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 10 === -UW BUDGET FIG 13GR: y- Convection Viscous Pressure Time-dep Total 0.0000e+00 -0.3158e-21 -0.7482e-10 0.3528e-11 0.3975e-22 -0.7129e-10 0.1426e-03 0.5190e-03 -0.5150e+00 -0.5649e-03 0.1393e-02 -0.5136e+00 0.4782e-03 0.2018e-01 -0.1052e+00 -0.1870e+00 0.1612e-01 -0.2558e+00 0.1006e-02 0.2009e+00 0.1023e+00 -0.4459e+00 0.7555e-01 -0.6715e-01 0.1726e-02 0.1068e+01 -0.4803e+00 -0.8708e+00 0.2365e+00 -0.4650e-01 0.2640e-02 0.3735e+01 -0.2643e+01 -0.1801e+01 0.5782e+00 -0.1299e+00 0.3747e-02 0.9324e+01 -0.7451e+01 -0.3083e+01 0.1173e+01 -0.3642e-01 0.5049e-02 0.1760e+02 -0.1479e+02 -0.4784e+01 0.2025e+01 0.4550e-01 0.6546e-02 0.2656e+02 -0.2265e+02 -0.7136e+01 0.3020e+01 -0.1978e+00 0.8240e-02 0.3385e+02 -0.2796e+02 -0.1008e+02 0.3969e+01 -0.2185e+00 0.1013e-01 0.3847e+02 -0.2963e+02 -0.1351e+02 0.4696e+01 0.2043e-01 0.1223e-01 0.4051e+02 -0.2829e+02 -0.1746e+02 0.5105e+01 -0.1314e+00 0.1452e-01 0.4104e+02 -0.2491e+02 -0.2159e+02 0.5201e+01 -0.2589e+00 0.1702e-01 0.4112e+02 -0.2104e+02 -0.2514e+02 0.5048e+01 -0.1168e-01 0.1972e-01 0.4014e+02 -0.1715e+02 -0.2790e+02 0.4726e+01 -0.1798e+00 0.2263e-01 0.3855e+02 -0.1352e+02 -0.2972e+02 0.4314e+01 -0.3691e+00 0.2575e-01 0.3698e+02 -0.1059e+02 -0.3050e+02 0.3867e+01 -0.2432e+00 0.2908e-01 0.3484e+02 -0.8018e+01 -0.3061e+02 0.3416e+01 -0.3701e+00 0.3263e-01 0.3271e+02 -0.6119e+01 -0.3017e+02 0.2986e+01 -0.5999e+00 0.3639e-01 0.3069e+02 -0.4764e+01 -0.2933e+02 0.2588e+01 -0.8121e+00 0.4038e-01 0.2874e+02 -0.3651e+01 -0.2815e+02 0.2231e+01 -0.8338e+00 0.4460e-01 0.2696e+02 -0.2891e+01 -0.2673e+02 0.1925e+01 -0.7392e+00 0.4904e-01 0.2537e+02 -0.2389e+01 -0.2518e+02 0.1670e+01 -0.5240e+00 0.5372e-01 0.2348e+02 -0.1895e+01 -0.2364e+02 0.1463e+01 -0.5859e+00 0.5863e-01 0.2187e+02 -0.1499e+01 -0.2224e+02 0.1299e+01 -0.5723e+00 0.6379e-01 0.2093e+02 -0.1361e+01 -0.2110e+02 0.1170e+01 -0.3599e+00 0.6920e-01 0.1958e+02 -0.1117e+01 -0.1995e+02 0.1070e+01 -0.4261e+00 0.7486e-01 0.1822e+02 -0.8699e+00 -0.1875e+02 0.9952e+00 -0.4049e+00 0.8077e-01 0.1714e+02 -0.1007e+01 -0.1760e+02 0.9402e+00 -0.5193e+00 0.8696e-01 0.1640e+02 -0.1055e+01 -0.1661e+02 0.8911e+00 -0.3665e+00 0.9341e-01 0.1534e+02 -0.9024e+00 -0.1558e+02 0.8383e+00 -0.3042e+00 0.1001e+00 0.1461e+02 -0.9725e+00 -0.1458e+02 0.7782e+00 -0.1700e+00 0.1072e+00 0.1352e+02 -0.9194e+00 -0.1368e+02 0.7089e+00 -0.3719e+00 0.1145e+00 0.1259e+02 -0.6548e+00 -0.1281e+02 0.6404e+00 -0.2359e+00 0.1221e+00 0.1190e+02 -0.5740e+00 -0.1205e+02 0.5795e+00 -0.1482e+00 0.1300e+00 0.1093e+02 -0.5923e+00 -0.1132e+02 0.5202e+00 -0.4600e+00 0.1383e+00 0.1029e+02 -0.4703e+00 -0.1056e+02 0.4593e+00 -0.2739e+00 0.1469e+00 0.9243e+01 -0.3322e+00 -0.9670e+01 0.3980e+00 -0.3609e+00 0.1558e+00 0.8160e+01 -0.2735e+00 -0.8629e+01 0.3272e+00 -0.4155e+00 0.1651e+00 0.7550e+01 -0.2379e+00 -0.7703e+01 0.2445e+00 -0.1464e+00 0.1748e+00 0.7060e+01 -0.2718e+00 -0.6915e+01 0.1598e+00 0.3329e-01 0.1849e+00 0.6442e+01 -0.2513e+00 -0.6206e+01 0.7317e-01 0.5754e-01 0.1953e+00 0.5981e+01 -0.2489e+00 -0.5501e+01 -0.2702e-01 0.2040e+00 0.2062e+00 0.5247e+01 -0.2222e+00 -0.4738e+01 -0.1305e+00 0.1562e+00 0.2175e+00 0.4521e+01 -0.7551e-01 -0.4036e+01 -0.2232e+00 0.1870e+00 0.2293e+00 0.3962e+01 -0.6395e-02 -0.3545e+01 -0.3031e+00 0.1075e+00 0.2416e+00 0.3620e+01 -0.8488e-01 -0.3136e+01 -0.3575e+00 0.4112e-01 0.2543e+00 0.3283e+01 -0.7142e-01 -0.2874e+01 -0.3858e+00 -0.4733e-01 0.2676e+00 0.2903e+01 -0.1873e-01 -0.2597e+01 -0.4081e+00 -0.1211e+00 0.2815e+00 0.2365e+01 -0.7972e-02 -0.2134e+01 -0.4215e+00 -0.1980e+00 0.2960e+00 0.1927e+01 0.1621e-01 -0.1594e+01 -0.4168e+00 -0.6764e-01 0.3110e+00 0.1458e+01 0.3172e-01 -0.1071e+01 -0.3925e+00 0.2619e-01 0.3268e+00 0.1034e+01 0.6328e-02 -0.5994e+00 -0.3506e+00 0.9031e-01 0.3433e+00 0.5873e+00 0.4421e-02 -0.1780e+00 -0.2967e+00 0.1171e+00 0.3605e+00 0.1995e+00 0.1893e-02 0.6698e-01 -0.2290e+00 0.3938e-01 0.3786e+00 -0.1113e+00 -0.7344e-02 0.2818e+00 -0.1580e+00 0.5137e-02 0.3975e+00 -0.2974e+00 0.6593e-02 0.4534e+00 -0.1072e+00 0.5540e-01 0.4174e+00 -0.3533e+00 -0.1779e-01 0.4565e+00 -0.8687e-01 -0.1439e-02 0.4383e+00 -0.3005e+00 -0.1344e-01 0.4472e+00 -0.7620e-01 0.5709e-01 0.4603e+00 -0.3098e+00 0.1679e-01 0.4298e+00 -0.6528e-01 0.7143e-01 0.4836e+00 -0.2021e+00 0.1595e-01 0.3818e+00 -0.6313e-01 0.1325e+00 0.5082e+00 -0.1244e+00 0.8233e-02 0.3639e+00 -0.5259e-01 0.1952e+00 0.5343e+00 -0.1023e+00 0.1998e-01 0.3229e+00 -0.3396e-01 0.2067e+00 0.5620e+00 -0.1188e+00 0.2525e-01 0.2072e+00 -0.1535e-01 0.9831e-01 0.5916e+00 -0.1117e+00 0.1046e-01 0.1603e+00 0.1069e-01 0.6979e-01 0.6232e+00 -0.1338e+00 0.7559e-02 0.1459e+00 0.2771e-01 0.4739e-01 0.6572e+00 -0.1051e+00 0.2662e-03 0.9903e-01 0.3556e-01 0.2981e-01 0.6939e+00 -0.5854e-01 0.6296e-02 0.5002e-01 0.4835e-01 0.4612e-01 0.7337e+00 -0.1643e-01 0.1155e-01 0.3033e-01 0.4541e-01 0.7086e-01 0.7772e+00 -0.1044e+00 0.1492e-01 0.2555e-01 0.3187e-01 -0.3206e-01 0.8251e+00 -0.6726e-01 0.1005e-02 0.2825e-01 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