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 Displacement thickness and momentum thickness 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Laminar Thin Boundary Layer Equations (d<<x) over flat plate Steady flow, constant r and m. Streamlines slightly divergent 2D Navier-Stokes Equations along x direction: Compared with 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Laminar Thin Boundary Layer Equations (d<<x) over flat plate Laminar thin boundary layer equations (d<<x) for flat plates pe external 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 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Turbulent Thin Boundary Layer Equations (d<<x) over flat plate 2D Thin Turbulent Boundary Layer Equation (d<<x) to flat plates: 0 Resulting from Reynolds Tensions (note the w term) 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

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 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

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 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Boundary Layer Separation Bidimensional (2D) Thin Boundary Layer (d<<x) Equations to flat plates: Close to the wall (y=0) u=v=0 : Similar results to turbulent boundary layer - close to the wall there is laminar/linear sub-layer region. 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Boundary Layer Separation Outside Boundary layer: Close to the wall (y=0) u=v=0 : Same sign 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 <–> U0 increases (Convergent outer streamlines): 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Boundary Layer Separation Zero pressure gradient: dpe/dx=0 <–> U0 constant (Paralell outer streamlines): y u Curvature of velocity profile is constant No separation of boundary layer Inflection point at the wall 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Boundary Layer Separation Favourable pressure gradient: dpe/dx<0 <–> U0 increases (Convergent outer streamlines): y No boundary layer separation Curvature of velocity profile remains constant 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Boundary Layer Separation Adverse pressure gradient: dpe/dx>0 <–> U0 decreases (Divergent outer streamlines): y Boundary layer Separation can occur Separated Boundary Layer Curvature of velocity profile can change P.I. 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

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

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

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

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

Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST 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: Blasius Solution to Laminar Boundary Layer Equation over a flat plate Boundary Condition: y=0 u=v=0 y=∞ u=U 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

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

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

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

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

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

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

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

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

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

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

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

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

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

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0 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 Reduction due to deficit momentum flow rate at BD 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST 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: Longitudinal momentum balance between the leading edge and a cross section at δ d -d LC x 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

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

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 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 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST 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) 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST 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. L=2m U=2m/s Large plate with neglectable thickness, lenght L=2m. Parallel and non-disturbed air flow. Zero pressure gradient over the flat plate. Transition to turbulent occurs for 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Exercise =1,2 kg/m3, =1,810-5 Pa.s (Rex)c =106. L=2m U=2m/s 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 Find xc: Boundary layer thickness at S1 Laminar Boundary layer at x1 and x2 – We can apply Blasius Solution 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST 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 Laminar Boundary layer at x1 and x2 – We can apply Blasius Solution 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Exercise =1,2 kg/m3, U=2m/s =1,810-5 Pa.s (Rex)c =106. y=(x) L=2m 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? 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Exercise =1,2 kg/m3, U=2m/s =1,810-5 Pa.s (Rex)c =106.  y1=? 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: We have the same flow rate between the streamline na Flow rate through a cross section of CL: Flow rate through section 2: Flow rate through section 1: 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST 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 0,0168m 0,0058m 0,0041m 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Exercise =1,2 kg/m3, =1,810-5 Pa.s (Rex)c =106. L=2m U=2m/s 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: Find the force per unit leght between sections The applied force between the leading edge and the cross section at 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 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Exercise =1,2 kg/m3, =1,810-5 Pa.s (Rex)c =106. L=2m U=2m/s f) True or False?: ”Under the conditions of the problem, if the plate was sufficiently long (L ), the boundary layer would eventually separate? 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. 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. 2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST