eISSN: 2577-8242 FMRIJ

Fluid Mechanics Research International Journal
Research Article
Volume 2 Issue 5

Laminar and turbulent analytical dam break wave modelling on dry-downstream open channel flow

Tarek Taha,1 Abdulrasheed O Abdul Lateef,2 Jaan H Pu3
1Graduate Student, Faculty of Engineering and Informatics, University of Bradford, United Kingdom
2 Former Graduate Student, Faculty of Engineering and Informatics, University of Bradford, United Kingdom
3Senior Lecturer, Faculty of Engineering and Informatics, University of Bradford, United Kingdom
Received: September 17, 2018 | Published: September 26, 2018

Correspondence: Jaan H Pu, Senior Lecturer, Faculty of Engineering and Informatics, University of Bradford, Bradford BD7 1DP, United Kingdom, Tel + (0) 1274234556, Email

Citation: Taha T, Lateef AOA, Pu JH. Laminar and turbulent analytical dam break wave modelling on dry-downstream open channel flow. Fluid Mech Res Int . 2018;2(5):177‒180. DOI: 10.15406/fmrij.2018.02.00036

Abstract

A dam break wave caused by the discontinuity in depth and velocity of a flow is resulted from instantaneous release a body of water from a channel and classified naturally as a rapidly varied unsteady flow. Due to its nature, it is hard to be accurately represented by analytical models. The aim of this study is to establish the modelling differences and complexity echelons between analytically simulated explicit laminar and turbulent dry bed dam break wave free surface profiles. An in-depth solution to the free surface profile has been provided and evaluated by representing the reported dam break flow measurements at various locations. The methodology adopted utilizes the free surface profile formulations presented by Chanson1,2, which are developed using the method of characteristics. In order to validate the results of the presented analytical models in illustrating the dam break wave under dry bed conditions, published experimental data provided by Schoklitsch3, Debiane4 and Dressler5 are used to compare and analyze the performance of the dam break waves under laminar and turbulent flow conditions.

Keywords: laminar model, turbulent model, free surface profile, open channel flow, dam break wave, dry bed, method of characteristics

Introduction

The means of generating power, agricultural irrigation, providing of flood control measures and drinking water can all be achieved by establishing a reservoir through the construction of dam. Dams are classified as an installation to ‘contain dangerous forces’ by the International Humanitarian Law 1977 on the ground that when a dam fails due to deplorable design or maintenance catastrophic events, it can cause homes, businesses and public spaces to be destroyed. Besides the displacement of people and property under such circumstances, the loss of life due to dam break disaster occurs all too common. Dam break waves are of interest to hydraulic engineers due to their practical relevance for fluid flow dynamic propagation as stated in Pu6, for morphological development as detailed by Zheng et al.,7 and for sediment transportation as concluded by Pu et al.8

Fluid flows are categorized respectively, according to the nature of their flow. Any fluid flow in which the fluid particles follow a smooth path, never interfering with one another and when the fluid velocity is constant at any given point is referred to as laminar flow. Understandably, any fluid flow in which the fluid particles follow an irregular and arbitrary path referred to as turbulent flow. In order to differentiate between the two, the dimensionless quantity referred to as Reynolds Number (Re) was developed, in which any fluid flow exhibiting a low Reynolds Number (less than 2300) is classified as a laminar flow, whilst any flow exhibiting a high Reynolds Number (greater than 4000) is classified as a turbulent flow. A flow experiencing a Reynolds Number between the two are classified as transitional flow. The Reynolds Number adopted in this study will represent the fluid properties of a reservoir flow, and hence is derived slightly different compared to the standard Reynolds Number and is herein referred to as ‘reservoir flow Reynolds Number’ (R), refer to its usage in Chanson1,2.

This study will explicitly develop an analytical free surface profile using both laminar and turbulent dam break wave concepts to identify their respective effect to dam break flow. The solutions originally developed via the Saint Venant approach, presented a simplified approach due to the neglecting the frictional factor and turbulence. A study conducted by Dressler9 provided evidence that accounting for the frictional and turbulent resistive effects by incorporating hydraulic resistances terms in the form of the Chezy function within the dam break flow governing equations produced highly complex equations that have no exact solution till present. During the initial stages of dam break wave as the flow propagates, the ideal fluid flow region simultaneously develops backwards (by depression wave), upstream (positive bore) in the reservoir flow region. At downstream, the front wave propagates due to the strong downward vertical acceleration of the fluid released. The dam break wave propagates with velocity decreases steadily due to the large resistance coefficient; hence the wave tip region develops rapidly as the wave tip velocity reduces. This reflects the inverse affiliation between time and the resistance coefficient. Additionally, the wave tip region is significantly complex to uphold due to the flow velocity condition in which the velocity must be uniformly distributed through-out the wave tip region. Deng et al.,10 stated the dam break simulation’s crucial fundamentals rely on the estimation of wave tip region, which aligns with the results they achieved.

The transitional phase into frictionally dominated in the wave front region in which a flow singularity by zero-water depth occurs, can consequently result in an explicitly unstable analytical solution of the dam break wave tip. The transitional interface connecting the main flow region and wave tip region is usually assumed in literatures to be first located in forward manner, then in backward towards upstream. Aligning with the studies of Ritter11 and Chanson1,2 the resistant coefficient was assigned with value zero just before the wave front starts (x<x1).Their computations are derived from the momentum and continuity equations, followed respectively by their forward and backward characteristics (method of characteristics) to establish the relationship between time and velocity of the wave tip. Considered frictional resistance within the wave tip region, their calculation results of the water depth at the transitional interface increases over time. In this study, the method of characteristics will be used to form the laminar and turbulent models for dam break wave, due to they are well-tested in Chanson1,2. Comparison of the presented laminar and turbulent models with published experimental data will also be used to validate the models.

Laminar modelling

During a laminar dam break wave the ideal fluid flow field yields the following basic equations. It is noted that the influencing factor of the wave tip region is not the flow velocity but the wave front celerity, as confirmed by the experimental studies of Whitham12.
d dt ( v+2c )=0  Along    x t =v+c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaadsgaa8aabaWdbiaadsgacaWG0baaamaabmaa paqaa8qacaWG2bGaey4kaSIaaGOmaiaadogaaiaawIcacaGLPaaacq GH9aqpcaaIWaGaaiiOaiaacckacaqGbbGaaeiBaiaab+gacaqGUbGa ae4zaiaabckacaGGGcGaaiiOamaalaaapaqaa8qacqGHciITcaWG4b aapaqaa8qacqGHciITcaWG0baaaiabg2da9iaadAhacqGHRaWkcaWG Jbaaaa@5377@                                                           (1)
d dt ( v2c )=0  Along    x t =vc MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaadsgaa8aabaWdbiaadsgacaWG0baaamaabmaa paqaa8qacaWG2bGaeyOeI0IaaGOmaiaadogaaiaawIcacaGLPaaacq GH9aqpcaaIWaGaaiiOaiaacckacaqGbbGaaeiBaiaab+gacaqGUbGa ae4zaiaabckacaGGGcGaaiiOamaalaaapaqaa8qacqGHciITcaWG4b aapaqaa8qacqGHciITcaWG0baaaiabg2da9iaadAhacqGHsislcaWG Jbaaaa@538C@                                                            (2)
Assuming that the flow resistance is dominant, and that both the inertial properties and velocity of the flow are negligible, then the momentum equation can be reduced into the diffusive wave equation.
d x + f 8 u 2 d =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiabgkGi2kaadsgaa8aabaWdbiabgkGi2kaadIha aaGaey4kaSYaaSaaa8aabaWdbiaadAgaa8aabaWdbiaaiIdaaaWaaS aaa8aabaWdbiaadwhapaWaaWbaaSqabeaapeGaaGOmaaaaaOWdaeaa peGaamizaaaacqGH9aqpcaaIWaaaaa@4347@                                                                                        (3)
The diffusive wave equation can be integrated to generate an equation that provides the shape of the wave tip region for laminar flow,
d= 6α u R ( x s x ) 3   at   x 1  x  x s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamizaiabg2da9maakeaapaqaa8qacaaI2aGaeqySde2aaSaaa8aa baWdbiaadwhaa8aabaWdbiaadkfaaaWaaeWaa8aabaWdbiaadIhapa WaaSbaaSqaa8qacaWGZbaapaqabaGcpeGaeyOeI0IaamiEaaGaayjk aiaawMcaaaWcpaqaa8qacaaIZaaaaOGaaiiOaiaacckacaWGHbGaam iDaiaacckacaGGGcGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaa k8qacqGHKjYOcaqGGaGaamiEaiaabccacqGHKjYOcaWG4bWdamaaBa aaleaapeGaam4CaaWdaeqaaaaa@5486@                                                        (4)
The transition between the ideal fluid flow region and the wave tip region yields two relationships, i.e. for the free-surface and velocity profiles, in which both must be continuous.
d 1 =  1 9 (2 x 1 t ) 2 = 6α u R ( x s x ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiza8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH9aqpcaGG GcWaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaiMdaaaGaaiikaiaaik dacqGHsisldaWcaaWdaeaapeGaamiEa8aadaWgaaWcbaWdbiaaigda a8aabeaaaOqaa8qacaWG0baaaiaacMcapaWaaWbaaSqabeaapeGaaG Omaaaakiabg2da9maakeaapaqaa8qacaaI2aGaeqySde2aaSaaa8aa baWdbiaadwhaa8aabaWdbiaadkfaaaWaaeWaa8aabaWdbiaadIhapa WaaSbaaSqaa8qacaWGZbaapaqabaGcpeGaeyOeI0IaamiEaaGaayjk aiaawMcaaaWcpaqaa8qacaaIZaaaaaaa@5072@                                                           (5)
v 1 =u= 2 3 ( 1+ x 1 t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamODa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH9aqpcaWG 1bGaeyypa0ZaaSaaa8aabaWdbiaaikdaa8aabaWdbiaaiodaaaWaae Waa8aabaWdbiaaigdacqGHRaWkdaWcaaWdaeaapeGaamiEa8aadaWg aaWcbaWdbiaaigdaa8aabeaaaOqaa8qacaWG0baaaaGaayjkaiaawM caaaaa@43F7@ (6)
Equation (6) provides the location of the transition between the frictional dominated flow region and the ideal fluid flow region. The conservation of mass can be attained by integrating equation (5) as follows
x1 2t 1 9 (2 x 1 t ) 2  dx= x1 x2 6α u R ( x s x ) 3  dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaybCaeqal8aabaqcLbmapeGaamiEaKqzacGaiGgMigdaaSWdaeaa peGaaGOmaiaadshaa0WdaeaapeGaey4kIipaaOWaaSaaa8aabaWdbi aaigdaa8aabaWdbiaaiMdaaaGaaiikaiaaikdacqGHsisldaWcaaWd aeaapeGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaOqaa8qaca WG0baaaiaacMcapaWaaWbaaSqabeaapeGaaGOmaaaakiaacckacaWG KbGaamiEaiabg2da9maawahabeWcpaqaaKqzadWdbiaadIhajugGai acOHjIXaaal8aabaWdbiaadIhajeaqcGaAyIOmaaqdpaqaa8qacqGH RiI8aaGcdaGcbaWdaeaapeGaaGOnaiabeg7aHnaalaaapaqaa8qaca WG1baapaqaa8qacaWGsbaaamaabmaapaqaa8qacaWG4bWdamaaBaaa leaapeGaam4CaaWdaeqaaOWdbiabgkHiTiaadIhaaiaawIcacaGLPa aaaSWdaeaapeGaaG4maaaakiaacckacaWGKbGaamiEaaaa@668B@                                                    (7)
Hence the wave front equation can be determined by
x s =( 3 2 u1 )t+  4 f u 2 ( 1 u 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiEa8aadaWgaaWcbaWdbiaadohaa8aabeaak8qacqGH9aqpdaqa daWdaeaapeWaaSaaa8aabaWdbiaaiodaa8aabaWdbiaaikdaaaGaam yDaiabgkHiTiaaigdaaiaawIcacaGLPaaacaWG0bGaey4kaSIaaiiO amaalaaapaqaa8qacaaI0aaapaqaa8qacaWGMbGaamyDa8aadaahaa Wcbeqaa8qacaaIYaaaaaaakmaabmaapaqaa8qacaaIXaGaeyOeI0Ya aSaaa8aabaWdbiaadwhaa8aabaWdbiaaikdaaaaacaGLOaGaayzkaa WdamaaCaaaleqabaWdbiaaikdaaaaaaa@4D45@                                                                 (8)
Equation (8) gives the dimensionless wave front location dependent on both the dimensionless time and wave front celerity. Due to this the entire free surface profile equation must satisfy conditions set up by equations (9) – (12), where outlined at Figure 1.
d= 1, at x  t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamizaiabg2da9iaabccacaaIXaGaaiilaiaacckacaWGHbGaamiD aiaacckacaWG4bGaaeiiaiabgsMiJkaabccacqGHsislcaWG0baaaa@4468@                                                                                   (9)
d=  1 9 (2 x t ) 2 ,attx( 3 2 u1 )t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamizaiabg2da9iaacckadaWcaaWdaeaapeGaaGymaaWdaeaapeGa aGyoaaaacaGGOaGaaGOmaiabgkHiTmaalaaapaqaa8qacaWG4baapa qaa8qacaWG0baaaiaacMcapaWaaWbaaSqabeaapeGaaGOmaaaak8aa caGGSaGaaGPaVlaaykW7peGaamyyaiaadshacaaMc8UaaGPaVlabgk HiTiaadshacqGHKjYOcaWG4bGaeyizIm6aaeWaa8aabaWdbmaalaaa paqaa8qacaaIZaaapaqaa8qacaaIYaaaaiaadwhacqGHsislcaaIXa aacaGLOaGaayzkaaGaamiDaaaa@57CE@                                                    (10)
d= 6α u R ( x s x ) d 0 3 at( 3 2 u1 )tx x s     MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamizaiabg2da9maakeaapaqaa8qacaaI2aGaeqySde2aaSaaa8aa baWdbiaadwhaa8aabaWdbiaadkfaaaWaaSaaa8aabaWdbmaabmaapa qaa8qacaWG4bWdamaaBaaaleaapeGaam4CaaWdaeqaaOWdbiabgkHi TiaadIhaaiaawIcacaGLPaaaa8aabaWdbiaadsgapaWaaSbaaSqaa8 qacaaIWaaapaqabaaaaaqaa8qacaaIZaaaaOGaaGPaVlaaykW7caWG HbGaamiDaiaaykW7caaMc8+aaeWaa8aabaWdbmaalaaapaqaa8qaca aIZaaapaqaa8qacaaIYaaaaiaadwhacqGHsislcaaIXaaacaGLOaGa ayzkaaGaamiDaiabgsMiJkaadIhacqGHKjYOcaWG4bWdamaaBaaale aapeGaam4CaaWdaeqaaOWdbiaacckacaGGGcGaaiiOaaaa@5F71@                                           (11)
d= 0 at  x s  x  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamizaiabg2da9iaabccacaaIWaGaaiiOaiaadggacaWG0bGaaiiO aiaadIhapaWaaSbaaSqaa8qacaWGZbaapaqabaGcpeGaeyizImQaae iiaiaadIhacaGGGcaaaa@44BB@                                                                                      (12)

Figure 1 Illustration of the Laminar Flow Regions with their Corresponding Solutions.

Turbulent modelling

During a turbulent dam break event, the ideal fluid flow field yields the following the basic equations. Similar to laminar flow, within its wave tip region the flow velocity does not vary rapidly, hence the influencing concern is not the flow velocity but the wave front celerity, as according to the experimental findings provided from Dressler5 and Whitham12. The wave tip region of the dam break flow under turbulent unsteady conditions for wide channels can be translated to
f= 1 d 1 4 ( 3.65x 10 5 k s + 2.5x 10 3 R e d u ) 1 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOzaiabg2da9maalaaapaqaa8qacaaIXaaapaqaa8qacaWGKbWd amaaCaaaleqabaWdbmaalaaapaqaa8qacaaIXaaapaqaa8qacaaI0a aaaaaaaaGcdaqadaWdaeaapeGaaG4maiaac6cacaaI2aGaaGynaiaa dIhacaaIXaGaaGima8aadaahaaWcbeqaa8qacqGHsislcaaI1aaaaO Gaam4Aa8aadaWgaaWcbaWdbiaadohaa8aabeaak8qacqGHRaWkdaWc aaWdaeaapeGaaGOmaiaac6cacaaI1aGaamiEaiaaigdacaaIWaWdam aaCaaaleqabaWdbiabgkHiTiaaiodaaaaak8aabaWdbiaadkfacaWG LbWdamaaBaaaleaapeGaamizaaWdaeqaaOWdbiaadwhaaaaacaGLOa GaayzkaaWdamaaCaaaleqabaWdbmaalaaapaqaa8qacaaIXaaapaqa a8qacaaI0aaaaaaaaaa@55C7@                                                                       (13)
The integration of the diffusive equation backward characteristic produces an equation that illustrates the shape of the wave tip region as follows
d= ( 9 32 ( 3.65x 10 5 k s + 2.5x 10 3 R e d u ) 1 4   u 2 ( x s x )) 4 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamizaiabg2da9iaacIcadaWcaaWdaeaapeGaaGyoaaWdaeaapeGa aG4maiaaikdaaaWaaeWaa8aabaWdbiaaiodacaGGUaGaaGOnaiaaiw dacaWG4bGaaGymaiaaicdapaWaaWbaaSqabeaapeGaeyOeI0IaaGyn aaaakiaadUgapaWaaSbaaSqaa8qacaWGZbaapaqabaGcpeGaey4kaS YaaSaaa8aabaWdbiaaikdacaGGUaGaaGynaiaadIhacaaIXaGaaGim a8aadaahaaWcbeqaa8qacqGHsislcaaIZaaaaaGcpaqaa8qacaWGsb Gaamyza8aadaWgaaWcbaWdbiaadsgaa8aabeaak8qacaWG1baaaaGa ayjkaiaawMcaa8aadaahaaWcbeqaa8qadaWcaaWdaeaapeGaaGymaa WdaeaapeGaaGinaaaaaaGccaGGGcGaamyDa8aadaahaaWcbeqaa8qa caaIYaaaaOWaaeWaa8aabaWdbiaadIhapaWaaSbaaSqaa8qacaWGZb aapaqabaGcpeGaeyOeI0IaamiEaaGaayjkaiaawMcaaiaacMcapaWa aWbaaSqabeaapeWaaSaaa8aabaWdbiaaisdaa8aabaWdbiaaiMdaaa aaaaaa@60E9@                                                                  (14)
The continuity of the flow properties during the transition between the ideal fluid and wave tip regions is preserved. Hence, the integration of the continuity equation yields the analytical solution below within the context of dimensionless wave front celerity.
t= 32 13 (1 u 2 ) 7 2 u 2 ( 3.65x 10 5 k s + 2.5x 10 3 R e d u ) 1 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiDaiabg2da9maalaaapaqaa8qacaaIZaGaaGOmaaWdaeaapeGa aGymaiaaiodaaaWaaSaaa8aabaWdbiaacIcacaaIXaGaeyOeI0YaaS aaa8aabaWdbiaadwhaa8aabaWdbiaaikdaaaGaaiyka8aadaahaaWc beqaa8qadaWcaaWdaeaapeGaaG4naaWdaeaapeGaaGOmaaaaaaaak8 aabaWdbiaadwhapaWaaWbaaSqabeaapeGaaGOmaaaakmaabmaapaqa a8qacaaIZaGaaiOlaiaaiAdacaaI1aGaamiEaiaaigdacaaIWaWdam aaCaaaleqabaWdbiabgkHiTiaaiwdaaaGccaWGRbWdamaaBaaaleaa peGaam4CaaWdaeqaaOWdbiabgUcaRmaalaaapaqaa8qacaaIYaGaai OlaiaaiwdacaWG4bGaaGymaiaaicdapaWaaWbaaSqabeaapeGaeyOe I0IaaG4maaaaaOWdaeaapeGaamOuaiaadwgapaWaaSbaaSqaa8qaca WGKbaapaqabaGcpeGaamyDaaaaaiaawIcacaGLPaaapaWaaWbaaSqa beaapeWaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaisdaaaaaaaaaaa a@5E85@                                                                    (15)
The wave front location is illustrated through equation (16) below, and Figure 2 demonstrates the full flow profile in the presented turbulent model.
x s =( 3 2 u1 )t+ 32 9 (1 u 2 ) 9 2 u 2 ( 3.65x 10 5 k s + 2.5x 10 3 R e d u ) 1 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiEa8aadaWgaaWcbaWdbiaadohaa8aabeaak8qacqGH9aqpdaqa daWdaeaapeWaaSaaa8aabaWdbiaaiodaa8aabaWdbiaaikdaaaGaam yDaiabgkHiTiaaigdaaiaawIcacaGLPaaacaWG0bGaey4kaSYaaSaa a8aabaWdbiaaiodacaaIYaaapaqaa8qacaaI5aaaamaalaaapaqaa8 qacaGGOaGaaGymaiabgkHiTmaalaaapaqaa8qacaWG1baapaqaa8qa caaIYaaaaiaacMcapaWaaWbaaSqabeaapeWaaSaaa8aabaWdbiaaiM daa8aabaWdbiaaikdaaaaaaaGcpaqaa8qacaWG1bWdamaaCaaaleqa baWdbiaaikdaaaGcdaqadaWdaeaapeGaaG4maiaac6cacaaI2aGaaG ynaiaadIhacaaIXaGaaGima8aadaahaaWcbeqaa8qacqGHsislcaaI 1aaaaOGaam4Aa8aadaWgaaWcbaWdbiaadohaa8aabeaak8qacqGHRa WkdaWcaaWdaeaapeGaaGOmaiaac6cacaaI1aGaamiEaiaaigdacaaI WaWdamaaCaaaleqabaWdbiabgkHiTiaaiodaaaaak8aabaWdbiaadk facaWGLbWdamaaBaaaleaapeGaamizaaWdaeqaaOWdbiaadwhaaaaa caGLOaGaayzkaaWdamaaCaaaleqabaWdbmaalaaapaqaa8qacaaIXa aapaqaa8qacaaI0aaaaaaaaaaaaa@672E@                                              (16)

Thus, the dimensionless depth satisfies the free surface profiles are
d = 1, at x  t    MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamizaiaabccacqGH9aqpcaqGGaGaaGymaiaacYcacaGGGcGaamyy aiaadshacaGGGcGaamiEaiaabccacqGHKjYOcaqGGaGaeyOeI0Iaam iDaiaacckacaGGGcGaaiiOaaaa@4877@                                                                                                (17)
d=  1 9 (2 x t ) 2 at tx( 3 2 u1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamizaiabg2da9iaacckadaWcaaWdaeaapeGaaGymaaWdaeaapeGa aGyoaaaacaGGOaGaaGOmaiabgkHiTmaalaaapaqaa8qacaWG4baapa qaa8qacaWG0baaaiaacMcapaWaaWbaaSqabeaapeGaaGOmaaaakiaa dggacaWG0bGaaGPaVlaacckacqGHsislcaWG0bGaeyizImQaamiEai abgsMiJkaaykW7daqadaWdaeaapeWaaSaaa8aabaWdbiaaiodaa8aa baWdbiaaikdaaaGaamyDaiabgkHiTiaaigdaaiaawIcacaGLPaaaaa a@5415@                                                                          (18)
d= ( 9 32 ( 3.65x 10 5 k s + 2.5x 10 3 R e d u ) 1 4   u 2 ( x s x )) 4 9 at( 3 2 u1 )tx  x s   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamizaiabg2da9iaacIcadaWcaaWdaeaapeGaaGyoaaWdaeaapeGa aG4maiaaikdaaaWaaeWaa8aabaWdbiaaiodacaGGUaGaaGOnaiaaiw dacaWG4bGaaGymaiaaicdapaWaaWbaaSqabeaapeGaeyOeI0IaaGyn aaaakiaadUgapaWaaSbaaSqaa8qacaWGZbaapaqabaGcpeGaey4kaS YaaSaaa8aabaWdbiaaikdacaGGUaGaaGynaiaadIhacaaIXaGaaGim a8aadaahaaWcbeqaa8qacqGHsislcaaIZaaaaaGcpaqaa8qacaWGsb Gaamyza8aadaWgaaWcbaWdbiaadsgaa8aabeaak8qacaWG1baaaaGa ayjkaiaawMcaa8aadaahaaWcbeqaa8qadaWcaaWdaeaapeGaaGymaa WdaeaapeGaaGinaaaaaaGccaGGGcGaamyDa8aadaahaaWcbeqaa8qa caaIYaaaaOWaaeWaa8aabaWdbiaadIhapaWaaSbaaSqaa8qacaWGZb aapaqabaGcpeGaeyOeI0IaamiEaaGaayjkaiaawMcaaiaacMcapaWa aWbaaSqabeaapeWaaSaaa8aabaWdbiaaisdaa8aabaWdbiaaiMdaaa aaaOWdaiaaykW7caaMc8+dbiaadggacaWG0bGaaGPaVlaaykW7daqa daWdaeaapeWaaSaaa8aabaWdbiaaiodaa8aabaWdbiaaikdaaaGaam yDaiabgkHiTiaaigdaaiaawIcacaGLPaaacaaMc8UaaGPaVlaadsha cqGHKjYOcaWG4bGaaeiiaiabgsMiJkaadIhapaWaaSbaaSqaa8qaca WGZbaapaqabaGcpeGaaiiOaaaa@7BD3@             (19)
d = 0 at  x s  x  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamizaiaabccacqGH9aqpcaqGGaGaaGimaiaacckacaWGHbGaamiD aiaacckacaWG4bWdamaaBaaaleaapeGaam4CaaWdaeqaaOWdbiabgs MiJkaabccacaWG4bGaaiiOaaaa@455E@                                                                                     (20)

Figure 2 Illustrations of the Turbulent Flow Regions with their Corresponding Solutions.

Laminar results and discussion

The laminar flow conditions usually used in the published analytical models to reflect flow condition in a large reservoir during the propagation of the dam break wave. To validate the proposed laminar model, this study is concentrated to the in-depth detailed measured data within dam break region to validate its dam break wave computation. In Figure 3 the results of laminar dam break wave model show a positive correlation to the published data. The analytically explicit free-surface profile solutions produces results illustrating accurate the wave front shape as referencing to the measured data provided by Debiane4. Also worth noting that different modelling results by various dimensionless velocities are presented in the figure, but make no obvious difference to computed results.

Figure 3 Laminar Dam Break Wave Model versus Debiane4 Measured Data.

Debiane4 also measured a set of data from their experimental study, which utilized the dimensionless reservoir flow Reynolds Number ‘R’ coefficient as 0.671 and the dimensionless time ‘t’ as 114. In Figure 4 comparison to Debiane4 data, the dimensionless ‘ x t  ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaacbmWdbiaa=Hhaa8aabaWdbiaa=rhaaaGaaiiOaiaa cEcaaaa@3A7D@  is captured within 7 x t    MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeyOeI0IaaG4naiabgsMiJoaalaaapaqaa8qacaWG4baapaqaa8qa caWG0baaaiabgsMiJkaacckacaGGGcaaaa@400A@ 3.1 due to majority of the wave propagation occurring in this segment during the experiment. The uniform flow velocity recorded by Debiane4 was 0.672m/s. Reflected in equations (4),(5) and (11) it can be observed that in order to estimate the wave front analytically a reduction factor α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaiigGiabeg7aHjaacMbiaaa@397E@  is required. The reduction factor has proved to be vital in explicitly computing an accurate free surface profile as presenting at Figure 4 and it was observed that adopting α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaiigGiabeg7aHjaacMbiaaa@397E@  as 10 increased the model accuracy significantly. In the figure, one can see that by adopting a reduction factor of 1 highly disturbed the accuracy of the model, while utilizing a reduction factor of 100 posed no significant increase in accuracy in comparison to using a reduction factor of 10.

Figure 4Laminar Flow Dam Break Wave Test using Different α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHbaa@37E5@ Reduction Factor.

Turbulent results and discussion

The proposed model is also used to produce the turbulent dam break wave results to represent the wave front details reported by experimental data provided by Dressler13 and Schoklitsch3. The analytical calculation of the turbulent dam break wave gives efficient solutions with the presences of bed friction as compared to numerical model, i.e. as presented in Pu et al.6,14. This study also adopts the conceptual model of Whitham12 in which the ideal fluid flow is used behind the frictionally dominated wave front region. Whitham12 provided evidence that the analytical dam break wave simulations are unstable at the wave tip, due to the flow singularity with zero water depth hence their study realistically limited to the in-depth detailed data and measurements available on dam break waves in horizontal channels. The published results achieved from the turbulent dam break wave model also indicates that analytical solutions dedicated to dam break waves are suited to flows of high reservoir flow Reynolds Number.2

In Figure 5 and Figure 6, the presented model shows a positive correlation with the reported experimental data. The results indicate that the model is dependent on the frictional factor and relative roughness. At the wave tip region located between the wave front start (x1) and the wave front end (xs), the inertia acceleration terms are significantly mynute and hence dominated by the flow resistance. Figure 5 illustrates the modelling results computed against the Schoklitsch3 data, which indicated the final wave front location and ideal fluid region is in agreement with published data. It is important to note that the reservoir flow Reynolds Number R was taken as 63000. Dressler9 further carried out another experiment, in which the results illustrated the difference between the wave front region of the free surface profile with and without a smooth invert. In order to investigate the measured test case by Dressler9, the dimensionless sand roughness height ‘ks’ has been used in the presented model as 0.35, the dimensionless time ‘t’ was taken as 55.6, and the dimensionless reservoir flow Reynolds Number ‘R’ as 340000 (as the same as conditions in Dressler9). The comparison results in Figure 6 shows that the proposed model reproduces the measured turbulent dam break data with good correspondence.

Figure 5 Turbulent Dam Break Wave Model versus Schoklitsch4 Measured Data.

Figure 6 Turbulent Dam Break Wave Model versus Dressler13 Measured Data.

Conclusion

This study has produced an analytical model to represent the dam break wave by utilizing the explicit modelling suggestions provided by Chanson1,2 for laminar and turbulent flow conditions. In our observation of turbulent modelling, fluids exhibiting low reservoir flow Reynolds Number (Debiane ‘R’=0.671) distorted the results significantly; while fluids exhibiting high reservoir flow Reynolds Number (Dressler ‘R’=340000) provided computation accurately when benchmarked by measurements. It was observed in the laminar dam break wave model that the reduction factor ‘α’ can impact the accuracy of results significantly. The laminar model calculation also indicated that a reduction factor higher than 10 presented no significant increase or decrease in simulation accuracy. It has been noted that the downstream boundary conditions of a dry bed dam break wave decrease to zero at the wave front as the resistance increases to infinity, and this can cause significantly difficult in dam break wave modelling, and approximation used in this studied model has capability to overcome this difficulty and provide precise calculation to the published dam break flow experiments. Generally, all measured data utilized in this study have pointed out that the proposed model is capable to represent different dam break flows under laminar or turbulent flow conditions.15

Acknowledgements

None.

Conflict of interest

Author declares there is no conflict of interest in publishing the article.

References

  1. Chanson H. A Simple Solution of the Laminar Dam Break Wave. Jounral of Applied Fluid Mechanics. 2008;1(1):1−10.
  2. Chanson H. A Simple Solution of the Turbulent Dam Break Wave. Journal of Hydraulic Research. 2009;47(1):41−49.
  3. Schoklitsch A. Uber Dammbruchwellen. Sitzungsberichte der k. Akademie de Wissenschanfern. 1917;126(IIA):1489−1514.
  4. Debiane J. Hydraulique des Ecoulements Laminaires à Surface Libre dans un Canal pour des Milieux Visqueux ou Viscoplastiques: Régimes Uniformes, Graduellement Varié, et Rupture de Barrage. France: Thèse de l’Université Joseph Fourier, Laboratoire de Rhéologie de Grenoble; 2000. 273p.
  5. Dressler RF. Comparison of Theories and Experiments for the Hydraulic Dam Break Wave. International Association of Scientific Hydrology Bulletin. 1954;3(38):319–328.
  6. Pu JH. Turbulence modelling of shallow water flows using kolmogorov appriach. Computers and Fluids. 2015;115(12):66−74.
  7. Zheng XG, Pu JH, Chen R, et al. A Novel Explicit Implicit Coupled Solution Method of SWE for Long-term River Meandering Process Induced by Dambreak. Journal of Applied Fluid Mechanics. 2016;9:2647−2660.
  8. Pu JH, Shao S, Huang Y. Numerical and experimental Turbulence Studies on Shallow Open Channel Flows. Journal of Hydro-environment Research. 2014;8(1):9−19.
  9. Dressler RF. Hydraulic Resistance Effect Upon the Dam-Break Functions. Journal of Research of the National Bureau of Standards. 1952;49(3):217−225.
  10. Deng X, Liu H, Lu S. Analytical Study of Dam-Break Wave Tip Region. Journal of Hydraulic Engineering. 2018;114(5):04018015.
  11. Ritter A. Die Fortpflanzung der Wasserwellen. Vereine Deutscher Ingenieure Zeitswchrift. 1892;36(2):947-954.
  12. Whitham GB. The Effects of Hydraulic Resistance in The Dam Break Problem. Proceedings of Royal Society, Series A. 1955;227(1170):399–407.
  13. Dressler RF, Pohle FV. Resistance Effects on Hydraulic Instability. Communications on Pure and Applied Mathematics. 1953;6:93−96.
  14. Pu JH, Lim SY. Efficient numerical computation and experimental study of temporally long equilibrium scour development around abutment. Enviromental Fluid Mechanics. 2014;14(1):69–86.
  15. Stoker J. Water Waves: The Mathematical Theory with Applications. New York: Interscience Publishers; 1957. 609p.
©2018 Taha et al. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and build upon your work non-commercially.
© 2014-2018 MedCrave Group, All rights reserved. No part of this content may be reproduced or transmitted in any form or by any means as per the standard guidelines of fair use.
Creative Commons License Open Access by MedCrave Group is licensed under a Creative Commons Attribution 4.0 International License.
Based on a work at https://medcraveonline.com
Best viewed in Mozilla Firefox | Google Chrome | Above IE 7.0 version | Opera |Privacy Policy