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 6 === V2 BUDGET FIG 13CR: y- Convection Viscous Pressure Time-dep Total 0.0000e+00 -0.5249e-10 -0.7547e-11 -0.5586e-10 0.0000e+00 -0.1159e-09 0.1426e-03 -0.4122e-06 0.7339e-02 -0.1911e-01 0.0000e+00 -0.1178e-01 0.4782e-03 -0.3431e-04 0.6837e-01 -0.1018e+00 0.0000e+00 -0.3350e-01 0.1006e-02 -0.1037e-02 0.2184e+00 -0.2651e+00 0.0000e+00 -0.4782e-01 0.1726e-02 -0.9064e-02 0.4094e+00 -0.3850e+00 0.0000e+00 0.1542e-01 0.2640e-02 -0.3543e-01 0.4363e+00 -0.4632e+00 0.0000e+00 -0.6227e-01 0.3747e-02 -0.6411e-01 0.1195e+00 -0.6683e-01 0.0000e+00 -0.1142e-01 0.5049e-02 0.1242e-01 -0.6629e+00 0.6523e+00 0.0000e+00 0.1886e-02 0.6546e-02 0.3836e+00 -0.1998e+01 0.1562e+01 0.0000e+00 -0.5242e-01 0.8240e-02 0.1177e+01 -0.3831e+01 0.2702e+01 0.0000e+00 0.4790e-01 0.1013e-01 0.2279e+01 -0.6054e+01 0.3677e+01 0.0000e+00 -0.9819e-01 0.1223e-01 0.3192e+01 -0.8554e+01 0.5416e+01 0.0000e+00 0.5467e-01 0.1452e-01 0.3320e+01 -0.1103e+02 0.7766e+01 0.0000e+00 0.5615e-01 0.1702e-01 0.2427e+01 -0.1326e+02 0.1089e+02 0.0000e+00 0.5246e-01 0.1972e-01 0.6312e+00 -0.1515e+02 0.1464e+02 0.0000e+00 0.1224e+00 0.2263e-01 -0.1451e+01 -0.1655e+02 0.1814e+02 0.0000e+00 0.1419e+00 0.2575e-01 -0.3315e+01 -0.1753e+02 0.2111e+02 0.0000e+00 0.2696e+00 0.2908e-01 -0.4931e+01 -0.1809e+02 0.2321e+02 0.0000e+00 0.1898e+00 0.3263e-01 -0.5815e+01 -0.1819e+02 0.2438e+02 0.0000e+00 0.3739e+00 0.3639e-01 -0.5924e+01 -0.1805e+02 0.2424e+02 0.0000e+00 0.2622e+00 0.4038e-01 -0.5808e+01 -0.1765e+02 0.2387e+02 0.0000e+00 0.4136e+00 0.4460e-01 -0.5484e+01 -0.1697e+02 0.2292e+02 0.0000e+00 0.4581e+00 0.4904e-01 -0.4772e+01 -0.1623e+02 0.2098e+02 0.0000e+00 -0.2979e-01 0.5372e-01 -0.4014e+01 -0.1537e+02 0.1987e+02 0.0000e+00 0.4887e+00 0.5863e-01 -0.3375e+01 -0.1438e+02 0.1813e+02 0.0000e+00 0.3698e+00 0.6379e-01 -0.2977e+01 -0.1342e+02 0.1628e+02 0.0000e+00 -0.1193e+00 0.6920e-01 -0.2598e+01 -0.1245e+02 0.1559e+02 0.0000e+00 0.5367e+00 0.7486e-01 -0.1870e+01 -0.1153e+02 0.1351e+02 0.0000e+00 0.1103e+00 0.8077e-01 -0.1341e+01 -0.1069e+02 0.1203e+02 0.0000e+00 -0.1463e-02 0.8696e-01 -0.1185e+01 -0.9878e+01 0.1157e+02 0.0000e+00 0.5043e+00 0.9341e-01 -0.8910e+00 -0.9131e+01 0.1002e+02 0.0000e+00 -0.1071e-02 0.1001e+00 -0.3200e+00 -0.8483e+01 0.9027e+01 0.0000e+00 0.2232e+00 0.1072e+00 -0.3179e-01 -0.7837e+01 0.8555e+01 0.0000e+00 0.6861e+00 0.1145e+00 0.2302e+00 -0.7215e+01 0.6820e+01 0.0000e+00 -0.1642e+00 0.1221e+00 0.6538e+00 -0.6714e+01 0.6236e+01 0.0000e+00 0.1764e+00 0.1300e+00 0.1119e+01 -0.6323e+01 0.5944e+01 0.0000e+00 0.7398e+00 0.1383e+00 0.1326e+01 -0.5918e+01 0.4365e+01 0.0000e+00 -0.2273e+00 0.1469e+00 0.1044e+01 -0.5471e+01 0.4582e+01 0.0000e+00 0.1549e+00 0.1558e+00 0.8713e+00 -0.5066e+01 0.4768e+01 0.0000e+00 0.5731e+00 0.1651e+00 0.5747e+00 -0.4710e+01 0.3895e+01 0.0000e+00 -0.2412e+00 0.1748e+00 0.3075e+00 -0.4421e+01 0.4148e+01 0.0000e+00 0.3401e-01 0.1849e+00 0.1790e+00 -0.4160e+01 0.4379e+01 0.0000e+00 0.3976e+00 0.1953e+00 0.7846e-01 -0.3881e+01 0.3168e+01 0.0000e+00 -0.6346e+00 0.2062e+00 0.3368e+00 -0.3558e+01 0.3311e+01 0.0000e+00 0.8938e-01 0.2175e+00 0.5182e+00 -0.3205e+01 0.2933e+01 0.0000e+00 0.2464e+00 0.2293e+00 0.6481e+00 -0.2848e+01 0.1740e+01 0.0000e+00 -0.4602e+00 0.2416e+00 0.7261e+00 -0.2538e+01 0.2010e+01 0.0000e+00 0.1973e+00 0.2543e+00 0.7050e+00 -0.2280e+01 0.1836e+01 0.0000e+00 0.2610e+00 0.2676e+00 0.6196e+00 -0.2057e+01 0.1257e+01 0.0000e+00 -0.1804e+00 0.2815e+00 0.5376e+00 -0.1804e+01 0.1705e+01 0.0000e+00 0.4386e+00 0.2960e+00 0.4319e+00 -0.1564e+01 0.1604e+01 0.0000e+00 0.4719e+00 0.3110e+00 0.2108e+00 -0.1346e+01 0.9446e+00 0.0000e+00 -0.1909e+00 0.3268e+00 -0.6964e-01 -0.1167e+01 0.1700e+01 0.0000e+00 0.4625e+00 0.3433e+00 0.5644e-01 -0.1057e+01 0.1381e+01 0.0000e+00 0.3798e+00 0.3605e+00 0.2993e+00 -0.9788e+00 0.4975e+00 0.0000e+00 -0.1820e+00 0.3786e+00 0.4587e+00 -0.9145e+00 0.8486e+00 0.0000e+00 0.3928e+00 0.3975e+00 0.5563e+00 -0.8446e+00 0.4961e+00 0.0000e+00 0.2079e+00 0.4174e+00 0.6207e+00 -0.7291e+00 0.9235e-01 0.0000e+00 -0.1607e-01 0.4383e+00 0.5743e+00 -0.5760e+00 0.2960e+00 0.0000e+00 0.2943e+00 0.4603e+00 0.3081e+00 -0.4791e+00 0.1308e+00 0.0000e+00 -0.4019e-01 0.4836e+00 0.3600e-01 -0.4298e+00 0.2506e+00 0.0000e+00 -0.1432e+00 0.5082e+00 0.2149e-01 -0.3802e+00 0.5012e+00 0.0000e+00 0.1425e+00 0.5343e+00 0.4709e-01 -0.3395e+00 0.1159e+00 0.0000e+00 -0.1766e+00 0.5620e+00 0.1746e-01 -0.2959e+00 0.6350e-01 0.0000e+00 -0.2149e+00 0.5916e+00 0.2224e-01 -0.2561e+00 0.3254e+00 0.0000e+00 0.9159e-01 0.6232e+00 0.3289e-01 -0.2338e+00 0.9775e-02 0.0000e+00 -0.1912e+00 0.6572e+00 0.9563e-01 -0.2136e+00 -0.5388e-02 0.0000e+00 -0.1233e+00 0.6939e+00 0.1579e+00 -0.1712e+00 0.1731e+00 0.0000e+00 0.1598e+00 0.7337e+00 0.1333e+00 -0.1149e+00 -0.9415e-01 0.0000e+00 -0.7579e-01 0.7772e+00 0.7754e-01 -0.7690e-01 -0.3234e-01 0.0000e+00 -0.3170e-01 0.8251e+00 0.1306e+00 -0.5459e-01 0.6435e-01 0.0000e+00 0.1403e+00 0.8783e+00 0.1405e+00 -0.2942e-01 -0.8689e-01 0.0000e+00 0.2415e-01 0.9381e+00 0.5867e-01 -0.1061e-01 -0.2579e-01 0.0000e+00 0.2227e-01 0.1006e+01 0.1672e-01 -0.3588e-02 0.1803e-01 0.0000e+00 0.3117e-01 0.1086e+01 0.3808e-02 -0.2487e-02 -0.6246e-02 0.0000e+00 -0.4925e-02 0.1180e+01 0.2227e-03 -0.8402e-03 -0.1094e-02 0.0000e+00 -0.1711e-02 0.1297e+01 0.9225e-04 -0.7022e-03 0.3529e-02 0.0000e+00 0.2919e-02 0.1451e+01 0.1877e-04 -0.9823e-03 -0.7053e-03 0.0000e+00 -0.1669e-02 0.1674e+01 -0.4040e-05 -0.1982e-03 -0.8267e-03 0.0000e+00 -0.1029e-02 0.2088e+01 0.4133e-06 -0.2129e-03 0.9228e-04 0.0000e+00 -0.1202e-03