
โข Downstream forces (+): ๐น๐ทโข Upstream Forces (-): ๐น๐
๐๐บ โ ๐๐ + ๐๐ = 0 EQ 7.2
EQ 7.3
EQ 7.4
๐๐บ + ๐๐ โ ๐๐ + ๐๐ + ๐๐ = 0
โข Steady/Unsteady (Time)โข Uniform/Nonuniform (Space)
๐๐บ + ๐๐ โ ๐๐ + ๐๐ + ๐๐ + ๐๐ก = 0
๐๐บ + ๐๐ โ ๐๐ + ๐๐ = ๐๐
๐๐บ + ๐๐ โ ๐๐ + ๐๐ = ๐๐ + ๐๐ก
SO WHAT IS NAVIER STOKES?
โข Unsteady or Steady?
โข Uniform or Nonuniform?
๐๐๐๐ฅ๐๐ก+ ๐๐ฅ๐๐๐ฅ๐๐ฅ= โ๐๐
๐๐ง
๐๐ฅโ๐๐
๐๐ฅ+ ๐น๐ฃ๐๐ ๐๐ฅ
๐๐บ + ๐๐ โ ๐๐ + ๐๐ = ๐๐ + ๐๐ก
๐๐๐ฅ๐๐ก+ ๐๐ฅ๐๐๐ฅ๐๐ฅ= โ๐๐๐ง
๐๐ฅโ1
๐โ๐๐
๐๐ฅ+ ๐น๐ฃ๐๐ ๐๐ฅ๐
๐๐ก + ๐๐ = โ๐๐บ โ ๐๐ + ๐๐
โThe difference between the driving & resisting forces is acceleration.โ
๐น๐ท โ ๐น๐
๐น๐ท โ ๐น๐ = ๐EQ 7.4
TODAYโS PLAN A
I. Force Classification
II. Overall Force Balance
III. Take-Homes from Summary of Force Magnitudes
IV. Reynolds Number as a Force Ratio
V. Froude Number as a Force Ratio
I. Froude Number & Hydraulic Jumps
FORCES & FLOW CLASSIFCATION
RANGE OF FORCE VALUES
โข Downstream forces (+): ๐น๐ทโข Upstream Forces (-): ๐น๐
โข Body Forces
โข Surface Forces
TODAYโS PLAN A
I. Force Classification
II. Overall Force Balance
III. Take-Homes from Summary of Force Magnitudes
IV. Reynolds Number as a Force Ratio
V. Froude Number as a Force Ratio
I. Froude Number & Hydraulic Jumps
FORCES & FLOW CLASSIFCATION
From Dingman (2008)
REYNOLDS NUMBER
โข Can also be shown that ๐ ๐ โ๐๐
๐๐(ratio of turbulent
forces to viscous forces):
โข Soโฆ is it fair to say that Reynolds number is sort of a ratio of driving forces to resisting forces?
โข No! Why?
๐ ๐ โก๐โ๐โ๐
๐=๐โ๐
๐Ratio of โinertial forcesโ to viscous forces
๐๐๐๐=
ฮฉ2 โ ๐ โ ๐2
๐3 โ ๐ โ ๐๐2
=ฮฉ2 โ ๐ โ ๐ โ ๐
3 โ ๐=ฮฉ2 โ ๐ โ ๐
3 โ ๐=ฮฉ2
3โ ๐ ๐
TODAYโS PLAN A
I. Force Classification
II. Overall Force Balance
III. Take-Homes from Summary of Force Magnitudes
IV. Reynolds Number as a Force Ratio
V. Froude Number as a Force Ratio
I. Froude Number & Hydraulic Jumps
FORCES & FLOW CLASSIFCATION
From Dingman (2008)
TODAYโS PLAN A
I. Force Classification
II. Overall Force Balance
III. Take-Homes from Summary of Force Magnitudes
IV. Reynolds Number as a Force Ratio
V. Froude Number as a Force Ratio
I. Froude Number & Hydraulic Jumps
FORCES & FLOW CLASSIFCATION
From Dingman (2008)
FROUDE NUMBER
โข What are Dimensions?
๐น๐ โก๐
๐ โ ๐ 12
Ratio of velocity to
Celerity (๐ถ๐๐ค = ๐ โ ๐)
[LT-1]
[LT-2 L]1/2
[LT-2 L]1/2 =LT-1
WHAT DOES THIS RATIO MEAN PHYSICALLY?
โข If ๐น๐ < 1, flow is subcritical
โ Meaning that the celerity is > then the velocity; โด waves can propagate upstream
โข If ๐น๐ = 1, flow is critical
โ Meaning that the celerity is = velocity (rare transitional point)
โข If ๐น๐ > 1, flow is supercritical
โ Meaning that the celerity is < then the velocity; โด waves cannot propagate
upstream
๐น๐ โก๐
๐ โ ๐ 12
THE HYDRAULIC JUMP
โข Transition from supercritical flow to subcritical flow is a hydraulic jump
โข For a hydraulic jump to occur, there has to be a transition from subcritical to supercritical flow!
โข Characterized by:
โ Development of large-scale turbulence
โ Surface waves
โ Spray
โ Energy dissipation
โ Air entrainment
โ River can not maintain supercritical state over too long a distanceโฆ
WHY CAN YOU SURF IN A RIVER?
โข Explain it in terms of Froude Numberโฆ
โข Which way is water going? (kayakers canโt answer!)
TODAYโS PLAN B
I. Energy Principle in 1D Flows
I. The Energy Equation (ยง 8.1.1)
II. Specific Energy (ยง 8.1.2)
ENERGY & MOMENTUM PRINCIPLES (Applied to 1D flows)
From Chanson (2004)
TODAYโS PLAN
I. Energy Principle in 1D Flows
I. The Energy Equation (ยง 8.1.1)
II. Specific Energy (ยง 8.1.2)
ENERGY & MOMENTUM PRINCIPLES (Applied to 1D flows)
From Dingman (2008), Chapter 8
TOTAL ENERGY ๐ (FROM ยง4.4)
โข The total energy โ of an element is the sum of its potential energy โ๐๐ธ & its kinetic energy โ๐พ๐ธ:
โข Recall that โ๐๐ธ is the sum of gravitational potential energy โ๐บ and pressure potential energy โ๐ :
โข โด total energy โ is:
From Dingman (2008), Chapter 8
โ = โ๐๐ธ + โ๐พ๐ธ
โ๐๐ธ = โ๐บ + โ๐
โ = โ๐บ + โ๐ + โ๐พ๐ธ
EQ 8.1
EQ 8.2
Energy quantities are expressed as energy
[F L] divided by weight [F], which is
called head [L]
DEFINITON DIAGRAM FOR ENERGY IN 1D
โข Used for derivation of the macroscopic one-dimensional energy equation
โข Flow through reach is steady
From Dingman (2008), Chapter 8
TODAYโS PLAN B
I. Energy Principle in 1D Flows
I. The Energy Equation (ยง 8.1.1)
II. Specific Energy (ยง 8.1.2)
ENERGY & MOMENTUM PRINCIPLES (Applied to 1D flows)
From Dingman (2008), Chapter 8
TOTAL MECHANICAL ENERGY AT A CROSS SECTION
โข Based on what Ryan showed us, using the channel bottom as a reference point, at cross section ๐ we can
write the integrated gravitational head (AKA elevation head), ๐ป๐บ๐:
โข Where ๐๐ is the elevation of the channel bottom, and the integrated pressure head ๐ป๐๐ is:
โข Kinetic energy head ๐ป๐พ๐ธ for a fluid element with velocity ๐ข is given by:
From Dingman (2008), Chapter 8
๐ป๐บ๐ = ๐๐ EQ 8.3
๐ป๐๐ = ๐๐ โ ๐๐๐ ๐0 EQ 8.4
โ๐พ๐ธ =๐ข2
2 โ ๐EQ 8.5
VELOCITY COEFFICIENT ๐ถ FOR ENERGY
โข Because velocity varies from point to point n a cross section, we need a fudge factor ๐ผ๐ to account for this
variation, so the velocity head ๐ป๐พ๐ธ๐ is:
โ๐พ๐ธ๐ =๐ผ๐ โ ๐๐
2
2 โ ๐
EQ 8.6
From Dingman (2008), Chapter 8
A CONVENIENT OUTCOME OF BOX 8.2
Velocity Head ๐ป๐พ๐ธ tends to be at
least an order of magnitude smaller than pressure head ๐ป๐
๐ผ โ 1.3 good enough typicallyโฆ
From Dingman (2008), Chapter 8
TOTAL HEAD ๐ฏ๐ AT A CROSS SECTION ๐
โข โThe total mechanical energy-per-weight, or total head ๐ป๐, at cross-section ๐, is the sum of the gravitational, pressure and velocity heads:
๐ป๐ = ๐ป๐บ๐ +๐ป๐๐ +๐ป๐พ๐ธ๐โ = โ๐บ + โ๐ + โ๐พ๐ธ
๐ป๐ = ๐๐ + ๐๐ โ ๐๐๐ ๐0 +๐ผ๐ โ ๐๐
2
2 โ ๐
From Dingman (2008), Chapter 8
EQ 8.6
But, this is just at one cross section ๐โฆ What
about for this diagram?
APPLY ๐ฏ๐ BETWEEN TWO CROSS SECTIONS
โข What is the change in in cross-sectional integrated energy from an upstream section (๐ = ๐) to a downstream section (๐ = ๐ท)?
โข Where โ๐ป is the energy lost
(converted to head) per weigh of fluid, or head loss.
From Dingman (2008), Chapter 8
๐๐ + ๐๐ โ ๐๐๐ ๐0 +๐ผ๐โ๐๐
2
2โ๐=๐๐ท + ๐๐ท โ ๐๐๐ ๐0 +
๐ผ๐ทโ๐๐ท2
2โ๐+ โ๐ป
EQ 8.8b
๐ป๐ = ๐๐ + ๐๐ โ ๐๐๐ ๐0 +๐ผ๐ โ ๐๐
2
2 โ ๐ EQ 8.6
๐ป๐บ๐ + ๐ป๐๐ + ๐ป๐พ๐ธ๐ = ๐ป๐บ๐ท + ๐ป๐๐ท + ๐ป๐พ๐ธ๐ท + โ๐ปEQ 8.8a
This is the ENERGY EQUATION!
THE ENERGY EQUATION APPLIES WHERE?
โข Depth can vary!
โข Assumed that pressure distribution was hydrostatic (i.e. streamlines no significantly curved)
โข We call this steady gradually varied flow
โข This is focus of Chapter 9
โข Weโll use it in flume on Thursday!
๐๐ + ๐๐ โ ๐๐๐ ๐0 +๐ผ๐โ๐๐
2
2โ๐=๐๐ท + ๐๐ท โ ๐๐๐ ๐0 +
๐ผ๐ทโ๐๐ท2
2โ๐+ โ๐ป
EQ 8.8b
SOME IMPORTANT IMAGINARY LINES
โข The line representing the total potential energy from section to section is called the piezometric head line
๐๐ธ โก โ๐ป๐ทโ๐ป๐
โ๐=โ๐ป
โ๐= EQ 8.9
โข The line representing the total head from section to section is called the energy grade line
โข The slope ๐๐ธ of the energy grade line is the energy slope:
WOULD THE ENERGY EQUATION APPLY HERE?
โข Hint, energy loss..
โข Okay.. If it does apply, what would the energy grade line look like for a hydraulic jump?
TODAYโS PLAN B
I. Energy Principle in 1D Flows
I. The Energy Equation (ยง 8.1.1)
II. Specific Energy (ยง 8.1.2)
ENERGY & MOMENTUM PRINCIPLES (Applied to 1D flows)
From Dingman (2008), Chapter 8
SPECIFIC ENERGY ๐ฏ๐บ๐
โข Similar to ๐ป๐, just that datum is changed so that mechanical energy is measured with respect to the channel bottom instead of horizontal datum
โข Upshot is that elevation head ๐ป๐บ๐ โ 0 in ๐ป๐ = ๐ป๐บ๐ +๐ป๐๐ +๐ป๐พ๐ธ๐ , such that:
๐ป๐๐ = ๐ป๐๐ +๐ป๐พ๐ธ๐
๐ป๐๐ = ๐๐ โ ๐๐๐ ๐0 +๐ผ๐ โ ๐๐
2
2 โ ๐EQ 8.10
SPECIFIC HEAD APPLICATIONโฆ
โข Consider a flow of discharge ๐ in a channel of constant width ๐:
โข Which can be substituted into our definition of specific
energy (๐ป๐๐ = ๐๐ โ ๐๐๐ ๐0 +๐ผ๐โ๐๐
2
2โ๐):
โข If ๐ and width ๐ are constant, specific head ๐ป๐ only depends on flow depth ๐. However, ๐ป๐ is a function of both ๐ and ๐2, which means it can be solved with two different positive values of depth ๐!
๐ = ๐ โ ๐๐ โ ๐๐
๐ป๐ = ๐ +๐ผ๐ โ ๐
2
2 โ ๐ โ ๐2 โ ๐2 EQ 8.12
From Dingman (2008), Chapter 8
ALTERNATE DEPTHS VS. CRITICAL DEPTH
โข Fancy way of saying there are two depth ๐ solutions for every specific head ๐ป๐, except at ๐ป๐๐๐๐ , which is the
critical depth (i.e. ๐ป๐๐๐๐ = ๐๐ถ).
โข To find ๐๐ถ, find where derivative of ๐ป๐ is
equal to zero:
๐ป๐ = ๐ +๐ผ๐ โ ๐
2
2 โ ๐ โ ๐2 โ ๐2
๐๐ป๐๐๐= 1 โ
๐2
๐ โ ๐2 โ ๐3= 0
EQ 8.12
EQ 8.13
๐๐ถ =๐2
๐ โ ๐2
1 3
EQ 8.14
๐๐ถ =๐ โ ๐๐ถ โ ๐๐ถ
2
๐ โ ๐2
1 3
=๐๐ถ2
๐EQ 8.15
WHY IS IT CRITICAL DEPTH?
โข If ๐๐ถ = 1 =๐๐ถ2
๐, notice the similarity with
the Froude Number ๐น๐:
โข The minimum value of ๐ป๐ occurs when ๐น๐2 = 1 (i.e. when ๐น๐ = 1) or when flow is critical!
โข ddd
๐๐ถ =๐๐ถ2
๐ EQ 8.15
๐น๐ โก๐
๐ โ ๐ 12
HOW TO READ SPECIFIC HEAD DIAGRAM
โข How to read ๐๐, ๐โฒ, and
๐โฒโฒ?
โข Is this a hydraulic jump?
TODAYโS PLAN B
I. Energy Principle in 1D Flows
I. The Energy Equation (ยง 8.1.1)
II. Specific Energy (ยง 8.1.2)
ENERGY & MOMENTUM PRINCIPLES (Applied to 1D flows)
From Dingman (2008), Chapter 8
THIS WEEKโS LAB
โข We will do a complete lab assignment and calculations for a hydraulic jump produced by a sluice gate
โข Youโll get to put the energy equation to use
โข We will then play with different configurations of the flume to induce different hydraulics. We will play with:
โ Width constrictions
โ Different spillways
โ Altering Q & S