J. Wind Eng. Ind. Aerodyn. 115 (2013) 112–120

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Journal of Wind Engineeringand Industrial Aerodynamics

0167-61

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/jweia

Computational modeling of the neutrally stratified atmospheric boundarylayer flow using the standard k–e turbulence model

Franjo Juretic, Hrvoje Kozmar n

Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lucica 5, 10000 Zagreb, Croatia

a r t i c l e i n f o

Article history:

Received 30 December 2011

Received in revised form

27 January 2013

Accepted 31 January 2013Available online 5 March 2013

Keywords:

Neutrally stratified atmospheric boundary

layer flow

Atmospheric turbulence

Reynolds stress

Computational modeling

Steady RANS

Standard k-e turbulence model

05/$ - see front matter & 2013 Elsevier Ltd. A

x.doi.org/10.1016/j.jweia.2013.01.011

esponding author. Tel.: þ385 1 6168 162; fa

ail address: hkozmar@fsb.hr (H. Kozmar).

a b s t r a c t

A novel approach, which focuses on the Reynolds stress in computational simulations of the

atmospheric boundary layer (ABL) flow in an empty domain using the k–e turbulence model, is

presented. A numerical setup mimics the experiments carried out in the boundary layer wind tunnel for

the rural, suburban, and urban terrain exposure. The method accounts for a decrease in turbulence

parameters with height, as observed in full scale. In addition, the paper presents analysis which shows

that the k–e turbulence model is capable of modeling decreasing turbulence parameters with height

and achieves satisfactory accuracy. It is supported with computational results which agree well with

the experimental results. In particular, the difference between the calculated and measured mean

velocity, turbulent kinetic energy and Reynolds stress profiles remains within 710% in most parts of

the computational domain.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The appropriate modeling of the atmospheric boundary layer(ABL) flow is a major prerequisite for a successful computationalstudy on wind loading of structures, air pollutant dispersion, windfarm design and pedestrian comfort. As wind-tunnel ABL simula-tions are already well established, using widely accepted Irwin(1981) and Counihan (1969a, 1969b, 1973) methods, theyincreasingly tend to be complemented with less expensive andtime consuming computational methods.

Even though a recent review (Tamura, 2008) proved theapplicability of Large Eddy Simulation (LES) to unsteady flowphenomena in various wind engineering problems, most simula-tions are still carried out using steady RANS codes available instandard commercial CFD software (O’Sullivan et al., 2011), as e.g.Fluent, OpenFOAM and others. This standard approach usuallyincludes k–e type turbulence closures, wall-functions based onequivalent sand-grain roughness modifications and neutrallystable atmospheric conditions. At the moment, the commonproblems with this approach, which still need to be resolved,are horizontal flow homogeneity and the simulation of turbulenceparameters.

ll rights reserved.

x: þ385 1 6156 940.

In the past, researchers proposed measures to improve hor-izontal homogeneity of the computationally simulated ABL flow(Richards and Hoxey, 1993; Blocken et al., 2007a; Franke et al.,2007; Hargreaves and Wright, 2007; Yang et al., 2008, 2009;O’Sullivan et al., 2011). In particular, Blocken et al. (2007a) arguedthat errors are caused by wall functions inconsistent with inflowboundary profiles. Riddle et al. (2004) indicated that errors couldbe reduced by using a second order turbulence closure model.Yang et al. (2009) developed new, more general and moreconsistent inflow profiles. Blocken et al. (2007a) suggested alter-native measures to reduce erroneous streamwise gradients,indicated that sensitivity tests in an empty computational domainare of critical importance, and advised to always report not onlythe inlet profiles but also the incident flow profiles obtained inthe empty domain because they characterize the real flow towhich the building models are subjected. In addition, Blockenet al. (2007b) indicated the ‘‘modified-for-roughness’’ wall func-tions currently implemented in many commercial CFD codes canbe unsuitable for simulation of the ABL flow. Several authors notethe importance of boundary conditions at the top of the domain(Richards and Hoxey, 1993; Blocken et al., 2007a; Franke et al.,2007; Hargreaves and Wright, 2007; Yang et al., 2009). Therefore,as Richards and Hoxey’s (1993) approach assumes a constantprofile of turbulent kinetic energy, which is not always consistentwith full-scale or wind-tunnel measurements, Yang et al. (2009)proposed a horizontally homogeneous turbulent kinetic energyprofile, varying with height, which remains preserved along the

Nomenclature

d displacement heightk(z) turbulent kinetic energypðzÞ kinematic air pressurepinlet inlet pressurepoutlet outlet pressureuðzÞ,vðzÞ,wðzÞ instantaneous velocity components in the x-, y-,

z-direction, respectivelyuðzÞ,vðzÞ,wðzÞ mean velocity components in the x-, y-, z-direc-

tion, respectivelyu0ðzÞ,v0ðzÞ,w0ðzÞ fluctuating velocity components in the x-, y-, z-

direction, respectivelyu0w0 ðzÞ Reynolds stressuref mean reference velocity at the reference height zref

uref_CFD calculated mean reference velocity at the referenceheight zref

uref_EXP measured mean reference velocity at the referenceheight zref

ut friction velocityx distance in the main flow directiony spanwise distance from the test section center planez vertical distance from the wind-tunnel floor and the

bottom of the computational domain

zc distance of the center in the near-wall cell fromthe wall

zref reference heightz0 aerodynamic surface roughness lengthA, B constantsCD constantCm, C1, C2, sk, se constants of the k–e turbulence modelD non-dimensional parameterIu,Iv,Iw turbulence intensity in the x-, y-, z-direction,

respectivelyP(z) production termPc turbulence production P in the near-wall cellResk(z) residual of turbulent kinetic energy k equationa power law exponente(z) turbulence dissipationec turbulence dissipation e in the near-wall cellk Von Karman constantm dynamic laminar viscositynt(z) kinematic turbulent (eddy) viscosityr air densitytxzðzÞ shear stresstw shear stress at the wall

F. Juretic, H. Kozmar / J. Wind Eng. Ind. Aerodyn. 115 (2013) 112–120 113

computational domain. Gorle et al. (2009) achieved flow homo-geneity and decreasing turbulent kinetic energy with height bymodifying the se constant of the standard k–e turbulence modelinto a function depending on height. Parente et al. (2011a, 2011b)added a source term to the transport equation of turbulent kineticenergy and dissipation to ensure equilibrium between produc-tion and dissipation, and suggested a novel rough wall functionformulation.

At the moment, it seems that several issues require furtherinvestigations: longitudinal flow homogeneity for suburban andurban terrains, Reynolds stress and wall-function treatment.Hence, in this study, a computational approach using the standardk–e turbulence model has been developed in order to correctlyreproduce the ABL wind-tunnel simulation developing aboverural, suburban, and urban type terrains. In addition to commonlyreproduced mean velocity and turbulent kinetic energy profiles,this work focuses on decreasing Reynolds stress profiles withheight as well. Hence, the paper also presents analysis of theapplicability of the k–e turbulence model in case of ABL flow withvarying turbulence parameters. The paper is organized as follows.Section 2 presents the wind-tunnel methodology and importantdata which was used to validate the computational methodology.The governing equations for the ABL flow and the analysis of thek–e model, when subjected to ABL flow, are given in Section 3.Section 4 describes the geometry and the developed methodologyfor prescribing boundary conditions. The comparison of thecomputational results with the measured data is given inSection 5. Concluding remarks and suggestions for future workare the content of Section 6.

2. Wind-tunnel experiments

Experimental simulations of rural, suburban, and urban ABLflow were carried out in a 1.80 m high, 2.70 m wide and 21 mlong test section of the low-speed boundary layer wind tunnel atthe Technische Universitat Munchen (TUM) to validate theperformed computations. This Gottingen type wind tunnel is

operable in a closed circuit and an open circuit mode with asuction configuration. Flow uniformity is achieved using a hon-eycomb, four sets of screens and a contraction. A zero pressuregradient can be obtained along the wind-tunnel test section byadjusting the ceiling height. Structural models are usually placedon a turntable, whose center is positioned 11.3 m downwind fromthe contraction. The blower is driven by a 210 kW electric motor,which allows velocity regulation from 1 m/s to 30 m/s. Moredetails on design of this wind tunnel can be found in (Kozmar,2011a).

The simulation technique was a modification of the originalCounihan (1969a, 1969b, 1973) method based on the use ofquarter-elliptic, constant-wedge-angle spires and a castellatedbarrier wall, followed by a fetch of surface roughness elements. Inparticular, the experimental results reported in (Kozmar, 2011b)were used for validation of the computational results obtained inthis study. These experiments were performed using truncatedvortex generators developed for part-depth ABL wind-tunnelsimulations, together with a castellated barrier wall and surfaceroughness as originally suggested by Counihan (1969a, 1969b,1973). Moreover, the performance of truncated vortex generatorswas further validated in comparison with classical full-sizeCounihan vortex generators for urban, suburban, and rural terrainexposures, (Kozmar, 2010, 2011c; Kozmar, 2012), respectively.

Velocities were measured using a triple hot-wire probe DAN-TEC 55P91 down to 10 cm height. Velocity signals were sampledat 1.25 kHz using a 12-bit digitizer Data Translation DT2821 witha record length of 150 s.

Computational simulations carried out in this study arecompared with the experimental results reported at the wind-tunnel scale, as presented in Figs. 1, 2, 3, and 4. In particular, theprofiles of longitudinal mean velocity uðzÞ, turbulent kineticenergy kðzÞ, and Reynolds stress �u’w’ðzÞ profiles were used forvalidation. Table 1 shows the parameters included in the pre-sentation of results.

Reported mean velocity profiles uðzÞ agree well with the powerlaw for exponents a¼0.16, 0.20, and 0.37 in rural, suburban, andurban ABL wind-tunnel simulations, respectively. Turbulence

Fig. 1. Recorded uðzÞ mean velocity profiles in (a) rural, (b) suburban and (c) urban ABL wind-tunnel simulations.

Fig. 2. Recorded turbulent kinetic energy kðzÞ profiles in (a) rural, (b) suburban and (c) urban ABL wind-tunnel simulations.

Fig. 3. Recorded �u’w’ðzÞ Reynolds stress profiles in (a) rural, (b) suburban and (c) urban ABL wind-tunnel simulations nondimensionalized using respective friction

velocities.

F. Juretic, H. Kozmar / J. Wind Eng. Ind. Aerodyn. 115 (2013) 112–120114

intensity profiles Iu, Iv, and Iw in longitudinal, lateral, and verticaldirection, respectively, which were used to calculate turbulentkinetic energy kðzÞ, proved to be in good agreement with full-scaleconditions compiled in ESDU 74031 (1974) for respective terrainexposures, as reported in (Kozmar, 2011b). Reynolds stress close

to surface is slightly larger than respective friction velocity, and itshows a trend of constant values in the near-ground region that issimilar to the Prandtl constant-flux-layer observed in full scale upto 100 m, Garratt (1992) and Holmes (2001). Therefore, in ESDU74031 (1974) it was reported that for rougher terrains this region

Fig. 4. Recorded turbulent kinetic energy kðzÞ to �u’w’ðzÞ Reynolds stress ratio in (a) rural, (b) suburban and (c) urban ABL wind-tunnel simulations.

Table 1Power-law exponent a, reference height zref, reference velocity uref , and friction

velocity ut in created ABL wind-tunnel simulations.

ABL simulation a zref (m) uref (m/s) ut (m/s)

Rural 0.16 0.202 14.97 1.11

Suburban 0.20 0.202 13.48 1.10

Urban 0.37 0.202 10.14 1.43

F. Juretic, H. Kozmar / J. Wind Eng. Ind. Aerodyn. 115 (2013) 112–120 115

of constant shear stress extends higher up in the atmosphere thanfor smoother terrains, that compares well with trends observedhere. In addition, Iyengar and Farell (2001) and Aubrun et al.(2005) observed this phenomenon in their wind-tunnel simula-tions as well. Apart from these parameters, turbulence integrallength scales and power spectra of velocity fluctuations wereadditionally used in Kozmar (2011b) to fully justify the similarityof created wind-tunnel ABL simulations with respective full-scaleconditions. Fig. 4 shows the ratio of turbulent kinetic energy kðzÞ

to �u0w0 ðzÞ Reynolds stress is nearly-constant with the averagevalue 4.76. This is not a universal constant, it is wind-tunneldependent and reported to have different values in previousstudies. Furthermore, this ratio is an important parameter fordetermining the constants of the k–e turbulence model. Inaddition, this implies that the constants of the k–e turbulencemodel need to be determined dependent on the measured data,which shall be available prior to the simulation. The next sectionpresents the governing equations of the ABL flow and analysis ofthe k–e model with respect to the measured data presented in thissection.

3. Governing equations

3.1. RANS and k–e equations

The turbulent flow in the wind tunnel is predominantly 1Dflow with the dominant flow velocity along the wind tunnel testsection (x-direction). For this case, the full set of RANS equationssimplifies to the following set of governing equations:

@

@zntðzÞ

@uðzÞ

@z�@pðzÞ

@x¼ 0, ð1Þ

@

@z

ntðzÞ

sk

@kðzÞ

@zþPðzÞ�eðzÞ ¼ 0, ð2Þ

@

@z

ntðzÞ

se@eðzÞ@zþC1

PðzÞeðzÞkðzÞ

�C2eðzÞ2

kðzÞ¼ 0: ð3Þ

The turbulent viscosity is calculated from turbulent kineticenergy kðzÞ and turbulence dissipation eðzÞ, (Jones and Launder,1972)

ntðzÞ ¼ CmkðzÞ2

eðzÞ : ð4Þ

The production term P(z) is modeled as follows:

PðzÞ ¼ �u0w0 ðzÞ@uðzÞ

@z¼ ntðzÞ

@uðzÞ

@z

� �2

, ð5Þ

by introducing the Boussinesq (1877) hypothesis �u0w0 ðzÞ ¼

�nt@uðzÞ=@z. The model has five constants, i.e., Cm, sk, se, C1, andC2. Their standard values (Jones and Launder, 1972) are Cm ¼ 0:09,sk ¼ 1, se ¼ 1:3, C1 ¼ 1:44 and C2 ¼ 1:92.

The wind-tunnel results presented in Fig. 4 show that the ratiobetween turbulent kinetic energy kðzÞ and dominant shear stressu’w’ðzÞ is fairly constant for all measurement points and for allterrain types, and it is used for determining the constant Cmaccording to the following expression, as given in Pope (2000)

1ffiffiffiffiffiffiCm

p ¼kðzÞ

u0w0 ðzÞ: ð6Þ

Eq. (6) is valid for homogeneous turbulence, when turbulencedissipation is equal to turbulence production. It is derived bymoving eðzÞ in Eq. (4) to the left-hand side and substituting it withthe expression given in Eq. (5). The product of ntðzÞ and Eq. (5) isequal to the square of the Reynolds stress present in Eq. (6). Thevalue of Cm was calculated based on wind-tunnel measurementscarried out in an empty test section without building models andas such will need to be further investigated for cases withstructures present. Therefore, the reported governing equationsare supposed to admit this behavior in order to reproduce windcharacteristics within ABL wind-tunnel simulation. In addition,the measurements show that u0w0 ðzÞ and kðzÞ are fairly constantnear the bottom wall, and their variation is linear in upper partsof the flow. Regarding the velocity profile, Dyrbye and Hansen(1997) indicated that velocity profiles throughout an entire depthof the ABL can be represented by a power-law solution, whileThuillier and Lappe (1964) reported that logarithmic law can beassumed to be valid within the inertial sublayer commonly takento extend to about 100 m height, e.g. Holmes (2001), Garratt(1992). Hence, in order to theoretically validate the behavior ofthe proposed mathematical model, four important cases wereidentified and studied. The first two cases assume logarithmicvelocity profile, with constant shear stress and with linearlyvarying shear stress, respectively. The second two cases assume

F. Juretic, H. Kozmar / J. Wind Eng. Ind. Aerodyn. 115 (2013) 112–120116

power-law velocity profile, with constant shear stress and withlinearly varying shear stress, respectively.

Fig. 5. Residual in turbulent kinetic energy kðzÞ calculated using the k–e turbu-

3.2. Logarithmic velocity profile

The first case provides insight into the behavior of the k–emodel in case of logarithmic velocity profile given in the form

uðzÞ

ut¼

1

kln

zþz0

z0, ð7Þ

where ut ¼ffiffiffiffiffiffiffiffiffiffiffitw=r

pis the shear velocity, k is von Karman constant

with the value of 0.41, and tw is a shear stress at the wall. Bysubstituting the derivative of the log-law velocity profile in theBoussinesq hypothesis yields the following expression for theturbulent viscosity:

ntðzÞ ¼ kðzþz0Þu0w0 ðzÞ

ut: ð8Þ

lence model in comparison with assumed analytical solution for the log-law

velocity profile.

3.3. Constant shear stress u0w0 ðzÞ¼ut2

This case has the analytical solution and it is the well-knownRichards and Hoxey (1993) solution. An equation for turbulentkinetic energy kðzÞ has the solution

kðzÞ ¼ut2ffiffiffiffiffiffi

Cmp ð9Þ

that is in agreement with the measured data which show thatshear stress and turbulent kinetic energy are constant higher upto some distance from the wall. The solution for turbulencedissipation is

eðzÞ ¼ ut3

kðzþz0Þ, ð10Þ

with the constraint imposed on the selected values for thecoefficients (Richards and Hoxey, 1993)

se ¼k2

ðC2�C1ÞffiffiffiffiffiffiCm

p : ð11Þ

3.4. Linear variation of shear stress

However, as shear stress and turbulent kinetic energy remainconstant only up to several tens of meters (e.g. ESDU 74031,1974), it seems important to check whether the turbulence modelresolves the case of decreasing turbulence parameters withheight, as observed in full scale. The variation of shear stresswith height, according to the measurements, can be assumed tobe governed by the function

u0w0 ðzÞ ¼ AzþB, ð12Þ

where A and B can be determined by fitting the line to themeasured data given in Fig. 3. According to Eq. (6), the turbulentkinetic energy shall have the following solution:

kðzÞ ¼AzþBffiffiffiffiffiffi

Cmp : ð13Þ

This enforces the solution for turbulence dissipation in theform

eðzÞ ¼ CmkðzÞ2

ntðzÞ¼

utðAzþBÞ

kðzþz0Þ, ð14Þ

When the solution for kðzÞ is substituted in Eq. (2) it results inthe following residual:

ReskðzÞ ¼kAðAzþBþAðzþz0ÞÞ

utsk

ffiffiffiffiffiffiCm

p : ð15Þ

By using the ut and z0 from the measured data, and finding thebest fit of the shear stress, it can be seen the expected residual inEq. (15) shall be less than 3% throughout the entire ABL simula-tion for all terrain types, as presented in Fig. 5. Resk(z) isnormalized using ut3=z0 to create a non-dimensional quantity.

3.5. Power law velocity profile

For taller constructions reaching deeply into the ABL, meanvelocity can be well represented by the power-law velocity profilegiven by the function

uðzÞ

uref¼

z

zref

� �a: ð16Þ

In this case, according to the Boussinesq approximation, theturbulent viscosity shall satisfy the following form:

ntðzÞ ¼zu0w0 ðzÞ

uðzÞa: ð17Þ

3.6. Constant shear stress u0w0 ðzÞ¼ut2

The profile for turbulent kinetic energy given in Eq. (9) stillsatisfies the Eq. (2) as that was also the case with the log-lawvelocity profile. The profile for turbulence dissipation is obtainedby substituting the profiles for turbulent viscosity (17) andturbulent kinetic energy (9) into Eq. (4), and it has the form

eðzÞ ¼ uðzÞau0w0 ðzÞ

z: ð18Þ

3.7. Linear variation of shear stress

As for the logarithmic law solution, the analysis accounting fora decrease in turbulence parameters with height is performed forthe power-law solution as well. The assumed profile for turbulentkinetic energy is given in Eq. (13), and the turbulence dissipationshall have the form given in Eq. (18), while the residual of the k

F. Juretic, H. Kozmar / J. Wind Eng. Ind. Aerodyn. 115 (2013) 112–120 117

equation is

ReskðzÞ ¼Au0w0 ðzÞ

uðzÞask

ffiffiffiffiffiffiCm

p �Au0w0 ðzÞ

uðzÞsk

ffiffiffiffiffiffiCm

p þzA2

uðzÞask

ffiffiffiffiffiffiCm

p : ð19Þ

For the power-law velocity profile, the residual of turbulentkinetic energy kðzÞ, given in Eq. (19) and calculated using themeasured data, is less than 2% throughout an entire ABL simula-tion for all terrain types, as presented in Fig. 6. Resk(z) in Fig. 6 isnormalized using ut3=z0 to create a non-dimensional quantity.

The above analysis shows that the k–e model is capable ofreproducing measured data with satisfactory accuracy, whensubjected to boundary conditions which enforce variation ofshear stress within the domain.

4. Numerical setup

Simulations were carried out by using OpenFOAM, an opensource, freely available CFD toolbox. The computational domainfor each terrain type was a 2D cut through the symmetry plane ofthe wind-tunnel test section. Calculations were performed at thewind-tunnel scale and results were sampled at 21.9 m downwindfrom the inlet. The geometry of the computational domain isshown in Fig. 7.

The slope of the ceiling was set to comply with values used inABL wind-tunnel simulations for all three different terrain types.The value of pressure is set to be constant with height at inlet andoutlet boundaries, and the streamwise gradient of the uðzÞ, kðzÞ,eðzÞ fields is set to zero at both inlet and outlet boundaries. Itneeds to be mentioned that these boundary conditions do notguarantee homogeneity of flow conditions in the longitudinaldirection because the height of the upper wall changes through-out the domain. However, the focus was to obtain agreement withthe experimental results at the sampling position. The similarapproach has been commonly used in wind tunnels, wheredifferent types of simulation hardware are employed to simulate

Fig. 6. Residual of turbulent kinetic energy kðzÞ calculated using the k–e turbu-

lence model in comparison with assumed analytical solution for the power-law

velocity profile.

Fig. 7. Geometry and dimensions

the ABL, and it is required to have a sufficient fetch to allow fordeveloping of the flow.

The walls are treated as rough walls, for which the followingwall-function was implemented into the research code

tw ¼uðzcÞutk

lnðzcþz0=z0Þ, ð20Þ

where the value of ut is calculated as ut ¼ffiffiffiffiffiffiCm

4p ffiffiffiffiffiffiffiffiffiffi

kðzcÞp

.The epsilon in the near-wall cell is set to

ec ¼ut3

kðzcþz0Þ, ð21Þ

while the normal gradient of turbulent kinetic energy is set tozero at the wall. The turbulence production P(z) in the wallfunction is set to

Pc ¼ twruðzcÞ, ð22Þ

as it has been commonly used in the OpenFOAM code.The values of z0 at the bottom wall were calculated to force the

code to re-create the same velocity and shear stress at the wall asthe measured values, using the following equation:

z0 ¼zc

eD�1, ð23Þ

where zc is the distance of the center in the near-wall cell fromthe wall, D is uðzcÞk=ut, and uðzcÞ is the measured average velocityat the given distance.

The coupled system is solved using second-order accuratediscretization procedure (Jasak, 1996 and Juretic, 2004) and theequations are coupled in the spirit of the SIMPLE (Semi-Implicitprocedure for Pressure-Linked Equations) algorithm (Patankarand Spalding, 1972). The equations are solved until the scaledresidual is reduced below 10�7 such that the solution error of thesystem of algebraic equations is negligible compared to thediscretization error, as well as the turbulence model error. Theconvection terms are approximated using the Gamma differen-cing (Jasak, 1996 and Jasak et al., 1999) scheme for all convectionterms.

The velocity at the reference point is regulated by adjustingthe pressure at the inlet, until the calculated velocity matched themeasured velocity within 0.1%. The pressure is adjusted by using

pinletþ ¼CD

2ðuref_EXP

2�uref_CFD

2Þ, ð24Þ

where uref_EXP is the measured velocity at the reference height,uref_CFD is the calculated velocity at the reference height and CD is

of the computational domain.

Table 2Aerodynamic surface roughness length z0 at the bottom and top walls of the

computational domain, and the angle of the top wall applied in the computational

domain.

Case z0 at the bottom wall (m) z0 at the top wall (m) a

Rural 1:70� 10�3 10�9 0.2

Suburban 4:00� 10�3 10�5 0.3

Urban 1:55� 10�2 5� 10�4 0.8

F. Juretic, H. Kozmar / J. Wind Eng. Ind. Aerodyn. 115 (2013) 112–120118

evaluated as

CD ¼2ðpinlet�poutletÞ

uref_CFD2

: ð25Þ

The pressure at the inlet is adjusted every 1000 iterations ofthe SIMPLE loop to stabilize the calculations and to allow the codeto converge before the next change of inlet pressure is applied.

The thickness of the near-wall grid layer is set approximatelythree times larger than the aerodynamic surface roughness lengthz0, to ensure that the distance of the first grid node from the wallis greater than z0. The discretization error is controlled bysystematically refining the mesh until the differences in thecalculated profiles were less than 1%.

5. Results and discussion

The calculations were performed with the same set of coeffi-cients for all three boundary layer types, i.e. Cm ¼ 0:044, C1 ¼ 1:44,C2 ¼ 1:92, sk ¼ 1, se ¼ 1:67. Therefore, Cm was selected to fit theratio between the measured shear stress and turbulent kineticenergy, given in Eq. (6), while se was calculated from Eq. (11).

Table 2 shows the angles of the top wall and the aerodynamicsurface roughness length z0 applied at the bottom and the upperwall for all terrain types.

The values of z0 at the bottom wall were calculated fromEq. (23) and adjusted until the best fit was obtained, and the

Fig. 8. Mean velocity profiles in (a) rural, (b) suburban and (c) urban ABL

Fig. 9. Reynolds stress profiles in (a) rural, (b) suburban and (c) urban AB

values at the top wall were adjusted to achieve the best fit ofshear stress with the experimental data.

5.1. Mean velocity profiles

Fig. 8 show the comparisons between the measured and thecalculated velocity profiles for all three types of boundary layers.Commonly, the lower 100–200 m of the ABL flow are a primarilyfocus of interest for a wind engineer, as this is the height range ofmost structures (Holmes, 2007). Therefore, the results in thisstudy are presented up to 1 m wind-tunnel scale that correspondsto 292 m, 273 m and 269 m full-scale in the rural, suburban andurban ABL simulations (Kozmar, 2011b), respectively.

The profiles recorded in the wind tunnel and calculated usingthe approach suggested in this study are in excellent agreementat z¼0.2 m, since the pressure at inlet is adjusted to match themeasured velocity at that height. The difference between thecalculated and the measured profiles is largest for the rural ABL.The authors believe that the error is mainly caused by the wallfunction which is not accurate in diffuser-like geometries aspresent in this case. Furthermore, the value of z0 cannot becalculated explicitly by using Eq. (23) at the top wall, and thevalues given in Table 2 are adjusted to get a reasonable fit. Inaddition, the accuracy is also affected by the errors in theequations for kðzÞ and eðzÞ which are high at the bottom wall,and they influence the velocity profile via the turbulent viscosity.

simulation nondimensionalized using respective reference velocities.

L simulation nondimensionalized using respective friction velocities.

Fig. 10. Turbulent kinetic energy profiles in (a) rural, (b) suburban and (c) urban ABL simulation.

F. Juretic, H. Kozmar / J. Wind Eng. Ind. Aerodyn. 115 (2013) 112–120 119

However, in all configurations the computational results remainwithin 710% of the experimental results in most parts ofthe flow.

5.2. Reynolds stress profiles

Fig. 9 presents Reynolds stress profiles in rural, suburban, andurban ABL simulation nondimensionalized using respective fric-tion velocities. The results show that shear stress reduces withheight and calculated profiles flatten near the bottom wall. Ingeneral, in full scale the turbulence parameters die out at theground surface, which is not the case in this study, as theboundary conditions applied at the wall for the used standardk–e turbulence model are not consistent with this behavior.

The differences between the calculated and the measuredresults are largest for the rural ABL, in the region near the bottomwall and it is attributed to the turbulence model which cannotresolve the jump in the shear stress observed in the measureddata due to the lack of source terms which sense the effects of theboundary roughness. Despite that, the discrepancy between themeasured and the calculated profiles remains within theacceptable 10% threshold in most parts of the flow.

5.3. Turbulent kinetic energy profiles

Comparisons between the measured and the calculated valuesof turbulent kinetic energy are given in Fig. 10 for all three typesof boundary layers.

All profiles show that kðzÞ reduces with height as observed inthe measured data, and the profiles are within 10% agreement inmost parts of the computational domain. The discrepancybetween the measured and the calculated data is largest at thebottom wall and it is mainly caused by the wall function and theturbulence model which are not valid for cases where the ratiobetween shear stress and turbulent kinetic energy is not constant.

6. Concluding remarks

A novel approach for determining the boundary conditions hasbeen developed to allow for accurate predictions of Reynoldsstress in computational simulations of ABL flow using the k–eturbulence model. A numerical setup attempted to mimic experi-ments carried out in the boundary layer wind tunnel for rural,suburban, and urban terrain exposure. Computational results

agree very well with experimental results. In particular, thedifference between the calculated and measured mean velocity,turbulent kinetic energy and Reynolds stress profiles is less than710% in most parts of the computational domain. The observeddiscrepancies are possibly due to the wall function which is notaccurate in diffuser-like geometries as present in this case, andthe uncertainty in the measurements data. Therefore, the accu-racy is also affected by the errors in the equations for kðzÞ and eðzÞwhich are highest at the bottom wall, as expected, and theyinfluence the velocity profile via the turbulent viscosity. Inaddition, the turbulence model cannot resolve the jump in theshear stress recorded in the measured data due to the lack ofsource terms which sense the effects of the boundary roughness.

It needs to be mentioned that this procedure is not yet amature approach for everyday use. Future work would attempt tomake the proposed procedure more appropriate for Computa-tional Wind Engineering (CWE) calculations by adding a set ofsource terms in the momentum equation as well as improvingwall-boundary conditions.

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