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Boundary layer Equations

Boundary layer Equations. Contents : Boundary Layer Equations; Boundary Layer Separation; Effect of londitudinal pressure gradient on boundary layer evolution Blasius Solution Integral parameters: Displacement thickness and momentum thickness.

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Boundary layer Equations

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  1. Boundary layer Equations • Contents: • Boundary Layer Equations; • Boundary Layer Separation; • Effect of londitudinal pressure gradient on boundary layer evolution • Blasius Solution • Integral parameters: Displacement thickness and momentum thickness Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  2. Streamlines slightly divergent • 2D Navier-Stokes Equations along x direction: Compared with Laminar Thin Boundary Layer Equations (d<<x) over flat plate • Steady flow, constant r and m. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  3. Laminar Thin Boundary Layer Equations (d<<x) over flat plate • Laminar thin boundary layer equations (d<<x) for flat plates peexternal pressure, can be calculated with Bernoulli’s Equation as there are no viscous effects outside the Boundary Layer Note 1. The plate is considered flat if d is lower then the local curvature radius Note 2. At the separation point, the BD grows a lot and is no longer thin Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  4. 0 0 Turbulent Thin Boundary Layer Equations (d<<x) over flat plate • 2D Thin Turbulent Boundary Layer Equation (d<<x) to flat plates: Resulting from Reynolds Tensions (note the w term) Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  5. Boundary Layer Separation • Boundary Layer Separation: reversal of the flow by the action of an adverse pressure gradient (pressure increases in flow’s direction) + viscous effects mfm: BL / Separation / Flow over edges and blunt bodies Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  6. Boundary Layer Separation • Boundary layer separation: reversal of the flow by the action of an adverse pressure gradient (pressure increases in flow’s direction) + viscous effects Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  7. Bidimensional (2D) Thin Boundary Layer (d<<x) Equations to flat plates: • Close to the wall (y=0) u=v=0 : Boundary Layer Separation • Similar results to turbulent boundary layer - close to the wall there is laminar/linear sub-layer region. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  8. Outside Boundary layer: Same sign • Close to the wall (y=0) u=v=0 : Boundary Layer Separation • The external pressure gradient can be: • dpe/dx=0 <–> U0 constant (Paralell outer streamlines): • dpe/dx>0 <–> U0 decreases (Divergent outer streamlines): • dpe/dx<0 <–> U0increases (Convergent outer streamlines): Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  9. y u Boundary Layer Separation • Zero pressure gradient: dpe/dx=0 <–> U0 constant (Paralell outer streamlines): Curvatureofvelocityprofileisconstant No separation of boundary layer Inflection point at the wall Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  10. y Boundary Layer Separation • Favourable pressure gradient: dpe/dx<0 <–> U0increases (Convergent outer streamlines): No boundary layer separation Curvature of velocity profile remains constant Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  11. y P.I. Separated Boundary Layer Boundary Layer Separation • Adverse pressure gradient: dpe/dx>0 <–> U0 decreases (Divergent outer streamlines): Boundary layer Separation can occur Curvature of velocity profile can change Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  12. Become zero with velocity Can not cause by itself the fluid stagnation (and the separation of Boundary Layer) Boundary Layer Separation • Sum of viscous forces: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  13. (Divergent outer streamlines) (Convergent outer streamlines) Viscous effects retarded Viscous effects reinforced Fuller velocity profiles Less full velocity profiles Decreases BL growth Increases BL growths Boundary Layer Separation • Effect of longitudinal pressure gradient: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  14. Decreases BL growth Increases BL growths Boundary Layer Separation • Effect of longitudinal pressure gradient: Less full velocity profiles Fuller velocity profiles Fuller velocity profiles – more resistant to adverse pressure gradients Turbulent flows (fuller profiles)- more resistant to adverse pressure gradients Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  15. Boundary Layer Sepaation Longitudinal and intense adverse pressure gradient does not cause separation => there’s not viscous forces Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  16. Boundary Condition: y=0 u=v=0 u=U y=∞ Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 • Bidimensional (2D) Thin Boundary Layer (d<<x) Equations to flat plates: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  17. Blasius hypothesis: with • Remark: e Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 The introdution of ηcorresponds to recognize that the nondimension velocity profile is stabilized. A and n are unknowns Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  18. Using current function: • Replace u/U=f(η) e at the boundary layer equation, choose n such that the resulting equation does not depend on x and A in order to simplify the equation. • . • Remark: Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 • Procedure: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  19. From: • results: Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  20. We will obtain: • Making n=1/2 and the equation comes: with Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 • Boundary Conditions: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  21. Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 • Graphical Solution: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  22. Shear stress at the wall • Friction coefficent Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 • Solution: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  23. Drag • Drag Coefficent Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 • Solution: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  24. Boundary layer thickness η=5 • Shear stress at y= Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 • Solution : Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  25. Déficit of flow rate due to velocity reduction at BD Ideal Fluid flow rate Real Flow rate U  Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 • Displacement thickness: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  26. Déficit of flow rate due to velocity reduction at BD Ideal Fluid flow rate Real Flow rate Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 • Displacement thickness : Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  27. Displacement thickness : Initial deviation of BD Deviation of outer streamlines δ δd q/U LC Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 Section where the streamline become part of boundary layer Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  28. ou d q/U δ LC Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 • Blasius Solution for displacement thickness: com Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  29. Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 • Momentum thickness: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  30. Momentum flow rate of uniform profile Reduction due to deficit of flow rate Reduction due to deficit momentum flow rate at BD Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 • Momentum flow rate through a section of BD: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  31. -d d δ LC x Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 • Longitudinal momentum balance between the leading edge and a cross section at x: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  32. or Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 • Blasius Solution to momentum thickness: with Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  33. Laminar Boundary Layer Equations • Contents: • Thin Boundary Layer Equations with Zero Pressure Gradient; • Boundary Layer Separation; • Effect of longitudinal pressure gradient on the evolution of Boundary Layer • Blasius Solution • Local Reynolds Number and Global Reynolds Number • Integral Parameters: displacement thickness and momentum thickness Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  34. Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 • Recommended study elements: • Sabersky – Fluid Flow: 8.3, 8.4 • White – Fluid Mechanics: 7.4 (sem método de Thwaites) Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  35. U=2m/s L=2m Exercise Large plate with neglectable thickness, lenght L=2m. Parallel and non-disturbed air flow. (=1,2 kg/m3, =1,810-5 Pa.s) with U=2 m/s. Zero pressure gradient over the flat plate. Transition to turbulent at Rex=106. =1,2 kg/m3, =1,810-5 Pa.s (Rex)c =106. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  36. U=2m/s L=2m Find xc: Laminar Boundary layer at x1 and x2 – We can apply Blasius Solution Exercise =1,2 kg/m3, =1,810-5 Pa.s (Rex)c =106. a) Find boundary layer thickness  at sections S1 and S2, at distance x1=0,75 m and x2=1,5 m of the leading edge Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  37. Laminar Boundary layer at x1 and x2 – We can apply Blasius Solution Exercise a) Find boundary layer thickness  at sections S1 and S2, at distance x1=0,75 m and x2=1,5 m of the leading edge Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  38. U=2m/s L=2m Exercise =1,2 kg/m3, =1,810-5 Pa.s (Rex)c =106. y=(x) b) Check that it is a thin boundary layer. A: Thin Blayer if /x<<1: Why/x at 2 is lower than /x at 1? Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  39. Exercise =1,2 kg/m3, =1,810-5 Pa.s (Rex)c =106. U=2m/s Streamline  y1=? x1=0,75m x2=1,5m L=2m d) Find the value of y1 at x1 of the streamline passing through the coordinates x2=1,5 and y2=. A: We have the same flow rate between the streamline and the plate at both cross sections Flow rate through a cross section of BD: Flow rate through section 2: Flow rate through section 1: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  40. Exercise =1,2 kg/m3, =1,810-5 Pa.s (Rex)c =106. U=2m/s Linha de corrente  y1=? x1=0,75m x2=1,5m L=2m d) Find the value of y1 at x1 of the streamline passing through the coordinates x2=1,5 and y2=. A: We have the same flow rate between the streamline and the plate at both cross sections Laminar BD: y1=0,0151m 0,0058m 0,0041m 0,0168m Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  41. U=2m/s L=2m Exercise =1,2 kg/m3, =1,810-5 Pa.s (Rex)c =106. e) Find the force per unit leght between sections S1 and S2. A: There are no other forces applied except that imposed by the resistance (Drag) of plate: The applied force between the leading edge and the cross section at x is: Laminar BD: Drag force to section 2: D0,2=0,0107N/m Drag force to section 1: D0,1=0,0076N/m Drag force between 1 and 2: D1,2=D0,2-D0,1=0,0031N/m Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

  42. U=2m/s L=2m Exercise =1,2 kg/m3, =1,810-5 Pa.s (Rex)c =106. f) True or False?: ”Under the conditions of the problem, if the plate was sufficiently long (L), the boundary layer would eventually separate? False: The BD will separate only with adverse pressure gradient. The drag forces will decrease with the velocity over the plate. The drga forces are not able to stop the fluid flow. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

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