Hi! I've been solving problems that apply Bernoulli's equations in *reasonable* situations. One comes from Serway's book. It applies the equation for the force exerted on a window pane in a stormy day. They give us the speed of the wind blowing outside, the area of the window pane, the hint that it's going horizontally and that the inside pressure is about the atmospheric. Solving this problem as if the air went along a stream line *through* the windowpane is what really troubles me. I can't imagine how the equation should be a reasonable approximation to create this textbook problem. The book even goes on to mention a real life incident about a skyscraper that had its windowpanes falling out because of bad design as an example of Bernoulli's equation in action. We all know that Bernoulli's equation is valid only for a stream line and the windowpane is blocking it, it can't be right. Is there any way to justify why even the solution manual for instructors explicitly states to take points 1 and 2 in the air just inside and outside the window pane??? Any thoughts? thanks
A building diverts and slows down the wind, resulting in slightly increased pressure on the upwind side, and slightly decreased pressure on the downwind side, and vortices at the edges. The slowing down of wind represents negative work done on the wind, which violates Bernoulli.
While technically true, you can use that violation of COE to your advantage. "Stagnation pressure" is what you get when you assume all of the air's kinetic energy is lost by smacking into something. Bernoulli's equation provides it to you if you re-arrange it.
Bernoulli's equation is similar to the first term of the series formula for impact pressure (the first term is the dynamic pressure), an approximation, but good enough in most cases since the next term is dynamic pressure x ((mach speed)^2) / 4. Wiki article about impact pressure: http://en.wikipedia.org/wiki/Impact_pressure The low pressure on the downwind side of the building violates Bernoulli, the stagnant zones on the upwind and downwind side of a building have the same air speed, zero, but the downwind side has lower pressure than the upwind side. Vortices that form around the edges of the building also create low pressure zones. Some buildings use a positive pressure ventilation system, so the pressure inside a building is slightly higher than ambient.
Ok thanks for the responses, I checked that wiki article... about impact pressure and stagnation pressure. what would be a correct explanation in the context of Bernouilli's equation? Also, the force is supposed to come from the pressure difference obtained just by rearranging Bernouilli's equation but as it is, it just seems like an incomplete explanation. I have to teach this to young students and I was assigned that problem to work out with them... the first thing they will find odd is that... So, I'm still not sure, Stagnation pressure seems to be also called total pressure and it's given by P_1+ 0.5D V_1^2 ... If I just take the pressure difference of this with the inside air pressure (which is set to atmospheric pressure) Iwould get exactly the same as applying blindly Bernoulli's equation... but in reality that can't be. Impact pressure, according to the wikipedia article, is just the difference of total minus static pressure, but this seems to be defined only for the same point. Again, this would mix up different points that are not on the same stream line... In the end... am I right in saying that Bernoulli's equation can't be used per se? (since it applies to the same stream line) but it turns out that the different terms on one side of Bernoulli's equation appear as measurable independent pressures that can be used to compute the net force on the windowpane... Would this be the right approach? (even when Serway's solution manual just explicitly applies it?) thanks
Stagnation pressure follows Bernoulli's equation for incompressable flows. That wiki article on stagnation pressure also contains the same equation as impact pressure for compressable flows, the equation with specific heat factor [itex]\gamma[/itex]. At low speeds, the effect of compressibility is small, which is what I was getting at when I mentioned that the first term of the stagnation or impact pressure series formula is the same as dynamic pressure (Bernoulli based), and the second term is ((mach speed)^2)/4. For example even at mach .4, the second term is ((.4)^2)/4 = .04, only a 4% difference, mostly an issue for higher speed aircraft, not buildings. The upwind side of the building isn't the issue, stagnation pressure is essentially the same as dynamic pressure at wind speeds, so Bernoulli can be used. It's the downwind side that can't be explained by Bernoulli. Stagnation or impact pressure is the pressure of the air "stopped" by the building (the air willl flow sideways and eventually around the edges, but there is a significant "stagnation" zone of air on the upwind side of the building). This "total" pressure is what a pressure sensing device would measure if placed inside the stagnation zone. The issue on the downwind side is there's also a stagnation zone, zero air speed, same speed as the upwind side, but lower pressure than the upwind side. I'm not sure how you adjust Bernoulli to account for this. The issue is that streamlines end when they hit the upwind side of the building, and on the downwind side, total energy of the wind has been reduced, which violates Bernoulli, which assumes that total energy (within a streamline) remains constant.