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
Rhone Switzerland Pfynnwald

Fig. 6 The Rhone River in Switzerland (source: Sebastian Schwindt 2014).

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 and the continuity equations [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 Even though one-dimensional (1d) approaches have largely been replaced by at least two-dimensional (2d) numerical models today, the 1d Strickler formula is still frequently used as a first approximation for boundary conditions.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:

\[ u = k_{st}\cdot S^{1/2} \cdot R_{h}^{2/3} \]

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 1948

  • \(S\) 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).

1d hydraulics parameters

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

\[ A = h \cdot 0.5\cdot (b + B) = h \cdot (b + h\cdot m) \]
\[ P = b + 2h\cdot (m^2 + 1)^{1/2} \]

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

\[ Q = u \cdot A = k_{st} \cdot S^{1/2}\cdot R_{h}^{2/3} \cdot A \]

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 usage of optional Keyword Arguments **kwargs 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).

Tip

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)