Coastal Sediment Bedforms and Understanding Bed Shear Stress and Shields Parameter

 Lecture 21

Bedforms

When shear stress is sufficient to initiate sediment transport, the bed begins to change, forming various bedforms depending on flow conditions. In uniform currents, small ripples form initially, which can grow into larger dunes. These dunes migrate downstream as sand is eroded from their crests and deposited on the lee side. At higher flow intensities, suspended sediment transport occurs, washing out the dunes. In oscillatory flows, more symmetric wave ripples or larger sand waves may develop.

Bedforms increase frictional resistance and turbulence, influencing total bed shear stress and enhancing suspended sediment transport. Numerous studies have attempted to relate key parameters like Froude Number, sediment and fluid properties, shear stress, bed roughness, dune size, and transport rates. Most equations used today are based on dimensional analysis, experiments, and simplified theoretical models.

 

 

Estimation of bed shear stress

The total bed shear stress (τ₀) consists of three components:

1.    Skin friction (τ₀ₛ): Grain-related friction.

2.    Form drag (τ₀_𝒇): Caused by ripple/dune formation.

3.    Sediment transport contribution (τ₀ₜ): Due to momentum transfer to mobilize grains.

     Hence, the total bed shear stress,     τ₀ = τ₀ₛ + τ₀_𝒇 + τ₀ₜ

The general equation, relating bed shear stress to depth mean velocity (Ū) is given by

τ₀   = ρ C_D Ū ²

The friction or shear velocity (u_∗​) is related to the bed shear stress (τ_0​) by:

u_* =√ (τ₀ / ρ)

where ρ is the fluid density. This parameter is useful for characterizing flow dynamics and shear stress in fluid mechanics.

Current skin friction bed shear stress

In the absence of bedforms:

·       The skin friction bed shear stress (τ₀​) is related to the bed slope (S_0​) by:

τ₀​=ρghS₀​

·       Substituting into the Manning equation:

 

V=1/n * h^2/3 S₀^1/2

·       The drag coefficient (CD​) is given by:

            C_D=gn² / h^1/3

In the presence of bedforms and tidal flows:

• Skin friction bed shear stress is determined by bed roughness, quantified by:
• Nikuradse roughness (k_s), or
• Roughness length (z₀​): the height above the bed where velocity tends to zero.
• A widely used equation for C_D​ is:

 

Hydraulically rough flow:

o    For u_∗ k_s/ν > 70 (common for coarse sands and gravels):

z₀=k_s /30

o    k_s​ is related to grain size (D) and is typically given as:

k_s=2.5D₅₀

 

Current generated ripples and dunes

 

Bedforms in Sandy Beds:

1.    Ripples:

o    Form for grain sizes ≤ 0.8 mm.

o    Wavelength (λr) and wave height (Δr) estimated by:

§  λr = 1000D₅₀

§  Δr = λr/7

o    Typical values: λr = 0.14 m, Δr = 0.016 m.

2.    Dunes & Sandwaves:

o    Larger than ripples, with wavelengths (λs) in tens of meters and wave heights (Δs) in few meters.

o    Dimensions depend on bed shear stress (τ_0s) and water depth (h).

o    Van Rijn (1984) equations:

§  λs = 7.3h

§  Δs depends on τ_0s relative to critical shear stress (τ_CR):

·       Δs = 0 for τ_0s < τ_CR  (no dunes form)

·       Δs = 0.11h (D₅₀/h)^0.3 (1 - e^-0.5Ts)(25 - T_s) for τ_CR < τ_0s < 26τ_CR, where T_s = (τ_0s - τ_CR)/τ_CR

·        Δs = 0 for τ_0s > 26τ_CR  (dunes are washed out).

 

Current total bed shear stress

 

1.    Shear Stress Ratio:

In the presence of bedforms, the ratio of total to skin friction shear stress typically ranges from 2 to 10. Calculating bedform drag is crucial.

2.    Bedform Roughness (z₀_f):

If bedform wavelength (λ) and wave height (Δ) are known, bedform roughness height (z₀f) can be estimated using:

z_0f =a_r Δr² / λ_r​

where a_r​ ranges from 0.3 to 3 (typical value = 1).

3.    Sheet Flow Roughness (z₀t):

Under sheet flow conditions, roughness increases due to turbulent momentum exchange. Wilson (1989) provides:

z_0t=5τ₀ /39g(ρ_s−ρ) ​​

where τ₀​ = bed shear stress, ρ_s​ = sediment density, and ρ = fluid density.

4.    Total Roughness (z₀):

The total roughness length is the sum of skin friction roughness (z₀_s), bedform roughness (z₀_f), and sheet flow roughness (z_0t):

z₀=z_0s+z_0f+z_0t​

5.    Total Drag Coefficient (C_D):

Use z₀​ in Equation for C_D ​, then apply it in Equation of τ₀ to estimate total bed shear stress.

 

The entrainment function (Shields parameter)

 

The Shield parameter (or Shields parameter) is a dimensionless number used in sediment transport studies to describe initiation of sediment motion due to fluid flow on a bed. It is defined as:

Where:

·         θ: Shields parameter (dimensionless)

·         τ₀​: Bed shear stress (the force per unit area exerted by the flow on the sediment bed)

·         ρ_s​: Density of the sediment particles

·         ρ: Density of the fluid (e.g., water)

·         g: Acceleration due to gravity

·         D: Characteristic particle diameter (typically the median grain size)

 

The Critical Shields Parameter is the threshold value of the Shields parameter at which sediment particles just begin to move due to fluid forces. It represents the critical condition for the initiation of sediment motion.

The analysis suggests that the critical entrainment function should be constant. However, Shields (1936) demonstrated that it depends on a form of Reynolds number (Re_∗​), based on the friction velocity:

Re_∗=ρu_∗D/μ

Shields plotted critical Shields parameter (θ_CR​) against Re_∗​, revealing a well-defined threshold band for sediment motion. Later, Soulsby and Whitehouse (1997) expressed this threshold in a more convenient form using a dimensionless particle size parameter (D_∗​).

·       s = ρ_s/ρ and

·       ν = kinematic viscosity of water = µ/ρ.

Above Equation can, therefore, be used to determine the critical shear stress (τ_CR) for any particle size (D).

On a flat bed, if the bed skin friction shear stress (τ_0s​) is known, the Shields parameter

θ_s=τ_0s/(g(ρ_s−ρ) D))

can be calculated to determine the sediment transport regime:

·  θ_s<θ_CR​: No transport occurs.

·  θ_CR≤θ_s≤0.8: Transport occurs with ripples or dunes.

·  θ_s>0.8: Transport occurs as sheet flow with a flat bed.