A Diffusion
Hydrodynamic Model
Authored by Theodore V. Hromadka II, Chung-Cheng Yen and Prasada Rao Chung-Cheng Yen and Prasada Rao
The Diffusion Hydrodynamic Model (DHM), as presented in the 1987 USGS publication, was one of the first computational fluid dynamics computational programs based on the groundwater program MODFLOW, which evolved into the
control volume modeling approach. Over the following decades, others developed similar computational programs that either used the methodology and approaches
presented in the DHM directly or were its extensions that included additional components and capacities. Our goal is to demonstrate that the DHM, which was developed in an age preceding computer graphics/visualization tools, is as robust as any of the popular models that are currently used. We thank the USGS for their
approval and permission to use the content from the earlier USGS report.
Published in London, UK
© 2020 IntechOpen
© PashaIgnatov / iStock
ISBN 978-1-83962-817-7
A Diffusion Hydrodynamic Model
A Diffusion
Hydrodynamic Model
Authored by Theodore V. Hromadka II, Chung-Cheng Yen and Prasada Rao
Published in London, United Kingdom
A Diffusion Hydrodynamic Model
http://dx.doi.org/10.5772/intechopen.90224
Authored by Theodore V. Hromadka II, Chung-Cheng Yen and Prasada Rao
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IN D E XE D
Meet the authors
Hromadka & Associates’ Principal and Founder, Theodore Hromadka II, PhD, PhD, PhD, PH, PE, has extensive scientific, engineering, expert witness, and litigation support experience.
His frequently referenced scientific contributions to the hydro- logic, earth, and atmospheric sciences have been widely pub- lished in peer-reviewed scientific literature, including 30 books and more than 500 scientific papers, book chapters, and gov- ernment reports. His professional engineering experience includes supervision and development of over 1500 engineering studies. He is currently a faculty member at the United States Military Academy at West Point, New York.
Chung-Cheng Yen received his Ph.D. degree from the University of California, Irvine, in 1985. He has more than 35 years of expe- rience in the field of water resource engineering, specializing in hydrology, hydraulics, dam breach, and groundwater modeling.
His work experience includes rainfall analysis, flood frequency analysis, rainfall-runoff modeling, detention basin flood routing analysis, drainage master plan, FEMA floodplain evaluations and mapping, dam breach analysis and flood inundation mapping, and the USACE risk and uncertainty analysis. Dr. Yen has conducted floodplain analyses using 2-D hydrodynamic models (such as DHM, FLO-2D, HEC-RAS 1D/2D, and XPSWMM), prepared hydrologic and hydraulic studies for government and private entities, and drainage master plans for various cities in southern California.
Prasada Rao is a Professor in the Civil and Environmental Engineering Department at California State University, Fuller- ton. His current research areas relate to surface and subsurface flow modeling and computational mathematics. He has worked extensively on developing innovative, hydraulic and hydrological modelling solutions to better predict surface flow phenomena along with its impact on groundwater levels. He has also worked on developing parallel hydraulic models for large scale applications. He has taught undergraduate and graduate level courses in hydraulics, hydrology, open channel flow, and hydraulic structures.
Contents
Preface XI
Chapter 1 1
Diffusion Hydrodynamic Model Theoretical Development by Theodore V. Hromadka II and Chung-Cheng Yen
Chapter 2 13
Verification of Diffusion Hydrodynamic Model by Theodore V. Hromadka II and Chung-Cheng Yen
Chapter 3 27
Program Description of the Diffusion Hydrodynamic Model by Theodore V. Hromadka II and Chung-Cheng Yen
Chapter 4 33
Applications of Diffusion Hydrodynamic Model by Theodore V. Hromadka II and Chung-Cheng Yen
Chapter 5 55
Reduction of the Diffusion Hydrodynamic Model to Kinematic Routing by Theodore V. Hromadka II and Chung-Cheng Yen
Chapter 6 63
Comparison of DHM Results for One- and Two-Dimensional Flows with Experimental and Numerical Data
by Theodore V. Hromadka II and Prasada Rao
Preface XIII
Chapter 1 1
Diffusion Hydrodynamic Model Theoretical Development by Theodore V. Hromadka II and Chung-Cheng Yen
Chapter 2 13
Verification of Diffusion Hydrodynamic Model by Theodore V. Hromadka II and Chung-Cheng Yen
Chapter 3 27
Program Description of the Diffusion Hydrodynamic Model by Theodore V. Hromadka II and Chung-Cheng Yen
Chapter 4 33
Applications of Diffusion Hydrodynamic Model by Theodore V. Hromadka II and Chung-Cheng Yen
Chapter 5 55
Reduction of the Diffusion Hydrodynamic Model to Kinematic Routing by Theodore V. Hromadka II and Chung-Cheng Yen
Chapter 6 63
Comparison of DHM Results for One- and Two-Dimensional Flows with Experimental and Numerical Data
by Theodore V. Hromadka II and Prasada Rao
Preface
The Diffusion Hydrodynamic Model (DHM), as presented in the 1987 USGS publication (https://pubs.er.usgs.gov/publication/wri874137), was one of the first computational fluid dynamics computational programs based on the groundwater program MODFLOW, which evolved into the control volume modeling approach.
In the DHM, overland flow effects are modeled by a two-dimensional unsteady flow hydraulic model based on the diffusion (non-inertial) form of the governing flow equations. The channel flow is modeled using a one-dimensional unsteady flow hydraulic model based on the diffusion type equation. DHM can simulate both approximate unsteady supercritical and subcritical flow (without the user predeter- mining hydraulic controls), backwater flooding effects, and escaping and returning flow from the two-dimensional overland flow model to the channel system. The model is also capable of treating such effects as backwater, drawdown, channel overflow, storage, and ponding.
Since 1987, others developed similar computational programs that either used the methodology and approaches presented in the DHM directly or were its exten- sions that included additional components and capacities. Later, the DHM itself was extended considerably to the version EDHM (Extended DHM), although the fundamental mechanics of the procedures were retained.
The original effort was funded by the USGS, and the authors acknowledge their support. The report submitted to the USGS is available online at https://pubs.
er.usgs.gov/publication/wri874137, and some of the relevant contract details are:
Water Resources Investigations Report: 87- 4137 Name of Contractor: Williamson and Schmid Principal Investigator: Theodore V. Hromadka II Contract Officer’s Representative: Marshall E. Jennings Short Title of Work: Diffusion Hydrodynamic Model Year Published: 1987
The time evolution of this document from the original 1987 USGS report to the present content in this book is summarized below.
Although the original report is available on the web as a pdf file, we were unable to locate the relevant computer files on our memory devices. Correspondence with the USGS also pointed only to the report that is on the web and not to any elec- tronic files available offline. Since some of the pages and figures in the pdf report lacked clarity, we started this book by retyping the entire report (as it is) along with the equations using MS Word. A few graduate students from the Civil and Environmental Engineering Department at California State University, Fullerton
Preface
The Diffusion Hydrodynamic Model (DHM), as presented in the 1987 USGS publication (https://pubs.er.usgs.gov/publication/wri874137), was one of the first computational fluid dynamics computational programs based on the groundwater program MODFLOW, which evolved into the control volume modeling approach.
In the DHM, overland flow effects are modeled by a two-dimensional unsteady flow hydraulic model based on the diffusion (non-inertial) form of the governing flow equations. The channel flow is modeled using a one-dimensional unsteady flow hydraulic model based on the diffusion type equation. DHM can simulate both approximate unsteady supercritical and subcritical flow (without the user predeter- mining hydraulic controls), backwater flooding effects, and escaping and returning flow from the two-dimensional overland flow model to the channel system. The model is also capable of treating such effects as backwater, drawdown, channel overflow, storage, and ponding.
Since 1987, others developed similar computational programs that either used the methodology and approaches presented in the DHM directly or were its exten- sions that included additional components and capacities. Later, the DHM itself was extended considerably to the version EDHM (Extended DHM), although the fundamental mechanics of the procedures were retained.
The original effort was funded by the USGS, and the authors acknowledge their support. The report submitted to the USGS is available online at https://pubs.
er.usgs.gov/publication/wri874137, and some of the relevant contract details are:
Water Resources Investigations Report: 87- 4137 Name of Contractor: Williamson and Schmid Principal Investigator: Theodore V. Hromadka II Contract Officer’s Representative: Marshall E. Jennings Short Title of Work: Diffusion Hydrodynamic Model Year Published: 1987
The time evolution of this document from the original 1987 USGS report to the present content in this book is summarized below.
Although the original report is available on the web as a pdf file, we were unable to locate the relevant computer files on our memory devices. Correspondence with the USGS also pointed only to the report that is on the web and not to any elec- tronic files available offline. Since some of the pages and figures in the pdf report lacked clarity, we started this book by retyping the entire report (as it is) along with the equations using MS Word. A few graduate students from the Civil and Environmental Engineering Department at California State University, Fullerton
did the retyping task, and we acknowledge their effort. While many figures were also redrawn, some of the figures (because of the complexity) were left as they were presented in the original report.
Our goal is to show the readers that the Diffusion Hydrodynamic Model, which was developed in an age preceding computer graphics/visualization tools, is as robust as any of the popular models that are currently used in the consulting industry. To this end, we wanted to enhance/revise the original report by adding a new chapter that compares the results of the DHM with current standard models, including HEC- RAS, TUFLOW, Mike 21, RAS 2D, WSPG, and OpenFOAM applied to a few com- plex flows and physical domain scenarios. Since we were building on the original USGS report, approval from the USGS was obtained to enhance the original report.
We thank the USGS for their approval and for permitting us to use the content from the earlier USGS report.
Specific major additions/deletions to the text in the original report are:
(1) The DHM Fortran source code was deleted. Since the source code (DHM21.
FOR) and its executable file for Windows environment (DHM21.EXE), along with the executable file for extended DHM (EDHM21.EXE) and the sample data file can be downloaded from www.diffusionhydrodynamicmodel.com, we did not see an advantage for again listing the source code and the data file.
(2) Chapter 6 has been added.
Minor formatting changes were made to the content in the original report to make it compatible with the publisher’s guidelines. We hope that this report, together with the resources present at the companion website, http://diffusionhydrodynamicmodel.
com, will motivate the readers to use DHM for their applications. The resources in the companion website include:
• DHM Program source code (DHM21.FOR) and its executable code (DHM21.
EXE).
• Executable code for the extended DHM (EDHM21.EXE).
• Sample input data files and related publications/presentations.
In this book, ample applications of DHM are included, which hopefully demon- strate the utility of this modeling approach in many drainage engineering problems.
The model is applied to a collection of one- and two-dimensional unsteady flows hydraulic problems including (1) one-dimensional unsteady flow problem, (2) rainfall-runoff model, (3) dam-break flow analysis, (4) estuary model, (5) channel floodplain interface model, (6) mixed flows in open channel, (7) overland flow, and (8) flow through a constriction. For selected applications, DHM results have been compared with those from other widely used hydraulic and CFD models.
Consequently, the diffusion hydrodynamic model promises to result in a highly useful, accurate, and simple to use computer model, which is of immediate use to practicing flood control engineers. Use of the DHM in surface runoff problems will result in a highly versatile and practical tool which significantly advances the current state-of-the-art flood control system and flood plain mapping analysis procedures, resulting in more accurate predictions in the needs of the flood control
system, and potentially proving a considerable cost saving due to reduction of conservation used to compensate for the lack of proper hydraulic unsteady flow effects approximation.
Theodore V. Hromadka II Department of Mathematical Sciences,
United States Military Academy, West Point, NY, USA Chung-Cheng Yen Tetra Tech, Irvine, CA, USA Prasada Rao Department of Civil and Environmental Engineering, California State University, Fullerton, CA, USA
presented in the original report.
Our goal is to show the readers that the Diffusion Hydrodynamic Model, which was developed in an age preceding computer graphics/visualization tools, is as robust as any of the popular models that are currently used in the consulting industry. To this end, we wanted to enhance/revise the original report by adding a new chapter that compares the results of the DHM with current standard models, including HEC- RAS, TUFLOW, Mike 21, RAS 2D, WSPG, and OpenFOAM applied to a few com- plex flows and physical domain scenarios. Since we were building on the original USGS report, approval from the USGS was obtained to enhance the original report.
We thank the USGS for their approval and for permitting us to use the content from the earlier USGS report.
Specific major additions/deletions to the text in the original report are:
(1) The DHM Fortran source code was deleted. Since the source code (DHM21.
FOR) and its executable file for Windows environment (DHM21.EXE), along with the executable file for extended DHM (EDHM21.EXE) and the sample data file can be downloaded from www.diffusionhydrodynamicmodel.com, we did not see an advantage for again listing the source code and the data file.
(2) Chapter 6 has been added.
Minor formatting changes were made to the content in the original report to make it compatible with the publisher’s guidelines. We hope that this report, together with the resources present at the companion website, http://diffusionhydrodynamicmodel.
com, will motivate the readers to use DHM for their applications. The resources in the companion website include:
• DHM Program source code (DHM21.FOR) and its executable code (DHM21.
EXE).
• Executable code for the extended DHM (EDHM21.EXE).
• Sample input data files and related publications/presentations.
In this book, ample applications of DHM are included, which hopefully demon- strate the utility of this modeling approach in many drainage engineering problems.
The model is applied to a collection of one- and two-dimensional unsteady flows hydraulic problems including (1) one-dimensional unsteady flow problem, (2) rainfall-runoff model, (3) dam-break flow analysis, (4) estuary model, (5) channel floodplain interface model, (6) mixed flows in open channel, (7) overland flow, and (8) flow through a constriction. For selected applications, DHM results have been compared with those from other widely used hydraulic and CFD models.
Consequently, the diffusion hydrodynamic model promises to result in a highly useful, accurate, and simple to use computer model, which is of immediate use to practicing flood control engineers. Use of the DHM in surface runoff problems will result in a highly versatile and practical tool which significantly advances the current state-of-the-art flood control system and flood plain mapping analysis procedures, resulting in more accurate predictions in the needs of the flood control
effects approximation.
Theodore V. Hromadka II Department of Mathematical Sciences,
United States Military Academy, West Point, NY, USA Chung-Cheng Yen Tetra Tech, Irvine, CA, USA Prasada Rao Department of Civil and Environmental Engineering, California State University, Fullerton, CA, USA
Chapter 1
Diffusion Hydrodynamic Model Theoretical Development
Theodore V. Hromadka II and Chung-Cheng Yen
Abstract
In this chapter, the governing flow equations for one- and two-dimensional unsteady flows that are solved in the diffusion hydrodynamic model (DHM) are presented along with the relevant assumptions. A step-by-step derivation of the simplified equations which are based on continuity and momentum principles are detailed. Characteristic features of the explicit DHM numerical algorithm are discussed.
Keywords: unsteady flow, conservation of mass, finite difference, explicit scheme, flow equations
1. Introduction
Many flow phenomena of great engineering importance are unsteady in
characters and cannot be reduced to a steady flow by changing the viewpoint of the observer. A complete theory of unsteady flow is therefore required and will be reviewed in this section. The equations of motion are not solvable in the most general case, but approximations and numerical methods can be developed which yield solutions of satisfactory accuracy.
2. Review of governing equations
The law of continuity for unsteady flow may be established by considering the conservation of mass in an infinitesimal space between two channel sections (Figure 1). In unsteady flow, the discharge, Q, changes with distance, x, at a rate
∂Q
∂x, and the depth, y, changes with time, t, at a rate∂∂yt. The change in discharge volume through space dx in the time dt is ∂∂Qx dxdt. The corresponding change in channel storage in space is Tdx ∂∂yt dt¼dx ∂∂At dt in which A¼Ty. Because water is incompressible, the net change in discharge plus the change in storage should be zero, that is
∂Q
∂x
dxdtþTdx ∂y
∂t dt¼ ∂Q
∂x
dxdtþdx ∂A
∂t
dt¼0
Diffusion Hydrodynamic Model Theoretical Development
Theodore V. Hromadka II and Chung-Cheng Yen
Abstract
In this chapter, the governing flow equations for one- and two-dimensional unsteady flows that are solved in the diffusion hydrodynamic model (DHM) are presented along with the relevant assumptions. A step-by-step derivation of the simplified equations which are based on continuity and momentum principles are detailed. Characteristic features of the explicit DHM numerical algorithm are discussed.
Keywords: unsteady flow, conservation of mass, finite difference, explicit scheme, flow equations
1. Introduction
Many flow phenomena of great engineering importance are unsteady in
characters and cannot be reduced to a steady flow by changing the viewpoint of the observer. A complete theory of unsteady flow is therefore required and will be reviewed in this section. The equations of motion are not solvable in the most general case, but approximations and numerical methods can be developed which yield solutions of satisfactory accuracy.
2. Review of governing equations
The law of continuity for unsteady flow may be established by considering the conservation of mass in an infinitesimal space between two channel sections (Figure 1). In unsteady flow, the discharge, Q, changes with distance, x, at a rate
∂Q
∂x, and the depth, y, changes with time, t, at a rate∂∂yt. The change in discharge volume through space dx in the time dt is ∂∂Qx dxdt. The corresponding change in channel storage in space is Tdx ∂∂yt dt¼dx ∂∂At dt in which A¼Ty. Because water is incompressible, the net change in discharge plus the change in storage should be zero, that is
∂Q
∂x
dxdtþTdx ∂y
∂t dt¼ ∂Q
∂x
dxdtþdx ∂A
∂t
dt¼0
Simplifying
∂Q
∂xþT∂y
∂t¼0 (1)
or
∂Q
∂xþ∂A
∂t ¼0 (2)
At a given section, Q = VA; thus Eq. (1) becomes
∂ðVAÞ
∂x þT∂y
∂t¼0 (3)
or
A∂V
∂xþV∂A
∂xþT∂y
∂t¼0 (4)
Because the hydraulic depth D = A/T and∂A¼T∂y, the above equation may be written as
D∂V
∂xþV∂y
∂xþ∂y
∂t¼0 (5)
The above equations are all forms of the continuity equation for unsteady flow in open channels. For a rectangular channel or a channel of infinite width, Eq. (1) may be written as
∂q
∂xþ∂y
∂t¼0 (6)
where q is the discharge per unit width.
Figure 1.
Continuity of unsteady flow.
A Diffusion Hydrodynamic Model
3. Equation of motion
In a steady, uniform flow, the gradient,dHdx, of the total energy line is equal to magnitude of the “friction slope” Sf ¼V2=C2R
, where C is the Chezy coefficient and R is the hydraulic radius. Indeed this statement was in a sense taken as the definition of Sf; however, in the present context, we have to consider the more general case in which the flow is nonuniform, and the velocity may be changing in the downstream direction. The net force, shear force and pressure force, is no longer zero since the flow is accelerating. Therefore, the equation of motion becomes
�γAΔh�τ0PΔx¼ρAΔx V∂V
∂xþ∂V
∂t
that is
τ0 ¼ �γR ∂h
∂xþV g
∂V
∂xþ1 g
∂V
∂t
�γR ∂H
∂x þ1 g
∂V
∂t
(7)
whereτ0is the same shear stress, P is the hydrostatic pressure, h is the depth of water,Δh is the change of depth of water,γis the specific weight of the fluid, R is the mean hydraulic radius, andρis the fluid density. SubstitutingγRτ0=CV22Rinto Eq. (7), we obtain
∂H
∂x þ1 g
∂V
∂t þ V2
C2R¼0 (8)
and this equation may be rewritten as
SeþSaþSf ¼0 (9)
where the three terms of Eq. (9) are called the energy slope, the acceleration slope, and the friction slope, respectively. Figure 2 depicts the simplified representation of energy in unsteady flow.
By substituting H¼V2g2þyþz and the bed slope So¼ �∂∂xzinto Eq. (8), we obtain
∂H
∂x ¼∂z
∂xþ∂y
∂xþV g
∂V
∂x
¼ �Soþ∂y
∂xþV g
∂V
∂x
¼ �1 g
∂V
∂t �Sf
(10) Diffusion Hydrodynamic Model Theoretical Development
DOI: http://dx.doi.org/10.5772/intechopen.93207
Simplifying
∂Q
∂x þT∂y
∂t¼0 (1)
or
∂Q
∂xþ∂A
∂t ¼0 (2)
At a given section, Q = VA; thus Eq. (1) becomes
∂ðVAÞ
∂x þT∂y
∂t¼0 (3)
or
A∂V
∂xþV∂A
∂xþT∂y
∂t¼0 (4)
Because the hydraulic depth D = A/T and∂A¼T∂y, the above equation may be written as
D∂V
∂xþV∂y
∂xþ∂y
∂t¼0 (5)
The above equations are all forms of the continuity equation for unsteady flow in open channels. For a rectangular channel or a channel of infinite width, Eq. (1) may be written as
∂q
∂xþ∂y
∂t¼0 (6)
where q is the discharge per unit width.
Figure 1.
Continuity of unsteady flow.
3. Equation of motion
In a steady, uniform flow, the gradient,dHdx, of the total energy line is equal to magnitude of the “friction slope” Sf ¼V2=C2R
, where C is the Chezy coefficient and R is the hydraulic radius. Indeed this statement was in a sense taken as the definition of Sf; however, in the present context, we have to consider the more general case in which the flow is nonuniform, and the velocity may be changing in the downstream direction. The net force, shear force and pressure force, is no longer zero since the flow is accelerating. Therefore, the equation of motion becomes
�γAΔh�τ0PΔx¼ρAΔx V∂V
∂xþ∂V
∂t
that is
τ0¼ �γR ∂h
∂xþV g
∂V
∂xþ1 g
∂V
∂t
�γR ∂H
∂xþ1 g
∂V
∂t
(7)
whereτ0is the same shear stress, P is the hydrostatic pressure, h is the depth of water,Δh is the change of depth of water,γis the specific weight of the fluid, R is the mean hydraulic radius, andρis the fluid density. SubstitutingγRτ0 =CV22Rinto Eq. (7), we obtain
∂H
∂xþ1 g
∂V
∂t þ V2
C2R¼0 (8)
and this equation may be rewritten as
SeþSaþSf¼0 (9)
where the three terms of Eq. (9) are called the energy slope, the acceleration slope, and the friction slope, respectively. Figure 2 depicts the simplified representation of energy in unsteady flow.
By substituting H¼V2g2þyþz and the bed slope So¼ �∂∂zxinto Eq. (8), we obtain
∂H
∂x ¼∂z
∂xþ∂y
∂xþV g
∂V
∂x
¼ �Soþ∂y
∂xþV g
∂V
∂x
¼ �1 g
∂V
∂t �Sf
(10)
Hence Eq. (8) can be rewritten as
ð11Þ
This equation may be applicable to various types of flow as indicated. This arrangement shows how the nonuniformity and unsteadiness of flows introduce extra terms into the governing dynamic equation.
4. Diffusion hydrodynamic model
4.1 One-dimensional diffusion hydrodynamic model
The mathematical relationships in a one-dimensional diffusion hydrodynamic model (DHM) are based upon the flow equations of continuity (2) and momentum (11) which can be rewritten [1] as
∂Qx
∂x þ∂Ax
∂t ¼0 (12)
∂Qx
∂t þ∂Qx2=Ax
∂x þgAx
∂H
∂xþSfx
¼0 (13)
Figure 2.
Simplified representation of energy in unsteady flow.
A Diffusion Hydrodynamic Model
where Qxis the flow rate; x,t are spatial and temporal coordinates, Axis the flow area, g is the gravitational acceleration, H is the water surface elevation, and Sfxis a friction slope. It is assumed that Sfxapproximated from Manning’s equation for steady flow by [1].
Qx¼1:486
n AxR2=3Sfx1=2 (14) where R is the hydraulic radius and n is a flow resistance coefficient which may be increased to account for other energy losses such as expansions and bend losses.
Letting mxbe a momentum quantity defined by mx¼ ∂Qx
∂t þ∂�Qx2=Ax�
∂x
!
=gAx (15)
then Eq. (13) can be rewritten as
Sfx¼ � ∂H
∂xþmx
� �
(16) In Eq. (15), the subscript x included in mxindicates the directional term. The expansion of Eq. (13) to two-dimensional case leads directly to the terms (mx, my) except that now a cross-product of flow velocities is included, increasing the com- putational effort considerably.
Rewriting Eq. (14) and including Eqs. (15) and (16), the directional flow rate is computed by
Qx¼ �Kx
∂H
∂xþmx
� �
(17) where Qxindicates a directional term and Kxis a type of conduction parameter defined by
Kx¼1:486 n
AxR2=3
∂H
∂xþmx
�� ��1=2 (18)
In Eq. (18), Kxis limited in value by the denominator term being checked for a smallest allowable magnitude (such as��∂∂HXþmX��1=2>10�3).
Substituting the flow rate formulation of Eq. (17) into Eq. (12) gives a diffusion type of relationship
∂
∂XKX
∂H
∂XþmX
� �
¼∂AX
∂t (19)
The one-dimensional model of Akan and Yen [1] assumed mX = 0 in Eq. (18).
The mXterm is assumed to be negligible when combined with the other similar terms—that is, they are considered as a sum rather than as individual directional terms that typically have more significance when examined individually. Addition- ally, the term “diffusion” routing indicates assuming that several convective and other components have a small contribution to the coupled mass and energy balance equations and therefore are neglected in the computational formulation to simplify the model accordingly. Thus, the one-dimensional DHM equation is given by Diffusion Hydrodynamic Model Theoretical Development
DOI: http://dx.doi.org/10.5772/intechopen.93207
Hence Eq. (8) can be rewritten as
ð11Þ
This equation may be applicable to various types of flow as indicated. This arrangement shows how the nonuniformity and unsteadiness of flows introduce extra terms into the governing dynamic equation.
4. Diffusion hydrodynamic model
4.1 One-dimensional diffusion hydrodynamic model
The mathematical relationships in a one-dimensional diffusion hydrodynamic model (DHM) are based upon the flow equations of continuity (2) and momentum (11) which can be rewritten [1] as
∂Qx
∂x þ∂Ax
∂t ¼0 (12)
∂Qx
∂t þ∂Qx2=Ax
∂x þgAx
∂H
∂xþSfx
¼0 (13)
Figure 2.
Simplified representation of energy in unsteady flow.
where Qxis the flow rate; x,t are spatial and temporal coordinates, Axis the flow area, g is the gravitational acceleration, H is the water surface elevation, and Sfxis a friction slope. It is assumed that Sfxapproximated from Manning’s equation for steady flow by [1].
Qx¼1:486
n AxR2=3Sfx1=2 (14) where R is the hydraulic radius and n is a flow resistance coefficient which may be increased to account for other energy losses such as expansions and bend losses.
Letting mxbe a momentum quantity defined by mx¼ ∂Qx
∂t þ∂�Qx2=Ax�
∂x
!
=gAx (15)
then Eq. (13) can be rewritten as
Sfx¼ � ∂H
∂xþmx
� �
(16) In Eq. (15), the subscript x included in mxindicates the directional term. The expansion of Eq. (13) to two-dimensional case leads directly to the terms (mx, my) except that now a cross-product of flow velocities is included, increasing the com- putational effort considerably.
Rewriting Eq. (14) and including Eqs. (15) and (16), the directional flow rate is computed by
Qx¼ �Kx
∂H
∂xþmx
� �
(17) where Qxindicates a directional term and Kxis a type of conduction parameter defined by
Kx¼1:486 n
AxR2=3
∂H
∂xþmx
�� ��1=2 (18)
In Eq. (18), Kxis limited in value by the denominator term being checked for a smallest allowable magnitude (such as��∂∂HXþmX��1=2>10�3).
Substituting the flow rate formulation of Eq. (17) into Eq. (12) gives a diffusion type of relationship
∂
∂XKX
∂H
∂XþmX
� �
¼∂AX
∂t (19)
The one-dimensional model of Akan and Yen [1] assumed mX= 0 in Eq. (18).
The mX term is assumed to be negligible when combined with the other similar terms—that is, they are considered as a sum rather than as individual directional terms that typically have more significance when examined individually. Addition- ally, the term “diffusion” routing indicates assuming that several convective and other components have a small contribution to the coupled mass and energy balance equations and therefore are neglected in the computational formulation to simplify the model accordingly. Thus, the one-dimensional DHM equation is given by
∂
∂XKX∂H
∂X¼∂AX
∂t (20)
where KXis now simplified as Kx¼
1:486 n AxR2=3
∂H
∂X
�� ��21 (21)
For a channel of constant width, WX, Eq. (20) reduces to
∂
∂XKX∂H
∂X¼WX∂H
∂t (22)
Assumptions other than mX= 0 in Eq. (19) result in a family of models:
mx¼
∂ðQx2=AXÞ
∂X
.gAX ðconvective acceleration modelÞ
∂QX
∂t =gAX ðlocal acceleration modelÞ
&
∂QX
∂t þ∂ðQx2=AXÞ
∂X
’,
gAX ðfully dynamic modelÞ
0 ðDHMÞ
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>>
>>
>>
>>
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>>
>>
>>
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(23)
4.2 Two-dimensional diffusion hydrodynamic model
The set of (fully dynamic) 2D unsteady flow equations consists of one equation of continuity
∂qx
∂x þ
∂qy
∂yþ∂H
∂t ¼0 (24)
and two equations of motion
∂qx
∂t þ ∂
∂x qx2
h
� �
þ ∂
∂y qxqy
h
� �
þgh Sfxþ∂H
∂X
� �
¼0 (25)
∂qy
∂t þ ∂
∂y qy2
h
" # þ ∂
∂y qxqy
h
� �
þgh Sfyþ∂H
∂y
� �
¼0 (26)
where qXand qyare flow rates per unit width in the x and y directions; Sfxand Sfy
represents friction slopes in x and y directions; H, h, and g stand for water surface elevation, flow depth, and gravitational acceleration, respectively; and x, y, and t are spatial and temporal coordinates.
The above equation set is based on the assumptions of constant fluid density without sources or sinks in the flow field and of hydrostatic pressure distributions.
The local and convective acceleration terms can be grouped, and Eqs. (25) and (26) are rewritten as
mZþ Sfzþ∂H
∂Z
� �
¼0, z¼x, y (27) A Diffusion Hydrodynamic Model
where mZrepresents the sum of the first three terms in Eqs. (25) or (26) divided by gh. Assuming the friction slope to be approximated by the Manning’s formula, one obtains, in the US customary units for flow in the x or y directions,
qZ ¼1:486
n h5=3Sfz1=2, z¼x, y (28) Eq. (28) can be rewritten in the general case as
qZ ¼ �KZ
∂H
∂Z�KZmZ, z¼x, y (29) where
KZ ¼1:486 n
h5=3
∂H
∂SþmS
1=2, z¼x, y (30)
The symbol s in Eq. (30) indicates the flow direction which makes an angle of θ¼ tan�1qy=qx
with the positive x direction.
The mzterm is assumed to be negligible [1–5] when combined with the other similar terms, i.e., they are considered as a sum rather than as individual directional terms that typically have more significance when examined individually. Additionally, the term “diffusion” routing indicates assuming that several convective and other components have a small contribution to the coupled mass and energy balance equa- tions and therefore are neglected in the computational formulation to simplify the model accordingly. Neglecting this term results in the simple diffusion model
qZ ¼ �KZ
∂H
∂Z, z¼x, y (31)
The proposed 2D DHM is formulated by substituting Eq. (31) into Eq. (24)
∂
∂XKx
∂H
∂Xþ ∂
∂yKy
∂H
∂y ¼∂H
∂t (32)
If the momentum term groupings were retained, Eq. (32) would be written as
∂
∂xKx∂H
∂xþ ∂
∂yKy∂H
∂y þS¼∂H
∂t (33)
where
S¼ ∂
∂xðKxmxÞ þ ∂
∂yKxmy
and Kxand Kyare also functions of mxand my, respectively.
5. Numerical approximation
5.1 Numerical solution algorithm
The one-dimensional domain is discretized across uniformly spaced nodal points, and at each of these points, at time (t) = 0, the values of Manning’s n, an Diffusion Hydrodynamic Model Theoretical Development
DOI: http://dx.doi.org/10.5772/intechopen.93207
∂
∂XKX∂H
∂X ¼∂AX
∂t (20)
where KX is now simplified as Kx¼
1:486 n AxR2=3
∂H
∂X
�� ��12 (21)
For a channel of constant width, WX, Eq. (20) reduces to
∂
∂XKX∂H
∂X¼WX∂H
∂t (22)
Assumptions other than mX = 0 in Eq. (19) result in a family of models:
mx¼
∂ðQx2=AXÞ
∂X
.gAX ðconvective acceleration modelÞ
∂QX
∂t =gAX ðlocal acceleration modelÞ
&
∂QX
∂t þ∂ðQx2=AXÞ
∂X
’,
gAX ðfully dynamic modelÞ
0 ðDHMÞ
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>>
>>
>>
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(23)
4.2 Two-dimensional diffusion hydrodynamic model
The set of (fully dynamic) 2D unsteady flow equations consists of one equation of continuity
∂qx
∂x þ
∂qy
∂y þ∂H
∂t ¼0 (24)
and two equations of motion
∂qx
∂t þ ∂
∂x qx2
h
� �
þ ∂
∂y qxqy
h
� �
þgh Sfxþ∂H
∂X
� �
¼0 (25)
∂qy
∂t þ ∂
∂y qy2
h
" # þ ∂
∂y qxqy
h
� �
þgh Sfyþ∂H
∂y
� �
¼0 (26)
where qXand qyare flow rates per unit width in the x and y directions; Sfxand Sfy
represents friction slopes in x and y directions; H, h, and g stand for water surface elevation, flow depth, and gravitational acceleration, respectively; and x, y, and t are spatial and temporal coordinates.
The above equation set is based on the assumptions of constant fluid density without sources or sinks in the flow field and of hydrostatic pressure distributions.
The local and convective acceleration terms can be grouped, and Eqs. (25) and (26) are rewritten as
mZþ Sfzþ∂H
∂Z
� �
¼0, z¼x, y (27)
where mZrepresents the sum of the first three terms in Eqs. (25) or (26) divided by gh. Assuming the friction slope to be approximated by the Manning’s formula, one obtains, in the US customary units for flow in the x or y directions,
qZ¼1:486
n h5=3Sfz1=2, z¼x, y (28) Eq. (28) can be rewritten in the general case as
qZ¼ �KZ
∂H
∂Z�KZmZ, z¼x, y (29) where
KZ¼1:486 n
h5=3
∂H
∂SþmS
1=2, z¼x, y (30)
The symbol s in Eq. (30) indicates the flow direction which makes an angle of θ¼ tan�1qy=qx
with the positive x direction.
The mzterm is assumed to be negligible [1–5] when combined with the other similar terms, i.e., they are considered as a sum rather than as individual directional terms that typically have more significance when examined individually. Additionally, the term “diffusion” routing indicates assuming that several convective and other components have a small contribution to the coupled mass and energy balance equa- tions and therefore are neglected in the computational formulation to simplify the model accordingly. Neglecting this term results in the simple diffusion model
qZ¼ �KZ
∂H
∂Z, z¼x, y (31)
The proposed 2D DHM is formulated by substituting Eq. (31) into Eq. (24)
∂
∂XKx
∂H
∂Xþ ∂
∂yKy
∂H
∂y ¼∂H
∂t (32)
If the momentum term groupings were retained, Eq. (32) would be written as
∂
∂xKx∂H
∂xþ ∂
∂yKy∂H
∂y þS¼∂H
∂t (33)
where
S¼ ∂
∂xðKxmxÞ þ ∂
∂yKxmy
and Kxand Kyare also functions of mxand my, respectively.
5. Numerical approximation
5.1 Numerical solution algorithm
The one-dimensional domain is discretized across uniformly spaced nodal points, and at each of these points, at time (t) = 0, the values of Manning’s n, an
elevation, and initial flow depth (usually zero) are assigned. With these initial conditions, the solution is advanced to the next time step (t +Δt) as detailed below
1.Between nodal points, compute an average Manning’s n and average geometric factors
2.Assuming mX= 0, estimate the nodal flow depths for the next time step (t + Δt) by using Eqs. (20) and (21) explicitly
3.Using the flow depths at time t and t +Δt, estimate the mid time step value of mXselected from Eq. (23)
4.Recalculate the conductivities KX using the appropriate mXvalues
5.Determine the new nodal flow depths at the time (t +Δt) using Eq. (19), and 6.Return to step (3) until KXmatches mid time step estimates.
The above algorithm steps can be used regardless of the choice of definition for mXfrom Eq. (23). Additionally, the above program steps can be directly applied to a two-dimensional diffusion model with the selected (mX, my) relations incorporated.
5.2 Numerical model formulation (grid element)
For uniform grid elements, the integrated finite difference version of the nodal domain integration (NDI) method [6] is used. For grid elements, the NDI nodal equation is based on the usual nodal system shown in Figure 3. Flow rates across the boundaryГare estimated by assuming a linear trial function between nodal points.
For a square grid of widthδ
qjГE¼ �½KXjГE�½HE�HC�∕δ (34) where
KxjГE
1:486 n h5=3
� �
ГE
.HE�HC δcosθ
��
��
��
��
1=2
;jHE�Hcj≥ ℇ
0 ; Hj E�HCj< ε
8>
<
>: (35)
In Eq. (35), h (depth of water) and n (the Manning’s coefficient) are both the average of their respective values at C and E, i.e., h¼ðhCþhEÞ=2 and n¼
nCþnE
ð Þ=2. Additionally, the denominator of KXis checked such that KXis set to zero if Hj E�HCjis less than a toleranceεsuch as 10�3ft.
The net volume of water in each grid element between time step i and i + 1 is ΔqCi¼q rj E þq rj w þq rj N þq rj S and the change of depth of water isΔHCi¼ ΔqCi∗Δt=δ2for time step i and i + 1 withΔt interval. Then the model advances in time by an explicit approach
HCiþ1¼ΔHCiþHCi (36) where the assumed input flood flows are added to the specified input nodes at each time step. After each time step, the hydraulic conductivity parameters of Eq. (35) are reevaluated, and the solution of Eq. (36) is reinitiated.
A Diffusion Hydrodynamic Model
5.3 Model time step selection
The sensitivity of the model to time step selection is dependent upon the slope of the discharge hydrograph (∂∂Qt) and the grid spacing. Increasing the grid spacing size introduces additional water storage to a corresponding increase in nodal point flood depth values. Similarly, a decrease in time step size allows a refined calculation of inflow and outflow values and a smoother variation in nodal point flood depths with respect to time. The computer algorithm may self-select a time step by increments of halving (or doubling) the initial user-chosen time step size so that a proper balance of inflow-outflow to control volume storage variation is achieved. In order to avoid a matrix solution for flood depths, an explicit time stepping algorithm is used to solve for the time derivative term. For large time steps or a rapid variation in the dam-break hydrograph (such as∂∂Qt is large), a large accumulation of flow volume will occur at the most upstream nodal point. That is, at the dam-break reservoir nodal point, the lag in outflow from the control volume can cause an unacceptable error in the computation of the flood depth. One method that offsets this error is the program to self-select the time step until the difference in the rate of volume accumulation is within a specified tolerance.
Due to the form of the DHM in Eq. (22), the model can be extended into an implicit technique. However, this extension would require a matrix solution process which may become unmanageable for two-dimensional models which utilize hun- dreds of nodal points.
6. Conclusions
The one- and two-dimensional flow equations used in the diffusion hydrody- namic model are derived, and the relevant assumptions are listed. These equations, which are the basis of the model, are based on the conservation of mass and momentum principles. The explicit numerical algorithm and the discretized equa- tions are also presented. The ability of the model to self-select the optimal time step is discussed.
Figure 3.
Two-dimensional finite difference analog.
Diffusion Hydrodynamic Model Theoretical Development DOI: http://dx.doi.org/10.5772/intechopen.93207
elevation, and initial flow depth (usually zero) are assigned. With these initial conditions, the solution is advanced to the next time step (t +Δt) as detailed below
1.Between nodal points, compute an average Manning’s n and average geometric factors
2.Assuming mX= 0, estimate the nodal flow depths for the next time step (t + Δt) by using Eqs. (20) and (21) explicitly
3.Using the flow depths at time t and t +Δt, estimate the mid time step value of mXselected from Eq. (23)
4.Recalculate the conductivities KXusing the appropriate mXvalues
5.Determine the new nodal flow depths at the time (t +Δt) using Eq. (19), and 6.Return to step (3) until KXmatches mid time step estimates.
The above algorithm steps can be used regardless of the choice of definition for mXfrom Eq. (23). Additionally, the above program steps can be directly applied to a two-dimensional diffusion model with the selected (mX, my) relations incorporated.
5.2 Numerical model formulation (grid element)
For uniform grid elements, the integrated finite difference version of the nodal domain integration (NDI) method [6] is used. For grid elements, the NDI nodal equation is based on the usual nodal system shown in Figure 3. Flow rates across the boundaryГare estimated by assuming a linear trial function between nodal points.
For a square grid of widthδ
qjГE¼ �½KXjГE�½HE�HC�∕δ (34) where
KxjГE
1:486 n h5=3
� �
ГE
.HE�HC δcosθ
��
��
��
��
1=2
;jHE�Hcj≥ ℇ
0 ; Hj E�HCj< ε
8>
<
>: (35)
In Eq. (35), h (depth of water) and n (the Manning’s coefficient) are both the average of their respective values at C and E, i.e., h¼ðhCþhEÞ=2 and n¼
nCþnE
ð Þ=2. Additionally, the denominator of KXis checked such that KX is set to zero if Hj E�HCjis less than a toleranceεsuch as 10�3ft.
The net volume of water in each grid element between time step i and i + 1 is ΔqCi¼q rj E þq rj w þq rj N þq rj S and the change of depth of water isΔHCi¼ ΔqCi∗Δt=δ2for time step i and i + 1 withΔt interval. Then the model advances in time by an explicit approach
HCiþ1¼ΔHCiþHCi (36) where the assumed input flood flows are added to the specified input nodes at each time step. After each time step, the hydraulic conductivity parameters of Eq. (35) are reevaluated, and the solution of Eq. (36) is reinitiated.
5.3 Model time step selection
The sensitivity of the model to time step selection is dependent upon the slope of the discharge hydrograph (∂∂Qt) and the grid spacing. Increasing the grid spacing size introduces additional water storage to a corresponding increase in nodal point flood depth values. Similarly, a decrease in time step size allows a refined calculation of inflow and outflow values and a smoother variation in nodal point flood depths with respect to time. The computer algorithm may self-select a time step by increments of halving (or doubling) the initial user-chosen time step size so that a proper balance of inflow-outflow to control volume storage variation is achieved. In order to avoid a matrix solution for flood depths, an explicit time stepping algorithm is used to solve for the time derivative term. For large time steps or a rapid variation in the dam-break hydrograph (such as∂∂Qt is large), a large accumulation of flow volume will occur at the most upstream nodal point. That is, at the dam-break reservoir nodal point, the lag in outflow from the control volume can cause an unacceptable error in the computation of the flood depth. One method that offsets this error is the program to self-select the time step until the difference in the rate of volume accumulation is within a specified tolerance.
Due to the form of the DHM in Eq. (22), the model can be extended into an implicit technique. However, this extension would require a matrix solution process which may become unmanageable for two-dimensional models which utilize hun- dreds of nodal points.
6. Conclusions
The one- and two-dimensional flow equations used in the diffusion hydrody- namic model are derived, and the relevant assumptions are listed. These equations, which are the basis of the model, are based on the conservation of mass and momentum principles. The explicit numerical algorithm and the discretized equa- tions are also presented. The ability of the model to self-select the optimal time step is discussed.
Figure 3.
Two-dimensional finite difference analog.
