Program for critical depth in trapezoidal channel

Principle. Critical flow requires $\dfrac{Q^2 T}{g A^3} = 1$. For a trapezoidal section with bed width $b$ and side slope $z$ (z H : 1 V):

$$A = (b + zy)y,\quad T = b + 2zy$$

Define $f(y) = Q^2 T - gA^3$. The critical depth is the root of $f(y) = 0$. $f > 0$ for $y < y_c$ and $f < 0$ for $y > y_c$, so the bisection method always converges.

Algorithm

1. Start.
2. Read $Q$, $b$, $z$. Set $g = 9.81$ and tolerance $\varepsilon = 10^{-6}$.
3. Set $y_{low} = 10^{-6}$ and $y_{high} = 1$.
4. While $f(y_{high}) > 0$, set $y_{high} = 2\,y_{high}$ (bracket the root).
5. Repeat: $y_{mid} = (y_{low} + y_{high})/2$.
   • If $f(y_{mid}) > 0$, set $y_{low} = y_{mid}$; otherwise set $y_{high} = y_{mid}$.
6. Stop repeating when $y_{high} - y_{low} < \varepsilon$.
7. Print $y_c = (y_{low} + y_{high})/2$.
8. Stop.

Program in C

#include <stdio.h>
#include <math.h>
#define G 9.81

/* f(y) = Q^2*T - g*A^3 ; zero at critical depth */
double f(double y, double Q, double b, double z)
{
    double A = (b + z * y) * y;
    double T = b + 2.0 * z * y;
    return Q * Q * T - G * A * A * A;
}

int main(void)
{
    double Q, b, z, lo, hi, mid, tol = 1e-6;
    int i;

    printf("Enter discharge Q (m3/s): ");
    scanf("%lf", &Q);
    printf("Enter bed width b (m): ");
    scanf("%lf", &b);
    printf("Enter side slope z (z H : 1 V): ");
    scanf("%lf", &z);

    lo = 1e-6;
    hi = 1.0;
    while (f(hi, Q, b, z) > 0.0)      /* expand until sign changes */
        hi *= 2.0;

    for (i = 0; i < 100 && (hi - lo) > tol; i++) {
        mid = 0.5 * (lo + hi);
        if (f(mid, Q, b, z) > 0.0)
            lo = mid;                 /* root is deeper */
        else
            hi = mid;                 /* root is shallower */
    }
    mid = 0.5 * (lo + hi);
    printf("Critical depth yc = %.4f m\n", mid);
    return 0;
}

Sample run: $Q = 6$, $b = 2.5$, $z = 2$ gives $y_c = 0.6912\ \text{m}$.

Setting $z = 0$ gives a rectangular channel and $b = 0$ gives a triangular channel, so the same program handles all three shapes.