Introduction to turbulence Modelling of turbulent flows: RANS and LES Turbulenzmodelle in der Str¨ omungsmechanik: RANS und LES Markus Uhlmann Institut f¨ ur Hydromechanik Karlsruher Institut f¨ ur Technologie www.ifh.kit.edu SS 2012 Lecture 0 1 / 15
Introduction to turbulence LECTURE 0: Recap of the lecture Fluid mechanics of turbulent flows 2 / 15
Introduction to turbulence Solution to wall-bounded flow problem: Determine the variation with wall-distance of the production P in fully-developed plane channel flow, valid for very small values of y . DNS by Jimenez et al., Re τ = 950 0 10 y 2 Result: −5 10 P ( y ) = O ( y 3 ) P y 3 u 3 τ /δ ν for y/h ≪ 1 y 4 −10 10 −4 −3 −2 10 10 10 y/h 3 / 15
The Reynolds number Introduction to turbulence Friction and dissipation Free shear flows There is a critical value of the Reynolds number Example: Reynold’s pipe flow experiment Re = inertial forces viscous forces Re Re = U D increases ν ↓ Re crit ≈ 2000 4 / 15
The Reynolds number Introduction to turbulence Friction and dissipation Free shear flows Consequences of turbulence Increased momentum transfer: higher wall friction friction factor (pipe flow) Reynolds number 5 / 15
The Reynolds number Introduction to turbulence Friction and dissipation Free shear flows Friction factor for rough pipes (Nikuradse’s data, from Pope “Turbulent flows”) friction factor f (pipe flow) smooth Reynolds number Re b 6 / 15
The Reynolds number Introduction to turbulence Friction and dissipation Free shear flows E k ( x , t ) ≡ 1 The kinetic energy equation 2 u · u � p � ◮ ∂E k = − ( E k u j ) ,j − ρu j + (2 νu i S ij ) ,j − 2 νS ij S ij ∂t ,j ◮ integration over an arbitrary volume V : � � � � p � d E k d V = − ( E k u · n ) d S − ρ u · n d S d t � �� � � �� � convection pressure work � � + ( u · τ ) · n d S − (2 νS ij S ij ) d V � �� � � �� � viscous work dissipation ◮ rate of dissipation: ε ≡ 2 νS ij S ij ≥ 0 7 / 15
The Reynolds number Introduction to turbulence Friction and dissipation Free shear flows Mean kinetic energy: contrib. from mean flow/turbulence Mean kinetic energy 1 1 2 � u ′ · u ′ � decomposition: � E k � = 2 � u � · � u � + � �� � � �� � ≡ ¯ ≡ k E (due to turbulence) (due to mean flow) Transport equations � � ∂ t ¯ � u j � ¯ j � + � u j �� p � /ρ − 2 ν � u i � ¯ E + E + � u i �� u ′ i u ′ S ij = −P − ¯ ε ,j � � � u j � k + 1 2 � u ′ i u ′ i u ′ j � + � u ′ j p ′ � /ρ − 2 ν � u ′ i S ′ ∂ t k + ij � = + P − ε ,j ◮ production term: P ≡ −� u ′ i u ′ j � � u i � ,j → exchange term ε ≡ 2 ν ¯ S ij ¯ ◮ dissipation: mean flow: ¯ S ij ; turb.: ε ≡ 2 ν � S ′ ij S ′ ij � 8 / 15
The Reynolds number Introduction to turbulence Friction and dissipation Free shear flows Types of free shear flows Flows developing far from solid boundaries ◮ wake ◮ jet ◮ mixing layer 9 / 15
The Reynolds number Introduction to turbulence Friction and dissipation Free shear flows Self-similarity round jet at Re = 10 5 ) ◮ collapse of profiles under correct scaling ◮ scaling (normalization) can be observed from experiments or precise numerical simulations ◮ self-similarity can also be inferred theoretically (analysis of boundary-layer equations) (Wygnanski & Fiedler 1969) 10 / 15
The Reynolds number Introduction to turbulence Friction and dissipation Free shear flows Self-similarity in different free shear flows Evolution of scales with axial distance x width velocity Re x − 1 round jet x cst. x − 1 / 2 x 1 / 2 plane jet x plane mixing layer x cst. x x 1 / 2 x − 1 / 2 plane wake cst. x 1 / 3 x − 2 / 3 x − 1 / 3 round wake 11 / 15
The Reynolds number Introduction to turbulence Friction and dissipation Free shear flows The plane mixing layer Re = 10 5 (Brown & Roshko 1974) ◮ largest scales are remarkably well organized ◮ similar to transitional structures 12 / 15
The Reynolds number Introduction to turbulence Friction and dissipation Free shear flows Other unbounded flow types Homogeneous shear flow Homogeneous-isotropic flow 13 / 15
The Reynolds number Introduction to turbulence Friction and dissipation Free shear flows Da Vinci on Turbulence “Observe the motion of the surface of the water, which resembles that of hair, which has two motions, of which one is caused by the weight of the hair, the other by the direction of the curls; thus the water has eddying motions, one part of which is due to the principal current, the other to random and reverse motion.” (cited by Lumley, Phys. Fluids A, 1992) 14 / 15
The Reynolds number Introduction to turbulence Friction and dissipation Free shear flows History of early turbulence research 1510 da Vinci: observes eddying motion in water 1854 Hagen, Darcy: observe two different laws for pressure drop in pipes 1851-70 St. Venant, Boussinesq: introduce concept of eddy viscosity 1883-94 Reynolds: transition criterion in pipes, flow decomposition, stresses 1922 Richardson: formulation of cascade process 1941 Kolmogorov: quantitative theory for the cascade at high Re number and: Prandtl, Taylor, von Karman, . . . 15 / 15
Conclusion Outlook Next lecture: Introduction to turbulence modelling The problem of computing turbulent flows Different approaches to the problem: DNS, LES, RANS Criteria for appraising models 1 / 1
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