Faculty of Mathematics, University of Waterloo, Canada EFMC12, September 9-13, 2018, Vienna, Austria
Problem description and background Conformal mapping Boundary conditions Rescaled equations Asymptotic expansion Fourier series decomposition
Problem description and background The unsteady, laminar, two dimensional flow of a viscous incompressible fluid past a cylinder has been investigated analytically and numerically subject to impermeability and slip conditions for small to moderately large Reynolds numbers Two geometries were considered: the circular cylinder and an inclined elliptic cylinder x y a U0 Figure 1: The flow setup for uniform right-to-left flow x y a b α U0
Problem description and background Some background information: There has been a lot work published for the no-slip case while very little for the slip case The no-slip condition is known to fail for: flows of rarified gases, flows within microfluidic / nanofluidic devices, and flows involving hydrophobic surfaces The widely used Beavers and Joseph [1967] semi-empirical slip condition was implemented in this study
Conformal mapping Boundary conditions In dimensionless form and in Cartesian coordinates the governing Navier-Stokes equations can be compactly formulated in terms of the stream function, ψ, and vorticity, ω: 2 ψ x 2 + 2 ψ y 2 = ζ ζ t = ψ ζ y x ψ ζ x y + 2 ( 2 ) ζ R x 2 + 2 ζ y 2 where R is the Reynolds number
Conformal mapping Boundary conditions Introduce the conformal transformation x + iy = H(ξ + iθ) which transforms the infinite region exterior to the cylinder to the semi-infinite rectangular strip ξ 0, 0 θ 2π The governing equations become: 2 ψ ξ 2 + 2 ψ θ 2 = M2 ζ M 2 ζ t = ψ ζ θ ξ ψ ζ ξ θ + 2 R For the circular cylinder: ( 2 ζ ξ 2 + 2 ζ θ 2 H(ξ + iθ) = exp(ξ + iθ), M 2 = exp(2ξ) while for the elliptic cylinder: H(ξ + iθ) = cosh[(ξ + ξ 0 ) + iθ], M 2 = 1 2 [cosh[2(ξ + ξ 0)] cos(2θ) where tanh ξ 0 = r with r denoting the aspect ratio )
Conformal mapping Boundary conditions The conformal transformation for the elliptic cylinder is illustrated below: CHAPTER 2 THE GOVERNING EQUATIONS 16 y x x + iy = cosh [(ξ + ξ 0 ) + iθ] 2π θ 0 ξ Figure 21: The conformal transformation where tanh ξ 0 = r, andserge r = D Alessio a is the ratio of the semi-minor to semi-major axes of b
Conformal mapping Boundary conditions The velocity components (u, v) can be obtained using u = 1 M ψ θ, v = 1 ψ M ξ and the vorticity is related to these velocity components through ζ = 1 [ M 2 ξ (Mv) ] θ (Mu) The surface boundary conditions include the impermeability and Navier-slip conditions given by u = 0, v = β v ξ at ξ = 0 where β denotes the slip length
Conformal mapping Boundary conditions In terms of ψ and ζ these conditions become ( ) ψ = 0, ψ ξ = βm0 4 M0 2 + β 2 sinh(2ξ ζ at ξ = 0 0) In addition, we have the periodicity conditions ψ(ξ, θ, t) = ψ(ξ, θ + 2π, t), ζ(ξ, θ, t) = ζ(ξ, θ + 2π, t) and the far-field conditions ψ e ξ sin θ (circular cylinder) ψ 1 2 eξ+ξ 0 sin(θ + α) (elliptic cylinder) ζ 0 as ξ
Rescaled equations Asymptotic expansion To adequately resolve the impulsive start and early flow development we introduce the boundary-layer coordinate z and rescale the flow variables according to 8t ξ = λz, ψ = λψ, ζ = ω/λ where λ = R The governing equations transform to 1 2 ω ω M 2 +2z z2 z 2 Ψ z 2 + λ2 2 Ψ θ 2 = M2 ω +2ω = 4t ω t λ2 M 2 2 ω θ 2 4t M 2 ( Ψ ω θ z Ψ z ) ω θ
Rescaled equations Asymptotic expansion Illustration of boundary-layer coordinates expanding with time: t 1 (left) and t 2 > t 1 (right) amline plots for R = 1, 000 at t = 1, 2, 5, 10 from top to bottom, respec nd β = 1 (right)
Rescaled equations Asymptotic expansion For small times and large Reynolds numbers both λ and t will be small Based on this we can expand the flow variables in a double series in terms of λ and t First we expand Ψ and ω in a series of the form Ψ = Ψ 0 + λψ 1 + λ 2 Ψ 2 + ω = ω 0 + λω 1 + λ 2 ω 2 + and then each Ψ n, ω n, n = 0, 1, 2,, is further expanded in a series Ψ n (z, θ, t) = Ψ n0 (z, θ) + tψ n1 (z, θ) + ω n (z, θ, t) = ω n0 (z, θ) + tω n1 (z, θ) +
Rescaled equations Asymptotic expansion We note that when performing a double expansion the internal orders of magnitudes between the expansion parameters should be taken into account Here, λ and t will be equal when t = 8/R, and thus, for a fixed value of R the procedure is expected to be valid for times that are of order 1/R provided that R is sufficiently large The following leading-order non-zero terms in the expansions have been determined: Ψ Ψ 00 + λ Ψ 10, ω λ ω 10 + λ 2 ω 20 This approximate solution will be used to validate the numerical solution procedure
Fourier series decomposition The flow variables are expanded in a truncated Fourier series of the form: Ψ(z, θ, t) = F 0(z, t) 2 ω(z, θ, t) = G 0(z, t) 2 + + N [F n (z, t) cos(nθ) + f n (z, t) sin(nθ)] n=1 N [G n (z, t) cos(nθ) + g n (z, t) sin(nθ)] n=1 The resulting differential equations for the Fourier coefficients are then solved by finite differences subject to the boundary and far-field conditions The computational parameters used were: z = 10, N = 25, z = 005, t = 001, ε = 10 6
For the circular cylinder the flow is completely characterized by the Reynolds number, R, and the slip length, β For the no-slip case (β = 0) comparisons in the drag coefficient, C D, were made with documented results: R Reference C D 40 Present (unsteady, t = 25) 1612 Dennis & Chang [1970] (steady) 1522 Fornberg [1980] (steady) 1498 D Alessio & Dennis [1994] (steady) 1443 100 Present (unsteady, t = 25) 1195 Dennis & Chang [1970] (steady) 1056 Fornberg [1980] (steady) 1058 D Alessio & Dennis [1994] (steady) 1077
ζ 0 5 0 t = 1 Comparison in surface vorticity distribution for R = 1, 000 and β = 05 Numerical - solid line Analytical - dashed line ζ 0 ζ 0 5 0 50 100 150 200 250 300 350 θ 5 0 5 0 50 100 150 200 250 300 350 θ 5 0 t = 05 t = 01 5 0 50 100 150 200 250 300 350 θ
Streamline plots at t = 15 for R = 500 and β = 0, 01, 05, 1 from top to bottom, respectively Figure 1: Streamline plots at t = 15 for R = 500 and β = 0, 01, 05, 1 from top to bottom, respectively
15 β = 0 β = 01 β = 05 β = 1 Time variation of C D for R = 500 and β = 0, 01, 05, 1 C D 1 05 0 0 5 10 15 t
0 02 04 The distribution of the pressure coefficient, P, at t = 15 for R = 500 and β = 0, 01, 05, 1 P * 06 08 1 12 14 16 18 β = 0 β = 01 β = 05 β = 1 2 0 50 100 150 200 250 300 350 θ
Streamline plots for R = 1, 000 at t = 1, 2, 5, 10 from top to bottom, respectively, with β = 0 (left) and β = 1 (right) Figure 1: Streamline plots for R = 1, 000 at t = 1, 2, 5, 10 from top to bottom, respe β = 0 (left) and β = 1 (right)
For the elliptic cylinder the flow is completely characterized by the Reynolds number, R, the slip length, β, the inclination, α, and the aspect ratio, r For the no-slip case (β = 0) comparisons in the drag and lift coefficients (C D, C L ) were made with documented results for R = 20 and r = 02: Dennis & Young D Alessio & Dennis Present [2003] (steady) [1994] (steady) (unsteady, t = 10) α C D C L C D C L C D C L 20 1296 0741 1305 0751 1382 0737 40 1602 0947 1620 0949 1786 0985 60 1911 0706 1931 0706 2228 0748
1 Comparison in C D, C L between present (solid line) and Staniforth [1972] (dashed line) no-slip results for the case R = 6, 250, r = 06 and α = 15 C D, C L 08 06 04 02 C D C L 0 0 02 04 06 08 1 t
Comparison in surface vorticity distributions for the case R = 1, 000, β = 05, α = 45 and r = 05 Numerical - solid line Analytical - dashed line ζ 0 ζ 0 ζ 0 20 0 20 0 50 100 150 200 250 300 350 θ 20 0 20 0 50 100 150 200 250 300 350 θ 20 0 t = 1 t = 05 t = 01 20 0 50 100 150 200 250 300 350 θ
Streamline plots for R = 500, r = 05, α = 45 and β = 0 at selected times t = 065, 075, 1, 3, 5, 9, 10 from top to bottom, respectively Figure 1: Streamline plots for R = 500, r = 05, α = 45 and β = 0 at selected times Serget = D Alessio 065, 075, 1, 3, 5, 9, 10 Flow from toppast bottom, a slippery respectively cylinder
25 2 Time variation in C D, C L for the case R = 500, r = 05, α = 45 and β = 0 C D, C L 15 1 05 0 C D C L 05 0 5 10 15 t
3 25 β = 0 β = 025 β = 05 Time variation in C D for the cases R = 500, r = 05, α = 45 and β = 0, 025, 05 C D 2 15 1 05 0 0 5 10 15 t
25 2 Time variation in C L for the cases R = 500, r = 05, α = 45 and β = 0, 025, 05 C L 15 1 05 0 β = 0 β = 025 β = 05 05 0 5 10 15 t
Streamline plots for R = 500, r = 05 and α = 45 at t = 3, 5, 7, 9, 10, 15 from top to bottom, respectively, with β = 025 (left) and β = 05 (right) Figure 1: Streamline plots for R = 500, r = 05, α = 45 at t = 3, 5, 7, 9, 10, 15 from top to bottom, respectively with β = 025 (left) and β = 05 (right)
Surface vorticity distributions at t = 15 for R = 500, r = 05 and α = 45 with β = 0, 025, 05 ζ 0 60 50 40 30 20 10 0 10 β = 0 β = 025 β = 05 20 30 0 50 100 150 200 250 300 350 θ
Slip flow past a cylinder was investigated analytically and numerically Circular and elliptic cylinders were considered over a small to moderately large Reynolds number range Excellent agreement between the analytical and numerical solutions was found Good agreement with previous studies for the no-slip case was also found The slip condition was observed to suppress flow separation and vortex shedding The key finding is a reduction in drag when compared to the corresponding no-slip case For more details see the papers: Acta Mechanica, 229, 3375-3392, 2018 Acta Mechanica, 229, 3415-3436, 2018