# Calculating resistance to flow in open channels

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Alternative Hydraulics Paper 2, 5 April 2010 Calculating resistance to flow in open channels Abstract The Darcy-Weisbach formulation of flow resistance has advantages over the Gauckler-Manning- Strickler form. It is more fundamental, and research results for it should be able to be used in practice. A formula for the dimensionless resistance coefficient was presented by Yen (1991, equation 30), which bridges between the limits of smooth flow and fully rough flow, to give a general formula for calculating resistance as a function of relative roughness and Reynolds number, applicable for a channel Reynolds number R> and relative roughness ε<0.05. In this paper, the author shows how a closely similar formula was obtained which is in terms of natural logarithms, which are simpler if further operations such as differentiation have to be performed. The formulae are shown to agree with predictions from the Gauckler-Manning-Strickler form, and with a modified form for open channels, have advantages that the equations are simpler and terms are more physically-obvious, as well as using dimensionless coefficients with a greater connectedness to other fluid mechanics studies. 1. Introduction The ASCE Task Force on Friction Factors in Open Channels (1963) expressed its belief in the general utility of using the Darcy-Weisbach formulation for resistance to flow in open channels, noting that it was more fundamental, and was based on more fundamental research. The Task force noted that the Manning equation could be used for fully rough flow conditions, however it presented a revealing figure for the variation of resistance with Reynolds number, which showed that with Manning s equation there is continual decay of resistance with Reynolds number, even in the limit of large values, so that one could deduce that it is fundamentally flawed.the recommendations of the Task Force have almost entirely been ignored, and the Gauckler-Manning-Strickler formulation continues to dominate, even though, with the exception of the Strickler formula, there are few general research results available, and a picture-book approach seems to prevail. The reason for the Darcy-Weisbach approach not being adopted may be that, even though the ASCE Task Force presented a number of experimental and analytical results, there was no simple path to follow for general problems. This document takes the two formulae that the Task Force presented for behaviour in the limits of a smooth boundary and fully rough flow, and obtains a general formula for resistance in open channels which plausibly interpolates between the smooth and rough limits, to give the resistance as a function of relative roughness and Reynolds number. The results are closely similar to the formula given by Yen (1991, equation 30).

2 2. Experimental results Consider the expression for the shear force τ on a pipe wall (e.g. 6.3 of White 2003) τ = 8 ρv2, where ρ is fluid density, the dimensionless Weisbach coefficient is a dimensionless resistance factor, and v is the mean longitudinal velocity in the pipe. The author has used this to obtain the long wave equations for a channel of arbitrary slope in terms of the coefficient (Fenton 2010). The ASCE Task Force on Friction Factors in Open Channels (1963) on p109 presented formulae for the Weisbach friction coefficient in two limits, fully rough and smooth flow. Consider their expressions, wherehereweusethesymbol: Fully rough: Smooth: 1 ³ a = c log 10 ε Ã 1 = c log 10 R o b where a, b, andc are numerical coefficients; ε = k s / (A/P ) is the relative roughness, which is the equivalent sand-grain diameter k s divided by the hydraulic radius A/P,inwhichA is cross-sectional area and P is wetted perimeter; and where R o =4Q/P ν,whereq is the discharge and ν is the kinematic viscosity, such that R o is the Reynolds number best suited to a circular pipe. In that case, if U is the mean velocity, and D is the pipe diameter, then R o =4U πd 2 /4/(πD) /ν = UD/ν. Below, we prefer to use the channel Reynolds number R = Q/P ν = UA/Pν = R o /4, in terms of the discharge per unit of wetted perimeter. Equation (1a) is explicit, giving the fully rough value in terms of relative roughness, however equation (1b) is implicit, as appears on both sides of the equation. As presented, neither of them can handle more general situations. A number of results of investigations were presented by the Task Force, some experimental, others based on analytical integration of logarithmic profiles over typical sections, providing values for the numerical coefficients a, b,andc. It presented the results shown in table 1.! (1a) (1b) c b a Bretting Colebrook Keulegan, trapezoidal Keulegan, trapezoidal Keulegan, wide channel Keulegan, wide channel Sayre & Albertson Thijsse Thijsse Tracy & Lester rectangular Tracy & Lester Tracy & Lester Zegzhda Table 1. Results presented by ASCE Task Force on Friction Factors in Open Channels (1963) 2

3 3. Explicit formula for smooth channels Here we seek to unify the results and to provide an explicit formula for valid in both smooth and rough limits. Prior to doing this, we obtain an explicit approximation to equation (1b), similar to that of Colebrook (1938, equation 14) for pipes. To do this we consider values of b and c from table 1, and adopt c =2.0 for simplicity and a value of b =3.0, representative of the values in the table. We write the Colebrook-type expression µ 1 = c 0 Ro log 10, (2) and solve equation (1b) numerically for for five values of R O =10 4, 10 5, 10 6, 10 7,and10 8, and use a least-squares method to find the optimal values of b 0 =7.90 and c 0 =1.79 in equation (2). Rounding slightly, a quite accurate explicit expression for for smooth boundaries is 4. General formula = log 2 10 b 0 µ Ro 8.. (3) Now rewriting the two limiting expressions (1a) and (3), using the adopted value of c =2., the channel Reynolds number R = Q/P ν = R o /4, and the representative value of a =12from the experimental and theoretical results of table 1 gives Fully rough: = 1 ³ ε log 2 10 (4a) 12. Smooth: = 1 µ Ã µ log 2 10 = 1! 0.9 R log 2 10, (4b) R where in the last equation we have taken one more step. Now, to incorporate both limiting behaviours in a single equation, from equations (4) we write the expression which simply combines the limiting forms: Ã = 1 µ! ε log (5) R In the limit of a fully rough flow, for large Reynolds number R, this gives equation (4a). In the other limit of a smooth boundary, ε =0, it gives equation (4b). There have been several presentations of such a simple uniting formula that encompasses such limiting behaviours, as described by Yen (1991, pages 14-15) for pipe flows and for wide channels, due to Churchill, Barr, and Swamee and Jain. Haaland (1983) showed that for pipes a similar equation, but with different exponents of the terms in the argument of the logarithm, gave good results for all intermediate values of roughness and Reynolds number also, agreeing well with the well-known implicit equation ofcolebrookandwhite(equation4,colebrook 1938). In fact, the equation developed here is almost exactly equation (30) of Yen (1991): = 1 4 log 2 10 µ ε R 0.9, (Yen 1991, eqn 30) with the exception that in our Reynolds number term the numerator appears in Yen s expression for wide channels as He stated, however, that the value would be smaller for narrower channels, when also the coefficient of 12 in the roughness term would be larger. It is an interesting coincidence that the author did all the calculations leading to (5), requiring certain arbitrary choices of numerical coefficients, before he discovered the existence of the Yen formula, with almost-identical numerical values. The author, however, did not succeed in arriving at the more-careful qualifications of Yen, who stated that it was applicable for R> and ε<0.05, which limitations we also accept. 3

4 Now we convert to natural logarithms for simplicity in case further operations such as differentiation have to be carried out, for example in calculating kinematicwavespeed,togivetheformulahererecommended for use in calculating the resistance coefficient: 1.33 = Ã µ ln 2 ε !, (6) R in which the term 1.33 is actually (ln 10) 2 / In case they are necessary, the derivatives are ε = Θ ln 3 Θ and R = R 1.9 Θ ln 3 Θ, (7) where Θ = ε/12. +(2./R) 0.9. We mention here another similar formula that can be deduced with similar limiting behaviours: µ ³ ε 12.. (8) = 1 10/ log R but which is not mathematically identical to equation (5). Haaland (1983) adopted this approach for pipes and found that it was within 1-2% of the implicit law of Colebrook and White, which is better known in the form of the curves on the Stanton-Moody diagram (see, for example, White 2003). Colebrook and White s original expression is accurate only to some 3-5%, even if equivalent roughness size were accurately known, and so the level of accuracy of the approximation is reasonable. However, in the case of channels, with different types of roughness and greater physical variability, in addition to the problem of defining equivalent sand roughness, equation (8) would not be expected to give similar high accuracy. In view of this, we choose the alternative where the roughness term is simpler, as in open channels it is more important than the Reynolds number term, so that we will recommend equation (5) or (6). However we mention the other form and in the following section compare its performance to give a feel for the effects of the two different arbitrary approximate expressions. 5. Results Equations (5) or (6) Equation (8) Relative roughness ε = k s /(A/P )= Smooth Channel Reynolds number Q/P/ν 10 8 Figure 1. Dependence of on relative roughness and channel Reynolds number Figure 1 shows on a modified Stanton-Moody plot the behaviour of the expressions presented here, in terms of the actual cross-sectional area and wetted perimeter of a channel, and not the equivalent pipe diameter. The abcissa is the channel Reynolds number R = Q/P/ν, and the parameter for the various 4

5 curves is the relative roughness expressed in terms of the hydraulic radius, ε = k s /(A/P ). It can be seen that the two families of curves, from equations (6) and (8), show some disagreement. However, as these have been obtained from smooth wall results with ε =0and completely rough results R,andare just a plausible means of incorporating both, it cannot be claimed in this case that one formulation is superior to the other. However, for channel applications, equation (6) is slightly simpler. We do not have general detailed experimental results for verification, however the spirit of the approximation is close to Haaland s approach, which worked very well for pipes. In most open channel applications the Reynolds number is large enough that its effect is small compared with that of relative roughness, nevertheless Yen s equation and our variant of it here is a way of incorporating it. The result, whether Yen s formula (30) or our slightly modifiedversionoftheyenformula, expression (6), is a formula of some generality which might be of benefit, showing theoretically at least how resistance depends on the hydraulic parameters, and practically giving a means of estimating the resistance. 6. Comparison with Manning s n Equation (10) Equation (11) /8 and gn (A/P ) 1/ Relative roughness = k s /(A/P ) Figure 2. Comparison of two variations of resistance coefficient with relative roughness Other formulations of the resistance include those of Chézy and Gauckler-Manning-Strickler. For them to agree for a steady uniform flow they are related by 8 = g C 2 = gn 2 (A/P ) 1/3 = g, (9) kst 2 (A/P )1/3 where C is the Chézy coefficient, n the Manning coefficient, and k St the Strickler coefficient, the latter two being in SI units. We can use equation (6) for for fully-rough flow, R with this equation to give the formula for /8: 8 = 1.33/8 ³ ln 2 ε. (10) 12. To compare predictions from the Gauckler-Manning-Strickler form we can use Strickler s 1923 result (Jaeger 1956, p30) for the empirical relationship for n in terms of the equivalent sand-grain diameter k s (in SI units): n = k1/6 s k1/6 s

6 Henderson (1966, p98) argued that due to bed armouring the coefficient of could be reduced by 10-20%. With power of hindsight, here we will take a value of (see also p107 of Sturm 2001). We then have for the Gauckler-Manning-Strickler term of equation (9), with g =9.8ms 2 gn 2 (A/P ) 1/3 = gε 1/ ε 1/3. (11) The approximations (10) and (11) are shown plotted on figure 2, and it can be seen that the two agree quite well, with the choice of coefficient we have made here. We interpret this as giving support for choosing the Darcy-Weisbach form, as it is apparently able to describe the variation of the resistance with relative roughness that the widely-used Gauckler-Manning-Strickler form does, but it involves a dimensionless coefficient, is simpler, can be related to more fundamental fluid mechanics results, and it can also include variation with Reynolds number if required. 7. Use of a modified coefficient Λ In Fenton (2010) it was shown that the long wave equations could also apply to channels on finite slopes. By introducing the symbol Λ: Λ = ³1+ S 2, (12) 8 where S is the downstream slope at a section, averaged around the wetted perimeter, the long wave momentum equation (Fenton 2010, eqn 13) becomes, ignoring lateral inflow and assuming unidirectional flow: Q t + µβ Q2 + ga η Q2 = ΛP x A x A 2, (13) where Q is the discharge, t is time, x is the horizontal streamwise co-ordinate, β is a Boussinesq momentum coefficient, and η is surface elevation. The resistance term contains the dimensionless coefficient Λ multiplied by the wetted perimeter around which the resistance acts, multiplied by the square of the mean velocity. This has a relatively simple physically-significant form, which can be compared with use of the Gauckler-Manning-Strickler form, when the two alternative forms of the resistance term are ΛP Q2 A 2 = gn2 P 4/3 Q 2 A 7/3, where the g is there only as an artifact gravity does not actually play a role in this term. The use of the dimensionless coefficient Λ is clearly simpler. In the case of steady uniform flow in a prismatic channel of constant slope S 0, the momentum equation (13) reduces to the Chézy-Darcy-Weisbach formula U 0 = Q r 0 g A 0 = S 0, (14) A 0 Λ P 0 where subscripts 0 denote uniform flow, and where U 0 is the mean flow velocity. The formula is written here in terms of the dimensionless resistance coefficient Λ, rather than Chézy s C which is dimensional, and additionally it allows for the effects of finite slope, as shown in equation (12). An interesting relationship emerges if equation (14) is re-written to give a formula for the Froude number F 0 : F0 2 = U 0 2 = S 0 B 0, (15) ga 0 /B 0 Λ P 0 where B 0 is the surface width. In wider channels, P 0 B 0, giving the relationship F 2 0 S 0 Λ, (16) 6

7 so that the square of the Froude number is approximately equal to the ratio of slope to the dimensionless resistance coefficient Λ. This shows how, now arguing in rough approximate terms, that the Froude number for a given stream is almost independent of flow, if we presume that neither Λ nor the geometry varies much. The simplicity of all formulae when using Λ is noteworthy. 8. Conclusions We have considered the Darcy-Weisbach form of resistance for flow in open channels, and presented a formulae by Yen that describes the variation of the dimensionless resistance coefficient with both relative roughness and Reynolds number, and have described the derivation of a variant (equation 6) that might be slightly simpler for theoretical studies, as it uses natural logarithms. The formula agrees with the Gauckler-Manning-Strickler form of resistance, but using the Darcy-Weisbach form gives simpler expressions, involves a dimensionless coefficient, and it can be related to more fundamental fluid mechanics results. If a modified form is used, Λ = /8, which is actually just g/c 2,whereC is the Chézy coefficient, and possibly supplemented by a slope term for the rarely-expected case of very steep channels, the long wave momentum equation has a simple and physically-revealing resistance term, and a simple relationship for the Froude number is obtained in terms of channel slope and that resistance coefficient. References ASCETaskForceonFrictionFactorsinOpenChannels (1963), Friction factors in open channels, J. Hydraulics Div. ASCE 89, Barnes, H. H. (1967), Roughness characteristics of natural channels, Water-Supply Paper 1849, U.S. Geological Survey. URL: Colebrook, C. F. (1938), Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws, J. Inst. Civ. Engrs 11, Fenton, J. D. (2010), The long wave equations, Technical report, Alternative Hydraulics Paper 1. URL: Haaland, S. E. (1983), Simple and explicit formulas for the friction factor in turbulent pipe flow, J. Fluids Engng 105, Henderson, F. M. (1966), Open Channel Flow, Macmillan, New York. Hicks, D. M. & Mason, P. D. (1991), Roughness Characteristics of New Zealand Rivers, DSIRMarine and Freshwater, Wellington. Jaeger, C. (1956), Engineering Fluid Mechanics, Blackie, London. Sturm, T. W. (2001), Open Channel Hydraulics, McGraw-Hill. White, F. M. (2003), Fluid Mechanics, fifth edn, McGraw-Hill, New York. Yen, B. C. (1991), Hydraulic resistance in open channels, in B. C. Yen, ed., Channel Flow Resistance: Centennial of Manning s Formula, Water Resource Publications, Highlands Ranch, USA, p

### Open channel flow Basic principle

Open channel flow Basic principle INTRODUCTION Flow in rivers, irrigation canals, drainage ditches and aqueducts are some examples for open channel flow. These flows occur with a free surface and the pressure

### Appendix 4-C. Open Channel Theory

4-C-1 Appendix 4-C Open Channel Theory 4-C-2 Appendix 4.C - Table of Contents 4.C.1 Open Channel Flow Theory 4-C-3 4.C.2 Concepts 4-C-3 4.C.2.1 Specific Energy 4-C-3 4.C.2.2 Velocity Distribution Coefficient

### Open Channel Flow. M. Siavashi. School of Mechanical Engineering Iran University of Science and Technology

M. Siavashi School of Mechanical Engineering Iran University of Science and Technology W ebpage: webpages.iust.ac.ir/msiavashi Email: msiavashi@iust.ac.ir Landline: +98 21 77240391 Fall 2013 Introduction

### CHAPTER 9 CHANNELS APPENDIX A. Hydraulic Design Equations for Open Channel Flow

CHAPTER 9 CHANNELS APPENDIX A Hydraulic Design Equations for Open Channel Flow SEPTEMBER 2009 CHAPTER 9 APPENDIX A Hydraulic Design Equations for Open Channel Flow Introduction The Equations presented

### Hydraulic losses in pipes

Hydraulic losses in pipes Henryk Kudela Contents 1 Viscous flows in pipes 1 1.1 Moody Chart.................................... 2 1.2 Types of Fluid Flow Problems........................... 5 1.3 Minor

### Chapter 8: Flow in Pipes

Objectives 1. Have a deeper understanding of laminar and turbulent flow in pipes and the analysis of fully developed flow 2. Calculate the major and minor losses associated with pipe flow in piping networks

### What is the most obvious difference between pipe flow and open channel flow????????????? (in terms of flow conditions and energy situation)

OPEN CHANNEL FLOW 1 3 Question What is the most obvious difference between pipe flow and open channel flow????????????? (in terms of flow conditions and energy situation) Typical open channel shapes Figure

### Chapter 9. Steady Flow in Open channels

Chapter 9 Steady Flow in Open channels Objectives Be able to define uniform open channel flow Solve uniform open channel flow using the Manning Equation 9.1 Uniform Flow in Open Channel Open-channel flows

### Experiment 3 Pipe Friction

EML 316L Experiment 3 Pipe Friction Laboratory Manual Mechanical and Materials Engineering Department College of Engineering FLORIDA INTERNATIONAL UNIVERSITY Nomenclature Symbol Description Unit A cross-sectional

### Pressure drop in pipes...

Pressure drop in pipes... PRESSURE DROP CALCULATIONS Pressure drop or head loss, occurs in all piping systems because of elevation changes, turbulence caused by abrupt changes in direction, and friction

### Chapter 13 OPEN-CHANNEL FLOW

Fluid Mechanics: Fundamentals and Applications, 2nd Edition Yunus A. Cengel, John M. Cimbala McGraw-Hill, 2010 Lecture slides by Mehmet Kanoglu Copyright The McGraw-Hill Companies, Inc. Permission required

### OPEN-CHANNEL FLOW. Free surface. P atm

OPEN-CHANNEL FLOW Open-channel flow is a flow of liquid (basically water) in a conduit with a free surface. That is a surface on which pressure is equal to local atmospheric pressure. P atm Free surface

### L r = L m /L p. L r = L p /L m

NOTE: In the set of lectures 19/20 I defined the length ratio as L r = L m /L p The textbook by Finnermore & Franzini defines it as L r = L p /L m To avoid confusion let's keep the textbook definition,

### Fluid Mechanics Prof. T. I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay. Lecture No. # 36 Pipe Flow Systems

Fluid Mechanics Prof. T. I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay Lecture No. # 36 Pipe Flow Systems Welcome back to the video course on Fluid Mechanics. In today

### 2.0 BASIC CONCEPTS OF OPEN CHANNEL FLOW MEASUREMENT

2.0 BASIC CONCEPTS OF OPEN CHANNEL FLOW MEASUREMENT Open channel flow is defined as flow in any channel where the liquid flows with a free surface. Open channel flow is not under pressure; gravity is the

### Darcy Friction Factor Formulae in Turbulent Pipe Flow

Darcy Friction Factor Formulae in Turbulent Pipe Flow Jukka Kiijärvi Lunowa Fluid Mechanics Paper 110727 July 29, 2011 Abstract The Darcy riction actor in turbulent pipe low must be solved rom the Colebrook

### Backwater Rise and Drag Characteristics of Bridge Piers under Subcritical

European Water 36: 7-35, 11. 11 E.W. Publications Backwater Rise and Drag Characteristics of Bridge Piers under Subcritical Flow Conditions C.R. Suribabu *, R.M. Sabarish, R. Narasimhan and A.R. Chandhru

### Chapter 2. Derivation of the Equations of Open Channel Flow. 2.1 General Considerations

Chapter 2. Derivation of the Equations of Open Channel Flow 2.1 General Considerations Of interest is water flowing in a channel with a free surface, which is usually referred to as open channel flow.

### FLUID FLOW Introduction General Description

FLUID FLOW Introduction Fluid flow is an important part of many processes, including transporting materials from one point to another, mixing of materials, and chemical reactions. In this experiment, you

### ANALYSIS OF OPEN-CHANNEL VELOCITY MEASUREMENTS COLLECTED WITH AN ACOUSTIC DOPPLER CURRENT PROFILER

Reprint from RIVERTECH 96 Proceedings from the1st International Conference On New/Emerging Concepts for Rivers Organized by the International Water Resources Association Held September 22-26, 1996, Chicago,

### M6a: Open Channel Flow (Manning s Equation, Partially Flowing Pipes, and Specific Energy)

M6a: Open Channel Flow (, Partially Flowing Pipes, and Specific Energy) Steady Non-Uniform Flow in an Open Channel Robert Pitt University of Alabama and Shirley Clark Penn State - Harrisburg Continuity

### ME 305 Fluid Mechanics I. Part 8 Viscous Flow in Pipes and Ducts

ME 305 Fluid Mechanics I Part 8 Viscous Flow in Pipes and Ducts These presentations are prepared by Dr. Cüneyt Sert Mechanical Engineering Department Middle East Technical University Ankara, Turkey csert@metu.edu.tr

### Experimentation and Computational Fluid Dynamics Modelling of Roughness Effects in Flexible Pipelines

Experimentation and Computational Fluid Dynamics Modelling of Roughness Effects in Flexible Pipelines Sophie Yin Jeremy Leggoe School of Mechanical and Chemical Engineering Daniel Teng Paul Pickering CEED

### Chapter 10. Open- Channel Flow

Updated: Sept 3 2013 Created by Dr. İsmail HALTAŞ Created: Sept 3 2013 Chapter 10 Open- Channel Flow based on Fundamentals of Fluid Mechanics 6th EdiAon By Munson 2009* *some of the Figures and Tables

### Evaluation of Open Channel Flow Equations. Introduction :

Evaluation of Open Channel Flow Equations Introduction : Most common hydraulic equations for open channels relate the section averaged mean velocity (V) to hydraulic radius (R) and hydraulic gradient (S).

### Topic 8: Open Channel Flow

3.1 Course Number: CE 365K Course Title: Hydraulic Engineering Design Course Instructor: R.J. Charbeneau Subject: Open Channel Hydraulics Topics Covered: 8. Open Channel Flow and Manning Equation 9. Energy,

### Heat Transfer Prof. Dr. Ale Kumar Ghosal Department of Chemical Engineering Indian Institute of Technology, Guwahati

Heat Transfer Prof. Dr. Ale Kumar Ghosal Department of Chemical Engineering Indian Institute of Technology, Guwahati Module No. # 04 Convective Heat Transfer Lecture No. # 03 Heat Transfer Correlation

### CHAPTER II UNIFORM FLOW AND ITS FORMULAS MODULE 1. This experiment was designed to observe the characteristics of uniform flow in

CHAPTER II UNIFORM FLOW AND ITS FORMULAS MODULE 1 2.1 Introduction and Objective This experiment was designed to observe the characteristics of uniform flow in the teaching flume and to utilize the common

### Experiment (13): Flow channel

Introduction: An open channel is a duct in which the liquid flows with a free surface exposed to atmospheric pressure. Along the length of the duct, the pressure at the surface is therefore constant and

### There are a number of criteria to consider when designing canals Below is a list of main criteria, not necessarily in order of importance:

Lecture 15 Canal Design Basics I. Canal Design Factors There are a number of criteria to consider when designing canals Below is a list of main criteria, not necessarily in order of importance: 1. Flow

### Practice Problems on Boundary Layers. Answer(s): D = 107 N D = 152 N. C. Wassgren, Purdue University Page 1 of 17 Last Updated: 2010 Nov 22

BL_01 A thin flat plate 55 by 110 cm is immersed in a 6 m/s stream of SAE 10 oil at 20 C. Compute the total skin friction drag if the stream is parallel to (a) the long side and (b) the short side. D =

### Hydraulic Jumps and Non-uniform Open Channel Flow, Course #507. Presented by: PDH Enterprises, LLC PO Box 942 Morrisville, NC 27560 www.pdhsite.

Hydraulic Jumps and Non-uniform Open Channel Flow, Course #507 Presented by: PDH Enterprises, LLC PO Box 942 Morrisville, NC 27560 www.pdhsite.com Many examples of open channel flow can be approximated

### LECTURE 1: Review of pipe flow: Darcy-Weisbach, Manning, Hazen-Williams equations, Moody diagram

LECTURE 1: Review of pipe flow: Darcy-Weisbach, Manning, Hazen-Williams equations, Moody diagram 1.1. Important Definitions Pressure Pipe Flow: Refers to full water flow in closed conduits of circular

### Hydraulics Laboratory Experiment Report

Hydraulics Laboratory Experiment Report Name: Ahmed Essam Mansour Section: "1", Monday 2-5 pm Title: Flow in open channel Date: 13 November-2006 Objectives: Calculate the Chezy and Manning coefficients

### STATE OF FLORIDA DEPARTMENT OF TRANSPORTATION DRAINAGE HANDBOOK OPEN CHANNEL. OFFICE OF DESIGN, DRAINAGE SECTION November 2009 TALLAHASSEE, FLORIDA

STATE OF FLORIDA DEPARTMENT OF TRANSPORTATION DRAINAGE HANDBOOK OPEN CHANNEL OFFICE OF DESIGN, DRAINAGE SECTION TALLAHASSEE, FLORIDA Table of Contents Open Channel Handbook Chapter 1 Introduction... 1

### Open Channel Flow 2F-2. A. Introduction. B. Definitions. Design Manual Chapter 2 - Stormwater 2F - Open Channel Flow

Design Manual Chapter 2 - Stormwater 2F - Open Channel Flow 2F-2 Open Channel Flow A. Introduction The beginning of any channel design or modification is to understand the hydraulics of the stream. The

### Civil Engineering Hydraulics Mechanics of Fluids. Flow in Pipes

Civil Engineering Hydraulics Mechanics of Fluids Flow in Pipes 2 Now we will move from the purely theoretical discussion of nondimensional parameters to a topic with a bit more that you can see and feel

### CIVE2400 Fluid Mechanics Section 2: Open Channel Hydraulics

CIVE400 Fluid Mechanics Section : Open Channel Hydraulics. Open Channel Hydraulics.... Definition and differences between pipe flow and open channel flow.... Types of flow.... Properties of open channels...

### EXAMPLES (OPEN-CHANNEL FLOW) AUTUMN 2015

EXAMPLES (OPEN-CHANNEL FLOW) AUTUMN 2015 Normal and Critical Depths Q1. If the discharge in a channel of width 5 m is 20 m 3 s 1 and Manning s n is 0.02 m 1/3 s, find: (a) the normal depth and Froude number

### Proceeding of International Seminar on Application of Science Matehmatics 2011 (ISASM2011) PWTC, KL, Nov, 1-3, 2011

Proceeding of International Seminar on Application of Science Matehmatics 2011 (ISASM2011) PWTC, KL, Nov, 1-3, 2011 INFLUENCE OF BED ROUGHNESS IN OPEN CHANNEL Zarina Md Ali 1 and Nor Ashikin Saib 2 1 Department

### Floodplain Hydraulics! Hydrology and Floodplain Analysis Dr. Philip Bedient

Floodplain Hydraulics! Hydrology and Floodplain Analysis Dr. Philip Bedient Open Channel Flow 1. Uniform flow - Manning s Eqn in a prismatic channel - Q, V, y, A, P, B, S and roughness are all constant

### Part 654 Stream Restoration Design National Engineering Handbook

United States Department of Agriculture Natural Resources Conservation Service Stream Restoration Design Chapter 6 Issued August 007 Cover photo: Stream hydraulics focus on bankfull frequencies, velocities,

### Urban Hydraulics. 2.1 Basic Fluid Mechanics

Urban Hydraulics Learning objectives: After completing this section, the student should understand basic concepts of fluid flow and how to analyze conduit flows and free surface flows. They should be able

### CEE 370 Fall 2015. Laboratory #3 Open Channel Flow

CEE 70 Fall 015 Laboratory # Open Channel Flow Objective: The objective of this experiment is to measure the flow of fluid through open channels using a V-notch weir and a hydraulic jump. Introduction:

### 2O-1 Channel Types and Structures

Iowa Stormwater Management Manual O-1 O-1 Channel Types and Structures A. Introduction The flow of water in an open channel is a common event in Iowa, whether in a natural channel or an artificial channel.

### The value of the wastewater flow used for sewer design is the daily peak flow. This can be estimated as follows:

This Section presents the theory of simplified sewer design. Firstly, in Section 2.1, the peak daily wastewater flow in the length of sewer being designed is described. Section 2.2 presents the trigonometric

### A LAMINAR FLOW ELEMENT WITH A LINEAR PRESSURE DROP VERSUS VOLUMETRIC FLOW. 1998 ASME Fluids Engineering Division Summer Meeting

TELEDYNE HASTINGS TECHNICAL PAPERS INSTRUMENTS A LAMINAR FLOW ELEMENT WITH A LINEAR PRESSURE DROP VERSUS VOLUMETRIC FLOW Proceedings of FEDSM 98: June -5, 998, Washington, DC FEDSM98 49 ABSTRACT The pressure

### Pipe Flow-Friction Factor Calculations with Excel

Pipe Flow-Friction Factor Calculations with Excel Course No: C03-022 Credit: 3 PDH Harlan H. Bengtson, PhD, P.E. Continuing Education and Development, Inc. 9 Greyridge Farm Court Stony Point, NY 10980

### Lecture 5 Hemodynamics. Description of fluid flow. The equation of continuity

1 Lecture 5 Hemodynamics Description of fluid flow Hydrodynamics is the part of physics, which studies the motion of fluids. It is based on the laws of mechanics. Hemodynamics studies the motion of blood

### A Comparison of Analytical and Finite Element Solutions for Laminar Flow Conditions Near Gaussian Constrictions

A Comparison of Analytical and Finite Element Solutions for Laminar Flow Conditions Near Gaussian Constrictions by Laura Noelle Race An Engineering Project Submitted to the Graduate Faculty of Rensselaer

### Dimensional analysis is a method for reducing the number and complexity of experimental variables that affect a given physical phenomena.

Dimensional Analysis and Similarity Dimensional analysis is very useful for planning, presentation, and interpretation of experimental data. As discussed previously, most practical fluid mechanics problems

### Spreadsheet Use for Partially Full Pipe Flow Calculations

Spreadsheet Use for Partially Full Pipe Flow Calculations Course No: C02-037 Credit: 2 PDH Harlan H. Bengtson, PhD, P.E. Continuing Education and Development, Inc. 9 Greyridge Farm Court Stony Point, NY

### NUMERICAL ANALYSIS OF OPEN CHANNEL STEADY GRADUALLY VARIED FLOW USING THE SIMPLIFIED SAINT-VENANT EQUATIONS

TASK QUARTERLY 15 No 3 4, 317 328 NUMERICAL ANALYSIS OF OPEN CHANNEL STEADY GRADUALLY VARIED FLOW USING THE SIMPLIFIED SAINT-VENANT EQUATIONS WOJCIECH ARTICHOWICZ Department of Hydraulic Engineering, Faculty

### 21. Channel flow III (8.10 8.11)

21. Channel flow III (8.10 8.11) 1. Hydraulic jump 2. Non-uniform flow section types 3. Step calculation of water surface 4. Flow measuring in channels 5. Examples E22, E24, and E25 1. Hydraulic jump Occurs

### Basic Equations, Boundary Conditions and Dimensionless Parameters

Chapter 2 Basic Equations, Boundary Conditions and Dimensionless Parameters In the foregoing chapter, many basic concepts related to the present investigation and the associated literature survey were

### Hydraulics Prof. A. K. Sarma Department of Civil Engineering Indian Institute of Technology, Guwahati. Module No. # 02 Uniform Flow Lecture No.

Hydraulics Prof. A. K. Sarma Department of Civil Engineering Indian Institute of Technology, Guwahati Module No. # 02 Uniform Flow Lecture No. # 04 Computation of Uniform Flow (Part 02) Welcome to this

### FLOW RATE MEASUREMENTS BY FLUMES

Fourteenth International Water Technology Conference, IWTC 14 010, Cairo, Egypt 1 FLOW RATE MEASUREMENTS BY FLUMES W. Dbrowski 1 and U. Polak 1 Department of Environmental Engineering, Cracow University

### Civil Engineering Hydraulics Open Channel Flow. Adult: Where s your costume? What are you supposed to be?

Civil Engineering Hydraulics Calvin: Trick or treat! Adult: Where s your costume? What are you supposed to be? Calvin: I m yet another resource-consuming kid in an overpopulated planet, raised to an alarming

### Note: first and second stops will be reversed. Bring clothing and shoes suitable for walking on rough ground.

Open Channel Page 1 Intro check on laboratory results Field Trip Note: first and second stops will be reversed Irrigation and Drainage Field Trip Bring clothing and shoes suitable for walking on rough

### Exact solution of friction factor and diameter problems involving laminar flow of Bingham plastic fluids

Journal of etroleum and Gas Exploration Research (ISSN 76-6510) Vol. () pp. 07-03, February, 01 Available online http://www.interesjournals.org/jger Copyright 01 International Research Journals Review

### Design Charts for Open-Channel Flow HDS 3 August 1961

Design Charts for Open-Channel Flow HDS 3 August 1961 Welcome to HDS 3-Design Charts for Open-Channel Flow Table of Contents Preface DISCLAIMER: During the editing of this manual for conversion to an electronic

### CHAPTER 4 OPEN CHANNEL HYDRAULICS

CHAPTER 4 OPEN CHANNEL HYDRAULICS 4. Introduction Open channel flow refers to any flow that occupies a defined channel and has a free surface. Uniform flow has been defined as flow with straight parallel

### Chapter 10. Flow Rate. Flow Rate. Flow Measurements. The velocity of the flow is described at any

Chapter 10 Flow Measurements Material from Theory and Design for Mechanical Measurements; Figliola, Third Edition Flow Rate Flow rate can be expressed in terms of volume flow rate (volume/time) or mass

### CHAPTER ONE Fluid Fundamentals

CHPTER ONE Fluid Fundamentals 1.1 FLUID PROPERTIES 1.1.1 Mass and Weight Mass, m, is a property that describes the amount of matter in an object or fluid. Typical units are slugs in U.S. customary units,

### Introduction to COMSOL. The Navier-Stokes Equations

Flow Between Parallel Plates Modified from the COMSOL ChE Library module rev 10/13/08 Modified by Robert P. Hesketh, Chemical Engineering, Rowan University Fall 2008 Introduction to COMSOL The following

### ME19b. SOLUTIONS. Feb. 11, 2010. Due Feb. 18

ME19b. SOLTIONS. Feb. 11, 21. Due Feb. 18 PROBLEM B14 Consider the long thin racing boats used in competitive rowing events. Assume that the major component of resistance to motion is the skin friction

### Basic Principles in Microfluidics

Basic Principles in Microfluidics 1 Newton s Second Law for Fluidics Newton s 2 nd Law (F= ma) : Time rate of change of momentum of a system equal to net force acting on system!f = dp dt Sum of forces

### LECTURE 9: Open channel flow: Uniform flow, best hydraulic sections, energy principles, Froude number

LECTURE 9: Open channel flow: Uniform flow, best hydraulic sections, energy principles, Froude number Open channel flow must have a free surface. Normally free water surface is subjected to atmospheric

### Natural Convection. Buoyancy force

Natural Convection In natural convection, the fluid motion occurs by natural means such as buoyancy. Since the fluid velocity associated with natural convection is relatively low, the heat transfer coefficient

### Viscous flow through pipes of various cross-sections

IOP PUBLISHING Eur. J. Phys. 28 (2007 521 527 EUROPEAN JOURNAL OF PHYSICS doi:10.1088/0143-0807/28/3/014 Viscous flow through pipes of various cross-sections John Lekner School of Chemical and Physical

### Investigation of Effect of Changes in Dimension and Hydraulic of Stepped Spillways for Maximization Energy Dissipation

World Applied Sciences Journal 8 (): 6-67, 0 ISSN 88-495 IDOSI Publications, 0 DOI: 0.589/idosi.wasj.0.8.0.49 Investigation of Effect of Changes in Dimension and Hydraulic of Stepped Spillways for Maximization

### 1. Fluids Mechanics and Fluid Properties. 1.1 Objectives of this section. 1.2 Fluids

1. Fluids Mechanics and Fluid Properties What is fluid mechanics? As its name suggests it is the branch of applied mechanics concerned with the statics and dynamics of fluids - both liquids and gases.

### Lecture 22 Example Culvert Design Much of the following is based on the USBR technical publication Design of Small Canal Structures (1978)

Lecture 22 Example Culvert Design Much of the following is based on the USBR technical publication Design of Small Canal Structures (1978) I. An Example Culvert Design Design a concrete culvert using the

### Lecture 24 Flumes & Channel Transitions. I. General Characteristics of Flumes. Flumes are often used:

Lecture 24 Flumes & Channel Transitions I. General Characteristics of Flumes Flumes are often used: 1. Along contours of steep slopes where minimal excavation is desired 2. On flat terrain where it is

### Fluids and Solids: Fundamentals

Fluids and Solids: Fundamentals We normally recognize three states of matter: solid; liquid and gas. However, liquid and gas are both fluids: in contrast to solids they lack the ability to resist deformation.

### Package rivr. October 16, 2015

Type Package Package rivr October 16, 2015 Title Steady and Unsteady Open-Channel Flow Computation Version 1.1 Date 2015-10-15 Author Michael C Koohafkan [aut, cre] Maintainer Michael C Koohafkan

### Applied Fluid Mechanics

Applied Fluid Mechanics 1. The Nature of Fluid and the Study of Fluid Mechanics 2. Viscosity of Fluid 3. Pressure Measurement 4. Forces Due to Static Fluid 5. Buoyancy and Stability 6. Flow of Fluid and

### MODELING FLUID FLOW IN OPEN CHANNEL WITH CIRCULAR CROSS SECTION DADDY PETER TSOMBE MASTER OF SCIENCE. (Applied Mathematics)

MODELING FLUID FLOW IN OPEN CHANNEL WITH CIRCULAR CROSS SECTION DADDY PETER TSOMBE MASTER OF SCIENCE (Applied Mathematics) JOMO KENYATTA UNIVERSITY OF AGRICULTURE AND TECHNOLOGY 2011 Modeling fluid flow

### Applied Fluid Mechanics

Applied Fluid Mechanics Sixth Edition Robert L. Mott University of Dayton PEARSON Prentkv Pearson Education International CHAPTER 1 THE NATURE OF FLUIDS AND THE STUDY OF FLUID MECHANICS 1.1 The Big Picture

### Basic Hydraulic Principles

CHAPTER 1 Basic Hydraulic Principles 1.1 General Flow Characteristics In hydraulics, as with any technical topic, a full understanding cannot come without first becoming familiar with basic terminology

### Hydraulics Guide. Table 1: Conveyance Factors (English Units)... 7 Table 2: Conveyance Factors (Metric Units)... 8

Table of Contents 1.1 Index of Tables... 1 1. Index of Figures... 1 1.3 Overview of Hydraulic Considerations... 1.4 Discharge Curves... 1.5 Conveyance Method... 5 1.6 Flow Velocity Considerations... 9

### These slides contain some notes, thoughts about what to study, and some practice problems. The answers to the problems are given in the last slide.

Fluid Mechanics FE Review Carrie (CJ) McClelland, P.E. cmcclell@mines.edu Fluid Mechanics FE Review These slides contain some notes, thoughts about what to study, and some practice problems. The answers

### Flow Loss in Screens: A Fresh Look at Old Correlation. Ramakumar Venkata Naga Bommisetty, Dhanvantri Shankarananda Joshi and Vighneswara Rao Kollati

Journal of Mechanics Engineering and Automation 3 (013) 9-34 D DAVID PUBLISHING Ramakumar Venkata Naga Bommisetty, Dhanvantri Shankarananda Joshi and Vighneswara Rao Kollati Engineering Aerospace, MCOE,

### OUTCOME 3 TUTORIAL 5 DIMENSIONAL ANALYSIS

Unit 41: Fluid Mechanics Unit code: T/601/1445 QCF Level: 4 Credit value: 15 OUTCOME 3 TUTORIAL 5 DIMENSIONAL ANALYSIS 3 Be able to determine the behavioural characteristics and parameters of real fluid

### Viscous flow in pipe

Viscous flow in pipe Henryk Kudela Contents 1 Laminar or turbulent flow 1 2 Balance of Momentum - Navier-Stokes Equation 2 3 Laminar flow in pipe 2 3.1 Friction factor for laminar flow...........................

### HEAT TRANSFER CODES FOR STUDENTS IN JAVA

Proceedings of the 5th ASME/JSME Thermal Engineering Joint Conference March 15-19, 1999, San Diego, California AJTE99-6229 HEAT TRANSFER CODES FOR STUDENTS IN JAVA W.J. Devenport,* J.A. Schetz** and Yu.

### CHAPTER 5 OPEN-CHANNEL FLOW

CHAPTER 5 OPEN-CHANNEL FLOW 1. INTRODUCTION 1 Open-channel flows are those that are not entirely included within rigid boundaries; a part of the flow is in contract with nothing at all, just empty space

### Sharp-Crested Weirs for Open Channel Flow Measurement, Course #506. Presented by:

Sharp-Crested Weirs for Open Channel Flow Measurement, Course #506 Presented by: PDH Enterprises, LLC PO Box 942 Morrisville, NC 27560 www.pdhsite.com A weir is basically an obstruction in an open channel

### Open Channel Flow Measurement Weirs and Flumes

Open Channel Flow Measurement Weirs and Flumes by Harlan H. Bengtson, PhD, P.E. 1. Introduction Your Course Title Here Measuring the flow rate of water in an open channel typically involves some type of

### ENV5056 Numerical Modeling of Flow and Contaminant Transport in Rivers. Equations. Asst. Prof. Dr. Orhan GÜNDÜZ

ENV5056 Numerical Modeling of Flow and Contaminant Transport in Rivers Derivation of Flow Equations Asst. Prof. Dr. Orhan GÜNDÜZ General 3-D equations of incompressible fluid flow Navier-Stokes Equations

### Mathematics and Computation of Sediment Transport in Open Channels

Mathematics and Computation of Sediment Transport in Open Channels Junping Wang Division of Mathematical Sciences National Science Foundation May 26, 2010 Sediment Transport and Life One major problem

### Abaqus/CFD Sample Problems. Abaqus 6.10

Abaqus/CFD Sample Problems Abaqus 6.10 Contents 1. Oscillatory Laminar Plane Poiseuille Flow 2. Flow in Shear Driven Cavities 3. Buoyancy Driven Flow in Cavities 4. Turbulent Flow in a Rectangular Channel

### CHAPTER 4 FLOW IN CHANNELS

CHAPTER 4 FLOW IN CHANNELS INTRODUCTION 1 Flows in conduits or channels are of interest in science, engineering, and everyday life. Flows in closed conduits or channels, like pipes or air ducts, are entirely

### Open Channel Flow Resistance 1

Open Channel Flow esistance 1 Ben Chie Yen, F.ASCE 2 Abstract: In 1965, ouse critically reviewed hydraulic resistance in open channels on the basis of fluid mechanics. He pointed out the effects of cross-sectional

### ANALYSIS OF FULLY DEVELOPED TURBULENT FLOW IN A PIPE USING COMPUTATIONAL FLUID DYNAMICS D. Bhandari 1, Dr. S. Singh 2

ANALYSIS OF FULLY DEVELOPED TURBULENT FLOW IN A PIPE USING COMPUTATIONAL FLUID DYNAMICS D. Bhandari 1, Dr. S. Singh 2 1 M. Tech Scholar, 2 Associate Professor Department of Mechanical Engineering, Bipin

### FLOW CONDITIONER DESIGN FOR IMPROVING OPEN CHANNEL FLOW MEASUREMENT ACCURACY FROM A SONTEK ARGONAUT-SW

FLOW CONDITIONER DESIGN FOR IMPROVING OPEN CHANNEL FLOW MEASUREMENT ACCURACY FROM A SONTEK ARGONAUT-SW Daniel J. Howes, P.E. 1 Charles M. Burt, Ph.D., P.E. 2 Brett F. Sanders, Ph.D. 3 ABSTRACT Acoustic

### Broad Crested Weirs. I. Introduction

Lecture 9 Broad Crested Weirs I. Introduction The broad-crested weir is an open-channel flow measurement device which combines hydraulic characteristics of both weirs and flumes Sometimes the name ramp