River Engineering

armor layer: 

• Line sampling followed by counting
• areal smapling followed by counting and sieving analysis
• photogrammetric analysis using basegrain

Subarmor layer: 

• volumetric samspling followed by a sieving analysis

Grain size:

• Longest dimension: a-axis
• Shortest dimension: c-axis
• Normal to a and c: b-axis

b-axis is …

• oriented in flow direction (armor layer)
• characteristic parameter to describe the hydraulic force acting on the grain
• represents the flow resistance ~roughness
• determines passage through sieve

Line Sampling

1. Line in flow direction
2. Count all grains that touch the line
3. Measure the b-axis of all grains ≥ 1 cm
4. Divide grain sizes in classes (1-2 cm; 2-3 cm; … > 200 cm)
5. Count number of grains for each class and list them in a protocol
6. Record at least 150 grains or until the distribution does not change anymore

rho_s≈ 2650 kg/m³ (solid matter)

n: the ratio of the pore volume V_p and the total volume V_tot (V_tot= V_s+ V_p) with V_s= sediment volume

n = V_p / V_tot

Density of sediment deposition r_l ≈ 1850 kg/m³(n≈ 0.3)

Angle of repose III corresponds to a threshold inclination that results from a loose fill up of the material

• Important for bank stability
• Depends on grain size and shape

Tilting angle phi describes the stability of a single grain in a river bed. Tested experimentally

Motion of a single grain is slipping/rolling or tilting – apply force and moments balance to determine threshold for incipient motion

Forces: 

• F_G
• F_drag
• F_lift
• F_B

Incipient motion of a grain is governed by stress versus resistance

Stress versus Resistance

• Stress varies in time and space
• Stress = non-dimensional shearstress θ
• Resistance varies in space
• Resistance = non-dimensional critical shear stress θc

1. No movement, Stress < resistance,Acting hydraulic forces too small for mobilization
2. Start of movement, Individual grains are mobilized, Stress < resistance
3. Start of transport, Several grains are mobilized, but not all, Stress = resistance
4. All grains are transported, Stress > resistance

Based on observations, we can express the incipient motion of uniform grains using the shields diagram  - non-dimensional bed shear stress q versus grain shear Reynolds number R

Tilting or rolling (kippen/abrollen): The threshold for incipient motion is reached when the moments of the acting forces with respect to Pare in equilibrium.

Slipping (abgleiten): The threshold for incipient motion is reached when the acting forces are larger than the retaining force in the tangent plane t-t.

Behaviour of grains on inclined plane, to:

• evaluate channel stability
• design embankment protection measures

Description via Force Balance

• F_Lift can be neglected due to small flow velocities

Increased Discharge Q ->top layer is coarsening, bc fine material is washed out

Reaching of Q_D -> protecting armor layer breaks up, self-stabilization not possible anymore

Embankment stability can also be derived based on a force balance and depends on the angle of repose y and degree of inclination g

Grain size distribution of armor layer can be determined using approaches by Gessler or Fehr: Fehr can only be used to get the maximum coarsening. Gessler can be used to assess multiple coarsening states

Definition bimodal mixture:

• characterized by fine and very coarse particles – sinking in versus slipping
• show a significant gap in the range of medium grain size fractions
• Mixture of fine and very coarse particles
• Important if the bed is characterized by finer material and larger blocks are used for bed stabilization

Grain size distribution according to Gessler

Grain size distribution of the armor layer can be derived as a function of the hydraulic load/ bed shear stress τ₀. The idea is to determine qi (the probability of a grain with d_mi to remain in place for a given hydraulic load) as a function of τ₀

Process:

1. Determine the shear velocity and grain shear Reynolds number for every grain size class i:
2. Using the results of R∗ for every grain size class i, the non-dimensional critical bed shear stress θ_c can
3. be determined and from that the ratio θ_c/θ_i.. By using the Gessler diagram the value qi can be estimated

Fehr

• easier approach to determine the grain size distribution of the armour layer at maximum coarsening
• factors 0.1 and 0.9 from experiments)

Stability according to Günther

For large grain shear Reynolds numbers R∗ = U∗d/ν > 10³, the critical non-dimensional bed shear stress for the armour layer based on Günter can be expressed:

Grains move if Stress = Resistance

• Maximum bed shear stress acts at channel center
• Mean bed shear stress acts across entire channel width
• θ_m>θ_c(bedmaterial) – Erosion across entire channel
• θ_max>θ_c (bedmaterial) – Erosion in central part of channel

Difference between tau and theta

Bed Shear Stress (τ)

• Definition: The force per unit area exerted by the flowing water on the riverbed.
• Physical Meaning: Determines the initiation of sediment movement and the ability of flow to transport particles.

Non-Dimensional Shear Stress (Shields Parameter, θ)

• Definition: A dimensionless number that compares fluid forces (shear stress) to gravitational forces acting on sediment particles.
• Physical Meaning: Used to predict sediment motion and determine the critical shear stress for particle movement.

• Q_D = discharge at the breakup of the armor layer
• Q_Dmax= discharge when erosion occurs across entire width.
  • With θ_m
• Q_Dmin= discharge when erosion occurs in channel center
  • With θ_max

Embankment/wall: shear stress distribution can be approximated to be linear and increase proportional with water depth h

• τ_Bmax at 0.2-0.3 h
• τ_Bmax ≈ 77% τ_0max = 0.77 rghJ

• Patterns caused by random erosion and deposition of grains
• Bedforms shape depends on acting stress
• Bedforms affect flow resistance

• Bedforms shape depends on the acting stress
• Lower regime: ripples dunes
• Upper regime: antidunes

Ripples:

• small compared to water depth, water surface not affected
• Ripple length Λ ≈ 1000 d
• maximum height Δ_max= 200 d
• steepness (Δ/Λ)_max≈ 0.2
• Maximum grain size diameter to form ripples: d= 0.6 mm
• Λ_max= 60 cm, Δ_max= 10 to 15 cm

Dunes

• Size of dunes scale with the water depth; water surface is affected
• F<1
• Average length (Yalin):
• Λ= (8-10)h
• (Δ/Λ)_max≈ 0.06
• According to Zanke(1982) for d> 0.2 mm:average dune height 0.15h< Δ <0.30h
• According to Yalin(1977); dune length Λ= 2πh≈ 6.3h (theory) /dune length 6.3h < Λ < 20h (observation)