1. TOWARDS A THEORETICAL MODEL OF LOCALIZED TURBULENT SCOUR TOWARDS A THEORETICAL MODEL OF LOCALIZED TURBULENT SCOUR by 1 Fabián A. Bombardelli and 2 Gustavo Gioia 1 Assistant Professor Department of Civil and Environmental Engineering, University of California, Davis 2 Department of Theoretical and Applied Mechanics, University of Illinois, Urbana- Champaign

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

3. Motivation I

4. Motivation II Applications Applications Erosion below dams Erosion below dams Scour below flip buckets Scour below flip buckets Scour downstream pipe outlets Scour downstream pipe outlets

5. Motivation III Notably large number of experimental evidence from last century: Notably large number of experimental 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.

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

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

8. Intermediate asymptotics I Barenblatt analysis Dimensional analysis (Buckingham Pi Theorem)

9. Intermediate asymptotics II Barenblatt analysis n Question: What happens with the function when the variable is very small or very large? Cases: n There is a limit, it is finite and non-zero: C n The limit is NOT finite COMPLETE SIMILARITY

10. Intermediate asymptotics III Barenblatt analysis n Third case: INCOMPLETE SIMILARITY – POWER LAWS!!! INTERMEDIATE LIMIT

11. Intermediate asymptotics III Barenblatt analysis n Example: velocity distribution in a turbulent flow in an open channel COMPLETE SIMILARITY – LAW OF THE WALL!!! INCOMPLETE SIMILARITY – POWER LAW!!!

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

13. Dimensional analysis and similarity 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 !!

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

15. 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

16. 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

17. Phenomenological theory of turbulence and bed shear stress Predicts nicely the scalings of Strickler, Manning and Blasius (Gioia and Bombardelli, 2002)

18. Phenomenological theory of turbulence and scour equation Kolmogorov-Taylor scaling Final result: α = 1 Shields stress

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

20. Validation with experiments 3D, axisymmetric case: Bombardelli and Gioia, 2005, submitted

21. Validation with experiments R measured (m) R computed (m)

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