3.1.30 · D5Compressible Flow & Aerodynamics
Question bank — Computational aerodynamics — panel method (intro), CFD overview
Before we begin, three symbols you will meet constantly — pinned here so nothing is used undefined:
- (phi) = the velocity potential, a scalar field whose slope gives velocity: . See Laplace's Equation & Potential Flow.
- (capital gamma) = circulation, the net "swirl" of the flow around the body — the quantity that produces lift via Kutta–Joukowski Theorem & Circulation.
- (lambda) = source strength, the volume of air per second a source spits out — see Sources, Sinks, Doublets, Vortices.
True or false — justify
Every prompt states a claim. Decide true/false, then give the reason — that reason is what's graded.
Laplace's equation is linear, so two solutions can be added to make a third.
True — linearity means no products of appear, so a uniform stream plus any number of sources/vortices is still a valid solution. This superposition is the panel method.
Adding viscosity to the flow keeps the equation linear.
False — the viscous Navier–Stokes Equations contain the convective term , which multiplies velocity by its own gradient; that nonlinearity kills superposition and forces us to use CFD.
A panel method can predict the drag on a 2-D airfoil in attached flow.
False — inviscid + irrotational + 2-D gives exactly zero drag (d'Alembert's Paradox); real drag lives in viscosity and the wake, which the potential model omits entirely.
Increasing the source strengths increases the lift.
False — lift is , set by circulation from vortices; sources only shape the body's thickness and carry zero net circulation.
The sum of all source strengths on a closed body is zero.
True — a closed body neither creates nor destroys mass, so total outflow must equal total inflow: . This is exactly why source-only bodies produce no lift.
Refining the mesh forever makes CFD steadily more accurate.
False — refinement helps until you hit mesh independence; past that, error is dominated by the turbulence model or geometry, not cell count, and cost keeps climbing for no gain.
If the CFD residuals drop to machine zero, the solution is physically correct.
False — residuals only measure that the discrete equations are satisfied; a converged run on a bad mesh or wrong turbulence model converges confidently to a wrong answer. Validate against experiment.
The Finite Volume Method conserves mass and momentum exactly between cells.
True — Finite Volume Method rewrites each law as surface fluxes via the divergence theorem, so the flux leaving one cell is the flux entering its neighbour — no leakage, which is why it captures shocks cleanly.
A doublet is a fundamentally different building block from a source.
False — a doublet is the limit of a source and equal-strength sink brought together as their spacing ; it is still a Laplace solution, just built from the source LEGO brick.
Spot the error
Each line contains a flawed statement. Name the flaw and correct it.
"We set velocity normal to the panel to at each control point."
Error — the boundary condition is zero normal velocity (), enforcing that no air flows through the solid wall; setting it to would let air pass through the body.
"The influence coefficient is the tangential velocity panel induces at panel ."
Error — is the normal velocity induced at by a unit source on ; it must be normal because the boundary condition it feeds is the flow-tangency (normal) condition.
"On its own midpoint a source panel induces no velocity."
Error — the self-influence is nonzero: a constant source sheet induces a normal velocity of at its own control point, which is why the diagonal of equals , not .
"We add vortex panels to make the body thicker."
Error — vortices supply circulation (hence lift); it's the sources that set thickness/shape. Vortices plus the Kutta condition fix .
" comes from Newton's second law directly."
Error — it comes from Bernoulli's Equation (energy conservation along a streamline), which relates the local tangential speed to the pressure coefficient.
"A pure source distribution can produce lift if the sources are strong enough."
Error — no amount of source strength creates circulation; a symmetric source layout is front-to-back symmetric, so and by Kutta–Joukowski regardless of magnitude.
"RANS resolves every turbulent eddy in the flow."
Error — RANS models all turbulence via time-averaging; it is DNS that resolves every eddy (at enormous cost). See Turbulence Modelling — RANS, LES, DNS.
Why questions
Explain the reason, not just the fact.
Why must the flow be irrotational before we can write ?
Because a gradient field automatically has zero curl; irrotationality () is the exact condition that lets velocity be written as the gradient of a single scalar potential.
Why do we discretise the surface into finitely many panels rather than solving for a continuous source distribution?
A continuous is an unknown function — infinitely many values; replacing it with constant strengths turns the problem into unknowns and linear equations a computer can invert.
Why does the panel influence matrix cost roughly to to handle?
Every panel influences every other, so is dense (): storage grows like and direct solution like — the reason panel counts plateau instead of going to millions.
Why does capturing shocks demand a conservative (flux-based) discretisation?
A shock is a discontinuity; only exact conservation of mass/momentum/energy across cell faces places the jump at the right strength and location — a non-conservative scheme smears or mispositions it.
Why is the Kutta condition needed at all?
Laplace's equation alone allows infinitely many circulation values; the Kutta condition (flow leaves the sharp trailing edge smoothly) picks the one physical , fixing the lift uniquely.
Why is panel method "80% of the answer for 1% of the cost" only in attached, low-speed flow?
Its assumptions (inviscid, incompressible, no separation) hold there; once viscosity, separation, or compressible shocks matter, those assumptions break and only CFD stays valid.
Why can't we simply superpose solutions of the full Navier–Stokes equations?
They are nonlinear, so the sum of two solutions is generally not a solution — superposition, the engine of the panel method, is unavailable.
Edge cases
Boundary and degenerate scenarios the topic quietly invites.
What is the source strength required on a perfectly symmetric body at zero angle of attack, for lift?
None for lift — a symmetric body at has zero circulation, so ; sources still shape it but contribute nothing to lift regardless of .
What happens to a source's induced velocity as you approach its singular point ?
The potential and its velocity blow up; that's why singularities are placed inside or on the surface, never in the flow region we care about, and why control points sit at panel midpoints.
If two panels are made nearly coincident (very fine mesh at a sharp edge), what goes wrong numerically?
Their influence rows become almost identical, making ill-conditioned (near-singular); the solved strengths become wildly noisy even though residuals look fine — a hidden accuracy trap.
At the trailing edge, what does the Kutta condition require of the velocity?
The flow must leave smoothly with a finite, single velocity (no infinite speed wrapping around the sharp corner); this selects and hence the lift.
In the limit of vanishing viscosity () for flow past a body, does the drag also vanish?
No — this is the subtle point behind Boundary Layer Theory & Skin Friction Drag: however small is, a thin boundary layer persists and can separate, so drag stays finite even as , unlike the idealised d'Alembert result.
For a doublet built from a source and sink, what happens to the net mass output as they merge?
It stays exactly zero — source and sink cancel — which is why a doublet models a closed (mass-conserving) body like a cylinder.
Recall One-line summary to lock in
Sources shape · vortices lift · Laplace is linear (superpose) · Navier–Stokes is not (CFD) · residuals ≠ correctness · d'Alembert ⇒ inviscid drag is zero. The single sentence ::: Lift needs circulation, drag needs viscosity, and convergence is not the same as being right.