27. Constraints on mcloud and vcloud from Θcl and mcloud

    The previous section discussed the constraints on Θcl and PE. This section begins the discussion of the other two important variables to the lurch, mcloud and vcloud. It will turn out that no significant constraints are available for either, but also that none are needed because their product, the momentum of the cloud, is constrained.
    We begin by fixing PE (via SL6A) and examining how vcloud varies in response to mcloud and Θcl . Only two cases are shown below, for mhead = 6 and 7 lb (each with dsnap = 2.2 in and PE = 300 ft-lb), because they are enough to show the general result.

For mhead = 6 lb
    The results for vlurch are shown here for general interest only. They range from -6.2 ft s-1 for heaviest and narrowest cloud to 0.9 ft s-1 for widest cloud (spherical). The effects of mcloud are much larger at small angles (range of 5 ft s-1) than at large angles (range of 0.010.05 ft s-1).

vbodyafter, ft s-1

mcl, lb Θcl   ~0 15 30 45 60 75 90 105 120 135 150 165 180
0.1 -1.34 -1.30 -1.15 -0.92 -0.64 -0.32 -0.01 0.27 0.51 0.70 0.82 0.88 0.90
0.2 -2.27 -2.20 -2.00 -1.67 -1.27 -0.83 -0.39 0.02 0.36 0.61 0.78 0.87 0.90
0.3 -2.98 -2.90 -2.64 -2.25 -1.76 -1.22 -0.68 -0.18 0.23 0.55 0.76 0.87 0.90
0.4 -3.58 -3.48 -3.19 -2.74 -2.17 -1.55 -0.92 -0.35 0.13 0.49 0.74 0.86 0.90
0.5 -4.11 -4.00 -3.67 -3.17 -2.54 -1.84 -1.14 -0.50 0.04 0.45 0.72 0.86 0.90
0.6 -4.58 -4.46 -4.11 -3.56 -2.87 -2.10 -1.34 -0.63 -0.04 0.40 0.70 0.86 0.90
0.7 -5.02 -4.89 -4.51 -3.92 -3.17 -2.35 -1.52 -0.76 -0.12 0.36 0.68 0.86 0.90
0.8 -5.42 -5.28 -4.88 -4.25 -3.45 -2.57 -1.69 -0.87 -0.19 0.33 0.67 0.85 0.90
0.9 -5.80 -5.66 -5.23 -4.56 -3.72 -2.78 -1.84 -0.98 -0.26 0.29 0.66 0.85 0.90
1.0 -6.16 -6.01 -5.56 -4.86 -3.97 -2.98 -2.00 -1.08 -0.32 0.26 0.64 0.85 0.91

    The results for vcloud are of direct interest here. They are nearly constant for a given mcloud. That means that for one PE and one mcloud, there is only one vcloud. This independence from Θcl is consistent with the previous section.

vcloud, ft s-1

mcl, lb Θcl   ~0 15 30 45 60 75 90 105 120 135 150 165 180
0.1 704 704 704 704 704 704 705 704 704 704 704 704 704
0.2 497 497 497 498 498 498 498 498 498 498 498 498 498
0.3 405 405 406 406 406 407 407 407 407 407 407 407 407
0.4 351 351 351 351 352 352 352 352 352 352 352 352 352
0.5 313 313 313 314 314 315 315 315 315 315 315 315 315
0.6 285 285 286 286 287 287 287 288 288 288 288 288 288
0.7 264 264 264 265 265 266 266 266 266 266 266 266 266
0.8 246 246 247 247 248 248 249 249 249 249 249 249 249
0.9 232 232 232 233 234 234 235 235 235 235 235 235 235
1.0 220 220 220 221 221 222 222 223 223 223 223 223 223

For mhead = 7 lb
    Here are the corresponding results for a 7-lb head. They are nearly the same as for the 6-lb head, that is, that one PE and one mcloud give only one vcloud.

vbodyafter, ft s-1

mcl, lb Θcl   ~0 15 30 45 60 75 90 105 120 135 150 165 180
0.1 -1.34 -1.29 -1.15 -0.93 -0.66 -0.36 -0.07 0.20 0.43 0.60 0.71 0.77 0.79
0.2 -2.22 -2.15 -1.95 -1.65 -1.27 -0.85 -0.43 -0.04 0.28 0.52 0.68 0.77 0.79
0.3 -2.89 -2.81 02.57 -2.20 -1.73 -1.22 -0.70 -0.23 0.16 0.46 0.66 0.76 0.79
0.4 -3.46 -3.36 -3.09 -2.66 -2.12 -1.53 -0.94 -0.39 0.06 0.41 0.64 0.76 0.79
0.5 -3.96 -3.85 -3.54 -3.07 -2.47 -1.81 -1.14 -0.53 -0.02 0.36 0.62 0.76 0.80
0.6 -4.41 -4.29 -3.96 -3.44 -2.78 -2.06 -1.33 -0.66 -0.10 0.32 0.60 0.75 0.80
0.7 -4.82 -4.70 -4.34 -3.77 -3.07 -2.28 -1.50 -0.78 -0.17 0.29 0.59 0.75 0.80
0.8 -5.20 -5.07 -4.69 -4.09 -3.34 -2.50 -1.66 -0.89 -0.24 0.25 0.58 0.75 0.80
0.9 -5.56 -5.42 -5.02 -4.38 -3.59 -2.70 -1.81 -0.99 -0.30 0.22 0.56 0.75 0.80
1.0 -5.90 -5.76 -5.33 -4.67 -3.82 -2.89 -1.95 -1.09 -0.36 0.19 0.55 0.74 0.80

vcloud, ft s-1

mcl, lb Θcl   ~0 15 30 45 60 75 90 105 120 135 150 165 180
0.1 668 668 668 668 668 668 668 668 668 668 668 668 668
0.2 472 472 472 472 472 473 473 473 473 473 473 473 473
0.3 385 385 385 385 385 386 386 386 386 386 386 386 386
0.4 333 333 333 333 334 334 334 334 334 334 334 334 334
0.5 297 297 297 298 298 299 299 299 299 299 299 299 299
0.6 271 271 271 272 272 272 273 273 273 273 273 273 273
0.7 250 250 251 251 252 252 252 253 253 253 253 253 253
0.8 234 234 234 235 235 236 236 236 236 236 236 236 236
0.9 220 220 220 221 222 222 223 223 223 223 223 223 223
1.0 208 208 209 209 210 211 211 211 211 211 211 211 211

Summary
    These results can be summarized very simply: They show that for a fixed PE, a fixed mass of cloud (mcloud) gives a fixed velocity of cloud (vcloud), independent of the width of the cloud (Θcl).

Ahead to Constraints on mcloud and vcloud from Θcl and PE
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