Bignose

Fred, welcome to our website. You should find here that we aren't disdainful of anyone's ideas. What we support more than anything else is the scientific method. That means, it isn't enough just to have a new idea. What you need to do is demonstrate, with objective, clear cut evidence, that your idea makes predictions that are at least as good as the currently accepted idea. So, what I'd like to see, then, is a brief synopsis of your idea and the predictions it makes. As well as a comparison to the current ideas and the predictions they make. And lastly, compare both of those to the current best knowledge. In effect, this is the scientific method in action. Lastly, if I may give you some advice on what not to do: Don't just tell us your idea is great. And don't just make disparaging comments about anyone who doesn't immediately embrace your idea. What wins scientific type people over is providing evidence. The more evidence you can provide that shows your idea's predictions match experimentally measured data, the better. That's what scientists want to see. Predictions from an idea that matches measurements.

Well, let's see. [math]\nabla p \ne \frac{dp}{dA}[/math]. What is true is: [math]\nabla p = \frac{\partial p}{\partial x} \boldsymbol{\delta_x} + \frac{\partial p}{\partial y} \boldsymbol{\delta_y} + \frac{\partial p}{\partial z} \boldsymbol{\delta_z}[/math] in a Cartesian coordinate system. It will have different definitions depending on the coordinate system picked. And the correct equation is:[math]\mathbf{F_L} = \int_{\partial V} \mathbf{n} \, p \, ds[/math], not what you wrote. You left out the normal vector on the RHS. both of these two equations as you wrote them are tensor rank incorrect. Both are vectors on the LHS, and you have only written scalars on the RHS. And then even if these two equations were not wrong, your third certainly wouldn't follow. This is just another example of what I've been railing on above. If this is demonstrating anything it is demonstrating a lack of knowledge. Again, what am I supposed to think from your posting these just plain incorrect equations? What I think is that you don't even know what the grad operator does (since you got its definition and tensor rank wrong) AND you can't even copy what I wrote all the way at the beginning of this thread! Please, please, please take the time to actually go and learn the mathematical tools used in fluid mechanics before you keep posting dead wrong stuff. So, yeah, I'm rejecting this, again, because it's wrong.

When the fixes don't actually fix the problem, or sometimes make it worse... then, yeah, I continue to reject them. I'll stop rejecting them when they actually, you know, work. Demonstrate something that works, and I promise I won't reject it. I think that's supremely fair, and it is how I try my best to treat everybody.

Yeah. I instantly reject equations that are dimensionally unsound or tensor rank unsound. Simply because there has never been a successful dimensionally unsound or tensor rank unsound. If you are seriously asking me to believe that you've found some of the first ones, then you need to present extraordinary evidence to support this really extraordinary claim. Evidence, by the way, that hasn't been forthcoming in either thread despite my asking for it. I also reject things when they are gross simplifications of the equations, and are at odds to what is already known. Again, if you really think that's wrong, you need to be presenting overwhelming extraordinary evidence that you're right and the body of knowledge in fluid mechanics is wrong. I reject elementary mathematical errors. Do I really need to defend this? And, lastly, I am getting really tired of writing out explanations for things -- which I guess in my opinion is not 'instantly' rejecting things because I am trying to take the time to explain where you made errors. But, I don't see you caring. So, I'm done. You demonstrate zero effort to understand what I write, so I'm not going to waste my time and write it anymore; I honestly am not sure I should have wasted my time to write this... If you really, really cared, you'd take the time to do some research and read about the great wealth of data and theories was have our there. So, I don't know what you're getting out of this. If you really cared about the subject matter, you sure as hell aren't showing it.

Yep. I'm not sure sure your indignity here is warranted since between your thread on drag and this thread, you've brushed aside plenty of my comments. If you actually want to demonstrate your fluid mechanics and mathematics knowledge, you could answer this question yourself.

This is a question that, per my above comment, I think that if you had a better grasp of fluid mechanics, you could answer it yourself. Considering what you've tried to do with the answers I've provided before -- specifically performing incorrect mathematics operations and misinterpretations of what the math actually says -- I don't think I want to provide an answer to this equation.

Rats. From the title I was hoping that this was going to be about The Who's album http://eil.com/images/main/The+Who+-+Odds+%26+Sods+-+Sealed+-+LP+RECORD-476893.jpg

Ender, I'm going to start at the bottom. As I wrote above, you have demonstrated very little to no knowledge of what the mathematics is actually representing. And, for that matter, little to no knowledge of mathematics itself. You've continuously posted equations that aren't dimensionally consistent, and tensor rank consistent. And it isn't like once or twice, but over and over. What exactly am I supposed to think? At the very, very best, you're demonstrating that you are incredibly sloppy with the mathematics. Or, as I've written above, that you are just symbol pushing. Pulling any equation from any source, taking shots in the dark hoping that the step is valid, and trying to turn something you don't understand into something you do. Sure, this assumption could be wrong. But you haven't demonstrated it at all. And, really, this latest post is just another example of this. You say you are calculating the "average velocity among [sic, I think you mean along] the top and bottom of the wing..." The problem is that it is a basic fluid mechanics assumption that the fluid velocity right at a solid boundary is equal to the velocity of that boundary. This assumption is only invalid if the fluid is exceptionally thin (the molecular density is such that the molecules can travel a very large distance before colliding with another molecule) or that the solid boundary is porous in some way. These are not valid assumptions for typical use of an airfoil. Therefore, there is no need to do any kind of complicated and frankly wrong calculation of the fluid velocity along the airfoil. Because that average fluid velocity is just the airfoil velocity. So, as I wrote above, either you are very sloppy here, or you are demonstrating lack of a rather elementary basic assumption of fluid mechanics. I just don't know what else to tell you except to repeat what I've written several times before that you need to put some time in and understand the basic mathematics and equations of fluid mechanics. It is my strongest assumption that you don't have enough background for it yet. Your elementary mathematics mistakes exhibit this. If you don't agree, then quite simply, stop posting some many mistakes. The last thing I want to say is that I hope you don't take this as a personal attack, because that is not how I intend it. We are all at different abilities and knowledge levels. The question you need to take back is: are you going to put in the time and work to gain the mathematical maturity to work through a modern fluid mechanics text and gain the knowledge and intuition of what the mathematics is describing? As before, I am willing to occasionally assist when I have time if you want to start on this path. But, I no longer want to assist in you fumbling, bumbling, and praying that you stumble on something correct. I've pointed out this lack of knowledge many times and the mistakes many times, and I see very little acknowledgement and correction of those mistakes. If you're not going to bother learning from these mistakes, I'm not going to bother pointing them out any more, because I don't think anyone is getting anything out of this. Look. This kind of curiosity and creativeness is needed in sciences, including fluid mechanics. It is craved for. There is lots of active research in fluid mechanics, lots of neat questions that need further study, including drag and lift. But, you can't just toss things out there and hope they are right. You need to understand the mathematics and what it is saying in order to contribute something meaningful. I hope you will put the time in to learn the math and the current state of fluid mechanics so that you can then apply this creativity. But, you really can't skip over that. It will take some time and effort. I really hope that you will take that time and make that effort.

Only in very special cases, so really, no. where was that? no one even used the nabla symbol until I did, so I don't see where you showed this at all. Also, this equation as it is posted isn't dimensionally sound or tensor rank sound. I know I am repeating myself here, but you really are just symbol pushing, and giving little to no thought about what the math is actually saying. You kept trying to turn an integral, the sum of a whole bunch of forces over very small areas, into a difference. Now you're asking if the change in pressure, as you travel along an airfoil, would be a constant. What in the mathematics would suggest that the change -- in every direction mind you since you just wrote [math]\nabla p[/math], not its projection in a certain direction -- would be the same. If you had given that some thought, you'd realize how unlikely that really would be. And this whole aside of using Bernoulli's equation where we are clearly discussing a viscous flow. This seems like you found an equation somewhere that said p + q = a constant. But, you obviously were missing under what conditions that was valid. Again, this smacks of symbol pushing. You're seemingly just trying to get it out of the way (and ever insistently into some kind of delta form), and not thinking about what the mathematics you are writing is actually saying. To do this work (and really, most of science) correctly, you need to take the time and work on gaining an understanding of how the math is actually describing the physics.

Why would p + q necessarily be constant? Again, that is true for an inviscid fluid, but that's not what we're talking about here. and, you do realize that [math]\nabla p[/math] is the change in pressure, right? It is how pressure changes with position. Your delta of the pressure of the top and bottom is a (exceptionally poor) approximation of this change in pressure.

Here's a critical mistake, then. Bernoulli's principle only works for inviscid fluids. Navier Stokes fluids most assuredly have viscosity, and it is the solution of the Navier Stokes equations that are needed to give correct pressure and velocity distributions to give correct lift and drag calculations. I don't see why p2 - p1 should be equal to q2 - q1. How is p2 - p1 the same as (p0 - q2) - (p0 - q1)? There is no real reason a change in pressure, [math]\nabla p[/math], should be directly proportional to a change in the 'dynamic pressure', [math]\nabla q = \nabla \rho u^2 [/math]. The pressure change and the velocity change are not so directly related. That's exactly what the Navier Stokes equations say: [math]\rho \frac{D \mathbf{u}}{D t} = \rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = \nabla p + \nabla \cdot \mathbf{T} + \mathbf{b}[/math] if you rearrange that for just [math]\nabla p[/math], you get a whole bunch of other stuff besides just [math]\nabla \rho \mathbf{u} \cdot \mathbf{u} = \nabla \rho u^2[/math].

Ender, I have two philosophical questions for you. 1) If the integral can just be replaced with a delta, why do you think I and many other authors write out the integral the first place? 2) If pressure can be so easily replaced by dynamic pressure which is merely a function of the fluid density and velocity, why is pressure explicitly included in the lift calculation and the Navier-Stokes equation? Why wouldn't it just be written in terms of density and velocity? In particular, the N-S equations would be a lot simpler because it would eliminate a whole field of unknowns. I'm not trying to be coy here, I'm trying to understand the root of your misconception, because what you are doing above is not valid, and I really don't understand why you think what you're doing is valid. I want to try to correct the error in your understanding, if you're willing.

Why do you keep insisting on evaluating the integral as just a delta? This is going to be a terrible approximation in all but the most special of cases. This is my same objection from 20+ posts ago.

No, you are missing my point. My point is that you can't bring the 'A' into the integrand like this because you are integrating over A. This invalidates all the other steps. [math]\frac{\int_A \mathbf{n} p \, dA}{qA}[/math] is what it really looks like and is not the same as what you wrote.

Yeah, I'm not sure that finding the Fibonacci sequence in the digits of pi is all that compelling. I am pretty sure you can find any sequence in pi if you want. Here's a webpage to search the first 200 million digits of pi for any number, for example: http://www.angio.net/pi/

Really? You're integrating over the area, and you just move the area into the denominator of the integrand like this? That isn't valid. You can't just willy nilly change the integrand on the variable you're actually integrating on. This needs to be corrected.

if you want, so long as you define the terms. It looks like it is just a re-write of what I put in post #2. I used [math]\partial V[/math] for the area, and ds. You are using A's. So long as they mean the same thing, it's, well, the same thing. I'd suggest you use the forum's LaTeX capabilities, rather than writing "integralA", though. The symbols help make it a lot clearer. There is a LaTeX primer thread stickied somewhere if you need help.

this isn't a perfect definition of what a vector is, but it'll be good enough for these purposes: a vector is a quantity that has a magnitude and a direction. Force does this, right? You have a magnitude -- how much force there is, and a direction -- the direction the force is applied in. But pressure doesn't have this. It has how much pressure there is... it has the magnitude. But it doesn't have a direction. Pressure applies, well, pressure equally in every direction. Hence it is not a vector. It doesn't have a direction. That's where the normal vector comes in. If you want to find out how much force the pressure applies on a spot on a surface, the pressure, over that infintesimal area, is applied normal to that area. And that's where the integral comes in. To find the total force on the object, you have to sum up all the infintesimal bits. Hence the integral I wrote in post #2. Note the form of that. The LHS is a vector, the lift force. The RHS is a vector because of the normal vector in there. The pressure in the integral on the RHS is a scalar, only n is a vector quantity on the RHS. Just like units, it is important to keep track of what is and isn't a vector quantity. The tensor rank of each side of the equation has to be equal, just like the units on each side of the equation has to be equal. Scalars have tensor rank 0, vectors have tensor rank 1, and tensors have rank 2 or greater.

It is complicated. It is often nice to break the lift into to parts: shear lift and spin lift. Shear lift occurs on an object where there is a gradient in the shear over the surface of that object. If there is angular motion of the fluid, in all likelihood there will be a shear gradient, but it isn't always true, and isn't always a very large effect. Spin lift occurs when the object itself is rotating (a really good example is the movement when a pitcher throws a breaking ball), which of course will then drive angular motion of the fluid. In short, you have to analyze every situation uniquely. You have to figure out if a rotating fluid flow is causing an object to more, a moving object causing a rotating fluid flow, or both. Therefore there is no straightforward simple answer to your question. The answer comes from analyzing the fluid flow with the conservation of mass, linear momentum, angular momentum, and energy equations.

I'm sorry, but this just still isn't right. Pressure doesn't point in a direction. Pressure actually applies equally in every direction. This is actually a big reason why pressure is a scalar. The normal vector points in the direction normal to the area. I am hesitant to actually address anything more, because, again this is fluid mechanics 101 basic stuff. I really feel like you are just symbol pushing, and not understanding at all what the equations you are writing actually mean. And, again, I'm sorry, but this isn't any way to actually form a meaningful idea. You need to actually understand what you're doing, and writing things like "the pressure is pointing in" shows you have very fundamental misunderstandings. Have you gotten yourself a fluid mechanics text? Like the one I suggested in the other thread? You really, really need to be reading through that to correct these wrong ideas you have. Before you try to tackle something as complicated as lift, I'd like to see that you have some of the basics down first.

Pressure is NOT a vector. Pressure is a scalar. The pressure force is F/A. This is fluid mechanics 101, again. And a serious lack of clear communications. You can't just toss these terms around however you like. That's fine. You explicitly asked what we thought about what you had so far. I gave you an answer in that so far you haven't done anything, but grossly simplifed the definition of how lift is calculated, and expressed my concern that this gross simplification would in all likelihood lead to erroneous predictions. If you didn't want this feedback, maybe you shouldn't have solicited it. Just like in your last thread, I really await the posting of the predictions of this forthcoming idea and comparing them to the predictions made by the best theories published today and measured experimental results. I do really hope you take some lessons learned from that previous thread and apply them here. Because, it is really important that you not only have a new way of calculating the lift coefficient, but that you have an accurate way of calculation the lift coefficient. New alone doesn't mean squat if it isn't accurate. And, in my opinion, during that pressure integral into just a difference, won't be very accurate. But, prove me wrong. Show your calculations and how they compare to measured values.

No. 1) F is a vector quantity, and neither A nor p are vectors, so this isn't right. 2) You have no account for the variation of pressure over the body. That's what the integral does, it adds up the amount of pressure over the entire body. your equation would only be valid for something like a flat plate. It is a gross simplification of the true problem. That is, there are times when the integral will evaluate to A*p, but that is going to be a fairly rare exception, not the rule. So, you're left with, again, the tricky question of how you are going to evaluate the integral of the normal pressure over the entire body. Also, I think it should be said that so far, you don't have a 'theory for the coefficient of lift'. So far, we're just using the terms as they are defined. i.e. you haven't brought anything new. If you have a new way of calculating it, then we might have something, but so far all you have is the definitions of terms...

I think it is a gross simplification of lift. Lift [math]=\int_{\partial V} \mathbf{n} \, p \, ds[/math] which represents the integral of the pressure force normal over the entire surface. Replacing this integral by just a simple difference won't work in general. And you have the exact same problem as I pointed out with drag, knowing the distribution of the pressure over the entire surface requires solution of the Navier-Stokes equations, which will only be analytic in some very, very special cases. You will either have to use CFD, a correlation for [math]C_L[/math], or some of the conformal mapping tricks aeronautical engineers were very, very good at (though largely supplanted by CFD today).

If it actually works, I don't know what advice you need from an Internet forum. If it really works, should be a piece of cake to publish the results, win multiple Nobel prizes with, and sell them for great sums of money.

So, I'm curious why you didn't try this with a few numbers: a=6, b = 9.6 --> sqrt(a+b) = 3.937003937; sqrt(a) + sqrt(b) - sqrt(2ab) = -5.145381508 a=13.3, b = 0.3 --> sqrt(a+b) = 3.687817783; sqrt(a) + sqrt(b) - sqrt(2ab) = 1.369749685 and so on... Clearly it doesn't hold except maybe in a few special cases.