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

More documents

Recommendations

Info

CHAPTER 4. NUMERICAL IMPLEMENTATION 50 Table 4.1: The notation for discretised values current node subscript P node above subscript N node below subscript S vertex above subscript n vertex below subscript s previous time step superscript t − 1 cell height ∆yp node-to-node dist. ∆yn,s time step ∆t 4.2 Volume Integral Form The finite volume discretisation method is convenient because quantities are conserved within each finite volume and are therefore necessarily conserved throughout the flow field. In order to employ the finite volume method, it is first necessary to integrate the differential equations governing the flow with respect to volume. The volume-integrated forms of the equations are pre- sented below. Note that, in channel flow geometry, integrating with respect to volume is equivalent to integrating with respect to y. 〈U〉 : k : ˜ε : ∂〈U〉 ∂k ∂t ∂〈P 〉 ∂〈U〉 dy + (ν + νt) ∂x ∂y ν+νt ∂k ∂t dy = − 1 ρ dy = ∂ ˜ε dy = ∂t ω : ∂ω dy = ∂t − Cε2f2 ν+νt σk ν+νt ∂ ˜ε σω σε ˜ε 2 (4.1) ∂y + Pkdy − εdy (4.2) ∂y + ˜ε Cε1f1 Pkdy k dy + k Edy + Y dy (4.3) ∂ω ∂y + γ ω Pkdy − k βω2dy (4.4)
CHAPTER 4. NUMERICAL IMPLEMENTATION 51 Before performing any dimensional analysis on the above equations, one must actually integrate with respect to volume, applying limits of 0 to 1 on the in- tegrations in x and z. It is also notable that the PASSABLE code uses a grid that is unit length in z but not unit length in x (∆x = 1). This detail must be remembered when working with the code, particularly when specifying volume-integrated source terms. Furthermore, in the PASSABLE implementation, Equations 4.1-4.4 are multiplied through by ρ. This convention is maintained in the subgrid. 4.3 Discretisation The process of discretisation involves approximating the governing differential equations by algebraic equations so that they can be solved by a com- puter. A fully implicit scheme is used in this work. This means that the previous time-step values of adjacent nodes are not used. In contrast, a fully explicit scheme would use the previous time-step values, but not the current time- step values of adjacent nodes. The fully implicit method ensures that the solution can converge at large time steps. Central differencing is used in this work to approximate derivatives. Other differencing schemes include PLDS 2 [51] and QUICK 3 [37]. These alternative schemes involve higher-order approximations. The QUICK scheme is also upstream-biased, in the sense that it makes use of more information on the 2 Power-Law Differencing Scheme 3 Quadratic Upstream Interpolation for Convective Kinetics
Page 1 and 2:
The UMIST-N Near-Wall Treatment App
Page 3 and 4:
Acknowledgements I would like to th
Page 5 and 6:
Contents 1 Introduction & Literatur
Page 7 and 8:
List of Figures 2.1 The log-law com
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
Page 57 and 58: CHAPTER 3. CHANNEL FLOW 47 y + is p
Page 59: CHAPTER 4. NUMERICAL IMPLEMENTATION
Page 63 and 64: 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
Page 117 and 118:
FIGURES 107 y ∗ y ∗ 600 500 400
Page 119 and 120:
FIGURES 109 y ∗ v2 y ∗ v2 600 5
Page 121 and 122:
FIGURES 111 〈U〉 + 〈U〉 + 20
Page 123 and 124:
FIGURES 113 〈U〉 + 〈U〉 + 20
Page 125 and 126:
FIGURES 115 U U ss U U ss 3.0 3.0 2
Page 127 and 128:
FIGURES 117 〈U〉 (Uτ ) ss k (U
Page 129 and 130:
Page 131 and 132:
FIGURES 121 〈U〉 (Uτ ) ss 〈U
Page 133 and 134:
FIGURES 123 k (U 2 τ ) ss k (U 2
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
FIGURES 129 ( ∂〈P 〉 ∂x ) (
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
show all

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

Create successful ePaper yourself

Delete template?

Save as template?