1d Hydraulics (Manning-Strickler Formula)

Get ready by cloning the exercise repository:

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

Theoretical background¶

The Gauckler-Manning-Strickler formula Kundu & Cohen, 2008 (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 Kundu & Cohen, 2008. Even though one-dimensional (1d) approaches have largely been replaced by at least two-dimensional (2d) numerical models today, the 1d Strickler formula is still frequently used as a first approximation for boundary conditions.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 & 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).

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 $k_{st}$.

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).

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 Akanbi & Katopodes, 1987 for this purpose (see also the the University of Stuttgart’s ILIAS platform).

Use a Newton-Raphson solution scheme Paine, 1992 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)