v = 0 Continuity Continuity Navier- -Stokes Stokes Navier - PowerPoint PPT Presentation

� Navier � Navier- -Stokes Equation Stokes Equation � Steady Parallel Flows � Steady Parallel Flows � Boundary Conditions � Boundary Conditions � Unsteady Parallel Flow � Unsteady Parallel Flow � Boundary Layer Flows � Boundary Layer Flows � Flow over a Flat Plate � Flow over a Flat Plate � Blasius � Blasius Solution Solution � Laminar Boundary Layer Laminar Boundary Layer � ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi ∇ v ⋅ = 0 Continuity Continuity Navier- -Stokes Stokes Navier ∂ v 1 + ⋅ ∇ = − ∇ + ν ∇ 2 v v P v ∂ ρ t Navier Navier Stokes Stokes ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 1

Incompressible Fluid Incompressible Fluid Cartesian Coordinates Cartesian Coordinates ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2 2 2 u u u u p u u u ρ + + + = ρ − + µ + + ( u v w ) g ( ) ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ x 2 2 2 t x y z x x y z y ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2 2 2 v v v v p v v v ρ + + + = ρ − + µ + + h ( u v w ) g ( ) ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ y 2 2 2 t x y z y x y z ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ u = 2 2 2 w w w w p w w w 0 ρ + + + = ρ − + µ + + ( u v w ) g ( ) ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ z 2 2 2 t x y z z x y z = = u ( y ), v 0 , w 0 ∂ ∂ ∂ u v w 4 Equations for 4 4 Equations for 4 + + = 0 ∂ ∂ ∂ x y z unknowns u,v,w and p unknowns u,v,w and p ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi Cartesian Coordinates Solid Walls Cartesian Coordinates Solid Walls ∂ 2 A viscous fluid sticks to its boundary, u u fluid =u u wall A viscous fluid sticks to its boundary, fluid = p d u wall ρ − + µ = g 0 ∂ x 2 x dy Fluid General Solution General Solution Free Surface Free Surface ∂ 1 p = − ρ − + + Shear Stress=0 Shear Stress=0 2 u ( g ) y Ay B u wall µ ∂ x 2 x ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 2

Cartesian Cartesian Between Two Fluids Between Two Fluids ∂ 2 p d u ∂ ρ − + µ = 1 p = − ρ − + + g 0 Velocity and shear stress are Velocity and shear stress are 2 u ( g ) y Ay B ∂ x 2 µ ∂ x x dy 2 x the same at the interface. the same at the interface. Fluid 2 Cylindrical Axial Cylindrical Axial Fluid 1 ∂ ∂ p 1 d dv 1 p ρ − + µ = = − ρ − + + u 1 = u 2 z 2 u 1 = u g ( r ) 0 v ( g ) r A ln r B ∂ µ ∂ 2 z z x z r dr dr 4 z u wall τ 1 τ = τ τ 2 Rotating Rotating 1 = 2 2 v 2 dp d v 1 dv v B = ρ θ + − = = + θ θ θ 0 v Ar θ 2 2 r dr r dr r dr r ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi y = u = ∂ ∂ 0 2 B.C. B.C. U Similarity Solution Similarity Solution u u = ν 0 ∂ ∂ 2 t y = ∞ u = 1 Let Let a t ~ t y ~ t y 0 y 1 = a = I.C. I.C. Navier- -Stokes Stokes 2 a 1 / 2 Navier t = u = 0 0 Viscous y u ( ) Similarity Similarity η = = f η Fluid x Variables ν Variables U 2 t 0 U o ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 3

1.0 0.8 = ′ ′ ′ + η = f ( 0 ) 1 f 2 f 0 NS NS t ν =4 B.C. B.C. 0.6 u/Uo ∞ = f ( ) 0 t ν =1 ′ = − η 2 f ce 0.4 t ν =0.25 0.2 t ν =0.062 2 η ( ) ∫ = − − η 2 η = − η f 1 e d 1 erf 1 t ν =0.0025 π 1 0 0.0 0 0.5 1 1.5 2 2.5 3 ⎛ ⎞ y y = ⎜ ⎟ u U erfc ⎜ ⎟ Solution Solution 0 ν ⎝ ⎠ 2 t Variation of velocity profile with time. Variation of velocity profile with time. ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi U U o o δ l δ l << / 1 Laminar Boundary Layer Laminar Boundary Layer ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 4

u Steady Two- Steady Two -D Flows D Flows Boundary Layer Boundary Layer = Distance 0 . 99 δ Thickness, δ at which U Thickness, ∂ ∂ ∂ ∂ ∂ 0 2 2 u u 1 p u u + = − + ν + u v ( ) ∞ ∂ ∂ ρ ∂ ∂ ∂ ⎛ − ⎞ 2 2 Displacement Displacement u x y x x y ∫ δ = ⎜ ⎟ * 1 dy ⎜ ⎟ Thickness Thickness ⎝ ⎠ U ∂ ∂ ∂ ∂ ∂ 2 2 0 v v 1 p v v 0 + = − + ν + u v ( ) ∂ ∂ ρ ∂ ∂ ∂ 2 2 ∞ Momentum Momentum ⎛ − ⎞ x y y x y u u ∫ ⎜ ⎟ θ = 1 dy ⎜ ⎟ Thickness Thickness U ⎝ U ⎠ 0 0 0 ∂ ∂ u v + = 0 Shape Shape δ ∂ ∂ * = x y H Factor Factor θ ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi ∂ ∂ ∂ 2 u u 1 dp u + = − + ν u v ∂ ∂ ρ ∂ 2 x y dx y ∂ ∂ u v + = 0 ∂ ∂ x y Ludwig Ludwig Prandtl Prandtl = = = at y 0 u 0 , v 0 Boundary Boundary = ∞ = Conditions at y u U Conditions o ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 5

U U U U U P = o P o o o o ∂ ∂ ∂ 2 atm u u u Blasius + = ν u v y ∂ ∂ ∂ 2 x y y δ U x y η y = o ν x ∂ ∂ δ u v ( x ) + = 0 ∂ ∂ x y x = = = 1 at y 0 u 0 , v 0 Boundary u Boundary = η f ' ( ) = ∞ = u / U at y u U Conditions Conditions o o U o ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi f ’=u/U o Numerical Numerical ∂ Boundary Layer u Boundary Layer U u U = η η η y = = η f ' ( ) Solution Solution o o δ f " ( ) Thickness, δ Thickness, ν ∂ ν U x y x o f 5 ν x δ = Blasius Blasius Equation Equation U 5 Experimental o f ” + = data Boundary Boundary ff " 2 f " ' 0 δ 1 − = 2 Layer Eq Layer Eq. . 5 Re x x 1 η = = = at 0 f 0 , f ' 0 Boundary Boundary f”(0)=0.332 du U τ = µ = µ o U f " ( 0 ) η = ∞ = ν Conditions Conditions at f ' 1 o dy x = y 0 ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 6

Friction Coefficient Friction Coefficient ∞ ⎛ − ⎞ ν U x ∫ Displacement Displacement δ = ⎜ ⎟ = * 1 dy 1 . 721 ⎜ ⎟ τ Thickness Thickness 2 f " ( 0 ) 0 . 664 ⎝ ⎠ U U = = = C 0 0 0 F 1 R R ρ 2 U ex ex o 2 ∞ ⎛ − ⎞ ν θ ∫ U U x Momentum Momentum = ⎜ ⎟ = 1 dy 0 . 664 ⎜ ⎟ Thickness Thickness Drag Coefficient U ⎝ U ⎠ U Drag Coefficient 0 0 0 0 D 4 f " ( 0 ) 1 . 328 δ = = = * Shape Shape C = = H 2 . 51 D 1 Factor Factor θ R R ρ 2 l U l l e e o 2 ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi � Navier � Navier- -Stokes Equation Stokes Equation � Steady Parallel Flows � Steady Parallel Flows � Boundary Conditions � Boundary Conditions � Unsteady Parallel Flow � Unsteady Parallel Flow � Boundary Layer Flows � Boundary Layer Flows � Flow over a Flat Plate � Flow over a Flat Plate � Blasius � Blasius Solution Solution � Laminar Boundary Layer � Laminar Boundary Layer ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 7