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Verification and Validation Database of American Society of Civil Engineers 2008
Copyright © American Society of Civil Engineers
VERIFICATION AND VALIDATION OF URBAN
ATMOSPHERIC FLOW MODELS
PART II: COMPARISON OF VARIOUS TURBULENCE
MODELS Andy Chan
1 & Hervé Morvan
2
1 School of Chemical and Environmental Engineering,
University of Nottingham Malaysia Campus,
Jalan Broga, 43500 Semenyih, Selangor Darul Ehsan, Malaysia.
2 Department of Mechanical Engineering, University of Nottingham,
University Park, Nottingham NG7 2RD, United Kingdom.
0. ABSTRACT
This second paper of the series compares the performance of different turbulent models
applied on different standard benchmark test on urban atmospheric flow problems
described in Part I. The k- model, the Reynolds-stress model and large-eddy simulation
are compared against available experimental and computational data. The details of each
turbulence model are described and their limitations and working conditions elaborated
and an assessment is made.
1. INTRODUCTION
One of the objectives of scientific inquiry rests on to develop a logical quantitative theory
that can give accurate prediction to a natural phenomenon. The study of urban
atmospheric flow problems is no exception. The problem, however, is that our
experience has told us there are no prospects of a simple, single, embracing theory that
can govern turbulent air flow (Pope 2002). In fact the study of the turbulence remains
one of the mysteries of modern physics. In spite of the existence of a robust theory, there
is no single simple methodology of solution to the full Navier-Stokes equations, due to its
high nonlinearity and variations of boundary conditions. No single physical problems in
physics invite the appearance of such a rich diversity of models to solve the equation
while no one understands for certain which one remains a good explanation.
Until recently, the main tool to understand turbulent fluid motions has been the statistical
models. With the advancement of computational resources, various newer models based
on the Navier-Stokes equations have been derived and solutions from them have been
made possible. Others, which have been developed earlier but could not be put in
practice, like the Direct Numerical Simulations (DNS) have been realised bit by bit.
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All these different models, obviously give different results, even for the same problems
and to a certain extent, proper analytical understanding has been procastinated. The fact
is that different models are given birth from a vastly different objective, perspective and
philosophy and it is inevitable that they differ from each other in results and predictions.
Different models also agree with experimental data at different extent. A model that
gives good agreement in one case can be very poor in a different situation. It is with this
objective that this paper is prepared, to illustrate the philosophies behind each model and
to describe and explain the performance of each model under the profiled benchmark
tests.
2. CHOICE OF TURBULENCE MODELS
It might be useful to give a brief account of the various turbulence models that are
frequently employed in practical engineering calculations of turbulent airflows. A more
in-depth study of various turbulence models can be found in most standard textbooks of
fluid mechanics (Pope 2002) and various review papers (Rodi 1997).
Before elaborating the technical details of various models, it is also important to outline
the various factors over the choice of model. After all there are so many models
available and it is thus important to know the criteria of selection over a wide range of
attributes.
As outlined in Pope (2002), a number of assessment criteria for model selection are
available which could include:
i) level of modeling,
ii) computational resources available,
iii) applicability and usability,
iv) accuracy…
It is important to note that this list is definitely subjective and non-exhaustive and any
extra criteria that are deemed necessary should be included.
2.1 Level of Modelling
In Direct Numerical Simulations (DNS), there is absolutely no ‘artificial’ modelling
involved. The velocity components are solved directly from the Navier-Stokes equations
at the Kolmogorov’s scales. The velocities are also obtained at instantaneous level and
statistics can be immediately extracted. At the other end of the spectrum, the mixing-
length model specifies various flow parameters: some physical, some even natural. This
model is deemed incomplete. Moreover, when using mixing length model, the velocity is
calculated to the mean values only. Velocity fluctuations or turbulence statistics
informations cannot be obtained directly.
2.2 Computational Resources
One important attribute to decide which turbulence model to use is the availability of
computational resources. Speaking based on the same computer hardware requirement,
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obviously the computational resources required increases with the required precision.
Yet some models like DNS requires many more times resources than conventional
models using the Reynolds-averaged Navier-Stokes equations (RANS) approach. In
some approach like DNS, the resources required is a function of the Reynolds number of
the flow, while simpler models only depend on the precision expected.
2.3 Applicability And Usability
Each model is designed and derived for a different application and definitely so models
are not applicable a particular flow phenomenon. In fact there is no single model that is
applicable to all flows, at least from an accuracy point of view. Mixing-length models
are in general confined to application of lower Reynolds numbers only because of their
derivation from the laminar principles. The various empirical constants of say, k-
models are tweaked from one application to another in order to match the experimental
data to varying degree of accuracy.
Some models, like DNS or Large-eddy simulations (LES) are applicable to a wider range
of applications due to their lower artificiality. On the other hand, they are usually not
commercially usable as their huge computational cost prohibits extensive engineering
applications, at least in present.
2.4 Accuracy
Accuracy is of course of fundamental importance. There are however a large number of
issues which could incur errors into the system. One very important subject is the
prescription of boundary condition in turbulent flows. As mentioned in Part I of the
paper, there are a lot of difficulties in the computational study of urban atmospheric flows
and all the mentioned issues incur inaccuracies into the solution to varying degree.
Different models incur different sort of inaccuracies according to their own
characteristics and it is important to know the model’s deficiencies before application.
3. TURBULENCE MODELS
3.1 Reynolds-Averaged Navier-Stokes Equations
Without going through trivialities which available in most standard texts, the Reynolds-
averaged Navier-Stokes (RANS) equations are given by
ij
ji
ii
x
P
x
uuU
Dt
DU
1''2
, (1)
following usual notations, where the capital letters refer to the mean flow parameters and
the minuscules + prime refer to the fluctuation parts. The key to solve the RANS
equations lies in closing the equations through modelling of the Reynolds-stress term
j
ji
x
uu
'', which is itself undeterminable. Modelling of different kinds have been
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developed to handle this particular term and relate this to various mean properties. We
shall only focus on the various models that are commonly used in engineering.
The k- model, developed by Jones & Launder (1972) is arguably the mostly commonly
used turbulence model in both engineering and research and is incorporated in many
commercial codes. The k- model is termed a two-equation model in which the Reynolds
stress term is solved by two additional turbulence quantities, k the turbulent kinetic
energy and its energy dissipation. The reason behind the development of these two
quantities is that a length-scale in the form of
23
kL can be derived which can eliminate
the specifications of the traditional mixing length.
The mean turbulent kinetic energy k of the system is simply 2
2
1iu and the governing
equation of the flux of k is
j
iji
j
T
j
j
j x
Uuu
x
kupuk
xDt
Dk''
'''' (2)
where the first bracket of the right-hand side represents the flux of energy and the
second term is usually referred to as the production of turbulent kinetic energy to
counter dissipation. T is the turbulent viscosity. Thus Equation (2) can be rewritten in
more compact form as
jxDt
Dk (3)
The equation for the dissipation can be written as
2
2
2
2
'2
'''2
''''2
''2
''2''
k
iT
k
j
k
i
j
iT
k
j
k
i
j
k
i
k
j
iT
ji
i
j
ikT
kjj
kTk
k
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
U
xx
U
x
uu
xx
p
x
uu
xDt
D
(4)
where
2
''
k
i
x
u.
Obviously the dissipation equation is far too cumbersome to deal with, even numerically.
Based on the Kolmogorov’s hypotheses of energy cascades, many of the terms can be
related to the energy transfer from the larger scale motions. Thus it is more useful to
relate them to large scale motions empirically through
kC
kC
xxDt
D
i
T
j
2
21
, (5)
where the model constants are obtained from Launder & Sharma (1974) from various
experimental measurements, though it is often tweaked slightly under different
applications,
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3.1,92.1~90.1,45.1~41.1,09.0~07.0 21 CCC . (6)
The entire mess is related by solving
2kCT
with 09.0C .
k- model is generally accurate for most simple flows, especially for two-dimensional
turbulence modeling. On the other hand, it is well-known that near-wall flow structures
cannot be handled well by this model, as elaborated in Part I of this paper (Murakami
1993). Moreover Kim et al. (1987) has shown that the various empirical constants
actually vary significantly near the wall-region due to changes in flow paradigm near
wall. Another well-known deficiency of the k- model is it over-predicts spreading in a
round jet or any area where flow enlargement is experienced (Pope 2002).
Various modifications have been proposed and made to the standard k- model for
various applications (Hanjalić & Launder 1980, Hanjalić 1994) but as Pope (2002) has
pointed many of these modifications’ overall performance in general flow predictions are
still inferior or inconclusive compared to the standard model.
One very common two-equation model developed from the k- is the k- model. In fact
many other two-equation models have been proposed with the intention to replace the
dissipation equation of the k- model due to its serious error discrepancies. Under this
philosophy, the two-equation model usually retains the k-equation and derives a new
equation to replace the -equation.
Based on dimensional principles, we can obtain a turbulent vorticity as
k
. (7)
Wilcox (1993) developed an equation for by
kk
Ck
CDt
D TT
211 2
21 . (8)
The k- model is another widely used two-equation model. This model is considered
superior in treatment of viscous near-wall region. The drawback is that it does have
difficulty in handling non-turbulent flows.
3.2 Reynolds Stress Models (RSM)
The major disadvantage of the previous two models is the necessity to incorporate the
artificiality of turbulent viscosity hypothesis, which many workers are uncomfortable
with. One way to circumvent the problem that does away with this is called the Reynolds
stress model, in which the Reynolds stress is handled without any specifications of
mixing length and is directly solved from the equation
The Reynolds stress equations are given by Chen & Jaw (2000) as.
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k
j
k
i
i
j
j
i
k
ikj
k
j
ki
k
jijikijk
kji
k
ji
x
u
x
u
x
u
x
up
x
Uuu
x
Uuu
x
uuuupuuu
xDt
uuD
''2
'''
'''''''''
'''''
(9)
which can again be compactly re-expressed as
ijijij
k
kijji
xDt
uuD
'' (10)
with each of the term replacing the corresponding brackets. kij is called the Reynolds-
stress flux, ij is the production tensor, ij is the pressure-rate-of-strain tensor and ij is
the dissipation tensor (Pope 2002).
Basically the above Reynolds stress equations are solved directly and the term ij is
modelled as a local function of Reynolds stress or shear. The major drawback is the high
computational cost involved in solving at least twelve equations in tandem (three velocity
equations, one pressure, six Reynolds stresses, the pressure-rate-of-strain tensor). In case
the energy equation is also needed, another four equations would need to be included.
To model the term ij , one common technique is to consider the so-called basic LRR-IP
model which is a combination of a proposal by Launder, Reece & Rodi (LRR) (Launder
et al. 1975) and the isotropisation of production (IP) by Naot et al. (1970). This can be
written as
ijijijjiRij Ckuu
kC
3
2
3
22 , (11)
where 6.0,8.1 2 CCR . (12)
Many other pressure-rate-of-strain models have been developed from the LRR-IP and
have been employed, like the LRR-QI HL model by Launder et al. (1975) and SSG
model by Speziale et al. (1991). Funnily enough, most newer models do not show
significant improvement when compared with the basic ones and LRR-IP is still the most
commonly employed model (Launder 1996, Pope 2002)
3.3 Large Eddy Simulations (LES)
In large eddy simulations (LES), the larger three-dimensional unsteady turbulent
structures are calculated explicitly from the governing equations without any degree of
modelling. The smaller-scale (subgrid scales) are modelled. Since larger scale motions
are directly represented, it is more reliable than any other previously described models for
the study of atmospheric systems or urban flows because in these situations, the smaller-
scale motions are less important in the practical sense. In fact the development of LES is
motivated by meteorological applications (Smagorinsky 1963, Mason 1994). The
drawback is obviously is the increased computational cost.
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The equations of motions of LES are derived by applying the filter to the Navier-Stokes
equations. The filter of LES is essentially the grid resolution such that the details of the
velocity field can be adequately resolved within the domain concerned. We shall just
illustrate the basic equation and the detailed technicalities of LES are given by Pope
(2002). Using the bar as representation of filtered variables, the equations of motion of
LES are
2
21
0
i
j
ji
jij
i
i
x
U
x
p
x
UU
t
U
x
U
(13)
The momentum equation can also be written as
u
ttD
D
xx
U
x
p
tD
UD
i
r
ij
i
j
j
j:
12
2
(14)
where ij is the residual stress tensor.
The closure of the nonlinear convective term in LES is usually handled by the
Smagorinsky model, which is analogous to the eddy-viscosity model in Reynolds stress,
ijr
r
ij S 2 and ijSSr SSSCSl 2:22 (15)
where r is the residual eddy viscosity, ls is the Smagorinsky lengthscale analogous to the
mixing length and Cs is the Smagorinsky’s coefficient proportional to the filter width .
The original Smagorinsky’s model is that the appropriate value of the Smagorinsky’s
constant varies in different flow regime and assigning it a constant value brings
inaccuracies within the domain. Recently Germano et al. (1991) proposed the dynamic
LES in which Smagorinsky’s value at each point in space is calculated based on the flow
domain and surrounding flow parameters.
Compared with RANS or RSM models, LES has the advantage of describing without
modelling the instantaneous large-scale turbulent flow structures, meaning that the
solutions are exact solutions of the Navier-Stokes equations. Therefore LES has been
receiving large attention in the aerodynamics and environmental sector lately. The
problem of LES, however is obviously its larger computational cost. Pope (2002) also
reported that LES are likely to be grid-dependent as it depends on a priori knowledge of
the flow structure, especially the subgrid scale motions. LES is also restrictively three-
dimensional and unsteady and unless three-dimensional unsteady solutions are truly
sought, sometimes the computational cost can be lavish.
In terms of accuracy, LES is reasonably accurate for most free shear flows (Piomelli
1993, Vreman et al. 1997) but since LES has only recently picked up its momentum,
fingers should still be kept crossed.
4. BENCHMARK RESULTS
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4.1 Lid Driven Cavity Test
Ghia et al. (1982) have presented numerical solutions for the lid-driven cavity flow for
various Reynolds number, and further validated by Ertürk et al. (2002). As mentioned in
Part I of the paper, the results of Ghia et al. (1982) are largely regarded as the standard
data for comparison purposes. Botella & Peyret (1998) used a Chebyshev collocation
method for the solution of the lid-driven cavity flow and obtained highly accurate
spectral solutions for the cavity flow for Reynolds numbers Re 9,000. They stated that
their numerical solutions exhibit a periodic behavior beyond this Re. Both of which can
validated with experiments at lower Reynolds numbers.
As mentioned in Part I of the paper, it is often useful to look and analyse the positions of
all the vortices developed, including the centre main vortex and the few corner vortices.
Figure 1 shows the location and approximate features of these vortex centres.
Figure 1: Flow structure within a lid-driven cavity (Ertürk et al. 2002)
Figure 2: Flow structure within lid-driven cavity under different Reynolds numbers
a) Re = 100, b) Re = 1000, and c) Re = 10000 (Ghia et al. 1982)
Figure 2 shows the computational results of Ghia et al. (1982) under different Reynolds
numbers, while Figure 3 shows the computational validation results of Ertürk et al.
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(2002). Comparisons are made with the location of the various vortices and the velocity
profile at the mid-section of the cavity as in Figure 3. The use of streamfunctions and
vorticity in this test case also favours comparison using streamline plots and vorticity
plots and vorticity plots have also been made.
Figure 3: Flow structures of lid-driven cavity flow at Re = 10000 and the horizontal
velocities at different Re by Ertürk et al. (2002)
It must be emphasised here that there had been many computational evidences that the
lid-driven cavity test cannot be extended for larger Reynolds number (Shankar &
Deshpande 2000). Many of the data show that instability and unsteadiness ensue beyond
Re > 15000 and disallows proper comparisons. The case is similar to vortex shedding of
flow past cylinder in large Reynolds number flows where comparison at any time-instant
is difficult.
4.2 Backward Facing Step Test
Driver & Seegmiller (1985) performed wind-tunnel experiments on the backward facing
step test and the data have been frequently used for comparison purposes. The setup is a
step-height to tunnel-exit-height ratio of 1:10 which would reduce free-stream pressure
changes at the expansion step. The tunnel is 12 times wider than the tunnel height to
reduce three-dimensional effect. The data are also available at NPARC Alliance website
(http://www.grc.nasa.gov/WWW/wind/valid/backstep/backstep.html).
To compare the results, the wind profile at each axial position and the length of the
separation zone is looked at. In general it is expected that the k- model would predict
the reattachment to occur too far upstream (Thangam 1992, Yoder & Georgiadis 1999).
Figure 4 shows the experimental data of the velocity profiles.
10
Figure 4: Velocity profiles of the backward facing step test at various axial positions
(Yoder & Georgiadis 1999)
4.3 T-Junction Test
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Sini et la. (1996) studied the T-junction problem and compiled extensive data-set for
comparison purposes. Most importantly the data show that the street canyon geometric
configurations exhibit and correspond to Oke’s (1988) flow regime. Figure 4 shows Sini
et al.’s (1996) computational results which confirmed with Chan et al. (2001) data. It is
noticed that the threshold value of width-to-height ratio W/H = 1.5 corresponds to the
change of skimming flow to wake interference flow whereas wake interference flow
(WIF) to isolated roughness flow (IRF) occurs at around W/H 8-9. These data are also in
agreement with 3-D simulations of Hunter et al. (1992). The streamlines illustrate how
the velocity field complies with the canyon geometry and flow regimes. Important points
to notice are the position of the vortex centres and the various configurations when these
vortex occur.
Figure 5: Streamlines and pollutant iso-concentration map of T-junction test by Sini et al.
(1996)
Sini et al. (1992) and Xie et al. (2005) studied the effect of wall heating under different
circumstance. Relatively few field measurements have been compiled due to the
difficulty in controlling the thermal effects in nature. In general, for the cases of ground
or leeward wall heating, the flow structure is quite similar to the isothermal case (Sini et
al. 1992) with slightly increased vortex intensity. When the windward wall is warmer, an
upward buoyant flux opposes the downward advection along the wall and thereby
dividing the flow structure into two-counter-rotating vortices (Figure 5). Due to the
formation of the extra vortices arising from the heated wall, it therefore appears that wall
heating can give rise to significant changes to flow regimes.
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Figure 6: Streamlines and pollutant iso-concentration map of T-junction test for different
cases of heating by Sini et al. (1996)
4.4 Indoor Ventilation Test
The simplicity of geometry of the indoor ventilation allows extensive measurements to be
taken. Data are compared for the velocity at different axial direction and the location of
the recirculating vortex. It is worthwhile to notice that the indoor ventilation test
resembles the backward step physically and many of the features are expected to be
similar. Nielsen (1990) and Nielsen et al. (1978) had performed extensive wind-tunnel
and numerical experiments and had compiled useful data for comparisons as illustrated in
Figure 7 and 8.
13
Figure 6: Velocity profiles of the indoor ventilation test (Nielsen 1990)
For the thermal case, temperature and the penetration depths are the two extra variables
to be compared with as in Figure 7 and 8. The temperature, velocity and the penetration
depths are compared for various Archimedes number, which is essentially the Grashof
number in convection. Data for comparisons are also available from Lemaire (1993).
14
Figure 7: Temperature distribution in the indoor ventilation test for various Archimedes
number (Nielsen 1990)
Figure 8: Penetration depth for the indoor ventilation test (Nielsen 1990)
4.5 Personal Exposure Test
Brohus (1997) documented the velocity profile and pollutant concentration profile for the
personal exposure test (Figure 9 and 10)
15
Figure 9: Velocity distribution for personal exposure test (Brohus 1997)
Figure 10: Contaminant concentration profile for personal exposure test (Brohus 1997)
4.6 Flow Round Cube
A seminal discussion on flow past a cube (or square cylinder) is available in Rodi (1997).
In general the phenomena of vortex shedding should be reproduced by all models. As
available in the literatures, the shedding produced by LES is not as regular as produced
by RSM or RANS models. Most models should be able to produce good agreement with
experimental data near the cylinder. On the other hand as expected different models
would produce different results in the outer domain. The standard k- model over-
predicts the separation zone length whereas the RSM models produce too short a region
(Murakami 1993, Murakami & Mochida 1995).
There have been data aplenty regarding the various flow parameters near a cube as listed
in the references mentioned. Centreline velocity comparisons have been mostly used to
study the flow structures and compare the features of different model. Figure 11 shows
the centreline mean velocities for flow past cube calculated using different model
(Murakami 1993) and compared with experimental data. Comparisons are focused along
velocities, pressure and vorticity distribution and their corresponding vortex-shedding
phenomena.
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Figure 11: Centreline velocity of flow past cube (Murakami 1993)
4.7 Flow Past a Two-Dimensional Hill
Yu et al. (2003) studied the flow past a two-dimensional topographical changes in the
form of a smooth hill and a steep cliff. The well-known shortcomings of the k- model
in over-predicting the development of turbulent kinetic energy near the sharp edges are
well exhibited. Aside from comparing with the various parameters, separation points and
recirculation regions are also looked at. Figure 12 shows the wind profile of flow pas a
2D hill for comparison purposes.
Figure 12: Wind profile along a 2D hill (Yu et al. 2003)
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17
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