River flow

A river runs through it - my garden. With increasing concern about floods, insurance companies are waking up ad asking questions. Council planning departments are asking a lot more about environmental impact too: bats, birds, bushes, rivers, run-off and trees.

The UK Environment Agency manages the waterways. It uses LIDAR, a radar measurement system, to generate flood maps and keeps a data-base of flows along rivers. But that doesn't help with individual properties. They can only tell me that the house is 33m above datum sea level and the worst case flow past it would be 41.2 cu.m/sec. I have their map from their 2015 online database** and it shows that the river will flood at both bends, about 200m above and 100m below my house where there are sharp turns winding around the hillside. That's fields, woodlands and two disused mill-ponds. The map also shows slight flooding over the bank opposite my house, which is a nature trail and dog-walk in woodland. No flooding shown at my side of the river bank.

Perhaps that's all I need to know, but I'm curious about that maximum flow value of 41.2 cu.m/sec thrown up by their computer modelling, presumably for the whole stretch of the river in my area. What would that look like?

I'm on a hill at 1:18 slope, according to the data from google maps. The river here is straight and the banks are vertical sheer, dressed stone or caissons, no significant protrusions and no necking. The bed is rough with slabs or rock jutting at random. In Summer it's a slow run that just covers your boot. In a winter storm it's a torrent about thigh deep and could sweep you away like a leaf.

So how high will it rise? That's a complicated calculation using the Navier-Stokes formulae for conservation of mass of a viscous liquid flowing down an inclined plane, corrected by the Poiseuille equation for flow against a surface that drags. It slices the river into laminar flows considered as ribbons moving past each other at different speeds, the fastest being at the top in the centre for an open channel like a river, the slowest at the bottom touching a bank. Then there's turbulence, seemingly unpredictable. This is the academic approach using differential calculus.

However the civil engineers in the USA have no use for this. They have found a practical solution based on experimental and empitical measurements called the Manning Equations* Q = (1.0/n)A(R2/3)(S1/2), Where

• Q = volumetric water flow rate passing through the stretch of channel, (m3/s)
• A = cross-sectional area of flow perpendicular to the flow direction, (m2 )
• S = bottom slope of channel, m/m (dimensionless),
• n = Manning roughness coefficient (empirical constant), dimensionless,
• R = hydraulic radius = A/P in m where
• A = cross-sectional area of flow as defined above,
• P = wetted perimeter of cross-sectional flow area, (m)

• A = cross-sectional area of flow perpendicular to the flow direction, (m2 )
• S = bottom slope of channel, m/m (dimensionless),
• n = Manning roughness coefficient (empirical constant), dimensionless,
• R = hydraulic radius = A/P in m where
• A = cross-sectional area of flow as defined above,
• P = wetted perimeter of cross-sectional flow area, (m)

So I know that the river's length (300m), width (9,2m) and depth (Z) are not going to change from minute to minute and I know the volume passing and the slope. That should be enough to get a rough solution to these equations. I just need to find a mathemetician working in the civil engineering field.

• * https://www.brighthubengineering.com/hydraulics-civil-engineering/52905-manning-equation-for-uniform-open-channel-flow-calculations/ thanks to Martin Brown for this link
• ** maps.environment-agency.gov.uk