CFD inflow conditions, wall functions and turbulence models for flows around obstacles

46 Pages·2013·6.03 MB·English

CFD inflow conditions, wall functions and turbulence
models for flows around obstacles
∗
Alessandro Parente
Universit´e Libre de Bruxelles, Belgium
March 19, 2013
Contents
1 Introduction 5
2 Theory 6
2.1 Inlet conditions and turbulence model . . . . . . . . . . . . . . . . . . . . . 7
2.2 Wall treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Generalization of the ABL model for the case of obstacles immersed in the
flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Applications 19
3.1 Empty fetch at wind-tunnel and full scale . . . . . . . . . . . . . . . . . . . 19
3.2 Flow around a ground-mounted building . . . . . . . . . . . . . . . . . . . 23
3.3 Flow over complex terrains, wind-tunnel and full-scale hills . . . . . . . . . 34
4 Influence of stability classes 40
∗The present lecture notes are based on the research work performed by several Authors in the field
of ABL flows at the von Karman Institute of Fluid Dynamics and at the Universit´e Libre de Bruxelles.
In particular, the contribution by Prof. Carlo Benocci, Prof. Jeroen van Beeck, Dr. Catherine Gorl´e and
Dr. Miklos Balogh should be acknowledged.
VKI - 1 -
LIST OF FIGURES LIST OF FIGURES
List of Figures
1 Computational domain with building models for CFD simulation of ABL
flows and indication of different parts in the domain for roughness modelling
Blocken et al. [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Law of the wall for smooth and sand-grain roughened surfaces as a function
+
of the dimensionless sand-grain roughness height kS Blocken et al. [1]. . . . 12
3 Turbulent kinetic energy profiles at the inlet and outlet sections of an empty
computational domain (see Figure 2b), Dashes: cell value for turbulent
dissipation rate and turbulent kinetic energy averaged over the first cell.
Short dashes: cell value for turbulent dissipation rate and kinetic energy.
Dots: turbulent dissipation rate and kinetic energy averaged over the first
cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Rough law of the wall implementation. . . . . . . . . . . . . . . . . . . . . 15
5 Configurations PS1 (a) and PS2 (b) for the definition of the building influ-
ence area (BIA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6 Variation of the turbulence model parameter Cµ for an undisturbed ABL
(top), using a prescribed region Beranek [2] (middle) and using an auto-
matic switching function (bottom). . . . . . . . . . . . . . . . . . . . . . . 18
7 Computational domain and main boundary conditions applied for the nu-
merical simulation of the unperturbed ABL at wind tunnel (a) and full (b)
scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
8 Profiles of velocity, turbulent kinetic energy and turbulent dissipation rate
at inlet and outlet section of the computational domain (Figure 7), ob-
tained when applying inlet conditions given by Equations 10-12 (a-c) and
Equations 10,15 and 12 (d-f). STD WF = Standard Wall Function; MOD
WF = Modified Wall Function [3]. . . . . . . . . . . . . . . . . . . . . . . 21
9 Wall shear stress as a function of the axial coordinate. Solid black line:
2
theoretical value, ρu∗. Dashed red line: Richards and Hoxey [4] profile.
Blue dots: Yang et al. [5] profile [3]. . . . . . . . . . . . . . . . . . . . . . . 23
10 Profiles of velocity (a), turbulent kinetic energy (b), turbulent dissipation
rate (c) and non-dimensional velocity gradient (d) at inlet and outlet section
of the computational domain, obtained with Equations 10, (23) and 12 [6]. 24
11 Profiles of velocity, turbulent kinetic energy and turbulent dissipation rate
at inlet and outlet section of the computational domain (Figure 7), ob-
tained when applying inlet conditions given by Equations 10-12 (a-c) and
Equations 10, (29) and 12 (d-f). Results obtained with the wall function
formulation by [3] [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
12 Wall shear stress as a function of the axial coordinate. Solid black line:
2
theoretical value, ρu∗. Dashed red line: Richards and Hoxey [4] profile.
Blue dots: Yang et al. [5] profile. . . . . . . . . . . . . . . . . . . . . . . . 26
13 Building geometry and location of measurement planes for the flow around
the obstacle [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
14 Computational domain and grid for the flow around the obstacle [7]. . . . . 26
VKI - 2 -
LIST OF FIGURES LIST OF FIGURES
15 Contour plots of non-dimensional velocity on the planes y = 0 (left) and
z = 0.035m (right). Experimental measurements are compared to the
results obtained applying the PS1 and PS2 model [3]. . . . . . . . . . . . . 29
16 Contour plots of non-dimensional turbulence kinetic energy on the planes
y = 0(left) and z = 0.035m (right). Experimental measurements are com-
pared to the results obtained applying the PS1 and PS2 model [3]. . . . . . 29
17 Experimental and numerical profiles of non-dimensional velocity upstream,
over and downstream of the obstacle. Solid line: experimental data [3].
Dashes: PS1 configuration. Short dashes: PS2 configuration . . . . . . . . 31
18 Experimental and numerical profiles of non-dimensional turbulent kinetic
energy upstream, over and downstream of the obstacle [3]. Solid line: ex-
perimental data. Dashes: PS1 configuration. Short dashes: PS2 configura-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
19 Experimental and numerical profiles of non-dimensional velocity over and
downstream of the obstacle. Solid line: experimental data [6]. Dashes:
UABL model. Short dashes: PS model. Dots: ASQ model. . . . . . . . . 31
20 Experimental and numerical profiles of non-dimensional turbulent kinetic
energy over and downstream of the obstacle. Solid line: experimental data
[6]. Dashes: UABL model. Short dashes: PS model. Dots: ASQ model. . . 32
21 Local hit rates for the non-dimensional turbulent kinetic energy, applying
the ASQ model (left) and the UABL model (right) [6]. . . . . . . . . . . . 33
22 Stream-wise velocity (top) and turbulent kinetic energy profiles (bottom)
in the symmetry plane against measurements obtained on the 3D hill at
laboratory scale using Fluent and OpenFOAM [8]. . . . . . . . . . . . . . . 35
23 Simulated wall shear stress along the symmetry of the domain against theo-
retical values, extracted from the inlet profile (Inlet τw) and against values
extracted from measured profiles (meas.).[8]. . . . . . . . . . . . . . . . . . 36
24 Building geometry and location of measurement planes for the flow around
the obstacle [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
25 Comparison of simulated and measured horizontal and vertical stream ve-
locity (Uh and W) and turbulent kinetic energy (k) along line-A, using the
comprehensive approach [6] with α = 3 [8]. . . . . . . . . . . . . . . . . . 39
26 Comparison of simulated and measured vertical profiles (U and k) at the
hill summit [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
27 Profiles of (a) velocity, (b) turbulent kinetic energy, (c) turbulent dissipa-
tion rate, (d) turbulent viscosity and e) temperature at the inlet and outlet
section of a 2D computational domain (60m high and 400m long ), and (f)
shear stress at the wall. Inlet conditions taken from [9]. . . . . . . . . . . . 42
VKI - 3 -
LIST OF TABLES LIST OF TABLES
List of Tables
1 Inlet conditions and turbulence model formulation. . . . . . . . . . . . . . 10
2 Fitting parameters for velocity and turbulent kinetic energy inlet profiles
according to Yang et al. [5], Parente et al. [6] and turbulent model parameters. 19
3 Fitting parameters for velocity and turbulent kinetic energy inlet profiles
according to Richards and Hoxey [4], Brost and Wyngaard [10] and turbu-
lent model parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Hit rate values for non-dimensional velocity and turbulent kinetic energy for
the prescribed BIA size approach., varying the turbulence model settings [3]. 28
5 Test cases and corresponding model settings for the numerical simulation
of the flow around a bluff-body [7], using the prescribed BIA size approach.
TM=Turbulence Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6 Test cases and corresponding model settings for the numerical simulation
of the flow around a bluff-body [7], using the automatic switch approach
BIA for the BIA. TM=Turbulence Model. . . . . . . . . . . . . . . . . . . 30
7 Hit rate values for non-dimensional velocity and turbulent kinetic energy
for the automatic switch approach, varying the turbulence model settings
[6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8 Hit rate values for non-dimensional velocity and turbulent kinetic energy
for the automatic switch approach, varying the turbulence model settings
[11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
9 Fitting parameters for velocity and turbulent kinetic energy inlet profiles
according to Parente et al. [6] and turbulent model parameters for the 3D
hill simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
10 Hit rate values for non-dimensional velocity and turbulent kinetic energy
for the 3D hill simulation, varying the turbulence model settings [8]. . . . . 37
11 Normalized errors in the prediction of the separation point, ϵSP , and wake
length, ϵWL, for the 3D hill. Negative and positive values sign the under-
and overestimation respectively[8]. . . . . . . . . . . . . . . . . . . . . . . . 37
12 Fitting parameters for velocity and turbulent kinetic energy inlet profiles
according to Parente et al. [6] and turbulent model parameters for the
Askervein hill simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
13 Hit rate values for non-dimensional velocity and turbulent kinetic energy
for the Askervein hill simulation, varying the turbulence model settings
within the wake region [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
VKI - 4 -
1 INTRODUCTION
1 Introduction
Computational Fluid Dynamics (CFD) is widely used to study flow phenomena in the
lower part of the atmospheric boundary layer (ABL), with applications to pollutant dis-
persion, risk analysis, optimization and siting of windmills and wind farms, and micro-
climate studies. Numerical simulations of ABL flows can be performed either by solving
the Reynolds-averaged Navier-Stokes (RANS) equations or by conducting large-eddy sim-
ulations (LES). It is generally acknowledged that LES, which explicitly accounts for the
larger spatial and temporal turbulent scales, can provide a more accurate solution for
the turbulent flow field, provided that the range of resolved turbulence scales is suffi-
ciently large and that the turbulent inflow conditions are well characterized Shah and
Ferziger [12], Lim et al. [13], Xie and Castro [14, 15]. For example, Xie and Castro [14]
presented a comparison of LES and RANS for the flow over an array of uniform height
wall-mounted obstacles: the Authors compared the results to available direct numerical
simulation (DNS) data, showing that LES simulations outperform RANS results within
the canopy. Dejoan et al. [16] compared LES and RANS for the simulation of pollutant
dispersion in the MUST field experiment and found that LES performs better in pre-
dicting vertical velocity and Reynolds shear stress, while the results for the stream-wise
velocity component are comparable. However, LES simulations are at least one order
of magnitude computationally more expensive than RANS [17] and the sensitivity to in-
put parameters such as inlet conditions, imply that, as for RANS, multiple simulations
are needed to quantify the resulting uncertainty in the output for realistic applications.
Hence, practical simulations of ABL flows are still often carried out solving the RANS
equations in combination with two-equation turbulence models. Consequently, investigat-
ing possible improvements to these models is worthwhile. In RANS simulations, the effect
of roughness on ABL flows is generally represented with the so-called sand-grain based
wall functions Cebeci and Bradshaw [18], based on the experiments conducted by Niku-
radze [19] for flow in rough, circular pipes covered with sand. Moreover, the upstream
turbulent characteristics of a homogeneous ABL flows are generally modeled using the
profiles suggested by Richards and Hoxey [4] for mean velocity, turbulent kinetic energy
and turbulent dissipation rate. However, this modelling approach can result in an unsat-
isfactory reproduction of the ABL for two main causes. The first cause of discrepancy
lies in the inconsistency between the fully developed ABL inlet profiles and the rough
wall function formulation Riddle et al. [20], Franke et al. [21], B. Blocken [22], Blocken
et al. [1], Hargreaves and Wright [23], Franke et al. [24]. Furthermore, the inlet profile
for the turbulence kinetic energy, k, proposed by Richards and Hoxey [4], assumes a con-
stant value with height, in conflict with wind-tunnel measurements Leitl [7], Xie et al.
[25], Yang et al. [5], where a variation of k with height is observed. A remedial measure
to solve the inconsistency between the sand-grain based rough wall function and the fully
developed inlet profiles was proposed by Blocken et al. [1]. It consists in the modifica-
tion of the wall law coefficients, namely the equivalent sand-grain roughness height ks
and the roughness constant Cs, to ensure a proper matching with the velocity boundary
conditions. This approach ensures the desired homogeneity of the velocity distribution
for the fully developed ABL, but it is code dependent and does not provide a general
solution to the problem. Moreover, the standard law of the wall for rough surfaces poses
limitations concerning the level of grid refinement that can be achieved at the wall. This
VKI - 5 -
2 THEORY
restriction becomes particularly relevant for applications requiring a high resolution near
the wall boundaries. An additional complicating factor is the necessity to apply different
wall treatments when a combination of rough terrains and smooth building walls must
be simulated. Concerning the inlet profile for turbulent kinetic energy, Yang et al. [5]
derived a new set of inlet conditions, with k decreasing with height. However, the ap-
plication of such a profile at the inlet boundary only provides an approximate solution
for the system of equations describing a fully developed ABL. In a recent work, Gorl´e
et al. [26] proposed a modification of the constant Cµ, and of the turbulent dissipation
Prandtl number, σϵ, to ensure homogeneity along the longitudinal ABL direction, when
the k profile of Yang et al. [5] is applied. Parente and Benocci [27], Parente et al. [3]
proposed a modification of the k − ϵ turbulence model compatible with the set of inlet
conditions proposed by Yang et al. [5]. Such a modification consisted in the generalization
of the model coefficient Cµ, which becomes a local function of the flow variables, and in
the introduction of two source terms in the transport equations for k and ϵ, respectively.
The limitation of such an approach consisted in the inlet profile adopted for turbulent ki-
netic energy, which does not satisfy all the governing simulations involved in the problem
Parente et al. [6]. Parente et al. [6] addresses the aforementioned aspects by proposing
a comprehensive approach for the numerical simulation of the neutral ABL. First, a new
profile for turbulent kinetic energy was derived from the solution of the turbulent kinetic
energy transport equation, resulting in a new set of fully developed inlet conditions for the
neutral ABL, which satisfies the standard k − ϵ model. This was accomplished through
the introduction of a universal source term in the transport equation for the turbulent
dissipation rate, ϵ, and the re-definition of the k − ϵ model coefficient Cµ as a function of
the flow variables. Second, for the purpose of solving the flow around obstacles immersed
in the flow, the modelling approach derived for the homogeneous ABL was generalized
with an algorithm for the automatic identification of the building influence area (BIA).
As a consequence, the turbulence model formulation is gradually adapted moving from
the undisturbed ABL to the region affected by the obstacle. Parente et al. [28, 3] also
proposed a novel implementation of a wall function, which incorporates both smooth-
and rough-wall treatments, employing a screening algorithm to automatically select the
desired formulation, i.e. rough or smooth, depending on the boundary surface properties.
Balogh et al. [8] extended the approach by Parente et al. [28, 3] to the simulation of flows
above complex terrains, i.e. wind-tunnel scale 3D hill model and Askervein Hill.
The present notes are organized as follows. The modelling approach for the numerical
simulation of neutral ABL flows is presented, by discussing the turbulence model formu-
lation, the different inlet profiles and the wall function. Applications are presented and
discussed for the flow over flat terrain, around ground mounted bluff bodies and over hills.
2 Theory
The standard k−ϵ model remains the most common option for the numerical simulation of
the homogeneous ABL. Such a family of models solves a transport equations for turbulent
kinetic energy, k, and for turbulent dissipation rate, ϵ:
[( ) ]
∂ ∂ ∂ µt ∂k
(ρk) + (ρkui) = µ + +Gk − Gb − ρϵ − YM (1)
∂ ∂xi ∂xj σk ∂xj
VKI - 6 -
2.1 Inlet conditions and turbulence model 2 THEORY
[( ) ]
2
∂ ∂ ∂ µt ∂ϵ ϵ ϵ
(ρϵ) + (ρϵui) = µ + + Cϵ1 (Gk + Cϵ3Gb) − Cϵ2ρ . (2)
∂ ∂xi ∂xj σϵ ∂xj k k
th
In Equations (1)-(2), ui is the i velocity component, ρ is the density, Cϵ1, Cϵ2 and Cϵ3 are
model constants, σk, and σϵ are the turbulent Prandtl numbers for k and ϵ, respectively, Gb
is the turbulent kinetic energy production due to buoyancy, YM represents the contribution
of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, Gk is
the generation of turbulence kinetic energy due to the mean velocity gradients, calculated
from the mean rate-of-strain tensor, Sij, as:
( )
2 √ 1 ∂ui ∂uj
Gk = µtS S = 2SijSij Sij = + . (3)
2 ∂xj ∂xi
For a steady ABL, under the hypothesis of zero vertical velocity, constant pressure along
vertical (z) and longitudinal (x) directions, constant shear stress throughout the boundary
layer and no buoyancy effects, the transport equations for turbulent kinetic energy k, and
turbulent dissipation rate ϵ, simplify to:
( )
∂ µt ∂k
+ Gk − ρϵ = 0 (4)
∂z σk ∂z
( )
2
∂ µt ∂ϵ ϵ ϵ
+ Cϵ1Gk − Cϵ2ρ = 0 (5)
∂z σϵ ∂z k k
( )2
∂u
Gk = µt . (6)
∂z
The model is completed by the momentum equations, which takes the form:
∂u
2
µt = τw = ρu
∗ (7)
∂z
where τw is the wall shear stress and u∗ is the friction velocity
√
τw
u∗ = . (8)
ρ
In Equations (4), (5) and (7) the laminar viscosity has been neglected with respect to the
turbulent one, µt, expressed as:
2
k
µt = ρcµ . (9)
ϵ
2.1 Inlet conditions and turbulence model
Fully developed inlet profiles of mean longitudinal velocity, turbulent kinetic energy
and dissipation rate under neutral stratification conditions are often specified following
Richards and Hoxey [4]:
( )
u∗ z + z0
U = ln (10)
κ z0
2
u
∗
k = √ (11)
Cµ
VKI - 7 -
2 THEORY 2.1 Inlet conditions and turbulence model
3
u
∗
ϵ = (12)
κ (z + z0)
where κ is the von Karman constant and z0 is the aerodynamic roughness length. It can
be shown that Equations (10)-(12) are analytical solutions of the standard k − ϵ model if
the turbulent dissipation Prandtl number, σ�ϵ, is defined as Richards and Hoxey [4]:
2
κ
σϵ = √ (13)
(Cϵ2 − Cϵ1) Cµ
or, equivalently Parente et al. [3, 6], Pontiggia et al. [29], if the following source term is
added to the dissipation rate equation:
( √ )
4
ρu
∗ (Cϵ2 − Cϵ1) Cµ 1
Sϵ (z) =
2 2 − . (14)
(z + z0) κ σϵ
A weakness of the formulation presented above is the assumption of a constant value for
the turbulent kinetic energy k in Equation (11). Indeed, experimental observations show
a decay of k with height Leitl [7], Xie et al. [25], Yang et al. [5]. Following this observation,
Yang et al. [5] analytically derived an alternative inlet condition for k:
√
k = C1ln (z + z0) + C2 (15)
where C1 and C2 are constants determined via experimental data fitting. The profile for k
expressed by Equation (15) is obtained directly as solution of the turbulent kinetic energy
transport equation, under the assumption of constant value for Cµ and local equilibrium
between production and dissipation:
√ du
ϵ (z) = Cµk . (16)
dz
Yang et al. [5] mentioned that the constant Cµ should be correctly specified in order to
ensure the correct level of turbulence kinetic energy throughout the domain. However, this
could be unnecessary, if the effect of a non-constant k profile on the momentum equation
is taken into account. Gorl´e et al. [26] generalized the expression of Cµ as a function of
z, by substituting Equations (9) and (16) into Equation (7):
2 2
∂u k ∂u k ∂u
2 2 2
µt = ρu
∗ → ρcµ = ρu∗ → ρcµ√ ∂u = ρu∗ (17)
∂z ϵ ∂z C
µk ∂z ∂z
and, then,
4
u
∗
Cµ = (18)
2
k
Equation (18) is simply the relation proposed by Richards and Hoxey [4] inverted, to
ensure consistency between the turbulence model and the k profile throughout the ABL
domain. From the point of view of the physical interpretation, the non-uniform k profile
and the definition of Cµ can be related to the large-scale turbulence present in ABL flows,
VKI - 8 -
2.1 Inlet conditions and turbulence model 2 THEORY
which can vary significantly with height. Bottema [30] indicated the relevance of large-
scale turbulence to several RANS models, pointing out the necessity for case and location
dependent model constants.
Using the k inlet profile by Yang et al. [5], together with Equations (10) and (12) for u
and ϵ, and employing Equation (18) for Cµ does not allow to close the system of Equations
(4)-(7), with the definition of an appropriate expression for σϵ. Only an approximate
solution Gorl´e et al. [26] can be found, using the constant value of Cµ obtained at the
wall adjacent cell. In alternative, the functional form of Cµ ((18)), by introducing an an
additional source term for the k transport equation Parente et al. [3], in addition to the
one expressed by Eq. 8 for the ε transport equation:
[ ]
∂k
ρu∗κ ∂ (z + z0)
∂z
Sk (z) = . (19)
σk ∂z
As a consequence, an arbitrary set of inlet conditions, including the ones by Yang et al.
[5] can be adopted at the inlet boundary, ensuring their conservation throughout the
computational domain.
An alternative approach is that of repeating the exercise by Yang et al. [5] considering
the functional variation of Cµ ((18)). In particular, assuming local equilibrium between
turbulence production and dissipation Equation (16), Equation (4) becomes:
( )
∂ µt ∂k
= 0 (20)
∂z σk ∂z
Substituting Equations (9), (16) and (18) into Equation (20), we get:
 
∂ (ρcµ k
ϵ2 ∂k) ∂ ρcµ√Ckµ2k duz ∂k ∂ ρuk24∗ √ukk24∗2k duz ∂k
= =  .
∂z σk ∂z ∂z σk ∂z ∂z  σk ∂z
du u∗ 1
Employing the analytical expression of the inlet velocity profile, = , (Equation
dz κ (z+z0)
(10)):
( ) ( )
( )
2 2
∂ ρu
∗ ∂k ∂ ρu∗ ∂k ∂ ρu∗κ ∂k
du = u∗ 1 = (z + z0) = 0 (21)
∂z σ
k dz ∂z ∂z σk κ (z+z
0) ∂z ∂z σk ∂z
which gives:
∂k
(z + z0) = const (22)
∂z
By integrating Equation (22), the following general solution for turbulent kinetic energy
profile is obtained:
k (z) = C1ln (z + z0) + C2 (23)
which differs from Equation (15) since the square root operator disappears. Similarly
to Equation (15), C1 and C2 are constants determined by fitting the equations to the
measured profile of k. For what concerns the profile of turbulent dissipation rate, the
VKI - 9 -
2 THEORY 2.1 Inlet conditions and turbulence model
Table 1: Inlet conditions and turbulence model formulation.
( )
u∗ z+z0
U = ln Equation (10)
κ z0
Inlet conditions
k (z) = C1ln (z + z0) + C2 Equation (23)
3
u∗
ϵ = Equation (12)
κ(z+z0)
2
k
µt
(= ρcµ ϵ √ ) Equation (9)
Turbulence Model ρu4
∗ (Cϵ2−Cϵ1) Cµ 1
Sϵ (z) =
(z+z0)2 κ2 − σϵ Equation (14)
4
u∗
Cµ =
k2 Equation (18)
equilibrium assumption (Equation (16)) and the relation for Cµ (Equation (18)) ensure
that Equation (12) remains valid. The full set of inlet conditions, the turbulence model
formulation and the wall function implementation are summarized in Table 1. The set of
inlet boundary conditions provided by Equations (10), (23) and (12) for velocity, turbulent
kinetic energy and dissipation rate, respectively, represents a consistent extension of the
formulation proposed by Richards and Hoxey [4] to the case of a non-constant turbulent
kinetic energy profile. Indeed, if Equation (23) for and Equation (18) for Cµ are used, the
transport equation for the turbulent dissipation rate is identically satisfied by the source
term Sϵ (z) (Equation (14)), which is independent of the specific form of the inlet profile.
In fact, the equilibrium assumption and the generalization of Cµ make the first term of
Equation (5) (which is the only one affected by the functional variation of k) universal
and equal to:
 2   √ √ 
∂
ρcµ√Ckµk duz ∂√Cµkduz  ∂  ρ uk24∗ k ∂ uk24∗ κ(zk+u∗z0)  ∂ ( ρu∗4 1 )
= = −
u∗
∂z σϵ ∂z ∂z σϵ
κ(z+z0) ∂z ∂z σϵ (z + z0)
( )
4 4
∂ ρu 1 ρu 1
∗ ∗
− = (24)
2
∂z σϵ (z + z0) σϵ (z + z
0)
It should be noted that, when employing the novel turbulent kinetic energy profile (Equa-
tion (23)), the source term in Equation (19) reduces to zero. Finally, it can be observed
that, assuming a constant profile for k, i.e. C1 = 0 in Equation (23), the proposed ap-
proach reduces to a formulation equivalent to the one proposed by Richards and Hoxey
[4], with the difference that the proper value of Cµ is automatically selected via Equation
(18).
The profile proposed by Yang et al. [5] requires the availability of experimental data to
determine the parameters C1 and C2 of Equation (23). This is not always guaranteed, es-
pecially for full-scale measurements. In this case, semi-empirical parameterizations avail-
able in the literature Brost and Wyngaard [10] can be applied for the turbulent quantities.
provided the following expressions for the mean squared fluctuating velocity components:
〈 〉
′
2
u z
= 5 − 4 (25)
2
u h
∗
〈 〉
′
2
v z
= 2 − (26)
2
u h
∗
VKI - 10