Solve Manning's Equation with Python Scipy library

from scipy.optimize import root

def manningC(d, args):
    Q, w,h,sSlopeL,sSlopeR,nMann,lSlope = args
    #left side slope can be different from right side slope
    area = ((((d*sSlopeL)+(d*sSlopeR)+w)+w)/2)*d
    # wet perimeter
    wPer = w+(d*(sSlopeL*sSlopeL+1)**0.5)+(d*(sSlopeR*sSlopeR+1)**0.5)
    #Hydraulic Radius
    hR = area/ wPer
    # following formula must be zero
    # manipulation of Manning's formula
    mannR = (Q*nMann/lSlope**0.5)-(area*hR**(2.0/3.0))
    return mannR
    
###### MAIN CODE
# the following are input data to our open channel manning calculation
# flow, width, height, left side slope, right side slope, 
# Manning coefficient, longitudinal slope
args0 = [2.5,2,.5,1.0,1.0,.015,.005]
initD = .00001  # initial water depth value

# then we call the root scipy function to the manningC
sol =root(manningC,initD, args=(args0,))    
# print the root found
print(sol.x)

Solving Manning's equation with Python

Manning's equation is a very common formula used in hydraulic engineering. It is an empirical formula that estimates the average velocity of open channel flow, based on a roughness coefficient.



The problem of solving Manning's formula is that it is an implicit formula, for the water depth variable, that defines the R (Hydraulic Radius).

A common approach for solving this equation is using numerical methods, as the Newton-Raphson method. An implementation for the resolution of the Manning Formula for pipes, by the Newton-Raphson approximation, is shown below.

def manningTub(depth, args):
    Q = args[0]
    diamMM = args[1]
    nMann = args[2]
    slope = args[3]
    diamM = diamMM / 1000.0
    angle = 2*acos(1-2*depth/diamM)
    area = diamM*diamM/8.0*(angle-sin(angle))
    radius = diamM/4*(1-sin(angle)/angle)
    mannR = (Q*nMann/slope**0.5)-(area*radius**(2.0/3.0))
    return mannR

def solve(f, x0, h, args):
    lastX = x0
    nextX = lastX + 10* h  # "different than lastX so loop starts OK
    while (abs(lastX - nextX) > h):  # this is how you terminate the loop - note use of abs()
        newY = f(nextX,args)  # just for debug... see what happens
        print "f(", nextX, ") = ", newY     # print out progress... again just debug
        lastX = nextX
        nextX = lastX - newY / derivative(f, lastX, h, args)  # update estimate using N-R
    return nextX

def derivative(f, x, h, args):
      return (f(x+h,args) - f(x-h,args)) / (2.0*h)  # might want to return a small non-zero if ==0

#### EXAMPLE CALL

xFound = solve(manningTub, initDepth, 0.001, [flowRate,diam,nMann,enSlope])    # call the solver