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9. Turbulent flows9.1 IntroductionThe characteristics of turbulent flows

(1) Turbulent flows are highly unsteady, three-dimensional.

(2) They contain a great deal of vorticity.

(3) Turbulence increases the rate at which conserved quantities are stirred. Itbrings fluids of differing momentum content into contact.

(4) Turbulent flows contain coherent structures.

(5) They fluctuate on a broad range of length and time scales.

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A classification scheme for the approaches to predicting turbulent flows.

(1) The use of correlations that give the friction factor as a function of the

Reynolds number or the Nusselt number of heat transfer as a function of the

Reynolds and Prandtl numbers.

(2) Integral equations which can be derived from the equations of motion. This

reduced the problem to one or more ordinary differential equations.

(3) Equations obtained by avberaging the equations of motion over time, over a

coordinate. One-point closure

.

(4) Two-point closure.

(5) Large eddy simulation (LES). This solves for the largest scale motions of theflow while approximating or modeling only the small scale motions.

(6) Direct numerical simulation (DNS)

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9.2 Direct numerical simulationTurbulence model is not used.

Direct numerical simulation (DNS) can be applied to

the flows which are

(1) unsteady

(2) three-dimensional

(3) composed of various sizes of eddies.

The number of grid points: Re9/4

Direct numerical simulation is possible only for flows at relatively low Reynolds

numbers and in geometrically simple domains.

DNS is too expensive to be employed very often and cannot be used as a design

tool.

Some examples of kinds of uses to which DNS has been put are

(1) Understanding the mechanisms of turbulence production, energy transfer, and

dissipation in turbulent flows

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(2) Simulation of the production of aerodynamic noise

(3) Understanding the effects of compressibility on turbulence

(4) Understanding the interaction between combustion and turbulence

(5) Controlling and reducing drag on a solid surface

The most important requirements placed on numerical methods for DNS arise

from the need to produce an accurate realization of a flow that contains a wide

range of length and time scales.

Second to fourth order Runge-Kutta, Adams-Bashforth,

The methods using spectrum (the Fourier decomposition of the velocity field) in

space.

Examples

Contours of the kinetic energy on a plane in the flow created by an oscillating grid

in a quiescent fluid; the grid is located at the top of the figure. Energetic packets

of fluid transfer energy away from the grid region. (Briggs et al., 1996)

The profile of the flux of turbulent kinetic energy compared with the predictions of

some commonly used turbulence models. (Briggs et al., 1996)

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9.3 Large eddy simulation (LES)

Schematic representation of turbulent motion and the time dependence of a

velocity component at a point (Ferziger and Peric, 2002)

The larger scale motions are generally much more energetic than the small scale

ones.

A simulation which treats the large eddies more exactly than the small ones may

make sense.

Large eddy simulation (LES)

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Turbulence similarity

The filtered velocity

''', dxxuxxGxu (9.1)

G: the filter kernel (a Gaussian, a box filter, a cutoff, etc.)

The Navier-Stokes equations are filtered.

Subgrid-scale Reynolds stress.-> modeled by Smagorinsky model etc.

9.4 Reynolds-averaged Navier-Stokes (RANS) equationsEvery variable can be written as the sum of an averaged value and a fluctuation

' (9.2)

Ensemble averaging ()

N

n

nN N 1

1lim (9.3)

Reynolds averaging

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The Navier-Stokes equations are averaged.

'''' uuuuu

(9.4)

Reynolds stresses

Turbulent scalar flux

The presence of the Reynolds stresses and turbulent scalar flux in the

conservation equations means that they contain more variables than there are

equations.

Closure requires use of some approximations in terms of the

mean quantities.-> turbulence model

The eddy-viscosity model

The effect of turbulence can be represented as an increased viscosity.

A minimum description of turbulence requires at least a velocity scale and alength scale.

An equation for the turbulent kinetic energy determines the velocity scale.

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The dissipation, turbulent kinetic energy, and the length scale are related

l

k2/3

(9.5)

One-equation model1

An exact equation for the dissipation can be derived from the Navier-Stokes

equations: Two-equation model 2