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 4 === 2K BUDGET FIG 13AR: (Budget for TWICE turb. k.e., i.e. sum of normal-stress budgets - "total" is error due to limited averaging time) y- Convection Viscous Pressure Time-dep. Total 0.0000e+00 -0.5249e-10 -0.5729e-10 -0.4635e-10 0.0000e+00 -0.1561e-09 0.1426e-03 0.9186e-02 -0.1865e+02 0.4602e+01 0.0000e+00 -0.1404e+02 0.4782e-03 0.3527e+00 -0.1759e+02 0.1257e+02 0.0000e+00 -0.4661e+01 0.1006e-02 0.3371e+01 -0.2628e+02 0.2211e+02 0.0000e+00 -0.8047e+00 0.1726e-02 0.1698e+02 -0.5126e+02 0.3005e+02 0.0000e+00 -0.4234e+01 0.2640e-02 0.5594e+02 -0.9042e+02 0.3287e+02 0.0000e+00 -0.1614e+01 0.3747e-02 0.1295e+03 -0.1629e+03 0.3281e+02 0.0000e+00 -0.5915e+00 0.5049e-02 0.2217e+03 -0.2549e+03 0.3045e+02 0.0000e+00 -0.2737e+01 0.6546e-02 0.2978e+03 -0.3241e+03 0.2536e+02 0.0000e+00 -0.9518e+00 0.8240e-02 0.3354e+03 -0.3557e+03 0.1970e+02 0.0000e+00 -0.5764e+00 0.1013e-01 0.3361e+03 -0.3515e+03 0.1329e+02 0.0000e+00 -0.2130e+01 0.1223e-01 0.3133e+03 -0.3201e+03 0.6606e+01 0.0000e+00 -0.1572e+00 0.1452e-01 0.2815e+03 -0.2833e+03 0.1353e+01 0.0000e+00 -0.4256e+00 0.1702e-01 0.2498e+03 -0.2497e+03 -0.1813e+01 0.0000e+00 -0.1714e+01 0.1972e-01 0.2210e+03 -0.2178e+03 -0.3599e+01 0.0000e+00 -0.3514e+00 0.2263e-01 0.1955e+03 -0.1926e+03 -0.3916e+01 0.0000e+00 -0.9691e+00 0.2575e-01 0.1738e+03 -0.1719e+03 -0.3416e+01 0.0000e+00 -0.1497e+01 0.2908e-01 0.1541e+03 -0.1515e+03 -0.3166e+01 0.0000e+00 -0.5685e+00 0.3263e-01 0.1365e+03 -0.1350e+03 -0.2492e+01 0.0000e+00 -0.1009e+01 0.3639e-01 0.1219e+03 -0.1206e+03 -0.2441e+01 0.0000e+00 -0.1199e+01 0.4038e-01 0.1082e+03 -0.1065e+03 -0.2352e+01 0.0000e+00 -0.6629e+00 0.4460e-01 0.9701e+02 -0.9516e+02 -0.2169e+01 0.0000e+00 -0.3180e+00 0.4904e-01 0.8817e+02 -0.8565e+02 -0.2595e+01 0.0000e+00 -0.6819e-01 0.5372e-01 0.7868e+02 -0.7661e+02 -0.1879e+01 0.0000e+00 0.1893e+00 0.5863e-01 0.7106e+02 -0.6907e+02 -0.1479e+01 0.0000e+00 0.5067e+00 0.6379e-01 0.6479e+02 -0.6250e+02 -0.1281e+01 0.0000e+00 0.1012e+01 0.6920e-01 0.5774e+02 -0.5647e+02 -0.4948e-01 0.0000e+00 0.1224e+01 0.7486e-01 0.5261e+02 -0.5120e+02 -0.3554e+00 0.0000e+00 0.1056e+01 0.8077e-01 0.4735e+02 -0.4629e+02 -0.1511e+00 0.0000e+00 0.9028e+00 0.8696e-01 0.4162e+02 -0.4187e+02 0.1052e+01 0.0000e+00 0.7962e+00 0.9341e-01 0.3742e+02 -0.3812e+02 0.9740e+00 0.0000e+00 0.2653e+00 0.1001e+00 0.3414e+02 -0.3481e+02 0.9667e+00 0.0000e+00 0.2910e+00 0.1072e+00 0.3064e+02 -0.3162e+02 0.1290e+01 0.0000e+00 0.3072e+00 0.1145e+00 0.2855e+02 -0.2870e+02 -0.5598e-01 0.0000e+00 -0.2029e+00 0.1221e+00 0.2734e+02 -0.2622e+02 -0.7424e+00 0.0000e+00 0.3813e+00 0.1300e+00 0.2613e+02 -0.2395e+02 -0.1195e+01 0.0000e+00 0.9791e+00 0.1383e+00 0.2517e+02 -0.2209e+02 -0.2782e+01 0.0000e+00 0.2975e+00 0.1469e+00 0.2334e+02 -0.2046e+02 -0.2168e+01 0.0000e+00 0.7164e+00 0.1558e+00 0.2049e+02 -0.1854e+02 -0.1361e+01 0.0000e+00 0.5866e+00 0.1651e+00 0.1808e+02 -0.1699e+02 -0.1329e+01 0.0000e+00 -0.2400e+00 0.1748e+00 0.1667e+02 -0.1600e+02 -0.2484e+00 0.0000e+00 0.4250e+00 0.1849e+00 0.1535e+02 -0.1502e+02 0.6138e+00 0.0000e+00 0.9493e+00 0.1953e+00 0.1444e+02 -0.1384e+02 -0.2720e+00 0.0000e+00 0.3238e+00 0.2062e+00 0.1365e+02 -0.1272e+02 -0.4173e-01 0.0000e+00 0.8881e+00 0.2175e+00 0.1283e+02 -0.1150e+02 -0.4329e+00 0.0000e+00 0.8950e+00 0.2293e+00 0.1201e+02 -0.1016e+02 -0.1511e+01 0.0000e+00 0.3355e+00 0.2416e+00 0.1101e+02 -0.8943e+01 -0.1004e+01 0.0000e+00 0.1058e+01 0.2543e+00 0.9565e+01 -0.7889e+01 -0.9383e+00 0.0000e+00 0.7373e+00 0.2676e+00 0.8257e+01 -0.6953e+01 -0.1051e+01 0.0000e+00 0.2535e+00 0.2815e+00 0.7253e+01 -0.6234e+01 -0.1998e+00 0.0000e+00 0.8189e+00 0.2960e+00 0.6227e+01 -0.5565e+01 -0.6253e-01 0.0000e+00 0.5994e+00 0.3110e+00 0.5227e+01 -0.4955e+01 -0.5759e+00 0.0000e+00 -0.3036e+00 0.3268e+00 0.4520e+01 -0.4359e+01 0.3750e+00 0.0000e+00 0.5355e+00 0.3433e+00 0.3990e+01 -0.3805e+01 0.1048e+00 0.0000e+00 0.2900e+00 0.3605e+00 0.3825e+01 -0.3350e+01 -0.7981e+00 0.0000e+00 -0.3239e+00 0.3786e+00 0.3707e+01 -0.2997e+01 -0.3230e+00 0.0000e+00 0.3867e+00 0.3975e+00 0.3325e+01 -0.2768e+01 -0.3593e+00 0.0000e+00 0.1975e+00 0.4174e+00 0.3013e+01 -0.2429e+01 -0.4238e+00 0.0000e+00 0.1601e+00 0.4383e+00 0.2738e+01 -0.2058e+01 -0.1029e+00 0.0000e+00 0.5772e+00 0.4603e+00 0.2287e+01 -0.1841e+01 -0.1651e+00 0.0000e+00 0.2811e+00 0.4836e+00 0.1567e+01 -0.1583e+01 0.7011e-01 0.0000e+00 0.5471e-01 0.5082e+00 0.1106e+01 -0.1263e+01 0.3744e+00 0.0000e+00 0.2174e+00 0.5343e+00 0.8520e+00 -0.1059e+01 0.2267e-01 0.0000e+00 -0.1841e+00 0.5620e+00 0.5048e+00 -0.9365e+00 -0.1063e-01 0.0000e+00 -0.4423e+00 0.5916e+00 0.4784e+00 -0.8316e+00 0.2296e+00 0.0000e+00 -0.1236e+00 0.6232e+00 0.4324e+00 -0.7082e+00 -0.9046e-01 0.0000e+00 -0.3662e+00 0.6572e+00 0.3054e+00 -0.5834e+00 -0.6952e-01 0.0000e+00 -0.3476e+00 0.6939e+00 0.3414e+00 -0.4819e+00 0.1510e+00 0.0000e+00 0.1048e-01 0.7337e+00 0.3441e+00 -0.3621e+00 -0.1061e+00 0.0000e+00 -0.1241e+00 0.7772e+00 0.2314e+00 -0.2528e+00 -0.6334e-01 0.0000e+00 -0.8470e-01 0.8251e+00 0.4104e+00 -0.1685e+00 0.1641e-01 0.0000e+00 0.2584e+00 0.8783e+00 0.3622e+00 -0.8665e-01 -0.1140e+00 0.0000e+00 0.1615e+00 0.9381e+00 0.1515e+00 -0.2663e-01 -0.2251e-01 0.0000e+00 0.1024e+00 0.1006e+01 0.3554e-01 -0.1087e-01 0.2536e-01 0.0000e+00 0.5003e-01 0.1086e+01 0.4537e-02 -0.6601e-02 -0.2645e-02 0.0000e+00 -0.4708e-02 0.1180e+01 0.5174e-03 -0.1626e-02 0.1054e-02 0.0000e+00 -0.5454e-04 0.1297e+01 0.1131e-03 -0.1967e-02 0.4742e-02 0.0000e+00 0.2889e-02 0.1451e+01 0.2568e-04 -0.2023e-02 -0.5956e-03 0.0000e+00 -0.2593e-02 0.1674e+01 -0.6816e-05 -0.3862e-03 -0.7136e-03 0.0000e+00 -0.1107e-02 0.2088e+01 0.5079e-07 -0.4494e-03 0.3189e-03 0.0000e+00 -0.1305e-03