Air on Cube

8/6/2019 Air on Cube

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Journal of Wind Engineering

and Industrial Aerodynamics 93 (2005) 115135

Numerical studies on air flow around a cube

Y. Gaoa, W.K. Chowb,

aDepartment of Building Engineering, Harbin Engineering University, Harbin, Heilongjiang, ChinabDepartment of Building Services Engineering, The Hong Kong Polytechnic University,

Hung Hom, Kowloon, Hong Kong, China

Received 21 March 2003; received in revised form 5 October 2004; accepted 15 November 2004

Abstract

In this paper, air approach flow moving towards a cube will be studied using computational

fluid dynamics (CFD). The Reynolds Averaging of NavierStokes (RANS) equation types of

k2 turbulence model are used. Some RANS predicted results are compared with different

upstream air speeds. Flow separation at the corner above the top of the cube, level ofseparation and reattachment are investigated. Reference is made to the experimental data on

wind tunnels reported in the literature.

A method similar to recirculation bubble promoter is used for different approach flow

speed distributions. Problems encountered in numerical simulations due to the sharp corner

are discussed with a view to obtaining better prediction on recirculation flow in regions above

the top of the cube. Correlations between the turbulent kinetic energy above the cube and the

recirculation bubble size are derived for different distributions of approach flow speed.

By limiting the longitudinal velocities in the first cell adjacent to the sharp edge of the cube

or rib, and making good use of the wall functions at the intersection cells of the velocity

components, positions of maximum turbulent kinetic energy and the flow separation and

reattachment can be predicted by a standard k2

model. The results agree with those obtainedin the experiments.

r 2004 Elsevier Ltd. All rights reserved.

Keywords: Standard k2 turbulence model; Turbulent kinetic energy; Upstream flow velocity; Flowseparation and reattachment

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www.elsevier.com/locate/jweia

0167-6105/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jweia.2004.11.001

Corresponding author. Tel.: +852 2766 5843; fax: +852 2765 7198.

E-mail address: [email protected] (W.K. Chow).
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1. Introduction

To enhance understanding of the physical characteristics of turbulent flow, it

would be useful to study the separation, reattachment, scale and direction of

backflow, and variation of pressure gradient of upstream air flowing around a cube

(or a rib). The results can be applied to the natural ventilation design of buildings

and the associated problem of fire spreading. Although a cube has a simple shape,

the air flows around it are found to be very complicated. This is due to the

interaction between the strain-rate tensor and vorticity tensor at the near-wall region

of the cube and the different flow speeds u, which vary with the height y to the power

1=m: Most of the works reported in the literature [114] are numerical studies of the

turbulent flow around a surface-mounted cube of higher Reynolds number (Re).There have been relatively few experimental works on wind tunnels.

Two cases of approach flow towards a cube as pointed out and discussed by

Baetke et al. [1] are considered (see Fig. 1):

Case A is a simple assumption with constant upstream flow. Lower wall surface isperfectly smooth so that no energy is lost and there is no shearing effect.

Case B is a more realistic scenario with values of Re varied due to the coarsesurface. The horizontal velocity u at vertical height y is given in terms of an

exponent 1/m, and two other constants C1 and C2:

u C1y1=m C2: (1)

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Nomenclature

u velocity component in x-directionv velocity component in y-direction

r air density

p pressure

P0 stagnant pressure

u0 bulk velocity in inflow boundary

u u=u0 non-dimensional velocity in x-directionk turbulent kinetic energy

Re Reynolds number

P01 stagnation pressure on y40.5H

P02 stagnation pressure on yo0.5HH height of cubic model

YB height of wind tunnel

XR reattachment length at the top of the cube

XF reattachment length behind the cube

All properties presented here are made non-dimensional by H and u0.

Y. Gao, W.K. Chow / J. Wind Eng. Ind. Aerodyn. 93 (2005) 115135116


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Experiments on separation and reattachment on the top of the cube were conducted

by Castro and Robins [2]. Numerical simulations with a standard k2 turbulencemodel for Case B with different ground roughness lengths were reported by Paterson

and Apelt [34]. The effects of the velocity at the leading edge were presented. Four

sets of non-dimensional turbulent kinetic energy k contours along a vertical

symmetry plane were compared. The local maxima ofkmoved from the leading edgeto the middle of the cube when the rough wall flow changed into smooth wall flow.

Setting the longitudinal velocity adjacent to the front edge to zero might work better

than using the wall functions.

Although k is important in understanding the properties of a flow field, there is a

lack of experimental data on distribution for k near a real building. Precise

measurements of the turbulent flow, including the measurement of k, around a

building model were reported by Murakami and coworkers [511]. Further

numerical simulations can be performed on those cases to compare with the

experimental data and their reported numerical works.

In the wind tunnel test, the distribution of k around the 2D square rib or the cubeas reported [510] are shown in Figs. 2a and 3a. The maximum turbulent kinetic

energy kmax was found above the centre of the roof. The velocity vector distributions

on the 2D square rib and the central vertical section of the cube are shown in Figs. 2b

and 3b. Values ofkare relatively small around the frontal corner of the roof. On the

other hand, k around the frontal corner of the roof was overestimated in using the

standard k2 model in Figs. 2c and 3c. This overestimation of kwas improved whenthe LK model and the MMK model was used as shown in Figs. 2d and e [10,11].

The large eddy simulation (LES) concept was first applied by Murakami et al. [8]

to calculate the flow over a periodic arrangement of cubes in a simulated

atmospheric boundary layer, i.e. applying periodic boundary conditions in the mainflow direction. Good agreement was found between numerical results and

experimental data in a wind tunnel.

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XR

H

H XF

YB

y

x

Fig. 1. Geometry of the problem. Case A: u 1; Case B: u y1=m

Y. Gao, W.K. Chow / J. Wind Eng. Ind. Aerodyn. 93 (2005) 115135 117


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Fig. 2. Turbulent kinetic energy k and velocity vectors around the 2D square rib [10]: (a) wind tunnel

experiment, (b) wind tunnel experiment, (c) standard k2

model, (d) LK model and (e) MMK model.

Fig. 3. Over-prediction ofkby the standard k2 model at the central section of cube [11]: (a) wind tunnelexperiment, (b) wind tunnel experiment and (c) standard k2 model.



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Various Reynolds stress models (RSM) were compared by Murakami [11] for the

flow field around a cube. Two reattachment lengths, XR and XF, on the roof and on

the floor, were compared. The results of the standard LES for XR and XF (0.6 and

1.4) agree fairly well with the experiment (0.7 and 1.2). However, the reattachmentlength on the roof was not predicted well using RSM models (XR41:0H; flow didnot reattach on the roof), and the reattachment lengths XF % 2:022:3 on the floorare much longer than the experimental value for all models [11].

A non-linear k2 model is presented [15] with distributions of the predicted karoundthe cube shown in Fig. 4. Values ofkat the edge of the upstream surface of the cube are

overpredicted for the ST and NL models as shown in Figs. 4a and b. However, the

production ofkin the KH model is suppressed. A possible explanation is that this is due

to the excessive production of turbulent kinetic energy k in using the standard k2model for the flows with impinging regions. As demonstrated in Fig. 4c, excessive

production of k near the impinging region is suppressed by the non-linear k2

model.However, the quadratic terms have no effect for suppressing the production term.

The turbulent velocity field of flow around a surface-mounted cube measured by

Laser Doppler Anemometry was reported by Hussein and Marinuzzi [13] and

Larousse et al. [16]. They presented an experimental investigation of the

inhomogeneous, three-dimensional flow around a surface-mounted cube of half-

channel height placed in a fully developed, turbulent channel flow (Re ffi 80; 000;based on the channel height, and bulk velocity u0). The inferred streamlines

downstream of the leading edge along the central vertical plane are shown in Fig. 5.

Generally, the mean flow did not reattach on the obstacle top side (the shear layeroriginating at the leading edge did not reattach downstream). The reattachment

point along the plane of symmetry was located at XF of 1.69 behind the cube.

As shown in Fig. 5, the path of the local k maxima is correlated with the location

of the local velocity maximum and hence the shear layer. The distribution ofkis also

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Fig. 4. Turbulent kinetic energy distributions around the cube along the x-axis [15]: (a) ST model, (b) NL

model and (c) KH model.



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related to the shear layer, from the very thin and concentrated distribution over the

cube to the rather broad expansion directly downstream of the reattachment. The

magnitude of k is fairly uniform in the recirculation. Further results were presented

by Martinuzzi and Havel [14].

Although little is available in the way of experimental results, at least three sets of

experiments [2,8,13] reported in the literature are worth discussing. A summary of

the experiments is listed in Table 1. Key points are:

In the wind tunnel experiments, for vertical height varying from YB of 10H, 5.2Hand 2H, the flow profile at the tunnel entrance is close to y1/3, y1/4 and y1/7. The

shape of the curve is very close to that of constant upstream flow for u 1: Asteeper velocity distribution curve will result where the 1/m is smaller and where

the boundary layer thickness is closer to the bottom surface. Flow separation will

be found at the upper part of the cube.

Bubble length XR on the top of the cube is varied. u varies from y to y1/7 to y1/3,

XR varies from XR41:0 (only separation, no reattachment) to XR 0:7; and then

to XR % 0:2: Castro and Robins [2] reported that no separation occurred for CaseB. This finding appears to be arguable and is perhaps due to the use of a

separation region which is too small.

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Fig. 5. (a) Inferred streamline pattern along the plane of symmetry [13] and (b) k profile distributions

along the plane of symmetry [13].



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The velocity pattern u is related to the Re number, YB and the scale of H, bottom

layer roughness and fluid viscosity. Figs. 25 show the flow separation at the frontal

corner of the cube, the fluid velocity vector after separation and the angle between

the cube surface, whether there will be reattachment on the top of the cube, and thelength and variation of XR. These can be related to the experiments on k profile on

the upper part of the cube by Murakami et al. [8] and Hussein and Martinuzzi [13].

2. Numerical experiments

Numerical experiments similar to those of Murakami and coworkers [511] were

carried out with the CFD package PHOENICS. Both homogenous and non-

homogenous grid systems were used. The same computational domain of 15.7H

(downstream length), 9.7H (lateral width) and 2.0H or 5.2H (vertical height) wasused for all the simulations. The grid system used was 57, 38, and 45 along the x-, y-,

and z-directions. The computing domain above the cube was extended by another

length H. For some cases, a domain extended to 4H above the cube was also tested.

Simulations were performed on a personal computer with central processor 2 GHz

with computing time from 5 to 20 h, depending on the grid system.

A comparison of upstream flow velocity profile is shown in Figs. 6 and 7. Because

of the different tunnel heights YB and Re used in different experiments, there is a

large difference in the velocity distribution at the tunnel entrance. The variation

pattern of du=dy is different, especially near the bottom layer. This also gives

stagnant pressures P0 along the cube which are very different from the values nearthe cube. The upstream flow velocity for different power m when YB 2H is shown

in Fig. 6, and the situation for YB 6:0H is shown in Fig. 7. In the figure,

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Table 1

Experimental conditions and results

Experiments u YB Re XR

Castro and Robins [2] p y1/3 * for Case B 1 10H 2 1041 105 no separation for Case B

for Case A XR 0:2 for Case AMurakami et al. [8] p y1/4 5.2H 7 1048.4 104 XR 0:7

XF 1:2Hussein and

Martinuzzi [13]

p y1/7 * 2H 8 104 XR41:0

XF 1:67

Note: No separation for Case B, or the bubble might have been so small that it was not found.

Velocity distribution curves at the wind tunnel entrance were stated by the authors. Expressions for

u y1=3 and u y1=7 are only approximations. Tangential velocity of upstream flow was not given [8,13].

The shape of the curves is slightly different from these expressions. A more accurate expression was

proposed by Hussein and Martinuzzi [13]:

uy 1:5574y4 1:07607y3 0:7365y2 1:3897y 0:62 for 0pyp1:



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u ln y 1 is close to u y1=2; which is generally taken as the upstream flowvelocity distribution of wind blowing through buildings under atmospheric

conditions.

When using a standard k2 model and wall function in both two- and three-

dimensional simulations, kmax is found at the sharp corner as shown in Figs. 2c, 3c,4a and b. It is different from the experimental results as in Figs. 2a, 3a and 5b where

kmax appears on the upper part of the cube. The separation and backflow should be

predicted as in Figs. 2b and 3b. These points are investigated to improve the

prediction.

In the literature [8], inflow boundary conditions with uy / y1=4 were summarized

in Table 2. The profiles of mean velocity u(y) and k(y) in the approach flow of the

channel given by experiment in Fig. 1.

The streamwise mean and fluctuation velocities measurement at different spanwise

positions have been studied in the literature [13]. The in-coming fully developed

conditions with uy / y1=7 were verified by comparing mean velocity and Reynoldsstress distributions at various location from Figs. 2 to 4. Present results for u02 are

consistent with other reported observations.

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0 0.5 10

0.5

1

Y

u

31yu =

41yu =

71yu =

101yu =34

y7607.1y557.1u =

62.0y3897.1y7365.0 2

+ +

Fig. 6. Upstream flow velocity with YB 2:0Hb:

0 1 20

1

2

3

4

5

6

Y

u

31yu =

41yu =

71yu =

101yu =

)1y +ln(u =

Fig. 7. Upstream flow velocity with YB 6:0Hb:



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Table 2

Comparison between Up Upmax ; Up 0and experiments

Case I: using wall function at the sharp corner II: remedial work of using wall

function at the sharp corner to

make Up 0

Characteristics of grid velocity at

the sharp corner

UB % 0; Up Upmax UB % 0; Up % 0

p % Vpmin ; VB VBmax p % Vpmax ; VB VBmax

Direction of grid velocity at

the sharp cornera arc cos

~Upj j~Up ~Vpj j

arc cos1 0

a arc cos0 90

No separation Largest separation

Position of kmax [source of results]


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Table 2 (continued)

Characteristics of

streamlines [source of

results]

Position of Umax in a

narrowed duct

Comments I: Incorrect calculation results due to

the incorrect use of wall function because

of the wrong position of staggered grids

at the sharp corner.

II: Other situations are the

same as in Case I. The results

resembled the real situations more

after the remedial work of making

UpE0, but the degree of separation

is slightly larger than in real

situations.


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If no information is available at all, the distributions for k and in inlet flows canbe estimated from the turbulence intensity Ti and a characteristic length L by

k 32 uyTi

2

;

C3=4mk3=2

;

0:07L:

Sensitivity analyses on grid systems were carried out to obtain converged results. It is

clear from Figs. 8 to 15 that different grid systems were assigned. The distribution of

k and velocity vectors with upstream flow velocity profile u y1=4 flowing through

the rib under two-dimensional situations are shown in Figs. 8 and 9. The results

shown in Fig. 8 were computed using wall functions in the calculation of the

longitudinal velocity adjacent to the front edge, which resulted in overestimation of k

at the windward sharp corner, inaccurate velocity field distribution, and no

separation and bubble. The results shown in Fig. 9 were computed with a zero

longitudinal velocity adjacent to the front edge. The results shown in Fig. 9 agree

well with the experimental results as shown in Fig. 2. For two-dimensional shapes,

there will usually be separation but no reattachment as in Figs. 2b and 9b. With the

same Re, kmax in Fig. 8a is 1.5 times that in Fig. 9a. For k-distribution as shown in

Fig. 9a, the predicted results agree well with experiments as in Fig. 2a.

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0.03

0.15

0.170.2

0.1

0.08

0.03

0.08

(a) (b)

Fig. 9. Results around the 2D rib using zero longitudinal velocity with u y1=4: (a) k profile and (b)

velocity vectors.

0.05

0.060.08

0.120.2

0.35

(a) (b)

Fig. 8. Results around the 2D rib using wall function with u y1=4: (a) k profile and (b) velocity vectors.



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Three-dimensional calculation results for upstream flow profile u y1=4 are shownin Fig. 10. The geometry and flow environment are the same as the experimental

results given by Murakami as shown in Fig. 3. The calculated results agree well with

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0.05

0.03

0.08

0.050.08

0.03

0.03

0.130.1

0.130.1

0.07

0.06

0.0

4

0.040.06

0.02

0.07

0.13

(a) (b)

(c)

(d) (e)

Fig. 10. Results around a cube with u y1=4: (a) k profile, (b) velocity vectors, (c) streamline, (d)

streamline at y 0:1H and (e) k profile at y 0:1H:

0.110.11 0.08

0.06

0.05

0.02

0.02

(a) (b)

(c)

Fig. 11. Results around a cube with u y1=5: (a) k profile, (b) streamline and (c) velocity vector.



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0.01

0.030.05

0.06

0.10.13

0.03

0.08

(a) (b)

(c)

Fig. 12. Results around a cube with u y1=6: (a) k profile, (b) streamline and (c) velocity vector.

0.09

0.12

0.070.050.03

0.02

0.030.02

(a) (b)

Fig. 13. Results around a cube with u y1=7: (a) k profile and (b) streamline.

(a) (b)

0.120.1

0.080.06

0.04

0.02

0.04

Fig. 14. Results around a cube with u y1=10: (a) k profile and (b) streamline.



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the experimental results. At the sharp corner, the velocity near the wall is treated in

the same way as in Fig. 9. The distribution ofk is shown in Fig. 10a. The position ofkmax is at 0.09Habove the cube, 0.26Hfrom the front edge of the cube; whereas the

position ofkmax in Fig. 3a is at 0.5Hfrom the edge of the cube. The velocity vectors

and streamlines at the upper middle of the cube are shown in Figs. 10b and c,

respectively. Weak bubbles are observed, XR is about 0.7. The streamlines and

distribution of k at horizontal plane y 0:1H are shown in Figs. 10d and e,respectively. There is separation but no reattachment on the two sides of the cube.

kmax is about 0.13 at the two sides of the cube, the same as the value above the

cube.

The results for flow profile u y1=5 are shown in Fig. 11. From Fig. 11a, it can be

seen that on the top of the cube, kmax occurred at about 0.2H above the cube and0.45Hfrom the front edge of the cube. Fig. 11b shows that separate flows above the

cube might not be reattached within the width H. The recirculation bubble only

occupies part of the regions above the cube. There will be reattachment for all

separate flows with incident profile of u y1=4; usually with separate flows and abubble size of about 0.7H. Velocity vector distribution is shown in Figs. 3b and 10b.

The results for u y1=6 are shown in Fig. 12. kmax is at about 0.4Habove the cube

and 0.3H from the front edge of the cube. XR is close to 1.0. Finer grids are used to

illustrate the backflow above the cube. It can be seen that more separation is

observed in Fig. 12 than in Fig. 11.

The streamline and k distributions for flow profile u y1=7 are shown in Fig. 13.There are larger eddies, but no reattachment, XR is larger than 1.0. The results for

u y1=10 with Dy 150

H grids are shown in Fig. 14. There is no backflow, but

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(a) (b)

0.130.210.2

7

0.1

0.070.05

0.02

(c)

Fig. 15. Results by wall function with u y1=4: (a) k profile, (b) velocity vectors and (c) streamline.



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stronger separation capability. As seen from Figs. 13a and 14a, despite stronger

separation, there is very little variation of kmax, and that only occurs towards the

centre of the cube.

Under the same inlet condition as for Figs. 810, the results predicted are notborne out using a logarithmic wall function under three-dimensional conditions, as

shown in Fig. 15. It is observed from Fig. 15a that kmax is twice the value in Fig. 10a,

i.e. 0.27, and it occurs at the frontal sharp corner. However, no separation is

observed as shown in the vectors diagram in Fig. 15b and streamline diagram

in Fig. 15c.

When u is given by y1=m with m lying from 3 to 4, experimental results indicate that

the upstream stagnation pressure P01 is between 0.6 and 0.75H. When the average

velocity u is near to Case A, or is in a fully developed situation, say with an m of

about 10, P02 is at the height of 0.3H.

3. Discussion on the predicted results

The experimental results obtained by Murakami et al. [7,10,11], Hussein and

Martinuzzi [13], and Martinuzzi and Havel [14] are shown in Figs. 2a, b, 3a and 5.

The measured distribution patterns of k adjacent to the rib and cube appear to be

reasonable. From the two-dimensional pattern in Fig. 2a, kmax is found in the middle

part of the rib top. Separation and backflow are shown above the rib as in Fig. 2b.

But for three-dimensional cases, the flow separation bubble above the cube is smalland the backflows are not so drastic that air can flow through the sides. The height of

wind tunnel as used for the results in Fig. 3b is about 5.2H. In the experiment on

wind tunnels with height 2H as in Fig. 5a, there are slight separate flows, compared

with Fig. 3b. The incident flow profiles are u y1=4 and u y1=7; respectively, for thetwo cases as shown in Table 1.

Attempts have been made to obtain better agreement with experiments by

correcting the standard k2 turbulence model, see, e.g., [7,10,11,15]. Their numericalresults are compared. Since air must flow above the rib in two-dimensional cases,

there would be sharp changes for the flow in front of the rib. Results of these two-

dimensional studies are easier to compare.kmax and XR should be related as kmax is found above the cube and rib. This is

another demonstration of separate flow.

The results predicted by the standard k2 model are shown in Fig. 2c. kmax wasfound at the sharp edge, not above the rib in the middle part as in Fig. 2. The results

of the modified LK model are shown in Fig. 2d and those of the modified MMK

model in Fig. 2e. These models gave better results but still deviated from the

experiments as in Fig. 2a. For example, the values of k at the sharp edge are still

over-predicted.

In this study, kmax was found above the cube as in Figs. 2b and 3b, indicating that

there is no recirculation bubble above the rib and the cube. Only in the experimentsshown in Figs. 2a, 3a and 5 are the positions of kmax found above the cube. Rib and

cube bubbles appear in these cases.

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In this study, with suitable treatment of the boundary condition of the sharp edge,

separate flow and recirculation size can be predicted by a standard k2 model. Theresults predicted by the modified LK and MMK models as shown in Figs. 2d and e

would not be the same as the results of the experiments. Note that the predicted kvalues at the shape edge are too big, and appear to have the same values as k above

the rib. That means the recirculation bubble is not as strong as in Fig. 2b, or there

may even be no recirculation. Therefore, modifying the turbulence model might not

be a good choice. In this paper, even a standard k2 model can predict results (as inFig. 9) which agree well with the experiments (as in Fig. 2).

For three-dimensional cases, predicted air flows around the cube are similar to

those found in the experiments. k-values measured by Murakami et al. [7,10,11] are

shown in Fig. 3a. kmax was found in the middle on the cube. The velocity

distributions are shown in Fig. 3b [10]. The separation for Fig. 3b is smaller than that

measured by Martinuzzi in Fig. 5a.

The results predicted by Murakami et al. [7,10,11] with standard k2 model forthree-dimensional simulation are shown in Fig. 3c. kmax was found at the upwind

sharp edge, which is different from the experiments. Efforts were made to modify

standard k2 models. Recently, Kimura and Hosoda [15] have been workingon similar studies. Figs. 4a and b are the same as Fig. 3a in giving predictions

which do not agree well with experiments. Modified standard k2 models suchas the KH model agreed well with the values of k distributions as measured

by Hussein and Martinuzzi [13] in Fig. 5b. Both experimental and predicted

results on distributions of k around the cube place kmax in the middle of the top ofthe cube.

In using the standard k2 model with proper boundary conditions near the sharpedge, the predicted results agree with the experiment. Incident flow profile of u

y1=m gives kmax at the top of the cube and the positions of kmax are correlated with

recirculation. This is clear from Figs. 9 to 14, which show the predicted results by

standard k2 model with different flow profiles.Turning to a more detailed discussion of the standard k2 model to study separate

flow and reattachmentoscillating vortices are found behind the cube. Computing

XF with transient and unstable properties matched better with the physics underlying

the LES model. XF was predicted to be 1.696 by Rodi et al. [12], reviewing relevantworks. For a wind tunnel height of 2H, the experimental XF value was 1.612.

However, the XF predicted by the standard k2 model was 2.182. Using the samemodel as shown in Figs. 10 and 11, XF lies between 2.2 and 2.3. For a wind tunnel of

height YBof 5.2H, XF was measured by Murakami [11] at 1.4, and XR at 0.7. The XFpredicted by LES was 1.2.

4. Predicted velocities at the sharp edge using the wall function

The uniform grids as in Fig. 16 are backward staggered for the Up-cell. At the wall,Uw-cell is at a distance of Dx=2 and VB-cell is Dy=2 from the scale node P. Thenear-wall values of the parallel velocity component, the turbulent kinetic energy k,

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and its dissipation rate are determined from the standard log-law wall function byLaunder and Spalding (1974).

For the wall bE0,

u 1

klnEy;

v 0: 2

For the wall bB0;

v 1

klnEx;

u 0: 3

The P-cell, U-cell and V-cell adjacent to the sharp edge are shown in Figs. 16a, c and

d. This might explain why the predicted Up amplified at the sharp edge.For the uniform grids as shown in Fig. 16bd, Dx Dy; the upper wall is bE0; and

the left wall is bB0: The values of the transport property are fI; J and fI 1; J;

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VB

VBB

BB

UB

B

W

WV

WU

VP

PUP

b

EV

UT T

EB

EEEUE

E

B(a)

n

Eb

B

w

(b)

P

VB

B

b

b

PUP E

(c) (d)

(I-1,J+1) U(i,J+1) (I,J+1)

V(I-1,j) V(I,j)

(I-1,J) U(i,J) (I,J)

Pbu

y/2

x/2

Bbv

Fig. 16. Computational cells around the sharp corner: (a) staggered grid arrangement, (b) the intersection

of a cell wall, (c) VB-cell and (d) Up-cell.



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etc. For the volume U(i,J), the upper wall is extended to W, and for the volume

V(I,j), the left wall is extend to n. It can be seen that for the staggered grids using the

wall function at the upper wall, U(i, J) would move Dx=2 to the left; for the left wall

V(I,j) would move Dy=2 upwards. These figures should be handled carefully to avoidgetting incorrect results at the windward sharp corner.

For the Up-cell as shown in Fig. 16d, only half of the cell face would be in contact

with the wall. When applying the wall function to fI; J; bE0 would move Dx=2 tothe left relative to the U-cell and become WbE0: The convective fluxes rvBb Dx=2Dz (for two-dimensional conditions, Dz 1) can be neglected across the half-face. If the grids are not right-angled coordinates, or are not at a uniform distance, it

will be difficult to evaluate the effective area for calculating the flux.

For the Up-cell, since rvBbDx=2 is not considered at the near-wall wb and v 0;

and Vp is approximated as Vpmin

; Up has reached its maximum value Upmax

: Errorscould result from neglecting the inflow ofrvBbDx=2; taking it as the wall in the Up-cell. IfvBb is very small, the effect would not be significant. At near-wall, vBb and VBare often very close to their maximum value in all flow fields, with very significant

effects.

When there is no heat source in the flow field and enthalpy is unchanged,

neglecting compressibility would not have a significant effect. Bernoullis equation

can be applied to the streamlines along the frontal corner for the stagnation pressure

P0, and static pressures PP and PB,

Pp

rp

2V2

p U2

p P

BrB

2V2

B U2

B P

0 constant: (4)

From the above equation, it can be seen that when P0 is constant, Vp would be Vpminwhen Up is Upmax : Numerical calculations indicate that when the scale of the Up-cellDx Dy4 1

20H; Upmax almost reaches the largest Umax value of the whole flow field.

Thus, using wall function for staggered grids when the effect of rupbDy=2 is notincluded, the value of Up would be amplified to Upmax ; approaching the largest valueof Umax in all flow fields.

Similarly, for the VB-cell as shown in Fig. 16c, the wall function extends upwards

from bB0 to become nbB0, neglecting the effect of outflow of mass rvpbDx=2 on VB-cell. The inflow velocity on the surface

Bon the left-hand side of the wall at near the

VB-cell would increase continuously until flowing out on the surface P. This is not

because the inflow velocity on the surface Brupb has already reduced when flowing

out on the surface P. For the VB-cell, VBhas reached VBmax ; and UB is approximatedas zero.

On the other hand, why Up was amplified with south extended wall can be

explained. The discretised u-momentum equation for the velocity U(i, J+1) at

location (i, J+1) shown as in Fig. 16 is given by (replacing i, J+1 with p):

apUp X

aiUi PP PEDy bp; (5)

bp is the momentum source term and is neglected here. The E, W, N and Sneighbours are involved in the summation

PaiUi; and the values of coefficients ai

may be calculated with any of the different methods suitable for convection

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diffusion problems.

Up PaiUi PP PEDyP

ai UE;W;N;S

PP PEDyPE;W;N;Si ai

: (6)

If the south area of the volume Up is wall, aS 0:

Upwall UE;W;N

PP PEDyPE;W;Ni ai

: (7)

Near the south wall, USo U; so UE;W;N4 UE;W;N;S; giving Upwall4Up:When US % 0; Upwall Up reaches maximum value, and Upwall Upmax ; Vpwall is

reduced to Vpmax :

5. Comparison between the enlarged value ofUp and the confined Up value

In the Up-cell, Up reaches it maximum value Upmax ; and Vp can be approximatedby Vpmin or zero. In the VB-cell, VB VBmax ; UB is also approximated as UBmin orzero. From this, there would be fluid flow at the left-hand side of the sharp corner at

a velocity of VBmax : It then moves to the right at a velocity ofUpmax after turning for901. The streamlines are shown as in Case I in Table 2.

For the whole flow field, the velocity and momentum at the top of the sharp corner

have become Upmax after turning for 901 from VBmax upwards. The change rate is thelargest at the sharp corner. Therefore, kmax also appears there as shown in Figs. 8a

and 15a.

Assuming Up % 0; then Vp Vpmax % VB: That means there would be backflow atthe top of the rib and cube. Since there would be flow velocity in the opposite

direction near the center of the backflow, kmax is found at the top of the rib and cube

due to the largest variation in velocity there. This method has been widely used in the

literature [14]. The results shown in Figs. 9a and 10a14a are also obtained by using

this method. Normal conditions are listed as Case II in Table 2.

In real situations in the P-cell, the vector should be ~Up ~Vp: The angle between it

and ~Up is:

a arc cos~Up

~Up ~Vp

; (8)~Up ~Vp

and a determine together whether there will be backflow, the degree ofseparation and the bubble length XR, as listed in Case III in Table 2. When ao45

;there is no separation. When 45oao70; there is only a little separation. Whena470; there is more separation.

The above indicates that the wall function at the sharp corner should be assigned

carefully when using staggered grids. This is very important at the windward sharpcorner, and in the first neighboring grid P at the solid sharp corners with strong

intersections ofru and rv:

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Recent studies on the value of k surrounding the windward sharp corner and the

separated backflow are useful for assessing the calculation results. This is very

obvious when determining kmax. The position of kmax can be used as an important

reference for validating the calculation results.For Case III as listed in Table 2, it can be seen that under two-dimensional

conditions, Up % 0:087Vp: Under three-dimensional conditions, Up % 0:26Vp: Thevelocity vectors and the distribution of kmax are closer to the situation of Case II

where Up is approximated as zero. Under two-dimensional conditions, approximat-

ing Up as zero basically agrees with the practical situations. Three-dimensional

simulations would be very close to the real situations.

6. Conclusion

The following can be concluded from this study:

The standard k2 turbulence model can predict well the separate andreattachment flow on the top surface of the cube and rib if the wall boundary is

treated properly. Assigning correct values ofUp in front of the cube and rib edge is

the key point in obtaining numerical results that agree better with the experiments.

Both experiments and measurements indicate that the prediction of therecirculation bubble is related to whether kmax is at the upper cube. If kmax is

found at the sharp edge, there will be no recirculation. Locations of kmax are affected by the incident flow profile u y1=m and Re:

When and are increased slightly, the location of kmax will move upward and

backward.

Air speed at the sharp edge is determined by interaction of ru and rv: In theabsence of more accurate information about the values of the velocity ~Up ~Vp

and a; an alternative that appears to work well is to set to zero. Reasonableexplanations for this have been presented in this paper.

Acknowledgement

This project was supported by the funding generated by the WKC at the Research

Centre for Fire Engineering of the Department of Building Services Engineering,

The Hong Kong Polytechnic University.

References

[1] F. Baetke, H. Werner, H. Wengle, Numerical simulation of turbulent flow over surface-mounted

obstacles with sharp edges and corners, J. Wind Eng. Ind. Aerodynamics 35 (1990) 129147.

[2] I. Castro, A. Robins, The flow around a surface-mounted cube in uniform and turbulent streams,

J. Fluid Mech. 79 (1977) 307335.

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[3] D. Paterson, C. Apelt, Simulation of flow past a cube in a turbulent boundary layer, J. Wind Eng.

Ind. Aerodynamics 35 (1990) 149176.

[4] D. Paterson, C. Apelt, Computation of wind flows over three-dimensional buildings, J. Wind Eng.

Ind. Aerodynamics 24 (1986) 193213.[5] S. Murakami, A. Mochida, 3-D Numerical simulation of airflow around a cube model by means of

the k2 model, J. Wind Eng. Ind. Aerodynamics 31 (1988) 283303.[6] S. Murakami, A. Mochida, Y. Hayashi, Examining the k2 model by means of a wind tunnel test and

large eddy simulation of the turbulence structure around a cube, J. Wind Eng. Ind. Aerodynamics 35

(1990) 87100.

[7] S. Murakami, Comparison of various turbulence models applied to bluff body, J. Wind Eng. Ind.

Aerodynamics 46 & 47 (1993) 2136.

[8] S. Murakami, A. Mochida, Y. Hayashi, S. Sakamoto, Numerical study on velocitypressure field and

wind force for bluff bodies by k2; ASM and LES, J. Wind Eng. Ind. Aerodynamics 4144 (1992)28412852.

[9] S. Murakami, A. Mochida, K. Hibi, Three-dimensional numerical simulation of air flow around a

cubic model by means of large eddy simulation, J. Wind Eng. Ind. Aerodynamics 25 (1987) 291305.[10] M. Tsuchiya, S. Murakami, A. Mochida, K. Kondo, Y. Ishida, Development of a new k2 model for

flow and pressure fields around bluff body, J. Wind Eng. Ind. Aerodynamics 67 & 68 (1997) 169182.

[11] S. Murakami, Overview of turbulence models applied in CWE1997, J. Wind Eng. Ind.

Aerodynamics 7476 (1998) 124.

[12] W. Rodi, J. Ferziger, M. Breuer, M. Pourquie, Status of large eddy simulation: results of a workshop,

J. Fluids Eng. 119 (1997) 119249.

[13] H. Hussein, R. Martinuzzi, Energy balance for turbulent flow around a surface mounted cube placed

in a channel, Phys. Fluids 8 (3) (1996) 764780.

[14] R. Martinuzzi, B. Havel, Turbulent flow around two interfering surface-mounted cubic obstacles in

tandem arrangement, J. Fluids Eng. 122 (2000) 2431.

[15] I. Kimura, T. Hosoda, Numerical simulation of flows around a surface-mounted cube by means of a

non-linear k2

model, Ninth International Symposium of Flow Visualization, Edinburgh, Scotland,UK, 2000.

[16] A. Larousse, R. Martinuzzi, C. Tropea, Flow around surface-mounted three-dimensional obstacles,

in: Turbulent Shear Flows, vol. 8, Springer, New York, 1993, pp. 127139.

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