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The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)

Application of nonlinear eddy viscosity model in simulations of

flows over bluff body
Jian Zhang a,b, Qingshan Yang b, Q. S. Li a
a

Dept of Civil and Architectural Engineering, City University of Hong Kong, Hong Kong
b
School of Civil Engineering, Beijing Jiaotong University, Beijing, 100044, China

ABSTRACT: Turbulent wind flows around building structures (bluff bodies) are investigated
numerically using Reynolds Averaged Navier-Stokes (RANS) turbulence models. A non-linear
viscosity assumption model, which can take into account the anisotropy of turbulence with less
numerical cost than LES method, is adopted herein as a turbulence treatment method. Model coefficients of the nonlinear terms are adjusted, aiming to model three-dimensional flows over
bluff bodies. Compared with the results of the linear eddy viscosity model and experimental
measurements, the nonlinear eddy viscosity model yields more reasonable results, especially for
the reattachment length and pressure coefficients. It is also noted that the nonlinear eddy viscosity model performs satisfactorily to reproduce complex turbulent flows around bluff bodies.
KEYWORDS: Nonlinear eddy viscosity model, CFD, Bluff body, Wind flow, RANS models
1 INTRODUCTION
Turbulent wind flows around building structures (bluff bodies) have been studied extensively by
wind tunnel tests and field measurements. Rapid developments of computer technology in recent
years have made it possible to adopt numerical methods, such as Large Eddy Simulation (LES)
or Reynolds Averaged Navier-Stokes (RANS) models, to simulate turbulent flows around bluff
bodies. Although LES is an effective tool to calculate turbulent flows around civil structures,
RANS turbulence models, such as eddy viscosity models or Reynolds Stress Models, are still
valid because they require less computer hardware sources than LES. However, it is imperative
to overcome the limitations of RANS models. From the viewpoint of computational efficiency
and practical application, nonlinear eddy viscosity models are attractive for simulations of flows
influenced by anisotropy of turbulence.
and the
series models,
The two-equation turbulence models, for instance, the
offer significant simplicity and numerical stability in many cases for predictions of flow fields.
The Linear Eddy Viscosity Models (LEVMs) construct eddy flow field using solvable turbulence
(Versteeg & Malalasekra, 2007). The Boussinesq hypothesis,
quantities, such as , and
model (Launder & Spalding, 1974), is generally regarded as the
together with the standard
conventional linear eddy viscosity concept. It is found that LEVMs can be successfully applied
to two-dimensional boundary layers, three-dimensional flow around simple bodies and thin shear
flows which can be treated as approximately equilibrium (Hellsten and Bezard, 2005). This is
because the LEVM is only able to provide principle stress component correctly in such flows.
Therefore, it is not applicable in flows where anisotropic turbulence eddy fluctuations essentially
influence the mean flow. Complex phenomena such as strong three-dimension, separations, recirculations and reattachments are common in wind flows over building structures. It is noted

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that the standard LEVMs have severe defects for simulations of complicated flow fields. Consequently, it is required to extend the applicability of the linear eddy viscosity models for capturing
anisotropic stress fields.
Recent studies have been mainly focused on the nonlinear relationship between the Reynolds
stress and mean flow quantities. The aim of adopting the Nonlinear Eddy Viscosity Model
(NLEVM) is to introduce mean flow strain rate (or vorticity) into the Reynolds stress expressions
to construct nonlinear stress terms. Quadratic, cubic and fifth order models are commonly used
in the NLEVM's, which are different from each another with the degrees of nonlinearity. Craft et
al. (1997) and Huang & Rajagopal (1996) developed individual quadratic NLEVM models to
simulate two-dimensional turbulent flow over a backward-facing step. Their results showed that
the two quadratic NLEVMs successfully predicted the reattachment length of flow over the step.
Meanwhile, the quadratic models demonstrated great robustness while CPU cost was not significantly increased. Other quadratic models have been proposed by Speziale (1987) based on the
strain rate and by Rubinstein (1990) based on renormalisation-group approach. These models
yielded satisfactory results for simple cases.
Jongen et al. (1998) observed that the sensitivity of quadratic models to the rotation is inadequate to predict fully developed channel flow subjected to rapid spanwise rotation. Craft et al.
(1995) argued that a cubic Reynolds stress expansion is needed for predicting complex flows
with large streamline curvature effects. Craft et al. (1995) adopted a third-order term to sensitize
the model to curvature effects to overcome the overestimation of turbulence energy in the impingement area. To capture non-equilibrium flows, Craft et al. (1997) developed a more complicated model, in which the second anisotropy invariant is added in Reynolds stress expansions to
form a three-equation of NLEVM. This new model was designed especially for low-Reynolds
number transitional and turbulent flows. Apsley & Leschziner (1998) derived a cubic model
based on successive iterative approximations to an algebraic Reynolds stress model. They validated the model by considering several flows including a high-lift airfoil flow and separating
flow in two-dimensional diffuser. However, they just used the original coefficient-formulas as
relationships between the coefficients and recalibrated their values to capture the low-Reynolds
number near-wall flow.
Applications of these NLEVMs to complex turbulent flows have displayed inspiring results
on flow anisotropy, strong streamline curvature, recirculation and adverse pressure gradient. To
evaluate the performance of the NLEVMs in predicting complex turbulent flows fields, this paper presents numerical simulations of wind flow over a bluff body with a quadratic NLEVM in
comparison with field measurement data. It should be noted that throughout the numerical simulations, realizability conditions are applied to the NLEVM. It will prove that the NLEVM can be
applied to the simulations of turbulent flows with strong anisotropy usually encountered in engineering practices.
2 THE NONLINEAR

TURBULENCE MODEL

2.1 Basic equations

The basic equations of continuity and momentum for steady incompressible flows are:

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The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012

(1)
Realizable

model is adopted in this study as follows:

(2)

where

and
are the spatial coordinates;
and
are the averaged wind velocities;
is the Reynolds stress;
is the averaged pressure;
is the density of fluid; is the
is the turbulent
Turbulent Kinetic Energy (TKE); is the turbulent energy dissipation rate;
is the molecular viscosity and
,
,
,
dynamic eddy viscosity,
are the model constants.
are expressed by the linear
For two-equation turbulence model, Reynolds stress tensors
constitutive equation:
(3)
2.2 The quadratic eddy viscosity assumption
Since the standard two-equation model does not take into account the anisotropy of the Reynolds
stresses. This shortcoming can be overcome to some extent by introducing a nonlinear constitutive expression as follows:
(4)
and
where
For three-dimensional flows around bluff bodies, the coefficients of Reynolds stress terms are
needed to be tuned by consideration of the anisotropy (Champagne, 1970). Firstly, Reynolds
stress components are extracted and simplified. According to zero-pressure gradient flow over a
smooth plate, wind velocity component can be expressed as follows:
(5)
Then, shear deformation invariant
ant

(6)

is deformation tensor invariant.

where
The normal Reynolds stress components are expressed as follows:

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(7)

In two-dimensional boundary flows, constants

,
,
and
vary with anisotropic flow characteristics. To satisfy the above mathematical constraints, dampis introduced into ~ as:
ing function
(8)
is positive, the following inequalities are thus satisfied:
It is well-known that the tensor
(9)
We refer to these three properties as realizability conditions. The normal Reynolds stress
components can be concluded based on the first realizability condition in equation
(9) as follows:

(10)

From
(10), constant

the
satisfies:

second

expression

of

equation
(11)

Based on the realizable

model, turbulent kinetic energy generation
and turbulent viscosity
are expressed below:
stress

, Reynolds shear
(12)

Reynolds
(11) and (12):

shear

can

stress

be

derived

by

equations
(13)

Then another

(14)

To make sure

expression is given as follows:

(15)
Based on the above derivation, the three-dimensional quadratic eddy viscosity model can be
expressed as:

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The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012

(16)

Turbulence modeling, aiming to wind engineering applications, should take the anisotropy of
turbulent flow into consideration. It is important to maintain the ease of use and computational
stable of the two equation models. Therefore, nonlinear expansions of the Boussinesq hypothesis--the quadratic expressions proposed by Craft et al. (1997)--have been extended in an attempt
to account for anisotropic turbulence and curvature related strain effects. Numerical tests of the
quadratic model will be carried out in the following section.
3 NUMERICAL DISCRETIZATIONS, MESH SCHEMES AND TEST CASE
3.1 Numerical Discretizations
The commercial package FLUENT has been used to solve the governing equations for mean
wind velocities and turbulent quantities. The equations are discretized by the finite volume
method on structure grids. The second-order upwind differencing scheme and SIMPLEC algorithm are used for convective terms and the pressure-velocity terms individually. The modified
is also modified based
quadratic model is incorporated in FLUENT. The turbulent viscosity
on the new expressions.
3.2 Boundary Conditions
For inlet condition, a fully developed velocity profile is adopted, which is expressed by a power
law:
(17)
(m/s) is the mean wind velocity at height (m),
(m/s) is the reference velocity at
where
is the roughness category coefficient.
height (m).
Turbulence intensity is defined in AIJ code (Architectural Institute of Japan, 2004):
(18)
where is the height along the fetch; ,
and
are parameters which can be found in AIJ
code. The inlet condition of turbulence kinetic energy (TKE) is defined as a function of turbulence intensity :
(19)
Dissipation rate
ship is:

can be determined by

, and the relation(20)

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where
is defined as a constant (0.09).
is defined independently of the terrain conditions of the site as:
Turbulence length scale
(21)
In summary, the inlet boundary conditions of TKE
below:

and dissipation rate (D)

are listed

(22)

The boundary conditions of the top and bilateral boundaries of the fetch are set as slip boundary conditions, which are expressed as:
(23)
The outflow face is assumed as fully developed outflow boundary condition, which can be
expressed as:
(24)
3.3 Numerical model and mesh schemes
To examine the performance of the quadratic eddy viscosity model, numerical simulations of
wind flow over a 6m cube are carried out based on the standard realizable
turbulence
model and the quadratic eddy viscosity model. The 6m cube is mounted in atmospheric boundary
layer (ABL) at a normal orientation to the incident wind with the computational domain shown
in Figure 1. Specific representations of the computational domain and mesh arrangements can be
seen in Figure 2.

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The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012

Figure 2. Grids scheme for 6m cube model.

Reynolds number based on the incident wind velocity and the height of the 6m cube model is
around 107. To resolve the flow field around the bluff body, the first-layer cell near the cube surmodel. The cells are nonface is set as 0.015 , which is appropriate for the realizable
uniformly distributed and the stretching ratio is 1.12 along the surfaces. 32 grids are placed on
the windward, roof and leeward surfaces of the cube ( -direction). 40 grids are located along the
lateral faces of the cube ( -direction). 30 grids with stretching ratio 1.2 are placed in the wake
zone of the computational domain.
4 NUMERICAL RESULTS
The computational results are compared with those by the linear eddy viscosity model (the realmodel) and the field measurement results (Richards et al., 2001).
izable

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Figure 4. Pressure distributions along the roof face.

Figure 3 shows comparisons of mean pressure coefficient distributions on the windward face.
It can be seen that the two turbulence models yield almost same results around the flow stagnation point. The pressure coefficients on the remaining part show relatively similar distribution
shapes with differences within approximately 10 percent of the measurement values. It is apparent that the CFD predicted pressure spike position is approximately 0.67m lower than that obtained by measurement results. Figure 4 shows the pressure distributions on the cube roof. The
quadratic nonlinear model predicted closer results with the measurements than the linear eddy
viscosity model. Both models over predicted the pressure at the front edge of the cube.

Figure 5. Pressure distributions along the leeward face.

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The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012

Figure 6. Pressure distributions along the side face.

The pressure distributions on the leeward face are showed in Figure 5. It is apparent that neither of the models could accurately predict the measurement results. The better approximations
are obtained by the quadratic model, although the errors are around 45 to 50 percent. The pressure distributions on the side face are displayed in Figures 6. The measurement results on the
side face shows a slow increase in negative magnitude as the wind velocity increases with height.
The quadratic model predicts more accurate values than the standard linear eddy viscosity model.
It has been proved that the size and strength of the roof recirculation zone is reduced as the flow
approaching the side walls of the cube. The mass of air flowing over the roof is also reduced. It
is therefore clear that the measurement results are affected to a much larger extent by threedimensional flow phenomenon, due to significant difference between the centre and edge negative pressures. This suggests that the roof recirculation zone in the full-scale measurement contains a much stronger vortex than that predicted by CFD, which causes the pressure difference
between the measurements and CFD results.
5 CONCLUSION
The quadratic eddy viscosity model, including new considerations of mathematic constraints,
was developed in this paper to predict turbulent flows around bluff bodies. This study shows that
adoption of the nonlinear eddy viscosity model yields better results than the standard linear eddy
viscosity model, especially for the pressure distributions on the windward, roof and side faces of
a 6m cube. Meanwhile, this nonlinear eddy viscosity model has great robustness to calculate turbulence anisotropy and is suitable for numerical simulation of complex flows around building
structures. It is indicated that the quadratic NLEVM exhibit its advantages in capturing the pressure variations on a low-rise building, compared with those predicted by the standard twoequation turbulence models.
6 ACKNOWLEDGEMENTS
The work described in this paper is fully supported by grants from National Natural Science
Foundation of China (Project No. 90815021 and 50778059) which are gratefully acknowledged.

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7 REFERENCES
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