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 8 === -UV BUDGET FIG 13ER: y- Convection Viscous Pressure Time-dep Total 0.0000e+00 -0.3174e-10 -0.1449e-10 -0.1274e-10 0.3490e-24 -0.5897e-10 0.1426e-03 0.9480e-04 0.2184e+00 -0.7955e+00 -0.2375e-06 -0.5770e+00 0.4782e-03 0.9783e-02 0.7779e+00 0.1629e+00 -0.8916e-05 0.9506e+00 0.1006e-02 0.1635e+00 0.1624e+01 -0.1942e+01 -0.8522e-04 -0.1546e+00 0.1726e-02 0.1145e+01 0.2553e+01 -0.5261e+01 -0.4382e-03 -0.1563e+01 0.2640e-02 0.4678e+01 0.2814e+01 -0.7322e+01 -0.1565e-02 0.1686e+00 0.3747e-02 0.1282e+02 0.1459e+01 -0.1384e+02 -0.4320e-02 0.4316e+00 0.5049e-02 0.2571e+02 -0.1565e+01 -0.2656e+02 -0.9757e-02 -0.2426e+01 0.6546e-02 0.4040e+02 -0.4938e+01 -0.3598e+02 -0.1871e-01 -0.5370e+00 0.8240e-02 0.5292e+02 -0.7236e+01 -0.4443e+02 -0.3136e-01 0.1225e+01 0.1013e-01 0.6098e+02 -0.7915e+01 -0.5632e+02 -0.4708e-01 -0.3302e+01 0.1223e-01 0.6441e+02 -0.7198e+01 -0.5725e+02 -0.6460e-01 -0.1011e+00 0.1452e-01 0.6441e+02 -0.5912e+01 -0.5740e+02 -0.8270e-01 0.1014e+01 0.1702e-01 0.6306e+02 -0.4712e+01 -0.6161e+02 -0.1004e+00 -0.3366e+01 0.1972e-01 0.6100e+02 -0.3741e+01 -0.5679e+02 -0.1170e+00 0.3530e+00 0.2263e-01 0.5821e+02 -0.3138e+01 -0.5506e+02 -0.1324e+00 -0.1181e+00 0.2575e-01 0.5599e+02 -0.2924e+01 -0.5547e+02 -0.1470e+00 -0.2552e+01 0.2908e-01 0.5333e+02 -0.2893e+01 -0.5023e+02 -0.1607e+00 0.5002e-01 0.3263e-01 0.5015e+02 -0.2976e+01 -0.4744e+02 -0.1738e+00 -0.4375e+00 0.3639e-01 0.4735e+02 -0.3129e+01 -0.4557e+02 -0.1866e+00 -0.1535e+01 0.4038e-01 0.4412e+02 -0.3200e+01 -0.4097e+02 -0.1992e+00 -0.2530e+00 0.4460e-01 0.4119e+02 -0.3237e+01 -0.3815e+02 -0.2117e+00 -0.4110e+00 0.4904e-01 0.3904e+02 -0.3236e+01 -0.3644e+02 -0.2245e+00 -0.8679e+00 0.5372e-01 0.3634e+02 -0.3146e+01 -0.3300e+02 -0.2373e+00 -0.5137e-01 0.5863e-01 0.3382e+02 -0.3077e+01 -0.3084e+02 -0.2499e+00 -0.3563e+00 0.6379e-01 0.3187e+02 -0.3015e+01 -0.2941e+02 -0.2623e+00 -0.8177e+00 0.6920e-01 0.2931e+02 -0.2853e+01 -0.2612e+02 -0.2740e+00 0.6081e-01 0.7486e-01 0.2701e+02 -0.2672e+01 -0.2447e+02 -0.2855e+00 -0.4139e+00 0.8077e-01 0.2499e+02 -0.2501e+01 -0.2310e+02 -0.2968e+00 -0.9042e+00 0.8696e-01 0.2242e+02 -0.2268e+01 -0.2000e+02 -0.3070e+00 -0.1500e+00 0.9341e-01 0.1998e+02 -0.2046e+01 -0.1804e+02 -0.3161e+00 -0.4180e+00 0.1001e+00 0.1841e+02 -0.1869e+01 -0.1685e+02 -0.3239e+00 -0.6343e+00 0.1072e+00 0.1610e+02 -0.1624e+01 -0.1440e+02 -0.3301e+00 -0.2552e+00 0.1145e+00 0.1412e+02 -0.1431e+01 -0.1258e+02 -0.3357e+00 -0.2256e+00 0.1221e+00 0.1288e+02 -0.1276e+01 -0.1168e+02 -0.3406e+00 -0.4103e+00 0.1300e+00 0.1189e+02 -0.1081e+01 -0.1061e+02 -0.3441e+00 -0.1463e+00 0.1383e+00 0.1158e+02 -0.9554e+00 -0.1019e+02 -0.3465e+00 0.8074e-01 0.1469e+00 0.1104e+02 -0.8449e+00 -0.1009e+02 -0.3480e+00 -0.2419e+00 0.1558e+00 0.1029e+02 -0.7070e+00 -0.9274e+01 -0.3488e+00 -0.4018e-01 0.1651e+00 0.9346e+01 -0.6127e+00 -0.8218e+01 -0.3488e+00 0.1662e+00 0.1748e+00 0.8279e+01 -0.5248e+00 -0.7535e+01 -0.3479e+00 -0.1284e+00 0.1849e+00 0.7030e+01 -0.4184e+00 -0.6713e+01 -0.3451e+00 -0.4465e+00 0.1953e+00 0.5919e+01 -0.3717e+00 -0.5048e+01 -0.3412e+00 0.1579e+00 0.2062e+00 0.5444e+01 -0.3440e+00 -0.5050e+01 -0.3364e+00 -0.2860e+00 0.2175e+00 0.5457e+01 -0.2830e+00 -0.5201e+01 -0.3298e+00 -0.3565e+00 0.2293e+00 0.4828e+01 -0.2533e+00 -0.4449e+01 -0.3234e+00 -0.1978e+00 0.2416e+00 0.4040e+01 -0.2260e+00 -0.3860e+01 -0.3157e+00 -0.3624e+00 0.2543e+00 0.3795e+01 -0.1718e+00 -0.3725e+01 -0.3060e+00 -0.4078e+00 0.2676e+00 0.3606e+01 -0.1423e+00 -0.3513e+01 -0.2961e+00 -0.3454e+00 0.2815e+00 0.3151e+01 -0.1141e+00 -0.2988e+01 -0.2845e+00 -0.2361e+00 0.2960e+00 0.2449e+01 -0.6564e-01 -0.2320e+01 -0.2719e+00 -0.2092e+00 0.3110e+00 0.1778e+01 -0.3070e-01 -0.1676e+01 -0.2593e+00 -0.1877e+00 0.3268e+00 0.1240e+01 0.6116e-03 -0.1025e+01 -0.2462e+00 -0.3008e-01 0.3433e+00 0.7148e+00 0.4411e-01 -0.6825e+00 -0.2320e+00 -0.1556e+00 0.3605e+00 0.2644e+00 0.4429e-01 -0.1950e+00 -0.2176e+00 -0.1039e+00 0.3786e+00 -0.2432e+00 0.4902e-01 0.3351e+00 -0.2011e+00 -0.6021e-01 0.3975e+00 -0.9051e+00 0.7604e-01 0.9001e+00 -0.1840e+00 -0.1130e+00 0.4174e+00 -0.1260e+01 0.7336e-01 0.1417e+01 -0.1670e+00 0.6347e-01 0.4383e+00 -0.1086e+01 0.6970e-01 0.1189e+01 -0.1491e+00 0.2353e-01 0.4603e+00 -0.7327e+00 0.7169e-01 0.8526e+00 -0.1318e+00 0.5968e-01 0.4836e+00 -0.3718e+00 0.4856e-01 0.3197e+00 -0.1147e+00 -0.1183e+00 0.5082e+00 -0.1682e+00 0.5663e-01 0.3252e+00 -0.9689e-01 0.1167e+00 0.5343e+00 -0.1689e+00 0.5683e-01 0.3104e+00 -0.8048e-01 0.1179e+00 0.5620e+00 -0.1540e+00 0.3206e-01 0.2250e+00 -0.6699e-01 0.3614e-01 0.5916e+00 -0.6131e-01 0.3046e-01 0.1477e+00 -0.5559e-01 0.6128e-01 0.6232e+00 -0.7890e-01 0.3356e-01 0.2634e+00 -0.4615e-01 0.1719e+00 0.6572e+00 -0.2194e+00 0.2711e-01 0.2743e+00 -0.4025e-01 0.4173e-01 0.6939e+00 -0.2024e+00 0.3030e-01 0.2825e+00 -0.3197e-01 0.7844e-01 0.7337e+00 -0.2276e+00 0.2528e-01 0.2685e+00 -0.2335e-01 0.4286e-01 0.7772e+00 -0.1724e+00 0.1768e-01 0.1208e+00 -0.1589e-01 -0.4976e-01 0.8251e+00 -0.1842e+00 0.1566e-01 0.1377e+00 -0.8280e-02 -0.3911e-01 0.8783e+00 -0.1802e+00 0.8785e-02 0.1373e+00 -0.3494e-02 -0.3758e-01 0.9381e+00 -0.7623e-01 0.1390e-02 0.3931e-01 -0.7535e-03 -0.3628e-01 0.1006e+01 -0.1135e-01 -0.7783e-03 0.1949e-02 0.1335e-03 -0.1004e-01 0.1086e+01 0.9407e-03 -0.1550e-03 -0.7247e-02 0.1975e-03 -0.6264e-02 0.1180e+01 0.9029e-04 -0.1600e-05 0.4291e-03 0.4674e-04 0.5646e-03 0.1297e+01 -0.3080e-04 0.6809e-05 -0.1033e-02 0.4484e-04 -0.1013e-02 0.1451e+01 -0.2709e-04 0.3826e-04 -0.3163e-02 0.3555e-04 -0.3117e-02 0.1674e+01 -0.6961e-05 0.7675e-05 0.1471e-03 -0.1762e-04 0.1301e-03 0.2088e+01 0.8248e-06 0.6583e-05 -0.1615e-02 -0.5411e-05 -0.1613e-02