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The Eighth Asia-Pacific Conference on Wind Engineering,December 10–14, 2013, Chennai, India

APPRAISAL OF CERTAIN LINEAR AND NONLINEAR k- TURBULENCE MODELS

S. Mandal1, P. Bhargava2 and C. S. P. Ojha2

1. Associate Prof., Dept. of Civil Engrg., IIT (BHU), Varanasi, UP, India, [email protected] 2. Professor, Dept. of Civil Engrg., IIT, Roorkee, UK, India. [email protected], [email protected]

ABSTRACT To simulate wind flow around bluff bodies, a large number of turbulence models have been proposed in the literature. Basically, these models can be classified into two categories, i.e., linear and nonlinear models. Such a classification is often used to interpret the possible interrelationship for stresses and strains within a flow field. Thus, it is expected that linear turbulence models should reflect a linear relationship between stresses and strains. In case of nonlinear models, attempts have been made to introduce more and more higher order terms in the linear models to enhance their nonlinear characteristics. Present study analyses a flow field data to ascertain the behaviour of linear and nonlinear models in terms of the stress and strain response. As the production and destruction terms in turbulence closure models also depend on a stress term, it also looks into the variations of production and destruction terms in turbulent transport equations in different zones of the turbulent flow field. Here, the variations of these terms have been also examined using linear and nonlinear turbulence models and the variations of these terms are linked with inherent nonlinearity in the turbulent models. To support the analysis, the stress strain relationship based on continuum approach has also been scrutinized using the data. Keywords: Computational wind engineering, linear and nonlinear k- turbulence models Introduction

Linear and nonlinear turbulence models have been used in the literature for simulating turbulent wind flow around bluff bodies. The earlier proposed linear models incorporate Boussinesq’s (1877) concept of eddy viscosity, which assumes an isotropic eddy diffusivity in modelling Reynolds stress tensor. These models use a linear relationship between stress and strain rate tensor. It has been observed in the literature (Speziale 1987) that normal Reynolds stresses are inaccurately predicted when these models are employed. To deal with turbulence anisotropy, the constitutive relation between the stress tensor and the strain rate tensor has been a subject of modifications by several investigators including Speziale (1987), Myong and Kasagi (1990), Acharya et al. (1994). These models are also referred in the literature as nonlinear models. Modification of stress strain relationship has an influence on the various terms, i.e., production, destruction of turbulence closure models (Mandal et al. 2001). In other words, these terms are also influenced by the inherent nonlinearity in the used stress models. The way this inherent nonlinearity may influence these terms in the different zone of turbulent flow also remains a subject of investigation. With this in background, the following objectives are set for the present study: (i) to examine the stress-strain relationship in case of linear as well as nonlinear models, and (ii) to ascertain the influence of nonlinearity on the variation of production and destruction terms of turbulence closure models. To achieve these objectives, the data available from Minson et al. (1995) are considered. Available governing equations used in the present study have been detailed in the full paper.

Results and discussion One of the objectives of the study is to see the nature of stress strain response in a turbulent flow field. For this purpose, five sections as shown in Fig. 1 are considered. At each of these sections, stresses and strain rates are computed at different points along the vertical. The detailed procedure of computing stress and strain rates using different models, linear, nlwot, nlwt and continuum approach has been explained in the full paper. Figures have been

Proc. of the 8th Asia-Pacific Conference on Wind Engineering – Nagesh R. Iyer, Prem Krishna, S. Selvi Rajan and P. Harikrishna (eds)Copyright c© 2013 APCWE-VIII. All rights reserved. Published by Research Publishing, Singapore. ISBN: 978-981-07-8011-1doi:10.3850/978-981-07-8012-8 136 558

Proc. of the 8th Asia-Pacific Conference on Wind Engineering (APCWE-VIII)

plotted to show the variation of equivalent turbulent stress versus equivalent turbulent strain rates using four types of models. It can be seen from these Figures that, in general, the stress strain relationship remains no more linear even with the use of linear models. Literature reflects that stress strain rate tensor should be linearly related in case of linear turbulence models. To ascertain this, the individual variation of stress and strain rate components are also plotted using different turbulence models. However, there is no evidence of linearity in stress strain response even with the use of linear models. This is found true for all other sections. Thus, the very basis of using the word “linear turbulence models” is not supported in view of their nonlinear stress strain characteristics. To quantify the extent of nonlinearity in the stress strain relations, use is made of correlation coefficient r2 (Table 2). It is known that for straight line or linear relationship, this coefficient is equal to unity. If there is departure in the linear relation, the r2 value is expected to drop. Thus, it can be reasonably assumed that a lower r2 value associated with a best fit line of stress strain spectra will be an acceptable indicator of the extent of nonlinearity inherent in the stress strain relationship. To study the variations of production and destruction terms of turbulence kinetic energy (k) and its rate of dissipation (ε), i.e., Pk, Dk, Pε , Dε, Tk ( = Pk - Dk) and Tε ( = Pε - Dε), data in respect of five sections are plotted. Fig. 2 shows the variations of these parameters at sections 1 typically. It can be noticed that the variation of each of these parameters are arbitrary and does not follow a well-defined trend along a given vertical. However, there is some trend, if the net area under these curves is considered. Unless stated, the word ‘area’ implies net area only.

The areas under different curves were computed and these have been used to rank the three approaches used in the study. Among three approaches, the highest area is represented symbolically as ‘H’, followed by the lowest area symbolically represented as ‘L’ and in between value has been represented as ‘M’. With these notations, Tables 3a and 3b are prepared based on five Figures plotted similar to Fig. 2. Table 1 also summarizes the relative ranking of different terms with the use of three approaches. To have a better insight into the relative ranking of these approaches, a numeric value has been assigned to the symbolic representations ‘H’, ‘M’, and ‘L’ as 3, 2, and 1, respectively. For each of the terms, a total of such numerical indices is evaluated considering all the five sections used in the study. It can be seen from Table 1 that numerical measure under linear approach is the highest followed by nonlinear approach without t terms. The nonlinear approach with t-terms appears to have a lowest numerical measure. Thus, the role of t-terms appears as a modifier of nonlinear models and it also reflects that addition of t-terms leads to more nonlinearity than that reflected by nonlinear models without t-terms. As a passing remark, it is also appropriate to mention the possible linkage between Pk and reattachment length. Several studies including Acharya et al. (1994) have suggested that reattachment length is decreased with the increase in magnitude of Pk term. This means that as the Pk terms are higher in case of linear models, the linear models should yield a smaller value of reattachment length compared with nonlinear models (Refer Table 4 reproduced from Acharya et al. 1994). This observation also finds support from the existing studies in the literature. In the present study, although sections considered are five in number, these have been selected in such a manner that the effect of turbulence should be better highlighted. The study clearly depicts that the linear turbulence models fail to indicate the linear behaviour between stresses and strain rates. This is based on the observation that neither the individual stress strain curves nor equivalent stress strain curve reflect linear behaviour. Also, the use of r2 serves a way of studying the nonlinearity between stress strain spectra. Different terms i.e.,

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Pk, Dk, Pε and Dε are also influenced by the nonlinearity and this is apparent from their variations in a vertical. The ranking of different models, based on the area of these terms, indicates a possible correlation between the area and nonlinearity. CONCLUSIONS Based on this study, the following conclusions can be made: (i) The linear relationship between stress and strain rate tensor is not observed in case of linear models. (ii) The variation of equivalent turbulent stress versus equivalent turbulent strain rate also does not reflect linearity for linear models. (iii) Use of t-terms in nonlinear models is found related to the nonlinearity of the turbulence models. (iv) Use of r2 is found a convenient way of measuring nonlinearity between stress and strain. (v) Magnitudes of Pk, Dk, Pε, Dε, Tk, and Tε vary in a vertical. Also, the area under the curve is found related to the nonlinearity of the stress models. References 1. Acharya, S., Dutta, S., Myrum, T. A. and Baker, R. S. (1994). “Turbulent flow past a surface mounted two-

dimensional rib.” J. Fluid Engrg., Trans. ASME, 116(6), 238-246. 2. Boussinesq, J. (1877). “Theorie de l ecoulement tourbillant”, Mem. Pre. Par. Div. Sav., 23, Paris. [Launder

and Spalding 1972]. 3. Mandal, S., Ojha, C. S. P., and Bhargava, P. (2001). “Significance of correlation coefficient in evaluating

Reynolds Stresses.” J. Wind Engineering, Japanese Association for Wind Engineering, 89(10), 317-320. 4. Minson, A. J., Wood, C. J., and Belcher, R. E. (1995). “Experimental velocity measurements for CFD

validation.” J. Wind Engrg. Industrial Aerodyn., 58, 205-215. 5. Myong, H. K., and Kasagi, N. (1990). “Prediction of anisotropy of the near wall turbulence with an

anisotropic low-Re-number k-ε turbulence model.” J. Fluids Engrg., Trans. ASME, 112(12), 521-524. 6. Speziale, C. G. (1987). “On nonlinear k-l and k-ε models of turbulence.” J. Fluid Mech., 178, 459-475. 7. Launder, B. E., and Spalding, D. B. (1974). “The numerical computation of turbulent flows.” Comp. Meths.

Appl. Mech. Engrg., 3, 269-289. Table 1 : Relative ranking of three approaches (terms in k-eqn) Term Sections

Pk Dk Tklin nlwot Nlwt lin nlwot nlwt Lin nlwot nlwt

Sec 1 H M L M H L H L M Sec 2 H M L L H M H M L Sec 3 H M L H L M M H L Sec 4 H M L M L H H M L Sec 5 H M L H L M H L M Total score 15 10 5 11 9 10 14 9 7 lin = 15+11+14 = 40; nlwot = 10+9+9 = 28; nlwt = 5+10+7 = 22 Table 2 : r2 values from individual stress strain response for xxσ~ components

Sections Approaches

Section 1 Section 2 Section 3 Section 4 Section 5

Linear 0.0154 0.9627 0.8547 0.0021 0.3898 nlwot 0.1706 0.0231 0. 0.0533 0.8285 nlwt 0.0139 0.8855 0.3749 0.0712 0.7393

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Fig. 1 : Definition sketch – II [location of computation sections for data of Minson et al. 1995].

Fig. 2 : Variation of Pk, Dk, Tk, Pε, Dε, and Tε at section 1.

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