High-Reynolds-number effects simulations for wind effects on a cooling tower model in a wind tunnel based on a statistical approach


The traditional high-Reynolds-number (Re) effects simulation method for wind effects on circular cylindrical models in wind tunnels is fundamentally a trial-and-error practice, which is strongly subjective and laborious. To this end, a mathematically well-founded high-Re effects simulation method is proposed based on the response surface (RS) optimization algorithm in this article. Having the advantages of good considerations of random errors and simplified calculations using the polynomial models, RS method has been widely employed in chemical industry. However, it is seldom employed in physical experiments. Using a case study of Weisweiler cooling tower, it has been proved that model tests using the new high-Re effects simulation method based on RS optimization algorithm can objectively and easily reproduce the mean and the fluctuating wind pressure distributions measured on the spot, and the model-scale spectral characteristics of the wind pressures obtained by using the new high-Re effects simulation method agree well with the full-scale ones. Besides, numerical analyses undertaken on a computational fluid dynamics platform also indicate the validity of the new method.

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Reprinted from Cheng et al. [1]

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Fig. 10


k :

Relative roughness

a :

Rib spacing

b :

Width of the rough zone

e :

Thickness of the rough zone

S A :

Square error caused by uncertain parameter

S e :

Square error caused by experiment

f A :

Dimension of freedom of square error caused by uncertain parameter

f e :

Dimension of freedom of square error caused by experiment

α :

Significance level

F A :

\(\frac{{S_{A} /f_{A} }}{{S_{e} /f_{e} }}\)

y :

Response of the model/outputs of the experiment

x i :

Significant uncertain parameter

β 0, β i, β ij, β ii :

Coefficients in Eq. (3)

R 2 :

Value R2

y R S (j):

Outputs of the RS

\(\overline{y}\) :

Mean of the outputs of the experiment

p :

Design parameter

{f A}:

Analytical structural response vector

{f E}:

Experimental model response vector


Lower bound of design space


Upper bound of design space

R :

Residual vector

n :


a 1(θ), b 1(θ), and β p(θ):

Coefficients in Eqs. (79)


Power law exponent

D :

Diameter of the throat

L u x :

Integral scale of turbulence

θ, θ′:

Circumferential positions

σ p 2 (z, θ):

Mean square of the fluctuating pressure sample

S p (z, θ, n):

Spectra of fluctuating pressures

R f (θ, θ′, n):

Circumferential coherence between fluctuating pressures

β 3 :

Coefficient in Eq. (10) (a constant = 25)


Mean wind speed at the gradient height

δ :

The gradient height

C 2 (θ, θ′):

Coefficient in Eq. (10)

K S :

Equivalent sand roughness

E :

Coefficient in Eq. (12) (a constant = 9.793)

C s :

An empirical dimensionless factor

K S + :

Re roughness


Friction velocity

υ :

Kinematic viscosity

u + :

\(\frac{u}{{u^{*} }}\)

κ :

Coefficient in Eq. (14) (a constant = 0.42)

y :

Wall distance of the inner-layer grid

y + :

Dimensionless distance normal to the wall surface determined empirically


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The authors gratefully acknowledge the financial supports from the National Natural Science Foundation of China (Grant No. 51908124) and the China Postdoctoral Science Foundation (Grant No. 2016M601793).

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Correspondence to L. Zhao.

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Cheng, X.X., Zhao, L. & Ge, Y.J. High-Reynolds-number effects simulations for wind effects on a cooling tower model in a wind tunnel based on a statistical approach. J Braz. Soc. Mech. Sci. Eng. 43, 99 (2021). https://doi.org/10.1007/s40430-021-02816-w

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  • Cooling tower
  • Wind tunnel model test
  • Reynolds number
  • Response surface method
  • Mean/fluctuating wind pressure distribution
  • Spectral characteristics
  • Numerical simulation