Design a gravity spillway with an upstream face sloping at $3$V:$2$H and a crest width of $35$ meters. This spillway must manage a $100-$year flood flow of $602$ cubic meters per second. Ensure the safety of nearby communities by maintaining the upstream water surface at a design discharge elevation of $856$ meters, with the reservoir bottom at $850$ meters. Downstream, the spillway feeds into a $35-$meter$-$wide stilling basin at an elevation of $842$ meters, which transitions to a culvert under a railway line.

Develop and present detailed profiles for the spillway crest, face, and the transition arc leading to the stilling basin. Design the stilling basin and determine its length. The culvert, downstream of the stilling basin, is made of multiple uniform rectangular concrete barrels with entrance characteristics K$=0\mathrm{.}061,$ M$=0\mathrm{.}75,$ c$=0\mathrm{.}042,$ Y$=0\mathrm{.}80$ and an entrance loss coefficient k$=0\mathrm{.}6.$ The culvert spans $75$ meters in length on a $5\%$ slope.

Calculate the required quantity and dimensions of the rectangular barrels to ensure that the headwater elevation remains below the water depth downstream of the hydraulic jump in the stilling basin. Note that a barrel height should be $60\%$ of its width. Furthermore, give an allowance of $200$mm for the concrete thickness of each barrel. Assume a Manning roughness for the concrete culverts of $0\mathrm{.}018$ and a tailwater elevation of $0\mathrm{.}8$ meters.