Energy dissipating structures generated by dipole-wall collisions at high Reynolds number

Energy dissipating structures generated by dipole-wall collisions at high Reynolds number Duncan Sutherland 1 Charlie Macaskill 1 David Dritschel 2 1. School of Mathematics and Statistics University of Sydney 2. School of Mathematical Sciences, University of St. Andrews ANZIAM 2012 Sutherland (Sydney) Influence Matrix ANZIAM 2012 1 / 16

Introduction: Dipole wall collisions We are interested in a vortex dipole colliding with a wall in a viscous fluid in a periodic channel We compare energy dissipation results with a recent paper by Nguyen van yen, et al Figure: Dipole colliding with the wall at times 0.3, 0.4, 0.5 Sutherland (Sydney) Influence Matrix ANZIAM 2012 2 / 16

Introduction: Channel domain y L y /2 u = v = 0 ω t = J (ω, ψ) + ν 2 ω 2 ψ = ω N L y /2 0 ψ y = u, ψ x = v u = v = 0 J (ω, ψ) = ω ψ x y ω ψ y x L x x M Sutherland (Sydney) Influence Matrix ANZIAM 2012 3 / 16

Introduction: Energy Dissipating Structures A recent study by Nguyen van yen, Schneider and Farge has indicated the existence of energy dissipating structures as the viscosity tends to zero Penalisation method: u t + (u )u = p + ν 2 u 1 η χu { 0 inside the fluid u = 0 χ = 1 elsewhere η is the penalisation parameter, chosen to be sufficiently small to ensure the velocity is zero outside of the fluid. These equations (in vorticity form) are then solved on a double-periodic domain using a Fourier-Fourier spectral method No-slip boundary conditions are approximated Navier boundary conditions with a slip length proportional to Re 1 Sutherland (Sydney) Influence Matrix ANZIAM 2012 4 / 16

Introduction: Key Results of Nguyen van yen, et al Energy, E dissipation far from the wall is proportional to Re 1, as for unbounded flows Prandtl 1904: Energy dissipation close to the wall is proportional to Re 1/2, found to hold reasonably throughout the rebound Enstrophy, Z dissipation during the collision is proportional to Re 1, hence de dt = 2ZRe 1 tends to a constant as viscosity vanishes Sutherland (Sydney) Influence Matrix ANZIAM 2012 5 / 16

Numerical Methods: Periodic direction and timestepping The periodic boundary conditions are dealt with using a Fourier Transform. The derivatives are calculated in spectral space and the nonlinear term is calculated in physical space The time equations are discretised using the Adams-Bashforth/Crank-Nicholson method This leaves N coupled Helmholtz problems for each Fourier wavenumber of ω and ψ Sutherland (Sydney) Influence Matrix ANZIAM 2012 6 / 16

Numerical Methods: Helmholtz problems and boundary conditions d 2 ω dy 2 κ2 ω = S n,n 1 d 2 ψ dy 2 k2 ψ = ω u = v = 0 or equivalently ψ = dψ dy Coordinate transform: y = sin(η) sin π/2+1 = 0 on y = ±1 to deal with the Chebyshev grid Compact finite difference discretisation of derivative operators αf i 1 + βf i + αf i+1 = af i 1 + bf i + af i+1 This gives rise to a block tridiagonal matrix problem which may be solved in O(M) operations Sutherland (Sydney) Influence Matrix ANZIAM 2012 7 / 16

Numerical Methods: Influence Matrix An influence matrix is used to treat the boundary conditions. Note the problem is linear, hence: ω n+1 = ω n 0 + α 1 ω 1 + α 2 ω 2 ψ n+1 = ψ n 0 + α 1 ψ 1 + α 2 ψ 2 = u n+1 = u n 0 + α 1 u 1 + α 2 u 2 Equipped with the following boundary conditions: ω 1 (±1) = 0, ω 1 (1) = 1, ω 1 ( 1) = 0, ω 2 (1) = 0, ω 2 ( 1) = 1,ψ i (±1) = 0 Then choose α 1,2 such that u(±1) = 0 This gives N 2 2 matrix problems of the form: ( ) ( ) ( u1 ( 1) u 2 ( 1) α1 u0 ( 1) = u 1 (1) u 2 (1) u 0 (1) α 2 ) Sutherland (Sydney) Influence Matrix ANZIAM 2012 8 / 16

Simulations ri 2 = (x x i ) 2 + y 2 2 ) ω = ω e ( 1) (1 i r2 i r0 2 e r2 i /r2 0 i=1 L x = 2π, L y = 0.9 ω e = 300, x 1 = 0.05, x 2 = 0.05, r 0 = 0.05 Re 704 1056 1407 2111 2814 N x 1024 2048 2048 2048 3072 N y 768 1536 1536 1536 2048 The initial condition must be selected such that it is consistent with the no-slip boundary conditions. Sutherland (Sydney) Influence Matrix ANZIAM 2012 9 / 16

Differences compared to Nguyen van yen, et al Our resolution is (apparently) lower, Nguyen van yen, et al demanded scales of Re 1 be resolved. (Work to get this resolution is in progress) Their test included higher Reynolds numbers up to 7880 They took L x = 1, we use L x = 2π. (Repeating the calculations on the shorter domain is in progress) Our initial condition will be slightly different, since we insure that the IC satisfies the boundary conditions exactly Sutherland (Sydney) Influence Matrix ANZIAM 2012 10 / 16

Dipole-Wall Collision Sutherland (Sydney) Influence Matrix ANZIAM 2012 11 / 16

Dipole-Wall Collision Sutherland (Sydney) Influence Matrix ANZIAM 2012 11 / 16

Dipole-Wall Collision Sutherland (Sydney) Influence Matrix ANZIAM 2012 11 / 16

Dipole-Wall Collision Sutherland (Sydney) Influence Matrix ANZIAM 2012 11 / 16

Dipole-Wall Collision Sutherland (Sydney) Influence Matrix ANZIAM 2012 11 / 16

Comparison: Energy Scaling Results Interior energy scaling Energy Dissipation 10 2.6 10 2.8 10 3 10 3.2 E(0.2) E(0) Re 1 Wall region energy scaling Energy Dissipation 10 2.8 10 3 E(0.495) E(0.39) Re 1/2 10 3.4 10 2.8 10 3 10 3.2 10 3.4 Re 10 2.8 10 3 10 3.2 10 3.4 Re The decay results are in agreement with Nguyen van yen, et al. Sutherland (Sydney) Influence Matrix ANZIAM 2012 12 / 16

Comparison: Enstrophy Scaling Results Enstrophy Dissipation 10 2 10 1.5 Z(0.495) Z(0.39) Re 1 10 2.8 10 3 10 3.2 10 3.4 Re This is very close to linear, which implies the energy dissipation rate tends to a constant as viscosity vanishes. If Z is weakly sublinear then global energy dissipation 2ZRe 1 0 as the viscosity vanishes. Sutherland (Sydney) Influence Matrix ANZIAM 2012 13 / 16

Further work to be done Conduct the tests at increased Reynolds number Increase the resolution (also helps with the higher Re) Nguyen van yen, et al tested dissipation in certain regions and examined Lagrangian trajectories Track and visualise the local energy dissipation Investigate the differences between the events at different Reynolds numbers Sutherland (Sydney) Influence Matrix ANZIAM 2012 14 / 16

References and Acknowledgements Nguyen van yen, R, Farge, M, and Schneider, K, (2011) Energy Dissipating Structures Produced by Walls in Two-Dimensional Flows at Vanishing Viscosity. PRL 106, 184502 Kramer, W. PhD thesis 2007 Schneider, K, (2005), Numerical simulation of the transient flow behaviours in chemical reactors using a penalisation method, Computers & Fluids, 34, pp 1223-1238. Clercx, H. J. H, and van Heijst G. J. F, (2002) Dissipation of kinetic energy in two-dimensional bounded flows, Physical Review E, 65, 066305 Lele, S. K. (1992), Compact finite difference schemes with spectral-like resolution. J. Comp. Phys. 103, pp 16-42. Kraichnan, R, H and Montgomery D, (1980) Two Dimensional Turbulence. Rep. Prog. Phys. 43, pp 548-619 Sutherland (Sydney) Influence Matrix ANZIAM 2012 15 / 16

Questions Sutherland (Sydney) Influence Matrix ANZIAM 2012 16 / 16