Transcript Slajd 1

Turbulent flow models
Katarzyna Miłkowska - Piszczek
Faculty of Metal Engineering and Industrial Computer Science
Department of Ferrous Metallurgy
Kraków 8.12.2010
Content
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Preface
CFD
Turbulence models
DNS
LES
K/ε
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Turbulent flow
In fluid dynamics , turbulence or
turbulent flow is a fluid regime
characterized 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
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The fundamental mathematical model are the non-isothermal
Navier-Stokes equations, governing the time-evolution of
velocity, pressure and temperature.
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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
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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.
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The second approach is called large-eddy simulation (LES). The
idea of LES is to apply a spatial averaging filter to the NavierStokes 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)
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compute the random motion of the large scale motion (and
model the small scale motion), which is referred to as large-eddy
simulation (abbreviated LES)
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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 mechanics
that 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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The Kolmogorov scale
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.
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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").
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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 :
Nh>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/Cη.
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 Reynoldsaveraged 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).
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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 motivation
for the approach of large-eddy simulation (LED).
Turbulent flow can be thought of as a superposition of locally
coherent structures, called eddies, of different sizes.
Today, the term 'eddy' is used more ambiguously; it is used to
characterise 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.
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Type of LES models
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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]
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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 the
decay rate of homogeneous, isotropic turbulence.
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are optimized by applying the model to various fundamental
flows such as flow in channel, pipes, jets, wakes.
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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 Nonlinearity, McGraw-Hill International (UK), 1991, Londyn.
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THANK YOU FOR YOUR ATTENCION !!!
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