3.1.30 · D1Computational aerodynamics — panel method (intro), CFD overv
Wing ke paas se behti hui hawa un equations ko follow karti hai jo haath se solve karna bahut mushkil hai, isliye hum problem ko chhote-chhote tukdon mein kaatke computer ko ek bada set of simple equations solve karne dete hain. Parent note mein jo kuch bhi hai — panels, sources, influence coefficients, meshes — sab kuch us "kaatne" ka alag-alag tarika hai.
Is page par kuch bhi assumed nahi hai. Agar parent note mein koi symbol, word, ya picture use hua hai, toh usse pehle yahan build karte hain. Upar se neeche padho; har block agle ko earn karta hai.
Kisi bhi symbol se pehle, yeh image apne dimaag mein rakho: hawa mein har point par ek chhota sa arrow hai jo bata raha hai ki hawa wahan kitni tez aur kis direction mein chal rahi hai. Space mein arrows ki ek carpet ko vector field kehte hain. Poora subject yahi hai: wing ke around us arrow carpet ko dhundo.
Definition Velocity fieldV V direction bhi hai, sirf size nahi". 2-D mein iske do numbers hote hain: kitni tez daayein (V x V y Picture: upar wali arrows ki carpet.
Kyun zaroorat hai: lift aur drag is baat se aate hain ki yeh arrows body ke around kaise mod khate hain.
x , y P x y point ko P ( x , y ) Picture: usual graph paper — right aur up.
Definition Freestream speedV ∞ α V ∞ bahut door ki hawa ki speed, jahan body ne usse abhi disturb nahi kiya. Chhota ∞ α angle of attack : aane wali hawa aur wing ke beech ki tilt.
Picture: left se aa rahe seedhe parallel arrows, α Kyun: yeh "input" wind hai; wing ka kaam ise modhna hai.
Parent ne kaha tha "incompressible ⇒ ∇ ⋅ V = 0 ⇒ V = ∇ ϕ ∇ ⋅
∇ ∇ koi number nahi hai. Yeh ek recipe hai jiska matlab hai "har direction mein slopes lo". Ise ek chhoti machine samjho jo measure karti hai ki koi quantity x y
∇ ⋅ V Bahar jaane wale arrows count karo minus andar aane wale arrows. Agar zyada nikle to zyada aaye, toh box fluid ka source hai — divergence positive hai.
∇ ⋅ V = ∂ x ∂ V x + ∂ y ∂ V y Incompressible matlab fluid squeeze nahi ho sakta, toh har box mein andar = bahar: ∇ ⋅ V = 0 Picture: figure mein left box — balanced arrows.
Definition Partial-slope symbol∂ ∂ x ∂ V x y x V x ∂ "slope jabki baaki variables frozen hain."
Kyun zaroorat hai: velocity point to point change hoti hai; slope change batata hai.
Definition Irrotational / curlFlow mein ek chhota sa paddle-wheel daalo. Agar woh ghoomta hai, toh flow mein wahan rotation (curl) hai. Agar woh sirf drift karta hai bina ghoomey, toh flow irrotational hai.
Picture: figure mein right box — wheel drift karta hai lekin ghoomta nahi.
Kyun: irrotational flow hame allow karta hai ki poori arrow-carpet ko ek single scalar "height map" se replace kar saken — potential ϕ
Yeh master trick hai. Har point par do numbers (V x , V y ek number ϕ slopes hote hain.
ϕ V = ∇ ϕ ϕ scalar (ek number), jaise kisi pahaad ki height.
V = ∇ ϕ sabse steep downhill direction mein point karta hai, aur uski length steepness hai.
Picture: ek pahaad ki contour lines; arrows unhe right angles par cross karte hain, sabse lamba jahan lines bheed jaati hain.
Kyun: ek number per point, do se kaafi aasaan hai — aur yeh automatically guarantee karta hai ki flow irrotational hai (pahaad mein koi built-in swirl nahi hoti).
Intuition Kyun "potential" apni jagah earn karta haiEk single height map ϕ ϕ ϕ
§2 ke do facts combine karo:
incompressible: ∇ ⋅ V = 0
irrotational: V = ∇ ϕ
Doosre ko pehle mein substitute karo:
∇ ⋅ ( ∇ ϕ ) = ∇ 2 ϕ = 0
∇ 2 ϕ ∇ 2 ϕ = ∂ x 2 ∂ 2 ϕ + ∂ y 2 ∂ 2 ϕ neighbours ke average se upar hai ya neeche . Ise zero karne ka matlab hai: koi bhi point bump ya dip nahi ho sakta — surface ek stretched drumhead ki tarah smooth hai.
Picture: ek wire loop par kheenchi soap film; har point apne aas-paas waalon ka average hai.
Kyun: Laplace's Equation & Potential Flow dekho. Yeh ideal flow ki the governing equation hai.
Linear do valid solutions ko add karne se ek aur valid solution milta hai.
Kyunki hum solutions add kar sakte hain, hum simple waalon ka ek box rakhte hain. Har ek ek valid ϕ
r θ r θ Picture: rings (r θ
Definition Source strengthλ λ fluid ka total volume jo per unit time pumped out hota hai , sab directions mein barabar spread hoke.
Picture: centre par ek garden sprinkler; bada λ Kyun: wing surface par hum har panel ka λ wall se skim kare aur kabhī usme se na guzre.
Γ γ Picture: ek whirlpool; Γ Kyun: lift ko swirl chahiye. Kutta–Joukowski Theorem & Circulation ke zariye, lift per length L ′ = ρ V ∞ Γ Γ = 0
ln r 2 π ln r e r 2 π 36 0 ∘ Kyun yeh tool: ln r unique radially-symmetric solution hai — koi doosra simple function aise nahi hai jo barabar spread bhi kare aur ∇ 2 ϕ = 0
Definition Panel aur control pointPanel = curved surface ke ek tukde ko approximate karne wala ek chhota seedha segment. Uska control point = uska midpoint, jahan hum rule enforce karte hain.
Picture: N Kyun: haath se smooth strength λ ( s ) N λ 1 … λ N
n ^ n ^ seedha point karta hua, uske perpendicular. Tangent uske along point karta hai.
Picture: har toothpick par, ek arrow bahar nikla hua, ek flat leta hua.
Kyun: wall condition wall ke through flow ke baare mein hai, matlab normal direction mein.
V ⋅ n ^ V ⋅ n ^ = ∣ V ∣ cos θ n ^ V Kyun yeh tool: hame sirf wall-ke-through component chahiye, aur dot product exactly "ek arrow kitna doosre ke along point karta hai" wali machine hai.
β i I ij β i i I ij i par feel ki gayi normal velocity, jo panel j se caused hoti hai. Subscripts padho "effect at i from j ".
Kyun: yeh numbers woh matrix fill karte hain jise computer invert karta hai.
Definition Pressure coefficientC p C p = 1 − ( V ∞ V t ) 2 V t tangential (sliding-along) speed. C p dimensionless pressure hai: jahan hawa speed up hoti hai (V t > V ∞ C p Bernoulli's Equation se aata hai (tez hawa = low pressure).
Kyun: surface ke around C p
λ Lagta sahi hai: strength power jaisi lagti hai. Pakad: lift Γ λ Fix: yaad karo L ′ = ρ V ∞ Γ
Definition Navier–Stokes, viscosityμ ρ stickiness hoti hai (viscosity μ squeeze kiya ja sakta hai (density ρ Navier–Stokes Equations ban jaati hai, jo nonlinear hain — tum solutions add nahi kar sakte ab, toh poora brick trick khatam aur humein volume mesh par Finite Volume Method chahiye. Yeh bhi dekho Boundary Layer Theory & Skin Friction Drag , Turbulence Modelling — RANS, LES, DNS , aur kyun inviscid theory zero drag predict karta hai: d'Alembert's Paradox .
Nonlinear solutions add nahi kiye ja sakte; superposition fail ho jaata hai.
divergence = 0 incompressible
linear so we may add solutions
elementary bricks source vortex stream
wall condition V dot n = 0
panels and control points
linear system A lambda = b
add viscosity and density
Navier Stokes then CFD mesh
V isme direction aur size dono hain — yeh ek vector hai, space mein har point par ek arrow.
∇ ⋅ V = 0 har chhote box mein utna hi fluid andar aata hai jitna bahar jaata hai — incompressible.
Irrotational ka matlab paddle-wheel picture se kya hai? ek chhota wheel drift karta hai lekin kabhi ghoomta nahi.
ϕ V = ∇ ϕ ek ek-number height map; velocity uski steepest slope hai.
∇ 2 ϕ = 0 koi bhi point apne neighbours se upar bump nahi kar sakta — ek smooth drumhead surface.
Linearity poora secret kyun hai? valid solutions add kiye ja sakte hain, isliye hum simple bricks ko complex flows mein stack karte hain.
Source strength λ per unit time pumped out fluid ka volume, barabar spread hoke.
Circulation Γ λ swirl, isliye lift, L ′ = ρ V ∞ Γ
Panel strengths kis single boundary condition se set hoti hain? zero normal velocity,
V ⋅ n ^ = 0 .
n ^ seedha wall ke bahar; dot wall-ke-through part pick karta hai use khatam karne ke liye.
C p dimensionless pressure 1 − ( V t / V ∞ ) 2
Viscosity aur density add karne se kya tootta hai? equations nonlinear ho jaate hain — solutions add nahi — isliye hume CFD chahiye.
