Vertical Wind Velocity Distribution in Typical Hilly Terrain

Wen-juan Lou1), * Hong-chao Liang2), Zheng-hao Li3), Li-gang Zhang4) and Rong Bian5)

1), 2), 3)

Institute of Structural Engineering, Zhejiang University, Hangzhou, Zhejiang, 310058, China

4), 5) State Grid Zhejiang Economic Research Institute, Hangzhou, Zhejiang, 310000,

China 2)

lianghczju@126.com

ABSTRACT

In this paper, the distribution of vertical wind velocity in typical hilly terrain was studied in consideration of varied influence factors, including the height and the slope of the hill and the length of the corresponding ridge. The vertical wind velocity field of a typical steep hill, measured by a wind tunnel test was established as a benchmark for verifying the CFD simulation method. Then the numerical method was used to study the distribution of vertical wind velocity, considering several topographic features. The numerical results indicated that the vertical wind velocity on the upwind slope, could be up to over 60 percent of the income flow wind speed and the maximum vertical wind velocity was located at about two thirds height of the hill. The profiles of the vertical wind velocity ratio at the top of the hill satisfied the exponential law perfectly and the equations were fitted to profiles with the change of hill height and slope. In the case of 90° wind angle, the maximum updraft velocity on upwind slope and mountaintop both increased with the length of ridge. While at 0° wind angle, the maximum updraft velocity on upwind slope decreased with the length of ridge. 1. INTRODUCTION

Vertical wind speed was unfavorable to transmission lines or other structures. Only in the flutter stability analysis of the large span bridge, wind attack angle was taken into consideration. General civil engineering structures tended to neglect the disadvantages of updraft wind in hilly terrain. Researches of hilly terrain wind field had been started since the 80's of last century. Jackson and Hunt (1975) analyzed the wind speed distribution of a two dimensional smooth hill. And developed an analytical method to predict speed-up over it. Since the model’s slope was smooth, vertical wind velocity component in the wind field was small, and it’s only regarded as disturb to the horizontal wind speed. Taylor (1984) made extensions to above formulas to calculate the velocity of wind speed at different heights. In Weng’s work (2000), the influence of roughness was introduced to the calculation process, which made the theoretical method became

more mature. Up to the 90's, considerable mountain measurements had been accomplished. Taylor et al (1987) analyzed and summarized those researches, and came to a conclusion that the theoretical calculation results of speed-up factor was quite different with the measured results at the near surface. The numerical simulation and wind tunnel tests were compared by Bitsuamlak (2004), It came to the conclusion that the numerical simulation results can match well with the experimental results in the upwind slope. Cao Shuyang (2012, 2014) and Yassin (2014) studied the effect of income roughness, hill slope and inflow turbulence to speed-up factor. Yassin concluded that two dimensional CFD simulations were compared well with the experimental results. Furthermore, Takeshi (1999) experimentally studied the typical steep hill though three-dimensional wind velocity measurement devices. The vertical wind speed was observed to be 28.5% of the horizontal wind speed of inflow on the upwind slope. However, the vertical wind speed had not been further studied in details.

The above researches improved the understanding of wind field in hilly terrain, and provided more mature wind tunnel test and numerical simulation methods. However, there were two shortcomings in those studies. Firstly most of the current studies focused on the horizontal wind velocity distribution, researches of vertical wind velocity distribution had been almost in blank state, let along studies about how distribution varied with the change of height, slope and length of ridge. Secondly most of current hill models had no ridge, but ridge length could influent the wind speed distribution significantly. For such reasons, this paper referred to the previous studies, carried out a wind tunnel test on a steep hill model with ridge, measured the vertical wind velocity component. Then numerically simulated the hill model, and studied the distribution of vertical wind velocity, and how the distribution varied with hill height, slope and length of ridge.

2. Details of Wind tunnel test

The cosine-squared cross section model was the most widely studied hill model previously, so in this paper the cross section of model was also given by as follow

2cos ( ),2

xy H x D

D (1)

Where the hill height H was 100 m and the hill bottom radius D was 150 m in the

experiment. Besides the length of the hill L was considered to be 300 m. The experiment was carried out at wind tunnel ZD-1 in Zhejiang University. The wind tunnel had an

section of 3 m×4 m. The geometrical scale of model was chosen as 1:500, so it could

be fitted in the wind tunnel with a block ratio less than 5%. The model was manufactured from polystyrene foam, and the surface of the model was stuck with synthetic fibers, which increased the surface roughness and reduced the Reynolds number effect near the surface of the hill (Lubitz W D, White B R 2007). Fig. 1 showed the photo of experiment model. The wind profile was simulated to an exponent law, and the exponent achieved was 0.15. Cobra Probe was used to measure three components of

the velocity vector. Cobra Probe was produced by TFI（Turbulence Flow

Instruments）in Australia, was used to measure the three-dimensional turbulent flow

signals. The probes were made of 4 pressure sensors and available for three-dimensional turbulent wind velocity and static pressure with the response frequency up to 2 kHz and the precision accuracy of velocity within ±0.5 m/s. Definition of wind angle and layout of measurement points were as shown in Fig.2. Where measuring points were divided into solid points and hollow points. The three-dimensional wind velocity was measured at the heights 10 m~300 m at solid points, and 10 m only at hollow points.

D50▲wind angle D4 D3 D2D1 C1

C2

C3

C4

C5

C6

B1

B2

B3

B4

B5

B6

A1

A2

A3

A4

A5

A6

E3

E5

Y

X

90▲wind angle

lenth of ridge

bottom radius

Fig.1 Photo of experiment model Fig.2 Definition of wind angle and layout of

measurement points

3. Numerical simulation and its verification

CFD analysis method was applied to investigate the distribution of vertical wind velocity. The numerical model chose the same geometrical scale as experiment model. The roughness height of ground and hill surface were set up to 0.5 m and 1 m.

Realizable k-epsilon model was chosen as turbulence model. To provide an self-maintenance of wind profile, the turbulent kinetic energy model was modified to Eq. (2)~(4), where essential height Z0 and essential velocity U0 were 10 m and 10m/s,

exponent α was 0.15, turbulence I(z) and turbulence integral scale Lu were defined by

Japanese Code (2004). Parameter Cμ and K in the equations were defined as 0.09 and

0.42.

0

0

( ) ( )Z

U z UZ

(2)

2( ) 0.5[ ( ) ( )]k z U z I z (3)

3 4 3 24 ( )

u

C k z

L (4)

-200 -150 -100 -50 0 50 100 150 2000

100

200

300

400

500

Experiment result

CFD result

He

igh

t (m

)

Disance (m)

-200 -150 -100 -50 0 50 100 150 2000

100

200

300

400

500

Experiment result

CFD result

Heig

ht (m

)

Distance (m)

5m/s

（a）profile of speed-up factor （b）profile of vertical velocity

Fig.3 Speed-up factor and vertical velocity profiles at middle position of the hill(test point A1~A6)

-200 -150 -100 -50 0 50 100 150 2000

100

200

300

400

500

Experiment result

CFD result

He

igh

t (m

)

Distance (m)

-200 -150 -100 -50 0 50 100 150 2000

100

200

300

400

500

5m/s

Experiment result CFD result

Heig

ht (m

)

Distance（m） （a）profile of speed-up factor （b）profile of vertical velocity

Fig.4 Speed-up factor and vertical velocity profiles at left side of the hill(test point C1~C6)

-300 -200 -100 0 100 200 3000

100

200

300

400

500 Experiment result

CFD result

Heig

ht (m

)

Distance (m)

-300 -200 -100 0 100 200 3000

100

200

300

400

500

Experiment result

CFD result

5m/s

He

igh

t (m

)

Distance (m) （a）profile of speed-up factor （b）profile of vertical velocity

Fig.5 Speed-up factor and vertical velocity profiles at cross section of the hill(test point A1,B1,C1,D1~D6)

As shown in Fig. 3~5, the speed-up factor, which was defined by Eq. (5), where

∆S(z) was the speed-up factor at a height z above the hill, U(z) was the horizontal wind

speed at same height, While U0(z) was the upstream speed of the hill. The speed-up was well-fitted with the experiment, especially at the crest and foot of hill. The vertical wind speed profiles fitted with experiment even better. And it can found that vertical wind velocity could up to about 60% of income velocity at point A2 and C2. Which implied that vertical wind speed can’t be ignored.

（ ） （ ）

（ ）（ ）

0

0

U z U zS z

U z (5)

Further comparison to previous study conducted by Takeshi etc (1999) was made to

test the feasibility of numerical simulation method. The model was simulated with the set up mentioned above, and comparison was made in Fig 8. Also the simulation result had a good agreement with Takeshi’s work.

0

100

200

300

400

0.1-0.1 0-0.2 0.2-0.4 -0.3

Test point

He

igh

t (m

)

Speed-up factor

CFD result

Takeshi' experiment

Income flow

0

100

200

300

400

0.40.20-0.2-0.4

CFD result

Takeshi' experiment

income flow

test point

Heig

ht (m

)

Speed-up factor

0

100

200

300

400

0.40.20-0.2-0.4

CFD result

Takeshi' experiment

Test point

Speed-up factor

Heig

ht (m

)

Income flow

Fig.6 Result compared with experiment result down by Takeshi et al

4. Characteristic and fitting of vertical wind velocity 4.1 Selection of fitting formula

Plenty of work had conducted to reveal the distribution of speed-up factor profile, but little researches had carried out to study the vertical wind velocity. This section introduced 2 typical methods to calculate speed-up factor in hilly terrain.

(1) Original Guidelines Original Guidelines were proposed by Taylor and Lee (1984), it used formulation as

below

max 1

max 1

/

exp( / )

S Bh L

S S Az L(6)

Where ΔS was speed-up factor, z was the height, A, B were fitting parameters,

which determined by the geometric characteristics of the hill. L1 means the horizontal distance from the peak to the 1/2 height of hill on the slope.

(2) New Guidelines Wensong Weng (2000) based on the original algorithm, proposed new Guidelines

as follow

3 1

max 11 2

1 0

( / )

1 2 1 1

max

= ln( )

(0, )ln ( ) A z L

S LB B

H L z

S zA A z L A e

S

(7)

Where B1, B2 were fitting parameters like A, B. Ai（i=1, 2, 3）were constant related

to the Surface roughness. Other parameters and values were the same as above. As discussed above, speed-up factor profile were commonly described in the

exponent form, so this paper tried also using exponent form to describe the distribution of vertical wind velocity ratio Rz, which was defined as

0

zz

WR

U (8)

Where Wz was the vertical wind velocity at height z, and U0 is the wind velocity at

the same height of income.

4.2. The effect of hill height

Model heights studied in this section varied from 50 m to 400 m, while their bottom radius D remained 1.5H, to provide them with a same slope. The distribution of vertical wind speed along the center line of upwind slope was shown in Fig. 7(a). When -x/D equaled to 0 and 1, respectively, representing the crest and upwind foot of the hill. It’s interesting that although hill height H varied, the maximum vertical wind speed appeared at almost the same place, about at two thirds height of upwind slope. And with the increase of hill height H, the position slightly moved upward. The value of maximum vertical wind velocity increased with the hill height, when the height H changed from 50 m to 400 m, the speed-up factor increased 1.88 times.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Rz

-x/D

H=50m

H=100m

H=200m

H=300m

H=400m 0 200 4000.4

0.8

Maxim

um

Rz o

n u

pw

ind s

lop

Hill height (m)

0.00 0.04 0.08 0.12 0.16

0

100

200

300

400

Heig

ht(

m)

Rz

H=50m

H=100m

H=200m

H=300m

H=400m 0 200 400

0.08

0.12

Rz

at c

rest

on

10

m

Hill height (m)

（a）Rz along upwind slope （b）Rz at crest

Fig.7 Vertical wind velocity ratio distribution under different hill heights Fig. 7(b) showed profiles of vertical wind velocity ratio at the crest of hill under

different hill heights. As shown in figure, the maximum vertical wind velocity appeared at

the height of 0.1H, which when the hill height increase, it increased at first and then decreased.

-7 -6 -5 -4 -3 -2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

z/H

ln(w/U0)

H=50m

H=100m

H=200m

H=300m

H=400m

Fig.8 Linearity between dimensionless height and logarithmic dimensionless vertical

velocity under different hill heights

As shown in Fig. 8, linearity, ln(w/U0)=az/H+b, was significant between dimensionless height z/H and logarithmic dimensionless vertical velocity above and below z/H=0.1. So Eq. (9) was fitted to describe the profile, while z/H=0.1 was regard as the demarcation point. In this fitting, more attention was paid to clarify the cases where hill heights were smaller than 300 meters, so that the equation could be simplified. The comparison between fitting and numerical simulation was made, and shown at Fig. 9. The equation result can match the CFD result perfectly when hill heights H were below 300 m, and become larger than the CFD result when H was larger than 300 m.

（ ） （ ）

（ ） （ ）

0 exp( / )

1, / 0.1 -1.45, / 0.1

-2.045, / 0.1 -1.80, / 0.1

w U az H b

a z H or z H

b z H or z H

(9)

0 0.05 0.10

50

100

150

200

250

300

350

400

Rz

Heig

ht(

m)

H=50m

CFD data

Fitting data

0 0.05 0.10

50

100

150

200

250

300

350

400

Rz

Heig

ht(

m)

H=100m

CFD data

Fitting data

0 0.05 0.10

50

100

150

200

250

300

350

400

Rz

Heig

ht(

m)

H=200m

CFD data

Fitting data

0 0.05 0.10

50

100

150

200

250

300

350

400

Rz

Heig

ht(

m)

H=300m

CFD data

Fitting data

0 0.05 0.10

50

100

150

200

250

300

350

400

Rz

Heig

ht(

m)

H=400m

CFD data

Fitting data

Fig.9 Comparison between fitting and numerical simulation under different hill

heights

4.2 The effect of hill slope

Models’ height H studied in this section were 100 m, and their bottom radius D varied from H to 4H.

0.0 0.2 0.4 0.6 0.8 1.0

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

4H2H

D=1.0H

D=1.5H

D=2.0H

D=2.5H

D=3.0H

D=4.0H

Rz

-x/D

0.4

0.6

Ma

xim

um

Rz o

n u

pw

ind

slo

p

Bottom radius(m)

0.00 0.04 0.08 0.12 0.16 0.20

0

100

200

300

400

4H2H

D=1.0H

D=1.5H

D=2.0H

D=2.5H

D=3.0H

D=4.0H

Heig

ht (m

)

Rz

0.00

0.08

0.16

Rz a

t cre

st

on1

0 m

Bottom radius(m)

（a）Rz along upwind slope （b）Rz at crest

Fig.10 Vertical wind velocity ratio distribution under different hill bottom radius

As shown in Fig. 10 (a), when the bottom radius increased, the maximum vertical wind speed decreased. It’s same that the maximum vertical velocity appeared at the fixed position on the upwind slope, which was about 2/3 H of the upwind slope. The maximum Rz can be over 0.65 when the hill was steep enough to H/D=1, and it decreased to about 0.3, when the slope changed to H/D=0.25. As shown in Fig. 10(b) the vertical wind speed at crest also decreased with the bottom radius increased. And when the slope H/D was less than 0.25, the vertical wind velocity almost can be neglected.

-7 -6 -5 -4 -3 -2 -1

0

1

2

3

4

z/H

ln(w/U0)

D=1.0H

D=1.5H

D=2.0H

D=2.5H

D=3.0H

D=4.0H

Fig.11 Linearity between dimensionless height and logarithmic dimensionless

vertical velocity under hill bottom radius

Still, the linearity between dimensionless height z/H and logarithmic dimensionless vertical velocity ln(w/U0) was significant, as shown in Fig. 11. The exponential law

function was still used to fit those profiles. When the hill bottom radius changed. The equation was little more complicated, since the parameter a and b changed significantly with the slope D/H changed. It can be expressed as follow

0

2

3 2

exp( / )

0.05( / ) 0.46( / ) 1.68

0.27( / ) 2.09( / ) 4.02( / ) 4.20

w U az H b

a D H D H

b D H D H D H

(10)

The fitting result were as follows:

0 0.05 0.1 0.150

100

200

300

400

Rz

Heig

ht(

m)

D=1H

CFD data

Fitting data

0 0.05 0.1 0.150

100

200

300

400

Rz

Heig

ht(

m)

D=1.5H

CFD data

Fitting data

0 0.05 0.1 0.150

100

200

300

400

Rz

Heig

ht(

m)

D=2H

CFD data

Fitting data

0 0.05 0.1 0.150

100

200

300

400

Rz

Heig

ht(

m)

D=2.5H

CFD data

Fitting data

0 0.05 0.1 0.150

100

200

300

400

Rz

Heig

ht(

m)

D=3H

CFD data

Fitting data

0 0.05 0.1 0.150

100

200

300

400

Rz

Heig

ht(

m)

D=4H

CFD data

Fitting data

Fig.12 Comparison between fitting and numerical simulation under different hill

bottom radius 4.3. The effect of length of ridge

This section focused on the influence of the length of ridge on vertical wind velocity distribution. Table 1 showed the cases of numerical simulation. Two wind direction angles were studied respectively. The hill height H was 100 m and the hill bottom radius D was 150 m for those cases.

Table 1 Cases of numerical simulation with different hill ridge

Wind angle Length of ridge

90° 0.0H, 0.5H, 1.0H, 2.0H, 3.0H, 4.0H, 5.0H, 8.0H 0° 0.0H, 1.0H, 3.0H, 5.0H, 8.0H

When the wind angle was 90°, where the wind direction was normal to the ridgeline,

the over-hill effect became more obvious with the increase of the ridge length at the middle of the ridge. As was shown in Table 2, with the increase of ridge length, the

maximum vertical wind velocity ratio Rz increased both at the hillside and crest, especially at the middle of ridge, the Rz doubled when the L became 8H.

Table 2 Updraft velocity ratio at the height of 10m under 90° wind angle

Ridge length L 0.0H 0.5H 1.0H 2.0H 3.0H 4.0H 5.0H 8.0H Maximum Rz on upwind slope

0.62 0.67 0.68 0.69 0.69 0.69 0.70 0.71

Maximum Rz at middle of ridge

0.13 0.19 0.23 0.26 0.27 0.28 0.28 0.29

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.50.16

0.18

0.20

0.22

0.24

0.26

0.28

0.30

R

z

x/L

L=0.5H

L=1.0H

L=2.0H

L=3.0H

L=4.0H

L=5.0H

L=8.0H

Fig.13 Updraft velocity ratio changes along the ridge at the height of 10m under 90°

wind angle

When wind flow over the ridge, the vertical wind velocity presents a certain distribution as shown in Fig. 13. While x, as shown in Fig. 2, meant the distance to the middle of hill. That made x/L present the location along ridge. When x/L was -0.5 or 0.5, it located at left or right crest of the hill. Since the over-hill effect was more obvious at the middle of ridge, velocity ratio was larger at middle of ridge than at crest of the hill. And the velocity ratio changed within 1/10L range around the crest.

When the wind direction angle was 0°, where the wind direction was perpendicular to the ridgeline, the maximum vertical wind speed was significantly decreased with the increase of the length of ridge, and the vertical wind speed ratio at crest was almost remain the same, as shown in Table 3

Tab 3 Updraft velocity ratio at the height of 10m under 0° wind angle

Length of ridge 0.0H 1.0H 3.0H 5.0H 8.0H Maximum Rz on upwind slope 0.62 0.62 0.50 0.36 0.36 Maximum Rz at middle of ridge 0.13 0.15 0.15 0.15 0.15

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.10

-0.05

0.00

0.05

0.10

0.15

Rz

x/L

L=1.0H

L=3.0H

L=5.0H

L=8.0H

Fig.14 Updraft velocity ratio changes along the ridge at the height of 10m under 0°

wind angle As shown in Fig. 14, x equated to -0.5 and 0.5, respectively, representing the

upwind and leeward crest of the hill. Vertical wind velocity in the top of hill near windward slope was updraft flow, in the middle region of the ridge was almost zero, when at the proximity of the leeward slope, it changed to downdraft flow. Same as 90° wind angle, the velocity ratio changed within 1/10L range around the crest.

5. Conclusions

This paper revealed the distribution of vertical wind speed in hill terrain. Studied the influence of topographic features including hill height, hill slope and length of ridge. Fitted the profiles of vertical wind velocity ratio at the crest with varied hill heights and hill slope. And came to conclusions as follows:

1. Vertical wind speed on the upwind side, could up to about 60 percent of the income flow speed and maximum vertical wind velocity located at about two thirds height of the hill on upwind slope.

2. Vertical wind velocity ratio profiles satisfied exponential law at the top of the hill and equations were fitted to profiles with the change of hill height and slope.

3. In the case of 90° wind angle, the maximum updraft velocity on upwind slope and mountaintop both increased with the length of ridge. While at 0° wind angle, the maximum updraft velocity on upwind slope decreased with the length of ridge. REFERENCES Jackson P S, Hunt J. (1975), ―Turbulent wind flow over a low hill,‖ Quarterly Journal of

The Royal Meteorological Society, 101(430), 929-955. Taylor P, Lee R. (1984), ―Simple guidelines for estimating wind speed variations due to

small scale topographic features,‖ Climatolog. Bull, 18, 3-32.

Wensong W, Peter A. T, John L. W. (2000), ―Guidelines for air flow over complex

terrain: model developments,‖ Journal of Wind Engineering and Industrial Aerodynamics, 169-186.

Taylor P A, Mason P J B E. (1987), ―boundary layer flow over low hills,‖ Boundary-Layer Meteorology, 1-2(39), 107-132.

Bitsuamlak G T, Stathopoulos T, Bedard C. (2004), ―Numerical evaluation of wind flow over complex terrain: Review,‖ Journal of Aerospace Engineering, 17(4), 135-145.

Wang T, Cao S, Ge Y. (2014), ―Effects of inflow turbulence and slope on turbulent boundary layers over two-dimensional hills,‖ Wind And Structures, 19(2), 219-232.

Cao S, Wang T, Ge Y, et al. (2012), ―Numerical study on turbulent boundary layers over two-dimensional hills — Effects of surface roughness and slope,‖ Journal of Wind Engineering and Industrial Aerodynamics, 104-106, 342-349.

Yassin M F, Al-Harbi M, Kassem M A. (2014), ―Computational Fluid Dynamics (CFD) Simulations on the Effect of Rough Surface on Atmospheric Turbulence Flow Above Hilly Terrain Shapes,‖ Environmental Forensics, 15(2), 159-174.

Takeshi I., Kazuki H. and Susumu O. (1999), ―A wind tunnel study of turbulent flow over

a three-dimensional steep hill‖, Journal of Wind Engineering and Industrial

Aerodynamics, 83(1–3), 95-107. Lubitz W D, White B R. (2007), ―Wind-tunnel and field investigation of the effect of local

wind direction on speed-up over hills,‖ Journal of Wind Engineering and Industrial Aerodynamics, 95(8), 639-661.

(2004)‖AIJ recommendations for loads on buildings,‖ Japan A I O.