How to calculate sink rate from wing measurements

Question

First of all, sorry to bother you when I am coming from a bird rather than aircraft background.

I have two questions that I'm struggling to get an answer for.

Firstly, how is sink rate calculated (estimated) given mass, wing area and wingspan? Assume air speed/density and the other variables are known.

I know others use these variables to calculate sink rate but I can't find how. I'd like to know the formula so I can try to calculate sink rate myself.

Secondly (though this should hopefully become more clear once I learn the answer to the first question), is sink rate affected by aspect ratio? I am assume it has a lot to do with wing loading, but does aspect ratio reduce it much?

Answer 1

Maximizing wing (+tail) area helps to minimize the sink rate, but not to maximize the still-air glide ratio. All things being equal, if wing area were fixed, increasing the aspect ratio should normally decrease the sink rate, but it makes sense that in the real world birds minimize sink rate by maximizing wing (and tail) area, even if aspect ratio is slightly decreased.

One formula for aspect ratio is (wingspan squared) / area. Birds generally have lifting tails, at least when broadly fanned, so it would seem reasonable to include the tail area in the calculation. I'm sure you've observed how raptors typically fan tails and sweep wings slightly forward, while opening the primary feathers as wide as possible, to minimize sink rate (maximize climb rate) in a thermal updraft. This configuration maximizes lift coefficient and also wing (and tail) area, but decreases aspect ratio somewhat.

For fixed wing area (which is not accurate for birds), min sink rate is achieved at max value of (lift coefficient cubed) / (drag coefficient squared), while flattest glide angle in still air is achieved at max ratio of lift coefficient / drag coefficient. Basically this means that minimizing drag is more important when you are trying to maximize still-air glide angle than when you are trying to minimize sink rate.

For a given wing area, increasing the aspect ratio will generally minimize the sink rate, but at the low Reynolds numbers experienced by smaller birds this is not always true.

Answer 2

1. Aeroplane.

   Force equilibrium,:$$ -D - W \cdot sin \text{ } \gamma = 0 \tag{1}$$ $$+L - W \cdot cos \text{ } \gamma = 0 \tag{2}$$

   With $\bar{\gamma}$ = -$\gamma$ being the positive glide angle, and with the lift and drag equations (with S = wing area): $$C_D \cdot \frac{1}{2} \rho V^2 \cdot S = W \cdot sin \text{ } \bar{\gamma} \tag{3}$$ $$C_L \cdot \frac{1}{2} \rho V^2 \cdot S = W \cdot cos \text{ } \bar{\gamma} \tag{4}$$

   Divide equations (3) and (4): $$tan \text{ } \bar{\gamma} = C_D/C_L \tag{5}$$

   In a glide, the glide angle is often small, and $cos \text{ } \bar{\gamma}$ can be approximated as 1. So now for airspeed and sink speed $\bar{C}$ follows: $$V ≈ \sqrt{\frac{W}{S} \cdot \frac{2}{\rho} \cdot \frac{1}{C_L}} \tag{6}$$ $$\bar{C} = V\cdot sin \text{ } \bar{\gamma} = V \cdot \frac{C_D}{C_L} \cdot cos \text{ } \bar{\gamma} ≈ \sqrt{\frac{W}{S} \cdot \frac{2}{\rho} \cdot \frac{{C_D}^2}{{C_L}^3}} \tag{7}$$

   $C_L$ and $C_D$ as function of angle of attack $\alpha$ need to be known, in order to find the maximum value of ${C_L}^3/{C_D}^2$ equating to the minimum sink rate. For the Reynolds numbers that flight takes place in, and for low subsonic speeds, $C_L$ follows from (4), and $C_D$ is approximated as $$C_D = C_{D0} + \frac{{C_L}^2}{\pi A e} \tag{8}$$ with A = aspect ratio = $b^2/S$ with $b$ = wing span. $C_{D0}$ needs to be established in a wing tunnel, or with CFD, or by referring to comparable craft in design books. As for the Oswald factor $e$.

2. Birds.

   Equation {7} remains valid for birds, but how much are the $C_L$ and $C_D$ values for birds in flight? There is a reference in Siegfried Hörner's book Fluid Dynamic Drag, page 7-22:

   A buzzard was ingeniously tested in free flight...by tracking (following) him in a sailplane. The L/D...ratio of that bird was determined using the calibrated characteristics of the plane as a measure.

   

   The measurements found a max buzzard $C_L$ of 1.3 at the lowest speed of 8.5 m/s, and a $C_{D0}$ of 0.009 with reference to wing area S. The pic above shows the buzzard data under (a) in the left lower corner. The sea gull data was measured on a plaster model in a wind tunnel, and Hörner places question marks on the drag numbers found, probably due to the feathers not being properly reproduced.

   If the bird sink rate needs to be computed from mass, wing area and wing span only, equation (8) will need to be re-worked to fit the measured data. The measured data line follows the $C_D$ vs. ${C_L}^2$ trend approximately.

Answer 3

The question involves Area and Aspect ratio.

From the Lift Equation we have:

Weight = Lift = Density × Area × Velocity$^2$ x Coefficient

Coefficient is Angle of Attack of a given airfoil. This data can be found at airfoiltools.

If you know your weight, design airspeed, fluid medium density (generally of air) and Coefficient at optimal angle of attack, Area can be calculated.

Sink rate (for a glider) will determined by the amount of drag produced at a given velocity to make Lift. Drag is equal to the amount of forward force (provided by gravity) to keep your velocity constant. This force is readily obtained from the glide ratio × the aircraft weight or can be deduced from the Lift/Drag ratio of the airfoil plus the form drag of the aircraft.

So now you test your sink rate at various airspeeds to build your sink rate vs velocity curve. We can see induced drag will be greater at lower speeds, and form drag is higher at greater speeds. The lowest drag value for the sum of the two is Vbg.

Too small a wing means too much speed is needed, too large is simply adding extra weight and drag.

If you get this far with a good strong rectangular Aspect Ratio 8 wing, glide ratio can be improved by increasing AR to 10 or more, at the expense of strength. This is more fine tuning. Gliding birds, such as albatross, have high AR, whereas acrobatic birds, such as hawks, favor lower AR strength.