OE44165 Sediment transport in dredging engineering
▪
Dr. G. H. Keetels 2 October 2023
OE44165
AdvectionDiffusion
Dr. G. H. Keetels
Advection-diffusion why important?
Governing equation for (fine) sediment transport
Applications:
• Environmental dispersion
• Dredging/Turbidity plumes
• Channel flow
• Sediment transport along underwater slopes
• Hopper sedimentation
• Breaching, remember from last lecture
Source: ESA
(near Rome)
What do we mean with fine sediment?
St=drag time scale / eddy
turnover time flow
Fine means, St<<1
Particles follow turbulent eddy
We can ignore inertia of particles
Learning objective
By the end of this lecture, the student will be able to
1. Describe the different terms in the advection-diffusion equation
2. Apply the Rouse distribution
3. Discuss the limitations of this approach
East
Y
West
Basic derivation: consider a control volume
ΔX
X
ΔY
Y
𝑤
East
𝑢𝑐
West
Basic derivation: consider a control volume
𝜕𝑢𝑐
𝑢𝑐 𝑤 + Δx
𝜕𝑥
X
𝑢 is the x-velocity of the particles [m/s] (note m3/m2=m, so volume flux per area)
c is concentration in [kg/m3]
Basic balance
𝜕𝑐
Δ𝑥Δ𝑦Δ𝑡 = uc
𝜕𝑡
Total change [kg]
Since
𝑢𝑐
𝐸
W Δ𝑦Δ𝑡
− 𝑢𝑐 𝐸 ΔyΔ𝑡
= Flux West – Flux East
𝜕𝑢𝑐
= 𝑢𝑐 𝑤 + Δx
𝜕𝑥
It follows that
𝜕𝑐 𝜕𝑢𝑐
+
=0
𝜕𝑡
𝜕𝑥
Change in time + Change due to advection=0
Extension to 3D
𝜕𝑐 𝜕𝑢𝑐 𝜕𝑣𝑐 𝜕𝑤𝑐
+
+
+
=0
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
Example laminar flow
▪ (2) Fluid Reversibility part 1 - YouTube
In turbulent flow
𝑢
Reynolds decomposition:
𝑢 = 𝑢 + 𝑢′
𝑢′
𝑢
𝑐 = 𝑐 + 𝑐′
𝑡
Instantaneous = mean + fluctuation
Mean could very slowly change in time compared to fluctuations
Also possible to consider ensemble averages (repeating experiments)
Note that
However
𝑢 = 𝑢 + 𝑢’
𝑢′2 ≠ 0
so
𝑢’=0
Variance and covariance
𝑢
Note that 𝑢 = 𝑢 + 𝑢’
However, the variance
Also, the covariance is
not necessarily zero!
𝑢′ 𝑐′
≠0
so
𝑢′2
𝑢′
𝑢’=0
≠0
𝑢
𝑡
𝑐
𝑐′
𝑐
𝑡
Applying Reynolds decomposition
𝜕𝑐 𝜕𝑢𝑐
+
=0
𝜕𝑡
𝜕𝑥
Gives
𝜕𝑐ҧ 𝜕𝑢ത 𝑐ҧ 𝜕𝑢′ 𝑐′
+
+
=0
𝜕𝑡
𝜕𝑥
𝜕𝑥
Closure problem: we want to solve for mean values,
but we need to provide the covariance between u an c
We could derive an evolution equation for the covariance, but
Then we need to solve for triple correlations and so forth
Solution: can we express the covariance in terms of the
mean variables?
Change of mean concentration = advection of mean concentration by mean velocity
+turbulent transport
Eddy diffusivity concept
𝑢′ 𝑐′
𝜕𝑐ҧ
= −𝜖
𝜕𝑥
𝑢′ 𝑐′ > 0
𝑢′ < 0 , 𝑐 ′ < 0
𝑢′ > 0, 𝑐 ′ > 0
𝜖
Turbulent diffusivity [m2/s]
Length scale x velocity scale of eddy
𝜕𝑐ҧ
<0
𝜕𝑥
Finally, we arrive at
𝜕𝑐ҧ 𝜕𝑢ത 𝑐ҧ
𝜕 𝜕𝑐ҧ
+
=
𝜖
𝜕𝑡
𝜕𝑥
𝜕𝑥 𝜕𝑥
Change in time+advection=turbulent diffusion
Usually, the bar is omitted for the sake of simplicity
𝜕𝑐 𝜕𝑢𝑐
𝜕 𝜕𝑐
+
=
𝜖
𝜕𝑡
𝜕𝑥
𝜕𝑥 𝜕𝑥
Basic solution for uniform velocity and diffusivity
Consider periodic boundary conditions at x=0 and x=L
and initial condition
gives
Translation of initial condition x decaying function in time, faster for larger wave numbers
Application to channel flow
𝑢𝑓
𝑐
𝑦
𝑥
𝐻
How can we estimate the diffusivity?
𝜕
𝑢
ത
𝜌𝑢′ 𝑤′ = −𝜌𝜈 𝑒
𝜕𝑦
Eddy viscosity [m2/s]
The same eddies that transport momentum also transport the particles
𝜈𝑒
𝜖=
𝜎
Turbulent Schmidt-Prandtl number
Typical profile of eddy viscosity in channel
Friction velocity
𝑦
𝜈𝑒 = 𝜅𝑢∗ 𝑦(1 − )
𝐻
Von Kármán constant (=0.4)
𝑢∗ =
𝜏/𝜌
𝜏 = 𝜌𝑢′ 𝑤′
Advection-diffusion in vertical direction
𝜕𝑐 𝜕𝑣𝑐
𝜕 𝜕𝑐
+
=
𝜖
𝜕𝑡
𝜕𝑦
𝜕𝑦 𝜕𝑦
Consider stationary concentration profile
𝜕𝑐
𝑣𝑐 = 𝜖
𝜕𝑦
And mean vertical velocity of particles equals terminal settling velocity in
stagnant water
𝑣𝑝∞ 𝑐
𝜕𝑐
=𝜖
𝜕𝑦
Solution Rouse profile
𝑐
𝐻−𝑦
𝑦𝑎
=
∙
𝑐𝑎
𝑦
𝐻 − 𝑦𝑎
𝑣𝑝∞ 𝜎
𝑃=
𝜅𝑢∗
𝑦𝑎
𝑃
𝑦
[-]
𝐻
Rouse number
Reference level
𝑐
[-]
𝑐𝑎
Limitations of Rouse I
Greimann, B. P., & Holly Jr, F. M. (2001). Two-phase flow analysis of
concentration profiles. Journal of hydraulic Engineering, 127(9), 753-762.
Limitations of Rouse II
More advanced two-phase flow analysis is essential for these cases!!
Why limited?
These deviation are controlled by the Stokes number
𝜏𝑝 𝑢 ∗
𝑆𝑡 =
𝐻
St=drag time scale / eddy
turnover time flow
Learning objective
By the end of this lecture, the student will be able to
1. Describe the different terms in the advection-diffusion equation
2. Apply the Rouse distribution
3. Discuss the limitations of this approach