The UMIST-N Near-Wall Treatment Applied to Periodic Channel Flow

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CHAPTER 4. NUMERICAL IMPLEMENTATION 56 Thus, while ω is infinite at the wall, finite values may be specified near the wall according to Equation 4.15. However, the gradient of ω near the wall is excessively large (approximately two orders of magnitude greater than gradients on ˜ε), so it is not practical to solve the ω transport equation in the near wall region. The approach recommended by Wilcox [91] is to specify ω according to the limiting behaviour suggested in Equation 4.15 within the first 7 to 10 cells, where y + < 2.5. Transport equations on 〈U〉 and k use specified values of ω in this region, and the transport equation of ω is only solved outside of this very near-wall region. 4.4.3 The Logarithmic Law of the Wall A wall function mesh incorporates a near-wall cell that is large enough to fully encompass the buffer layer and extend into the region where the log law is valid. This is shown schematically in Figure 4.2. Because flows of low Reynolds number are considered in this thesis, the buffer layer is very large. The near-wall node location was set to y/δ = 20% or y + = 36.0. The main grid of PASSABLE uses vertexes centred between nodes, so the near-wall node is positioned far from the centre of its cell. The remaining grid was made up of 18 nodes. An expansion factor of 1.00 was used for these 18 cells. The total number of grid cells used with wall functions was thus much less than with the low-Reynolds-number solution. Numerically, the log law is implemented by severing the link to the wall in the near-wall cell, and then manipulating the 〈U〉, k, ε source terms in the near-wall cell to reflect the log law. A full discussion of the numerical implementation of log laws in CFD can be found in the book of Versteeg &
CHAPTER 4. NUMERICAL IMPLEMENTATION 57 Malalasekera [83]. U + P nience: wall function applied Figure 4.2: The high-Reynolds-number grid is calculated from the log law (Equation 2.39), repeated here for conve- U + P 1 = κ ln Ey + p (4.16) where y + p is obtained from the C 1/4 µ k 1/2 velocity scale suggested by Launder & Spalding [34]: y + p = C1/4 µ k 1/2 p yp ν yp is the distance of the near-wall cell node from the wall. Wall shear stress is obtained from Equation 2.44. It is − τw ρ τw = ρC1/4 µ k 1/2 p 〈U〉 P U + P (4.17) (4.18) from Equation 4.18 may be added to the x-momentum equation as a
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The UMIST-N Near-Wall Treatment App
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Acknowledgements I would like to th
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Contents 1 Introduction & Literatur
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List of Figures 2.1 The log-law com
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5.32 k vs y + snapshots through tim
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Chapter 1 Introduction & Literature
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CHAPTER 1. INTRODUCTION & LITERATUR
Page 15 and 16: CHAPTER 1. INTRODUCTION & LITERATUR
Page 27 and 28: Chapter 2 Turbulence Models The Rey
Page 29 and 30: CHAPTER 2. TURBULENCE MODELS 19 The
Page 31 and 32: CHAPTER 2. TURBULENCE MODELS 21 2.2
Page 33 and 34: CHAPTER 2. TURBULENCE MODELS 23 way
Page 35 and 36: CHAPTER 2. TURBULENCE MODELS 25 y i
Page 37 and 38: CHAPTER 2. TURBULENCE MODELS 27 The
Page 39 and 40: CHAPTER 2. TURBULENCE MODELS 29 In
Page 41 and 42: Chapter 3 Channel Flow Channel flow
Page 43 and 44: CHAPTER 3. CHANNEL FLOW 33 of x and
Page 45 and 46: CHAPTER 3. CHANNEL FLOW 35 varies o
Page 47 and 48: CHAPTER 3. CHANNEL FLOW 37 u 2 +
Page 49 and 50: CHAPTER 3. CHANNEL FLOW 39 3.4.1 Em
Page 51 and 52: CHAPTER 3. CHANNEL FLOW 41 profile
Page 53 and 54: CHAPTER 3. CHANNEL FLOW 43 Figure 3
Page 55 and 56: CHAPTER 3. CHANNEL FLOW 45 The prof
Page 57 and 58: CHAPTER 3. CHANNEL FLOW 47 y + is p
Page 59 and 60: CHAPTER 4. NUMERICAL IMPLEMENTATION
Page 65: CHAPTER 4. NUMERICAL IMPLEMENTATION
Page 77 and 78: CHAPTER 5. RESULTS 67 Table 5.1: Co
Page 79 and 80: CHAPTER 5. RESULTS 69 The solution
Page 81 and 82: CHAPTER 5. RESULTS 71 match the DNS
Page 83 and 84: CHAPTER 5. RESULTS 73 Integrating w
Page 85 and 86: CHAPTER 5. RESULTS 75 Figures 5.11,
Page 87 and 88: CHAPTER 5. RESULTS 77 gradient of k
Page 89 and 90: CHAPTER 5. RESULTS 79 The log law o
Page 91 and 92: CHAPTER 5. RESULTS 81 maxima). This
Page 93 and 94: Chapter 6 Conclusions & Suggestions
Page 95 and 96: CHAPTER 6. CONCLUSIONS & SUGGESTION
Page 97 and 98: Bibliography [1] Addad, Y., Applica
Page 99 and 100: BIBLIOGRAPHY 89 [14] Craft, T. J.,
Page 101 and 102: BIBLIOGRAPHY 91 [30] Kim J., Moin P
Page 103 and 104: BIBLIOGRAPHY 93 [47] Niederschulte,
Page 105 and 106: BIBLIOGRAPHY 95 [64] Rotta, J. C.,
Page 107 and 108: BIBLIOGRAPHY 97 [82] Tu, S. W. & Ra
Page 109 and 110: Figures 99
Page 111 and 112: FIGURES 101 − 〈uv〉 + 1.0 0.9
Page 113 and 114: FIGURES 103 k + 5 4 3 2 1 0 10 0 +
Page 115 and 116: FIGURES 105 1.4 〈vv〉 + 1.2 1.0
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FIGURES 107 y ∗ y ∗ 600 500 400
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FIGURES 109 y ∗ v2 y ∗ v2 600 5
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FIGURES 111 〈U〉 + 〈U〉 + 20
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FIGURES 113 〈U〉 + 〈U〉 + 20
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FIGURES 115 U U ss U U ss 3.0 3.0 2
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FIGURES 117 〈U〉 (Uτ ) ss k (U
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FIGURES 121 〈U〉 (Uτ ) ss 〈U
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FIGURES 123 k (U 2 τ ) ss k (U 2
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FIGURES 129 ( ∂〈P 〉 ∂x ) (
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The UMIST-N Near-Wall Treatment Applied to Periodic Channel Flow

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