1 00:00:10,500 --> 00:00:14,010 PROFESSOR: So let's begin to examine our assumptions having 2 00:00:14,010 --> 00:00:17,860 to do with the transport of respiratory droplets 3 00:00:17,860 --> 00:00:21,700 in a well-mixed room by thinking about effects of fluid flow 4 00:00:21,700 --> 00:00:25,310 and transport going beyond a well-mixed room. 5 00:00:25,310 --> 00:00:26,980 So to begin thinking about this problem, 6 00:00:26,980 --> 00:00:30,550 it's instructive to think of examples of simpler flows, 7 00:00:30,550 --> 00:00:32,920 in particular the canonical problem of flow 8 00:00:32,920 --> 00:00:33,940 past an object-- 9 00:00:33,940 --> 00:00:35,860 for example, a cylinder. 10 00:00:35,860 --> 00:00:40,270 Here I've shown a cylinder placed in a uniform flow field 11 00:00:40,270 --> 00:00:45,830 with increasing speed and fixed object size. 12 00:00:45,830 --> 00:00:48,850 So as you can see, at low speeds, 13 00:00:48,850 --> 00:00:52,930 the streamlines are fairly reversible and simple looking. 14 00:00:52,930 --> 00:00:55,570 And at high speeds, you end up with very complicated 15 00:00:55,570 --> 00:00:57,710 turbulent flows. 16 00:00:57,710 --> 00:01:00,220 A very simple parameter controls the transition 17 00:01:00,220 --> 00:01:03,190 between these flow regimes, which is the Reynolds number. 18 00:01:09,810 --> 00:01:12,789 This is a dimensionless quantity, 19 00:01:12,789 --> 00:01:15,630 which is a property of the fluid, which 20 00:01:15,630 --> 00:01:20,860 includes the kinetic viscosity, for example, of air. 21 00:01:20,860 --> 00:01:25,840 And I'll just write this, kinematic viscosity. 22 00:01:29,020 --> 00:01:31,420 And it has a measure of the flow speed. 23 00:01:31,420 --> 00:01:34,600 So we'll call that U. So the magnitude of the background 24 00:01:34,600 --> 00:01:41,110 flow here is U. And it has some information about the geometry, 25 00:01:41,110 --> 00:01:42,890 in particular the size of the object. 26 00:01:42,890 --> 00:01:46,400 So for example, we could define this based on, 27 00:01:46,400 --> 00:01:50,720 well, let's just say, some length scale L, which could be, 28 00:01:50,720 --> 00:01:53,810 for example, the radius of the cylinder. 29 00:01:53,810 --> 00:01:59,060 And in the case of this cylinder that I've shown here, 30 00:01:59,060 --> 00:02:04,090 if the Reynolds number is less than around 10, 31 00:02:04,090 --> 00:02:08,050 you're in this regime here of so-called creeping 32 00:02:08,050 --> 00:02:12,960 flow where you can see the streamlines are 33 00:02:12,960 --> 00:02:14,070 very simple looking. 34 00:02:14,070 --> 00:02:15,480 And they are reversible. 35 00:02:15,480 --> 00:02:17,520 So they kind of smoothly conform to the object. 36 00:02:17,520 --> 00:02:20,000 And you can run the flow in the backwards direction. 37 00:02:20,000 --> 00:02:23,060 You'll get exactly the same streamlines. 38 00:02:23,060 --> 00:02:25,350 But once you get to a Reynolds number of around 10, 39 00:02:25,350 --> 00:02:27,020 then you start to see different effects. 40 00:02:27,020 --> 00:02:30,730 And that's because the physical interpretation of this quantity 41 00:02:30,730 --> 00:02:36,430 is the ratio of inertia, which is the tendency for fluid 42 00:02:36,430 --> 00:02:38,620 to want to keep moving in the same direction 43 00:02:38,620 --> 00:02:40,300 due to its mass that has been set 44 00:02:40,300 --> 00:02:43,940 into motion, relative to viscosity 45 00:02:43,940 --> 00:02:44,900 or to viscous stresses. 46 00:02:48,079 --> 00:02:49,760 And that's the tendency for the fluid 47 00:02:49,760 --> 00:02:51,320 to have some friction with itself. 48 00:02:51,320 --> 00:02:53,660 If you start to move a piece of the fluid, 49 00:02:53,660 --> 00:02:55,770 other elements of fluid nearby are pulling back. 50 00:02:55,770 --> 00:02:56,690 There's some friction. 51 00:02:56,690 --> 00:02:58,640 And it impedes the motion. 52 00:02:58,640 --> 00:03:02,690 So whenever inertia is getting bigger than viscous stress, 53 00:03:02,690 --> 00:03:04,430 we start to have a tendency for the fluid 54 00:03:04,430 --> 00:03:06,260 to want to keep going in the same direction. 55 00:03:06,260 --> 00:03:08,260 And it can lead to very complicated flows. 56 00:03:08,260 --> 00:03:10,640 In particular here, the first thing that starts to happen 57 00:03:10,640 --> 00:03:15,200 is the fluid kind of whizzes past the cylinder 58 00:03:15,200 --> 00:03:18,710 and then starts to have a recirculation on the back side. 59 00:03:18,710 --> 00:03:22,140 So this happens around Reynolds number equals 10. 60 00:03:22,140 --> 00:03:26,880 So if you have Reynolds number greater than around 10, 61 00:03:26,880 --> 00:03:30,370 you have some vortices are created. 62 00:03:30,370 --> 00:03:34,240 So it's no longer an irrotational flow. 63 00:03:34,240 --> 00:03:37,540 It has some obvious closed streamlines. 64 00:03:37,540 --> 00:03:39,760 If we increase the Reynolds number further, 65 00:03:39,760 --> 00:03:42,910 like in this situation here, we get to Reynolds number bigger 66 00:03:42,910 --> 00:03:45,910 than around 90. 67 00:03:45,910 --> 00:03:48,880 Then those vortices themselves are starting to spin fast. 68 00:03:48,880 --> 00:03:50,350 And they also have some inertia. 69 00:03:50,350 --> 00:03:52,660 And they start to separate. 70 00:03:52,660 --> 00:03:55,760 Also the fluid is pulling those vortices away. 71 00:03:55,760 --> 00:03:58,980 So we have vortex shedding. 72 00:04:02,610 --> 00:04:05,630 And that happens in an unsteady fashion. 73 00:04:05,630 --> 00:04:07,090 So what happens is there's-- 74 00:04:07,090 --> 00:04:10,290 one of these two vortices goes unstable first and peels away 75 00:04:10,290 --> 00:04:11,540 and starts to move downstream. 76 00:04:11,540 --> 00:04:13,120 The other one kind of takes its place. 77 00:04:13,120 --> 00:04:15,580 And it's almost like a flapping flag kind of motion, 78 00:04:15,580 --> 00:04:17,990 where a train of vortices is released. 79 00:04:17,990 --> 00:04:23,430 So this is an unsteady situation. 80 00:04:23,430 --> 00:04:27,290 And this is called a vortex sheet. 81 00:04:27,290 --> 00:04:29,140 But it's basically an array of vortices 82 00:04:29,140 --> 00:04:31,890 that are being released in a time-dependent fashion. 83 00:04:31,890 --> 00:04:33,890 And at first, that's a fairly regular process 84 00:04:33,890 --> 00:04:36,260 when the Reynolds number is on the order of 100. 85 00:04:36,260 --> 00:04:38,300 But if you keep increasing the Reynolds number, 86 00:04:38,300 --> 00:04:40,640 that process becomes more and more chaotic 87 00:04:40,640 --> 00:04:44,720 until there's a transition to turbulence, which is a fully 88 00:04:44,720 --> 00:04:50,890 chaotic, heavily-mixed flow. 89 00:04:50,890 --> 00:04:53,460 And that happens through an instability 90 00:04:53,460 --> 00:04:55,980 around Reynolds number of a 2,000, 91 00:04:55,980 --> 00:04:58,909 where you have a turbulent wake. 92 00:04:58,909 --> 00:05:01,520 So behind the object or the obstacle, 93 00:05:01,520 --> 00:05:04,250 there is a steady stream of turbulence 94 00:05:04,250 --> 00:05:05,750 where, as I've tried to sketch here, 95 00:05:05,750 --> 00:05:08,670 you have vortices and eddies of all sizes. 96 00:05:08,670 --> 00:05:10,340 So there are eddies like I've shown here 97 00:05:10,340 --> 00:05:13,470 that are at the scale of the cylinder, but then much, 98 00:05:13,470 --> 00:05:14,780 much smaller ones, too. 99 00:05:14,780 --> 00:05:19,330 So it's a very complicated, time-dependent flow field. 100 00:05:19,330 --> 00:05:20,960 So we can see here, the Reynolds number 101 00:05:20,960 --> 00:05:22,940 has a big effect on the types of flows 102 00:05:22,940 --> 00:05:24,980 that are generated and obviously also 103 00:05:24,980 --> 00:05:29,510 on mixing as you go from low to high Reynolds number. 104 00:05:29,510 --> 00:05:32,840 So let's think about how that would change 105 00:05:32,840 --> 00:05:35,750 in the setting of indoor air. 106 00:05:35,750 --> 00:05:40,360 So let's look at flow. 107 00:05:40,360 --> 00:05:43,590 Let's look at airflow in a room. 108 00:05:47,080 --> 00:05:50,770 So let's think about the different scenarios 109 00:05:50,770 --> 00:05:51,360 we could have. 110 00:05:51,360 --> 00:05:55,440 So maybe our first scenario would be 111 00:05:55,440 --> 00:05:57,050 we have all the windows closed. 112 00:05:57,050 --> 00:05:58,390 There's no movement in the room. 113 00:05:58,390 --> 00:05:59,880 It's essentially a still room. 114 00:05:59,880 --> 00:06:01,510 But as we've discussed earlier, there's 115 00:06:01,510 --> 00:06:03,800 still some air change with the outside. 116 00:06:03,800 --> 00:06:06,200 So air is still leaking to the outside. 117 00:06:06,200 --> 00:06:10,330 There's still a little bit of instability and movement 118 00:06:10,330 --> 00:06:12,760 also from thermal effects, which we'll talk about shortly. 119 00:06:12,760 --> 00:06:15,120 So there is a little bit of flow in the room. 120 00:06:15,120 --> 00:06:17,800 And in particular, let's just ask ourselves 121 00:06:17,800 --> 00:06:21,130 what happens if we have a little bit of air exchange 122 00:06:21,130 --> 00:06:24,940 with the outside, which is this flow rate Q that we've 123 00:06:24,940 --> 00:06:32,690 discussed, but in the case of natural ventilation 124 00:06:32,690 --> 00:06:33,590 with closed windows. 125 00:06:37,300 --> 00:06:41,240 And in that situation, we've used as an estimate 126 00:06:41,240 --> 00:06:45,050 that the air change time, or air change rate, 127 00:06:45,050 --> 00:06:50,040 might be on the order of 0.3 air changes per hour, 128 00:06:50,040 --> 00:06:52,740 which corresponds to 3 hours' time 129 00:06:52,740 --> 00:06:55,190 to have the room air changed. 130 00:06:55,190 --> 00:06:57,050 So that's a pretty slow pace. 131 00:06:57,050 --> 00:07:02,030 If you consider a room whose height is 2.7 meters, 132 00:07:02,030 --> 00:07:06,140 just to put a number on it, and you have that air change rate, 133 00:07:06,140 --> 00:07:09,590 then the average velocity in the room 134 00:07:09,590 --> 00:07:16,550 is on the order of 1 meter per hour. 135 00:07:16,550 --> 00:07:18,150 So that's a pretty slow pace. 136 00:07:18,150 --> 00:07:21,120 So to go 1 meter, it's going take a whole hour, so very 137 00:07:21,120 --> 00:07:21,620 slow. 138 00:07:21,620 --> 00:07:24,620 So you think, not much interesting is happening. 139 00:07:24,620 --> 00:07:26,290 But if you calculate the Reynolds number 140 00:07:26,290 --> 00:07:29,380 for this situation, the Reynolds number 141 00:07:29,380 --> 00:07:34,409 is actually 110 with those numbers. 142 00:07:34,409 --> 00:07:36,450 And that's already putting us in the regime 143 00:07:36,450 --> 00:07:39,510 of not only forming vortices, but also having some vortex 144 00:07:39,510 --> 00:07:40,440 shedding. 145 00:07:40,440 --> 00:07:43,050 So even when a room is rather still, and there's 146 00:07:43,050 --> 00:07:47,520 just some very gentle movement of air out the windows, 147 00:07:47,520 --> 00:07:50,230 or cracks around the windows and other places 148 00:07:50,230 --> 00:07:51,690 where the room may not be tight, or 149 00:07:51,690 --> 00:07:54,480 from some other minor movement going on in the room, 150 00:07:54,480 --> 00:07:58,860 we already expect to see some unsteady vortices and movement 151 00:07:58,860 --> 00:08:00,920 of air in that room. 152 00:08:00,920 --> 00:08:05,840 But the situation gets more-- 153 00:08:05,840 --> 00:08:11,990 gets much stronger if we now move to having ventilation. 154 00:08:11,990 --> 00:08:16,460 So let's imagine that we have an HVAC unit on top-- maybe, 155 00:08:16,460 --> 00:08:18,470 let's say, an air conditioner, which 156 00:08:18,470 --> 00:08:22,820 is blowing air into the room perhaps from somewhere above. 157 00:08:22,820 --> 00:08:25,820 So now we have a flow which is going into the room. 158 00:08:25,820 --> 00:08:28,400 It still has to leave somewhere, let's say, 159 00:08:28,400 --> 00:08:30,110 through an outlet vent. 160 00:08:33,659 --> 00:08:35,909 And in this case, the flow rate might be a lot higher. 161 00:08:35,909 --> 00:08:39,360 So let's imagine, now, we have lambda a would be, 162 00:08:39,360 --> 00:08:41,840 let's say, 8 air changes per hour. 163 00:08:44,980 --> 00:08:47,260 And this would be looking at a typical case 164 00:08:47,260 --> 00:08:48,700 of mechanical ventilation. 165 00:08:51,480 --> 00:08:53,440 And what we really care about here, by the way, 166 00:08:53,440 --> 00:08:54,910 is the total flow rate. 167 00:08:54,910 --> 00:08:57,030 So we're not interested in just the outdoor air. 168 00:08:57,030 --> 00:08:59,340 So what I really want, I should mention, 169 00:08:59,340 --> 00:09:03,520 is lambda bar a, where that is the total air change. 170 00:09:03,520 --> 00:09:08,250 So lambda a bar, we'll define as the air change due to just 171 00:09:08,250 --> 00:09:11,250 the fresh air-- so that was the Q over V-- 172 00:09:11,250 --> 00:09:19,390 plus the air change due to the filtration flows. 173 00:09:19,390 --> 00:09:22,350 So there is sort of this recirculating flow-- 174 00:09:22,350 --> 00:09:23,820 and we've written that as Q-- 175 00:09:23,820 --> 00:09:27,510 the outdoor airflow plus the filtration airflow 176 00:09:27,510 --> 00:09:28,510 divided by volume. 177 00:09:28,510 --> 00:09:30,240 So I just want to make sure we throw that in there, 178 00:09:30,240 --> 00:09:32,610 because you can get some flow going also by having a HEPA 179 00:09:32,610 --> 00:09:34,260 filtration unit in your room. 180 00:09:34,260 --> 00:09:36,060 And there's circulation going on from that. 181 00:09:36,060 --> 00:09:38,320 And that does contribute to mixing. 182 00:09:38,320 --> 00:09:41,910 And so if we now calculate what is the Reynolds number, 183 00:09:41,910 --> 00:09:46,520 we're now getting up to around 2,000. 184 00:09:46,520 --> 00:09:49,550 And that's interesting, because that is already 185 00:09:49,550 --> 00:09:51,480 getting to the regime of turbulent flow. 186 00:09:51,480 --> 00:09:55,430 So just the velocities and flows that are generated in the air 187 00:09:55,430 --> 00:10:01,130 from typical air conditioning or even gentle fans 188 00:10:01,130 --> 00:10:03,290 will lead to turbulent flows in the room. 189 00:10:03,290 --> 00:10:05,640 And turbulent flows are very effective at mixing. 190 00:10:05,640 --> 00:10:08,370 So that's one reason we might expect to see strong mixing. 191 00:10:08,370 --> 00:10:11,230 So what this actually looks like is-- 192 00:10:11,230 --> 00:10:16,340 imagine there's a person in the room. 193 00:10:16,340 --> 00:10:19,650 There could be a table and a chair, 194 00:10:19,650 --> 00:10:21,770 some kind of obstacles in the room. 195 00:10:21,770 --> 00:10:25,100 And even if nobody's moving, just the flow through the room 196 00:10:25,100 --> 00:10:29,780 is causing some turbulent wakes and vortex shedding. 197 00:10:29,780 --> 00:10:33,530 And there can be some circulations, such 198 00:10:33,530 --> 00:10:37,430 that you have some fairly significant inertial effects 199 00:10:37,430 --> 00:10:40,310 leading to mixing in that system. 200 00:10:40,310 --> 00:10:45,710 Now, we can also ask ourselves, what about other ways 201 00:10:45,710 --> 00:10:48,500 that momentum is imparted to the fluid in a room 202 00:10:48,500 --> 00:10:49,730 with people in it? 203 00:10:49,730 --> 00:10:52,250 Well, it could be, for example, human movement. 204 00:10:52,250 --> 00:10:55,450 So what if we have-- 205 00:10:55,450 --> 00:10:56,910 let's think about what the Reynolds 206 00:10:56,910 --> 00:10:57,990 numbers might be for that. 207 00:10:57,990 --> 00:11:00,650 So if we have human movement-- 208 00:11:00,650 --> 00:11:02,550 let's say, for example, I'm moving my arm. 209 00:11:02,550 --> 00:11:03,470 Or I'm moving my head. 210 00:11:03,470 --> 00:11:05,840 I'm not even talking about running or really moving fast. 211 00:11:05,840 --> 00:11:08,970 But let's just think about this kind of a motion. 212 00:11:08,970 --> 00:11:11,690 I might be moving with a velocity 213 00:11:11,690 --> 00:11:14,990 anywhere from 10 centimeters, or 0.1 meters, 214 00:11:14,990 --> 00:11:16,940 to 1 meters per second, right? 215 00:11:16,940 --> 00:11:19,740 So I could easily go 10 centimeters in 1 second. 216 00:11:19,740 --> 00:11:21,740 But I could maybe go a little faster. 217 00:11:21,740 --> 00:11:23,960 And I'm moving a part of my body, which 218 00:11:23,960 --> 00:11:29,000 might have a length that scales from, let's say, 10 centimeters 219 00:11:29,000 --> 00:11:29,730 to 1 meter. 220 00:11:29,730 --> 00:11:33,620 So I'm moving maybe my arm or my hand or my head. 221 00:11:33,620 --> 00:11:35,960 So if we take this range of values here and ask, 222 00:11:35,960 --> 00:11:38,240 now what's the Reynolds number, then 223 00:11:38,240 --> 00:11:44,230 the Reynolds number is around 10 to the 3 to 10 to the 5. 224 00:11:44,230 --> 00:11:45,910 So and you know this. 225 00:11:45,910 --> 00:11:48,280 If you have a room where there's some smoke or some dust 226 00:11:48,280 --> 00:11:50,080 particles that you can see in the sunlight, 227 00:11:50,080 --> 00:11:52,720 just take your hand and move it like this, even fairly gently. 228 00:11:52,720 --> 00:11:55,540 You're going to see very complicated turbulent flows 229 00:11:55,540 --> 00:11:57,680 in the wake of that movement. 230 00:11:57,680 --> 00:11:59,540 So basically, anything going on in the room 231 00:11:59,540 --> 00:12:02,930 in terms of movement is leading to substantial complexity 232 00:12:02,930 --> 00:12:05,750 in the flow fields, which contributes to mixing. 233 00:12:05,750 --> 00:12:09,050 Another one to think about, which we will come back to, 234 00:12:09,050 --> 00:12:14,210 is human respiration, so the fact that you're breathing. 235 00:12:14,210 --> 00:12:17,340 So we will come back to this term more carefully. 236 00:12:17,340 --> 00:12:20,390 But we can still think about it in terms of the Reynolds number 237 00:12:20,390 --> 00:12:21,480 right now. 238 00:12:21,480 --> 00:12:25,340 So if you imagine just the flow that is leaving your mouth-- 239 00:12:25,340 --> 00:12:27,320 so in this case your length scale 240 00:12:27,320 --> 00:12:29,480 is on the order of 1 centimeter. 241 00:12:29,480 --> 00:12:31,370 So maybe the area of your mouth that's open 242 00:12:31,370 --> 00:12:33,650 is maybe several centimeters. 243 00:12:33,650 --> 00:12:36,080 And the velocity of your breathing 244 00:12:36,080 --> 00:12:38,180 depends on how heavily you're breathing 245 00:12:38,180 --> 00:12:39,920 and what your activities are. 246 00:12:39,920 --> 00:12:43,340 But it's on the order-- so if I write this as the velocity U-- 247 00:12:43,340 --> 00:12:44,720 I should have written this as a U 248 00:12:44,720 --> 00:12:46,470 as well, from actually over there. 249 00:12:46,470 --> 00:12:51,380 So if my velocity scale is around 0.5 to 2 meters 250 00:12:51,380 --> 00:12:52,170 per second-- 251 00:12:52,170 --> 00:12:56,310 that's a typical respiratory velocity-- 252 00:12:56,310 --> 00:13:00,560 then now we find the Reynolds number is on the order of 10 253 00:13:00,560 --> 00:13:04,800 to the 3 to 10 to the 4. 254 00:13:04,800 --> 00:13:10,500 So again, all these activities of breathing, motion, anything 255 00:13:10,500 --> 00:13:12,660 humans are doing in the room, even just 256 00:13:12,660 --> 00:13:15,780 sitting there and breathing is leading to Reynolds numbers 257 00:13:15,780 --> 00:13:19,540 locally that are on the order of thousands or tens of thousands, 258 00:13:19,540 --> 00:13:23,400 which means that we are seeing turbulent flows in the vicinity 259 00:13:23,400 --> 00:13:24,210 of those motions. 260 00:13:24,210 --> 00:13:25,920 So that doesn't mean that those flows are 261 00:13:25,920 --> 00:13:27,410 enough to mix the entire room. 262 00:13:27,410 --> 00:13:30,600 But certainly in the vicinity of a person, 263 00:13:30,600 --> 00:13:33,840 there's a lot of mixing just from the natural movement 264 00:13:33,840 --> 00:13:36,580 and respiration of that person. 265 00:13:36,580 --> 00:13:39,730 And when you add into that the mechanical ventilation, 266 00:13:39,730 --> 00:13:42,300 there can be very significant inertial mixing 267 00:13:42,300 --> 00:13:45,120 going on in the air of a room. 268 00:13:45,120 --> 00:13:48,300 So in this simulation, courtesy of Saint-Gobain Ceramics 269 00:13:48,300 --> 00:13:51,810 and Plastics, we see an office space containing a number 270 00:13:51,810 --> 00:13:53,220 of workers sitting in cubicles. 271 00:13:53,220 --> 00:13:55,230 And one of them is an infected person 272 00:13:55,230 --> 00:13:56,970 emitting infectious aerosols. 273 00:13:56,970 --> 00:14:00,540 And the simulation shows you how those aerosol particles 274 00:14:00,540 --> 00:14:04,080 are transmitted through the room by forced convection, where 275 00:14:04,080 --> 00:14:07,290 the orange and white squares represent 276 00:14:07,290 --> 00:14:11,190 ducts of inflow and outflow in the mechanical ventilation 277 00:14:11,190 --> 00:14:12,640 system. 278 00:14:12,640 --> 00:14:17,160 This video shows the effect of motion 279 00:14:17,160 --> 00:14:20,610 and also respiration leading to turbulent flows 280 00:14:20,610 --> 00:14:23,940 by forcing the air to move at a high Reynolds number. 281 00:14:23,940 --> 00:14:26,760 So in the case of motion, we see a turbulent wake 282 00:14:26,760 --> 00:14:28,410 behind the moving person. 283 00:14:28,410 --> 00:14:30,870 And for each breath, we see a turbulent plume 284 00:14:30,870 --> 00:14:35,370 emitted by the momentum transferred from the breath. 285 00:14:35,370 --> 00:14:38,670 These images are taken by a schlieren imaging method that 286 00:14:38,670 --> 00:14:40,950 looks at differences in the density of the air 287 00:14:40,950 --> 00:14:44,760 and visualizes the texture of the patterns that 288 00:14:44,760 --> 00:14:46,880 are formed in the density.