7. Natural Convection#
Coupling of Navier-Stokes and heat transport
Change in temperature leads to gravity forces
Temperature is convected by fluid velocity
Rayleigh-Benard benchmark example
from ngsolve import *
import netgen.geom2d
import netgen.meshing
from ngsolve.webgui import Draw
import math
from ngs_templates.NavierStokes import *
from ngs_templates.ConvectionDiffusion import *
ngsglobals.msg_level = 0
geo = netgen.geom2d.SplineGeometry()
geo.AddRectangle( (0,0), (0.06, 0.01), bcs=['b','r','t','l'])
mesh = Mesh(geo.GenerateMesh(maxh=0.002))
timestep = 0.1
navstokes = NavierStokes (mesh, nu=1.04177e-6, order=3, timestep = timestep,
inflow="", outflow="", wall="l|r|b|t", uin=(0,0) )
Tinitial = 293.5-50*y+y*(0.01-y)*1e3*sin(20/0.06*x*math.pi)
convdiff = ConvectionDiffusionEquation (mesh, order=3, lam=1.38e-7, \
wind = navstokes.velocity, dirichlet="b|t", udir=Tinitial, timestep=timestep)
convdiff.SetInitial(Tinitial)
T0 = 293
beta = 2.07e-4
navstokes.AddForce ( (1-beta*(convdiff.concentration-T0))*(0, -9.81))
navstokes.SolveInitial()
scenev = Draw (navstokes.velocity, mesh, "velocity", vectors = { "grid_size" : 40 })
scenet = Draw (convdiff.concentration, mesh, "temp")
t, tend = 0, 1000
cnt = 0
with TaskManager():
while t < tend:
navstokes.DoTimeStep()
convdiff.DoTimeStep()
t = t+timestep
cnt = cnt+1
if cnt % 10 == 0:
print ('t = ', t, end='\r')
scenev.Redraw()
scenet.Redraw()
t = 0.9999999999999999
t = 2.0000000000000004
t = 3.0000000000000013
t = 4.000000000000002
t = 4.999999999999998
t = 5.999999999999995
t = 6.999999999999991
t = 7.999999999999988
t = 8.999999999999984
t = 9.99999999999998
t = 10.999999999999977
t = 11.999999999999973
t = 12.99999999999997
t = 13.999999999999966
t = 14.999999999999963
t = 15.99999999999996
t = 16.99999999999997
t = 17.999999999999986
t = 19.0
t = 20.000000000000014
t = 21.00000000000003
t = 22.000000000000043
t = 23.000000000000057
t = 24.00000000000007
t = 25.000000000000085
t = 26.0000000000001
t = 27.000000000000114
t = 28.000000000000128
t = 29.000000000000142
t = 30.000000000000156
t = 31.00000000000017
t = 32.000000000000185
t = 33.0000000000002
t = 34.00000000000021
t = 35.00000000000023
t = 36.00000000000024
t = 37.000000000000256
t = 38.00000000000027
t = 39.000000000000284
t = 40.0000000000003
t = 41.00000000000031
t = 42.00000000000033
t = 43.00000000000034
t = 44.000000000000355
t = 45.00000000000037
t = 46.000000000000384
t = 47.0000000000004
t = 48.00000000000041
t = 49.000000000000426
t = 50.00000000000044
t = 51.000000000000455
t = 52.00000000000047
t = 53.00000000000048
t = 54.0000000000005
t = 55.00000000000051
t = 56.000000000000526
t = 57.00000000000054
t = 58.000000000000554
t = 59.00000000000057
t = 60.00000000000058
t = 61.0000000000006
t = 62.00000000000061
t = 63.000000000000625
t = 64.00000000000064
t = 65.00000000000058
t = 66.00000000000053
t = 67.00000000000047
t = 68.00000000000041
t = 69.00000000000036
t = 70.0000000000003
t = 71.00000000000024
t = 72.00000000000018
t = 73.00000000000013
t = 74.00000000000007
t = 75.00000000000001
t = 75.99999999999996
t = 76.9999999999999
t = 77.99999999999984
t = 78.99999999999979
t = 79.99999999999973
t = 80.99999999999967
t = 81.99999999999962
t = 82.99999999999956
t = 83.9999999999995
t = 84.99999999999945
t = 85.99999999999939
t = 86.99999999999933
t = 87.99999999999928
t = 88.99999999999922
t = 89.99999999999916
t = 90.9999999999991
t = 91.99999999999905
t = 92.99999999999899
t = 93.99999999999893
t = 94.99999999999888
t = 95.99999999999882
t = 96.99999999999876
t = 97.9999999999987
t = 98.99999999999865
t = 99.9999999999986
t = 100.99999999999854
t = 101.99999999999848
t = 102.99999999999842
t = 103.99999999999837
t = 104.99999999999831
t = 105.99999999999825
t = 106.9999999999982
t = 107.99999999999814
t = 108.99999999999808
t = 109.99999999999802
t = 110.99999999999797
t = 111.99999999999791
t = 112.99999999999785
t = 113.9999999999978
t = 114.99999999999774
t = 115.99999999999768
t = 116.99999999999763
t = 117.99999999999757
t = 118.99999999999751
t = 119.99999999999746
t = 120.9999999999974
t = 121.99999999999734
t = 122.99999999999729
t = 123.99999999999723
t = 124.99999999999717
t = 125.99999999999712
t = 126.99999999999706
t = 127.999999999997
t = 128.99999999999696
t = 129.9999999999969
t = 130.99999999999685
t = 131.9999999999968
t = 132.99999999999673
t = 133.99999999999667
t = 134.99999999999662
t = 135.99999999999656
t = 136.9999999999965
t = 137.99999999999645
t = 138.9999999999964
t = 139.99999999999633
t = 140.99999999999628
t = 141.99999999999622
t = 142.99999999999616
t = 143.9999999999961
t = 144.99999999999605
t = 145.999999999996
t = 146.99999999999594
t = 147.99999999999588
t = 148.99999999999582
t = 149.99999999999577
t = 150.9999999999957
t = 151.99999999999565
t = 152.9999999999956
t = 153.99999999999554
t = 154.99999999999548
t = 155.99999999999542
t = 156.99999999999537
t = 157.9999999999953
t = 158.99999999999525
t = 159.9999999999952
t = 160.99999999999514
t = 161.99999999999508
t = 162.99999999999503
t = 163.99999999999497
t = 164.9999999999949
t = 165.99999999999486
t = 166.9999999999948
t = 167.99999999999474
t = 168.99999999999469
t = 169.99999999999463
t = 170.99999999999457
t = 171.99999999999451
t = 172.99999999999446
t = 173.9999999999944
t = 174.99999999999434
t = 175.9999999999943
t = 176.99999999999423
t = 177.99999999999417
t = 178.99999999999412
t = 179.99999999999406
t = 180.999999999994
t = 181.99999999999395
t = 182.9999999999939
t = 183.99999999999383
t = 184.99999999999378
t = 185.99999999999372
t = 186.99999999999366
t = 187.9999999999936
t = 188.99999999999355
t = 189.9999999999935
t = 190.99999999999343
t = 191.99999999999338
t = 192.99999999999332
t = 193.99999999999326
t = 194.9999999999932
t = 195.99999999999315
t = 196.9999999999931
t = 197.99999999999304
t = 198.99999999999298
t = 199.99999999999292
t = 200.99999999999287
t = 201.9999999999928
t = 202.99999999999275
t = 203.9999999999927
t = 204.99999999999264
t = 205.99999999999258
t = 206.99999999999253
t = 207.99999999999247
t = 208.9999999999924
t = 209.99999999999235
t = 210.9999999999923
t = 211.99999999999224
t = 212.99999999999218
t = 213.99999999999213
t = 214.99999999999207
t = 215.999999999992
t = 216.99999999999196
t = 217.9999999999919
t = 218.99999999999184
t = 219.9999999999918
t = 220.99999999999173
t = 221.99999999999167
t = 222.99999999999162
t = 223.99999999999156
t = 224.9999999999915
t = 225.99999999999145
t = 226.9999999999914
t = 227.99999999999133
t = 228.99999999999127
t = 229.99999999999122
t = 230.99999999999116
t = 231.9999999999911
t = 232.99999999999105
t = 233.999999999991
t = 234.99999999999093
t = 235.99999999999088
t = 236.99999999999082
t = 237.99999999999076
t = 238.9999999999907
t = 239.99999999999065
t = 240.9999999999906
t = 241.99999999999054
t = 242.99999999999048
t = 243.99999999999042
t = 244.99999999999037
t = 245.9999999999903
t = 246.99999999999025
t = 247.9999999999902
t = 248.99999999999014
t = 249.99999999999008
t = 250.99999999999002
t = 251.99999999998997
t = 252.9999999999899
t = 253.99999999998985
t = 254.9999999999898
t = 255.99999999998974
t = 256.99999999998994
t = 257.99999999999017
t = 258.9999999999904
t = 259.9999999999906
t = 260.99999999999085
t = 261.9999999999911
t = 262.9999999999913
t = 263.99999999999153
t = 264.99999999999176
t = 265.999999999992
t = 266.9999999999922
t = 267.99999999999244
t = 268.99999999999267
t = 269.9999999999929
t = 270.9999999999931
t = 271.99999999999335
t = 272.9999999999936
t = 273.9999999999938
t = 274.99999999999403
t = 275.99999999999426
t = 276.9999999999945
t = 277.9999999999947
t = 278.99999999999494
t = 279.99999999999517
t = 280.9999999999954
t = 281.9999999999956
t = 282.99999999999585
t = 283.9999999999961
t = 284.9999999999963
t = 285.99999999999653
t = 286.99999999999676
t = 287.999999999997
t = 288.9999999999972
t = 289.99999999999744
t = 290.99999999999767
t = 291.9999999999979
t = 292.9999999999981
t = 293.99999999999835
t = 294.9999999999986
t = 295.9999999999988
t = 296.99999999999903
t = 297.99999999999926
t = 298.9999999999995
t = 299.9999999999997
t = 300.99999999999994
t = 302.00000000000017
t = 303.0000000000004
t = 304.0000000000006
t = 305.00000000000085
t = 306.0000000000011
t = 307.0000000000013
t = 308.00000000000153
t = 309.00000000000176
t = 310.000000000002
t = 311.0000000000022
t = 312.00000000000244
t = 313.0000000000027
t = 314.0000000000029
t = 315.0000000000031
t = 316.00000000000335
t = 317.0000000000036
t = 318.0000000000038
t = 319.00000000000404
t = 320.00000000000426
t = 321.0000000000045
t = 322.0000000000047
t = 323.00000000000495
t = 324.0000000000052
t = 325.0000000000054
t = 326.0000000000056
t = 327.00000000000585
t = 328.0000000000061
t = 329.0000000000063
t = 330.00000000000654
t = 331.00000000000676
t = 332.000000000007
t = 333.0000000000072
t = 334.00000000000745
t = 335.0000000000077
t = 336.0000000000079
t = 337.00000000000813
t = 338.00000000000836
t = 339.0000000000086
t = 340.0000000000088
t = 341.00000000000904
t = 342.00000000000927
t = 343.0000000000095
t = 344.0000000000097
t = 345.00000000000995
t = 346.0000000000102
t = 347.0000000000104
t = 348.00000000001063
t = 349.00000000001086
t = 350.0000000000111
t = 351.0000000000113
t = 352.00000000001154
t = 353.00000000001177
t = 354.000000000012
t = 355.0000000000122
t = 356.00000000001245
t = 357.0000000000127
t = 358.0000000000129
t = 359.00000000001313
t = 360.00000000001336
t = 361.0000000000136
t = 362.0000000000138
t = 363.00000000001404
t = 364.00000000001427
t = 365.0000000000145
t = 366.0000000000147
t = 367.00000000001495
t = 368.0000000000152
t = 369.0000000000154
t = 370.00000000001563
t = 371.00000000001586
t = 372.0000000000161
t = 373.0000000000163
t = 374.00000000001654
t = 375.00000000001677
t = 376.000000000017
t = 377.0000000000172
t = 378.00000000001745
t = 379.0000000000177
t = 380.0000000000179
t = 381.00000000001813
t = 382.00000000001836
t = 383.0000000000186
t = 384.0000000000188
t = 385.00000000001904
t = 386.00000000001927
t = 387.0000000000195
t = 388.0000000000197
t = 389.00000000001995
t = 390.0000000000202
t = 391.0000000000204
t = 392.00000000002063
t = 393.00000000002086
t = 394.0000000000211
t = 395.0000000000213
t = 396.00000000002154
t = 397.00000000002177
t = 398.000000000022
t = 399.0000000000222
t = 400.00000000002245
t = 401.0000000000227
t = 402.0000000000229
t = 403.00000000002314
t = 404.00000000002336
t = 405.0000000000236
t = 406.0000000000238
t = 407.00000000002404
t = 408.0000000000243
t = 409.0000000000245
t = 410.0000000000247
t = 411.00000000002495
t = 412.0000000000252
t = 413.0000000000254
t = 414.00000000002564
t = 415.00000000002586
t = 416.0000000000261
t = 417.0000000000263
t = 418.00000000002655
t = 419.0000000000268
t = 420.000000000027
t = 421.0000000000272
t = 422.00000000002746
t = 423.0000000000277
t = 424.0000000000279
t = 425.00000000002814
t = 426.00000000002836
t = 427.0000000000286
t = 428.0000000000288
t = 429.00000000002905
t = 430.0000000000293
t = 431.0000000000295
t = 432.00000000002973
t = 433.00000000002996
t = 434.0000000000302
t = 435.0000000000304
t = 436.00000000003064
t = 437.00000000003087
t = 438.0000000000311
t = 439.0000000000313
t = 440.00000000003155
t = 441.0000000000318
t = 442.000000000032
t = 443.00000000003223
t = 444.00000000003246
t = 445.0000000000327
t = 446.0000000000329
t = 447.00000000003314
t = 448.00000000003337
t = 449.0000000000336
t = 450.0000000000338
t = 451.00000000003405
t = 452.0000000000343
t = 453.0000000000345
t = 454.00000000003473
t = 455.00000000003496
t = 456.0000000000352
t = 457.0000000000354
t = 458.00000000003564
t = 459.00000000003587
t = 460.0000000000361
t = 461.0000000000363
t = 462.00000000003655
t = 463.0000000000368
t = 464.000000000037
t = 465.00000000003723
t = 466.00000000003746
t = 467.0000000000377
t = 468.0000000000379
t = 469.00000000003814
t = 470.00000000003837
t = 471.0000000000386
t = 472.0000000000388
t = 473.00000000003905
t = 474.0000000000393
t = 475.0000000000395
t = 476.00000000003973
t = 477.00000000003996
t = 478.0000000000402
t = 479.0000000000404
t = 480.00000000004064
t = 481.00000000004087
t = 482.0000000000411
t = 483.0000000000413
t = 484.00000000004155
t = 485.0000000000418
t = 486.000000000042
t = 487.00000000004223
t = 488.00000000004246
t = 489.0000000000427
t = 490.0000000000429
t = 491.00000000004314
t = 492.00000000004337
t = 493.0000000000436
t = 494.0000000000438
t = 495.00000000004405
t = 496.0000000000443
t = 497.0000000000445
t = 498.00000000004474
t = 499.00000000004496
t = 500.0000000000452
t = 501.0000000000454
t = 502.00000000004565
t = 503.0000000000459
t = 504.0000000000461
t = 505.0000000000463
t = 506.00000000004655
t = 507.0000000000468
t = 508.000000000047
t = 509.00000000004724
t = 510.00000000004746
t = 511.0000000000477
t = 512.0000000000479
t = 513.0000000000481
t = 514.0000000000483
t = 515.0000000000485
t = 516.0000000000488
t = 517.000000000049
t = 518.0000000000492
t = 519.0000000000495
t = 520.0000000000497
t = 521.0000000000499
t = 522.0000000000501
t = 523.0000000000504
t = 524.0000000000506
t = 525.0000000000508
t = 526.000000000051
t = 527.0000000000513
t = 528.0000000000515
t = 529.0000000000517
t = 530.000000000052
t = 531.0000000000522
t = 532.0000000000524
t = 533.0000000000526
t = 534.0000000000529
t = 535.0000000000531
t = 536.0000000000533
t = 537.0000000000535
t = 538.0000000000538
t = 539.000000000054
t = 540.0000000000542
t = 541.0000000000545
t = 542.0000000000547
t = 543.0000000000549
t = 544.0000000000551
t = 545.0000000000554
t = 546.0000000000556
t = 547.0000000000558
t = 548.000000000056
t = 549.0000000000563
t = 550.0000000000565
t = 551.0000000000567
t = 552.000000000057
t = 553.0000000000572
t = 554.0000000000574
t = 555.0000000000576
t = 556.0000000000579
t = 557.0000000000581
t = 558.0000000000583
t = 559.0000000000585
t = 560.0000000000588
t = 561.000000000059
t = 562.0000000000592
t = 563.0000000000595
t = 564.0000000000597
t = 565.0000000000599
t = 566.0000000000601
t = 567.0000000000604
t = 568.0000000000606
t = 569.0000000000608
t = 570.000000000061
t = 571.0000000000613
t = 572.0000000000615
t = 573.0000000000617
t = 574.000000000062
t = 575.0000000000622
t = 576.0000000000624
t = 577.0000000000626
t = 578.0000000000629
t = 579.0000000000631
t = 580.0000000000633
t = 581.0000000000636
t = 582.0000000000638
t = 583.000000000064
t = 584.0000000000642
t = 585.0000000000645
t = 586.0000000000647
t = 587.0000000000649
t = 588.0000000000651
t = 589.0000000000654
t = 590.0000000000656
t = 591.0000000000658
t = 592.000000000066
t = 593.0000000000663
t = 594.0000000000665
t = 595.0000000000667
t = 596.000000000067
t = 597.0000000000672
t = 598.0000000000674
t = 599.0000000000676
t = 600.0000000000679
t = 601.0000000000681
t = 602.0000000000683
t = 603.0000000000686
t = 604.0000000000688
t = 605.000000000069
t = 606.0000000000692
t = 607.0000000000695
t = 608.0000000000697
t = 609.0000000000699
t = 610.0000000000701
t = 611.0000000000704
t = 612.0000000000706
t = 613.0000000000708
t = 614.000000000071
t = 615.0000000000713
t = 616.0000000000715
t = 617.0000000000717
t = 618.000000000072
t = 619.0000000000722
t = 620.0000000000724
t = 621.0000000000726
t = 622.0000000000729
t = 623.0000000000731
t = 624.0000000000733
t = 625.0000000000736
t = 626.0000000000738
t = 627.000000000074
t = 628.0000000000742
t = 629.0000000000745
t = 630.0000000000747
t = 631.0000000000749
t = 632.0000000000751
t = 633.0000000000754
t = 634.0000000000756
t = 635.0000000000758
t = 636.000000000076
t = 637.0000000000763
t = 638.0000000000765
t = 639.0000000000767
t = 640.000000000077
t = 641.0000000000772
t = 642.0000000000774
t = 643.0000000000776
t = 644.0000000000779
t = 645.0000000000781
t = 646.0000000000783
t = 647.0000000000786
t = 648.0000000000788
t = 649.000000000079
t = 650.0000000000792
t = 651.0000000000795
t = 652.0000000000797
t = 653.0000000000799
t = 654.0000000000801
t = 655.0000000000804
t = 656.0000000000806
t = 657.0000000000808
t = 658.0000000000811
t = 659.0000000000813
t = 660.0000000000815
t = 661.0000000000817
t = 662.000000000082
t = 663.0000000000822
t = 664.0000000000824
t = 665.0000000000827
t = 666.0000000000829
t = 667.0000000000831
t = 668.0000000000833
t = 669.0000000000836
t = 670.0000000000838
t = 671.000000000084
t = 672.0000000000842
t = 673.0000000000845
t = 674.0000000000847
t = 675.0000000000849
t = 676.0000000000852
t = 677.0000000000854
t = 678.0000000000856
t = 679.0000000000858
t = 680.0000000000861
t = 681.0000000000863
t = 682.0000000000865
t = 683.0000000000867
t = 684.000000000087
t = 685.0000000000872
t = 686.0000000000874
t = 687.0000000000877
t = 688.0000000000879
t = 689.0000000000881
t = 690.0000000000883
t = 691.0000000000886
t = 692.0000000000888
t = 693.000000000089
t = 694.0000000000892
t = 695.0000000000895
t = 696.0000000000897
t = 697.0000000000899
t = 698.0000000000902
t = 699.0000000000904
t = 700.0000000000906
t = 701.0000000000908
t = 702.0000000000911
t = 703.0000000000913
t = 704.0000000000915
t = 705.0000000000917
t = 706.000000000092
t = 707.0000000000922
t = 708.0000000000924
t = 709.0000000000927
t = 710.0000000000929
t = 711.0000000000931
t = 712.0000000000933
t = 713.0000000000936
t = 714.0000000000938
t = 715.000000000094
t = 716.0000000000942
t = 717.0000000000945
t = 718.0000000000947
t = 719.0000000000949
t = 720.0000000000952
t = 721.0000000000954
t = 722.0000000000956
t = 723.0000000000958
t = 724.0000000000961
t = 725.0000000000963
t = 726.0000000000965
t = 727.0000000000967
t = 728.000000000097
t = 729.0000000000972
t = 730.0000000000974
t = 731.0000000000977
t = 732.0000000000979
t = 733.0000000000981
t = 734.0000000000983
t = 735.0000000000986
t = 736.0000000000988
t = 737.000000000099
t = 738.0000000000992
t = 739.0000000000995
t = 740.0000000000997
t = 741.0000000000999
t = 742.0000000001002
t = 743.0000000001004
t = 744.0000000001006
t = 745.0000000001008
t = 746.0000000001011
t = 747.0000000001013
t = 748.0000000001015
t = 749.0000000001017
t = 750.000000000102
t = 751.0000000001022
t = 752.0000000001024
t = 753.0000000001027
t = 754.0000000001029
t = 755.0000000001031
t = 756.0000000001033
t = 757.0000000001036
t = 758.0000000001038
t = 759.000000000104
t = 760.0000000001043
t = 761.0000000001045
t = 762.0000000001047
t = 763.0000000001049
t = 764.0000000001052
t = 765.0000000001054
t = 766.0000000001056
t = 767.0000000001058
t = 768.0000000001061
t = 769.0000000001063
t = 770.0000000001065
t = 771.0000000001068
t = 772.000000000107
t = 773.0000000001072
t = 774.0000000001074
t = 775.0000000001077
t = 776.0000000001079
t = 777.0000000001081
t = 778.0000000001083
t = 779.0000000001086
t = 780.0000000001088
t = 781.000000000109
t = 782.0000000001093
t = 783.0000000001095
t = 784.0000000001097
t = 785.0000000001099
t = 786.0000000001102
t = 787.0000000001104
t = 788.0000000001106
t = 789.0000000001108
t = 790.0000000001111
t = 791.0000000001113
t = 792.0000000001115
t = 793.0000000001118
t = 794.000000000112
t = 795.0000000001122
t = 796.0000000001124
t = 797.0000000001127
t = 798.0000000001129
t = 799.0000000001131
t = 800.0000000001133
t = 801.0000000001136
t = 802.0000000001138
t = 803.000000000114
t = 804.0000000001143
t = 805.0000000001145
t = 806.0000000001147
t = 807.0000000001149
t = 808.0000000001152
t = 809.0000000001154
t = 810.0000000001156
t = 811.0000000001158
t = 812.0000000001161
t = 813.0000000001163
t = 814.0000000001165
t = 815.0000000001168
t = 816.000000000117
t = 817.0000000001172
t = 818.0000000001174
t = 819.0000000001177
t = 820.0000000001179
t = 821.0000000001181
t = 822.0000000001183
t = 823.0000000001186
t = 824.0000000001188
t = 825.000000000119
t = 826.0000000001193
t = 827.0000000001195
t = 828.0000000001197
t = 829.0000000001199
t = 830.0000000001202
t = 831.0000000001204
t = 832.0000000001206
t = 833.0000000001208
t = 834.0000000001211
t = 835.0000000001213
t = 836.0000000001215
t = 837.0000000001218
t = 838.000000000122
t = 839.0000000001222
t = 840.0000000001224
t = 841.0000000001227
t = 842.0000000001229
t = 843.0000000001231
t = 844.0000000001234
t = 845.0000000001236
t = 846.0000000001238
t = 847.000000000124
t = 848.0000000001243
t = 849.0000000001245
t = 850.0000000001247
t = 851.0000000001249
t = 852.0000000001252
t = 853.0000000001254
t = 854.0000000001256
t = 855.0000000001259
t = 856.0000000001261
t = 857.0000000001263
t = 858.0000000001265
t = 859.0000000001268
t = 860.000000000127
t = 861.0000000001272
t = 862.0000000001274
t = 863.0000000001277
t = 864.0000000001279
t = 865.0000000001281
t = 866.0000000001284
t = 867.0000000001286
t = 868.0000000001288
t = 869.000000000129
t = 870.0000000001293
t = 871.0000000001295
t = 872.0000000001297
t = 873.00000000013
t = 874.0000000001302
t = 875.0000000001304
t = 876.0000000001306
t = 877.0000000001309
t = 878.0000000001311
t = 879.0000000001313
t = 880.0000000001315
t = 881.0000000001318
t = 882.000000000132
t = 883.0000000001322
t = 884.0000000001324
t = 885.0000000001327
t = 886.0000000001329
t = 887.0000000001331
t = 888.0000000001334
t = 889.0000000001336
t = 890.0000000001338
t = 891.000000000134
t = 892.0000000001343
t = 893.0000000001345
t = 894.0000000001347
t = 895.000000000135
t = 896.0000000001352
t = 897.0000000001354
t = 898.0000000001356
t = 899.0000000001359
t = 900.0000000001361
t = 901.0000000001363
t = 902.0000000001365
t = 903.0000000001368
t = 904.000000000137
t = 905.0000000001372
t = 906.0000000001374
t = 907.0000000001377
t = 908.0000000001379
t = 909.0000000001381
t = 910.0000000001384
t = 911.0000000001386
t = 912.0000000001388
t = 913.000000000139
t = 914.0000000001393
t = 915.0000000001395
t = 916.0000000001397
t = 917.00000000014
t = 918.0000000001402
t = 919.0000000001404
t = 920.0000000001406
t = 921.0000000001409
t = 922.0000000001411
t = 923.0000000001413
t = 924.0000000001415
t = 925.0000000001418
t = 926.000000000142
t = 927.0000000001422
t = 928.0000000001424
t = 929.0000000001427
t = 930.0000000001429
t = 931.0000000001431
t = 932.0000000001434
t = 933.0000000001436
t = 934.0000000001438
t = 935.000000000144
t = 936.0000000001443
t = 937.0000000001445
t = 938.0000000001447
t = 939.000000000145
t = 940.0000000001452
t = 941.0000000001454
t = 942.0000000001456
t = 943.0000000001459
t = 944.0000000001461
t = 945.0000000001463
t = 946.0000000001465
t = 947.0000000001468
t = 948.000000000147
t = 949.0000000001472
t = 950.0000000001475
t = 951.0000000001477
t = 952.0000000001479
t = 953.0000000001481
t = 954.0000000001484
t = 955.0000000001486
t = 956.0000000001488
t = 957.000000000149
t = 958.0000000001493
t = 959.0000000001495
t = 960.0000000001497
t = 961.00000000015
t = 962.0000000001502
t = 963.0000000001504
t = 964.0000000001506
t = 965.0000000001509
t = 966.0000000001511
t = 967.0000000001513
t = 968.0000000001515
t = 969.0000000001518
t = 970.000000000152
t = 971.0000000001522
t = 972.0000000001525
t = 973.0000000001527
t = 974.0000000001529
t = 975.0000000001531
t = 976.0000000001534
t = 977.0000000001536
t = 978.0000000001538
t = 979.000000000154
t = 980.0000000001543
t = 981.0000000001545
t = 982.0000000001547
t = 983.000000000155
t = 984.0000000001552
t = 985.0000000001554
t = 986.0000000001556
t = 987.0000000001559
t = 988.0000000001561
t = 989.0000000001563
t = 990.0000000001565
t = 991.0000000001568
t = 992.000000000157
t = 993.0000000001572
t = 994.0000000001575
t = 995.0000000001577
t = 996.0000000001579
t = 997.0000000001581
t = 998.0000000001584
t = 999.0000000001586
t = 1000.0000000001588