A Dissertation Entitled An Initial Study to Determine a Friction-Factor Model for Ground Vegetation by Peter Martin Kenney Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Engineering ____________________________ Advisor: Dr. Theo G. Keith, Jr. ___________________________ Graduate School The University of Toledo December 2009 The University of Toledo College of Engineering I HEREBY RECOMMEND THAT THE DISSERTATION PREPARED UNDER MY SUPERVISION BY Peter Martin Kenney ENTITLED An Initial Study to Determine a Friction-Factor Model for Ground Vegetation BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN ENGINEERING _________________________________________________________________ Dissertation Advisor: Dr. Theo G. Keith, Jr. Recommendation concurred by ________________________________ Dr. Rodman R. Linn Committee ________________________________ Dr. Tsun-Ming (Terry) Ng On ________________________________ Final Examination Dr. Abdollah A. Afjeh ________________________________ Dr. K. Cyril Masiulaniec _________________________________________________________________ Dean, College of Engineering Copyright © 2009 This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. An Abstract of An Initial Study to Determine a Friction-Factor Model for Ground Vegetation By Peter Martin Kenney Submitted as partial fulfillment of the requirements for The Doctor of Philosophy in Engineering The University of Toledo December 2009 Fluid drag data for flows through ground vegetation is needed by those studying the atmospheric boundary layer near the earth in wind turbine design, by those performing drainage calculations for fields awash from a great river, by CFD modelers of the lowest levels of a forest fire, and by agriculturalists investigating the wind overturning of crops. All of these researchers lament that there is little plant-drag data available. This work presents an initial study into a plant friction-factor model for a forest-fire simulation CFD program; but hopefully will be useful to the other professionals mentioned. The model develops a friction factor as a function of Reynolds number for various percentage cover of plants. Artificial plant arrangements were mounted in a clear acrylic frame, backlit from underneath, and, using a mirror, are photographed from sixty feet away. From the photographic data, fractal dimension and an average gap diameter can be determined as a function of percentage coverage. Then the arrangement is placed in a wind tunnel and drag forces on the system measured for upstream velocities ranging up to thirty- eight miles an hour. Most arrangements had axes of greatest and least flow resistance. Averaging the friction-factor values computed for these extremes a friction-factor model is developed for three different artificial plants. These individual models are, in turn, combined into a single model using the fractal dimension data for the plants. Vorticity and flow through the plant canopy causing plant oscillation are not considered. iv For my parents Lura Jean Koons Kenney Robert Sherman Kenney v Acknowledgements Many people helped make this project possible. Their assistance is gratefully acknowledged; but, as always any oversights or errors are this person’s alone. First and foremost, my advisor, Dr. Theo G. Keith, has patiently mentored this work and made available funds to support it. Thank you for your wise counsel and friendship. Dr. Rodman R. Linn provided the original suggestion for this model. Frustrating times during in developing the experimental apparatus were shortened thanks to the knowledge and experience of Dr. Terry Ng. This project would not have been possible without the skills of Randy Reihing, John Jaegly, and Tim Grivanos in the MIME Department’s shop. Early on this project, it was thought to use cut-and-dried natural vegetation in testing. The debris from such would have left the insides of the wind tunnel less than tidy and the collected cuttings were never used. Nonetheless, recognition and thanks go to Lois and William Brodbeck (City View Farm, Ottawa Lake, MI) who kindly allowed collection of vegetation samples from their farmstead and to John Jaeger (Director of Natural Resources) and Denise Gehring (Director of Environmental Programs) of the Metropolitan Park District of the Toledo Area who issued a collection permit and helped in choosing and identifying species. New Way Air Bearings made available to the project a Boxway Air Slide at a generous academic discount. vi Table of Contents Abstract iv Dedication v Acknowledgements vi Table of Contents vii List of Figures ix 1 Prologue 1 2 Nomenclature 7 3 Introduction 10 4 Theory 13 4.1 Types of Flow 13 4.2 Percentage Cover 14 4.3 Fractal Dimension 15 4.4 Lacunarity 16 4.5 Characteristic Dimension 18 4.6 Dimensional Analysis 19 4.7 Preferred Axis of Flow 21 5 Experimental Procedure 22 5.1 Photographic Data 22 5.2 Force and Velocity Measurements 23 vii 6 Results 28 6.1 Photographic Results 28 6.2 Wind-Tunnel Friction-Factor Results 31 6.2.1 Initial Data Reduction 31 6.2.2 Friction-Factor Curves and Model Development 32 7 Discussion 45 8 Conclusions 47 9 Future Work 48 10 References 51 Appendix A—Vegetation Density Indices 54 Appendix B—Fractals and Lacunarity 71 Appendix C—Mathcad Image Processing 144 Appendix D—Mathcad Friction-Factor Calculations 192 Appendix E—Equation Fitting 229 viii List of Figures Figure 1 Terms in Bernoulli’s Equation 1 Figure 2 Moody’s 1944 diagram 3 Figure 3 Types of flow through vegetation 13 Figure 4 Percentage Cover 14 Figure 5 Results for fractal dimension calculation 16 Figure 6 Lacunarity vs. gliding-box size 18 Figure 7 Silhouettes of three plants investigated 22 Figure 8 Experimental setup for taking plan-view photographs 23 Figure 9 Test apparatus 24 Figure 10 Plots of fractal dimension, lacunarity, and mean gap diameter 27 Figure 11 Plots of fractal dimension, lacunarity, and mean gap diameter 29 Figure 12 Identical arrangements differing only in orientation produced 30 different values of percentage cover Figure 13 Initial plot of friction factor versus Reynolds number 31 Figure 14 Friction-factor vs Reynolds-number curve after adjusting 31 the calculated friction factors to coincide with the average Reynolds numbers Figure 15 Improved certainty interval after discarding first data sampling 31 Figure 16 Effective-friction-factor vs. Reynolds-number curves for 33 the three plants investigated Figure 17 Variance in effective friction factor vs. percentage cover 35 for a Reynolds number of 4 104 ix Figure 18 The same data as presented in Figure 17 except identical 1 36 plant arrangements at varying orientations to the flow have not be averaged Figure 19 Plots of the slopes of the curves 38 Figure 20 Plots of the intercepts of the curves 39 Figure 21 Individual models for each plant 41 Figure 22 Plots of individual models overlaid on data for Re = 4104 42 Figure 23 Points used to determine individual model coefficients 43 should be taken from the linear range Figure 24 Coefficients for combined model 44 Figure 25 Goldenrod with senescent leaves at base 45 Figure 26 In-field wind tunnels 49 x Chapter 1—Prologue1 In 1738, Dutch-Swiss mathematician Daniel Bernoulli (1700-1782) published the principle named after him in his Hydrodynamica. This principle states that the fluid frictional head loss (pressure loss) between positions 1 and 2 of Figure 1 in a closed conduit is expressed by V 2 p V 2 p h 1 1 z 2 2 z (1) l 2g g 1 2g g 2 where hl is the fluid friction or energy head loss between positions subscripted 1 and 2, V is the mean velocity, g is the acceleration of gravity, p is the fluid static pressure, is the fluid density, and z is the elevation of the pipe. [1, 2] Equation 1 cannot be used to predict head loss in piping design. Instead a system must be constructed and all terms in the equation measured. Thus, engineering design requires a relationship that predicts hl as a function of the fluid, the velocity, the pipe diameter, and the type of pipe material. [1] hl = (p1 – p2)/( g) D V L Figure 1—Terms in (Bernoulli’s) Equation 1 [after 1] 1 This section relies heavily on Reference 1. It should give the reader a feeling of what is to be presented here in a slightly different arena. 1 In 1845, a native of Saxony (eastern Germany) named Julius Weisbach (1806-1871) proposed a relation for this fluid head loss which, in the form used today, is fL V 2 p pD hl or f 2 (2) D 2g g LV 2 where p = p1 – p2, L is the pipe length, D is the pipe diameter, and f is a friction factor. The friction factor is a complex function of the pipe relative roughness, /D, pipe diameter, fluid kinematic viscosity, , and velocity of the flow. Weisbach deduced the influence of roughness, diameter, and velocity on f, but the professional community apparently ignored his conclusions. As discussed below, the friction factor was perfected by Henry Darcy (1803-1858) in 1857 and Equation 2 is referred to as the Darcy-Weisbach equation. [1] Weisbach’s relation was based on eleven of his own measurements and 51 from pub- lished data by Claude Couplet (1642-1722), Charles Bossut (1730-1799), Pierre Du Buat (1734-1809), Gaspard Riche de Prony (1755-1839), and Johann Eytelwein (1764-1848). [1] Weisbach’s work covered most engineering mechanics and set the standard for all later engineering texts—except in France the then center for hydraulic research. The French may have believed that it offered no improvement over Prony’s work. However, Prony’s work was not dimensionally independent; whereas, Weisbach’s friction factor was a dimensionless number. At the time (before calculators), however, Prony’s required fewer mathematical operations.