Numerical simulation of the flow across an asymmetric street intersection

J. Franke�, W. Frank

�

ABSTRACT: A numerical simulation of the flow across an asymmetric street junction formed by four ringsof buildings has been performed with the commercial flow solver FLUENT V6.1 using the Reynolds averagedNavier-Stokes equations with six different turbulence models. The turbulence models comprised four lineartwo-equation models and a differential Reynolds stress model with two formulations for the pressure-strain.The simulations were done for two directions of the approach flow using three systematically refined grids foreach case. The results on the finest grid were shown to differ only slightly from the next coarser grid. Theseresults were then compared with velocity and turbulence measurements that are available from the CEDVALdatabase. The best results for the velocity are obtained with the realizable and standard

� � �model while the

Reynolds stress models perform best for the Reynolds stresses and slightly worse for velocity.

1 INTRODUCTION

The detailed knowledge of the interaction of the atmo-spheric boundary layer flow with clusters of obstacleslike buildings, trees or other obstructions is impor-tant for several reasons. Understanding and predictingthe forces and loadings on the obstructions requiresknowledge of the complex flow patterns around theobstacles and within the streets. This information isalso needed to assess the mechanical wind effects onpedestrians and to determine the dispersion of pollu-tants or contaminants.

Besides the experimental investigation of theflow within the built environment in boundarylayer wind tunnels, Computational Fluid Dynam-ics (CFD) is increasingly used for the numericalsimulation of the flow around clusters of build-ings. [Lien & Yee (2004)] computed the flow overa matrix of cubes solving the Reynolds AveragedNavier Stokes (RANS) equations with the standardand [Kato & Launder (1993)]

� � �turbulence model.

While there was a good agreement between mea-sured and computed velocities, the turbulent kineticenergy was underpredicted. A similar case has beenstudied by [Cheng et al. (2003)]. Besides the RANSequations with the standard

� � �turbulence model

they used Large Eddy Simulation (LES) with differ-ent subgrid scale models. While RANS computa-tions solve for the statistically steady flow field di-rectly, LES computes the unsteady flow field formedby the larger eddies and only models the influence ofthe unresolved eddies. [Cheng et al. (2003)] report

much better agreement for the LES with the mea-suerments than for the RANS with standard

� � �

model. LES was also used by [Hanna et al. (2002)]for the flow over a matrix of cubes with reason-able agreement between simulation results and exper-iments. While LES leads in general to better predic-tions than RANS its costs are currently too high forengineering applications. Therefore industrial scenar-ios are normally treated with the RANS approach,e.g. [Ferreira et al. (2002), Richards et al. (2002),Ketzel et al. (2002), Westbury et al. (2002)]. None ofthe cited works uses higher-order turbulence modelsand many did not analyse the influence of the grid res-olution on the results.

In the present work we therefore analyse the per-formance of the RANS approach with six turbulencemodels, including two differential Reynolds stressmodels, for the prediction of the flow over a mod-erately complex cluster of four rings of buildingsforming an asymmetric street junction. For this casevelocity and turbulence measurements are availablefor three directions of the approach flow from theCEDVAL database (www.mi.uni-hamburg.de/cedval)[Leitl (2000)]. The simulations are performed withthe commercial flow solver FLUENT V6.1 for two di-rections of the approach flow, using three systemati-cally refined grids in each case. The grid dependanceof the solutions is analysed only qualitatively. Thecomputational results for the mean velocities and theturbulence on the finest grid are then compared withthe experimental data.

Dr.-Ing. Jorg Franke, Department of Fluid- and Thermodynamics, University of Siegen, e-mail franke@ift.uni-siegen.de�Prof. Dr.-Ing. Wolfram Frank, Department of Fluid- and Thermodynamics, University of Siegen, e-mail frank@ift.uni-

siegen.de

EACWE4 — The Fourth European & African Conference on Wind EngineeringJ. Naprstek & C. Fischer (eds); ITAM AS CR, Prague, 11-15 July, 2005, Paper #138

1

2 NUMERICAL METHOD

2.1 Basic equations

The basic equations describing a turbulent, neutrallystratified flow are the Navier-Stokes equations withconstant density � and dynamic viscosity � . Whenthese equations are averaged over time one arrives atthe RANS equations for a statistically steady turbu-lent flow. These are mass conservation,

� � �� � � � (1)

and momentum conservation,

� � � � � � � �� � � � � �

� � � � �� � � �

� � � � � � � � � � � � �� � � � (2)

Here,� �

denotes the velocity components of themean, i.e. time averaged velocity vector and

�is

the mean pressure. � � � � �are the Reynolds stresses

containing the time averaged correlations of velocityfluctuations � �

. The Reynolds stresses have to be de-termined to close the system of equations (1) and (2).

In this work we use the linear eddy viscosity ap-proximation for the Reynolds stresses and differentialsecond-moment modelling, where additional equa-tions are solved for each of the Reynolds stresses.Contrary to that the deviatoric part of the Reynoldsstresses is approximated by analogy with the molecu-lar stresses in linear eddy viscosity modelling,

� � � � � � � ! # � � � � %

� � �� � � �

� � �� � � � (3)

Here,! '� � � � �

is the turbulent kinetic energy,# � �the Kronecker delta function and � % the turbulent

or eddy viscosity. The turbulent viscosity is computedfrom the turbulent kinetic energy

!and a measure of

its dissipation, for each of which an additional trans-port equation is solved. Therefore these models arecalled two-equation turbulence models.

The modelled equation for the turbulent kineticenergy

!is

� � � ! � � �� � �

�� � � ) *

� !� � � � , * � / * � (4)

, * is the production of!

and defined as

, * � � � � � � � � �� � � � (5)

For all two-equation models that are used in thiswork, , * is computed from

, * � % 3 � (6)

where 3 is the modulus of the mean rate-of-straintensor,

3 � 3 � � 3 � � 3 � � 4

�� � �

� � � �� � �

� � � � (7)

In Equation (4) ) * denotes the diffusion coef-ficient of

!and

/ * its dissipation. These quanti-ties have different definitions for the different two-equation turbulence models that are used. Altogetherfour models are used of which three belong to theclass of

! � 8models and the fourth is the shear stress

transport! � ;

model of [Menter (1994)]. All! � 8

models solve a modelled transport equation for thedissipation

8in addition to Equation (4).

� � � 8 � � �� � �

�� � � ) >

� 8� � � � , > � / > � (8)

Here, ) > is the diffusion coefficient of8

and , >and

/ > its production and dissipation, respectively.Their definitions are listed in Table 1 for the two-equation turbulence models which are briefly intro-duced in the following.

Table 1 Definitions for ? A C turbulence modelsmodel

/ * ) * ) > , > / > � %SKE � 8 � � � % D F * � � � % D F > G ' , * 8 D ! G � � 8 � D ! � G I ! � D 8RNG � 8 J � � � � % � J � � � � % � G ' , * 8 D ! K G � � � G I M O � 4 � M D M Q � D � 4 � S M O � U � 8 � D ! � G I ! � D 8RKE � 8 � � � % D F * � � � % D F > � G ' 3 8 � G � 8 � D � ! � W X 8 � � G I ! � D 8

Table 2 Constants for ? A C turbulence modelsmodel F * F > G ' G � G I Z [SKE 1.3 1.0 1.44 1.92 0.09 0.4187 9.793RNG – – 1.42 1.68 0.0845 0.4187 9.793RKE 1.0 1.2 \ ^ ` K � � b M D � M � e � U

1.94 D K g Q � g j ! k D 8 U

0.4187 9.793

The Fourth European & African Conference on Wind Engineering, Paper #138

2

� Standard � � �(SKE) model

The SKE model of [Launder & Spalding (1972)]is still widely used in CWE despite its stagnationpoint anomaly, which leads to an overprediction of�

in stagnant flow regions. Despite this shortcom-ing it yields reasonable results for the flow withinclusters of obstacles, e.g. [Lien & Yee (2004)].The constants used in this model are given in Ta-ble 2.� ReNormalization Group (RNG) � � �

modelThe RNG model of [Yakhot & Orszag (1986)]mainly differs from the SKE model in the equationfor

�. The parameter � in the dissipation term of

�is defined as

� � � � � (9)

with from Equation (7). The constants are � � � � � �

and � � � � � �. This modification leads to a

smaller� � in regions of large strain rate and there-

fore to lower values of�, thus alleviating the stag-

nation point anomaly.Finally the coefficient

�in the diffusion coeffi-

cients for�

and�, see Table 1, has the value� � � � ! �

.� Realizable � � �(RKE) model

The RKE model of [Shih et al. (1995)] takes two ofthe three realizability conditions for the Reynoldsstresses into account. As the Reynolds stressesform a real valued tensor all its invariants must benon negative, e.g. [Schumann (1977)]. The RKEmodel fulfills the condition that the normal stressesare non negative and that the Schwarz inequalityholds. The first condition leads to a limitation ofthe production of

�at high rate of strain and there-

fore reduces the stagnation point anomaly. To ful-fill the realizability conditions " $ is not constantbut defined as given in Table 2.

% � � � � � is aconstant and

'is defined as

' ( * ( * , . ( * . ( * � (10)

where . ( *is the mean rate-of-rotation tensor,

. ( * �

�/ 1 (

/ 2 * � / 1 */ 2 ( � (11)

% 6is defined as

% 6 7 8 : < =�

� > ? : : < = 7 8 ( * * @ @ (

( * ( * � (12)

The dissipation of�

is also changed, cf. Table 1.A B � D is the kinematic viscosity. Additionally," E is no longer constant but a function of � which

is computed from Equation 9. The remaining con-stants are given in Table 2.

As has been said before, the fourth two-equationmodel also solves the transport equation for

�, Equa-

tion 4. The second equation is now for the specificdissipation

F H � � �.

� Shear Stress Transport (SST) � � Fmodel

The SST model of [Menter (1994)] combines thestandard

� � Fmodel of [Wilcox (1988)] with

the SKE model and also takes the transport ef-fects of the principal turbulent shear stress into ac-count through a modified turbulent viscosity for-mulation. It has been shown to perform very wellfor adverse pressure gradient aerodynamic flowsbut was not often applied for building aerodynam-ics, see e.g. [Westbury et al. (2002)]. The blend-ing of the standard

� � Fand the SKE model leads

to many equations that are necessary for a detaileddescription. Due to space limitations we do notprovide the detailed model here but refer to theoriginal work of [Menter (1994)] and the manualof [FLUENT (2003)].

All linear two-equation models are known tohave shortcomings and deficiencies wherever stresstransport and normal stresses are important, lead-ing to poor predictions in separating and reattach-ing flows [Hanjalic (1999)]. While non linear two-equation models are able to render improved results,full second-moment modelling is the most generalapproach for the closure of Equations (1) and (2).Within that approach a transport equation is solvedfor each of the Reynolds stresses. These modelledequations are

/ O D 1 @ P ( P * R/ 2 @

S U W

// 2 @ B , B Y

Z @/ P ( P *

/ 2 @ (13)

� P ( P * / 1 */ 2 @ , P * P @

/ 1 (/ 2 @

] ^ U *

, _/ P (

/ 2 * ,/ P *

/ 2 (a U W

� �� D � e ( * �

where the model of [Lien & Leschziner (1994)]for the turbulent diffusive transport has been alreadytaken into account, as well as the approximation of thedissipation tensor with the scalar dissipation

�. " ( *

denotes the convective term andf ( *

the shear stress

J. Franke, W. Frank

3

generation. Both terms do not need further modellingas does the pressure-strain term � � �

. We use two dif-ferent models for � � �

, a linear and a quadratic model.

� Linear Pressure-Strain Model (LRR-IP)This model is based on the proposal of Laun-der, Reece and Rodi [Launder et al. (1975)]. Thepressure-strain term is decomposed as

� � � � � � � � � � � � � � � � � � � (14)

where � � � � � is the slow pressure-strain term, � � � � �the rapid pressure-strain term and � � � �

the wall-reflection term. � � � � � is modelled according to[Rotta (1951)],

� � � � � � � � � � � � � � � � (15)

where� � �

is the Reynolds stress anisotropy tensordefined as

� � � � � � � � � � � � � � � � � � �. The rapid part

is modelled as proposed by [Fu et al. (1987)],

� � � � � � � � � � ! � � � " � � � ! # # � " # #� � � � � (16)

Finally, � � � � is approximated as proposed by

[Gibson & Launder (1978)]

� � � � � ' � ( ) �� * " ( ) -.

��

/� � # � 1 3 # 3 1 � � �

(17)

� �5

� � � � # 3 � 3 # � � � # 3 � 3 # �

� � � � # 1 � � 3 # 3 1 � � � � �� � � # � � 3 � 3 #

� �� � � # � � 3 � 3 # �

Here, 3 �are the components of the unit normal vec-

tor to the wall,*

is the wall normal distance and 'the von Karman constant. Its value and the one of

" . are the same as for the SKE model, cf. Table 2.This model for the wall-reflection term isknown to be inadequate for impinging flows[Hanjalic (1999), Murakami (1998)]. Thereforesimulations are performed with and without inclu-sion of � � � �

. The grid convergence study of theLRR-IP model is done without � � � �

. One simula-tion with � � � �

, denoted LRR-IP+, is perormed onthe fine grid for comparison.� Quadratic Pressure-Strain Model (SSG):The model of Speziale, Sarkar and Gatski (SSG)[Speziale et al. (1991)] uses an approximation forthe slow part of the pressure strain which isquadratic in

� � �, and for the rapid part a quasi-linear

model. The entire pressure-strain follows from

� � � � � � � 5 � � / � 9

� ! # # � � �(18)

5 � � � � � � # � # � � /� � 1 ; � 1 ; � � �

� � = � � � � 9 � / � � � � � � � �

/ � � > � � � � # = � # � � # = � # � /� � 1 ; = 1 ; � � �

� � 5 � � � � � # A � # � � # A � # � �

Both, the LRR-IP and the SSG model use Equa-tion (8) to compute the scalar dissipation

�. The pro-

duction and dissipation term are computed as in theSKE model with the same constants. However, theproduction term C # is now computed exactly from! � �

as C # � ! � � � �. In this form it is also used in the

transport equation for�, Equation (4), where the co-

efficient H # , which is also used in the diffusion term ofEquation (14), is now H # � � � 9 �

. L N is computed as inthe SKE model with the same value for " . . The equa-tion for

�is only necessary to compute

�in the wall

adjacent cells. In the rest of the domain�

is computeddirectly from the normal stresses. Near the wall

�is

needed to compute the Reynolds stresses. Assuminga logarithmic distribution of the velocity and equilib-rium of the turbulence and neglecting convection anddiffusion in Equation (14), the stresses are

� � N � � � / � � O 9 � � � ; � � � � � � 5 Q � (19)

� � R � � � � � � > > � � � N � ; � � � � � � > > �in a local coordinate system with V the tangential,

3 the normal and W the binormal coordinate.The logarithmic velocity distribution is also used

to compute the tangential velocityX N in the wall ad-

jacent cells with all the presented turblence models.As part of the walls in the computational domain arerough the formulation of the standard wall functionwith roughness is used for all turbulence models.

X N � [\ � � �

/'

^ _ ` � � [ b ;L

� e g � hi � (20)

Here, \ is the wall shear stress, b ; the nor-

mal distance from the wall and � [ � " � ) -. � � ) �is

the friction velocity in the case of local equilib-rium.

e gdepends on the non dimensional rough-

ness height� hi � � � [ � i � L , where

� iis the dimen-

sional roughness height.e g

is computed from thefollowing formulas which have been derived frommeasurements in sand-grain roughened pipe flow[Cebeci & Bradshaw (1977)].

The Fourth European & African Conference on Wind Engineering, Paper #138

4

� � �� � � � � � � � �

� � � � �� � � � � �� ! � ! � $ & � � ' ) � + � , � � - /0 � � � � 2 � - 4 4 5 7 � � � � 9 � � � : � �

� � � 0 4 $ & � � 5 � � � ; : � (21)

For the constant & the default value & � � � is

used in the present work. Finally, the standard treat-ment for

�,

=and

>with the wall function approach

is used [FLUENT (2003)]. Thus the boundary condi-tions at rough and smooth solid walls are described.The other boundary conditions applied are describedin subsection 2.3.

2.2 Numerical approximations

FLUENT uses a finite volume method to solve thetransport equations described in the previous subsec-tion. The fluxes through the surfaces of the controlvolumes are numerically integrated with the midpointrule. The local values of flow variables at the surfacecentres are approximated with a second order upwindscheme. Gradients at the surface centres are approx-imated with central differencing. For the coupling ofpressure and velocity the SIMPLE algorithm is used.

The stopping criterion for the scaled? � norm of

the residuals of all equations is4 � � A . In FLUENT scal-

ing of the continuity residual is done with the largestabsolute value of the continuity residual in the firstfive iterations. For the other equations the

? � normof the residual is divided by the sum of B D F D over allcells, where F D is the value of the solution variable Fin cell H and B D its coefficient in the algebraic equationthat results from discretizing the differential equation.The stopping criterion is reached for all simulationsonly on the coarsest of the three grids, except for theSSG model which diverged. On all meshes iterativeconvergence is therefore also checked with the aid ofpressure, velocity and turbulence quantity histories attwo monitoring positions within the junction.

2.3 Computational domains, boundary conditionsand grids

In Figure 1 the geometry of the street interscetion isshown. Two of the ring shaped buidlings partly havea slanted roof while all other roofs are flat. The heightof the building with flat roof is I � J � K K . The crosssection of the streets is quadratic with edge length I .This cluster of buildings is located in the box shapedcomputational domain shown in Figure 2(a) for the

� M and in Figure 2(b) for the� : M approach flow. For

the� : M direction the buildings are rotated by that an-

gle in clockwise direction around the z-axis. Thusthe minimum distance from the domain boundaries

is smaller in this case but still large enough to have asmall blockage ratio. In fact the distance to the inflow,outflow and top boundary can be chosen smaller butwe want to have only a small influence of the bound-ary conditions on the solution.

X

Y

Z

60H = 60

90

250250

130

Figure 1 Geometry of the buildings and the street canyons.All sizes are in N N .

X Y

Z(a)

30H

10H

10H 10H

15H

(b)

27.8H9.3H

9.2H8.3H

15H

Figure 2 Computational domain with distance of the builtarea from the boundaries. (a) P M , (b) Q R M approach flow.

Symmetry boundary conditions are used at thetop and at the sides of the domain. At the outflowconstant static pressure is prescribed. The walls ofthe buidlings and the streets between the buidlingsare treated as smooth walls while the ground around

J. Franke, W. Frank

5

the intersection is treated as rough wall with a hy-drodynamic roughness length of � � � � � � , in-ferred from the measurements of the approach flow.The measured approach flow corresponds to a neu-trally stratified boundary layer flow over a rough wallwith the cited roughness length and a friction veloc-ity � � � � � � � � � � � � . In addition to these data theexponent in the power law approximation of the ve-locity profile is provided by the CEDVAL database as

� � � �. In Figure 3(a) the two resulting velocity profiles

are shown together with the measurements. While thepower law approximates the measured velocity profilevery well, the logarithmic distribution leads to muchtoo high velocities. Therefore the power law is cho-sen for the prescription of the velocity at the inflowboundary.

U [m/s]

z/H

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

Log law

Power law

(a)

k [m2/s2]

z/H

0 0.25 0.5 0.75 10

1

2

3

4

5

6

7

8Exp.SKE-FRNG-FRKE-FSST-FLRR-IP-FSSG-FInlet

(b)

Figure 3 Inflow boundary conditions for velocity � (a) andturbulent kinetic energy � (b). The coloured lines are therespective profiles on the fine grid at � � � � away fromthe inlet.

Also shown in Figure 3(a) are the computed!

ve-locity profiles after

" � # � % from the inlet plane forthe � ' approach flow case which are the same as forthe

� ( ' case. The profiles are nearly identical for allturbulence models. Below � � # � �

the computa-tions have smaller velocities than the measurementsand the power law at the inflow. This momentum lossis attributed to the usage of a cell height at the wallwhich is too low for the applied value of the rough-ness height

* +, see discussion below.

A similar behaviour can be observed for the turbu-lent kinetic energy

*, see Figure 3(b). There the com-

puted*

also decreases considerably below � � # ,- � % from the constant value prescribed at the inlet.The inflow profile for

*(and for

.or

/) is com-

puted from � � and � � according to the proposal of[Richards & Hoxey (1993)]. The resulting turbulent

kinetic energy is smaller than the measured values, cf.Figure 3(b). In case of the Reynolds stress model theassumption of isotropic turbulence is used to computethe individual stress components from the constant*. While this contradicts the well known anisotropy

of the normal stresses and the relation1 � 3 � �

4�

in boundary layer flows, the influence on the veloc-ity distribution in the junction is small as is shownin subsection 3.1. But for the normal stresses a bet-ter agreement with the measurements is obtained inthe junction when measured values are used at the in-let. Then also the turbulent kinetic energy matches theexperiments at

" � # � % from the inlet plane above� � # , - � % . The influence of the higher

*at the inlet

on the results of the two-equation models is currentlyinvestigated.

The given hydrodynamic roughness length � � hasto be converted into the roughness height

* +to be used

with the wall function, Equation (21). By equating(20) in the fully rough regime (

* 7+ 8 ( � ) with the log-arithmic velocity distribution in terms of � � one findsthe relation

* + , � � � � . Thus we use* + � - �

on the ground around the built area, while treating allother walls as smooth. The large roughness heightwould require a cell height of

9 � 8 � * + � � � � #at the wall to place the first computational locationhigher than the roughness height. This contradicts theneed for a good resolution of the boundary layer evenwhen the wall function approach is applied. There-fore we decided to use lower cell heights at the wall,the smallest being

9 � , � � � % # � � � � % * +on the fine

grid. For the medium and the coarse grid this heighthas always been doubled. We are currently investigat-ing the effect of using

* + � � � % 9 � .

x/H

z/H

-5 -4 -3 -2 -1 0 1 2 3 4 50

1

2(a)

x/H

z/H

-5 -4 -3 -2 -1 0 1 2 3 4 50

1

2(b)

Figure 4 Resolution of the buildings’ height with (a) themedium grid and (b) the fine grid.

The grids are all blockstructured consisting onlyof hexahedral cells. The fine (F) grid has 2 335 360cells. The other two grids have been generated byomitting every second node, leading to 291 920 cells

The Fourth European & African Conference on Wind Engineering, Paper #138

6

for the medium (M) and 36 490 cells for the coarse(C) grid. While the cell volume change is kept be-low 2.5 for the fine grid, its maximum increases to3.5 on the medium and 5.5 on the coarse grid. This isone reason why no Richarson extrapolation is used toexamine the grid convergence. The resolution of thebuildings’ height with the medium and the fine grid isshown on the previous page in Figure 4(a) and 4(b),respectively.

x/H

y/H

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6(a)

x/H

y/H

-8 -6 -4 -2 0 2 4 6 8-8

-6

-4

-2

0

2

4

6

8(b)

Figure 5 Resolution of the crossing with medium grid. (a)� � , (b) � � � approach flow. Red colored is the surface meshon the rough ground around the crossing, black colored thesurface mesh on the smooth ground between the buildingsand blue colored the surface mesh on the smooth buildingroofs.

With the medium grid 10 cells are used to resolve� and with the fine grid 20. The resolution as well asthe total number of cells is the same for both approach

flow directions. For the� � � approach flow only the

topology of the grid is changed, as can be seen in Fig-ure 5(b). To better resolve the boundary layers on thebuilding walls the grid lines are perpendicular to thewalls up to a normal distance of � .

3 RESULTS

The computational results are linearly interpolated onthe measurement locations which are located in sixplanes within the crossing. Two planes have con-stant � � , namely � � � � �

and � � � � � � , cf.Figure 4. The other planes are for constant � � � or

� � � � , respectively. The primed coordinates (and vec-tor and tensor components) are used for the

� � � caseand are aligned with the streets. They are obtainedfrom the

�and � coordinate by clockwise rotation of� � � around the -axis. The respective positions are

� � � � � � � � � � � � � � � � � � � � � � � �, see Figure 5(a).

3.1 Grid dependance and hit rate of the solutions

First a qualitative comparison of the results on the dif-ferent grids is presented. As there are in total 1706measurement locations for the � case and 1528 forthe

� � � case the computed results are plotted againstthe measured ones at all positions where data areavailable. As 2D laser doppler anemometry has beenused in the experiments, only the velocity

!and the

stress " " are available at all positions. In Figure 6the corresponding results are shown for the RKE tur-bulence model. For the other models similar resultsare obtained. The computed velocities are normalisedwith

! � � � �at

� � � � � from the inlet, cf. Fig-ure (3)(a). For the experimental velocities the corre-sponding value from the approach flow measurementsis used. It can be seen that the values on the mediumand fine grid differ only little, with a few exceptions.For the turbulence quantities similar results are ob-tained.

As already stated in subsection 2.3, a grid conver-gence study by Richardson extrapolation is not usedas the expansion ratio differs considerably betweenthe grids and only linear interpolation to the measure-ment locations is used. Therefore the quantitative as-sessment of grid convergence is based on the hit rate &which is recommended by the [VDI (2003)] for codevalidation. The hit rate is defined as

& �

'(

) * + - ) � (22)

where'

is the total number of measurements and

J. Franke, W. Frank

7

UEXP / UEXP,REF

US

IM/U

SIM

,RE

F

-0.5 0 0.5 1 1.5-0.5

0

0.5

1

1.5

RKE-CRKE-MRKE-F

(a)

0°

U’EXP / UEXP,REF

U’ S

IM/U

SIM

,RE

F

-0.5 0 0.5 1 1.5-0.5

0

0.5

1

1.5

RKE-CRKE-MRKE-F

(b)

29°

Figure 6 Grid convergence of normalised � and � � veloc-ity for the RKE model. (a) � � , (b) � � � approach flow.

� �

for � � �� � � � � � � � � � � � � � " # % & # () otherwise

* (23)

Here, + denotes the normalised results of thecomputation and , the normalised experimental re-sults.

# %is the absolute value of the relative error and# (

the allowed deviation which has different valuesfor the examined quantities. For the velocity compo-nents we use

# ( ) * . / and for the Reynolds stresses# ( ) * / . The resulting hit rates for the velocity andnormal Reynolds stress components are listed in Ta-ble 3 and Table 4 for the ) � and

. 3 � approach flow,respectively.

For both cases the hit rate of the velocity com-ponents converges in general with grid refinement.Some models display oscillating behaviour which canbe attributed to the bad convergence on the fine grid

where the values at the monitoring points have largeoscillations. This is especially the case for the

. 3 �approach flow. The hit rate of the normal Reynoldsstress components displays a similar behaviour withonly small changes between the medium and fine grid.The results on the fine grid can therefore be regardedas grid independent.

Table 3 Hit rate 5 (in %) of velocity and normal Reynoldsstress components for the � � approach flow.

turb. model 6 � 6 � 6 � 6 � � 6 � � 6 7 7SKE-C 50 38 14 72 62 42SKE-M 58 56 14 77 69 34SKE-F 59 57 15 78 69 39RNG-C 43 36 16 37 30 59RNG-M 54 53 16 55 40 65RNG-F 51 53 18 60 40 60RKE-C 51 39 14 63 53 42RKE-M 59 56 16 68 52 46RKE-F 61 55 15 64 51 49SST-C 43 36 14 29 24 54SST-M 51 54 15 36 30 56SST-F 48 45 18 42 38 51SSG-C – – – – – –SSG-M 58 59 16 70 57 72SSG-F 58 59 16 71 58 74LRR-IP-C 51 38 19 51 41 65LRR-IP-M 62 60 17 72 60 59LRR-IP-F 68 63 18 73 64 60LRR-IP+F 60 57 21 72 83 64LRR-IP+F, ES 61 60 21 73 81 75

Table 4 Hit rate 5 (in %) of velocity and normal Reynoldsstress components for the � � � approach flow.

turb. model 6 � 6 � 6 � 6 � � 6 � � 6 7 7

SKE-C 47 29 21 55 7 25SKE-M 57 53 23 49 3 20SKE-F 59 54 24 56 3 25RNG-C 50 43 16 66 25 61RNG-M 51 45 20 66 26 61RNG-F 49 47 17 64 20 52RKE-C 48 27 23 59 17 34RKE-M 57 55 21 63 8 37RKE-F 59 56 20 65 10 39SST-C 43 30 20 52 52 52SST-M 49 46 19 54 44 42SST-F 50 39 16 53 37 34SSG-C 44 32 19 68 38 69SSG-M 52 46 14 65 14 65SSG-F 54 50 15 62 13 64LRR-IP-C 45 33 18 67 39 66LRR-IP-M 51 51 16 67 15 60LRR-IP-F 50 41 16 63 14 64

The Fourth European & African Conference on Wind Engineering, Paper #138

8

The quantitative analysis of the hit rates showsthat for the � � case the best results are obtainedwith the differential Reynolds stress models. Here,LRR-IP+F, ES refers to the LRR-IP model with wall-reflection term, Equation (17), and the Experimen-tal Stresses as inflow condition. Comparing the re-sults of this model with the ones of the LRR-IP+ andthe LRR-IP model, the best results for the velocitiesare obtained with the LRR-IP model while the nor-mal stresses are best predicted with the LRR-IP+, ESmodel. The inclusion of the wall-reflection thereforeleads to a better prediction of the turbulence at thecost of the velocity prediction. The best results forthe normal stresses are obtained when using the mea-sured stresses as inflow condition.

The results with the SSG model are similar to theones with the LRR-IP+ model except for � � whereworse agreement is obtained and for � � �

which isclose to the one of LRR-IP+, ES. Concerning the re-sults for � � all models have bad results with a max-imum of 21 % of the computed values agreeing withthe measurements within 25 %. The reason for thebad agreement is visualised in subsection 3.2.

Comparing the two-equation models, the SKE andthe RKE model perform equally well with results forthe velocities only slightly worse than with the differ-ential Reynolds stress models. The SKE model alsohas remarkably good results for the normal stresses,except for � � �

which is best predicted with the RNGmodel. But this model performs worse for the ve-locities. The least accurate results are obtained withthe SST model. This is also true for the velocitiesin the

� � � case but not for the normal stresses wherethe SKE model performs worst, see Table 4. But forthe velocities the best results are obtained with theSKE model, followed by the RKE and then the SSGmodel. The LRR-IP model is even outperformed bythe RNG model. For the normal stresses the situationis inversed as the differential Reynolds stress mod-els have the best results. All models predict � � � � verybadly, as well as � � � again. The reason for the latteris displayed in subsection 3.3.

Resuming the results we can state that all modelsperform reasonably well in predicting the

� � and� � velocities, especially in the � � case, but fail forthe � � velocity. When averaging the results for allvelocities for the � � and

� � � case we get the followingranking of the models: the SKE model performs best,slightly better than the RKE model. Then the LRR-IP, SSG, RNG and SST model follow. Performing thesame averaging for the normal stresses we have thefollowing order: SSG, LRR-IP, RNG, RKE, SKE andSST. Thus the best results for the velocities are ob-

tained with linear two-equation models, namely theSKE and the RKE model.

3.2 Mean velocities and turbulence for � � approachflow

After presenting the main quantitative results of thesimulations in the previous subsection we here wantto present the flow field in the junction at two of thesix measurement planes. These measurement planesare chosen as the worst results based on the previ-ously introduced hit rate are obtained there. For thetwo measurement planes parallel to the ground this isthe one at � � � � � � . In Figure 7 the experimentalvelocity vectors from

�and

�are dispplayed. The

flow enters the intersection through the street facingthe approach flow and the crossroad from the positive

� � direction. At the entrance into the crossroad arecirculation region is formed. The fluid then followsthe crossroad until it is blocked by the fluid comingfrom the street entering the junction. Therefore theflow from the crossroad changes direction and leavesthe intersection by the street parallel to the

� � -axiswhich faces the outlet plane of the computational do-main. In this street a weak recirculation region can beobserved behind the corner of the upper right build-ing. Though this is merely a 2D picture of the 3Dflow within the intersection, the flow from the streetentering the intersection has high enough momentumto deflect the flow from the crossroad. This high mo-mentum flow is itself deflected in negative � � direc-tion in the crossroad, with a small recirculation zoneat the corner of the lower left building. Another recir-culation region can be observed at � � � � � wherethe flow leaves the crossroad and is deflected by theexternal flow around the buildings.

X/H

Y/H

-3 -2 -1 0 1 2 3-5

-4

-3

-2

-1

0

1

2

3

4

5

6

UEXP,REF

EXP.

Figure 7 Velocity vectors from � and � at � � " � # .Experimental results.

J. Franke, W. Frank

9

X/H

Y/H

-3 -2 -1 0 1 2 3-5

-4

-3

-2

-1

0

1

2

3

4

5

6

USIM,REF

SKE

X/H

Y/H

-3 -2 -1 0 1 2 3-5

-4

-3

-2

-1

0

1

2

3

4

5

6

USIM,REF

RNG

X/H

Y/H

-3 -2 -1 0 1 2 3-5

-4

-3

-2

-1

0

1

2

3

4

5

6

USIM,REF

RKE

X/H

Y/H

-3 -2 -1 0 1 2 3-5

-4

-3

-2

-1

0

1

2

3

4

5

6

USIM,REF

SST

Figure 8 Velocity vectors from � and � at � � � � � � � .Two-equation model results.

The corresponding numerical results with the two-equation models are displayed in Figure 8. None ofthe models predicts the recirculation zone when theflow enters the crossroad from the positive � � di-rection. The deflection of this flow towards the streetin positive

� � direction is only captured by the SKEand RKE model, where the last one is the only modelthat shows the experimentally observed flow reversalabove � � � � � � . The RNG and the SST model pre-dict that the flow goes through the entire crossroadand does not enter the street in positive

� � direction.This is due to the fact that the flow into the junctionfrom the street facing the inlet plane has lower mo-mentum than in the experiment. This is already obvi-ous from the first velocity vectors at the entrance intothis street. While in the experiment these have nearlyequal length, representative of a block profile, all sim-ulations have smaller velocities close to the walls, thesmallest for the SST model. Thus the prediction ofthe flow near the walls seems to be responsible for thelow momentum that then leads to a flow field which

is qualitatively different from the one observed in theexperiments.

Due to the bad prediction of the flow within thecrossroad all models predict too large a recirculationregion behind the corner of the upper right building.Another common feature of the simulations is that thevelocity of the external flow at � � � � � is under-predicted. This is also the case for the simulationswith the differential Reynolds stress models which aredisplayed in Figure 9.

X/H

Y/H

-3 -2 -1 0 1 2 3-5

-4

-3

-2

-1

0

1

2

3

4

5

6

USIM,REF

LRR-IP

X/H

Y/H

-3 -2 -1 0 1 2 3-5

-4

-3

-2

-1

0

1

2

3

4

5

6

USIM,REF

SSG

X/H

Y/H

-3 -2 -1 0 1 2 3-5

-4

-3

-2

-1

0

1

2

3

4

5

6

USIM,REF

LRR-IP+

X/H

Y/H

-3 -2 -1 0 1 2 3-5

-4

-3

-2

-1

0

1

2

3

4

5

6

USIM,REF

LRR-IP+, ES

Figure 9 Velocity vectors from � and � at � � � � � � � .Differential Reynolds stress model results.

An excellent prediction of the flow is obtainedwith the LRR-IP and the SSG model which both showthe flow reversal above � � � � � � . But the LRR-IP model better predicts several other features of theflow than the SSG model. While both models do notcompute the recirculation region at the entrance of theflow into the crossroad from the positive � � direc-tion, the LRR-IP model better predicts the strengthand direction of the flow. The same is true for the flowin the crossroad below � � � � � � � . When using thewall-reflection term, Equation (17), with the LRR-IP

The Fourth European & African Conference on Wind Engineering, Paper #138

10

model worse results are obtained that compare wellwith the ones of the SKE model, cf. Figure 8. Theuse of the experimental normal stresses as boundarycondition has obviously no influence on the velocitiesin this plane.

Y/H

U/U

RE

F

-5 -4 -3 -2 -1 0 1 2 3 4-0.5

0

0.5

1SKERNGRKESSTLRR-IPSSGEXP

z/H=0.5x/H=0.17

Y/H

uu

/UR

EF2

-5 -4 -3 -2 -1 0 1 2 3 40

0.05

0.1SKERNGRKESSTLRR-IPSSGEXP

z/H=0.5x/H=0.17

Figure 10 Comparison of � velocity and � � normalReynolds stress at � � � � � � � � � � � � � � � .

Next, a comparison between the computed andmeasured

velocity and the � � normal Reynolds

stress at the position� � � � � � � within the mea-

surement plane at � � � � � � is shown in Figure 10.For the SKE, RKE, SSG and LRR-IP model we ob-tain an excellent agreement with the measurementswhich deteriorates towards the ends of the crossroad.The RNG and SST model predict too low velocitiesaround � � � � which is a result of the low momen-tum of the flow entering the junction from the nega-tive

� � direction. These models also significantlyunderpredict � � . For the other models the results arebetter, largely capturing the qualitative distribution of

� � . The magnitude of the maximum value is best pre-dicted with the SKE model.

The velocity vectors from

and in the mea-surement plane at � � � � �

are shown in Figure11 for the experiment and the two-equation models.This is the measurement plane where with all modelsthe worst results for the velocity have been ob-tained. It cuts the recirculation region in the cross-road at the corner of the lower left building. Fromthe experiments one sees that the entire flow is di-rected away from the bottom of the crossroad. Thereis also hardly any flow from above � � � �

into thecrossroad. These features are not captured by any ofthe models used, cf. Figure 12 for the differential

Reynolds stress models. All models predict a largerflow from above � � � �

into the crossroad and arecirculation region, roughly in the lower right part ofthe crossroad’s cross section. This recirculation zoneis largest and strongest for the RKE, SSG and LRR-IP model which showed the best agreement with themeasurements in the plane at � � � � � � , see Figure7, 8 and 9. The inclusion of the wall-reflection termleads to a smaller recirculation zone with again no vis-ible influence of the boundary condition of the normalReynolds stresses. The smallest recirculation zone isobtained with the SST model. These results indicatea correlation between the prediction of the flow in theplanes � � � � � � and � � � � �

. The better theresults are for the � � � � � � plane the worse they arefor the � � � � �

plane.

X/H

Z/H

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

EXP

UEXP,REF y/H=-1

X/H

Z/H

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

SKE

USIM,REF y/H=-1

X/H

Z/H

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

RNG

USIM,REF y/H=-1

X/H

Z/H

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

RKE

USIM,REF y/H=-1

X/H

Z/H

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

SST

USIM,REF y/H=-1

Figure 11 Velocity vectors from � and � at � � � � # � .Experimental and two-equation model results.

J. Franke, W. Frank

11

X/H

Z/H

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

SSG

USIM,REF y/H=-1

X/H

Z/H

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

LRR-IP

USIM,REF y/H=-1

X/H

Z/H

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

LRR-IP+

USIM,REF y/H=-1

X/H

Z/H

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

LRR-IP+, ESUSIM,REF y/H=-1

Figure 12 Velocity vectors from � and � at � � � � � � .Differential Reynolds stress model results.

Finally, a comparison between the computed andmeasured � velocity and the � � normal Reynoldsstress at the position

� � � within the measure-ment plane at � � � � �

is shown in Figure 13.Above � � � �

all models have negative � veloc-ities due to the aforementioned flow into the cross-road whereas the experiments show � � � . Insidethe crossroad the simulation results show the recircu-lation region while the experiment has only positivevelocities. For � � we obtain the best agreement withthe SKE model. While the RKE and even the RNGmodel also produce reasonable results, the differentialReynolds stress models and the SST model computetoo low a � � without even reproducing the distinctpeak at � � � �

. So while in general the differentialReynolds stress models show a better agreement for

� � with the experimental results, as has been shownwith the hit rate in Table 3, they perform less goodthan most of the other models at this position.

For the � � approach flow in general we obtain anexcellent qualitative agreement with the experimentsin the plane � � � � � � for the SSG and LRR-IP

model, good agreement for the SKE, RKE and LRR-IP+ model and no agreement for the RNG and SSTmodel. For the plane � � � � � � which is notshown here, all models yield good agreement. On theother hand all models’ performance is weak for the

� � � const. planes.

W / UREF

Z/H

-0.2 0 0.2 0.40

0.5

1

1.5

2

2.5y/H=-1x/H=0

ww / UREF2

Z/H

0 0.02 0.04 0.060

0.5

1

1.5

2

2.5y/H=-1x/H=0

Figure 13 Comparison of � velocity and � � normalReynolds stress at � � � � � � � � . For colourlegend see Figure 10.

3.3 Mean velocities and turbulence for � � � approachflow

In the � � � case the same measurement planes areused but now in the primed coordinates. For the twoplanes � � � const. which are not shown here, thequalitative and quantitative results of all models arebetter than in the ! � case. The lower hit rate pre-sented in Table 4 is therefore due to the results in the

" $ � � const. planes. There again " $ � � & ( hasthe worst results in terms of the hit rate. The exper-imental and numerical velocity vectors in this planeare shown in Figure 14. In the experiments the flowis directed towards the roofs for ) $ � � * ! and intothe crossroad for ) $ � � , ! up to � � � . ( 0 2 . An-other feature of the flow inside the crossroad is thatthe 3 $ component is nearly always negative. This isnot captured by any of the models which all predictpositive 3 $ velocities for ) $ � � * ! . Also all mod-els except the SKE and RKE model display flow innegative ) $ � � direction for ) $ � � , ( 0 2 in contrast tothe experiments. These characteristics of the simula-tions explain the bad agreement with the experimen-tally observed 3 $ components.

For the 6 $ component the results are better. Espe-cially the RKE and SSG model predict upward flowfor ) $ � � 7 ! like in the experiment. Also the flowdirection above � � � . ( 0 � 2 is fairly good predicted.This can be also seen in Figure 15 where the 6 $ ve-locity and ; ; $ normal Reynolds stress are displayed

The Fourth European & African Conference on Wind Engineering, Paper #138

12

at � � � � � � within the plane � � � � � � .

x’/H

z/H

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

EXP

UEXP,REFy’/H=-1

x’/H

z/H

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

SKE

USIM,REFy’/H=-1

x’/H

z/H

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

RNG

USIM,REFy’/H=-1

x’/H

z/H

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

RKE

USIM,REFy’/H=-1

x’/H

z/H

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

SST

USIM,REFy’/H=-1

x’/H

z/H

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

LRR-IP

USIM,REFy’/H=-1

x’/H

z/H

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

SSG

USIM,REFy’/H=-1

Figure 14 Velocity vectors from � � and � � at � � � � � � � .

W’ / UREF

z/H

-0.2 0 0.2 0.40

0.5

1

1.5

2

2.5y’/H=-1x’/H=0

ww’ / UREF2

z/H

0 0.05 0.10

0.5

1

1.5

2

2.5y’/H=-1x’/H=0

Figure 15 Comparison of � � velocity and � � � normalReynolds stress at � � � � � � � " � � � � $ . For colourlegend see Figure 10.

Up to % � � & � ' ) + all models predict too lowvelocities with the SST and RNG model even hav-ing negative velocities close to the ground. The bestagreement is obtained with the SSG, LRR-IP andRKE model. Above % � � & � ' ) + the agreement withthe experiments is good although a stronger down-ward movement is observed experimentally.

For the normal stress all models display too lowvalues within the crossroad. There the SKE model isclosest to the experiments but predicts too high a peakvalue. This peak is reproduced by all models exceptfor the SSG model albeit at a larger % � � than in theexperiments.

So for the / 0 2 approach flow case we find in gen-eral the same results as in the � 2 case. While thequalitative and quantitative prediction of the flow inthe % � � � const. planes is very good for all mod-els, even better than in the � 2 case, the results for the

� � � � const. planes are worse.

4 CONCLUSIONS

Six turbulence models comprising four linear two-equation models and two differential Reynolds stressmodels were used to simulate the flow from two ap-proach flow directions across an asymmetric street in-tersection formed by four rings of buildings whichpartly have a slanted roof. Three systematically re-fined grids were used in each case to assess the griddependance of the results and the simulations on thefinest grid were found to be nearly grid independent.The predictions were compared with available windtunnel data from the CEDVAL database. The agree-ment between the predicted and measured velocitycomponents in the measurement planes parallel to theground is in general very good. There the standard

J. Franke, W. Frank

13

and the realizable � � � model perform slightly bet-ter than the differential Reynolds stress models whenboth approach flow directions are taken into account.The worst results are obtained with the renormaliza-tion group � � � and the shear stress transport � � �

model. Greater discrepancy is obtained for the veloc-ity components in planes perpendicular to the cross-road with all models.

The normal turbulent stresses are best predictedwith the two differential Reynolds stress models. Butthe standard and the realizable � � � model yield againreasonable agreement with the measurements. Theresults therefore indicate that these two linear two-equation models are accurate enough for the predic-tion of at least the mean velocities in a cluster of build-ings with moderate complexity.

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