Turbulence Nutshell Slides

8/3/2019 Turbulence Nutshell Slides

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Turbulence Modeling in a Nutshell

Gerald Recktenwald

March 26, 2008

Associate Professor, Mechanical and Materials Engineering Department Portland State University,Portland, Oregon, [email protected]


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Turbulence is a Hard Problem

Unsteady

Many length scales Energy transfer between scales: Large eddies break up into small eddies

Steep gradients near the wall

ME 4/548: Turbulence Modeling in a Nutshell page 1


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Engineering Model: Flow is Steady-in-the-Mean (1)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t

u



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Engineering Model: Flow is Steady-in-the-Mean (2)

Reality:

Turbulent flows are unsteady: fluctuations at a point are caused byconvection of eddies of many sizes. As eddies move through the flowthe velocity field at a fixed point changes.

Model:

When measured with a slow sensor (e.g. Pitot tube) the velocity

at a point is apparently steady. Treat flow variables (velocitycomponents, pressure, temperature) as time averages (or ensembleaverages). These averages are steady (ignorning ensemble averagingof periodic flows).



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Engineering Model: Enhanced Transport (1)

Turbulent eddies enhance mixing: e.g. pollutants spread more rapidly in aturbulent flow than a laminar flow.

Transport in turbulent flow is much greater than in laminar flow As a result of enhanced local transport, mean profiles tend to be more

uniform except near walls.

Near walls, gradients of velocity, temperature (and other scalars (e.g.,chemical concentration) are steep.

Visualize two-layer: high viscosity in core of pipe and low viscositynear the wall Cartoon view only!



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Engineering Model: Enhanced Transport (2)

Turbulence models are a way to account for enhanced mixing whiletreating the flow as steady-in-the-mean

Apparent effect of turbulence is to increase the effective viscosity,thermal conductivity, and diffusivity.

Enhanced transport coefficients are properties of the flow, not realthermophysical transport coefficients.

Most commonly used turbulence models provide a way to compute theeffective transport coefficients, e.g. the turbulence viscosity.



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Reynolds Decomposition

u = U + u v = V + v w = W + w

Turbulence kinetic energy

k =1

2uku

k =1

2

uu + vv + ww

(1)



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k

model

Effective viscosityeff = + t

To relate t to the Reynolds stresses, and assume that

Vt

k

so that

t = C mk1/2

(2)



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k

model

Solve two additional field equations. One for k and one for , theturbulence dissipation rate.

The turbulence is assumed to be isotropic |u| = |v| = |w| then

k =3

2u

u

isotropic turbulence

|u| = |v| = |w| is a consequence of isotropy, not the definition of it.Additional boundary conditions

TI = |u|

U

and turbulence length scale


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Turbulence Nutshell Slides