3次元マントル対流計算基本コード (その2)
ベンチマークテストの結果

Blankenbach et al. (1989) とのベンチマークテスト

2次元定常熱対流シミュレーションの結果と比較する。

• Case 1a (constant viscosity; Ra=10⁴)

  (64x1x64 mesh)

  mesh Nu Vrms q1(=q3) q2(=q4) 16x16 4.726885 41.755217 7.695426 0.887088 32x32 4.840006 42.560118 7.966605 0.664927 64x64 4.872745 42.785441 8.036044 0.608154 128x128 4.881429 42.844667 8.053536 0.593695 Benchmark Standard 4.884409 (0.000010) 42.864947 (0.000020) 8.059384 (0.000003) 0.588810 (0.000003)

• Case 1b (constant viscosity; Ra=10⁵)

  (64x1x64 mesh)

  mesh Nu Vrms q1(=q3) q2(=q4) 16x16 9.235550 181.18346 14.424512 3.441283 32x32 10.082608 189.19219 17.764322 1.425553 64x64 10.394786 191.98526 18.731196 0.898253 128x128 10.495490 192.87381 18.990358 0.767465 Benchmark Standard 10.534095 (0.000010) 193.21454 (0.00010) 19.079440 (0.000040) 0.722751 (0.000020)

• Case 1c (constant viscosity; Ra=10⁶)

  (64x1x64 mesh)

  mesh Nu Vrms q1(=q3) q2(=q4) 16x16 13.505592 675.69469 16.63581 9.836338 32x32 18.963003 780.80943 31.21463 6.941879 64x64 20.884342 814.68028 41.44228 2.477731 128x128 21.604074 827.43096 44.72368 1.255997 Benchmark Standard 21.972465 (0.000020) 833.98977 (0.00020) 45.96425 (0.00030) 0.877170 (0.000010)

• Case 2a (temperature-dependent viscosity)

  (64x1x64 mesh)

  mesh Nu Vrms q1 q2 q3 q4 16x16 10.3634 372.2759 17.42173 2.25932 14.1505 7.036967 32x32 10.4780 457.4738 20.41169 1.16135 20.2221 4.248769 64x64 10.1666 475.5292 18.33101 1.03954 25.3614 1.557456 128x128 10.0862 478.9647 17.72910 1.01498 26.6611 0.743177 Benchmark Standard 10.0660 (0.00020) 480.4334 (0.1000) 17.53136 (0.00400) 1.00851 (0.00020) 26.8085 (0.0100) 0.497380 (0.000100)

• Case 2b (temperature- and depth-dependent viscosity)

  (160x1x64 mesh)

  mesh Nu Vrms q1 q2 q3 q4 16x16 7.3772 218.823 15.6704 0.93118 13.8681 4.65131 40x16 7.4020 201.478 18.4789 0.37860 14.5474 3.51882 32x32 7.3417 196.915 23.3237 0.33166 15.7209 2.32090 80x32 7.2761 171.085 30.6337 0.17729 14.7261 1.28403 64x64 7.0691 180.646 20.5302 0.22174 14.9697 1.15836 160x64 7.0450 175.343 20.9644 0.18048 14.7092 0.79348 128x128 6.9731 174.691 19.1649 0.18980 14.4228 0.77270 320x128 6.9605 172.806 19.0478 0.17873 14.3514 0.66355 Benchmark Standard 6.9299 (0.0005) 171.755 (0.020) 18.4842 (0.0100) 0.17742 (0.00003) 14.1682 (0.0050) 0.61770 (0.00005)

Busse et al. (1993) とのベンチマークテスト

3次元定常熱対流シミュレーションの結果と比較する。

• Case 1a (constant viscosity)

  (64x32x64 mesh)

  mesh 16x8x16 32x16x32 64x32x64 128x64x128 Benchmark Standard Nu 3.7373 3.6019 3.5549 3.5419 3.5374(0.0005) Vrms 42.047 41.382 41.104 41.026 40.999(0.004) w(0,0,0.5) 122.919 120.451 117.880 116.964 116.625(0.030) w(0,b,0.5) 20.044 33.693 38.566 39.994 40.500(0.030) T(0,0,0.5) 0.79161 0.80376 0.80262 0.80169 0.80130(0.00005) T(0,b,0.5) 0.57314 0.60324 0.61434 0.61761 0.61876(0.00005) Q(0,0) 7.68656 7.08517 6.82042 6.74082 6.7127(0.0500) Q(a,0) 1.85083 1.60300 1.53359 1.51460 1.5080(0.0500) Q(0,b) 2.75670 2.96542 3.10916 3.15671 3.1740(0.0500) Q(a,b) 0.74943 0.70104 0.70778 0.71225 0.7140(0.0500)

• Case 2 (mildly temperature-dependent viscosity)

  (64x64x64 mesh)

  mesh 16x16x16 32x32x32 64x64x64 128x128x128 Benchmark Standard Nu 3.0870 3.0503 3.0419 3.0399 3.0393(0.0050) Vrms 35.350 35.161 35.130 35.124 35.13(0.05) w(0,0,0.5) 154.076 162.020 164.814 165.631 165.9(1.0) w(0,b,0.5) -25.662 -26.446 -26.650 -26.703 -26.72(0.1) w(a,b,0.5) -59.973 -58.731 -58.360 -58.260 -58.23(0.1) T(0,0,0.5) 0.88286 0.89803 0.90323 0.90474 0.90529(0.0010) T(0,b,0.5) 0.51736 0.50170 0.49724 0.49606 0.49565(0.0010) T(a,b,0.5) 0.25785 0.24417 0.24052 0.23957 0.23925(0.0010) Q(0,0) 6.09608 5.90091 5.85018 5.83783 5.834(0.015) Q(a,0) 1.80145 1.73607 1.71942 1.71508 1.714(0.015) Q(a,b) 0.78762 0.77281 0.76952 0.76869 0.768(0.015)

Ogawa et al. (1991) とのベンチマークテスト

1.7×0.5×1の箱で、η=η_texp(-ET) なる形の温度依存粘性をもつ流体の熱対流シミュレーション。以下の表で、Ra_tは粘性率η_tで定義したRayleigh数、r=exp(E) は上面と下面での粘性率コントラスト。

• Case 1 (Ra_t=10⁵, r=1, WL-3D flow)

  (64x32x64 mesh)

  code present Ogawa et al. mesh 32x16x32 64x32x64 128x64x128 24x14x22 Nu 9.816 10.071 10.101 9.72

• Case 4 (Ra_t=10³, r=10², WL-2D roll)

  (64x32x64 mesh)

  code present Ogawa et al. mesh 32x16x32 64x32x64 128x64x128 24x14x22 Nu 4.060 4.073 4.077 4.13

• Case 10 (Ra_t=10³, r=10³, WL-3D)

  (64x32x64 mesh)

  code present Ogawa et al. mesh 32x16x32 64x32x64 128x64x128 24x14x22 Nu 4.961 4.932 4.921 4.96

• Case 16 (Ra_t=10³, r=3.2x10³, WL-3D)

  (64x32x64 mesh)

  code present Ogawa et al. mesh 32x16x32 64x32x64 128x64x128 44x18x30 Nu 5.284 5.271 5.271 5.37

• Case 17 (Ra_t=10², r=3.2x10⁴, SL-3D)

  (64x32x64 mesh)

  code present Ogawa et al. mesh 32x16x32 64x32x64 128x64x128 44x18x30 Nu 3.561 3.631 3.659 3.61

• Case 18 (Ra_t=32, r=10⁵, SL-3D)

  (64x32x64 mesh)

  code present Ogawa et al. mesh 32x16x32 64x32x64 128x64x128 44x18x30 Nu 3.119 3.174 3.197 3.17

参考文献

• Blankenbach, B. and Busse, F. and Christensen, U. and Cserepes, L. and Gunkel, D. and Hansen, U. and Jarvis, G. and Koch, M. and Marquatt, G. and Moore, D. and Olson, P. and Schmeling, H. and Schnaubelt, T., A benchmark comparison for mantle convection codes, Geophys. J. Int., 98, 23-38, 1989.
• Busse, F. H. and Christensen, U. and Clever, R. and Cserepes, L. and Gable, C. and Giannandrea, E. and Guillow, L. and Houseman, G. and Nataf, H.-C. and Ogawa, M. and Parmentier, M. and Sotin, C. and Travis, B., 3D convection at infinite Prandtl number in Cartesian geometry -- a benchmark comarison, Geophys. Astrophys. Fluid Dyn., 75, 39-59, 1993.
• Ogawa, M. and Schubert, G. and Zebib, A., Numerical simulations of three-dimensional thermal convection in a fluid with strongly temperature-dependent viscosity, J. Fluid Mech., 233, 299-328, 1991.