Exercises — Computational aerodynamics — panel method (intro), CFD overview

Equation hai Laplace's equation: ${\nabla }^2\varphi =0$ Yahan $\varphi$ ("phi") velocity potential hai — ek single scalar field jiska slope (gradient) har jagah velocity deta hai, $\vec{V}=\nabla \varphi$. Linearity ka matlab hai: agar ${\varphi }_1$ aur ${\varphi }_2$ dono isko solve karte hain, toh ${\varphi }_1+{\varphi }_2$ bhi karta hai. Yahi superposition hai. Isse hum ek freestream aur bahut saare chote sources ko add kar sakte hain aur phir bhi ek valid flow milta hai — yahi panel method ka poora engine hai.

• (a) Uniform stream — air saari ek hi direction mein $V_{\infty }$ speed se bahti hai; door ki hawa ko represent karta hai.
• (b) Source — air ko sabhi directions mein equally bahar dhakelta hai; body ki thickness dene ke liye use hota hai.
• (c) Vortex — air ko ek point ke around chakkar lagata hai; yahi akela "brick" hai jo circulation produce karta hai, aur isliye lift.
• (d) Doublet — ek source aur sink ko ek saath pack kar do; yahi combination exactly cylinder ke around flow model karta hai.

KYA karte hain: ratio plug in karo. KYUN: $C_p$ local speed ko freestream se compare karta hai — tez air ka matlab hai kam pressure (Bernoulli's Equation yahi kehta hai). $C_p=1-(1.4{)}^2=1-1.96=-0.96$ Negative value matlab pressure yahan freestream se neeche hai — yeh ek suction region hai, exactly wahi jo ek wing ki upper surface par hota hai jahan flow tez hoti hai.

$V_t=0$ ke saath: $C_p=1-0^2=1$ Kyunki ${\left(V_t/V_{\infty }\right)}^2\ge 0$ hamesha hota hai, jo term hum subtract karte hain wo kabhi negative nahi hoti, isliye $C_p$ kabhi $1$ se zyada nahi ho sakta. Maximum $C_p=1$, sirf wahan milta hai jahan flow ruk jaaye. Baaki har point par $C_p<1$ hoga.

KYA: freestream vector ko normal par project karo. KYUN: boundary condition jo hum ultimately enforce karte hain wo hai zero normal velocity; yahan hum measure karte hain ki akela freestream ise kitna violate karta hai, jo exactly panel system ke right-hand side $b_i$ hai. ${\vec{V}}_{\infty }$ ka component $\hat{n}$ ke along hai: $V_{\infty }\cos \beta =V_{\infty }\cos 60^{\circ }=0.5V_{\infty }$ Toh freestream akela $0.5V_{\infty }$ diwar ke through drive karta hai. Sources ko exactly itna cancel karna padega: $b_i=-V_{\infty }\cos {\beta }_i=-0.5V_{\infty }$.

Sum zero kyun hai: body solid aur closed hai — iske andar koi air create ya destroy nahi hoti. Har bit air jo koi source spray karta hai, use kahin na kahin kisi sink (negative source) ko swallow karna padta hai. Isliye strengths cancel ho jaati hain: $\sum {\lambda }_j=0$. Yeh sirf mass conservation hai. Drag ke liye consequence: ek pure source distribution mein koi circulation nahi hoti aur, apne pressure pattern mein fore-aft symmetric hone ke kaaran, 2-D inviscid flow mein zero net drag produce karta hai. Yahi exactly d'Alembert's Paradox hai: ideal fluid mein body par bilkul bhi drag nahi hota. Real drag viscosity aur wake se aata hai, jo Boundary Layer Theory & Skin Friction Drag mein rehte hain, is model mein nahi.

KYA: right-hand side banao, phir $2\times 2$ system solve karo. KYUN: diagonal $\frac{1}{2}$ panel ka khud par effect hai; off-diagonal $\frac{1}{4}$ neighbour ka influence hai. Solve karne se pata chalta hai ki dono control points par flow tangency enforce karne ke liye strengths kaise adjust hoti hain. Right-hand sides: $b_1=-\cos 60^{\circ }=-0.5,b_2=-\cos 120^{\circ }=+0.5$ System: $0.5{\lambda }_1+0.25{\lambda }_2=-0.5$ $0.25{\lambda }_1+0.5{\lambda }_2=0.5$ Symmetry use karne ke liye subtract karo. Pehle ko 2 se multiply karo: ${\lambda }_1+0.5{\lambda }_2=-1$. Doosre ko 2 se multiply karo: $0.5{\lambda }_1+{\lambda }_2=1$. Solve: Pehle se, ${\lambda }_1=-1-0.5{\lambda }_2$. Substitute karo: $0.5(-1-0.5{\lambda }_2)+{\lambda }_2=1\Rightarrow -0.5+0.75{\lambda }_2=1\Rightarrow {\lambda }_2=2$ Phir ${\lambda }_1=-1-0.5(2)=-2$. $\boxed{{\lambda }_1=-2,{\lambda }_2=2}$ Dhyan do ${\lambda }_1+{\lambda }_2=0$ — mass automatically conserved hai, exactly jaisa L3.1 ne predict kiya tha.

KYA: $L^{\prime }=\rho V_{\infty }\Gamma$ apply karo. KYUN: ideal flow mein lift poori tarah circulation se set hoti hai; yeh formula vortex strength (jise Kutta condition ne fix kiya) se ek force tak ka bridge hai. $L^{\prime }=\rho V_{\infty }\Gamma =1.225\times 50\times 12=735\text{N/m}$ Source panels ne sirf body ki thickness shape ki aur flow tangency enforce ki; unhe $\sum {\lambda }_j=0$ milta hai aur koi circulation produce nahi hoti, isliye $L^{\prime }$ mein exactly $0$ add hota hai. Saari lift vortex part se aayi.

$C_p=1-(V_t/V_{\infty }{)}^2$ use karke:

• Point 1: $V_t=0\Rightarrow C_p=1-0=1.000$ → stagnation point (flow ruka, sabse zyada pressure).
• Point 2: $V_t=65\Rightarrow C_p=1-(65/50{)}^2=1-1.69=-0.690$ → suction (tez flow, kam pressure — typical upper surface).
• Point 3: $V_t=40\Rightarrow C_p=1-(40/50{)}^2=1-0.64=0.360$ → mild overpressure (freestream se dheema — typical lower surface aage ki taraf). Yeh pattern (upar suction, neeche overpressure) hi surface ke around integrate karne par net lift produce karta hai.

Cost factor: cost $\propto N^3$, aur $N$ $4\times$ badha, toh $\text{factor}=4^3=64.$ Solve 64 times zyada expensive ho jaata hai. Accuracy plateau kyun hoti hai: panel method inviscid Laplace problem solve karta hai. Jab mesh geometry resolve karne ke liye fine enough ho jaata hai, toh remaining error discretisation error nahi hoti — yeh model error hai: wo physics jo tumne chhor di (viscosity, boundary layer, separation, compressibility). Koi bhi panels in cheezein recover nahi kar sakti. Aage badhne ke liye tumhe Navier–Stokes Equations-based Finite Volume Method CFD par switch karna hoga.

(a) Code tuta nahi hai — yeh exactly waisa behave kar raha hai jaisa ek inviscid method ko karna chahiye. 2-D inviscid, incompressible flow mein, d'Alembert's Paradox zero drag guarantee karta hai. $0$ ka prediction model ke liye sahi hai. (b) Missing physics: viscosity. Yahan real drag skin-friction drag hai surface par patti viscous layer se (dekho Boundary Layer Theory & Skin Friction Drag), saath mein koi pressure/wake drag bhi. Ise capture karne ke liye tum Navier–Stokes Equations ko ek volume mesh par Finite Volume Method aur turbulence model ke saath solve karte ho (Turbulence Modelling — RANS, LES, DNS), ya panel solution par boundary-layer correction bolt karo. Measured $C_d=0.008$ ek purely viscous effect hai jo ideal model dekh nahi sakta.

Incompressible flow mein maximum possible $C_p$ kya hai, aur kahan? ::: $C_p=1$, stagnation point par. Closed body par $\sum {\lambda }_j=0$ kyun hona chahiye? ::: Mass conservation — andar koi air create nahi hoti. Lift ko circulation se jodne wala formula? ::: $L^{\prime }=\rho V_{\infty }\Gamma$. $N$ mein dense direct panel solve ki cost scaling? ::: Lagbhag $N^3$. Inviscid airfoil zero drag kyun dikhata hai? ::: d'Alembert's paradox — koi viscosity nahi, koi wake nahi.