Understanding Coastal Sediment Bed Load Transport Equations

 Lecture 22

Bed load transport equations

Following Shields’ work, several bedload transport equations have been developed, relating transport to the entrainment function and its critical value. These are often expressed using the dimensionless bedload transport rate factor (Φ):

Where:

• q_b​: Volumetric bedload transport rate per unit width (m³/m/s)
• s: Relative density of sediment (ρ_s/ρ)
• g: Acceleration due to gravity
• D: Particle diameter

Meyer-Peter and Müller given by

A more recent formula is that of Neilsen (1992), given by

Example:

Calculate the bedload sediment transport rate in a tidal current given the following data: Depth mean current u = 2.0 m/s, grain size D₅₀ = 0.4 mm, water depth h = 10 m, sea water density ρ = 1027 kg/m³ (@ 10° C and salt content 35 ppt), sediment density ρ_s = 2650 kg/m³ and kinematic viscosity ν = 1.36 × 10-6 m²/s.

Solution

calculate the roughness height (height of the irregularities on a surface that influence the flow of fluid) and skin friction drag coefficient (resistance or drag force caused by the friction between a fluid (e.g., air or water) and the surface of an object in motion)

k_s = 2.5D₅₀ = 2. 5 x 0.0004 = 0.001             [D₅₀ = 0.4 mm = 0.0004 m]

z_o = k_s / 30 = 0.001/30 = 3.33x10^-5 m

Now skin friction shear stress and shear velocity

·       The equation for z₀​ is strictly valid for hydraulically rough flow, defined as:

                                   u_∗k_s / ν>70

·       In this case, the flow is not hydraulically rough because:

u_∗k_s /ν=50.7(which is less than 70)

• Despite the flow not being hydraulically rough, the error introduced by using the equation is only about 1% in the calculation of the drag coefficient (C_D​).  50.7 which is near the threshold of 70)
• This small error is acceptable for practical purposes, making the equation sufficiently accurate in this context.

 

The Shields parameter and critical Shields parameter

Total load transport formulae

• Bedload: Sediment moving along the riverbed.
• Suspended Load: Fine particles carried within the water column.
• Total Load: Combination of bedload and suspended load.

Note: Suspended load rarely occurs alone, except with very fine silts. Most transport involves bedload or a mix of both.

 

Ackers and White formula (White 1972) and in revised form in Ackers (1993)

Initially, bedload (coarse material) and suspended load (fine material) were studied separately. Ackers and White developed transitional relationships for intermediate grain sizes using three dimensionless parameters:

1.    G_gr: Sediment transport parameter based on stream power.

o    For bedload, it depends on flow velocity (u) and net shear force on grains.

o    For suspended load, it relates to total stream power.

2.    F_gr: Particle mobility number, representing shear stress/immersed grain weight.

o    Critical value (A_gr) indicates inception of motion.

3.    D_*: Dimensionless particle size number.

These parameters help model sediment transport across varying grain sizes.

The equations are then as follows:

• q_t​ = volumetric total transport rate per unit width (m³/s/m).

·         Index n: Reflects grain size influence.

o    n=1 for fine grains.

o    n=0 for coarse grains.

o    n=f(logD_∗​) for transitional grain sizes.

• The values for n, m, A_gr and C are as follows:

·         Grain Size (D_∗​):

o    D_∗​>60: Coarse sediment (D₅₀>2 mm).

§  n=0, m=7, A_gr​=1.8, C=0.025.

o    1<D_∗​<60: Transitional/fine sediment (D₅₀=0.06−2 mm).

§  n=1−0.56logD_∗​.

§  m=1.67+6.83/D_∗​

§  A_gr​=0.14+0.23/D_∗​​

§  logC=2.79 logD_∗​−0.98(logD_∗​)² − 3.46

·         Use D_35​ when a range of sediment sizes is present.

 

Van Rijn (1984)

Van Rijn developed a sediment transport theory for rivers using fundamental physics and empirical data. His full method is detailed in van Rijn (1993), with simplified equations provided for practical use.

With parameter ranges from h = 1 to 20 m, u = 0.5 to 5.0 m/s, in fresh water @15 °C.

Example:

Calculate the total load sediment transport rate in a tidal current, using the Ackers and White method and the van Rijn method, given the following data:

Depth mean current u = 2.0 m/s, grain size D = 0.4 mm, water depth h = 10 m, sea water density ρ = 1027 kg/m³ (@ 10 °C and salt content 35 ppt), sediment density ρ_s = 2650 kg/m³ and kinematic viscosity ν = 1.36 × 10^-6 m²/s.

Solution:

Ackers and White method:

Need to calculate the form drag contribution to the total bed shear stress

k_s = 2.5D₅₀ = 2. 5 x 0.0004 = 0.001             [D₅₀ = 0.4 mm = 0.0004 m]

z_o = k_s / 30 = 0.001/30 = 3.33x10^-5 m

λr = 1000D₅₀ = 1000x0.0004 = 0.4

Δr = λr/7 = 0.057

Now find the additional roughness height due to bedforms

Take a_r = 1

For ripples, z_0f = 0.057²/0.4 = 8.12 × 10^-3 m.

For dunes, z_0f = 0.253²/73 = 0.88 × 10^-3 m.

Add the two contributions to obtain the total

z_0f = 8.12 × 10^-3 + 0.88 × 10^-3 = 9 × 10^-3 m

Now find the total roughness height and calculate the total drag coefficient C_D:

The particle mobility number F_gr

Finally,

Second, the Van Rijn method:

q_s =3.57 ×10^-3 m³ /s/m

Hence,

 q_t = q_b + q_s = 4. 26 x 10^-3 m³/s/m