Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

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Appendix A Non-conservative form of Navier±Stokes equations To derive the Navier±Stokes equations in their non-conservative form, we start with the conservative form. Conservation of mass: Conservation of momentum: Conservation of energy: @… E† @t ‡ @…uj E† @ @t ‡ @… ui† ˆ @xi @ @t ‡ @ui @xi @… ui† @t ‡ @…uj ui† @x j ÿ @ @x i @x j k @T @x i @ ‡ ui Rewriting the momentum equation with terms di€erentiated as @u i @t ‡ u i @ @t ‡ @u j @x j @ ‡ uj @x j @x i ˆ 0 …A:1† ÿ @ ij ‡ @xj @p ˆ 0 …A:2† @xi ‡ @…ujp† ÿ @xj @… ijuj† ˆ 0 …A:3† @xj ‡ u j @ui @xj ÿ @ ij ‡ @xj @p ˆ 0 …A:4† @xi and substituting the equation of mass conservation (Eq. A.1) into the above equation gives the reduced momentum equation @u i @t ‡ u j @ui @xj ÿ 1 @ ij ‡ @xj 1 @p ˆ 0 …A:5† @xi Similarly as above, the energy equation (Eq. A.3) can be written with di€erentiated terms as E @ @t ‡ @u j @x j @ ‡ uj @x j ‡ @E @t ‡ u @E j @x j ÿ @ @x i k @T @x i ‡ @…ui p† ÿ @xi @… ijuj† ˆ 0 …A:6† @xi
292 Appendix A Again substituting the continuity equation into the above equation, we have the reduced form of the energy equation @E @t ‡ u @E j @x j 1 @ ÿ @xi k @T @x i ‡ 1 @…ui p† ÿ @xi 1 @… ijuj† @xi …A:7† Some authors use Eqs. (A.1), (A.5) and (A.7) to study compressible ¯ow problems. However these non-conservative equations can result in multiple or incorrect solutions in certain cases. This is t<strong>ru</strong>e especially for high-speed compressible ¯ow problems with shocks. <strong>The</strong> reader should note that such non-conservative equations are not suitable for simulation of compressible ¯ow problems.
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The Finite Element Method Fifth edi
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The Finite Element Method Fifth edi
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Dedication This book is dedicated t
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3.7 The performance of two- and sin
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Volume 2: Solid and structural mech
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xiv Preface to Volume 3 this algori
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2 Introduction and the equations of
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10 Introduction and the equations o
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12 Introduction and the equations o
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14 Convection dominated problems We
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16 Convection dominated problems wh
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18 Convection dominated problems ch
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20 Convection dominated problems i.
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22 Convection dominated problems 2
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24 Convection dominated problems 2.
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26 Convection dominated problems ar
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28 Convection dominated problems Th
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30 Convection dominated problems co
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32 Convection dominated problems 2.
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34 Convection dominated problems ha
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36 Convection dominated problems an
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38 Convection dominated problems ac
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40 Convection dominated problems An
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42 Convection dominated problems U
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44 Convection dominated problems (a
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46 Convection dominated problems sp
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48 Convection dominated problems ag
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50 Convection dominated problems To
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52 Convection dominated problems C
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56 Convection dominated problems an
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60 Convection dominated problems Ma
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62 Convection dominated problems 47
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3 A general algorithm for compressi
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66 A general algorithm for compress
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92 Incompressible laminar ¯ow Cons
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94 Incompressible laminar ¯ow and,
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96 Incompressible laminar ¯ow V/V
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98 Incompressible laminar ¯ow The
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100 Incompressible laminar ¯ow 1.5
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102 Incompressible laminar ¯ow p 1
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104 Incompressible laminar ¯ow �
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106 Incompressible laminar ¯ow x 2
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108 Incompressible laminar ¯ow Thi
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114 Incompressible laminar ¯ow (a)
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116 Incompressible laminar ¯ow (b)
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118 Incompressible laminar ¯ow wit
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120 Incompressible laminar ¯ow The
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122 Incompressible laminar ¯ow Fig
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124 Incompressible laminar ¯ow and
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126 Incompressible laminar ¯ow U U
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128 Incompressible laminar ¯ow C L
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130 Incompressible laminar ¯ow are
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132 Incompressible laminar ¯ow sha
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134 Incompressible laminar ¯ow Fig
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136 Incompressible laminar ¯ow 28.
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138 Incompressible laminar ¯ow 71.
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140 Incompressible laminar ¯ow 114
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142 Incompressible laminar ¯ow 153
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144 Free surfaces, buoyancy and tur
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170 Compressible high-speed gas ¯o
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7 Shallow-water problems 7.1 Introd
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220 Shallow-water problems reduced
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222 Shallow-water problems Approxim
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224 Shallow-water problems These si
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226 Shallow-water problems h = 2 η
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228 Shallow-water problems Surface
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230 Shallow-water problems FL: 578
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232 Shallow-water problems B Fig. 7
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234 Shallow-water problems Time = 0
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236 Shallow-water problems Fig. 7.1
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238 Shallow-water problems where no
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Page 255 and 256: 8 Waves Peter Bettess 8.1 Introduct
Page 257 and 258: 244 Waves the familiar form where
Page 259 and 260: 246 Waves of 10 nodes or thereabout
Page 261 and 262: 248 Waves and the Astley wave envel
Page 263 and 264: 250 Waves agreement with the analyt
Page 265 and 266: 252 Waves Fig. 8.3 General wave dom
Page 267 and 268: 254 Waves Relative errors (%) 10 8
Page 269 and 270: 256 Waves (The indirect boundary in
Page 271 and 272: 258 Waves where m is the number of
Page 273 and 274: 260 Waves Fig. 17.6 of Volume 1. La
Page 275 and 276: 262 Waves to using such coordinate
Page 277 and 278: 264 Waves r/a 5 4 3 2 1 0 0 2 4 6 8
Page 279 and 280: 266 Waves 8.16 Three-dimensional ef
Page 281 and 282: 268 Waves wave elevations in shallo
Page 283 and 284: 270 Waves integrals. Preliminary re
Page 285 and 286: 272 Waves 39. R.J. Astley. FE mode
Page 287 and 288: 9 Computer implementation of the CB
Page 289 and 290: 276 Computer implementation of the
Page 303: 290 Computer implementation of the
Page 307 and 308: 294 Appendix B As usual the domain
Page 309 and 310: 296 Appendix B 5. While the piecewi
Page 311 and 312: Appendix C Edge-based ®nite elemen
Page 313 and 314: Appendix D Multigrid methods It is
Page 315 and 316: Appendix E Boundary layer±inviscid
Page 317 and 318: 304 Appendix E In Fig. E.2, Cp is t
Page 320 and 321: Author index Page numbers in bold r
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Page 324 and 325: Nakos, D.E. 146, 165 Narayana, P.A.
Page 326: Wu, J. 67, 90; 102, 108, 112, 137;
Page 329 and 330: 316 Subject index Blurring of the s
Page 331 and 332: 318 Subject index Convected coordin
Page 333 and 334: 320 Subject index Elements ± cont.
Page 335 and 336: 322 Subject index Flows: buoyancy d
Page 337 and 338: 324 Subject index Incompressible St
Page 339 and 340: 326 Subject index Metals: hot, 120
Page 341 and 342: 328 Subject index Prescribed tracti
Page 343 and 344: 330 Subject index Shock interaction
Page 345 and 346: 332 Subject index Theorem ± cont.
Page 347: 334 Subject index Wave ± cont. ste
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