Please wait...
uu*=1κln(yu*ν)+Bthe fraction with numerator u and denominator u sub * end-sub end-fraction equals the fraction with numerator 1 and denominator kappa end-fraction l n open paren the fraction with numerator y u sub * end-sub and denominator nu end-fraction close paren plus cap B is the friction velocity. (von Kármán constant). (constant for smooth walls).
) conditions. This allows the velocity field to be defined by a scalar velocity potential ( ), satisfying Laplace's equation: Problem: Flow Past a Rotating Cylinder (The Magnus Effect) A uniform free-stream flow with velocity U∞cap U sub infinity end-sub flows past an infinitely long cylinder of radius . The cylinder rotates with a circulation strength Γcap gamma Determine the stream function
The flow is modeled by superimposing three elementary potential flows: a uniform stream, a doublet (to represent the geometry of the solid cylinder), and a line vortex (to represent circulation). advanced fluid mechanics problems and solutions
Superimpose the complex potentials for uniform flow, a doublet (to represent the cylinder geometry), and a vortex (to represent circulation):
to convert the partial differential equations into an ordinary differential equation: ) conditions
The stagnation point lifts off the surface and moves into the fluid core. No stagnation points exist on the cylinder wall. Key Mathematical Reference Summary Flow Regime Governing Equation Primary Solution Methodology Navier-Stokes ( -direction simplification) Direct Integration via Boundary Conditions Boundary Layer Flow Prandtl Boundary Layer Equations Similarity Transformations (Blasius Variable) Inviscid/Irrotational Flow Laplace Equation ( Complex Potential Superposition
L=−∫02πp(θ)sinθ(Rdθ)cap L equals negative integral from 0 to 2 pi of p open paren theta close paren sine theta space open paren cap R space d theta close paren Substitute into the integral. Constant and symmetric terms ( p∞p sub infinity end-sub ) integrate to zero over a full cycle. The only surviving term contains Superimpose the complex potentials for uniform flow, a
A uniform stream ( U ) flows in the positive ( x )-direction. A source of strength ( m ) (volume flow rate per unit length) is located at the origin. (a) Derive the stream function ( \psi ) and velocity potential ( \phi ). (b) Find the stagnation point location. (c) Determine the width of the half-body far downstream (i.e., the asymptotic half-width).
Insight: This profile shows that turbulent velocity changes rapidly near the wall and flattens out towards the center, differing significantly from the parabolic profile of laminar flow. Problem C: Normal Shock Waves (Compressible Flow) An supersonic flow (