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Turbulence and CFD: Beyond k modeling

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Although two-equation closures such as the k model are arguably the most popular

turbulence model for industrial applications of CFD, they have significant limitations

and/or deficiencies. These are summarized below, together with some alternativeapproaches:

1. The use of wall functions is convenient but fundamentally not sound for complexflows where the underlying assumptions are violated, e.g. for impinging flows. In

the 1990s extensive effort was given to the development of so called lowReynolds number (LRN) models, which implement carefully designed damping

functions to ensure the correct near-wall behaviour of important parameters

such as the Reynolds shear stress, which behaves as 3)( + yuv in the near-wall

region.

2. The kequation is sometimes used together with other turbulence variables. Forexample, in the k closure, instead of the dissipation rate, one uses the

quantity k/= , which has units of ]/1[ T , i.e. represents an inverse turbulence

time scale. In form, this equation set is very similar to the k model, and uses

an eddy viscosity model (EVM) relation to reconstruct the Reynolds stress, where

the eddy viscosity is now given by /kCt= . The k closure has the

remarkable and useful property of being applicable all the way to the wall.

3. Most two-equation closures still suffer from the inherent limitation of using anEVM relation for the Reynolds stress terms, e.g.

kx

U

x

Uuu ij

i

j

j

itji

3

2 +

+

=

Recall that this constitutive relation was modeled in a similar way to that ofmolecular transport, i.e. it is a gradient transport model. For complex flows,

where there is more than one important strain-rate, this model fails. It is also

deficient for flows with strong curvature and buoyancy.

One alternative to EVM closures is an approach which solves the Reynolds Stress

Transport Equations (RSTE). These equations are formidable, since the Reynoldsstress is a second-order tensor quantity. In order to closethese equations, many of

the terms must be represented by simpler model relations. Nonetheless, the

framework of the transport equations does allow the complex coupling betweenthe Reynolds stress terms and the mean velocity field to be much better

represented. These models are especially applicable to flows with significant

anisotropy, including near-wall flows.

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A symbolic representation of the RSTE is as follows:

ijijijijjij

jji Puux

Uuut

++=

+

D

where

ion termn/dissipatdestructiotheis

tion termredistributheis

termproductiontheis

termdiffusiontheis

ij

ij

ij

ij

P

D

Low Reynolds number or near-wall formulations have also been developed for

the RSTE. As well, algebraic models can be derived from these equations byneglecting the net transport.

4. The challenge of modeling the Reynolds stress becomes problematic for manycomplex flows. (Recall that the Reynolds stress represents the contribution of the

turbulent motions to the mean transport.) Furthermore, in some cases onerequires specific information about the large-scale turbulent motions, e.g. in

applications with unsteady or separated flows. This has led to the development of

Large Eddy Simulation (LES) methods which calculate the large-scale motionsand model the smaller-scale motions. These represent very challenging

simulations since the unsteady large-scale motions must be calculated in three-

dimensions in a time-accurate manner. On the other hand, the expectation is that

the residual stress (or subgrid-scale) terms can be adequately represented bysimpler models, since the small-scale motions are more isotropic. In LES the

large-scale variable is referred to as the filtered velocity field, iU . LES typically

uses different calculationmethods than do RANS models, although many CFDcodes try to include this option.

5. Currently, the most accurate way to predict a turbulent flow is to use a DirectNumerical Simulation (DNS) in which all scales of motion are resolved by an

unsteady, three-dimensional calculation. Examples of this approach include thepipe flow simulations of Wu and Moin (2008) (J Fluid Mech, Vol 608, pp. 81-

112) and the backward-facing step study used as a benchmark for the project.

Even with massive computing resources, these types of simulations can only

consider flows with simple geometry and low Reynolds number. From anengineering viewpoint, they are not practical since they discard the vast majority

of the information being calculated. However, they presently have been used asrefined experiments for exploring flow physics and calibrating turbulence models.