Shield's diagram and its application

Shields diagram

Shields (1936) showed by experiments that the start of motion of non-cohesive sediment depends on two dimensionless numbers:

• Shields parameter (dimensionless critical shear stress): $\tau_* = \dfrac{\tau_c}{(\gamma_s - \gamma)\,d}$
• Boundary (shear) Reynolds number: $Re_* = \dfrac{u_*\,d}{\nu}$, with $u_* = \sqrt{\tau_0/\rho}$ the shear velocity

where $\tau_c$ is the critical bed shear stress, $\gamma_s$ and $\gamma$ the specific weights of sediment and water, $d$ the grain size and $\nu$ the kinematic viscosity.

 tau*
 0.1 |\
     | \
     |  \                     ___________  0.056
 0.05|   '.                _.-'
     |     '.          _.-'
 0.03|       '-..__..-'      <- minimum
     |    (smooth)  (transition) (rough)
     +------+--------+---------+------------> Re*
           2        10        400
  above curve: bed moves ; below curve: no motion

Important points (based on $Re_*$):

The curve separates the region of no motion (below) from the region of sediment motion (above).

Application in design

To find the critical shear stress for a given grain size $d$:

1. Assume $\tau_*$ (for coarse material 0.056 to 0.06).
2. Find $\tau_c = \tau_*(\gamma_s - \gamma)d$ and $u_* = \sqrt{\tau_c/\rho}$.
3. Compute $Re_* = u_*d/\nu$, read the correct $\tau_*$ from the diagram and repeat until it agrees.

To check or design a channel:

4. Compute the actual bed shear stress $\tau_0 = \gamma R S_0$ (or $\gamma y S_0$).
5. If $\tau_0 < \tau_c$ the bed is stable; if $\tau_0 > \tau_c$ the bed material moves.
6. For design, choose the bed material (or a stable size $d$) so that $\tau_c \ge \tau_0$. For fully rough flow, the size that just stays stable is $d_{cr} \approx 10\,R S_0$ (with $\gamma_s = 2.65\gamma$).