TOWARDS A THEORETICAL MODEL OF LOCALIZED TURBULENT SCOUR

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TOWARDS A THEORETICAL MODEL OF LOCALIZED
TURBULENT SCOUR

• by 1Fabián A. Bombardelli and
• 2Gustavo Gioia
• 1Assistant Professor
• Department of Civil and Environmental
  Engineering, University of California, Davis
• 2Department of Theoretical and Applied Mechanics,
  University of Illinois, Urbana-Champaign

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Outline

• Motivation
• Intermediate asymptotics. Dimensional analysis
• Methodology for the case of jet-induced erosion
• Application of dimensional analysis
• Imposing of the incomplete similarity
• Derivation of an expression for the turbulent
  shear stress on the bed using the
  phenomenological theory of turbulence
• Derivation of the equation and the similarity
  exponent
• Validation of results with available measurements
  

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Motivation I
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Motivation II

• Applications
• Erosion below dams
• Scour below flip buckets
• Scour downstream pipe outlets

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Motivation III

• Notably large number of experimental evidence
  from last century
• Schoklitsch (1932)
• Veronese (1937)
• Eggenberger and Muller (1944)
• Hartung (1959)
• Franke (1960)
• Kotoulas (1967)
• Chee and Padiyar (1969)
• Chee and Kung (1974)
• Machado (1980)
• Mason and Arumugam (1985)
• Yuen (1984)
• Bormann and Julien (1991)
• Stein et al. (1993)
• Chen and Lu (1995)
• DAgostino and Ferro (2004)

• Drawbacks of some of the formulas
• They often lack dimensional homogeneity.
• They often have been the result of mangled
  attempts at dimensional analyses.
• They are often predicated on limited experimental
  data.
• They sometimes disregard the importance of the
  bed particle size.

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Motivation IV

• Questions
• Can we improve existing dimensional analyses?
• Can we obtain a completely theoretical expression
  for the maximum scour depth?
• Can we interpret physically the exponents of the
  equation through the theory of turbulence?

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Outline

• Motivation
• Intermediate asymptotics. Dimensional analysis
• Methodology for the case of jet-induced erosion
• Application of dimensional analysis
• Imposing of the incomplete similarity
• Derivation of an expression for the turbulent
  shear stress on the bed using the
  phenomenological theory of turbulence
• Derivation of the equation and the similarity
  exponent
• Validation of results with available measurements
  

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Intermediate asymptotics IBarenblatt analysis
Dimensional analysis (Buckingham Pi Theorem)
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Intermediate asymptotics IIBarenblatt analysis

• Question What happens with the function when the
  variable is very small or very large?

• Cases
• There is a limit, it is finite and non-zero C

COMPLETE SIMILARITY

• The limit is NOT finite

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Intermediate asymptotics IIIBarenblatt analysis

• Third case

INTERMEDIATE LIMIT
INCOMPLETE SIMILARITY POWER LAWS!!!
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Intermediate asymptotics IIIBarenblatt analysis

• Example velocity distribution in a turbulent
  flow in an open channel

COMPLETE SIMILARITY LAW OF THE WALL!!!
INCOMPLETE SIMILARITY POWER LAW!!!
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Outline

• Motivation
• Intermediate asymptotics. Dimensional analysis
• Methodology for the case of jet-induced erosion
• Application of dimensional analysis
• Imposing of the incomplete similarity
• Derivation of an expression for the turbulent
  shear stress on the bed using the
  phenomenological theory of turbulence
• Derivation of the equation and the similarity
  exponent
• Validation of results with available measurements
  

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Dimensional analysis and similarity
What happens with P when d/R tends to 0? We
assume INCOMPLETE SIMILARITY on d/R !!
Partial result. It depends only on one exponent
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Outline

• Motivation
• Intermediate asymptotics. Dimensional analysis
• Methodology for the case of jet-induced erosion
• Application of dimensional analysis
• Imposing of the incomplete similarity
• Derivation of an expression for the turbulent
  shear stress on the bed using the
  phenomenological theory of turbulence
• Derivation of the equation and the similarity
  exponent
• Validation of results with available measurements
  

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Phenomenological theory of turbulence and bed
shear stress
Based on two tenets a) The production of TKE
occurs at large scales b) The rate of production
of TKE is independent of viscosity
Large scales
Small scales
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Phenomenological theory of turbulence and bed
shear stress
We surmise that the excess of energy of the jet
converts to TKE
The eddy close to the wall belongs to the
inertial sub-range
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Phenomenological theory of turbulence and bed
shear stress
Predicts nicely the scalings of Strickler,
Manning and Blasius (Gioia and Bombardelli, 2002)
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Phenomenological theory of turbulence and scour
equation
Kolmogorov-Taylor scaling
Shields stress
Final result a 1
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Outline

• Motivation
• Intermediate asymptotics. Dimensional analysis
• Methodology for the case of jet-induced erosion
• Application of dimensional analysis
• Imposing of the incomplete similarity
• Derivation of an expression for the turbulent
  shear stress on the bed using the
  phenomenological theory of turbulence
• Derivation of the equation and the similarity
  exponent
• Validation of results with available measurements
  

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Validation with experiments
3D, axisymmetric case Bombardelli and Gioia,
2005, submitted
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Validation with experiments
R computed (m)
R measured (m)
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Conclusions

• Dimensional analysis is a powerful technique but
  it does not provide the values of the exponents.
  The phenomenological theory of turbulence is the
  key to address the dynamics.
• The exponents are driven by the Kolmogorov-Taylor
  scaling, signaling the effect of momentum
  transfer (clear physical meaning).
• The dimensional analysis in terms of the power of
  the jet is crucial in exposing the correct
  factors that govern the scour problem.
• The final expression for scour is purely
  theoretical and agrees with data and existing
  formulas.