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GY2311/GY2312 Lectures 6-7 Fluid Flows Uniform flows Boundary layers

GY2311/GY2312 Lectures 6-7 Fluid Flows Uniform flows Boundary layers. DEPARTMENT OF GEOGRAPHY. Boundary layers. Boundary layers in air. Schematic of velocity variation in river channels (isovels – lines of equal velocity).

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GY2311/GY2312 Lectures 6-7 Fluid Flows Uniform flows Boundary layers

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  1. GY2311/GY2312 Lectures 6-7 Fluid Flows Uniform flows Boundary layers DEPARTMENT OF GEOGRAPHY

  2. Boundary layers Boundary layers in air Schematic of velocity variation in river channels (isovels – lines of equal velocity).

  3. Effect of different bed roughness on vertical velocity profiles in rivers • Studying boundary layers provides information on • Local flow velocities (as opposed to mean flow velocity) • Local shear stresses (as opposed to mean boundary shear stress) • Estimates of bed roughness

  4. u Laminar flow

  5. Turbulent flow

  6. Comparison of laminar and turbulent velocity profiles

  7. Laminar, newtonian fluid Water Laminar, Non-newtonian fluid Ice Turbulent, newtonian fluid Water, Air Velocity distributions in water, wind and ice.

  8. A simple thought experiment

  9. The results of the experiment

  10. Calculus Differential calculus is concerned with rates of change of continually varying quantities.

  11. Y=mX+c, but c = 0 so Y = mX or The results of the experiment • This tells us that • The velocity gradient induced by a shear stress is directly proportional to the shear stress, but inversely proportional to viscosity (since du/dy=t.1/m); • As m increases, a larger shear stress must be applied to induce the same velocity gradient; • Any shear stress > 0 causes instantaneous deformation.

  12. t = rg(d-y)S The velocity profile for laminar flow t = rgRS

  13. The velocity profile of laminar Newtonian flow (water) uy=velocity at height y r=density of flow g=accel gravity S=bed slope d=flow depth m=molecular viscosity

  14. The velocity profile of a laminar non-newtonian fluid (ice) ub=basal ice velocity a=constant (from Glen’s Law)

  15. The velocity profile of a laminar non-newtonian fluid (ice)

  16. The velocity profile of a laminar non-newtonian fluid (ice)

  17. tv tv tv Shear stresses in laminar flows In laminar flow, the shear stress is generated by the viscous forces between neighbouring ‘sliding’ layers. Laminar flow, therefore, has a viscous shear stress (tv=mdu/dy)

  18. Shear stresses in turbulent flows In turbulent flows, the shear stress is generated by turbulent mixing Turbulent flow, therefore, has an additional turbulent shear stress (tt=hdu/dy) where h is the eddy viscosity tt

  19. Turbulent shear stresses due to movement of fluid across planes parallel to the direction of movement Viscous shear stresses across planes parallel to the direction of movement Shear stresses in turbulent flows In turbulent flows, therefore, the total shear stress (tT) is In turbulent flows, tt>>tv

  20. The law of the wall Approaches to understanding turbulent velocity profiles 1) Empirical/experimental (measurement of velocity profiles in water/air) 2) Semi-theoretical equations adopting simplifying assumptions

  21. The structure of the turbulent velocity profile

  22. Hydraulically smooth and rough flow

  23. Semi-theoretical approachesthe law of the wall uy=velocity at height y u*=shear velocity (t/r)0.5 k=Von Karman’s constant (0.4) y=height above bed yo=height above bed at which u = 0

  24. Turbulent velocity profile

  25. Velocity profiles in rivers

  26. Velocity profile Can derive estimate of shear stress and bed roughess from gradient and intercept of regression line Log e height Velocity

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