Outline Viscous Flow Turbulence Mixing Length Models One-Equation - PowerPoint PPT Presentation

ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi Outline  Viscous Flow  Turbulence  Mixing Length Models  One-Equation Models  Two-Equation Models  Stress Transport Models  Rate-Dependent Models ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 1

    Viscous                     p p G G ( ( u u ) ) Mass ( ( u u ) ) 0 0   kl kl kl kl kl kl i i , , j j t t Fluids 1 1 d d u u     d d ( ( u u u u ) ) Momentum  dt  dt           f f τ τ τ  τ  T T       τ τ u u d d kl kl k k , , l l l l , , k k 2 2 i i , , j j ij ij ij ij 1 1       ( ( u u u u ) ) Energy               e e   τ τ : : u u q q h h kl kl k k , , l l l l , , k k 2 2 Material Frame-             p p F F ( ( d d ) ) q q h h Entropy Indifference                 ( ( ) ) 0 0 kl kl kl kl kl kl ij ij T T T T ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi                           u  u  ( ( p p u u ) ) 2 2 d d 0 0 u i  u i    u u U U u u U U 0 0 i i i i kl kl i i , , i i kl kl kl kl i i i i i i       p p P P p p           p  P p  p  p  3 3 2 2 0 0 P 0 0 Navier- Stokes Reynolds Equation         2 2 u u u u p p u u             ( ( i i u u i i ) ) i i               2 2 u u u u   j j         U U U U 1 1 P P U U . . t t x x x x x x x x i i j j             i i U U i i i i j j i i j j j j j j             t t x x x x x x x x x x j j i i j j j j j j   u u i  i  0 0     Turbulent Stress         T T u u u u   x x ij ij i i j j i i ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 2

u  u  Eddy dU dU Eddy Velocity Velocity Scale Scale                   T  T  T T u u v v c c   u u 21 21 T T Viscosity dy dy Viscosity   Length Length Scale Scale     T T     U U U U 1 1 Kinematic c  c  Speed Speed of of Sound Sound         ij ij           j j     u u u u ( ( i i ) ) u u u u     c   c i i j j T T k k k k ij ij       x x x x 3 3 Viscosity j j i i     Mean Mean Free Free Path Path Mixing         U U U U T T Free Shear Flows               T T l 2 l 2 | | | | T T v v T T ~  ~  c c   21 21 m m         Length y y y y y y m m 0 0 T T   Near Wall Flows U U     y y       l 2 l 2 | | | | m m T T m m   y y ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi Local Equilibrium Mixing length Production = Dissipation Hypothesis Short Comings of Mixing Length  Eddy viscosity vanishes when velocity Mixing   T  T  0 0 gradient is zero Reattachment Point Length  Lack of transport of turbulence scales Maximum Heat Experiment  Estimating the mixing length Transfer ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 3

Modeled k-equation Eddy Viscosity     1 1 / / 2 2 c  c  k k   T T       Exact k-equation dk dk k k   ( ( T T ) )                     dt dt x x x x d d u u u u u u u u P P U U               i i i i u u ( ( i i i i ) ) u u u u i i j j k k j j           k k i i j j       dt dt 2 2 x x 2 2 x x           k k               j j           Diffusion Diffusion Convective Convective Turbulence Turbulence Diffusion Diffusion Pr Pr oduction oduction     U U   Transport Transport 3 3 / / 2 2 U U U U k k       j j   i i i i ( ( ) ) c c               2 2 u u u u u u u u         T T D D i i i i i i i i       x x x x x x               x x x x x x x x 2 2 j j i i j j                   j j j j j j j j                     Dissipatio Dissipatio n n Pr Pr oduction oduction Dissipatio Dissipatio n n Viscous Viscous Diffusion Diffusion ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi K-equation           dz dz z z U U     2 2 ( ( T T ) ) z z [ [ c c T T ( ( ) )       3 3 / / 2 2 1 1 dk dk U U k k         dt dt y y y y k k y y       ( ( Bk Bk Max Max ) ) ak ak c c             z z     D D       dt dt y y y y         Pr Pr oduction oduction       Diffusion Diffusion               Dissipatio Dissipatio n n Pr Pr oduction oduction k k Diffusion Diffusion     c c ] ] S S 2 2   z z   Short Comings of One-Equation Models       T T Secondary Secondary  Lack of transport of turbulence length scale Source Source Dissipatio Dissipatio n n  Estimating the length scale ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 4

z  z   k k          2 2           / / k k Time Time Scale Scale d d u u u u u u               ( ( u u ' ' ) ) 2 2 i i i i k k   j j       dt dt x x x x x x x x 2  2  j j k k l l l l                         k k / / Frequency Frequency Scale Scale   Generation Generation by by Diffusion Diffusion vortex vortex stretching stretching 2  2  k k / / Vorticity Vorticity Scale Scale           2 2 2 2 u u u u     2 2 i i i i                 u u u u x x x x x x x x         i i i i Dissipatio Dissipatio n n Rate Rate       k k     l l     k k     l l     x x x x Viscous Viscous distructio distructio n n j j j j ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi   U U i  i  E(k) Universal Equilibrium Mass 0 0   x x i i ε ε     dU dU 1 1 P P Momentum           i i u u u u i i j j       dt dt x x x x i i j j Inertia 2 2 c c k k Subrang Kolmogorov     U U U U 2 2           j j             u u u u ( ( i i ) ) k k e T T i i j j T T     ij ij   k x x x x 3 3 j j i i ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 5