Analysis of Pile Scour and Associated
Hydrodynamic Forces using Proper
Orthogonal Decomposition
Niels G. Jacobsen (Deltares), Greta van Velzen (Deltares) and
Jørgen Fredsøe (TU Denmark)
7th Dutch OpenFOAM Users Group
17 december 2014
Outline
• Short on other Deltares and OpenFoam
• Motivation and model description
• Test case
• Results – flat bed (onset of scour)
• Results – scoured bed
• Conclusion
17 december 2014
Deltares and OpenFoam
Wave and Structures
• Interaction with porous structure
• Slamming forces on structures
• Knowledge about waves2Foam
Sediment Transport and Scour/Morphology
• Sediment transport and resulting bed level change
• Moving meshes (either 3 or 4 connected meshes)
• Finite area mesh
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Motivation and Model
Description
Motivation
• Modelling scour at large scales could increase knowledge on scale
effects
• The modelling is, however, still very time consuming (Dixen et al.,
2013)
• How can the modelling approach be accelerated?
• Is it valid to decouple hydrodynamics and scour development?
• I.e. average out the vortex shedding in the hydrodynamics (e.g.
Roulund et al., 2005 and Dixen et al., 2013).
• Can be addressed by answering: What are the transient effects?
• Presentation based on paper accepted at ICSE-7 in Perth 2014.
17 december 2014
Numerical Model (Overview)
• Transient solution to the Reynolds Averaged Navier-Stokes
equations
• 𝑘 − 𝜔 turbulence closure with wall function (Jacobsen et al., 2012)
• Sediment transport is modelled with bed load and suspended load
(Jacobsen et al., 2014)
• Bedload and reference concentration concept. Both following
Engelund and Fredsøe (1976)
• No response of the bed (immobile bed)
• Using OpenFoam® v. 1.6-ext.
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POD Analysis – I
• With POD analysis main coherent features (modes) can be
extracted (in excess of the average)
• Define the fluctuating field of some properties (𝜏𝑏 , Δℎ𝑏 /Δ𝑡, Δℎ𝑠 /Δ𝑡):
𝒙′ = 𝒙 − 𝒙
• Construct the square matrix 𝑨𝑁×𝑁
𝑨𝑁×𝑁 = 𝑼𝑇 𝑼 = 𝒙1′ 𝒙′2 … 𝒙′𝑁
𝑇
𝒙1′ 𝒙′2 … 𝒙′𝑁
• 𝑨𝑁×𝑁 has 𝑁 eigenvalues (𝜆𝑖 ) and eigenvectors (𝒆𝑖,𝑁×1 ) and the
modes are a linear combination of the fluctuating fields:
𝒆𝑖 𝑼𝑇
𝝓𝑖 =
𝒆𝑖 𝑼𝑇
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2
POD Analysis – II
• Example of POD-mode
• When does a mode matter?
Determined by its weights, which is a
projection:
𝒘𝑖 (𝑡) = 𝝓𝑖 𝒙′(𝑡)
and the relative importance:
𝝓1
𝝓1
Relative importance =
𝜆𝑖
𝑗
𝝓2
𝝓2
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𝜆𝑗
Test Case
Flow and Sediment Properties
• Inlet: Equilibrium conditions with V = 29.1 cm/s (also turbulence)
• Sediment is 0.18 mm and surface roughness is 0.45 mm
• Far-field Shields number is 0.045,
i.e. studying clear-water scour
• A total of 554,000 computational cells
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Shape of the Sediment Bed
• Three scour depths are investigated:
• 𝑆 𝐷 = 0.0 (flat bed)
• 𝑆 𝐷 = 0.5 (intermediate scour)
• 𝑆 𝐷 = 1.0 (developed scour)
• The bed is kept immobile
• Simulation of 250 s (or equal
to ~150 shed vortices)
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Results – Flat Bed
Vortex Shedding and Erosion/Deposition
• Effect of incoming shear flow in vortex shedding
• Drag coefficients in expected range
• Example of transient erosion/deposion pattern (in m/s)
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Average Bed Properties
𝜏𝑏 /𝜌
Δℎ𝑏 /Δ𝑡
Δhs /Δ𝑡
Mean
Std
• Stagnation in 𝜏𝑏 /𝜌 around 𝑦 = 0
• Std. of Δℎ𝑏 and Δℎ𝑠 only considerable for 0 < 𝑥
• Also, the skewness of Δℎ𝑏 and Δℎ𝑠 is small, i.e. averaging can be
allowed (on a flat bed)
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POD Modes – Bed Shear Stress
• Symmetric in magnitude
• Asymmetric in direction
• The modes describe 75% of the
agitating force (in excess of the
average)
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POD Modes – Erosion/Deposition
Δℎ𝑏 Δ𝑡
Δℎ𝑠 Δ𝑡
• Modes are either symmetric or
asymmetric.
• Erosion/deposition (in excess of
average) most prominent in the
wake.
Δℎ𝑏 Δ𝑡
Δℎ𝑏 Δ𝑡
Δℎ𝑠 Δ𝑡
Δℎ𝑠 Δ𝑡
17 december 2014
• Asymmetric modes have some
upstream response.
• Can this be linked to modes in bed
shear stresses?
POD Modes – Matching the Frequency
•
•
•
•
FFT on POD-weights
Only near-bed frequencies observed
Modes are linked using a correlation analysis on weights
For larger 𝜃, the far-field frequencies can be important
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Results – Scoured Bed
Mean Velocity Fields - Horizontal
• Effect from scour hole on mean
flow
• Mean horizontal velocity
normalised by 𝑉.
• Horseshoe vortex becomes a
separation bubble
• Lee-side flow is somewhat
confused
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Mean Velocity Field - Vertical
• Effect from scour hole on mean
vertical flow
• Mean vertical velocity normalised
by 𝑉
• Down-flow increases with 𝑆/𝐷
• Lee-side upwelling increases with
𝑆/𝐷
• Important for advection of
suspended sediment out of the
scour hole
17 december 2014
Summary of Findings of Scoured Bed
• The skewness of Δℎ𝑏 /Δ𝑡 and Δℎ𝑠 /Δ𝑡 stays small at 𝑆 𝐷 = 0.5
• For 𝑆 𝐷 = 1 the skewness becomes important relative to mean
change, i.e. average formulation will carry a larger error
• The near-bed shedding frequencies still dominate the near-bed
processes
• The upstream separation on the scour edge could explain the
variation in the location of maximum scour:
• ±45o at the initial scour development
• Almost constant depth from -45o to +45o at later stages
17 december 2014
Conclusion
Conclusion
• Under clear water scour conditions only the lower part of the
shedding affects near-bed processes
• The patterns in erosion/deposition is linked to the vortex shedding
• Up to 𝑆 𝐷 = 0.5 sediment transport can be calculated based on
average quantities (the skewness is small)
• This is not the case for 𝑆 𝐷 = 1.0, but mainly in the wake
• For scour predictions in uni-directional flow, average quantities
could therefore be used
17 december 2014
References
Dixen, M., Sumer, B.M. and Fredsøe, J. (2013). Numerical and experimental investigation of flow
and scour around a half-buried sphere. Coastal Engineering, 73, 84-105.
Engelund, F. and Fredsøe, J. (1976). Sediment Transport Model for Straight Alluvial Channels.
Nordic Hydrology, 7(5), 293-306.
Jacobsen, N.G., Fuhrman, D.R., and Fredsøe, J. (2012). A Wave Generation Toolbox for the OpenSource CFD Library: OpenFoam®. International Journal for Numerical Methods in Fluids, 70(9),
1073-1088.
Jacobsen, N.G., Fredsøe, J. and Jensen, J.H. (2014). Formation and development of a breaker bar
under regular waves. Part 1: Model description and hydrodynamics. Coastal Engineering, 88,
182-193.
Jacobsen, N.G., van Velzen, G. and Fredsøe, J. (2014). Analysis of Pile Scour and Associated
Hydroynamic Forces using Proper Orthogonal Decomposition. Proceedings for International
Conference on Scour and Erosion, 12 pages.
Roulund, R., Sumer, B.M., Fredsøe, J. and Michelsen, J. (2005). Numerical and experimental
investigation of flow and scour around a circular pile. Journal of Fluid Mechanics, 534, 351-401.
17 december 2014