Simple Equations to Calculate Fall Velocity and Sediment Scale Parameter

Transcription

1 TECHNICAL NOTES Simple Equations to Calculate Fall Velocity and Sediment Scale Parameter John P. Ahrens, Aff.ASCE 1 Abstract: This paper investigates and compares four simple, continuous equations that can potentially make useful estimates of fall velocity. These estimates span the range of laminar, transitional, and turbulent flow regimes. All equations take advantage of the fact that the fall velocity Reynolds number is a function of only one variable, the Archimedes buoyancy index. Adjusting the can make a significant improvement in the fit of these equations to data. In the commonly used beach equilibrium profile model, h A p x 2/3, the sediment scale parameter A p, is usually tabulated as a function of grain diameter. It will be shown that the normalized sediment scale parameter A p /d 1/3 is also a function of only the Archimedes buoyancy index. This research indicates that the Archimedes buoyancy index is the fundamental independent variable for both the fall velocity Reynolds number and the normalized sediment scale parameter. Calculating the Archimedes buoyancy index is facilitated by some simple equations, which in turn allow easy computation of fall velocity and sediment scale parameter. DOI: / ASCE X :3 146 CE Database subject headings: Grain size; Settling velocity; Sediment. Introduction In recent decades, there has been a consistent trend of increased use by coastal engineers of the terminal settling velocity for sand, usually referred to as the fall velocity. Hallermeier 1981 defines fall velocity as, A sediment grain in a less dense, viscous fluid attains a terminal settling velocity...as the gravitational force is balanced by the hydrodynamic drag force on the grain. Increased interest in the variable reflects acknowledgment of the fundamental physical importance of the fall velocity as a logical way to characterize sediment. Until recently, a modest disadvantage of the fall velocity as a variable has been that there was no single, continuous, accurate equation to calculate it over a wide range of conditions. Four simple continuous relations for calculating the fall velocity will be discussed and compared herein. This paper will follow the approach suggested by Yalin s 1977 and Hallermeier s 1981 research, i.e., the Reynolds number R, associated with a falling particle is a function of only the Archimedes buoyancy index, A. The Reynolds number is given by R wd/v (1) where w fall velocity; d characteristic diameter of the particle; and v kinematic viscosity of water. The Archimedes buoyancy index is given by 1 Coastal Consultant, 6702 Springfield Dr., Mason Neck, VA j.ahrens@worldnet.att.net Note. Discussion open until October 1, Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this technical note was submitted for review and possible publication on February 28, 2002; approved on November 8, This technical note is part of the Journal of Waterway, Port, Coastal, and Ocean Engineering, Vol. 129, No. 3, May 1, ASCE, ISSN X/2003/ /$ A gd 3 /v 2 (2) where ( s )/ ; s and density of the sediment particle and the density of the fluid, respectively; and g acceleration of gravity. Following Hallermeier 1981 the variable, A can be used to define flow regimes laminar A 39, transitional 39 A 10 4, and turbulent A For reference, the following approximations are useful; when A 39, R 2; and when A 10 4, R 100. In the commonly used beach equilibrium profile model h A p x 2/3 (3) Dean 1987 showed that the sediment scale parameter A p can be predicted as a function of fall velocity where h water depth; and x distance from the shoreline. This finding helps tie beach equilibrium profiles to the basic physics of the sediment. Dean 1987 notes that, This result reinforces the greater relevance of fall velocity in sediment transport processes. Because A p is a function of the fall velocity, then it is also a function of the Archimedes buoyancy index; this paper will give a simple method to calculate A p as a function of A. Background Rubey s Fall Velocity Equation Rubey 1933 developed a very simple equation to predict fall velocity based on equating the buoyant weight of a particle to the sum of the viscous and turbulent flow resistance. The existence of this almost forgotten research was brought to the writer s attention by Prof. Yoshimi Goda personal communication, Rubey s 1933 equation is R 6 2 2A/3 6 (4) 146 / JOURNAL OF WATERWAY, PORT, COASTAL AND OCEAN ENGINEERING ASCE / MAY/JUNE 2003

2 Table 1. Predictive Ability of Original Four Equations by Flow Regime data flow regime Laminar N 8 Transitional N 72 Turbulent N 35 Total N 115 a Hallermeier b Rubey c Cheng d Ahrens e Chang and Liou Rubey b Eq. 4 Cheng c Eq. 7 Root-Mean Square Percent Error Ahrens d Eqs. 9 and 10 Chang and Liou e suggested Eq The form of Eq. 4 has been greatly simplified by the use of the Archimedes buoyancy index. Rubey s equation does a good job of fitting the Hallermeier 1981 data set in the laminar flow regime but underpredicts fall velocities, typically by about 20%, in the turbulent flow regime Table 1. Cheng s Fall Velocity Equation Cheng 1997 shows two general relationships for drag for sediment particles falling in a fluid; they may be written as C D 4 gd/3w 2 4A/3R 2 (5) and C D a/r 1/c b 1/c c (6) Coefficients a, b, and c are dimensionless numbers which have approximately the following role: coefficient a is important at low Reynolds numbers, laminar flow; coefficient b is important at high Reynolds numbers, turbulent flow; and coefficient c was determined by fitting to data, not Hallermeier s 1981, with Reynolds numbers, in the range 1 R Cheng 1997 gives the following values for these constants: a 32, b 1.0, and c 1.5, that are appropriate for his data set. Combining Eqs. 5 and 6 and solving for the positive root of the quadratic equation gives Cheng s 1997 fall velocity equation R A 2/3 5 3/2 (7) Eq. 7 can be used as a simple, continuous equation to estimate fall velocities over a wide range of Reynolds numbers, but it is not good in fitting Hallermeier s 1981 data in the laminar or transitional flow regimes, Table 1. Ahrens Fall Velocity Equation Ahrens 2000 starts with the basic fall velocity relation R C L A C T A (8) and determines the C L and C T. The C L and C T were determined using a trial and error procedure to minimize error compared to 52 data of Hallermeier 1981 in the quartz density range, i.e C L and C T are associated with the laminar and turbulent flow regimes, respectively. The following relations were developed for these : C L tanh 12A 0.59 exp A (9) C T 1.06 tanh 0.016A 0.50 exp 120/A (10) Coefficients given by Eqs. 9 and 10 are for somewhat angular shaped grains but ignores the exact grain shape consistent with Hallermeier It can be seen in Ahrens 2000 that the above approach fits the quartz data subset quite well and even fits the nonquartz data subset well. Chang and Liou s Fall Velocity Equation Chang and Liou 2001 give the following relation to calculate the fall velocity: R A / 1 A 1 (11) For small values of A, laminar flow, Eq. 11 has the limiting form R A/ and for large values of A turbulent flow, Eq. 11 has the limiting form, R A /. Chang and Liou 2001 suggest the following values for the 30.22, 0.463, and Method of Analysis Coefficients given above for the fall velocity prediction equations will be adjusted to minimize the root-mean square RMS error when compared to the complete 115 observation Hallermeier 1981 data set, i.e., both quartz and nonquartz subsets. Percent error is defined as percent error predicted observed /observed *100 Table 1 compares the ability of the four equations to fit the data, where N is the number of observations in the various flow regime subsets of Hallermeier s 1981 data. Adjusted Coefficients for Fall Velocity Equations Equations with adjusted to minimize error are given below and will be referred to as the recommended versions. The recommended version of Rubey s 1933 equation is JOURNAL OF WATERWAY, PORT, COASTAL AND OCEAN ENGINEERING ASCE / MAY/JUNE 2003 / 147

3 Table 2. Comparison of the Recommended Continuous Equations and Segmented Equations by Flow Regime data flow regime Laminar N 8 Transitional N 72 Turbulent N 35 Total N 115 a Hallermeier b Rubey c Cheng d Ahrens e van Rijn f Not applicable. Rubey b Eq. 16 Cheng c Eq. 17 Root-Mean Square Percent Error Ahrens d Eqs. 9 and 14 Chang and Liou e Eq. 11 in Eq. 15 segmented equations van Rijn e segmented equations NA f NA f R A (12) In adjusting the in Rubey s equation from Eqs. 4 12, the very good fit for laminar flow was lost in obtaining a better overall fit, especially in the turbulent flow region. The adjusted for Cheng s 1997 equation are: a 21.2; b 1.03; and c 1.53 and the recommended equation is R A 1/ (13) The adjusted for Ahrens 2000 equation, Eqs. 8 and 9 are C T 1.01 tanh 0.016A 0.50 exp 115/A (14) No adjustment of the laminar coefficient C L was required, but a small adjustment was required for the turbulent coefficient C T. The adjusted for Chang and Liou s 2001 equation, Eq. 11 are 24.6, 0.477, and 17.9 (15) A useful comparison can be made between the ability of the four continuous equations and the segmented equations of Hallermeier 1981 and van Rijn 1993 to predict Reynolds numbers in the various flow regimes. Table 2 compares the continuous and segmented relations for the laminar A 39, transitional 39 A 10 4, and turbulent 10 4 A flow regimes for Hallermeier s 1981 data. Comparison of Tables 1 and 2 show that adjusting the has made a substantial improvement in Rubey s 1933 and Cheng s 1977 equations. Table 2 indicates that the simpler equations of Cheng 1997 and Chang and Liou 2001, when their are adjusted to the Hallermeier 1981 data set, fit the data almost as well as the more complex relationship of Ahrens Table 2 also shows the surprising finding that often the continuous relations fit the data in the various flow regimes somewhat better than the segmented relations fit to data in a specific flow regime. Predicting Limiting Values A further measure of the usefulness of continuous equations is how well they approach accepted limiting values. The limiting values for fall velocity in the laminar and turbulent flow regimes are widely accepted within relatively narrow ranges, e.g., refer to the discussion of drag for falling particles at extreme Reynolds numbers by Cheng The physics of falling particles is well enough understood so that approximate limiting values can be safely assumed in the laminar range at A 1.0 and in the turbulent range at A This range is about one-half an order of magnitude smaller than the smallest value in the Hallermeier 1981 data set and about one and a half orders of magnitude greater than highest values in the Hallermeier data set. The range 1.0 A 10 8 includes particle sizes from about cm to about 2.5 cm. Table 3 compares the four continuous equations with a reference value in both the laminar and turbulent region. The reference value is calculated using R A/18 in the laminar range at A 1.0 and using R 1.05 (A) in the turbulent range at A 10 8 ; these relationships are the same as the segmented relationships of Hallermeier 1981 for the corresponding flow regimes. Sediment Scale Parameter Dean 1987 shows that an alternative way to predict the sediment scale parameter A p is as a function of the fall velocity rather than just a function of grain diameter. Usually the sediment scale parameter is recognized in the relationship h A p x 2/3. Both Yalin 1978 and Hallermeier 1981 have shown that the fall velocity Reynolds number is a function of only the Archimedes buoyancy index. Together these findings indicate that the underlying causative variable for the sediment scale parameter is the Archimedes buoyancy index. Fig. 1 shows the sediment scale parameter A p normalized by the cube root of the grain diameter d 1/3 plotted as a function of the Archimedes buoyancy index. The data shown in Fig. 1 are 100 recommended values of A p and d given by Dean Fig. 1 shows a very strong functional relationship between A p /d 1/3 and A. A simple prediction equation is given by A p /d 1/ exp 1.24/A 1/3 (16) which fits the data extremely well, R Using the Haller- 148 / JOURNAL OF WATERWAY, PORT, COASTAL AND OCEAN ENGINEERING ASCE / MAY/JUNE 2003

4 Table 3. Comparison of Continuous Equations Ability to Approach Logical Limiting Values Limiting Value Characteristics Continuous Equations Values and Percent Deviation from Reference Values Flow regime Reference value Rubey a Eq. 12 Cheng b Eq. 13 Ahrens c Eqs. 9 and 14 Chang and Liou d Eq. 15 in Eq. 11 Laminar A 1.0 R R % R % R % R % Turbulent A 10 8 R 10,500 R 9, % R 11, % R 10, % R 8, % a Rubey b Cheng c Ahrens d Chang and Liou meier 1981 definition of turbulent flow A 10,000, we have for turbulent flow conditions A p 2.1d 1/3. This corresponds to coarse sand with grain diameters of 0.90 d 1.09 mm. For the data of Dean 1999 shown in Fig. 1, the term A 1/3 can be thought of as a dimensionless particle diameter Cheng Conclusions and Recommendations Coefficients for four continuous equations to predict the fall velocity Reynolds number are adjusted to minimize the error in predicting data tabulated by Hallermeier By comparing the error analysis in Tables 1 and 2, it is seen that adjusting the results in considerable improvement in the predictive ability of two equations. Cheng s 1997 equation, as modified to fit the Hallermeier 1981 data set, Eq. 13, combines simplicity with accuracy. The are constants rather than functions of the Archimedes buoyancy index as are the of Ahrens Eq. 13 fits Hallermeier s 1981 data better than the segmented relationships of either Hallermeier 1981 or van Rijn 1993, Table 2. Fig. 2 shows Eq. 13 plotted with the Hallermeier 1981, data, and it can be seen that the equation follows the trend of that data quite well. Eq. 13 also does a good job following the laminar and turbulent limits, as shown in Table 3. Fig. 2 shows the laminar limit and the turbulent limit plotted over the range 1.0 A This range is somewhat greater than the range of observed data as shown in Fig. 2 and probably should be regarded as the upper limit for the use of Eq. 13. Eq. 13 provides a simple way to make useful estimates of the fall velocity; using Eqs. 9 and 14 provides a more cumbersome but slightly more accurate method to calculate fall velocities A finding related to the research on fall velocity is that the normalized sediment scale parameter is also only a function of the Archimedes buoyancy index. Eq. 16 is a simple equation approximating this functional relationship. This finding shows that the Archimedes buoyancy index is a logical link between fall velocity and equilibrium beach profiles and indicates the surprising importance of the Archimedes buoyancy index variable in coastal engineering. Fig. 1. Predicted values of dimensionless sediment scale parameter using Eq. 16 compared to data of Dean JOURNAL OF WATERWAY, PORT, COASTAL AND OCEAN ENGINEERING ASCE / MAY/JUNE 2003 / 149

5 Fig. 2. Cheng s fall velocity Eq. 13 compared to data and limiting values given by Hallermeier 1981 Appendix. Mass Density of Water An ancillary problem related to calculating the Archimedes buoyancy index and, therefore, the fall velocity and the sediment scale parameter is using the correct value of kinematic viscosity and the mass density of water. Kinematic viscosity can be calculated as shown in Ahrens The following equation can be used to make accurate estimates of the mass density for water over the range of 0 30 C: g/cm 3 c 0 c 1 Ť c 2 Ť 2 (17) where Ť is the temperature in degrees Celsius and the value of the are as follows: c ; c ; and c ; and for salt water: c ; c ; and c The Eq. 17 approach is extremely accurate. Predicted values using Eq. 17 with the above always differ by less than 0.01% from values tabulated in Newman 1977 for the mass density of water. Notation The following symbols are used in this paper: A Archimedes buoyancy index, A gd 3 /v 2 ; A p sediment scale parameter, h A p x 2/3 ; d characteristic diameter of sediment; g acceleration of gravity; h water depth; N number of observations within given data category; R Reynolds number, R wd/v; Ť temperature in degrees Celsius; v kinematic viscosity of water w fall velocity; x distance from shoreline; relative density of sediment, ( s )/ density of fluid; and density of sediment. s References Ahrens, J. P A fall-velocity equation. J. Waterw., Port, Coastal, Ocean Eng., 126 2, Chang, H.-K., and Liou, J.-C Discussion of a free-velocity equation, by John P. Ahrens. J. Waterw., Port, Coastal, Ocean Eng., 127 4, Cheng, N.-S Simplified settling velocity formula for sediment particle. J. Hydraul. Eng., 123 2, Dean, R. G Coastal sediment processes: Toward engineering solutions. Proc., Coastal Sediments 87, ASCE, Reston, Va., Dean, R. G Beach nourishment: A limited review and some recent results. Proc., Coastal Engineering 1998, ASCE, Reston, Va., Hallermeier, R. J Terminal settling velocity of commonly occurring sand grains. Sedimentology, 28, Newman, J. N Marine hydrodynamics, MIT Press, Cambridge, Mass. Rubey, W. W Settling velocities of gravel, sand, and silt particles. Am. J. Sci., , van Rijn, L. C Principles of sediment transport in rivers, estuaries, and coastal seas, Aqua Publications, Amsterdam, The Netherlands. Yalin, M. S Mechanics of sediment transport, Pergamon, Oxford, U.K. 150 / JOURNAL OF WATERWAY, PORT, COASTAL AND OCEAN ENGINEERING ASCE / MAY/JUNE 2003