# Turbulent flow models

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Turbulent flow models Katarzyna Miłkowska - Piszczek

Faculty of Metal Engineering and Industrial Computer Science

Department of Ferrous Metallurgy

Kraków 8.12.2010

Content

1. Preface2. CFD 3. Turbulence models 4. DNS5. LES6. K/ε

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Turbulent flow

In fluid dynamics , turbulence or turbulent flow is a fluid regimecharacterized by chaotic,

stochastic property changes.

This includes low momentum diffusion ,high momentum convection and rapid variation of pressure and velocity in space and time.

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Turbulent flow

• The fundamental mathematical model are the non-isothermalNavier-Stokes equations, governing the time-evolution of velocity, pressure and temperature.

• The phenomenon of turbulence reveals that their solutions can become very complex if a critical parameter e.g., the Reynolds number or the Rayleigh number, becomes large

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Preface

A proper numerical resolution of the random motion of all scales of

~u, ~p, and ~ T (called Direct Numerical Simulation) is feasible

only for a very limited number of flows.

Thus the major task in turbulence modeling is to reduce the complexity of the Navier-Stokes equations in a manner which is appropriate to the needs of science and engineering. The goal is to develop models that are computationally simpler

than the Navier-Stokes equations but "whose predictions are close

to those of the Navier-Stokes equations".

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Preface

• The first approach is a statistical approach which is based on a statistical averaging procedure for the Navier-Stokes equations. The objective is to obtain a set of equations for the statistical mean values for ~u, ~p, and ~ T, which requires an empirical modelling of the terms involving statistical fluctuations.

• The second approach is called large-eddy simulation (LES). The idea of LES is to apply a spatial averaging filter to the Navier-Stokes equations in order to extract the large-scale structures of ~u, ~p, and ~ T, and to attenuate their small-scale structures.Then only the random motion of the large scales is resolved and the effects of the small scales on the large scales are modelled.

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Preface

Then the turbulent state of motion is simply the phenomenological

aspect of this complexity. The complexity of the solution has two

aspects, viz.,

its randomness

its vast and continuous range of scales the turbulence problem is

how to describe and how to reduce this complexity in a manner

which is appropriate to the needs of science and engineering.

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Preface

compute the random motion of all scales, which is referred to as direct numerical simulation (abbreviated DNS)

compute the random motion of the large scale motion (and model the small scale motion), which is referred to as large-eddysimulation (abbreviated LES)

predict mean flow field, pressure and temperature (in a statistical sense), referred toas statistical turbulence modelling or Reynolds averaged CFD (called RANS)

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Classes of turbulence models

RANS-based models

Linear eddy-viscosity models o Algebraic models o One and two equation models

Non-linear eddy viscosity models and algebraic stress models

Reynolds stress transport models

Large eddy simulations Detached eddy simulations and other hybrid models Direct numerical simulations

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CFD

Computational fluid dynamics (CFD) is a branch of fluid mechanicsthat uses numerical method and algorithms to solve and analyze problems that involve fluid flows.

Computers are used to perform the calculations required to simulate

the interaction of liquids and gases with surfaces defined by boundary conditions.

With high-speed supercomputers, better solutions can be achieved. Ongoing research, however, yield software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows.

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A direct numerical simulation (DNS) is a simulation in computational fluid dynamics in which the Navier-Stokes equations are numerically solved without any turbulence model. This means that the whole range of spatial and temporal scales of the turbulence must be resolved. All the spatial scales of the turbulence must be resolved in the computational mesh, from the smallest dissipative scales ( Kolomogorov scales) , up to the integral scale L, associated with the motions containing most of the kinetic energy.

The Kolmogorov scale is given by: where ν is the kinematic viscosity , ε is the rate of kinetic energy dissipation.

The Kolmogorov scale

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Dissipation of turbulent kinetic energy

By a production mechanism P the large eddies are generated. These are unstable and break up into successively smaller and smaller eddies, i.e. their energy is transferred to smaller and smaller scales by inviscid processes.

At the smallest scales the energy is dissipated into heat by molecular viscosity. This process is called dissipation of turbulent kinetic energy or simply dissipation.

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The Kolmogorov thesis

Kolomogorov postulated that for very high Re the small scale

turbulent motions are statistically isotropic (i.e. no preferential

spatial direction could be discerned).

In general, the large scales of a flow are not isotropic, since they are

determined by the particular geometrical features of the boundaries

(the size characterizing the large scales will be denoted as L).

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The Kolmogorov thesis

A turbulent flow is characterized by a hierarchy of scales through which the energy cascade takes place. Dissipation of kinetic

energy takes place at scales of the order of Kolmogorov length η, while

the input of energy into the cascade comes from the decay of the

large scales, of order L.

These two scales at the extremes of the cascade can differ by several

orders of magnitude at high Reynolds numbers. In between there is

a range of scales (each one with its own characteristic length r) that

has formed at the expense of the energy of the large ones.

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The Kolmogorov thesis

Since eddies in this range are much larger than the dissipative eddies that exist at Kolmogorov scales, kinetic energy is

essentially not dissipated in this range, and it is merely transferred to

smaller scales until viscous effects become important as the order of

the Kolmogorov scale is approached.

Within this range inertial effects are still much larger than viscous

effects, and it is possible to assume that viscosity does not play a

role in their internal dynamics (for this reason this range is called

"inertial range").15

The Kolmogorov thesis

At very high Reynolds number the statistics of scales in the range

η≤ r ≤ L are universally and uniquely determined by the scale r and

the rate of energy dissipation .

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DNS

The integral scale depends usually on the spatial scale of the boundary conditions. To satisfy these resolution requirements,

the number N of points along a given mesh direction with

increments h, must be :

N h > L

so that the integral scale is contained within the computational domain, and also h ≤ η so that the Kolmogorov scale can be resolved.

Since ε ≈ u' ³ / L where u' is the root mean square (RMS) of the velocity

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DNS

The previous relations imply that a three-dimensional DNS requires

a number of mesh points satisfying N³

where Re is the turbulent Reynolds number

the memory storage requirement in a DNS grows very fast with the

Reynolds number. In addition, given the very large memory necessary, the integration of the solution in time must be done by

an explicit method.

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DNS

C is here the Courant number :

Combining these relations, and the fact that h must be of the order

of , the number of time-integration steps must be proportional to

L / C η .

By other hand, from the definitions for Re, η and L given above, it

follows that

and consequently, the number of time steps grows also as a power

law of the Reynolds number

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DNS

One can estimate that the number of floating-point operations required to complete the simulation is proportional to the

number of mesh points and the number of time steps, and in conclusion, the number of operations grows as Re³ .

Therefore, the computational cost of DNS is very high, even at low

Re. For the Reynolds numbers encountered in most industrial applications, the computational resources required by a DNS

would exceed the capacity of the most powerful computer currently available.

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DNS

Also, direct numerical simulations are useful in the development of

turbulence models for practical applications, such as sub-grid scale

models for LES and models for methods that solve the Reynolds-averaged Navier-Stokes equations (RANS).

This is done by means of "a priori" tests, in which the input data for

the model is taken from a DNS simulation, or by "a posteriori" tests,

in which the results produced by the model are compared with those

obtained by DNS.

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Property of turbulent flow

A major property of turbulent flows is that they appear to be chaotic

or random.

Randomness is a consequence of the interaction of the singular perturbation parameter Re resp. Ra and the non-linearity of the Navier-Stokes equations

In a fluid flow experiment, there are unavoidably inaccuracies and

perturbations in initial conditions, boundary conditions

(e.g., differential heating, surface roughness) and material

properties, i.e. viscosity and thermal diffusivity (due to impurities of

the fluid). 22

The scales of turbulent flow

A second characteristic feature of a turbulent flow is its large variety

of scales. Understanding of the different scales of motion in turbulent flows and the processes among them, being a

motivationfor the approach of large-eddy simulation (LED).Turbulent flow can be thought of as a superposition of locallycoherent structures, called eddies, of different sizes.

Today, the term 'eddy' is used more ambiguously; it is used tocharacterise the scales of structures in the flow field: Large eddies refer to large structures, small eddies refer to

small structures in the flow field

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LES

Large eddy simulation is a mathematical model for

turbulence used in computational fluid dynamic.

LES grew rapidly and is currently

applied in a wide variety of engineering applications, including combustion,

acoustics, and simulations of the

atmospheric boundary layer. 24

Type of LES models

• Smagorinsky-Lilly model • Dynamic subgrid-scale model • RNG-LES model • Wall-adapting local eddy-viscosity (WALE) model • Kinetic energy subgrid-scale model • Near-wall treatment for LES models

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K/ε model

The k/ε model is the most widespread turbulence model, but it

suffers from several well-known deficiencies.

A successful improvement of the standard k/ ε model is the so called

k- ε -υ² model.

This model requires resolving the near-wall region, which is infeasible

for three-dimensional problems of practical relevance.

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K/ε model

The model solves the turbulent closure problem by adding two

transport equations to the Reynolds-averaged Navier- Stokes equations:

one for the turbulent kinetic energy k [m² / s²]

one for the rate of turbulent dissipation ε [m² / s³]

A turbulent velocity scale is then given by: υ = √ k [ m /s]

and a turbulent time scale as Г = k/ ε [s]27

K/ε model

For the case of an incompressible flow, the transport equations are

given by

where the production term P is

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K/ε model

The turbulent kinematic viscosity is then modeled as :

The standard constans values of the model :

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K/ε model

The coeffcients are determined by demanding that this turbulence

model should satisfy experimental data for certain simple standard

flow cases. coeffcient is obtained by considering the log-law region

of a turbulent boundary layer. The is usually fixed from

calibrations with homogeneous shear flows, and is usually

determined from thedecay rate of homogeneous, isotropic turbulence.

, are optimized by applying the model to various fundamental

flows such as flow in channel, pipes, jets, wakes.30

K/ε model

Because the standard K/ε model is derived under the assumption of a

high (local) turbulent Reynolds number, regions of low turbulent

Reynolds number, such as close to the wall, are poorly modeled.

In those regions the destruction-of-dissipation term is singular at the

wall since ε is finite and the turbulent kinetic energy k is zero.

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K/ε model

Two-equation turbulence models widely used in industrial CFD applications although their shortcomings are well known.

The model coefficients in turbulence modeling are usually kept constant in turbulent flows with different geometry and at

different Reynolds numbers.

The use of these asymptotic constraints on the model constants

provides a formally-consistent model.

The k-ε model constants have assumed different values depending

on the applications.

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RANS based turbulence models

Linear eddy viscosity models Two equation models

• k-epsilon models – Standard k-epsilon model – Realisable k-epsilon model – RNG k-epsilon model – Near-wall treatment

• k-omega models – Wilcox's k-omega model – Wilcox's modified k-omega model – SST k-omega model – Near-wall treatment

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Reference list

1. R.W .Lewis. , K. Ravindran and A.S. Usmani, „Finite Element Solution of Incompressible Flows Using an Explicit Segregated Approach”, Archives of Computational Methods in Engineering, Vol. 2, 4, 69–93 (1995).

2. A.T. Patera, „ A spectral element method for fluid dynamics: Laminar flow in a channel expansion”, Journal of Computationing Physics 54, 468-488 (1984).

3. R.Peyret, T.D. Taylor, „Computational Methods for Fluid Flow”, Springer-Verlag New York Inc., 1983, USA.

4. O.C. Zienkiewicz, „The finite element method” Fourth Edition Volume 1 Basic Formulation and Linear Problems, McGraw-Hill International (UK), 1989, Londyn.

5. O.C. Zienkiewicz, „The finite element method” Fourth Edition Volume 2 Solid and Fluid Mechanics Dynamics and Non-linearity, McGraw-Hill International (UK), 1991, Londyn.

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THANK YOU FOR YOUR ATTENCION !!!

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