**Introduction to the Boundary Layer concept** • Content: • Introduction to the Bounday Layer concept; • Constraint and unconstraint boundary layers • Free shear flows and wakes • Laminar and turbulent boundary layers; • Boundary layer separation; • Thin boundary layer equations • Longitudinal pressure gradient effects on the boundary layer growth. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

**Introduction to the Boundary Layer** Movies (6), 88, 89 MFM: BL, Impulsive Started Flow, Overview MFM: BL, BL Concepts,Viscous effects near boundaries Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

**y** U U x d(x) Boundary layer width: u(d)=0,99 U Boundary layer: significant Introduction to the Boundary Layer • Boundary layer: flow region in the vicinity of a wall where viscous/diffusive effects and energy dissipation are significative. Outer invisicid flow Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

**Limit strealine of Boundary layer** y Streamlines U U x Introduction to the Boundary Layer • Streamlines over a flat plat 1. The streamlines moves away slowly from the wall. Why? 2. This separation of streamlines is most intense outside the boundary layer. Why? Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

**Introduction to the Boundary Layer** • Notes about boundary layer: 1. The boundary layer may be laminar or turbulent 2. Thin boundary layer if d(x)<<x 3. Boundary layer confined: cannot grow free (ex: tube or between plates) Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

**External ** Flow d(x) Fully Developed Velocity Profiel Boundary Layer Entrance Region 2. After the union of all boundary layers, all the flow is boundary layer flow . In turbulent flow, the eddy dimension is limited by d. Introduction to the Boundary Layer • Constrained Boundary Layer: 1. Entrance region: the velocity increases at the center line (to keep the mass flow rate) and the pressure falls (Bernoulli’s Equation)–> dp/dx<0. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

**Introduction to the Boundary Layer** • Boundary layer in external flows (unconstrained) 1. d is not limited, it grows with the distance to the leading edge x (beginning). 2. Nondimensional velocity profile can stabilize Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

**Introduction to the Boundary Layer** • Shear flows (longitudinal convective transport of momentum affected by diffusion): • Free Shear Flows: ex: free jet • Wake: flow zone resulting from the joining of the boundary layers on the two faces of the plate Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

**x – Start of Boundary Layer (BL)** Laminar Flow • Start of BL: Very high • Long plate: Re increases Critical Re (5105) decreases Transition to turbulent Introduction to the Boundary Layer • Transition from laminar to turbulent: Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

**Introduction to the Boundary Layer** • Regions of turbulent boundary layer: • Linear sub-layer or laminar sub-layer; • Transition layer; • Logaritmical profile zone; • Outer zone (turbulent vorticity and non turbulent outer flow). mfm – BL/ Instability, Transition and Turbulence: Boundary Layer transition Fully turbulent BL flow Instability and transition in pipe and duct flow Fully turbulent duct flow Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

** Streamlines slightly divergents** • 2D Navier-Stokes Equations at x direction: Compared with Laminar Thin Boundary Layer Equations (d<<x) over flat plate • Steady flow, r e m constants. 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) to flat plates peexternal pressure, can be calculated bu Bernoulli’s Equation because there is not viscous effects outer 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 no longer thin Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

**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

**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

**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

**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

**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

**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

**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

**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

**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

**(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

**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

**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

**V=0 (stagnation point)** No reversal of the flow From pressure forces Boundary Layer Separation • No viscous forces – no separation of Boundary Layer: (ds displacement over a streamline) Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

**MECÂNICA DOS FLUIDOS II** • Contents: • Boundary Layer; • Boundary Layer thickness; • Limiting line of Boundary Layer; • Deviation of streamlines at Boundary Layer; • Thin Boundary Layer; • Constrained and unconstrained boundary layer; • Free Shear flows; • Wakes; • Thin boundary layer equations. Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

**Separação da Camada limite ** • Contents: • Boundary Layer Separation: conditions to separation • Adverse, favourable and zero pressure gradient; • Effects of pressure gradient on the Boundary layer evolution Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

**MECÂNICA DOS FLUIDOS II** • Bibliography : • Sabersky – Fluid Flow: 8.1, 8.2 • White – Fluid Mechanics: 7.1, 7.3, 7.5 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST