Bouras, I.

Ingham, D.

Ma, L.

Pourkashanian, M.

English

White Rose Research Online

This is a repository copy of Response to the discussion on “An improved k – ω turbulence model for the simulations of the wind turbine wakes in a neutral atmospheric boundary layer flow” by Y Yang.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/141717/Version: Accepted VersionArticle:Bouras, I., Ma, L. orcid.org/0000-0002-3731-8464, Ingham, D. et al. (1 more author) (2019) Response to the discussion on “An improved k – ω turbulence model for the simulations of the wind turbine wakes in a neutral atmospheric boundary layer flow” by Y Yang. Journal of Wind Engineering and Industrial Aerodynamics, 184. pp. 456-457. ISSN 0167-6105 https://doi.org/10.1016/j.jweia.2018.12.005Article available under the terms of the CC-BY-NC-ND licence (https://creativecommons.org/licenses/by-nc-nd/4.0/).eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse This article is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs (CC BY-NC-ND) licence. This licence only allows you to download this work and share it with others as long as you credit the authors, but you can’t change the article in any way or use it commercially. More information and the full terms of the licence here: https://creativecommons.org/licenses/ Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Response to the discussion on さAﾐ ｷﾏヮヴﾗ┗WS ﾆ に ＼ デ┌ヴH┌ﾉWﾐIW ﾏﾗSWﾉ aﾗヴ デｴW ゲｷﾏ┌ﾉ;デｷﾗﾐゲ ﾗa デｴW ┘ｷﾐS デ┌ヴHｷﾐW ┘;ﾆWゲ ｷﾐ ; ﾐW┌デヴ;ﾉ ;デﾏﾗゲヮｴWヴｷI Hﾗ┌ﾐS;ヴ┞ ﾉ;┞Wヴ aﾉﾗ┘ざ H┞ Y Y;ﾐｪ Ioannis Bouras, Lin Ma*, Derek Ingham, Mohamed Pourkashanian Energy 2050, Faculty of Engineering, The University of Sheffield, S10 2TN, England, UK  The authors appreciate the comments made by the discusser on potential mistaking of the source term for the turbulence eddy dissipation rate 0 ｷﾐ デｴW I;ﾉI┌ﾉ;デｷﾗﾐ ﾗa デｴW WSS┞ aヴWケ┌WﾐI┞ ＼く Fﾗヴ デｴW ┘ﾗヴﾆ ヮヴWゲWﾐデWS ｷﾐ ﾗ┌ヴ ヮ;ヮWヴが デｴW ゲﾗ┌ヴIW デWヴﾏ aﾗヴ ＼ ┘;ゲ ﾗHデ;ｷﾐWS H┞ ﾏﾗSｷa┞ｷﾐｪ デｴW ゲﾗ┌ヴIW デWヴﾏ aﾗヴ 0 デｴ;デ ┘;ゲ proposed by Chen and Kim (1987), using the concept of the k に ＼ ﾏﾗSWﾉ ;ﾐS ｷﾐ ; SｷﾏWﾐゲｷﾗﾐ;ﾉﾉ┞ consistent manner. The following sections explain the source term that was employed in the modified k に ＼ ﾏﾗSWﾉ ｷﾐ ﾗ┌ヴ ヮ;ヮWヴく Despite the fact that the standard k に 0 model (Launder and Spalding, 1972) has been validated in a wide range of applications, it still fails to predict accurately flows with strong adverse pressure gradients (e.g. backward facing step). One of the applications that the standard k に 0 model fails to predict correctly, as many researchers have proven such as Crespo et al. (1985), Kasmi and Masson (2008), is the simulation of wind turbine wakes. Chen and Kim (1987) have concluded that the problem with the standard k に 0 model is due to the fact that the transport equation for the eddy dissipation rate is highly empirical and it does not represent the energy transfer from the large to small scales effectively. Consequently, Chen and Kim (1987) added a second time scale in the transport equation of the eddy dissipation rate to overcome this problem and they compared their model with the standard k に 0 model in many different applications (e.g. backward facing step, swirling flows, turbulent boundary layer). The time scale that has been added is as follows: ܵఌ ൌ ఌସܥ  (1) ݇ߩ௞ଶܩ The term ܩ௞ is the production of turbulence and is calculated as follows: ܩ௞ ൌ  (௧ܵଶ (2ߤ The term ܵ is the strain rate and is defines as follows: ܵ ൌ ටʹ ௜ܵ௝ ௜ܵ௝ (3)  Kasmi and Masson (2008) have employed the term given in equation (1) in the vicinity of the wind turbine and they showed that the results for 3 different wind turbines are significantly improved in relation to the standard k に 0 model, which completely fails to predict the wind turbine wake characteristics. Unfortunately, the main issue with the Kasmi and Masson (2008) model is that it is empirical constant dependent, as some researchers have pointed out, because they validated their model with only one specific value of the relative inlet turbulent kinetic energy. This is because the time scale added by Chen and Kim (1987) in the transport equation for the eddy dissipation rate in the standard k に 0 model becomes a function of the eddy viscosity, which is a function of the empirical constant ܥఓ which defines the relative turbulent kinetic energy of the field. In order to clarify the validity of the previous statement, the equation (1) is transformed as follows, in the standard k に 0 model: ܵఌ ൌ ଶߝఌସ݇ଷܥఓଶܥߩ ൬ටʹ ௜ܵ௝ܵ௜௝൰ସ (4)  As seen in equation (4), the extra term is strongly dependent on the empirical constant ܥఓ, which defines the relative turbulent kinetic energy of the flow field. In our paper, we have proven that similar problems to the standard k に 0 model arise with the standard k に ＼ model (Wilcox, 1998) because the transport equation for the eddy frequency is also highly empirical and guided by physical reasoning. As in the standard k に 0 model, the turbulent kinetic energy and the relative velocity at the rear of the wind turbine are both highly overpredicted by the standard k に ＼ model. The standard k に ＼ model uses the eddy frequency as the second determining variable. The standard k に ＼ model (Wilcox, 1998) uses the following transport equation for the eddy frequency: ߲߲ݐ ሺ߱ߩሻ ൅ ௜ݔ߲߲ ሺݑ߱ߩ௜ሻ ൌ ௜ݔ߲߲ ൤൬ߤ ൅ ఠ൰ߪ௧ߤ ௜൨ݔ߲߲߱ ൅ ߱݇ ௞ܩ െ  (௜߱ଶ (5ߚߩ The relationship between the eddy frequency and eddy dissipation rate is given by: ߱ ൌ כஶߚ݇ߝ  (6)  Consequently, the source term of equation (1), for the standard k に ＼ model is simply transformed as follows: ܵఠ ൌ כஶߚఌସܥߩ ߱ଶ ൬ටʹ ௜ܵ௝ ௜ܵ௝൰ସ (7)  The empirical constant ߚஶכ  was given the value of unity for the simulation of both wind turbines investigated in the paper. The results have been compared with those obtained from the standard k に ＼ model for various freestream wind speeds and turbulence levels. The modified k に ＼ model showed superiority to the standard k に ＼ model when compared with the available experimental data for 2 different small wind turbines for both near and far wake regions. This is probably the reason why some researchers used slightly different versions for the k に ＼ model in order to limit the production of turbulent kinetic energy by making it a function of the dissipation. In particular, Menter (1992) used a stress limiter modification in order to control the magnitude of the eddy viscosity. The definition of the eddy viscosity was given by: ௧ߥ ൌ ݇߱ഥ (8)  ഥ߱ ൌ ݔܽ݉ ቐ߱ǡ ʹ௟௜௠ඨܥ ௜ܵ௝ ௜ܵ௝ߚஶכ ቑ (9)  Researchers, such as Huang (1999), showed that the above modification leads to better results for incompressible flows. Other researchers, such as Speziale et al. (1992), used the so に called cross diffusion term given by: ܵఠ ൌ ௗ߱ߪ ௜ݔ߲߲݇  (௜ (10ݔ߲߲߱ ௗߪ ൌ ۔ەۓ Ͳǡ ௜ݔ߲߲݇ ௜ݔ߲߲߱ ൑ Ͳߪௗ଴ ௜ݔ߲߲݇ ௜ݔ߲߲߱ ൐ Ͳۙۘۗ (11)  The source term in equation (10) has been proven to improve the results for wall bounded flows (Hellsten, 2005). All things considered, employment of source terms or modifications to the standard k に ＼ model can be easily made to improve the accuracy of the results. In our paper, the source term, originally proposed by Chen and Kim (1987), has been employed in the k に ＼ model after a transformation and it has been proven that the results are improved significantly for the simulations of 2 different small wind turbine wakes. References Chen, Y.S., Kim, S.W., 1987. Computation of turbulent flow using an extended turbulence closure model. NASA Contractor Report, NASA CR-179204. Crespo, A., Manuel, F., Moreno, D., Fraga, E., Hernandez, J., 1985. Numerical analysis of wind turbine wakes. In: Proceedings of Delphi Workshop on Wind Energy Applications, Delphi, Greece, pp. 15に25. Hellsten A., 2005. New advanced k に ＼ turbulence model for high に lift aerodynamics. AIAA Journal, 43(9), 1857 に 1869. Huang P. 1999. Physics and Computations of Flows with Adverse Pressure Gradients. In: Salas M.D., Hefner J.N., Sakell L. (eds) Modeling Complex Turbulent Flows. ICASE/LaRC Interdisciplinary Series in Science and Engineering, Vol 7. Springer, Dordrecht. Kasmi A., Masson C., 2008. An extended k に 0 model for turbulent flow through horizontal に axis wind turbines. Journal of Wind Engineering and Industrial Aerodynamics. 96, 103 に 122. Launder B., Spalding D., 1972. Lectures in Mathematical Models of Turbulence. Academic Press, London, England. Menter F., 1992. Improved two に equation k に ＼ turbulence models for aerodynamic flows. NASA TM に 108854. Speziale C., Abid R., Anderson E., 1992. A critical evaluation of two に equation models for near wall turbulence. AIAA Journal, 30(2), 324 に 331. Wilcox D., 1998. Turbulence modeling for CFD. DCW Industries, Inc., La Canada, California.  

Response to the discussion on “An improved k – ω turbulence model for the simulations of the wind turbine wakes in a neutral atmospheric boundary layer flow” by Y Yang

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Response to the discussion on “An improved k – ω turbulence model for the simulations of the wind turbine wakes in a neutral atmospheric boundary layer flow” by Y Yang

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