DRAG POLARS
Slope of a Straight Line
· Let (x1, y1) and (x2, y2) be any two points on a straight line in Cartesian 2-space
· The slope of the line is defined as (y2 – y1) / (x2 – x1)
· Example: Let y = 2x+1
o If x1 = 2, then y1 = 5
o If x2 = 6, then y2 = 13
o Slope of the line is (y2 – y1) / (x2 – x1). = (13 - 5) / (6 - 2) = 8 / 4 = 2.
· The slope of a straight line is everywhere the same
· For any point (x, y) on a straight line drawn through the origin, the slope of the line is just y / x, , since (x1, y1) can always be chosen as the origin (where x1 = y1 = 0)
Tangent Line to a Curve
· Straight line from the origin drawn tangent to a curve
· If (x1, y1), (x2, y2), (x3, y3) as shown above are points on the tangent line, the slope of the tangent line is y1/x1 = y2/x2 = y3/x3 = &c.
· When determining the slope of straight line from a plotted graph (as opposed to an equation), the slopes will vary slightly from point to point due to inaccuracies in
o the graphical reproduction of the line
o reading the coordinates of the points you choose on the graphical reproduction
Drag Polar:
· Definition: a plot of cL vs. cD for an airplane in SS SL flight for various airspeeds / AOAs
· A straight line drawn from the origin of a drag polar tangent to the curve has slope (cL / cD)MAX = (L/D)MAX
· Why?
o A straight line drawn from the origin to any point (cD, cL) on the curve—including the tangent point—has slope cL / cD
o For the airspeed / AOA corresponding to that point
o Any line steeper than the tangent line misses the curve
o Any line shallower than the tangent line has a smaller slope, thus intersects the curve where the ratio cL / cD = L / D is smaller
Important Points
· The axes of drag polars plots used in AS310 have different scales
o x is labeled 0.0, 0.02, 0.04, 0.06, …
o y is labeled 0.0, 0.2, 0.4, 0.6, …
· The origin may not be located at the intersection of the x and y axes as drawn on the graph.
· This facilitates plotting the drag polar in a restricted portion of Cartesian 2-space, making the diagram smaller and more readable.
Given a drag polar plot, find (L/D)MAX = GRBEST as follows:
1. Locate the origin.
2. From the origin, “draw” a tangent line to the curve
3. Locate 5 points on the tangent line, and record cD (x-axis) and cL (y-axis)
· for each point, pick either cL or cD on a major grid line
· pick the points (cL, cD) along the full length of the cure
4. For each point located in step 3, calculate the quotient cL / cD (All quotients theoretically should be the same. If the various cL / cD values do not lie very close together, you have made an error somewhere)
5. Average the quotients calculated in step 4, omitting any outliers.
cD (x-axis) / cL (y-axis) / cL / cD0.01 / 0.195 / 19.5
0.02 / 0.4 / 20.0
0.03 / 0.58 / 19.3
0.04 / 0.78 / 19.5
0.05 / 0.975 / 19.5
Figure 4.5. B767 Low Speed Drag Polar, Clean Configuration
0.04 / 0.565 / 14.1
0.06 / 0.87 / 14.5
0.08 / 1.15 / 14.4
0.10 / 1.45 / 14.5
0.11 / 1.61 / 14.6
Figure 4.14. B767 Drag Polar at Flaps-15, Gear Up
Glide Distances and Glide Angles at Different Flap Setting
· GR= (L/D)MAX for the 767 is approximately 19.5 clean and 14.4 at Flaps 15
· Extending flaps increases wing camber and hence increases lift
· However, it also increases drag faster than lift
· Hence GR = L/D goes down, so glide distance goes down and glide angle gets steeper
Example 1: Glide Distances from 40,000’ Clean and at Flaps 15
· Clean GD = GR (AA) = 19.5 (40,000 feet) / (6076 feet/NM) = 128.4 NM
· Flaps 15 GD = GR (AA) = 14.4 (40,000 feet) / (6076 feet/NM) = 94.8NM
Example 2: Glide Angles Clean and at Flaps 15
· Recall that GR = GD / AA = L / D = 1 / tan a, where a is the glide angle
· GR = 1 / tan a Þtan a = 1 / GR Þ a = tan-1 ( 1 / GR)
· Clean glide angle a = tan-1 ( 1 / GR) = tan-1 (1 / 19.5) = 2.94o rounded (2.9335673446)
· Flaps 15 glide angle a = tan-1 ( 1 / GR) = tan-1 (1 / 14.4) = 3.97o rounded (3.972495941)
Important Observations:
· From the Total Drag Curve, one can find the airspeed VBG corresponding to (L/D)MAX, i.e. the airspeed corresponding to best glide ratio. One cannot find (L/D)MAX from a DT plot.
· From the drag polar, one can find (L/D)MAX = GRBEST. One cannot find the airspeed VBG corresponding to this glide ratio.
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