Transport processes Part 4 Ron Zevenhoven bo Akademi University - PDF document

Transport processes (TRP) Transport processes – Part 4 Ron Zevenhoven Åbo Akademi University Thermal and Flow Engineering / Värme- och strömningsteknik tel. 3223 ; ron.zevenhoven@abo.fi VST rz18 Transport processes (TRP) VST rz18

3/42 4 Transport processes (TRP) Except when Re = ∞: inviscid flow) Chapters 7-8-9 (not part of this course) VST rz18 4/42 4 Transport processes (TRP) VST rz18

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 6/42 4 5/42 4

7/42 4  (viscous effects neglected: ”inviscid”)   v  v   y  z   y z       v v      rot v v x z     z x    v  v   y  x     x y   in Cartesian coordinate s Transport processes (TRP) Steady state: ∂ ../ ∂ t = 0 v not 0  rot v = 0 for streamline - sign because h↑ and g↓ VST rz18 8/42 Example rot v • Assume a flow field v = (v x ,v y ,v z ) = (k·y,0,0), Transport processes (TRP) with a constant y. v y • For this case    v  v    y z    y z    0 0   0  x         v v         rot v v x z  0 0   0      z x         0 k k      v  v   y  x     x y   • gives a vector with a non-zero component in z-direction. VST rz18

9/42 4 Transport processes (TRP)    ( x , y )        d dx dy   x y    v dx v dy y x VST rz18 10/42 4 Transport processes (TRP)       0 for any scalar      v 0 for any vector v F    (Similarly , for an electric field E and electric force F : E elec voltage for an electric charge q ) elec elec elec q VST rz18

11/42 4 Transport processes (TRP)    i . e . 2 0   ( x , y )      v v ( x , y ) v ( x , y ) v  x   y y y x         continuity : v dx dx etc . x      x y y y y VST rz18 12/42 4 Transport processes (TRP) r θ VST rz18

13/42 4 Transport processes (TRP) note: constant A = A(4.23)·C(4.25) VST rz18 14/42 4 Transport processes (TRP) VST rz18

Transport processes (TRP) Transport processes (TRP) ½ρv 2 +p+ρgh = constant VST rz18 VST rz18 16/42 4 15/42 4

Transport processes (TRP) Transport processes (TRP) paradox D’Alembert VST rz18 VST rz18 18/42 4 17/42 4

Creeping flow z=h Transport processes (TRP) v x , v y , v z z z=0 y x  p • The expression (4.38) gives, with v z << v y and << v x    v μ               v v v   v v v p p    y  y  y  x x x and               x y z μ x x y z μ y which with ∂v x /∂y and ∂v x /∂x << ∂v x /∂z, and similarly ∂v y /∂y and ∂v y /∂x << ∂v y /∂z simplifies to (4.39, 4.40)       v   v p p  y  x and       z μ x z μ y VST rz18 20/42 4 Transport processes (TRP)   p 6 µ p 6 µ     v and v x , mean y , mean  2  2 x x h h 2 ph     potential flow function 6 µ VST rz18

21/42 4 Re << 1 Transport processes (TRP) VST rz18 22/42 A classroom exercise - 4 • An inviscid incompressible fluid flow can be Transport processes (TRP) described by a two-dimensional stream function ψ(x,y) and potential Φ, for –L ≤ x ≤ L, –L ≤ y ≤ L . With velocity v ∞ at x = L, y = L, the velocity potential Φ is given by : Φ = v ∞ · x· y /L. • Give the expression for the velocity vector v(x,y) = (v x ,v y ) and for the stream function ψ(x,y). VST rz18

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 24/42 4 23/42 4

Transport processes (TRP) Transport processes (TRP) @ y ≥ δ(x) : v y = 0 VST rz18 VST rz18 26/42 4 25/42 4

Transport processes (TRP) Transport processes (TRP)  ~  V  x VST rz18 VST rz18 28/42 4 27/42 4

29/42 4 Transport processes (TRP) VST rz18 30/42 4 Transport processes (TRP) y   , V (or V ) free flow velocity (Fig. 4.4)      A ( x ) f ( )  df ( )      v x V f ' ( ) V  d VST rz18

31/42 4  df ( )      v x V f ' ( ) V  d Transport processes (TRP) 3 2 d f d f    f 2   d 3 d 2 see next 4 slides VST rz18 Boundary layers – Blasius /1 • The starting point for Blasius’ analysis are Prandtl’s boundary layer equations, which with dp/dx ≈ 0 (or Transport processes (TRP) at least dp/dx << the other terms) are with boundary conditions v ∞ =v ∞ (x) in the undisturbed flow, v x =v y =0 at x=0, v x =v ∞ at y=∞. • Considering a two-dimensional flow (i.e. symmetry in third dimension) described by stream function ψ(x,y) and introducing a dimensionless variable η(x,y) = y/√( ν · x/v ∞ ) ~ y/δ gives a function f(η): VST rz18

Boundary layers – Blasius /2 • Producing from this the terms for the Prandtl equations gives result Transport processes (TRP) where f´ = ∂f/∂η, f´´ = ∂ 2 f/∂η 2 and noting that ∂f/∂x = ∂f/∂η·∂η/∂x = f´·∂η/∂x and ∂f/∂y = ∂f/∂η·∂η/∂y = f´·∂η/∂y • Using this in Prandtl’s equation gives finally with boundary conditions VST rz18 Boundary layers – Blasius /3 • A numerical solution was produced by Howarth (1938) Transport processes (TRP) VST rz18

Boundary layers – Blasius /4 • Note that the equation (4.63), however, defines η(x,y) as which differs Transport processes (TRP) by a factor 2 from the expression used by Blasius. • This then gives instead of f´´´+ f·f´´ = 0 the expression 2·f´´´+ f·f´´ = 0 with boundary conditions f(0) = f´(0) = 0 and f´(∞) = 1. • The numerical solution for this is given in the table:      v v d      y     Note : x dy dy v dy v ( ) x y   x y dx 0 0 0 VST rz18 36/42 4  v 0 y Transport processes (TRP)     v v     y Note : x dy dy   x y 0 0  d      v dy v ( ) x y dx 0 use Leibniz VST rz18

Transport processes (TRP) Transport processes (TRP) V v y   b 0   b b 3 1     3   b b 4 2     4 2 VST rz18 VST rz18 38/42 4 37/42 4

39/42 4 Transport processes (TRP)     v v V         x x  ( 2 3 ) 2 3 3      y y 2 V 2 µV          (@ 0 )  0  VST rz18 40/42 A classroom exercise - 5 • Blasius’ boundary layer analysis describes the Transport processes (TRP) velocity profile (v x ,v y ) in a laminar boundary layer with a function f(η) where η = ½y√(v ∞ /xν) = ψ(x,y)/√(v ∞ xν), with kinematic viscosity ν, position x along the surface on which the boundary layer builds up, position y from the surface, and undisturbed flow (v ∞ , 0). See course material § 4.3.2 + added material. (continues) VST rz18

41/42 A classroom exercise - 5 • Using the analytical solution by Howarth, given in the Transport processes (TRP) table for η, f´(η)= ∂f/∂η and f´´(η)= ∂²f/∂η², show that the thickness of the boundary layer, defined by v x /v ∞ = 0.99, can be approximated by  5 v x    with Re x x ν Re x VST rz18 Sources used 42/42 4 (besides course book Hanjalić et al. ) • Beek, W.J., Muttzall, K.M.K., van Heuven, J.W. ”Transport phenomena” Wiley, 2nd edition (1999) Transport processes (TRP) • R.B. Bird, W.E. Stewart, E.N. Lightfoot ”Transport phenomena” Wiley, New York (1960) • * C.J. Hoogendoorn ”Fysische Transportverschijnselen II”, TU Delft / D.U.M., the Netherlands 2nd. ed. (1985) • * C.J. Hoogendoorn, T.H. van der Meer ”Fysische Transport- verschijnselen II”, TU Delft /VSSD, the Netherlands 3nd. ed. (1991) • J.R. Welty, C.E. Wicks, R.E. Wilson. “Fundamentals of momentum, heat and mass transfer” Wiley New York (1969) * Earlier versions of Hanjalić et al. book but in Dutch VST rz18