3D Unsteady Flow: approximation to Ekman layer (P.R. Spalart, J.Fluid Mech. 205, 319) Free-stream velocity components are U,inf = U,0 cos(ft) and W,inf = U,0 sin(ft). U and W (and all six Reynolds stresses) are non-zero, but are functions of y and t only, not of x and z (so V = 0 by continuity). A program capable of computing steady flow over infinite swept wings (z coordinate along generators and d[ ]/dz=0) could be adapted: Ud[ ]/x becomes d[ ]/dt; Vd[ ]/dy and Wd[ ]/dz are zero - for different reasons - and the program can be stepped in time instead of marching in x. Notation is explained in the JFM paper, especially para. 3.1: the defining Reynolds number, Rl, is U,0 [2/(f*nu)]^(1/2) where f is in rad/sec not Hz: it is independent of time. The length scale,delta = (Utau)/f, is quite close to the physical thickness: y- = y/delta. Rt = (Utau)*(delta)/nu. Phi0 (phi* in the JFM paper, eq. 18) is related to the direction of the surface shear stress, alpha = lim tan^1 (W/U) by eq. 18: it is not relevant to the computations. Rl Utau Phi0 Rt 766.51428 0.05580 0.30867 914.74451 === Part 9 === -VW BUDGET FIG 13FR: y- Convection Viscous Pressure Time-dep Total 0.0000e+00 0.2791e-11 -0.1362e-09 -0.1497e-10 -0.5107e-24 -0.1484e-09 0.1426e-03 -0.4016e-04 -0.2570e-01 0.6845e-01 -0.2012e-05 0.4270e-01 0.4782e-03 0.5245e-03 -0.8920e-01 -0.6017e-01 -0.7723e-04 -0.1489e+00 0.1006e-02 0.4594e-02 -0.1744e+00 0.2490e+00 -0.7482e-03 0.7845e-01 0.1726e-02 0.1699e-01 -0.2563e+00 0.3102e+00 -0.3892e-02 0.6702e-01 0.2640e-02 0.1546e-01 -0.2601e+00 0.1780e+00 -0.1395e-01 -0.8056e-01 0.3747e-02 -0.8752e-01 -0.1017e+00 0.2837e+00 -0.3819e-01 0.5633e-01 0.5049e-02 -0.4028e+00 0.2540e+00 0.3422e+00 -0.8412e-01 0.1093e+00 0.6546e-02 -0.9697e+00 0.7274e+00 0.4633e+00 -0.1546e+00 0.6635e-01 0.8240e-02 -0.1755e+01 0.1205e+01 0.6943e+00 -0.2450e+00 -0.1007e+00 0.1013e-01 -0.2745e+01 0.1574e+01 0.1804e+01 -0.3443e+00 0.2883e+00 0.1223e-01 -0.3899e+01 0.1739e+01 0.2712e+01 -0.4404e+00 0.1107e+00 0.1452e-01 -0.5218e+01 0.1778e+01 0.3708e+01 -0.5241e+00 -0.2549e+00 0.1702e-01 -0.6846e+01 0.1773e+01 0.6078e+01 -0.5914e+00 0.4136e+00 0.1972e-01 -0.8729e+01 0.1666e+01 0.7728e+01 -0.6419e+00 0.2348e-01 0.2263e-01 -0.1067e+02 0.1514e+01 0.9633e+01 -0.6777e+00 -0.2003e+00 0.2575e-01 -0.1274e+02 0.1411e+01 0.1255e+02 -0.7014e+00 0.5285e+00 0.2908e-01 -0.1464e+02 0.1311e+01 0.1393e+02 -0.7153e+00 -0.1114e+00 0.3263e-01 -0.1648e+02 0.1202e+01 0.1586e+02 -0.7211e+00 -0.1430e+00 0.3639e-01 -0.1822e+02 0.1153e+01 0.1817e+02 -0.7206e+00 0.3753e+00 0.4038e-01 -0.1887e+02 0.1074e+01 0.1840e+02 -0.7149e+00 -0.1080e+00 0.4460e-01 -0.1845e+02 0.9792e+00 0.1811e+02 -0.7054e+00 -0.6014e-01 0.4904e-01 -0.1764e+02 0.9449e+00 0.1762e+02 -0.6927e+00 0.2330e+00 0.5372e-01 -0.1632e+02 0.8986e+00 0.1621e+02 -0.6771e+00 0.1106e+00 0.5863e-01 -0.1529e+02 0.9071e+00 0.1498e+02 -0.6600e+00 -0.6060e-01 0.6379e-01 -0.1476e+02 0.9387e+00 0.1440e+02 -0.6415e+00 -0.6474e-01 0.6920e-01 -0.1355e+02 0.9263e+00 0.1338e+02 -0.6210e+00 0.1366e+00 0.7486e-01 -0.1215e+02 0.9491e+00 0.1160e+02 -0.5995e+00 -0.2023e+00 0.8077e-01 -0.1092e+02 0.9946e+00 0.1069e+02 -0.5780e+00 0.1930e+00 0.8696e-01 -0.9865e+01 0.9874e+00 0.9508e+01 -0.5559e+00 0.7455e-01 0.9341e-01 -0.9397e+01 0.9638e+00 0.8863e+01 -0.5335e+00 -0.1041e+00 0.1001e+00 -0.9192e+01 0.9533e+00 0.9033e+01 -0.5117e+00 0.2819e+00 0.1072e+00 -0.9158e+01 0.9316e+00 0.8602e+01 -0.4893e+00 -0.1131e+00 0.1145e+00 -0.9472e+01 0.9371e+00 0.9192e+01 -0.4671e+00 0.1897e+00 0.1221e+00 -0.1001e+02 0.9297e+00 0.9862e+01 -0.4450e+00 0.3376e+00 0.1300e+00 -0.1041e+02 0.8857e+00 0.9695e+01 -0.4217e+00 -0.2545e+00 0.1383e+00 -0.1056e+02 0.8546e+00 0.1027e+02 -0.3988e+00 0.1654e+00 0.1469e+00 -0.1045e+02 0.8416e+00 0.1024e+02 -0.3756e+00 0.2547e+00 0.1558e+00 -0.1000e+02 0.7986e+00 0.9466e+01 -0.3511e+00 -0.8682e-01 0.1651e+00 -0.9392e+01 0.7420e+00 0.9114e+01 -0.3265e+00 0.1377e+00 0.1748e+00 -0.9088e+01 0.7137e+00 0.8789e+01 -0.3019e+00 0.1132e+00 0.1849e+00 -0.9088e+01 0.6568e+00 0.8705e+01 -0.2766e+00 -0.2934e-02 0.1953e+00 -0.8929e+01 0.6195e+00 0.8727e+01 -0.2517e+00 0.1653e+00 0.2062e+00 -0.8196e+01 0.5926e+00 0.7994e+01 -0.2265e+00 0.1646e+00 0.2175e+00 -0.7304e+01 0.5362e+00 0.7138e+01 -0.2007e+00 0.1691e+00 0.2293e+00 -0.6311e+01 0.5047e+00 0.6068e+01 -0.1766e+00 0.8491e-01 0.2416e+00 -0.5104e+01 0.4658e+00 0.4943e+01 -0.1539e+00 0.1511e+00 0.2543e+00 -0.4807e+01 0.4175e+00 0.4577e+01 -0.1319e+00 0.5553e-01 0.2676e+00 -0.5029e+01 0.3958e+00 0.4897e+01 -0.1118e+00 0.1520e+00 0.2815e+00 -0.4822e+01 0.3559e+00 0.4500e+01 -0.9270e-01 -0.5960e-01 0.2960e+00 -0.4028e+01 0.3212e+00 0.3922e+01 -0.7340e-01 0.1417e+00 0.3110e+00 -0.3314e+01 0.2840e+00 0.3192e+01 -0.5517e-01 0.1070e+00 0.3268e+00 -0.2959e+01 0.2345e+00 0.2686e+01 -0.3706e-01 -0.7470e-01 0.3433e+00 -0.2898e+01 0.1868e+00 0.2696e+01 -0.1845e-01 -0.3369e-01 0.3605e+00 -0.2765e+01 0.1711e+00 0.2716e+01 -0.3566e-02 0.1184e+00 0.3786e+00 -0.2518e+01 0.1522e+00 0.2239e+01 0.9154e-02 -0.1184e+00 0.3975e+00 -0.2118e+01 0.1342e+00 0.1986e+01 0.2096e-01 0.2344e-01 0.4174e+00 -0.1824e+01 0.1225e+00 0.1762e+01 0.2890e-01 0.8947e-01 0.4383e+00 -0.1427e+01 0.1016e+00 0.1231e+01 0.3604e-01 -0.5905e-01 0.4603e+00 -0.1063e+01 0.8726e-01 0.1081e+01 0.4197e-01 0.1470e+00 0.4836e+00 -0.8901e+00 0.7990e-01 0.9513e+00 0.4281e-01 0.1839e+00 0.5082e+00 -0.6202e+00 0.6338e-01 0.5574e+00 0.4416e-01 0.4478e-01 0.5343e+00 -0.3746e+00 0.4409e-01 0.4886e+00 0.4349e-01 0.2015e+00 0.5620e+00 -0.2557e+00 0.3158e-01 0.3363e+00 0.3948e-01 0.1517e+00 0.5916e+00 -0.2992e+00 0.2444e-01 0.2971e+00 0.4010e-01 0.6238e-01 0.6232e+00 -0.2484e+00 0.1719e-01 0.3510e+00 0.4147e-01 0.1613e+00 0.6572e+00 -0.1683e+00 0.2066e-01 0.2634e+00 0.3877e-01 0.1545e+00 0.6939e+00 -0.1896e+00 0.1250e-01 0.1137e+00 0.3663e-01 -0.2679e-01 0.7337e+00 -0.9769e-01 0.7826e-02 0.9187e-01 0.3204e-01 0.3404e-01 0.7772e+00 -0.6886e-01 0.5773e-02 0.3440e-01 0.2651e-01 -0.2179e-02 0.8251e+00 -0.3210e-01 0.6802e-03 -0.4412e-01 0.2260e-01 -0.5294e-01 0.8783e+00 -0.1213e-02 -0.1029e-02 -0.2739e-01 0.1482e-01 -0.1481e-01 0.9381e+00 -0.1202e-01 -0.1139e-02 -0.1413e-02 0.5236e-02 -0.9333e-02 0.1006e+01 -0.4379e-02 -0.4470e-03 0.2729e-03 0.2974e-03 -0.4256e-02 0.1086e+01 -0.4100e-03 -0.9257e-04 0.6353e-02 -0.2501e-03 0.5600e-02 0.1180e+01 0.3289e-03 -0.1458e-04 -0.4936e-03 -0.7877e-04 -0.2581e-03 0.1297e+01 0.1251e-04 -0.7144e-05 0.6648e-03 -0.2389e-04 0.6462e-03 0.1451e+01 -0.3039e-05 -0.2379e-05 0.3862e-02 0.6388e-05 0.3863e-02 0.1674e+01 0.7040e-05 -0.5600e-05 0.1377e-03 0.1833e-04 0.1575e-03 0.2088e+01 0.5519e-06 0.1329e-05 0.9073e-03 0.1031e-05 0.9102e-03