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Varied Flow in a Trapezoidal Channel <Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>Water flows along a gently sloped trapezoidal channel with a known initial depth. As the flow progresses along the channel, the water depth eventually reaches a uniform depth that no longer changes with distance along the channel; this is known as the normal depth.
This is known as gradually varied flow. The governing differential equation is 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where S0 is the channel slope, Sf is the energy gradient (or the rate at which energy is lost via friction) as given by the Manning formula, Fr is the Froude number, and x is the distance along the channel.The normal depth (i.e. when y(x) no longer changes with x) is given when S0 = Sf.Given a trapezoidal channel, this application derives formulae for Sf and Fr. These are substituted into the differential equation, which is then numerically solved. Finally, the water surface profile alone the channel is plotted.Trapezoidal channels have several advantages over rectangular channels. Primarily, the wetted perimeter is small compared to the flow area; this reduces energy losses due to viscous drag, and thus maximizes flow.<Text-field style="Heading 1" layout="Heading 1">Differential Equation for Gradually Varied Flow in a Trapezoidal Channel</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnWidth of water 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-sectional area of 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the complete differential equation 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style="Heading 1" layout="Heading 1">Parameters</Text-field>Bottom 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Slope of channel 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Channel 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Coefficient in Manning Equation (1 for SI, 1.49 for FPS)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi9RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lKnJlZmVyZW5jZUdRJTY2MTBGJy8lMmRpc3BsYXlfcmVmZXJlbmNlR1ElNjYwOUYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZDLyUpc3RyZXRjaHlHRkMvJSpzeW1tZXRyaWNHRkMvJShsYXJnZW9wR0ZDLyUubW92YWJsZWxpbWl0c0dGQy8lJ2FjY2VudEdGQy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlItSSNtbkdGJDYmUSIxRidGOUY7Rj4tRjY2L1EiOkYnRjlGO0Y+RkFGREZGRkhGSkZMRk5GUEZTLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJStleGVjdXRhYmxlR0ZDRjk=Gravitational constantLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi9RKSZBc3NpZ247RicvRjNRJ25vcm1hbEYnLyUqcmVmZXJlbmNlR1ElNjYxMEYnLyUyZGlzcGxheV9yZWZlcmVuY2VHUSU2NjA5RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUi1JI21uR0YkNiZRJTkuODFGJ0Y5RjtGPi1GNjYvUSI6RidGOUY7Rj5GQUZERkZGSEZKRkxGTkZQRlMvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2V4ZWN1dGFibGVHRkNGOQ==FlowrateLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEkMC4yRidGOS1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrZXhlY3V0YWJsZUdGPUY5Maximum channel 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Water depth at maximum channel length (this will be the boundary condition on the differential equation)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The differential equation is reduced toLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjZGVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==<Text-field style="Heading 1" layout="Heading 1">Critical Depth and Normal Depth</Text-field>Normal depth (the constant water depth that the flow reaches at some point along the channel).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Critical 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The relative values of yn and yc determine the flow profile
<Text-field style="Heading 1" layout="Heading 1">Numerical Solution of the Differential Equation</Text-field>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 of the water surface profile along the 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 of the water surface profile and the slope of the channel 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