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

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3.1.2 The k-ω Model . . . . . . . . . . . . . . . . . . . . . . 34 3.2 Flow Characterisation . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Nondimensionalisation . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Steady Channel Flow Data . . . . . . . . . . . . . . . . . . . . 37 3.4.1 Empirical Profile for U + . . . . . . . . . . . . . . . . . 39 3.4.2 Empirical Profile for − 〈uv〉 + . . . . . . . . . . . . . . 39 3.4.3 Empirical Profile for k + . . . . . . . . . . . . . . . . . 40 3.4.4 Empirical Profile for 〈u 2 〉 + . . . . . . . . . . . . . . . . 41 3.4.5 Empirical Profile for 〈v 2 〉 + . . . . . . . . . . . . . . . . 42 3.4.6 Near-Wall Behaviour . . . . . . . . . . . . . . . . . . . 43 3.5 Local Nondimensionalisation . . . . . . . . . . . . . . . . . . . 45 4 Numerical Implementation 48 4.1 The Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 Volume Integral Form . . . . . . . . . . . . . . . . . . . . . . 50 4.3 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 54 4.4.1 Wall Boundaries on k-ε . . . . . . . . . . . . . . . . . . 55 4.4.2 Wall Boundaries on k-ω . . . . . . . . . . . . . . . . . 55 4.4.3 The Logarithmic Law of the Wall . . . . . . . . . . . . 56 4.4.4 The Subgrid Approach . . . . . . . . . . . . . . . . . . 60 4.5 Under-Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 64 5 Results 66 5.1 Steady Flow Results . . . . . . . . . . . . . . . . . . . . . . . 68 5.2 Prescribed Periodic Pressure Gradient . . . . . . . . . . . . . 71 5.3 Prescribed Periodic Bulk Flow Rate . . . . . . . . . . . . . . . 75 6 Conclusions & Suggestions for Future Work 83 v
List of Figures 2.1 The log-law compared to experiments in channel flow . . . . . 30 4.1 The low-Reynolds-number grid . . . . . . . . . . . . . . . . . 49 4.2 The high-Reynolds-number grid . . . . . . . . . . . . . . . . . 57 4.3 The subgrid mesh, adapted from Gant [21] . . . . . . . . . . . 60 3.1 Reichardt’s law to estimate U + . . . . . . . . . . . . . . . . . 100 3.2 Reichardt’s law applied to − 〈uv〉 + . . . . . . . . . . . . . . . 101 3.3 A revised profile for − 〈uv〉 + . . . . . . . . . . . . . . . . . . . 102 3.4 A profile for k + . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.5 A profile for 〈uu〉 + . . . . . . . . . . . . . . . . . . . . . . . . 104 3.6 A profile for 〈vv〉 + . . . . . . . . . . . . . . . . . . . . . . . . 105 3.7 Near-wall behaviour of flow parameters . . . . . . . . . . . . . 106 3.8 y ∗ vs. y + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.9 U ∗ superimposed on U + . . . . . . . . . . . . . . . . . . . . . 108 3.10 y ∗ v2 vs. y + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.11 U ∗ v2 superimposed on U + . . . . . . . . . . . . . . . . . . . . . 110 5.1 〈U〉 + vs y/δ in the steady flow case . . . . . . . . . . . . . . . 111 5.2 k + vs y/δ in the steady flow case . . . . . . . . . . . . . . . . 112 5.3 〈U〉 + vs y + in the steady flow case . . . . . . . . . . . . . . . 113 5.4 k + vs y + in the steady flow case . . . . . . . . . . . . . . . . . 114 vi
Page 1 and 2: The UMIST-N Near-Wall Treatment App
Page 3 and 4: Acknowledgements I would like to th
Page 5: Contents 1 Introduction & Literatur
Page 9 and 10: 5.32 k vs y + snapshots through tim
Page 11 and 12: Chapter 1 Introduction & Literature
Page 13 and 14: 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
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CHAPTER 3. CHANNEL FLOW 47 y + is p
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CHAPTER 4. NUMERICAL IMPLEMENTATION
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CHAPTER 5. RESULTS 67 Table 5.1: Co
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CHAPTER 5. RESULTS 69 The solution
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CHAPTER 5. RESULTS 71 match the DNS
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CHAPTER 5. RESULTS 73 Integrating w
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CHAPTER 5. RESULTS 75 Figures 5.11,
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CHAPTER 5. RESULTS 77 gradient of k
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CHAPTER 5. RESULTS 79 The log law o
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CHAPTER 5. RESULTS 81 maxima). This
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Chapter 6 Conclusions & Suggestions
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CHAPTER 6. CONCLUSIONS & SUGGESTION
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Bibliography [1] Addad, Y., Applica
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BIBLIOGRAPHY 89 [14] Craft, T. J.,
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BIBLIOGRAPHY 91 [30] Kim J., Moin P
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BIBLIOGRAPHY 93 [47] Niederschulte,
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BIBLIOGRAPHY 95 [64] Rotta, J. C.,
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BIBLIOGRAPHY 97 [82] Tu, S. W. & Ra
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Figures 99
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FIGURES 101 − 〈uv〉 + 1.0 0.9
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FIGURES 103 k + 5 4 3 2 1 0 10 0 +
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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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