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A novel and complete velocity profile law is presented for turbulent open channel flows. The law embeds the linear law in the viscous sublayer; the quartic law due to turbulent bursting that must satisfy instantaneous mass conservation, the modified log-wake law in the outer region and the effects of wall roughness. Specifically, an arctangent law smoothly connects the linear law with the log law of the wall. Combining the arctangent law with the modified log-wake law, a continuous velocity profile is proposed for smooth boundary conditions. A complete velocity profile is obtained after subtracting a new roughness function. Finally, the proposed laws have been validated with data for hydraulically smooth, transitional and rough turbulent regimes.