I recently got involved in a discussion of biplanes and induced drag. It reinforced the impression that I had that many (to include formally trained engineers) were perpetuating some misconceptions about induced drag, about the price you have to pay to get lift.
The biggest misconception is the idea that to minimize induced drag, you must maximize aspect ratio, you must favor long, skinny wings. This misconception is reinforced by the commonly cited formula:
\[C_{d_i} = \frac{C_l^2}{\pi e AR}\]
Yes, the larger the aspect ratio, the smaller the induced drag coefficient, but when you're dealing with non-dimensional coefficients, you're distancing yourself from the real world parameters that truly make or break a design.
\[\frac{D_i}{\frac{1}{2}\rho V^2 S} = \frac{(\frac{L}{\frac{1}{2}\rho V^2 S})^2}{\pi e \frac{b}{c}}\]
\[D_i=\frac{L^2}{\frac{1}{2}\rho V^2 bc \pi e \frac{b}{c}}\]
\[D_i=\frac{L^2}{\frac{1}{2}\pi e\rho V^2 b^2}\]
Induced drag is proportional to the square of lift (and therefore the square of weight), and inversely proportional to the square of the span.
\[D_i \varpropto \frac{W^2}{b^2}\]
The chord doesn't enter into it, aspect ratio doesn't matter! Of course, area matters to profile drag, which drives designers to narrow chords, which results in high aspect ratios. But the aspect ratio, per se, isn't the important bit!
So how does this relate to biplanes? Suppose we divide the load between two wings:
\[D_{i_{biplane}} \varpropto 2 \frac{(W/2)^2}{b^2} = \frac{2}{4}\frac{W^2}{b^2} \varpropto \frac{1}{2} D_i\]
That's right, twice the wings, half the induced drag (assuming you don't shrink the span). Some of those flying Venetian blinds from the early days of aviation were not the work of cranks.
A carefully designed biplane can offer an overall drag reduction, but it does tend to have a narrow performance envelope because of the need to balance profile and induced drag, and to carefully manage interference effects.
Examples:
A few weeks ago, I saw a page on recent drone work along these lines, but of course as I write this, I can't find it.
Symbols:
The biggest misconception is the idea that to minimize induced drag, you must maximize aspect ratio, you must favor long, skinny wings. This misconception is reinforced by the commonly cited formula:
\[C_{d_i} = \frac{C_l^2}{\pi e AR}\]
Yes, the larger the aspect ratio, the smaller the induced drag coefficient, but when you're dealing with non-dimensional coefficients, you're distancing yourself from the real world parameters that truly make or break a design.
\[\frac{D_i}{\frac{1}{2}\rho V^2 S} = \frac{(\frac{L}{\frac{1}{2}\rho V^2 S})^2}{\pi e \frac{b}{c}}\]
\[D_i=\frac{L^2}{\frac{1}{2}\rho V^2 bc \pi e \frac{b}{c}}\]
\[D_i=\frac{L^2}{\frac{1}{2}\pi e\rho V^2 b^2}\]
Induced drag is proportional to the square of lift (and therefore the square of weight), and inversely proportional to the square of the span.
\[D_i \varpropto \frac{W^2}{b^2}\]
The chord doesn't enter into it, aspect ratio doesn't matter! Of course, area matters to profile drag, which drives designers to narrow chords, which results in high aspect ratios. But the aspect ratio, per se, isn't the important bit!
So how does this relate to biplanes? Suppose we divide the load between two wings:
\[D_{i_{biplane}} \varpropto 2 \frac{(W/2)^2}{b^2} = \frac{2}{4}\frac{W^2}{b^2} \varpropto \frac{1}{2} D_i\]
That's right, twice the wings, half the induced drag (assuming you don't shrink the span). Some of those flying Venetian blinds from the early days of aviation were not the work of cranks.
A carefully designed biplane can offer an overall drag reduction, but it does tend to have a narrow performance envelope because of the need to balance profile and induced drag, and to carefully manage interference effects.
Examples:
Symbols:
- \(C_{d_i}\): Coefficient of induced drag
- \(C_l\): Coefficient of lift
- \(\pi\): A mathematical constant, approximately 3.14159
- \(e\): An efficiency factor, 1.00 for elliptical lift distributions
- \(AR\): Aspect ratio, for a rectangular wing, the span divided by the chord
- \(D_i\): Induced drag
- \(L\): Lift
- \(\rho\): Density
- \(V\): Velocity
- \(S\): Wing area
- \(b\): Wing span
- \(c\): Wing chord
- \(W\): Weight
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