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 5 === U2 BUDGET FIG 13BR: "Pressure" is redistribution plus transport y- Convection Viscous Pressure Time-dep Total 0.0000e+00 0.1156e-20 0.1556e-10 0.1025e-11 0.8352e-23 0.1659e-10 0.1426e-03 0.9649e-02 -0.9581e+01 0.9587e+00 -0.3075e-03 -0.8613e+01 0.4782e-03 0.3714e+00 -0.4906e+01 0.1806e+01 -0.3420e-02 -0.2732e+01 0.1006e-02 0.3535e+01 -0.6456e+01 0.2450e+01 -0.1525e-01 -0.4857e+00 0.1726e-02 0.1768e+02 -0.2200e+02 0.1679e+01 -0.4554e-01 -0.2686e+01 0.2640e-02 0.5782e+02 -0.5603e+02 -0.2463e+01 -0.1078e+00 -0.7843e+00 0.3747e-02 0.1330e+03 -0.1245e+03 -0.8728e+01 -0.2170e+00 -0.3810e+00 0.5049e-02 0.2266e+03 -0.2115e+03 -0.1670e+02 -0.3821e+00 -0.1945e+01 0.6546e-02 0.3032e+03 -0.2767e+03 -0.2640e+02 -0.5978e+00 -0.4792e+00 0.8240e-02 0.3396e+03 -0.3035e+03 -0.3583e+02 -0.8397e+00 -0.6042e+00 0.1013e-01 0.3379e+03 -0.2944e+03 -0.4415e+02 -0.1074e+01 -0.1743e+01 0.1223e-01 0.3124e+03 -0.2600e+03 -0.5118e+02 -0.1271e+01 -0.1141e+00 0.1452e-01 0.2780e+03 -0.2214e+03 -0.5575e+02 -0.1412e+01 -0.6221e+00 0.1702e-01 0.2446e+03 -0.1868e+03 -0.5797e+02 -0.1494e+01 -0.1674e+01 0.1972e-01 0.2154e+03 -0.1556e+03 -0.5875e+02 -0.1520e+01 -0.4950e+00 0.2263e-01 0.1901e+03 -0.1319e+03 -0.5784e+02 -0.1501e+01 -0.1149e+01 0.2575e-01 0.1689e+03 -0.1128e+03 -0.5611e+02 -0.1454e+01 -0.1417e+01 0.2908e-01 0.1501e+03 -0.9526e+02 -0.5413e+02 -0.1390e+01 -0.6670e+00 0.3263e-01 0.1332e+03 -0.8149e+02 -0.5142e+02 -0.1323e+01 -0.1032e+01 0.3639e-01 0.1185e+03 -0.6968e+02 -0.4858e+02 -0.1261e+01 -0.9824e+00 0.4038e-01 0.1046e+03 -0.5884e+02 -0.4560e+02 -0.1205e+01 -0.1056e+01 0.4460e-01 0.9263e+02 -0.5057e+02 -0.4201e+02 -0.1158e+01 -0.1110e+01 0.4904e-01 0.8279e+02 -0.4381e+02 -0.3836e+02 -0.1124e+01 -0.5134e+00 0.5372e-01 0.7275e+02 -0.3784e+02 -0.3477e+02 -0.1097e+01 -0.9622e+00 0.5863e-01 0.6465e+02 -0.3320e+02 -0.3110e+02 -0.1078e+01 -0.7289e+00 0.6379e-01 0.5788e+02 -0.2920e+02 -0.2755e+02 -0.1066e+01 0.6167e-01 0.6920e-01 0.5073e+02 -0.2571e+02 -0.2419e+02 -0.1058e+01 -0.2306e+00 0.7486e-01 0.4480e+02 -0.2289e+02 -0.2099e+02 -0.1058e+01 -0.1327e+00 0.8077e-01 0.3935e+02 -0.2040e+02 -0.1810e+02 -0.1073e+01 -0.2174e+00 0.8696e-01 0.3412e+02 -0.1813e+02 -0.1553e+02 -0.1091e+01 -0.6294e+00 0.9341e-01 0.2959e+02 -0.1618e+02 -0.1305e+02 -0.1104e+01 -0.7415e+00 0.1001e+00 0.2590e+02 -0.1450e+02 -0.1085e+02 -0.1115e+01 -0.5642e+00 0.1072e+00 0.2200e+02 -0.1285e+02 -0.8972e+01 -0.1114e+01 -0.9323e+00 0.1145e+00 0.1936e+02 -0.1145e+02 -0.7297e+01 -0.1102e+01 -0.4954e+00 0.1221e+00 0.1731e+02 -0.1041e+02 -0.6177e+01 -0.1091e+01 -0.3674e+00 0.1300e+00 0.1553e+02 -0.9404e+01 -0.5323e+01 -0.1076e+01 -0.2765e+00 0.1383e+00 0.1415e+02 -0.8567e+01 -0.4479e+01 -0.1042e+01 0.5999e-01 0.1469e+00 0.1247e+02 -0.7965e+01 -0.3532e+01 -0.9970e+00 -0.2648e-01 0.1558e+00 0.1052e+02 -0.7131e+01 -0.2690e+01 -0.9560e+00 -0.2590e+00 0.1651e+00 0.8959e+01 -0.6364e+01 -0.1902e+01 -0.9238e+00 -0.2312e+00 0.1748e+00 0.7745e+01 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-0.2652e+00 0.3786e+00 0.1963e+00 -0.1086e+01 0.4980e+00 0.9296e-01 -0.2983e+00 0.3975e+00 0.1057e+00 -0.1016e+01 0.4928e+00 0.1060e+00 -0.3119e+00 0.4174e+00 0.1884e+00 -0.8809e+00 0.5040e+00 0.9579e-01 -0.9277e-01 0.4383e+00 0.3326e+00 -0.7675e+00 0.3579e+00 0.9367e-01 0.1661e-01 0.4603e+00 0.5247e+00 -0.6917e+00 0.2206e+00 0.1054e+00 0.1589e+00 0.4836e+00 0.4193e+00 -0.5397e+00 0.1888e+00 0.1095e+00 0.1779e+00 0.5082e+00 0.2510e+00 -0.4107e+00 0.1081e+00 0.1107e+00 0.5916e-01 0.5343e+00 0.2259e+00 -0.3460e+00 0.5608e-01 0.1236e+00 0.5962e-01 0.5620e+00 0.1027e+00 -0.3046e+00 0.5822e-01 0.1295e+00 -0.1421e-01 0.5916e+00 0.4721e-01 -0.2929e+00 0.1770e-01 0.1090e+00 -0.1191e+00 0.6232e+00 0.1241e+00 -0.2454e+00 -0.2944e-01 0.7640e-01 -0.7439e-01 0.6572e+00 0.3715e-01 -0.1839e+00 -0.4512e-02 0.4181e-01 -0.1094e+00 0.6939e+00 -0.2142e-01 -0.1652e+00 -0.1375e-01 0.2744e-01 -0.1729e+00 0.7337e+00 0.1093e+00 -0.1320e+00 -0.5087e-01 0.2841e-01 -0.4518e-01 0.7772e+00 0.8067e-01 -0.9401e-01 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