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A Diffusion Hydrodynamic ModelAuthored by Theodore V. Hromadka II,

Chung-Cheng Yen and Prasada Rao

Authored by Theodore V. Hromadka II, 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

ISBN 978-1-83962-817-7

iffusion H

ydrodynamic M

A Diffusion Hydrodynamic Model

Published in London, United Kingdom

Supporting open minds since 2005

A Diffusion Hydrodynamic Modelhttp://dx.doi.org/10.5772/intechopen.90224Authored by Theodore V. Hromadka II, Chung-Cheng Yen and Prasada Rao

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BOOKCITATION

IVATE ANALYTICS

IN D E X E 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 1Diffusion Hydrodynamic Model Theoretical Developmentby Theodore V. Hromadka II and Chung-Cheng Yen

Chapter 2 13Verification of Diffusion Hydrodynamic Modelby Theodore V. Hromadka II and Chung-Cheng Yen

Chapter 3 27Program Description of the Diffusion Hydrodynamic Modelby Theodore V. Hromadka II and Chung-Cheng Yen

Chapter 4 33Applications of Diffusion Hydrodynamic Modelby Theodore V. Hromadka II and Chung-Cheng Yen

Chapter 5 55Reduction of the Diffusion Hydrodynamic Model to Kinematic Routingby Theodore V. Hromadka II and Chung-Cheng Yen

Chapter 6 63Comparison of DHM Results for One- and Two-Dimensional Flows with Experimental and Numerical Databy Theodore V. Hromadka II and Prasada Rao

Contents

Preface XIII

Chapter 1 1Diffusion Hydrodynamic Model Theoretical Developmentby Theodore V. Hromadka II and Chung-Cheng Yen

Chapter 2 13Verification of Diffusion Hydrodynamic Modelby Theodore V. Hromadka II and Chung-Cheng Yen

Chapter 3 27Program Description of the Diffusion Hydrodynamic Modelby Theodore V. Hromadka II and Chung-Cheng Yen

Chapter 4 33Applications of Diffusion Hydrodynamic Modelby Theodore V. Hromadka II and Chung-Cheng Yen

Chapter 5 55Reduction of the Diffusion Hydrodynamic Model to Kinematic Routingby Theodore V. Hromadka II and Chung-Cheng Yen

Chapter 6 63Comparison of DHM Results for One- and Two-Dimensional Flows with Experimental and Numerical Databy 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 RaoDepartment of Civil and Environmental Engineering,

California State University,Fullerton, CA, USA

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 RaoDepartment of Civil and Environmental Engineering,

California State University,Fullerton, CA, USA

Chapter 1

Diffusion Hydrodynamic ModelTheoretical DevelopmentTheodore V. Hromadka II and Chung-Cheng Yen

Abstract

In this chapter, the governing flow equations for one- and two-dimensionalunsteady flows that are solved in the diffusion hydrodynamic model (DHM) arepresented along with the relevant assumptions. A step-by-step derivation of thesimplified equations which are based on continuity and momentum principles aredetailed. Characteristic features of the explicit DHM numerical algorithm arediscussed.

Keywords: unsteady flow, conservation of mass, finite difference, explicit scheme,flow equations

1. Introduction

Many flow phenomena of great engineering importance are unsteady incharacters and cannot be reduced to a steady flow by changing the viewpoint of theobserver. A complete theory of unsteady flow is therefore required and will bereviewed in this section. The equations of motion are not solvable in the mostgeneral case, but approximations and numerical methods can be developed whichyield solutions of satisfactory accuracy.

2. Review of governing equations

The law of continuity for unsteady flow may be established by considering theconservation 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 ∂y

∂t. The change in dischargevolume through space dx in the time dt is ∂Q

� �dxdt. The corresponding change in

channel storage in space is Tdx ∂y∂t

� �dt ¼ dx ∂A

� �dt in which A ¼ Ty. Because water is

incompressible, the net change in discharge plus the change in storage should bezero, that is

∂Q∂x

� �dxdtþ Tdx

∂y∂t

� �dt ¼ ∂Q

� �dxdtþ dx

∂A∂t

� �dt ¼ 0

Chapter 1

Diffusion Hydrodynamic ModelTheoretical DevelopmentTheodore V. Hromadka II and Chung-Cheng Yen

Abstract

In this chapter, the governing flow equations for one- and two-dimensionalunsteady flows that are solved in the diffusion hydrodynamic model (DHM) arepresented along with the relevant assumptions. A step-by-step derivation of thesimplified equations which are based on continuity and momentum principles aredetailed. Characteristic features of the explicit DHM numerical algorithm arediscussed.

Keywords: unsteady flow, conservation of mass, finite difference, explicit scheme,flow equations

1. Introduction

Many flow phenomena of great engineering importance are unsteady incharacters and cannot be reduced to a steady flow by changing the viewpoint of theobserver. A complete theory of unsteady flow is therefore required and will bereviewed in this section. The equations of motion are not solvable in the mostgeneral case, but approximations and numerical methods can be developed whichyield solutions of satisfactory accuracy.

2. Review of governing equations

The law of continuity for unsteady flow may be established by considering theconservation 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 ∂y

∂t. The change in dischargevolume through space dx in the time dt is ∂Q

� �dxdt. The corresponding change in

channel storage in space is Tdx ∂y∂t

� �dt ¼ dx ∂A

� �dt in which A ¼ Ty. Because water is

incompressible, the net change in discharge plus the change in storage should bezero, that is

∂Q∂x

� �dxdtþ Tdx

∂y∂t

� �dt ¼ ∂Q

� �dxdtþ dx

∂A∂t

� �dt ¼ 0

Simplifying

∂Q∂x

þ T∂y∂t

¼ 0 (1)

∂Q∂x

þ ∂A∂t

¼ 0 (2)

At a given section, Q = VA; thus Eq. (1) becomes

∂ VAð Þ∂x

þ T∂y∂t

¼ 0 (3)

A∂V∂x

þ V∂A∂x

þ T∂y∂t

¼ 0 (4)

Because the hydraulic depth D = A/T and ∂A ¼ T∂y, the above equation may bewritten as

D∂V∂x

þ V∂y∂x

þ ∂y∂t

¼ 0 (5)

The above equations are all forms of the continuity equation for unsteady flowin open channels. For a rectangular channel or a channel of infinite width, Eq. (1)may be written as

∂q∂x

þ ∂y∂t

¼ 0 (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 tomagnitude of the “friction slope” S f ¼ V2= C2R

� �, where C is the Chezy coefficient

and R is the hydraulic radius. Indeed this statement was in a sense taken as thedefinition of Sf; however, in the present context, we have to consider the moregeneral case in which the flow is nonuniform, and the velocity may be changing inthe downstream direction. The net force, shear force and pressure force, is nolonger zero since the flow is accelerating. Therefore, the equation of motionbecomes

�γAΔh� τ0PΔx ¼ ρAΔx V∂V∂x

þ ∂V∂t

� �

that is

τ0 ¼ �γR∂h∂x

þ Vg∂V∂x

þ 1g∂V∂t

� �

�γR∂H∂x

þ 1g∂V∂t

� �(7)

where τ0 is the same shear stress, P is the hydrostatic pressure, h is the depth ofwater, Δh is the change of depth of water, γ is the specific weight of the fluid, R isthe mean hydraulic radius, and ρ is the fluid density. Substituting τ0

γR =V2

C2Rinto

Eq. (7), we obtain

∂H∂x

þ 1g∂V∂t

C2R¼ 0 (8)

and this equation may be rewritten as

Se þ Sa þ Sf ¼ 0 (9)

where the three terms of Eq. (9) are called the energy slope, the accelerationslope, and the friction slope, respectively. Figure 2 depicts the simplifiedrepresentation of energy in unsteady flow.

By substituting H ¼ V2

2g þ yþ z and the bed slope So ¼ � ∂z∂x into Eq. (8), we

obtain

∂H∂x

¼ ∂z∂x

þ ∂y∂x

þ Vg∂V∂x

¼ �So þ ∂y∂x

þ Vg∂V∂x

¼ � 1g∂V∂t

� S f

Diffusion Hydrodynamic Model Theoretical DevelopmentDOI: http://dx.doi.org/10.5772/intechopen.93207

Simplifying

∂Q∂x

þ T∂y∂t

¼ 0 (1)

∂Q∂x

þ ∂A∂t

¼ 0 (2)

At a given section, Q = VA; thus Eq. (1) becomes

∂ VAð Þ∂x

þ T∂y∂t

¼ 0 (3)

A∂V∂x

þ V∂A∂x

þ T∂y∂t

¼ 0 (4)

Because the hydraulic depth D = A/T and ∂A ¼ T∂y, the above equation may bewritten as

D∂V∂x

þ V∂y∂x

þ ∂y∂t

¼ 0 (5)

The above equations are all forms of the continuity equation for unsteady flowin open channels. For a rectangular channel or a channel of infinite width, Eq. (1)may be written as

∂q∂x

þ ∂y∂t

¼ 0 (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 tomagnitude of the “friction slope” S f ¼ V2= C2R

� �, where C is the Chezy coefficient

and R is the hydraulic radius. Indeed this statement was in a sense taken as thedefinition of Sf; however, in the present context, we have to consider the moregeneral case in which the flow is nonuniform, and the velocity may be changing inthe downstream direction. The net force, shear force and pressure force, is nolonger zero since the flow is accelerating. Therefore, the equation of motionbecomes

�γAΔh� τ0PΔx ¼ ρAΔx V∂V∂x

þ ∂V∂t

� �

that is

τ0 ¼ �γR∂h∂x

þ Vg∂V∂x

þ 1g∂V∂t

� �

�γR∂H∂x

þ 1g∂V∂t

� �(7)

where τ0 is the same shear stress, P is the hydrostatic pressure, h is the depth ofwater, Δh is the change of depth of water, γ is the specific weight of the fluid, R isthe mean hydraulic radius, and ρ is the fluid density. Substituting τ0

γR =V2

C2Rinto

Eq. (7), we obtain

∂H∂x

þ 1g∂V∂t

C2R¼ 0 (8)

and this equation may be rewritten as

Se þ Sa þ Sf ¼ 0 (9)

where the three terms of Eq. (9) are called the energy slope, the accelerationslope, and the friction slope, respectively. Figure 2 depicts the simplifiedrepresentation of energy in unsteady flow.

By substituting H ¼ V2

2g þ yþ z and the bed slope So ¼ � ∂z∂x into Eq. (8), we

obtain

∂H∂x

¼ ∂z∂x

þ ∂y∂x

þ Vg∂V∂x

¼ �So þ ∂y∂x

þ Vg∂V∂x

¼ � 1g∂V∂t

� S f

Hence Eq. (8) can be rewritten as

ð11Þ

This equation may be applicable to various types of flow as indicated. Thisarrangement shows how the nonuniformity and unsteadiness of flows introduceextra 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 hydrodynamicmodel (DHM) are based upon the flow equations of continuity (2) and momentum(11) which can be rewritten [1] as

∂xþ ∂Ax

∂t¼ 0 (12)

∂tþ ∂ Qx

2=Ax� �

∂xþ gAx

∂H∂x

þ Sfx

� �¼ 0 (13)

Figure 2.Simplified representation of energy in unsteady flow.

where Qx is the flow rate; x,t are spatial and temporal coordinates, Ax is the flowarea, g is the gravitational acceleration, H is the water surface elevation, and Sfx is afriction slope. It is assumed that Sfx approximated from Manning’s equation forsteady flow by [1].

Qx ¼1:486n

AxR2=3Sfx1=2 (14)

where R is the hydraulic radius and n is a flow resistance coefficient which maybe increased to account for other energy losses such as expansions and bend losses.

Letting mx be a momentum quantity defined by

mx ¼ ∂Qx

∂tþ ∂ Qx

2=Ax� �

!=gAx (15)

then Eq. (13) can be rewritten as

Sfx ¼ � ∂H∂x

� �(16)

In Eq. (15), the subscript x included in mx indicates the directional term. Theexpansion 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 iscomputed by

Qx ¼ �Kx∂H∂x

� �(17)

where Qx indicates a directional term and Kx is a type of conduction parameterdefined by

Kx ¼ 1:486n

AxR2=3

∂H∂x þmx�� 1=2 (18)

In Eq. (18), Kx is limited in value by the denominator term being checked for a

smallest allowable magnitude (such as ∂H∂X þmX�� 1=2 > 10�3).

Substituting the flow rate formulation of Eq. (17) into Eq. (12) gives a diffusiontype of relationship

∂XKX

∂H∂X

� �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 similarterms—that is, they are considered as a sum rather than as individual directionalterms that typically have more significance when examined individually. Addition-ally, the term “diffusion” routing indicates assuming that several convective andother components have a small contribution to the coupled mass and energy balanceequations and therefore are neglected in the computational formulation to simplifythe model accordingly. Thus, the one-dimensional DHM equation is given by

Hence Eq. (8) can be rewritten as

ð11Þ

This equation may be applicable to various types of flow as indicated. Thisarrangement shows how the nonuniformity and unsteadiness of flows introduceextra 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 hydrodynamicmodel (DHM) are based upon the flow equations of continuity (2) and momentum(11) which can be rewritten [1] as

∂xþ ∂Ax

∂t¼ 0 (12)

∂tþ ∂ Qx

2=Ax� �

∂xþ gAx

∂H∂x

þ Sfx

� �¼ 0 (13)

Figure 2.Simplified representation of energy in unsteady flow.

where Qx is the flow rate; x,t are spatial and temporal coordinates, Ax is the flowarea, g is the gravitational acceleration, H is the water surface elevation, and Sfx is afriction slope. It is assumed that Sfx approximated from Manning’s equation forsteady flow by [1].

Qx ¼1:486n

AxR2=3Sfx1=2 (14)

where R is the hydraulic radius and n is a flow resistance coefficient which maybe increased to account for other energy losses such as expansions and bend losses.

Letting mx be a momentum quantity defined by

mx ¼ ∂Qx

∂tþ ∂ Qx

2=Ax� �

!=gAx (15)

then Eq. (13) can be rewritten as

Sfx ¼ � ∂H∂x

� �(16)

In Eq. (15), the subscript x included in mx indicates the directional term. Theexpansion 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 iscomputed by

Qx ¼ �Kx∂H∂x

� �(17)

where Qx indicates a directional term and Kx is a type of conduction parameterdefined by

Kx ¼ 1:486n

AxR2=3

∂H∂x þmx�� 1=2 (18)

In Eq. (18), Kx is limited in value by the denominator term being checked for a

smallest allowable magnitude (such as ∂H∂X þmX�� 1=2 > 10�3).

Substituting the flow rate formulation of Eq. (17) into Eq. (12) gives a diffusiontype of relationship

∂XKX

∂H∂X

� �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 similarterms—that is, they are considered as a sum rather than as individual directionalterms that typically have more significance when examined individually. Addition-ally, the term “diffusion” routing indicates assuming that several convective andother components have a small contribution to the coupled mass and energy balanceequations and therefore are neglected in the computational formulation to simplifythe model accordingly. Thus, the one-dimensional DHM equation is given by

∂XKX

∂H∂X

¼ ∂AX

∂t(20)

where KX is now simplified as

Kx ¼1:486n AxR2=3

∂H∂X

�� 12 (21)

For a channel of constant width, WX, Eq. (20) reduces to

∂XKX

∂H∂X

¼ WX∂H∂t

Assumptions other than mX = 0 in Eq. (19) result in a family of models:

∂ðQx2=AXÞ∂X

.gAX ðconvective acceleration modelÞ

∂t=gAX ðlocal acceleration modelÞ

&∂QX

∂tþ ∂ðQx

2=AXÞ∂X

’,gAX ðfully dynamic modelÞ

0 ðDHMÞ

8>>>>>>>>>><>>>>>>>>>>:

4.2 Two-dimensional diffusion hydrodynamic model

The set of (fully dynamic) 2D unsteady flow equations consists of one equationof continuity

∂qx∂x

þ∂qy∂y

þ ∂H∂t

¼ 0 (24)

and two equations of motion

∂qx∂t

þ ∂

∂xqx

� �þ ∂

� �þ gh Sfx þ ∂H

� �¼ 0 (25)

∂qy∂t

þ ∂

" #þ ∂

� �þ gh Sfy þ ∂H

� �¼ 0 (26)

where qX and qy are flow rates per unit width in the x and y directions; Sfx and Sfyrepresents friction slopes in x and y directions; H, h, and g stand for water surfaceelevation, flow depth, and gravitational acceleration, respectively; and x, y, and tare spatial and temporal coordinates.

The above equation set is based on the assumptions of constant fluid densitywithout 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

mZ þ Sfz þ ∂H∂Z

� �¼ 0, z ¼ x, y (27)

where mZ represents the sum of the first three terms in Eqs. (25) or (26) dividedby 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 ¼ 1:486n

h5=3Sfz1=2, z ¼ x, y (28)

Eq. (28) can be rewritten in the general case as

qZ ¼ �KZ∂H∂Z

� KZmZ, z ¼ x, y (29)

KZ ¼ 1:486n

∂H∂S þmS�� 1=2 , z ¼ x, y (30)

The symbol s in Eq. (30) indicates the flow direction which makes an angle of

θ ¼ tan �1 qy=qx� �

with the positive x direction.

Themz term is assumed to be negligible [1–5] when combined with the othersimilar terms, i.e., they are considered as a sum rather than as individual directionalterms that typically have more significance when examined individually. Additionally,the term “diffusion” routing indicates assuming that several convective and othercomponents have a small contribution to the coupled mass and energy balance equa-tions and therefore are neglected in the computational formulation to simplify themodel accordingly. Neglecting this term results in the simple diffusion model

qZ ¼ �KZ∂H∂Z

, z ¼ x, y (31)

The proposed 2D DHM is formulated by substituting Eq. (31) into Eq. (24)

∂XKx

∂H∂X

þ ∂

∂yKy

∂H∂y

¼ ∂H∂t

If the momentum term groupings were retained, Eq. (32) would be written as

∂xKx

∂H∂x

þ ∂

∂yKy

∂H∂y

þ S ¼ ∂H∂t

S ¼ ∂

∂xKxmxð Þ þ ∂

∂yKxmy� �

and Kx and Ky are also functions of mx and my, respectively.

5. Numerical approximation

5.1 Numerical solution algorithm

The one-dimensional domain is discretized across uniformly spaced nodalpoints, and at each of these points, at time (t) = 0, the values of Manning’s n, an

∂XKX

∂H∂X

¼ ∂AX

∂t(20)

where KX is now simplified as

Kx ¼1:486n AxR2=3

∂H∂X

�� 12 (21)

For a channel of constant width, WX, Eq. (20) reduces to

∂XKX

∂H∂X

¼ WX∂H∂t

Assumptions other than mX = 0 in Eq. (19) result in a family of models:

∂ðQx2=AXÞ∂X

.gAX ðconvective acceleration modelÞ

∂t=gAX ðlocal acceleration modelÞ

&∂QX

∂tþ ∂ðQx

2=AXÞ∂X

’,gAX ðfully dynamic modelÞ

0 ðDHMÞ

8>>>>>>>>>><>>>>>>>>>>:

4.2 Two-dimensional diffusion hydrodynamic model

The set of (fully dynamic) 2D unsteady flow equations consists of one equationof continuity

∂qx∂x

þ∂qy∂y

þ ∂H∂t

¼ 0 (24)

and two equations of motion

∂qx∂t

þ ∂

∂xqx

� �þ ∂

� �þ gh Sfx þ ∂H

� �¼ 0 (25)

∂qy∂t

þ ∂

" #þ ∂

� �þ gh Sfy þ ∂H

� �¼ 0 (26)

where qX and qy are flow rates per unit width in the x and y directions; Sfx and Sfyrepresents friction slopes in x and y directions; H, h, and g stand for water surfaceelevation, flow depth, and gravitational acceleration, respectively; and x, y, and tare spatial and temporal coordinates.

The above equation set is based on the assumptions of constant fluid densitywithout 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

mZ þ Sfz þ ∂H∂Z

� �¼ 0, z ¼ x, y (27)

where mZ represents the sum of the first three terms in Eqs. (25) or (26) dividedby 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 ¼ 1:486n

h5=3Sfz1=2, z ¼ x, y (28)

Eq. (28) can be rewritten in the general case as

qZ ¼ �KZ∂H∂Z

� KZmZ, z ¼ x, y (29)

KZ ¼ 1:486n

∂H∂S þmS�� 1=2 , z ¼ x, y (30)

The symbol s in Eq. (30) indicates the flow direction which makes an angle of

θ ¼ tan �1 qy=qx� �

with the positive x direction.

Themz term is assumed to be negligible [1–5] when combined with the othersimilar terms, i.e., they are considered as a sum rather than as individual directionalterms that typically have more significance when examined individually. Additionally,the term “diffusion” routing indicates assuming that several convective and othercomponents have a small contribution to the coupled mass and energy balance equa-tions and therefore are neglected in the computational formulation to simplify themodel accordingly. Neglecting this term results in the simple diffusion model

qZ ¼ �KZ∂H∂Z

, z ¼ x, y (31)

The proposed 2D DHM is formulated by substituting Eq. (31) into Eq. (24)

∂XKx

∂H∂X

þ ∂

∂yKy

∂H∂y

¼ ∂H∂t

If the momentum term groupings were retained, Eq. (32) would be written as

∂xKx

∂H∂x

þ ∂

∂yKy

∂H∂y

þ S ¼ ∂H∂t

S ¼ ∂

∂xKxmxð Þ þ ∂

∂yKxmy� �

and Kx and Ky are also functions of mx and my, respectively.

5. Numerical approximation

5.1 Numerical solution algorithm

The one-dimensional domain is discretized across uniformly spaced nodalpoints, 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 initialconditions, 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 geometricfactors

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 ofmX selected from Eq. (23)

4.Recalculate the conductivities KX using the appropriate mX values

5.Determine the new nodal flow depths at the time (t + Δt) using Eq. (19), and

6.Return to step (3) until KX matches mid time step estimates.

The above algorithm steps can be used regardless of the choice of definition formX from Eq. (23). Additionally, the above program steps can be directly applied to atwo-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 nodaldomain integration (NDI) method [6] is used. For grid elements, the NDI nodalequation is based on the usual nodal system shown in Figure 3. Flow rates across theboundary Г are estimated by assuming a linear trial function between nodal points.

For a square grid of width δ

q ГEj ¼ � KX ГEj½ � HE �HC½ �∕δ (34)

KxjГE

1:486n

h5=3� �

. HE �HC

δ cos θ

��1=2

; jHE �Hcj≥ℇ

0 ; HE �HCj j< ε

In Eq. (35), h (depth of water) and n (the Manning’s coefficient) are both theaverage of their respective values at C and E, i.e., h ¼ hC þ hEð Þ=2 and n ¼nC þ nEð Þ=2. Additionally, the denominator of KX is checked such that KX is set tozero if HE �HCj j is less than a tolerance ε such as 10�3 ft.

The net volume of water in each grid element between time step i and i + 1 isΔqC

i ¼ q rEj þ q rwj þ q rNj þ q rSj and the change of depth of water is ΔHCi ¼

ΔqCi ∗Δt=δ2 for time step i and i + 1 with Δt interval. Then the model advances in

time by an explicit approach

HCiþ1 ¼ ΔHC

i þHCi (36)

where the assumed input flood flows are added to the specified input nodes ateach time step. After each time step, the hydraulic conductivity parameters ofEq. (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 ofthe discharge hydrograph (∂Q

∂t ) and the grid spacing. Increasing the grid spacing sizeintroduces additional water storage to a corresponding increase in nodal point flooddepth values. Similarly, a decrease in time step size allows a refined calculation ofinflow and outflow values and a smoother variation in nodal point flood depths withrespect to time. The computer algorithm may self-select a time step by incrementsof halving (or doubling) the initial user-chosen time step size so that a properbalance of inflow-outflow to control volume storage variation is achieved. In orderto avoid a matrix solution for flood depths, an explicit time stepping algorithm isused to solve for the time derivative term. For large time steps or a rapid variation inthe dam-break hydrograph (such as ∂Q

∂t is large), a large accumulation of flowvolume will occur at the most upstream nodal point. That is, at the dam-breakreservoir nodal point, the lag in outflow from the control volume can cause anunacceptable error in the computation of the flood depth. One method that offsetsthis error is the program to self-select the time step until the difference in the rate ofvolume accumulation is within a specified tolerance.

Due to the form of the DHM in Eq. (22), the model can be extended into animplicit technique. However, this extension would require a matrix solution processwhich 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 andmomentum principles. The explicit numerical algorithm and the discretized equa-tions are also presented. The ability of the model to self-select the optimal time stepis discussed.

Figure 3.Two-dimensional finite difference analog.