360. Online publications featuring online calculations [250121]

[Unsteady flow, flood routing, computational hydraulics, amplitude and phase portraits] 034 36034.  Muskingum-Cunge portraits revisited [240309] [Versión en Español - Under construction, 240630]

ABSTRACT:   A comprehensive review of the Muskingum-Cunge method's amplitude and phase portraits is accomplished. Expressions for the amplitude convergence ratio R1 and phase convergence ratio R2 are expressed as a function of the following basic numerical parameters: (a) spatial resolution L/Δx; (b) Courant number C; and (c) weighting factor X. An online calculator Online Muskingum-Cunge Convergence Ratios is used for convenience. For practical applications, input may be expresssed in terms of relevant hydraulic variables, such as mean velocity, flow depth, channel slope, rating exponent, and flood wave time-of-rise. For this case, an online calculator Online Muskingum-Cunge Convergence Ratios Practical is developed.

[Surface-water hydrology, river hydraulics] 033 36033.  Rating curves revisited [230808] [Versión en Español]

ABSTRACT. A review of rating curve concepts is accomplished. The curve depicts the relationship between discharge and stage at a given point in a stream or river. Under steady uniform flow, the rating is unique, i.e., there is one value of discharge for each value of stage. However, the situation is complicated under unsteady flow conditions. Depending on the flow, the rating may actually be nonunique, typically showing a histeresis, i.e., a nonsingular value of discharge as a function of stage, and vice versa. Further complications arise if the flow is actually able to move its bed, as it is typically the case in alluvial channels in the absence of geologic controls. In medium-sized to large rivers, the flow acts to minimize the changes in stage, effectively reducing the variability in stage between low flows and high flows. Nature accomplishes this feat by increasing form friction during low flows, by way of bedforms, such as ripples and dunes, and by obliterating the bedforms during high flows. This effectively reduces form friction to negligible amounts, leaving grain friction as the only acting type of friction. Lastly, unsteady sediment transport may also cause alterations in rating curves. Some very active rivers may be aggrading; yet, others may be degrading. Active rivers may substantially change their cross section and/or course/alignment during major floods. Invariably, shifts in rating are the net result of these geomorphic processes.

[Unsteady flow, Vedernikov number, catchment routing] 032 36032.  Comparison between kinematic and dynamic hydraulic diffusivities using script ONLINEOVERLAND [220708] [Versión en Español]

ABSTRACT: Two existing formulations for the coefficient of hydraulic diffusivity in diffusion wave modeling of catchment dynamics are evaluated. While the kinematic hydraulic diffusivity is independent of the Vedernikov number, the dynamic hydraulic diffusivity is dependent on the Vedernikov number. We use the script ONLINEOVERLAND, an analytical tool capable of considering either the kinematic or dynamic approaches to model hydraulic diffusivity. For this purpose, we use the reference test problem. Testing of the model varying the drainage area, plane slopes, rating exponent, and channel slope show the model's consistency in simulating catchment response under a variety of flow conditions. Calculated outflow hydrographs correctly show the type of catchment response, showing either superconcentrated, concentrated, or subconcentrated catchment flow, depending on the input data. The reference test problem is confirmed to depict a kinematic wave, effectively featuring zero wave diffusion. Therefore, the differences in outflow hydrograph between the two hydraulic diffusivities are shown to be negligible. For the general problem, which includes all types of shallow-water waves, i.e., kinematic, diffusion, and mixed kinematic-dynamic, the use of the dynamic hydraulic diffusivity is recommended.

[Unsteady flow, roll waves, Vedernikov number, channel design] 031 36031.  Effect of cross-sectional shape on free-surface channel instability [220424] [Versión en Español]

ABSTRACT:   A study of the effect of cross-sectional shape on free-surface channel hydrodynamic instability is accomplished. At the outset, the rating exponent β, Froude number F, and Vedernikov number V are identified as the controlling variables. A steep, lined channel is specified for the analysis. The selected design discharge is Q = 100 m3/s, with bottom slope S = 0.06 and Manning's n = 0.025, closely resembling the flow conditions of the Huayñajahuira river, in La Paz, Bolivia, where roll waves have been shown to recur with worrisome regularity. The testing program considers the variation of the bottom width b in the range 5 ≥ b ≥ 1, at 1-m intervals, and the side slope z in the range 0.25 ≥ z ≥ 0, at 0.05 intervals.

[Surface-water hydrology, unit hydrographs] 030 36030.  Comparison between Clark's original and Clark's Ponce unit hydrograph [201224]

ABSTRACT:   The theoretical basis for Clark's original 1945 and Clark's Ponce 1989 methods of catchment routing are explained and compared. It is shown that Ponce's method consistently provides a somewhat longer time base and a correspondingly smaller peak discharge than Clark's original methodology. This is a direct consequence of Ponce's use of a continuous time-area derived unit hydrograph, in lieu of the discrete hyetograph used by Clark. However, the differences in peak discharge are consistent with the methodologies used and do not appear to be significant.

[Flood routing, unsteady flow] 029 36029.  A dimensionless convection-diffusion-dispersion equation of flood waves [201213] [Versión en Español]

ABSTRACT:   The coefficients of the dimensionless partial differential equation of convection-diffusion-dispersion of flood waves are derived, and shown to be functions of the Froude and Vedernikov numbers only. The Froude number is the ratio of mean velocity to relative dynamic wave celerity. The Vedernikov number is the ratio of relative kinematic wave celerity to relative dynamic wave celerity. The third-order convection-diffusion-dispersion equation can be used to analyze flood propagation problems where both diffusion and dispersion are deemed to be significant.

[Fluvial geomorphology] 028 36028.  The equiiibrium shape of self-formed channels in noncohesive alluvium [201107]

ABSTRACT: The Lane et al. (1959) theory for the equilibrium shape of self-formed channels in noncohesive alluvium has been revisited, with all assumptions and derivations clearly stated. The results are used to estimate self-formed top-width/maximum-depth ratios as a function of: (1) the friction angle of the noncohesive material forming the bed, and (2) the lift-to-drag force ratio acting on the particles. The findings may be used as a point-of-start in the study of unsteady alluvial channel morphology.

[DO sag curve, water quality] 027 36027.  Differential equation for DO sag curve [200726] [Versión en Español]

ABSTRACT:   The differential equation for the dissolved oxygen sag curve (DO sag curve) is derived. The solution of this differential equation can be shown to be essentially the same as that of the well known Streeter-Phelps equation (Streeter and Phelps, 1925). Unlike the latter, the differential equation derived herein can be solved numerically and, therefore, does not require integration. Moreover, the differential equation is valid for all deoxygenation and oxygenation constants, unlike the Streeter-Phelps equation, which is undefined when these constants are equal. Two online calculators: (a) single case, and (b) general case, round up the analysis.

[Flood routing, unsteady flow] 026 36026.  Analytical verification of Muskingum-Cunge flood routing method [200620] [Versión en Español]

ABSTRACT:   A verification of the Muskingum-Cunge flood routing method is accomplished by comparing theoretically calculated peak outflow and travel time with those generated using the constant-parameter Muskingum-Cunge method. The remarkably close agreement between analytical and numerical results underscores the utility of Muskingum-Cunge routing as a viable and accurate method for practical applications in flood hydrology.

[Hydrography, hydrology] 025 36025.  The true source of the Missouri river [200315]

ABSTRACT: The source of a large river system, for example, the Missouri river, is often taken as the location of the uppermost spring in the farthest tributary. Under this optic, the source of the Missouri river would be Brower's Spring, located in the headwaters of Hell Roaring Creek, in southwestern Montana, near the border with Idaho. It is shown here that the true source of the Missouri river is at a nearby location, on the U.S. Continental Divide, at 44° 33' 27.2" N and 111° 28' 9.55" W, at an elevation of 2,864 m (9,396 ft). By comparison, Brower's Spring is at 44° 33' 0.74" N and 111° 28' 25.2" W, at an elevation of 2,684 m (8,806 ft). The distance between the two points is calculated to be 886.64 m (2,908.9 ft).

[Open-channel hydraulics, urban drainage, flood control] 024 36024.  Design of a stable channel on a steep slope using the exponent of the rating curve [200219]

ABSTRACT:   The design of a lined channel, with a steep slope, to be hydraulically stable is governed by the well-known Vedernikov criterion. However, it can be shown that this depends on the shape of the cross section, whether trapezoidal, rectangular, or triangular. For a given section, there is a unique relationship between the exponent β of the rating curve Q - A (discharge vs flow area), and the value of V /F, in which V = Vedernikov number, and F = Froude number. In this work we use the onlinechannel15b calculator to calculate the value of β and the corresponding Vedernikov number for a rectangular, trapezoidal, or triangular cross section. Three series of tests are carried out in a hypothetical channel, keeping constant discharge Q, Manning's n, and bottom slope S, and varying the value of the side slope z: (a) 0.25; (b) 0.5, and (c) 1. It is concluded that when the bottom width b is reduced, the Vedernikov number V is reduced more quickly to values less than 1 for the lower values of z in the range 0.25 ≤ z ≤ 1.

[Urban drainage, open-channel hydraulics, unsteady flow] 023 36023.  The control of roll waves in channelized rivers [190611] [Versión en Español]

ABSTRACT:   The theoretical foundations and relevant experience with open-channel flow instability are examined with the objective of controlling roll waves. The latter recur with relative frequency in channelized rivers where the Vedernikov number V has increased above the threshold V = 1 due to the channelization. Several examples show conclusively the existence of a definite relation between β, the exponent of the discharge-area rating, and V. As the cross-sectional shape departs from rectangular, β decreases accordingly, resulting in a drop in the value of V. A sufficient decrease in β will cause a drop in V below the threshold of flow instability V = 1.

[Water Chemistry, Environmental Hydrology] 022 36022.  The Properties of Water [190607] [Versión en Español]

ABSTRACT: This article explores the properties of water, including physical, chemical, and biological properties. Most of the properties of water span more than one field, such as physics and chemistry, or chemistry and biology, or biology and physics. Understanding the nature of water requires a thorough interdisciplinary approach to science.

[Sedimentation Engineering] 021 36021.  Froude number for initiation of motion calculated online [180417] [Versión en Español]

ABSTRACT:   The classical Shields criterion for initiation of motion is expressed in terms of the Froude number and associated mean velocity required for initiation of motion in a sand-bed channel. To solve the problem exactly, an iterative algorithm is developed to calculate these values using an online calculator.

[Sustainability of groundwater] 020 36020.  How much water could be pumped from an aquifer and still remain sustainable? [180206] [Versión en Español]

ABSTRACT: The concepts of safe yield and sustainable yield of groundwater are analyzed and compared in the context of a hydrologic balance. It is surmised here that vertical recharge, i.e., the recharge originating in local precipitation, is the only recharge that may be tapped for capture by groundwater to avoid encroachment on established rights. A methodology to evaluate vertical recharge is developed and tested. The methodology is based on L'vovich's cybernetic hydrologic balance. This coefficient represents the fraction of precipitation that reaches the water table; therefore, it may be used to evaluate and assess sustainable groundwater yield.

[Surface-water hydrology] 019 36019.  Why is the cybernetic hydrologic balance better suited for yield hydrology than the conventional approach? [180115]

ABSTRACT: A comparison between the conventional approach to the hydrologic balance and L'vovich's catchment wetting approach, referred herein as the cybernetic approach, reveals fundamental conceptual differences. The conventional approach is seen to be mostly suited to event hydrology, particularly for applications of flood hydrology and related urban hydrology. On the other hand, the cybernetic approach is suited to yield hydrology, i.e., for determinations of the availability of water resources on an annual basis.

[Flood routing] 018 36018.  The Thomas problem with online computation [161209] [Versión en Español]

ABSTRACT:   An online calculator has been developed and tested using the Muskingum-Cunge method to solve the classical Thomas problem of flood routing. The calculator can vary peak inflow, time base, and channel length. The choice for peak inflow qp (cfs/ft) is: (a) 200, (b) 500, and (c) 1,000. The choice for time base Tb (hr) is: (a) 48, (b) 96, and (c) 192. The choice for channel length L (mi) is: (a) 200, and (b) 500. The results are in agreement with analytical results of the Thomas problem.

[Open-channel hydraulics] 017 36017.  The inherently stable channel with online computation [160530] [Versión en Español]

ABSTRACT: The inherently stable channel is reviewed, elucidated, and calculated online. Theoretically, such a channel will become neutrally stable when the Froude number reaches infinity. Thus, constructing an inherently stable channel provides an unrealistically high factor of safety against roll waves. This suggests the possibility of designing instead a conditionally stable cross-sectional shape, for a suitably high but realistic Froude number such as F = 25, for which the risk of roll waves would be so small as to be of no practical concern.

[Computational Hydraulics] 016 36016.  Muskingum-Cunge amplitude and phase portraits with online computation [160430] [Versión en Español]

ABSTRACT: A comprehensive review of the amplitude and phase portraits of the Muskingum-Cunge method of flood routing is accomplished. Expressions for the amplitude and phase convergence ratios are developed as a function of: (a) spatial resolution L/Δx; (b) Courant number C; and (c) weighting factor X. It is concluded that the Muskingum-Cunge routing model is a good representation of the physical prototype, provided: (1) the spatial resolution is sufficiently high, (2) the Courant number is around 1, and (3) the weighting factor is high enough in the range 0.0 ≤ X ≤ 0.5. Two online calculators of the convergence ratios are developed and tested.

[Open-channel hydraulics] 015 36015.  Design of channel transitions [151117] [Versión en Español]

ABSTRACT:   The hydraulic design of a channel transition is described and explained. The calculation of an inlet transition between canal and flume is shown by an example, originally presented by Hinds (1928) and subsequently cited by Chow (1959). The example is reproduced with detailed explanation and minor corrections for rounding accuracy. An online calculator is provided.

[Channel morphology and sediment transport] 014 36014.  The Lane relation revisited, with online calculation [150223] [Versión en Español]

ABSTRACT: A new Lane relation of fluvial hydraulics is derived from basic principles of sediment transport. It is expressed as follows: Qs (ds/R)1/3 ∝ γ Qw So Unlike the original Lane relation, this new relation is dimensionless. An online calculator is developed to solve the sediment transport equation arising from the new Lane relation.

[Open-channel Hydraulics] 013 36013.  Chow, Froude, and Vedernikov [140624] [Versión en Español]

ABSTRACT: The concepts of Froude and Vedernikov numbers are reviewed on the occasion of the 50th anniversary of the publication of Ven Te Chow's Handbook of Applied Hydrology. While the Froude number (F) is standard fare in hydraulic engineering practice, the Vedernikov number (V) remains to be recognized by many practicing engineers. A comprehensive description of the variation of β, the altogether important exponent of the discharge-flow area rating (β - 1 = V/F) , is accomplished to recognize the contributions of Professor Ven Te Chow to the hydraulic engineering profession. Two online calculators are presented to round up the experience.

[Unsteady Open-channel Flow] 012 36012.  Runoff diffusion reexamined [140611] [Versión en Español]

ABSTRACT:   The concept of runoff diffusion is reexamined. Diffusion is inherent to reservoirs and it is always produced in flow through reservoirs. In channel flow, diffusion is produced in the absence of kinematic wave conditions, i.e., under diffusion wave conditions, provided the Vedernikov number is less than 1. In catchment runoff, diffusion is produced: (1) for all wave types, when the time of concentration exceeds the effective rainfall duration, a condition which is usually associated with midsize and large basins, or (2) for all effective rainfall durations, when the wave is a diffusion wave, which is usually associated with a sufficiently mild catchment slope.

[Unsteady Open-channel Flow] 011 36011.  The dynamic hydraulic diffusivity reexamined [140429] [Versión en Español]

ABSTRACT: The concept of hydraulic diffusivity and its extensions to the dynamic regime are examined herein. Hayami (1951) originated the concept of hydraulic diffusivity in connection with the propagation of flood waves. Dooge (1973) extended Hayami's hydraulic diffusivity to the realm of dynamic waves. Subsequently, Dooge et al. (1982) expressed the dynamic hydraulic diffusivity in terms of the exponent of the discharge-area rating. Lastly, Ponce (1991) expressed it in terms of the Vedernikov number, further clarifying the mechanics of flood wave propagation.

[Open-channel Hydraulics] 010 36010.  The limiting contraction ratio revisited [140409] [Versión en Español]

ABSTRACT: Henderson's formulations of the energy-based and momentum-based limiting contraction ratios are reviewed (Henderson 1966). Henderson's explicit energy-based equation is found to be correct, however, his implicit momentum-based equation is found to be incorrect. A new explicit momentum-based equation is derived, rendering the implicit formulation unnecessary. An online calculator enables the calculation of the limiting contraction ratio for both energy and momentum formulations.

[Evaporation] 009 36009.  The Penman-Monteith Method [140312] [Versión en Español]

ABSTRACT:   The Penman-Monteith combination method for the calculation of evaporation is reviewed and clarified. Unlike the original Penman model, in the Penman-Monteith model the mass-transfer evaporation rate is calculated based on physical principles. An illustrative example is worked out to show the computational procedure. An online calculation using ONLINE PENMAN-MONTEITH gives the same answer.

[Open-channel Hydraulics] 008 36008.  Comparison of sharp-crested weirs for discharge measurement in open-channel flow [131104]

ABSTRACT: This document provides a tabular comparison of several sharp-crested weirs for discharge measurement in open-channel flow. The following weirs are considered: (1) V-notch, fully contracted; (2) V-notch, partially contracted; (3) Cipolletti; (4) rectangular; (5) standard contracted rectangular; and (6) standard suppressed rectangular. Descriptions follow the USBR Water Measurement Manual.

[Rainfall-runoff transform] 007 36007.  Comparison of two types of Clark unit hydrographs [131006] [Versión en Español]

ABSTRACT:  Clark's original unit hydrograph and Ponce's somewhat improved version are explained and compared. Clark's procedure routes, through a linear reservoir, the discrete time-area-derived unit-runoff hyetograph, while Ponce's procedure routes the continuous time-area-derived unit hydrograph. Since the unit hydrograph has a longer time base than the unit-runoff hyetograph, Ponce's procedure provides a somewhat smaller peak discharge than Clark's. The difference, however, does not appear to be substantial.

[Flood hydrology] 006 36006.  Creager and flood wave diffusion [130821] [Versión en Español]

ABSTRACT:   The Creager curves are reinterpreted in light of the theory of flood wave diffusion. Experience shows that greater flood wave diffusion corresponds with larger drainage areas. Thus, the trend of the Creager curves admirably reflects the flood wave diffusion that is likely to be present in the real world.

[Evapotranspiration] 005 36005.  Evapotranspiration using the Shuttleworth-Wallace formula [130516]

ABSTRACT: An online calculator of the Shuttleworth-Wallace method for calculating evapotranspiration from sparse crops is developed.   The method can be used to complement evapotranspiration calculations based on the Penman-Monteith method.

[Gradually varied flow] 004 36004.  Gradually varied flow profiles using critical slope and online calculators [130302] [Versión en Español]

ABSTRACT:   Gradually varied flow water-surface profiles are expressed in terms of the critical slope Sc. In this way, the flow-depth gradient dy/dx is shown to be strictly limited to values outside the range encompassed by Sc and So, in which So is the bed slope. This new approach improves and completes the definition of flow-depth-gradient ranges in the analysis of water-surface profiles. Online calculators are provided to round up the experience.

[Water balance] 003 36003.  Water balance using catchment wetting [100728] [Versión en Español] ABSTRACT:   The concept of catchment wetting due to L'vovich (1979) enables a better water balance than that possible with conventional methods. Given precipitation and streamflow data, and using an appropriate baseflow separation technique, L'vovich's method enables the calculation of the matrix of precipitation/runoff/surface_runoff/baseflow/wetting/vaporization. This provides a clearer understanding of all the components of the water balance for a given gaged catchment. [Unit hydrograph theory and practice] 002 36002.  Cascade and convolution: One and the same [090525] [Versión en Español] ABSTRACT:   The methods of cascade of linear reservoirs and unit hydrograph convolution are shown to be one and the same when the cascade parameters are used to calculate the unit hydrograph of the convolution. In the absence of gaged data, the cascade parameters may be estimated based on geomorphology. Once the parameters are established, the composite flood hydrograph is uniquely determined. [Unit hydrograph theory and practice] 001 36001.  A general dimensionless unit hydrograph [090521] [Versión en Español] ABSTRACT:   A general dimensionless unit hydrograph (GDUH) based on the cascade of linear reservoirs is formulated and calculated online. The GDUH is shown to be solely a function of the Courant number and the number of linear reservoirs. Since the GDUH is independent of the basin drainage area and the unit hydrograph duration, it is applicable on a global basis. Each GDUH ia a function only of the basin's prevailing runoff diffusion properties. The model's two-parameter feature provides increased flexibility for simulating a wide range of diffusion effects.