TOWARDS A THEORETICAL MODEL OF LOCALIZED TURBULENT SCOUR TOWARDS A THEORETICAL MODEL OF LOCALIZED...

TOWARDS A THEORETICAL TOWARDS A THEORETICAL MODEL OF LOCALIZED MODEL OF LOCALIZED

TURBULENT SCOURTURBULENT SCOUR

by by 11Fabián A. Bombardelli and Fabián A. Bombardelli and 22Gustavo GioiaGustavo Gioia

11Assistant ProfessorAssistant ProfessorDepartment of Civil and Environmental Department of Civil and Environmental

Engineering, University of California, DavisEngineering, University of California, Davis

22Department of Theoretical and Applied Department of Theoretical and Applied Mechanics, University of Illinois, Urbana-Mechanics, University of Illinois, Urbana-

ChampaignChampaign

• MotivationMotivation

• Intermediate asymptotics. Dimensional Intermediate asymptotics. Dimensional analysisanalysis

• Methodology for the case of jet-induced Methodology for the case of jet-induced erosion:erosion:• Application of dimensional analysisApplication of dimensional analysis• Imposing of the incomplete similarityImposing of the incomplete similarity• Derivation of an expression for the turbulent Derivation of an expression for the turbulent

shear stress on the bed using the shear stress on the bed using the phenomenological theory of turbulencephenomenological theory of turbulence

• Derivation of the equation and the similarity Derivation of the equation and the similarity exponentexponent

• Validation of results with available Validation of results with available measurements measurements

OutlineOutline

Motivation IMotivation I

Motivation IIMotivation II• ApplicationsApplications

• Erosion below damsErosion below dams• Scour below flip bucketsScour below flip buckets• Scour downstream pipe outletsScour downstream pipe outlets

Motivation IIIMotivation III• Notably large number of experimental Notably large number of experimental

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

Drawbacks of some of the formulas:

1) They often lack dimensional homogeneity.

2) They often have been the result of mangled attempts at dimensional analyses.

3) They are often predicated on limited experimental data.

4) They sometimes disregard the importance of the bed particle size.

Motivation IVMotivation IV

er

s

edegeheq dghqKDR

• Questions:Questions:• Can we improve existing dimensional analyses?Can we improve existing dimensional analyses?• Can we obtain a completely theoretical Can we obtain a completely theoretical

expression for the maximum scour depth?expression for the maximum scour depth?• Can we interpret physically the exponents of Can we interpret physically the exponents of

the equation through the theory of turbulence? the equation through the theory of turbulence?







OutlineOutline

Intermediate asymptotics IIntermediate asymptotics IBarenblatt analysisBarenblatt analysis

mlkrqp xxx

X

xxx

y

321

4

321

4321 ,,, Xxxxfy

Dimensional analysis (Buckingham Pi Theorem)

1

Intermediate asymptotics IIIntermediate asymptotics IIBarenblatt analysisBarenblatt analysis

11 0?,lim orwhen

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

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

The limit is NOT finiteThe limit is NOT finite

rqp

orxxxCyC 321

01

lim

COMPLETE SIMILARITY

Intermediate asymptotics IIIIntermediate asymptotics IIIBarenblatt analysisBarenblatt analysis

11?, 11 orwhen

Third case:Third case:

43211 ... XxxxKyK mrlqkp

INCOMPLETE SIMILARITY – POWER LAWS!!!

INTERMEDIATE LIMIT

Intermediate asymptotics IIIIntermediate asymptotics IIIBarenblatt analysisBarenblatt analysis

*

*** Re,,Re,

uu

uy

y

u

y

u

Example: velocity distribution in a Example: velocity distribution in a turbulent flow in an open channelturbulent flow in an open channel

COMPLETE SIMILARITY – LAW OF THE WALL!!!(Re)

** (Re)

uy

Ay

u

y

u

INCOMPLETE SIMILARITY – POWER LAW!!!







OutlineOutline

Dimensional analysis and Dimensional analysis and similaritysimilarity

sdghqKR )25/(2)25/(1)25/(2)25/(2

s

R

dRgP ,2/52/3

.;0 constR

d s

qhgPdgRPVariables s ;,,,,,:

Partial result. It depends only on one exponent

What happens with P when d/R tends to 0?

We assume INCOMPLETE SIMILARITY on d/R !!







OutlineOutline

Phenomenological theory of Phenomenological theory of turbulence and bed shear turbulence and bed shear

stressstressBased 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


stressstress

l

ul3

~R

V 3

~3/1

R

lVul

3/1

R

dVud

We surmise that the excess of energy of the jet converts to TKE 2~ RMqhgP

3/1

~

R

hgqV

The eddy close to the wall belongs to the inertial sub-range


stressstress

3/1

R

dVud

nt vv

duV~Predicts nicely the scalings of Strickler, Manning and Blasius (Gioia and Bombardelli, 2002)

Phenomenological theory of Phenomenological theory of turbulence and scour turbulence and scour

equationequation

R

Pd

3/23/1~

dgR

Pd s

3/23/1~

Kolmogorov-Taylor scaling

s

dghqR 3/23/13/23/2~ Final result: α = 1

Shields stress






• Validation of results with available Validation of results with available measurementsmeasurements

OutlineOutline

Validation with Validation with experimentsexperiments

3D, axisymmetric case: Bombardelli and Gioia, 2005, submitted

Validation with Validation with experimentsexperiments

0.01

0.1

1

10

0.01 0.1 1 10

Z measured (m)

Z c

om

pu

ted

(m

)

Our formula

Eggenberger and Mullerformula

Perfect agreement line

D'Agostino data (1994)

R measured (m)

R computed (m)

ConclusionsConclusions• Dimensional analysis is a powerful technique Dimensional analysis is a powerful technique

but it does not provide the values of the but it does not provide the values of the exponents. The phenomenological theory of exponents. The phenomenological theory of turbulence is the key to address the turbulence is the key to address the dynamics.dynamics.

• The exponents are driven by the Kolmogorov-The exponents are driven by the Kolmogorov-Taylor scaling, signaling the effect of Taylor scaling, signaling the effect of momentum transfer (clear physical meaning).momentum transfer (clear physical meaning).

• The dimensional analysis in terms of the The dimensional analysis in terms of the power of the jet is crucial in exposing the power of the jet is crucial in exposing the correct factors that govern the scour problem.correct factors that govern the scour problem.

• The final expression for scour is purely The final expression for scour is purely theoretical and agrees with data and existing theoretical and agrees with data and existing formulas.formulas.

TOWARDS A THEORETICAL MODEL OF LOCALIZED TURBULENT SCOUR TOWARDS A THEORETICAL MODEL OF LOCALIZED...

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