Computational aerodynamics — panel method (intro), CFD overview: Question bank

3.1.30 · D5 · Physics › Compressible Flow & Aerodynamics › Computational aerodynamics — panel method (intro), CFD overv

Shuru karne se pehle, teen symbols jo baar baar milenge — yahan pin kiye hain taaki kuch undefined na rahe:

$ϕ$ (phi) = velocity potential, ek scalar field jiska slope velocity deta hai: $V = \nabla ϕ$ . Dekho Laplace's Equation & Potential Flow.
$Γ$ (capital gamma) = circulation, body ke around flow ka net "swirl" — woh quantity jo Kutta–Joukowski Theorem & Circulation ke zariye lift produce karti hai.
$λ$ (lambda) = source strength, volume of air per second jo ek source bahar phenkta hai — dekho Sources, Sinks, Doublets, Vortices.

True or false — justify

Har prompt ek claim batata hai. True/false decide karo, phir reason do — wahi reason grade hoti hai.

Laplace's equation

\nabla^{2} ϕ = 0

linear hai, isliye do solutions ko add karke teesra banaya ja sakta hai.

True — linearity ka matlab hai ki

ϕ

ke koi products nahi aate, isliye ek uniform stream plus kisi bhi number of sources/vortices phir bhi ek valid solution hai. Yahi superposition hai panel method.

Flow mein viscosity add karne se equation linear rehti hai.

False — viscous Navier–Stokes Equations mein convective term

(V \cdot \nabla) V

hota hai, jo velocity ko uske apne gradient se multiply karta hai; woh nonlinearity superposition ko khatam kar deti hai aur hume CFD use karne par majboor karti hai.

Ek panel method attached flow mein 2-D airfoil ka drag predict kar sakta hai.

False — inviscid + irrotational + 2-D exactly zero drag deta hai (d'Alembert's Paradox); real drag viscosity aur wake mein hota hai, jo potential model bilkul miss kar deta hai.

Source strengths

λ_{j}

badhane se lift badhti hai.

False — lift hai

L^{'} = ρ V_{\infty} Γ

, jo vortices se milne wali circulation se set hoti hai; sources sirf body ki thickness aur shape banate hain aur zero net circulation rakhte hain.

Ek closed body par sab source strengths ka sum zero hota hai.

True — ek closed body na mass create karta hai na destroy, isliye total outflow total inflow ke barabar hona chahiye:

\sum_{j} λ_{j} = 0

. Yahi reason hai ki sirf-source bodies lift produce nahi karti.

Mesh ko baar baar refine karne se CFD steadily aur accurate hoti jaati hai.

False — refinement tab tak help karta hai jab tak mesh independence nahi aa jaati; uske baad, error turbulence model ya geometry se dominate hoti hai, cell count se nahi, aur cost bina kisi fayde ke badhti rehti hai.

Agar CFD residuals machine zero tak drop ho jayein, toh solution physically correct hai.

False — residuals sirf yeh measure karte hain ki discrete equations satisfy hui hain ya nahi; ek galat mesh ya wrong turbulence model par converged run confidently ek galat answer tak converge karta hai. Experiment ke saath validate karo.

Finite Volume Method cells ke beech mass aur momentum exactly conserve karta hai.

True — Finite Volume Method har law ko divergence theorem ke zariye surface fluxes ke roop mein likhta hai, isliye ek cell se nikalne wala flux uske neighbour mein jaata hai — koi leakage nahi, yahi reason hai ki yeh shocks ko cleanly capture karta hai.

Ek doublet source se fundamentally alag building block hai.

False — ek doublet ek source aur equal-strength sink ko ek saath laane ki limit hai jab unka spacing

\to 0

ho jaata hai; yeh abhi bhi ek Laplace solution hai, bas source LEGO brick se bana hua hai.

Spot the error

Har line mein ek flawed statement hai. Galti batao aur correct karo.

"Hum har control point par panel ke normal velocity ko

V_{\infty}

set karte hain."

Error — boundary condition zero normal velocity hai (

V \cdot \overset{n}{^} = 0

), yeh enforce karta hai ki koi bhi air solid wall se through nahi flow karti; ise

V_{\infty}

set karne se air body ke through pass ho jaayegi.

"Influence coefficient

I_{ij}

woh tangential velocity hai jo panel

j

panel

i

par induce karta hai."

Error —

I_{ij}

woh normal velocity hai jo

i

par unit source on

j

ke zariye induce hoti hai; yeh normal honi chahiye kyunki jis boundary condition ko yeh feed karta hai woh flow-tangency (normal) condition hai.

"Apne midpoint par ek source panel koi velocity induce nahi karta."

Error — self-influence nonzero hai: ek constant source sheet apne control point par

λ_{i} /2

ki normal velocity induce karta hai, isliye

A

ka diagonal

0

nahi balki

1/2

hota hai.

"Hum body ko thicker banane ke liye vortex panels add karte hain."

Error — vortices circulation (aur isliye lift) supply karte hain; sources hi thickness/shape set karte hain. Vortices plus Kutta condition

Γ

fix karte hain.

C_{p} = 1 - (V_{t} / V_{\infty})^{2}

directly Newton's second law se aata hai."

Error — yeh Bernoulli's Equation (streamline ke along energy conservation) se aata hai, jo local tangential speed

V_{t}

ko pressure coefficient se relate karta hai.

"Ek pure source distribution lift produce kar sakta hai agar sources kaafi strong hon."

Error — koi bhi source strength circulation create nahi karti; ek symmetric source layout front-to-back symmetric hoti hai, isliye

Γ = 0

aur Kutta–Joukowski ke zariye

L^{'} = 0

magnitude ki parwah kiye bina.

"RANS flow mein har turbulent eddy ko resolve karta hai."

Error — RANS time-averaging ke zariye saari turbulence ko model karta hai; DNS hai jo har eddy ko resolve karta hai (bahut zyada cost par). Dekho Turbulence Modelling — RANS, LES, DNS.

Why questions

Sirf fact nahi, reason explain karo.

Flow irrotational kyun hona chahiye pehle hum

V = \nabla ϕ

likh sakein?

Kyunki ek gradient field automatically zero curl rakhta hai; irrotationality (

\nabla \times V = 0

) wahi exact condition hai jo velocity ko ek single scalar potential ke gradient ke roop mein likhne deti hai.

Hum surface ko finitely many panels mein kyun discretise karte hain instead of ek continuous source distribution solve karne ke?

Ek continuous

λ (s)

ek unknown function hai — infinitely many values; ise

N

constant strengths se replace karne par problem

N

unknowns aur

N

linear equations mein convert ho jaati hai jo computer invert kar sakta hai.

Panel influence matrix

A

ko handle karne mein roughly

N^{2}

N^{3}

cost kyun lagti hai?

Har panel baaki sab ko influence karta hai, isliye

A

dense hai (

N \times N

): storage

N^{2}

ki tarah badhti hai aur direct solution

N^{3}

ki tarah — yahi reason hai ki panel counts plateau karte hain instead of millions mein jaane ke.

Shocks capture karne ke liye conservative (flux-based) discretisation kyun zaroori hai?

Shock ek discontinuity hai; sirf mass/momentum/energy ka exact conservation across cell faces jump ko sahi strength aur location par rakhta hai — ek non-conservative scheme use ise smear ya misposition kar deta hai.

Kutta condition ki zaroorat kyun hai?

Laplace's equation akela infinitely many circulation values allow karta hai; Kutta condition (flow sharp trailing edge se smoothly nikle) woh ek physical

Γ

chunti hai, lift ko uniquely fix karti hai.

Panel method "1% cost mein 80% answer" sirf attached, low-speed flow mein kyun hai?

Uske assumptions (inviscid, incompressible, no separation) wahan hold karte hain; jab viscosity, separation, ya compressible shocks matter karte hain, toh woh assumptions toot jaate hain aur sirf CFD valid rehta hai.

Hum full Navier–Stokes equations ke solutions simply superpose kyun nahi kar sakte?

Woh nonlinear hain, isliye do solutions ka sum generally ek solution nahi hota — superposition, jo panel method ka engine hai, available nahi hai.

Edge cases

Boundary aur degenerate scenarios jo topic quietly invite karta hai.

Zero angle of attack par perfectly symmetric body ke liye lift ke liye kitna source strength chahiye?

Lift ke liye koi nahi — ek symmetric body

α = 0

par zero circulation rakhti hai, isliye

L^{'} = 0

; sources phir bhi isse shape karte hain lekin lift mein kuch contribute nahi karte

α

ki parwah kiye bina.

Jab source ke singular point

r \to 0

par jaate hain toh induced velocity ka kya hota hai?

Potential

ϕ = \frac{λ}{2 π} ln r

aur uski velocity blow up ho jaati hai; isliye singularities andar ya surface par rakhi jaati hain, flow region mein kabhi nahi jis ki hume parwah hai, aur isliye control points panel midpoints par hote hain.

Agar do panels almost coincident ho jayein (sharp edge par bahut fine mesh), numerically kya galat hota hai?

Unki influence rows almost identical ho jaati hain, jisse

A

ill-conditioned (near-singular) ho jaata hai; solved strengths bahut noisy ho jaati hain chahe residuals theek lagein — ek hidden accuracy trap.

Trailing edge par, Kutta condition velocity ke baare mein kya require karti hai?

Flow ko smoothly nikalna chahiye ek finite, single velocity ke saath (sharp corner ke around infinite speed nahi); yeh

Γ

aur isliye lift select karta hai.

Vanishing viscosity (

μ \to 0

) ki limit mein body past flow ke liye, kya drag bhi vanish ho jaata hai?

Nahi — yeh Boundary Layer Theory & Skin Friction Drag ke peechhe ka subtle point hai:

μ

kitna bhi chhota ho, ek thin boundary layer rehti hai aur separate ho sakti hai, isliye drag finite rehta hai chahe

μ \to 0

ho, unlike idealised d'Alembert result ke.

Source aur sink se bane doublet ke liye, jab woh merge hote hain toh net mass output ka kya hota hai?

Woh exactly zero rehta hai — source

+ λ

aur sink

- λ

cancel ho jaate hain — isliye ek doublet ek closed (mass-conserving) body jaise cylinder ko model karta hai.

Recall Lock in karne ke liye ek-line summary

Sources shape karte hain · vortices lift dete hain · Laplace linear hai (superpose) · Navier–Stokes nahi (CFD) · residuals ≠ correctness · d'Alembert ⇒ inviscid drag zero hai. Ek sentence ::: Lift ko circulation chahiye, drag ko viscosity chahiye, aur convergence ka matlab sahi hona nahi hai.