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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Boundary layer Equations u 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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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Laminar Thin Boundary Layer Equations ( <

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Laminar Thin Boundary Layer Equations ( <

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Turbulent Thin Boundary Layer Equations ( <

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Boundary Layer Separation u 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

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Boundary Layer Separation u Boundary layer separation: reversal of the flow by the action of an adverse pressure gradient (pressure increases in flow’s direction) + viscous effects

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Boundary Layer Separation Bidimensional (2D) Thin Boundary Layer ( <

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Boundary Layer Separation u Outside Boundary layer: u The external pressure gradient can be: o dp e /dx=0 U 0 constant ( Paralell outer streamlines): o dp e /dx>0 U 0 decreases ( Divergent outer streamlines): o dp e /dx U 0 increases ( Convergent outer streamlines ): u Close to the wall (y=0) u=v=0 : Same sign

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Zero pressure gradient: dp e /dx=0 U 0 constant (Paralell outer streamlines): y u Inflection point at the wall No separation of boundary layer Boundary Layer Separation Curvature of velocity profile is constant

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Favourable pressure gradient: dp e /dx U 0 increases ( Convergent outer streamlines ): y Curvature of velocity profile remains constant No boundary layer separation Boundary Layer Separation

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Adverse pressure gradient: dp e /dx>0 U 0 decreases ( Divergent outer streamlines): Curvature of velocity profile can change Boundary layer Separation can occur y P.I. Boundary Layer Separation Separated Boundary Layer

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Sum of viscous forces: Become zero with velocity Can not cause by itself the fluid stagnation (and the separation of Boundary Layer) Boundary Layer Separation

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Effect of longitudinal pressure gradient: (Convergent outer streamlines) (Divergent outer streamlines) Viscous effects retardedViscous effects reinforced Fuller velocity profiles Less full velocity profiles Decreases BL growthIncreases BL growths Boundary Layer Separation

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Effect of longitudinal pressure gradient: Fuller velocity profiles Less full velocity profiles Decreases BL growthIncreases BL growths Fuller velocity profiles – more resistant to adverse pressure gradients Turbulent flows (fuller profiles)- more resistant to adverse pressure gradients Boundary Layer Separation

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Boundary Layer Sepaation Longitudinal and intense adverse pressure gradient does not cause separation => there’s not viscous forces

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0 Bidimensional (2D) Thin Boundary Layer ( <

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Blasius hypothesis: with The introdution of η corresponds to recognize that the nondimension velocity profile is stabilized. A and n are unknowns Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0 u Remark: e

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0 u Procedure: oUsing current function: o Remark: oReplace 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..

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0 o u results: o o o o u From:

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0 u We will obtain: u Boundary Conditions: oMaking n=1/2 and the equation comes: with

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Graphical Solution: Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Solution: Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0 oShear stress at the wall o Friction coefficent

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Solution: Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0 o Drag o Drag Coefficent

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Solution : Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0 o Boundary layer thickness η=5 o Shear stress at y=

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Displacement thickness: Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0 U Ideal Fluid flow rate Real Flow rate Déficit of flow rate due to velocity reduction at BD

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Displacement thickness : Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0 Ideal Fluid flow rate Real Flow rate Déficit of flow rate due to velocity reduction at BD

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0 u Displacement thickness : Initial deviation of BD δ Deviation of outer streamlines Section where the streamline become part of boundary layer δdδd q/U LC

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0 u Blasius Solution for displacement thickness: δ dd q/U LC com ou

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0 u Momentum thickness:

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0 u Momentum flow rate through a section of BD: Momentum flow rate of uniform profile Reduction due to deficit of flow rate Reduction due to deficit momentum flow rate at BD

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0 u Longitudinal momentum balance between the leading edge and a cross section at x: δ dd -d-d LC x

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0 u Blasius Solution to momentum thickness: with or

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Laminar Boundary Layer Equations u 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

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST u Recommended study elements: –Sabersky – Fluid Flow: 8.3, 8.4 –White – Fluid Mechanics: 7.4 (sem método de Thwaites) Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dp e /dx=0

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST =1,2 kg/m3, =1,8 Pa.s (Re x ) c =10 6. Exercise L=2m U=2m/s Large plate with neglectable thickness, lenght L=2m. Parallel and non-disturbed air flow. ( =1,2 kg/m3, =1,8 Pa.s) with U=2 m/s. Zero pressure gradient over the flat plate. Transition to turbulent at Re x =10 6.

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST a) Find boundary layer thickness at sections S 1 and S 2, at distance x 1 =0,75 m and x 2 =1,5 m of the leading edge Exercise Find x c : Laminar Boundary layer at x 1 and x 2 – We can apply Blasius Solution =1,2 kg/m3, =1,8 Pa.s (Re x ) c =10 6. L=2m U=2m/s

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Exercise Laminar Boundary layer at x 1 and x 2 – We can apply Blasius Solution a) Find boundary layer thickness at sections S 1 and S 2, at distance x 1 =0,75 m and x 2 =1,5 m of the leading edge

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Exercise b) Check that it is a thin boundary layer. =1,2 kg/m3, =1,8 Pa.s (Re x ) c =10 6. L=2m U=2m/s A: Thin Blayer if /x<<1: Why /x at 2 is lower than /x at 1? y= (x)

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Exercise d) Find the value of y 1 at x 1 of the streamline passing through the coordinates x 2 =1,5 and y 2 = . =1,2 kg/m3, =1,8 Pa.s (Re x ) c =10 6. L=2m U=2m/s Streamline x 2 =1,5m x 1 =0,75m y 1 =? 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:

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Exercise d) Find the value of y 1 at x 1 of the streamline passing through the coordinates x 2 =1,5 and y 2 = . =1,2 kg/m3, =1,8 Pa.s (Re x ) c =10 6. L=2m U=2m/s Linha de corrente x 2 =1,5m x 1 =0,75m y 1 =? A: We have the same flow rate between the streamline and the plate at both cross sections Laminar BD: 0,0168m 0,0058m0,0041m y 1 =0,0151m

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Exercise e) Find the force per unit leght between sections S 1 and S 2. L=2m U=2m/s =1,2 kg/m3, =1,8 Pa.s (Re x ) c =10 6. 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: D 0,2 =0,0107N/m Drag force to section 1: D 0,1 =0,0076N/m Drag force between 1 and 2: D 1,2 =D 0,2 -D 0,1 = 0,0031N/m

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2004Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Exercise f) True or False?: ”Under the conditions of the problem, if the plate was sufficiently long (L ), the boundary layer would eventually separate? L=2m U=2m/s =1,2 kg/m3, =1,8 Pa.s (Re x ) c =10 6. 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.

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