Introduction to Hydrodynamic Instability 黃美嬌 台灣大學機械系 計算熱流研究室

1. Introduction to Hydrodynamic Instability 黃美嬌 台灣大學機械系 計算熱流研究室 Computational Thermo-fluid Research Lab (CTRL) presented at 東海應數 on Nov. 18, 2004 CTRL

2. OUTLINE Overview of hydrodynamic instability Examples: Rayleigh-Benard Instability Taylor (Dean) Instability CTRL

3. CTRL Hydrodynamic Instability • Which type, laminar or turbulent, is more likely to occur? • laminar when the Reynolds number is very low • turbulent at larger Reynolds number • Reynolds number = UL/n = (L2/n)(L/U)-1 • The equations of hydrodynamics allow some flow patterns. • Given a flow pattern , is it stable? • If the flow is disturbed, will the disturbance gradually die down, or will the disturbance grow such that the flow departs from its initial state and never recovers?

4. Free-shear flows: mixing layers, wakes, jets, etc smaller critical Reynolds number less sensitive to the form of the basic flow inviscid instability leads to coherent structures not affected by viscosity if it is small enough Wall-bounded flows: boundary layers, pipe flows, etc basic flows without inflexion point viscosity plays a role sensitive to the form of the basic flow CTRL

5. Hydrodyanmics ~ n/k = momentum/thermal diffusivities (m2/sec) ~ incompressible viscous Newtonian flows ~ mass,momentum,energy conservation ~ negligible viscous dissipation heat CTRL

6. z x linear stability analysis+normal mode disturbance: Kelvin-Helmholtz (inviscid) instablility: CTRL

7. Kelvin-Helmholtz instablility A long rectangular tube, initially horizontal, is filled with water above colored brine. The fluids are allowed to diffuse for about an hour, and the tube then quickly tilted six degrees, setting the fluids into motion. The brine accelerates uniformly down the slope, while the water above similarly accelerates up the slope. Sinusoidal instability of the interface occurs after a few seconds, and has here grown nonlinearly into regular spiral rolls. CTRL

8. Kelvin-Helmholtz (inviscid) instablility: ~ instability due to heavy fluid on the upside ~ instability due to shear ~ instability due to an rapid downward vertical acceleration and heavy fluid rests below ~ instability for all cases CTRL

9. Wall Shear Flows ~ inviscidly unconditionally stable (Rayleigh analysis) ~ viscously unstable (Orr-Sommerfeld analysis) ~ unstable in labs as Re > 2000 CTRL

10. Vortex Shedding CTRL

11. Hope Bifurcation velocity signals vortex shedding behind a vertical plate CTRL

12. T fluid T+DT Rayleigh-Benard Instability: Low DT: motionless, pure thermal conduction Higher DT: steady convection roll Even Higher DT: unsteady, turbulent • Driving force: buoyancy • Damping force: viscous dissipation CTRL

13. W Centrifugal Instability: Low W: laminar, concentric streamlines Higher W: steady convection roll Even Higher W: unsteady, turbulent • Driving force: centrifugal force • Damping force: viscous dissipation CTRL

14. d Görtler Instability ~ instability due to an imbalance between the centrifugal force and the restoring normal pressure gradient ~ concave walls, e.g. lower side of airfoils; turbine blades CTRL

15. Görtler number Görtler Vortex (streamwise vorticity) R = radius of curvature CTRL

16. Curves of marginal stability based on the parallel and non-parallel stability theory and experimental data. (temporary instability) CTRL Blasius boundary layer ~ uniform flow over an semi-infinite flat plate ~ Tollmien-Schlichting waves ~ temporary/spatial growth

17. Surface tension instability CTRL

18. Examples: • Rayleigh-Benard instability • Taylor instability CTRL

19. TL z H x TH steady stationary solution § Rayleigh-Benard Convection under Boussinesq approximation CTRL

20. Linear stability analysis: CTRL

21. characteristic length = H characteristic time = H2/k characteristic velocity = k/H CTRL

22. ~ characteristic frequency of gravity wave Rayleigh number: Ra = the relative importance of buoyancy effects compared to momentum and thermal diffusive effects. CTRL

23. free and constant-temperature surfaces: • rigid surfaces: Given Pr and Ra, if there exists any mode (kx,ky) such that its (eigenvalue) w has positive imaginary part, then the system is linearly unstable. Normal mode approach: CTRL

24. free and constant-temperature surfaces: CTRL

25. j=3 Ra j=1 j=2 k CTRL

26. H CTRL

27. As Ra increases, more and more unstable modes are inspired and the flow transition to turbulence via successive bifurcations. • rigid surfaces: • ~ no analytical solution yet • ~ numerical solutions available and show CTRL

28. T fluid T+DT Rayleigh-Benard Instability: Low DT: motionless, pure thermal conduction Higher DT: steady convection roll Even Higher DT: unsteady, turbulent • Driving force: buoyancy • Damping force: viscous dissipation CTRL

29. Lorenz: modes with j = 1 X and Y : rising warm fluid and descending cold fluid Z : distortion of the vertical temperature profile from linearity CTRL

30. Lorenz equations: Fixed points: (0,0,0) = pure (stationary) conduction state ~ the only fixed point if Ra<Racr ~ always exists ~ stable if Ra<Racr ~ unstable if Ra>Racr CTRL

31. ~ pitchfork bifurcation at ~ stable if Fixed points: ~ exists only if Ra>Racr ~ steady convection roll CTRL

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37. W Taylor Instability: inviscid fluid CTRL

38. base solution axis-symmetric disturbance ~ Angular momentum per unit mass of a fluid element about the axis (z-axis) remains constant. CTRL

39. z r motion in the r-z plane： r-direction：pressure force + centrifugal force z-direction：pressure force CTRL

40. Centrifugal force = ~ potential-energy-like Consider two fluid particles originally located at r1 and r2 respectively and later interchange their locations at later time. CTRL

41. The change in the kinetic energy is azimuthal kinetic energy is released instability possible in the r-z motion CTRL

42. Linear stability analysis: Normal mode approach + axis-symmetric disturbance: CTRL

43. ~ classical Sturm-Liouville eigenvalue problem Rayleigh quotient: CTRL

44. W1 W2 Couette Flow • Cylinders rotate in the same direction. • Cylinders rotate in different directions. CTRL

45. 0.25 r CTRL

46. r CTRL

47. Taylor number narrow gap approximation: Viscous damping CTRL

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49. Hydrodynamic instability ~ free shear ~ wall effect ~ buoyancy-induced ~ stratification effect ~ centrifugal-force induced ~ surface tension ~ others CTRL

50. Figure 2: The instability generated by increasing flow rate, as seen with the naked eye. Figure 3: The same instability visualized with a strobe lamp. Figure 1: A stable fluid chain. CTRL