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Velocity Scaling of High-Speed Turbulent Boundary Layer Flows With Velocity scaling of high-speed turbulent boundary layer flows with wall heat transfer by Khaled Younes A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Applied Science in Mechanical Engineering Waterloo, Ontario, Canada, 2021 c Khaled Younes 2021 Author's Declaration I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract The law of the wall has been a staple in collapsing the mean velocity profiles of seemingly very different turbulent channel and boundary layer flows into one single, semi-analytical function. Accordingly, it has equipped engineers and scientists with unique predictive ca- pabilities over the randomness of turbulence. Its universality and widespread success for incompressible canonical flows has further encouraged research efforts that seek transfor- mations that scale any type of flow, whether canonical or not, to match with the law of the wall. In this work, the problem of scaling non-adiabatic boundary layer flows is tackled in the high-speed, compressible context. Starting from a generalization of the two most successful transformations to date, the Van Driest and Trettel and Larsson, the shortcomings of each transformation are high- lighted. Following that, and contrary to most classical approaches, an extension is carried out using the conservation of energy, from which a potential velocity transformation is ob- tained. In turn, by basing the analysis on energy as opposed to momentum, the resulting transformation explicitly accounts for the non-adiabatic condition at the wall. The transformation is assessed by comparison against the law of the wall in the case of hypersonic, non-adiabatic boundary layer flows. It is found to yield perfect collapse for weakly-to-moderately cooled walls. After factoring in the higher-order fluctuations in the turbulence terms, the transformation is also found to produce excellent agreement for strongly cooled walls. The findings demonstrate that the physics of non-adiabatic walls cannot be fully cap- tured by solely relying on the conservation of momentum. In addition, in high-speed flows with large heat transfer at the wall, the higher-order turbulence terms cannot be neglected. Ahead of full-scale implementation, the transformation has to be rigorously tested using broader data sets. This is left for future work. iii Acknowledgements I would like to start by thanking my supervisor, Professor Jean-Pierre Hickey, for all his dedicated help, tireless commitment, and countless hours spent in shaping up this thesis. Prof. Hickey not only supported me in completing the scientific aspect of this work, but he has also motivated me to keep pushing the envelope and pursue the ideas further. It is for this reason that I show him my utmost gratitude. I would also like to thank my dear friends, Kieran Strobel and Matthew Yao, for making themselves available whenever I needed a brainstorming buddy. Their critical analysis has helped clarify my thoughts and ensured that I perfected my understanding of this topic. iv Dedication This work is dedicated to my loving parents, Mohamad & Houeida, my siblings, Ab- dallah, Faysal, and Aya, and my dear grandfather, Khalil. v Table of Contents List of Figures viii List of Tables xi 1 Introduction1 1.1 Objectives of the Thesis............................6 2 Detailed Background8 2.1 Conservation of Momentum..........................9 2.2 Law of the Wall................................. 11 2.3 The Van Driest Transformation........................ 12 2.4 The Trettel and Larsson Transformation................... 13 3 Mathematical Framework 15 3.1 General Analysis................................ 15 3.2 Conservation of Momentum Revisited..................... 17 3.2.1 Averaging in Turbulence........................ 17 3.2.2 Higher-Order Terms.......................... 18 3.3 Conservation of Energy............................. 20 3.3.1 Velocity Transformation Based on Energy.............. 23 3.3.2 Wall-Normal Coordinate........................ 24 3.3.3 Remarks on Higher-Order Terms................... 25 vi 4 Results 27 4.1 Scaling with Conservation of Momentum................... 28 4.2 Scaling with Conservation of Energy..................... 32 4.2.1 Without the Higher-Order Terms................... 33 4.2.2 Accounting for the Higher-Order Terms................ 35 4.3 Discussion.................................... 37 4.3.1 Scaling the Reynolds Stress...................... 38 4.3.2 Implications on Wall Modeling..................... 40 5 Conclusions and Recommendations 41 References 44 APPENDICES 51 A Nomenclature 52 B Supplementary Derivations for Momentum 55 B.1 Simplifying the Stress Balance......................... 55 B.2 Difference in Averaging............................. 56 B.3 Proof for Alpha................................. 57 C Supplementary Derivations for Energy 58 C.1 Scale Analysis on the Energy Equation.................... 58 C.2 Simplifying the Energy Equation....................... 59 C.3 Higher-Order Terms in Sigma......................... 62 D Supplementary Figures 64 vii List of Figures 1.1 Adiabatic wall temperature as a function of the free stream Mach number, p γ−1 2 3 calculated using Tr = T1 1 + 2 M1 P r [1] with T1 = 298 K, γ = 1:4, and P r = 0:7...................................3 1.2 Scaled velocity profiles with wall heat transfer from the experimental data of Danberg [2] at M1 = 6:7. The law of the wall is plotted with κ = 0:41 and C = 5:2. (···): viscous sublayer; (− −): log layer............3 1.3 The Trettel and Larsson transformation as applied to the data set of Zhang et al [3]. The law of the wall is plotted with κ = 0:41 and C = 5:2. Bq is the non-dimensional heat transfer at the wall.................5 1.4 The Van Driest transformation as applied to the data set of Zhang et al [3]. The law of the wall is plotted with κ = 0:41 and C = 5:2. For legend, see Fig. 1.3......................................6 4.1 Scaled velocity profiles from the data set of Zhang et al [3]. Top figure shows the results with β =µ/µ ¯ w along with the Trettel and Larsson (−·) p transformed profiles; bottom figure shows the results with β = ρ/ρ¯ w along with the Van Driest (···) transformed profiles................. 30 4.2 Scaled velocity profiles using a linear fitting function in the buffer layer for β. For case legend, see Fig. 4.1 or Table 4.1.................. 31 4.3 Scaled velocity profiles using an exponential fitting function for β. For case legend, see Fig. 4.1 or Table 4.1......................... 31 4.4 Scaled velocity profiles with refined bounds for the linear profile of β in the buffer layer. The bounds are extrapolated directly from DNS for each case and detailed in Fig. D.1. For legend, see Fig. 4.1 or Table 4.1........ 32 viii 4.5 Scaled velocity profiles using the proposed transformation from conservation of energy with α = 1 and σ = 1 for all the cases. For legend, see Fig. 4.1 or Table 4.1..................................... 34 4.6 Scaled velocity profiles using the proposed transformation from conservation of energy with α = 0:85 and σ = 1 for the strongly cooled cases. The rest of the cases are scaled with α = 1 and σ = 1. For legend, see Fig. 4.1 or Table 4.1..................................... 34 4.7 Scaled velocity profiles using the proposed transformation from conservation of energy with α = 0:85 and σ = 0:75 for the strongly cooled cases. The rest of the cases are scaled with α = 1 and σ = 1. For legend, see Fig. 4.1 or Table 4.1.................................... 35 4.8 Scaled velocity profiles using the proposed transformation from conservation of energy with α = 0:85 and σ = 0:75 for the strongly cooled cases. The rest of the cases are scaled with α = 1 and σ = 1. The law of the wall is plotted with κ = 0:39 and C = 5:2. For legend, see Fig. 4.1 or Table 4.1.. 36 4.9 Morkovin's scaling of the Reynolds stresses vs: the wall-normal coordinate. For legend, see Fig. 4.1 or Table 4.1...................... 38 4.10 Morkovin's scaling of the Reynolds stresses vs: the semi-local coordinate. For legend, see Fig. 4.1 or Table 4.1...................... 39 4.11 Morkovin's scaling of the Reynolds stresses vs: the proposed wall-normal coordinate. For legend, see Fig. 4.1 or Table 4.1................ 39 4.12 The triple correlation as a function of the wall-normal distance in the inner layer region. For legend, see Fig. 4.1 or Table 4.1............... 40 D.1 Non-dimensional shear stress vs y+ for the data set of Zhang et al [3]. The colour outline is the same as in Table 4.1. The refined start and end bounds, + respectively, for each case are as follows: y = [15; 90] for M1 = 5:84; −Bq = + 0:14 and M1 = 13:64; −Bq = 0:19, y = [10; 50] for M1 = 7:87; −Bq = + 0:06, and y = [8; 40] for M1 = 5:86; −Bq = 0:02 and M1 = 2:5; −Bq = 0. du¯ (−):µ ¯ /τ ;(−·): −ρ¯u0v0/τ ......................... 64 dy w w D.2 Temperature profile in the channel simulations of Trettel [4], where Mb is the bulk Mach number of the channel. The profiles are differentiable with no inflection points................................ 65 ix D.3 Temperature profile in the boundary layer simulations of Zhang et al [3]. The colour outline is the same as in Table 4.1. A marked inflection point is seen in the near-wall region for the cooled cases................ 65 x List of Tables 3.1 Summary of the triple correlation terms dropped from the log layer in the energy equation. Note that it is assumed that cp = cpw = const:, otherwise more terms will result.............................
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