# 1d Hydraulics (Manning-Strickler Formula)¶

Goals

Write a basic script and use loops. Write a function and parse optional keyword arguments (`**kwargs`

).

Requirements

*Python* libraries: *math* (standard library). Read and understand how loops and Functions work in Python.

Get ready by cloning the exercise repository:

```
git clone https://github.com/Ecohydraulics/Exercise-ManningStrickler.git
```

## Theoretical background¶

The Gauckler-Manning-Strickler formula [KC08] (or *Strickler formula* in Europe) relates water depth and flow velocity of open channel flow based on the assumption of one-dimensional (cross-section-averaged) flow characteristics. The *Strickler formula* results from a heavy simplification of the Navier-Stokes equations and the Continuity equation [KC08]. Even though one-dimensional (1d) approaches have largely been replaced by at least two-dimensional (2d) numerical models today, the 1d Strickler formula is still frequently used as a first approximation for boundary conditions.

The basic shape of the *Strickler formula* is:

where:

\(u\) is the cross-section-averaged flow velocity in (m/s)

\(k_{st}\) is the

*Strickler*coefficient in*fictional*(m\(^{1/3}\)/s) corresponding to the inverse of Manning’s \(n_m\).\(k_{st}\) \(\approx\) 20 (\(n_m \approx\) 0.05) for rough, complex, and near-natural rivers

\(k_{st}\) \(\approx\) 90 (\(n_m \approx\) 0.011) for smooth, concrete-lined channels

\(k_{st}\) \(\approx\) 26/\(D_{90}^{1/6}\) (approximation based on the grain size \(D_{90}\), where 90% of the surface sediment grains are smaller, according to Meyer-Peter and Müller [MPM48]

\(S\) is the hypothetic energy slope (m/m), which can be assumed to correspond to the channel slope for steady, uniform flow conditions.

\(R_{h}\) is the hydraulic radius in (m)

The hydraulic radius \(R_{h}\) is the ratio of wetted area \(A\) and wetted perimeter \(P\). Both \(A\) and \(P\) can be calculated as a function of the water depth \(h\) and the channel base width \(b\). Many channel cross-sections can be approximated with a trapezoidal shape, where the water surface width \(B=b+2\cdot h\cdot m\) (with \(m\) being the bank slope as indicated in the figure below).

Thus, \(A\) and \(P\) result from the following formulas:

Finally, the discharge \(Q\) (m³/s) can be calculated as:

## Calculate the discharge¶

Write a script that prints the discharge as a function of the channel base width \(b\), bank slope \(m\), water depth \(h\), the slope \(S\), and the *Strickler* coefficient \(k_{st}\).

Tip

Use `import math as m`

to calculate square roots (`m.sqrt`

). Powers are calculated with the `**`

operator (e.g., \(m^2\) corresponds to `m**2`

).

## Functionalize¶

Cast the calculation into a function (e.g., `def calc_discharge(b, h, k_st, m, S): ...`

) that returns the discharge \(Q\).

## Flexibilize¶

Make the function more flexible through the implementation of (optional) keyword arguments so that a user can optionally either provide the \(D_{90}\) (`D90`

), the *Strickler* coefficient \(k_{st}\) (`k_st`

), or *Manning’s* \(n_m\) (`n_m`

).

Tip

In the code, only use *Manning’s* \(n_m\) and parse `kwargs.items()`

to find out the `kwargs`

provided by a user.

## Invert the function¶

The backward solution to the *Manning-Strickler* formula is a non-linear problem if the channel is not rectangular. This is why an iterative approximation is needed and here, we use the *Newton-Raphson* scheme [AK87] for this purpose (see also the the University of Stuttgart’s ILIAS platform).

Absolute Values

The absolute value of a parameter can be easily accessed through the built-in `abs()`

method in *Python3*.

Use a Newton-Raphson solution scheme [Pai92] to interpolate the water depth `h`

for a given discharge `Q`

of a trapezoidal channel.

Write a new function

`def interpolate_h(Q, b, m, S, **kwargs):`

Define an initial guess of

`h`

(e.g.,`h = 1.0`

) and an initial error margin (e.g.,`eps = 1.0`

)Use a

`while`

loop until the error margin is negligible small (e.g.,`while eps > 10**-3:`

) and calculate the :wetted area

`A`

(see above formula)wetted perimeter

`P`

(see above formula)current discharge guess (based on

`h`

):`Qk = A ** (5/3) * sqrt(S) / (n_m * P ** (2 / 3))`

error update

`eps = abs(Q - Qk) / Q`

derivative of

`A`

:`dA_dh = b + 2 * m * h`

derivative of

`P`

:`dP_dh = 2 * m.sqrt(m ** 2 + 1)`

function that should become zero

`F = n_m * Q * P ** (2 / 3) - A ** (5 / 3) * m.sqrt(S)`

its derivative:

`dF_dh = 2/3 * n_m * Q * P ** (-1 / 3) * dP_dh - 5 / 3 * A ** (2 / 3) * m.sqrt(S) * dA_dh`

water depth update

`h = abs(h - F / dF_dh)`

Implement an emergency stop to avoid endless iterations - the Newton-Raphson scheme is not always stable!

Return

`h`

and`eps`

(or calculated discharge`Qk`

)