Complex functions in the theory of 2D flow

Transcription

1 Complex functions in the theory of D flow Martin Scholtz Institute of Theoretical Physics Charles University in Prague scholtz@utf.mff.cuni.cz Faculty of Transportation Sciences Czech Technical University in Prague May st, Ondřejov

2 Overview

3 Definition of D flow Two-dimensionality Layer of the fluid is thick enough to neglect boundary effects The flow can be foliated The velocity field in each slice is the same v = (v x, v y, ) Stationarity t v = Inviscid fluid Irrotational velocity field v = Incompressibility v =

4 Potential Irrotationality Existence of the potential φ v = In components v x = x φ v = φ v y = y Lines of constant potential are orthogonal to velocity dφ = ( i φ)dx i = v i dx i = v dx φ = const. v

5 Stream function Incompressibility Existence of the vector potential ψ v = v = ψ For D flow ψ = (,, ψ), ψ stream function In components v x = y ψ v y = x ψ (v i = ɛ ij j ψ) ψ is constant along the stream lines dψ = ( i ψ)dx i = ɛ ij v j v i dt = v ψ = const.

6 ΣΣ (Summa summarum) Stream function v x = y ψ v y = x ψ Potential v x = x φ v y = y φ Both satisfy the Laplace equation Cauchy-Riemann relations ψ = φ = x φ = y ψ y φ = x ψ

7 We introduce the complex potential F = φ + i ψ Complex coordinates z = x + i y F is holomorphic function Related to velocity by z = x i y Cauchy-Riemann conditions v = F z F z =

8 Circulation and flux Circulation Γ = v dr = γ γ v x dx + v y dy Flux Q = v n dr = v x dy v y dx γ γ Observation v dz = v x dx + v y dy + i(v x dy v y dx) Complex flux Γ + i Q = γ v dz

9 Recapitulation Description of the flow in complex terms coordinates z = x + i y z = x i y velocity v = v x + i v y v = v x i v y potential v = z F z F = flux Γ + i Q = v dz γ Notice: v dz = df = γ γ unless there are singularities in the region surrounded by γ

10 Any holomorphic function F generates some flow Uniform flow F (z) = C z = ū z = u e iθ z u θ No singularities Γ + i Q =

11 Sources and sinks Potential of the form F (z) = k log z, k R Velocity Flux Hence: v = z F = k z = k x + i y x + y Γ + i Q = k Γ = γ dz z = π i k Q = π k

12 Sources and sinks I Martin Scholtz, Weber-Vesel y Potential F (z) = Q log z π I I source: Q > sink: Q <

13 Dipole Source at the point A with the flux Q > Sink at the point B with the flux Q A ε/ α B ε/ ɛ distance of A and B Potential F (z) = Q π log z ɛ eiα z + ɛ eiα In the limit ɛ, Q ɛ = M = constant F (z) = M π e iα z

14 Dipole Martin Scholtz, Weber-Vesel y

15 Line vortex Potential Velocity Flux Hence: F (z) = i k log z, k R v = i k z = k (y i x) r Γ + i Q = i k Q = γ dz z = π k Γ = π k

16 Line vortex I Potential F (z) = Martin Scholtz, Weber-Vesel y Γ log z πi

17 Potential F (z) = v z + (v u) R z + Γ π i log z v complex velocity at infinity R real radius of the u complex velocity on the boundary o the Γ real circulation about the

18 Martin Scholtz, Weber-Vesel y Examples

19 Potential of a line vortex located at z F = Corresponding stream function Γ π i log(z z ) ψ = I(F ) = Γ 4π log [ (x x ) + (y y ) ] Continuously distributed line vortices ψ(r) = γ(r ) log ( r r ) dr 4π C γ(r) vortex density Velocity v i (r) = ɛ ij π C γ(r ) x j x j r r dr

20 Linear vortex density Along the vortex panel we choose γ(t) = α + β t, t [, ] α =, β =

21 Non-penetration condition Motivation: modelling the flow past the Velocity cannot penetrate the boundary of the v n v t We impose condition v n = v t

22 Mathematica package Arbitrary shape of the Discretization of the N segments Each segment is modelled by vortex panel with parameters α i, β i α i + β i = α i+ for i < N α N + β N = α (Joukowski-Kutta condition) Solving the no-penetration condition Calculation of the pressure, force,... ρ v + P = ρ v t + P c p = P P ρ v = ( ) vt v

23 Clark Y profile Black arrow net force exerted on the profile Red dashed line dimensionless pressure coefficients c p below and above the profile

24 The END Thank you for attention