Lecture notes on Boundary layer Theory

boundary layer theory applications, boundary layer theory mass transfer coefficient, boundary layer theory multiple choice questions, boundary-layer linear stability theory
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Published Date:21-07-2017
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- Prof. Dr. Norbert Ebeling Boundary Layer Theory Lecture notesProf. Dr. N. Ebeling Boundary Layer Theory - 1 - Contents : 1) General fluid mechanics / Newton fluids 1.1) Euler's law of hydrostatics 1.2) Friction 1.3) Dimensionless numbers 1.4) Laminar flow in a tube 2) Conservation equations 2.1) Mass balance for ρ = const. 2.2) Euler's and Bernoulli's equations 2.3) Navier-Stokes equations 3) Boundary layers 3.1) Boundary layers on a flat plate 3.2) Friction forces on a plate 3.3) Boundary layer on an obstacle 4) Potential and stream functions 5) Law of Kutta-Joukowski 6) Exact calculation of the Boundary layer thickness 6.1) Conservation of mass (continuity equation) 6.2) Navier-Stokes and Blasius equations 6.3) Friction 7) Thermal Boundary layer 8) Mass Transfer Boundary layer equation 9) Turbulent Boundary layer 10) Burbling 11) Bibliography 12) AcknowledgmentProf. Dr. N. Ebeling Boundary Layer Theory - 2 - 1) General fluid mechanics / Newton Fluids General definitions  i e x u  x   j e y v general definitions  y   k e z w z  dy (dz) dm = ρ i dx i dy i dz dx Du Acceleration : dF = dmi i Dt Du ∂u ∂u ∂u ∂u = + u i + v i + w i Dt ∂t ∂x ∂y ∂z ∂u stationary : = 0 ∂t Du ∂u frequently : = u i Dt ∂x volume force: f (e.g. g ) iProf. Dr. N. Ebeling Boundary Layer Theory - 3 - 1.1) Euler's law of hydrostatics ↓dF 1 x↓ f x ↑dF 2 ρ i f i dV + dF = dF x 1 2 ∂F ρ i f i dx i dy i dz = i dx x ∂x dF = dp i dy i dz ∂p ρ i f = x ∂x Prof. Dr. N. Ebeling Boundary Layer Theory - 4 - 1.2) Friction ∂u Moving fluid : ( Couette - flow ) τ = +η i ∂y Newton fluid ∂u Schlichting : τ = μ i ∂y 1.3) Dimensionless numbers : Reynolds number inertial force Re friction force ∂τ τ + i dy ∂y  ∂τ F = i dy i dx i dz ( ) FR  τ  ∂y dAProf. Dr. N. Ebeling Boundary Layer Theory - 5 - dm   ∂u ρ i dx i dy i dz i u i ∂x Re ∂τ i dx i dy i dz ∂y   ∂u u ∂τ ∂ ∂u ∞ ; = η i   ∂x d ∂y ∂y ∂y   ∂τ ∂ u   or any comparable ∞ η i   speed v else ∂y ∂y d   V ρ i v i ρ i v i d d Re = = v η η i 2 d η v i d with ν = Re = ρ ν laminar flow : high friction forces, low inertial forces avoided by friction υ decidingProf. Dr. N. Ebeling Boundary Layer Theory - 6 - ascending force F A C = A p i s s 1 F 2 Bernoulli : R p → i ρ i u C = ∞ w 2 p i s C ( or ζ ) analogous w d dp Pipe : ζ = i or λ ( ) R ρ dx 2 i u m 2 v Fr = Gravity influence : g i d Froude - number Prof. Dr. N. Ebeling Boundary Layer Theory - 7 - 1.4) Laminar flow in a tube ν extremely high nearly no initial forces, no influence of dm or ρ 64 Hagen-Poisseulle : ζ = R Re d dp 64 i η 64 i ν i = = ρ dx v i ρ i d v i d 2 i v 2 Derivation : dp   2 2 p i π i r - p + dx i πr - τ i 2πr i dx = 0   dx   r dp du − i = τ = - η i 2 dx dr Integration with u (r = R) = 0 leads to : 2 2   R dp r   u(r) = i i - 1     4η dx R      Prof. Dr. N. Ebeling Boundary Layer Theory - 8 - R V = u (r) i 2π i dr ∫ 0 4 π i R dp   V = i -   8 i η dx   2 V R Δp u = = i 2 π i R 8 i η l u i ρ i 2R ( ) Re = η Δp d ξ = i 2 1 i ρ i u l 2 64 ξ = laminar Re 2) Conservation equations Important conservation equations for describing continuous flow ( cartesian coordinates ) : 2.1) Mass balance for ρ = const. ∂u ∂v + = 0 ∂x ∂y Prof. Dr. N. Ebeling Boundary Layer Theory - 9 - u i Δy i Δz = u i Δy i Δz + v i Δx iΔz 1 2 2 u - u i Δy = + v i Δx ( ) 1 2 2 Δu Δv + = 0 Δx Δy 2.2) Euler's and Bernoulli's equations Eulers equation ( one direction, pipe ): Du ∂u dV i ρ i = ρ i dV i u i = + dF x Dt ∂x ∂u ∂p ρ i u i = - ∂x ∂x Integration : W = F • l leads to Bernoulli's equation Prof. Dr. N. Ebeling Boundary Layer Theory - 10 - Mechanical energy balance : Bernoulli incl. hydrostatics ∂u -∂p ρ i u i = + ρ i f x ∂x ∂x 2 2 2 2 u ρ i = p - ρ i g i h 2 1 1 1 Euler (2 directions ): Du Dt     ∂u ∂u ∂p ρ i u i + v i = -   ∂x ∂y ∂x   ↑v v leads to a higher value of u 2.3) Navier - Stokes - equation Bernoulli and Euler neglect friction ∂u τ =η i ∂y ∂τ dF = idy i dx i dz ( ) R ∂yProf. Dr. N. Ebeling Boundary Layer Theory - 11 - 2 2   ∂u ∂ u f = η i +   R 2 2 ∂y ∂x   Navier - Stokes - Equations ( Can be simplified in a boundary layer (later)) Du ∂p ρ i = ρ i f - + f x Rx Dt ∂x 2 2    ∂u ∂u ∂p ∂ u ∂ u ρ u i + v i = ρ i f - + η i +     x 2 2 ∂x ∂y ∂x ∂y ∂x     2 2    ∂v ∂v  ∂p ∂ v ∂ v ρ v i + u i = ρ i f - + η i +     y 2 2 ∂y ∂x ∂y ∂y ∂x     3) Introduction to Boundary layers 3.1) Boundary layers on a flat plate No influence of the viscosity but directly on the wall Boundary layer phenomena : ( Schlichting )Prof. Dr. N. Ebeling Boundary Layer Theory - 12 - Thickness of a boundary layer, laminar on a plate u ∞ → δ = f x ( ) u u ∂u ∂τ ∞ ∞ ; η i 2 ∂x x ∂y δ 2 2 ∂u ∂ u ∂τ ρ i u i = η i = 2 ∂x ∂y ∂y inertial force = friction force ( Navier -Stokes ) 2 u u ∞ ∞ ρ i η i 2 x δ ν i x δ u ∞ ν i x δ = 5 i 99 (x) u ∞Prof. Dr. N. Ebeling Boundary Layer Theory - 13 - Dimensionless : δ 5 x 99 (x) = i l l Re δ is arbitrary 99 A non - arbitrary value : displacement thickness ∞ U i δ (x) = U - u(x,y) i dy ( ) i ∫ y = 0 1 δ i δ i ≈ 99 3 3.2) Friction forces on a plate : low value high valueProf. Dr. N. Ebeling Boundary Layer Theory - 14 -   ∂u τ (x) = η i w   ∂y   w η i x u ρ ∞ τ η i with δ w δ u ∞ 3 η i ρ i u ∞ τ w x F F W W ζ = c = = w ρ E 2 i u b i l ∞ 2 S (Surface) l F = b i τ x i dx ( ) W W ∫ 0 l 1 − 3 2 F b i μ i ρ i u i x i dx W ∞ ∫ 0 1 3 2 F b i μ i ρ i u i 2 i l W ∞ 3 b i 2 i η i ρ i u i l ∞ c w 2 ρ 4 2 b i u i i l ∞ 4Prof. Dr. N. Ebeling Boundary Layer Theory - 15 - 1,1328 c = w Re l c w Re 3.3) Boundary layer on an obstacle : Navier - Stokes : Far away from the obstacle (stream line) : dU l dp U i = - i no friction ( ) dx ρ dx dU dp and are related to Bernoulli dx dxProf. Dr. N. Ebeling Boundary Layer Theory - 16 - 4) Potential and Stream functions For describing vortex streams ( and comparable ) : ∂γ ∂v 1 ω = = 1 ∂t ∂x ∂γ ∂u 2 ω = = - 2 ∂t ∂y 1 ∂v ∂u ω = -   2 ∂x ∂y   Circulation : Γ = w i ds ∫ Potential streams (no friction ) : no rotation ∂v ∂u ω = 0 ; - = 0 ∂x ∂y Mass balance ; conservation equation : ∂u ∂v + = 0 ∂x ∂yProf. Dr. N. Ebeling Boundary Layer Theory - 17 - Stream function (definition ) : ∂Ψ ∂Ψ u = ; v = - ∂y ∂x Conservation equation :   ∂ ∂Ψ ∂ ∂Ψ   + - = 0     ∂x ∂y ∂y ∂x     No rotation : 2 2 ∂ Ψ ∂ Ψ + = 0 2 2 ∂x ∂y Potential function : ∂φ ∂φ u = ; v = ∂x ∂y Potential streams 1 ∂v ∂u ∂v ∂u ω = i - ; - = 0   2 ∂x ∂y ∂x ∂y   Prof. Dr. N. Ebeling Boundary Layer Theory - 18 - Streams without any rotation : ∂v ∂u - = 0 ∂x ∂y also conservation equation : ∂u ∂v + = 0 ∂x ∂y 2 ∂ ∂v ∂ u ∂ ∂u ∂ ∂v      - - + = 0       2 ∂y ∂x ∂y ∂x ∂x ∂x ∂y       2 2 ∂ u ∂ u + = 0 2 2 ∂x ∂y Insert in Navier - Stokes :  ∂u ∂u ∂p ρ i u i + v i = - + 0   ∂x ∂y ∂x   p = f u, v ( ) leads to Bernoulli for v = 0 - no friction - no rotation - no friction

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