Lecture Notes #10 - Boundary Layer Theory (Laminar).pdf -...

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Unformatted text preview: MECH 561/610 Lecture 10 Boundary Layer Theory Mostafa Najafiyazdi Department of Mechanical Engineering McGill University mostafa.najafi[email protected] Boundary Layer ● Original definition ○ layer of flow next to the wall affected by wall viscous effects 2 Boundary Layer ● General definition ○ Any layer of flow between two zones which is affected by strong gradients of velocity, temperature, mass concentration, etc. 3 Boundary Layer 4 Wall Boundary Layer ● boundary layer coordinate system ○ ○ ○ Y = 0 is the wall Y-axis is normal to the wall X-axis is always tangent to the wall ● High Reynolds number ● As , viscous region near wall becomes thinner 5 Wall Boundary Layer ● ● Assume flow velocity U away from the wall (inviscid flow velocity) Assume boundary layer thickness = δ(x) ● Scales: ○ ○ ○ ○ Tangential velocity u ⇒ Scale U (same as inviscid) Tangential distance x ⇒ Scale L (same as inviscid) Normal distance y ⇒ Scale δ (boundary layer thickness) Normal velocity v ⇒ Scale ?? ⇒ Assume V 6 Incompressible Wall Boundary Layer ● Continuity Equation (incompressible) ● General rule (incompressible flow) ○ never drop a term from the continuity equation Scale for v 7 Incompressible Wall Boundary Layer (cont.) ● Y-momentum equation (steady state): 8 Incompressible Wall Boundary Layer (cont.) ● Y-momentum equation (steady state): ● High Reynolds number ⇒ Convection dominant ○ ● ● Pressure scales as: Compare terms when: It means that: Pressure is constant across wall B.L. 9 Incompressible Wall Boundary Layer (cont.) ● X-momentum equation (steady state): ● Compare terms when: 10 Incompressible Wall Boundary Layer (cont.) ● Three possibilities: 0 0 ⇒ Inviscid Flow 0 0 No-slip BC cannot be satisfied! 11 Incompressible Wall Boundary Layer (cont.) ● Option #1: 0 0 0 0 ⇒ Inviscid Flow ⇒ 1st order fro u ⇒ 1 BC for u is required ⇒ u = U at y→ ∞ ⇒ No-slip BC cannot be satisfied! 12 Incompressible Wall Boundary Layer (cont.) ● Option #2: 13 Incompressible Wall Boundary Layer (cont.) ● Option #3: 0 0 ⇒ 2nd order for u ⇒ 2 BC for u is required ⇒ u = U at y→ ∞ ⇒ No-slip BC can be satisfied: u(y=0) = 0 14 Flow over Flat Plate: Blasius B.L. ● ● ● ● ● ● Flow over thin flat plate Uniform stream of velocity u0 Uniform pressure p = p0 Steady state Incompressible flow Plate is perfectly aligned with flow ○ ● No disturbances in inviscid flow Take coordinate system x-y ○ ○ Y normal to the flat plate X along the flat plate starting from leading edge 15 Flow over Flat Plate: Blasius B.L. (cont.) ● Boundary Conditions ● Farfield ● No-slip ● Leading Edge Inviscid velocity profile 16 Flow over Flat Plate: Blasius B.L. (cont.) 0 First solved by H. Blasius (1908) when he was a student of Prandtl. 17 Blasius Boundary Layer Solution ● Self similar variables: ● Stream function solution: ● Non-dimensional velocities 18 Blasius Boundary Layer Solution (cont.) ● Y-Momentum Equation ● Boundary conditions 19 Alternative Derivation ● Non-dimensionalization: 20 Alternative Derivation (cont.) ● Stretching variables 21 Alternative Derivation (cont.) ● Unchanged groups: ● Solution format: 22 Alternative Derivation (cont.) 23 Alternative Derivation (cont.) 24 Blasius Boundary Layer Properties ● ● B.L. thickness ⇒ point where the u velocity is 0.99u0 u/u0 = 0.99 occurs at ξ≈ 4.9 ● Friction coefficient: ● Drag coefficient for plate of length L: 25 Blasius Boundary Layer Properties (cont.) ● Vertical to tangential velocity ratio ● As V doesn’t go to zero! ● Why? ○ ○ ○ ○ Due to shear force ⇒ dissipates x-momentum Dissipation is from inner parts towards outer parts x-momentum should diffuse in y direction v is non-zero 26 Displacement Thicknesses ● Definition: The distance wall should be displaced such that an inviscid flow would have the same flow rate as the viscous flow has 27 Momentum Thickness ● Definition: The distance wall should be displaced such that an inviscid flow would have the same momentum as the viscous has 28 Boundary Layer Transition & Separation 29 Boundary Layer Transition & Turbulence 30 Boundary Layer Transition & Turbulence (cont.) 31 Thermal Boundary Layer ● What if flow temperature T is different than wall temperature Tw? ○ ○ ○ Incompressible flow Newtonian fluid with constant transport properties Neglecting gravity 0 ○ Steady state flow 32 Thermal Boundary Layer 33 Thermal Boundary Layer (cont.) ● Dimensional analysis 34 Thermal Boundary Layer (cont.) ● ● ● High Reynolds No. Low Eckert No. Moderate Prandtl No. 0 0 0 0 35 Thermal Boundary Layer (cont.) 36 Flow Over a Cylinder with Boundary Layer 37 Flow Over a Cylinder: Reynolds No. effect 38 Flow Over a Cylinder Flow separation 39 Flow Separation 40 Flow Separation 41 Flow Separation 42 More on Boundary Layer Solutions ● Non-flat Plate ● Curved geometries ● Adverse pressure gradient ● Infinite boundary layers ● Unsteady boundary layers Read : - chapter 16, 20, incompressible flow, Panton - Part B, Boundary Layer Theory, H. Schilichting, et al. 43 ...
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