Modelling of turbulent flows: RANS and LES Turbulenzmodelle in der - - PowerPoint PPT Presentation

Introduction to turbulence

Modelling of turbulent flows: RANS and LES

Turbulenzmodelle in der Str¨

• mungsmechanik: RANS und LES

Markus Uhlmann

Institut f¨ ur Hydromechanik Karlsruher Institut f¨ ur Technologie www.ifh.kit.edu

SS 2012 Lecture 0

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Introduction to turbulence

LECTURE 0: Recap of the lecture Fluid mechanics of turbulent flows

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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. Result: P(y) = O(y3) for y/h ≪ 1

P u3

τ /δν

DNS by Jimenez et al., Reτ = 950 10

−4

10

−3

10

−2

10

−10

10

−5

10 y2 y3 y4

y/h

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Introduction to turbulence The Reynolds number Friction and dissipation Free shear flows

There is a critical value of the Reynolds number

Example: Reynold’s pipe flow experiment

Re increases ↓ Re = inertial forces viscous forces Re = U D ν Recrit ≈ 2000

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Introduction to turbulence The Reynolds number Friction and dissipation Free shear flows

Consequences of turbulence

Increased momentum transfer: higher wall friction

friction factor (pipe flow) Reynolds number

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Introduction to turbulence The Reynolds number Friction and dissipation Free shear flows

Friction factor for rough pipes

friction factor (pipe flow)

f

(Nikuradse’s data, from Pope “Turbulent flows”) smooth

Reynolds number Reb

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Introduction to turbulence The Reynolds number Friction and dissipation Free shear flows

The kinetic energy equation Ek(x, t) ≡ 1

2u · u

◮ ∂Ek

∂t = − (Ekuj),j − p ρuj

• ,j

+ (2νuiSij),j − 2νSijSij

◮ integration over an arbitrary volume V:

d dt

• EkdV

= −

• (Eku · n) dS
• convection

− p ρ u · n

• dS
• pressure work

+

• (u · τ) · n dS
• viscous work

−

• (2νSijSij) dV
• dissipation

◮ rate of dissipation: ε ≡ 2νSijSij ≥ 0

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Introduction to turbulence The Reynolds number Friction and dissipation Free shear flows

Mean kinetic energy: contrib. from mean flow/turbulence

Mean kinetic energy

decomposition: Ek = 1 2u · u

• ≡ ¯

E (due to mean flow) + 1 2u′ · u′

• ≡ k

(due to turbulence)

Transport equations

∂t ¯ E +

• uj ¯

E + uiu′

iu′ j + ujp/ρ − 2νui ¯

Sij

• ,j

= −P − ¯ ε ∂tk +

• ujk + 1

2u′

iu′ iu′ j + u′ jp′/ρ − 2νu′ iS′ ij

• ,j

= +P − ε

◮ production term: P ≡ −u′ iu′ j ui,j → exchange term ◮ dissipation: mean flow: ¯

ε ≡ 2ν ¯ Sij ¯ Sij; turb.: ε ≡ 2νS′

ijS′ ij

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Introduction to turbulence The Reynolds number Friction and dissipation Free shear flows

Types of free shear flows

Flows developing far from solid boundaries

◮ wake ◮ jet ◮ mixing layer

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Introduction to turbulence The Reynolds number Friction and dissipation Free shear flows

Self-similarity

◮ 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)

round jet at Re = 105)

(Wygnanski & Fiedler 1969) 10 / 15

Introduction to turbulence The Reynolds number Friction and dissipation Free shear flows

Self-similarity in different free shear flows

Evolution of scales with axial distance x

width velocity Re round jet x x−1 cst. plane jet x x−1/2 x1/2 plane mixing layer x cst. x plane wake x1/2 x−1/2 cst. round wake x1/3 x−2/3 x−1/3

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Introduction to turbulence The Reynolds number Friction and dissipation Free shear flows

The plane mixing layer

Re = 105 (Brown & Roshko 1974)

◮ largest scales are remarkably well organized ◮ similar to transitional structures

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Introduction to turbulence The Reynolds number Friction and dissipation Free shear flows

Other unbounded flow types

Homogeneous shear flow Homogeneous-isotropic flow

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Introduction to turbulence The Reynolds number 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)

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Introduction to turbulence The Reynolds number Friction and dissipation Free shear flows

History of early turbulence research

1510 da Vinci:

• bserves eddying motion in water

1854 Hagen, Darcy:

• bserve 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, . . .

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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

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