L=∫02π(12ρ2U∞ΓπRsinθ)sinθRdθcap L equals integral from 0 to 2 pi of open paren one-half rho the fraction with numerator 2 cap U sub infinity end-sub cap gamma and denominator pi cap R end-fraction sine theta close paren sine theta space cap R space d theta Simplify the integrands:
Advanced Fluid Mechanics: Problems and Solutions for Engineers and Physicists
Uniform Flow U_∞ ──> ┌─────────┐ ──>│ ┌───┐ │──> │ ↻│ │ │ ──>│ └───┘ │──> └─────────┘ Circulation (Γ) Step 1: Superimpose Elementary Flow Fields advanced fluid mechanics problems and solutions
ψuniform=U∞rsinθ,ϕuniform=U∞rcosθpsi sub uniform end-sub equals cap U sub infinity end-sub r sine theta comma space phi sub uniform end-sub equals cap U sub infinity end-sub r cosine theta A line source at the origin emitting a volume flow rate per unit length has:
p2=100[2(1.4)(1.503)2−0.41.4+1]=100[5.92522.4]≈100⋅2.469=246.9 kPap sub 2 equals 100 open bracket the fraction with numerator 2 open paren 1.4 close paren open paren 1.503 close paren squared minus 0.4 and denominator 1.4 plus 1 end-fraction close bracket equals 100 open bracket 5.9252 over 2.4 end-fraction close bracket is approximately equal to 100 center dot 2.469 equals 246.9 kPa Final Answer Shock angle: Downstream Mach number: Downstream pressure: 4. Potential Flow Theory: Rankine Half-Body Problem Statement The circulation increases velocity on one side and
Combine three elementary flows: Uniform flow , Doublet (to create the cylinder shape), and a Point Vortex (to add rotation). Stream Function ( ): In polar coordinates:
), the inertial terms in the Navier-Stokes equations can be neglected (Stokes flow / creeping flow). Potential flow describes an ideal
Potential flow describes an ideal, inviscid, and irrotational flow. A typical problem involves the stream function from a source and finding streamlines in a spiral vortex.
This model explains the Magnus Effect . The circulation increases velocity on one side and decreases it on the other, creating a pressure difference and resulting in lift ( ), known as the Kutta-Joukowski theorem . 3. Boundary Layer Theory & Separation
ψtotal=U∞rsinθ−κrsinθ−Γ2πlnrpsi sub t o t a l end-sub equals cap U sub infinity end-sub r sine theta minus the fraction with numerator kappa and denominator r end-fraction sine theta minus the fraction with numerator cap gamma and denominator 2 pi end-fraction l n r To make the cylinder surface ( ) a streamline where