whereU*is the shear velocity.
Since in fully turbulent flow 0 "V^2 , equation (8.1) leads to the well-
known Chézy equation for uniform flow:
VC(RS 0 )1/2 (8.2)(the dimensions of Care L1/2T^1 ).
The ‘coefficient’ Ccan be expressed as
C(8g/)1/2 (8.3)whereis the friction coefficient in the Darcy–Weisbach equation:
hf(L/D)V^2 /2gLV^2 /8gR. (8.4)can be expressed from boundary layer theory as
1/1/22 log 6 R
k
2^ /7 (8.5)
wherekis the roughness ‘size’ and (11.6v/U*) is the thickness of the
laminar sublayer.
Another frequently used expression is the Manning equation, using a
constantnwhich is a function of roughness:
V(1/n)R2/3S1/2 0 (8.6)(i.e.CR1/6/n; the dimensions of nare T L^ 1/3). According to Strickler,
n≈0.04d1/6, where dis the roughness (sediment) size (in metres) (Worked
example 8.1).
From equations (8.1), (8.2) and (8.3) it follows that
U*V(/8)1/2. (8.7)From Bernoulli’s equation it follows that for a general non-prismatic
channel and non-uniform flow
S 0 dy/dx (Q^2 /gA^3 )[Bdy/dx(∂A/∂b)(db/dx)]Sf 0and thus
dy/dx[S 0 Sf(Q^2 /gA^3 )(∂A/∂b)(db/dx)]/(1 Fr^2 ). (8.8)For a prismatic channel, db/dx0 and equation (8.8) reduces to
dy/dx(S 0 Sf)/(1 Fr^2 ). (8.9)