Exercise set

Aerodynamic Lift, Drag, and Polar Exercises

Worked aerodynamics exercises for dynamic pressure, lift and drag coefficients, drag polar, induced drag, similarity, pressure loads and validation.

Branch: Aerospace Engineering
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.1.0 · reviewed

These exercises focus on aerodynamic force and coefficient practice: dynamic pressure, lift, drag, drag polars, induced drag, similarity, pressure loads, center of pressure, vortex shedding and wind-tunnel validation. Stall speed and aircraft operating margins are handled in a separate specialist exercise set.

Use the results as screening calculations. Design release still needs configuration control, calibrated balances, Reynolds and Mach review, wall and support corrections, uncertainty budgets, CFD or flight-test comparison and traceable acceptance criteria.

How to use these exercises

Use the set as an aerodynamic data-release review. Exercises 1 to 4 establish dynamic pressure, force normalization and lift-to-drag ratio. Exercises 5 to 8 interpret drag-polar terms, induced drag and Oswald efficiency. Exercises 9 to 13 check Reynolds number, Mach number, pressure coefficient, panel load and center of pressure. Exercises 14 to 18 add vortex shedding, roughness, ground effect, similarity and final polar release gates.

Before calculating, name the configuration, reference area, coordinate convention, freestream condition, Reynolds number, Mach number, support correction, wall correction and data source. A lift or drag coefficient is only useful if another engineer can reproduce the normalization. The engineering comment below each exercise identifies the correction, validity range or acceptance evidence needed before the number can support design release.

Release Evidence Notes

Aerodynamic coefficient evidence must state reference area, freestream condition, coordinate convention and correction method. A correct number with the wrong reference area or dynamic pressure is not a valid aerodynamic result.

The evidence package should separate raw force data, corrected coefficient data and approved aerodynamic model data. Raw balance readings need tare, calibration and support corrections. Coefficients need reference dimensions, density, speed and sign convention. A released polar needs fit range, residuals, similarity bounds and uncertainty. Skipping these distinctions turns a calculation into an undocumented assumption.

Release evidence should also state intended use. A low-speed wind-tunnel polar may support preliminary sizing but not a compressibility-sensitive pressure distribution, flutter load case or transonic drag-rise prediction. The acceptable Reynolds and Mach mismatch depends on the physics being released.

Engineering Boundary Notes

These exercises use low-order incompressible or mildly compressible models. Shock effects, separated flow, transonic buffet, aeroelastic deformation and high-angle-of-attack hysteresis require dedicated test or simulation evidence. Treat pass results as data-quality screens, not as proof that the full aircraft envelope is cleared.

The main boundary is validity range. A drag polar fitted near cruise should not be used at high lift, stall onset, strong ground effect or separated-flow conditions without evidence. The second boundary is model support: CFD, wind-tunnel and flight-test data can disagree for legitimate reasons, so residuals and correction methods should be visible in the release record.

Common Release Mistakes

mixing wing area, frontal area and model reference area;
using true speed where equivalent dynamic pressure is required;
fitting a drag polar outside the measured lift-coefficient range;
quoting Reynolds similarity while Mach similarity is violated;
integrating pressure taps without panel area and sign convention;
accepting a wind-tunnel result without calibration and blockage review.

Another common mistake is reporting $L/D$ without the operating condition. A high lift-to-drag ratio at one lift coefficient, Reynolds number and configuration does not define climb, range or endurance by itself. Performance use requires weight, thrust, altitude and mission condition as well.

Do not hide correction quality inside a single final coefficient. Balance tare, wall interference, blockage, support interference, roughness, transition fixing and pressure-tap lag can each control the result. If one correction is unvalidated, the data may need a narrower validity envelope rather than a broad release.

Scenario Map

Scenario	Main calculation	Release decision
Force coefficients	dynamic pressure, lift and drag	Normalize and compare data.
Drag polar	$C_{D0}$ , induced factor and $L/D$	Select operating point or refit data.
Similarity	Reynolds, Mach and scale	Accept or reject test condition.
Pressure loads	pressure coefficient and panel force	Validate local load evidence.
Wind-tunnel release	uncertainty, calibration and corrections	Release data or retest.

Validation Package Checklist

reference area, span, chord and coordinate convention;
density, speed, dynamic pressure, Reynolds number and Mach number;
balance calibration, tare, blockage and support corrections;
pressure-tap map, panel areas and sign convention;
drag-polar fit range and residuals;
transition state, surface roughness and configuration-control record;
CFD, wind-tunnel or flight-test comparison where used for release;
intended-use limits for Reynolds, Mach, lift range and flow separation;
uncertainty budget and acceptance decision.

A complete validation package should make the aerodynamic data point reproducible. Another engineer should be able to see how the coefficient was normalized, what corrections were applied, where the model is valid and which retest, refit or restriction follows if a residual gate fails.

Exercise 1: Dynamic Pressure

Air density is $1.18\ \text{kg/m}^3$ and speed is $72\ \text{m/s}$ . Compute dynamic pressure.

Solution

q=\dfrac{1}{2}\rho V^2=\dfrac{1}{2}(1.18)(72^2)=3059\ \text{Pa}

Engineering Comment

Most aerodynamic force coefficients depend directly on the dynamic pressure used for normalization.

Plausibility Check

At around $70\ \text{m/s}$ near sea-level density, a few kilopascals is reasonable.

Exercise 2: Lift Coefficient from Force

A wind-tunnel model has reference area $S=1.6\ \text{m}^2$ . Measured lift is $4100\ \text{N}$ at $q=3059\ \text{Pa}$ . Compute $C_L$ .

Solution

C_L=\dfrac{L}{qS}=\dfrac{4100}{3059(1.6)}=0.838

Engineering Comment

The coefficient is only meaningful if $S$ is the declared reference area for the model and report.

Plausibility Check

Subsonic wings often operate with $C_L$ around one, so $0.838$ is plausible.

Exercise 3: Drag Coefficient from Force

At the same condition, measured drag is $310\ \text{N}$ . Compute $C_D$ .

Solution

C_D=\dfrac{D}{qS}=\dfrac{310}{3059(1.6)}=0.0634

Engineering Comment

Drag coefficients are sensitive to tare, support interference and surface finish.

Plausibility Check

Drag coefficient is much smaller than lift coefficient at a lifting operating point.

Exercise 4: Lift-to-Drag Ratio

Using $C_L=0.838$ and $C_D=0.0634$ , compute lift-to-drag ratio.

Solution

\dfrac{L}{D}=\dfrac{C_L}{C_D}=\dfrac{0.838}{0.0634}=13.2

Engineering Comment

$L/D$ is an aerodynamic efficiency measure, but performance decisions also need thrust, weight and mission condition.

Plausibility Check

The lift coefficient is about thirteen times the drag coefficient.

Exercise 5: Drag Polar Value

A parabolic drag polar is:

C_D=C_{D0}+kC_L^2

with $C_{D0}=0.024$ , $k=0.045$ and $C_L=0.80$ . Compute $C_D$ .

Solution

C_D=0.024+0.045(0.80^2)=0.024+0.0288=0.0528

Engineering Comment

The polar should not be used beyond the lift range where it was fitted.

Plausibility Check

Induced drag roughly doubles the zero-lift drag at this lift coefficient.

Exercise 6: Best Lift-to-Drag Ratio

For $C_{D0}=0.024$ and $k=0.045$ , estimate the lift coefficient for maximum $L/D$ :

C_{L,*}=\sqrt{\dfrac{C_{D0}}{k}}

Solution

C_{L,*}=\sqrt{\dfrac{0.024}{0.045}}=0.730

Engineering Comment

This optimum comes from the simplified polar, not from a full aircraft performance map.

Plausibility Check

The value is below one, which is typical for efficient cruise-like conditions.

Exercise 7: Induced Drag Factor

A wing has aspect ratio $AR=9.5$ and Oswald efficiency $e=0.82$ . Compute:

k=\dfrac{1}{\pi e AR}

Solution

k=\dfrac{1}{\pi(0.82)(9.5)}=0.0409

Engineering Comment

Higher aspect ratio and better span efficiency reduce induced drag.

Plausibility Check

A value near $0.04$ is reasonable for a moderate-to-high aspect-ratio wing.

Exercise 8: Oswald Efficiency from Polar Fit

A fitted induced-drag factor is $k=0.050$ and aspect ratio is $AR=8.0$ . Estimate $e$ .

Solution

e=\dfrac{1}{\pi AR k}=\dfrac{1}{\pi(8.0)(0.050)}=0.796

Engineering Comment

Do not interpret $e$ as generic efficiency; it is a finite-wing correction for this polar model.

Plausibility Check

The estimate is below one and close to a typical aerodynamic value.

Exercise 9: Reynolds Number

Air density is $1.20\ \text{kg/m}^3$ , speed is $55\ \text{m/s}$ , chord is $0.75\ \text{m}$ and dynamic viscosity is $1.8\times10^{-5}\ \text{Pa s}$ . Compute Reynolds number.

Solution

Re=\dfrac{\rho V c}{\mu}=\dfrac{(1.20)(55)(0.75)}{1.8\times10^{-5}}=2.75\times10^6

Engineering Comment

Reynolds mismatch can change transition, separation and drag.

Plausibility Check

Meter-scale aircraft testing at tens of meters per second often gives millions.

Exercise 10: Mach Number

Aircraft speed is $95\ \text{m/s}$ and speed of sound is $340\ \text{m/s}$ . Compute Mach number.

Solution

M=\dfrac{V}{a}=\dfrac{95}{340}=0.279

Engineering Comment

Compressibility is modest here, but Mach should still be reported for similarity and drag-rise screening.

Plausibility Check

$95\ \text{m/s}$ is well below one third of the speed of sound.

Exercise 11: Pressure Coefficient

Freestream static pressure is $101.3\ \text{kPa}$ , local surface pressure is $99.7\ \text{kPa}$ and dynamic pressure is $3.2\ \text{kPa}$ . Compute $C_p$ .

Solution

C_p=\dfrac{p-p_\infty}{q}=\dfrac{99.7-101.3}{3.2}=-0.50

Engineering Comment

Negative $C_p$ indicates suction relative to freestream static pressure.

Plausibility Check

The pressure drop is half the dynamic pressure.

Exercise 12: Local Panel Load

A pressure panel has area $0.18\ \text{m}^2$ , $C_p=-0.50$ and $q=3.2\ \text{kPa}$ . Estimate suction force magnitude on the panel.

Solution

|\Delta p|=|C_p|q=0.50(3200)=1600\ \text{Pa}

F=1600(0.18)=288\ \text{N}

Engineering Comment

Panel load integration must use the correct local normal direction and panel area.

Plausibility Check

Kilopascal pressure over a fraction of a square meter gives hundreds of newtons.

Exercise 13: Center of Pressure

Lift is $4100\ \text{N}$ and pitching moment about the leading edge is $-820\ \text{N m}$ . Using magnitude, estimate center-of-pressure distance from the leading edge.

Solution

x_{cp}=\dfrac{|M_{LE}|}{L}=\dfrac{820}{4100}=0.200\ \text{m}

Engineering Comment

Moment sign convention must be documented before using center-of-pressure movement in a stability discussion.

Plausibility Check

The moment is one fifth of lift per meter, so the arm is $0.2\ \text{m}$ .

Exercise 14: Vortex Shedding Frequency

A strut has diameter $d=0.08\ \text{m}$ in flow $V=30\ \text{m/s}$ . With Strouhal number $St=0.2$ , estimate shedding frequency.

Solution

f=\dfrac{St V}{d}=\dfrac{0.2(30)}{0.08}=75\ \text{Hz}

Engineering Comment

Compare this frequency with structural modes before accepting the geometry.

Plausibility Check

Small diameter and fast flow produce tens of hertz.

Exercise 15: Roughness Drag Penalty

Clean drag coefficient is $0.0528$ . Surface contamination adds $\Delta C_D=0.0065$ . Compute percentage drag increase.

Solution

\Delta=\dfrac{0.0065}{0.0528}=0.123=12.3\%

Engineering Comment

Small coefficient changes can matter for range, climb and thermal load.

Plausibility Check

$0.0065$ is about one eighth of $0.0528$ .

Exercise 16: Ground-Effect Induced-Drag Relief

Out-of-ground-effect induced drag coefficient is $0.0288$ . Ground effect reduces induced drag by $18\%$ . Compute revised induced drag coefficient.

Solution

C_{D_i,new}=0.0288(1-0.18)=0.0236

Engineering Comment

Ground effect is configuration and height dependent; do not use this relief outside the validated height range.

Plausibility Check

The revised value is slightly below the original.

Exercise 17: Similarity Release Gate

A wind-tunnel test matches Reynolds number within $4\%$ but Mach number is $0.18$ while the flight condition is $0.42$ . Is a compressibility-sensitive pressure distribution release justified?

Solution

No. Reynolds similarity is close, but Mach mismatch is large:

\Delta M=0.42-0.18=0.24

For compressibility-sensitive pressure data, the test condition is not sufficient without correction or supporting evidence.

Engineering Comment

Similarity requirements must match the physics being released.

Plausibility Check

The test Mach is less than half the flight Mach.

Exercise 18: Aerodynamic Polar Release Gate

A drag-polar package has correct reference area, $C_L$ residual within $2\%$ , drag residual within $7\%$ against a $5\%$ limit, Reynolds mismatch $4\%$ , Mach mismatch $0.24$ for a compressibility-sensitive case and complete balance calibration. Decide release status.

Solution

Drag residual fails:

7\%>5\%

Mach similarity also fails for the intended use:

\Delta M=0.24

The polar should not be released for the compressibility-sensitive condition.

Engineering Comment

Release the data only after refitting, retesting or narrowing the validity envelope.

Plausibility Check

Two validation gates fail even though area and calibration evidence are present.

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