Ship Resistance Simulations with OpenFOAM

Transcription

1 Ship Resistance Simulations with OpenFOAM Kevin Maki University of Michigan Ann Arbor, MI USA 6th OpenFOAM Workshop June 2011 The Pennsylvania State University State College, PA, USA Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 1 / 22

2 Introduction simulate flow around ship moving steadily in calm water high Reynolds number 106 model scale, 109 full scale longest wave is 2πF 2 L in deep water the Kelvin angle is approximately 19 deg seek primarily the force on the body, also position of body Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 2 / 22

3 Outline 1 Introduction 2 Governing Equations Volume of Fluid Momentum and Continuity Equations 3 Numerics interfoam solver 4 Solution Settings 5 Wigley Hull Tutorial Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 3 / 22

4 Interface tracking versus interface capturing OpenFOAM has both interface capturing and interface tracking solvers most ship hydrodynamics solvers use interface capturing, volume-of-fluid, level set, or a combination, to solve ship hydrodynamics problems. Can a ship move (at model or full scale) and not generate a breaking wave? Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 4 / 22

5 Volume of Fluid use scalar indicator function to represent the phase of the fluid in each cell (was γ in versions 1.5, is α in versions 1.6). α = α(x, t) (1) µ(x, t) = µ water α + µ air (1 α) (2) ρ(x, t) = ρ water α + ρ air (1 α) (3) The density and viscosity are material properties of the fluids. Dα Dt = 0 (4) α + u α t = 0 (5) α + uα t = 0 (6) Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 5 / 22

6 Volume-of-fluid with compression the α function transitions from 1 to 0 over an infinitesimal thickness. This leads to difficulty in approximating the gradient of α, and results in smearing of the interface. One remedy, is to use a modified governing equation. The modification should return solutions of the original equation for the time evolution of the interface, but help by keeping the interface crisp. α + uα + wα = 0 t u is the physical velocity field, and w is an artificial velocity field that is directed normal to and towards the interface. α t + uα + w(α(1 α)) = 0 the user can specify the relative magnitude of the artificial velocity (using calpha) Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 6 / 22

7 Momentum, dynamic pressure Full Reynolds-averaged momentum equations for the velocity U and pressure P in a fluid with density ρ and dynamic viscosity µ ρu t [ ] + UU = P + ρg + (µ + µ t )( U + U ) Express the pressure in terms of a hydrostatic component, and the remainder or that due to dynamic or non-zero velocity p P = ρg x }{{} + p }{{} hydrostatic dynamic Governing equation in terms of dynamic pressure ρu t [ ] + UU = p g x ρ + (µ + µ t )( U + U ) Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 7 / 22

8 Momentum, viscous stress See Henrik Rusche s Thesis, pg 156 [ ] µ eff ( U + U ) = (µ eff U) + (µ eff U ) Final form of the momentum equation: ρu t = (µ eff U) + U µ eff + µ eff ( U) = (µ eff U) + U µ eff + UU = p g x ρ + (µ eff U) + U µ eff Note, ρ is zero away from the interface, and VERY large along the interface. Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 8 / 22

9 Boundary conditions Body U = 0 p/ n = 0 α/ n = 0 Inlet Centerplane: symmetryplane Top U/ n = 0 p = 0 α = 0 U = U p/ n = 0 { 1 if z < 0 α = 0 otherwise Outlet U/ n = 0 p = 0 α/ n = 0 Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 9 / 22

10 simulation. interfoam VOF for interface capturing PISO for pressure velocity coupling unknowns: z p_rgh p dynamic pressure p P y total pressure (P = p + ρg x) alpha1 α Space domain volume x fraction U U velocity vector phi S f U f velocity flux rhophi S f ρ f U f mass flux t t gh g x P hydrostatictime pressure domain over density at cell center ghf g x f Figurehydrostatic 2.1: Discretisationpressure of the solutionover domain density at face center f P S f d N Figure 2.2: Parameters in finite volume discretisation Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 10 / 22

11 interfoam algorithm 1 solve transport equation for volume fraction 2 generate linear systems for momentum components U, V, W, using convection and viscous terms only 3 (optional) solve for momentum components using old values of pressure gradient and density gradient 4 form the pressure Poisson equation, and solve (may loop over this for non-orthogonal correction update) 5 update velocity with pressure gradient 6 update face flux with pressure contribution 7 update turbulence quantities PISO loop over steps 4-6. Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 11 / 22

12 Momentum prediction Total momentum equation: ρu + UU = p g x ρ + (µ eff U) + U µ eff t in prediction, form linear systems using convection and viscous terms only: ρu t + UU (µ eff U) U µ eff = 0 [A] U {U} = {b U } [A] V {V } = {b V } [A] W {W } = {b W } if you solve for momentum prediction: {U} = [A] 1 U [{b U} p i g x ρ i] Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 12 / 22

13 Pressure correction start with semi-discrete momentum equation look at equation for a single cell [A] U {U} = [{b U } p i g x ρ i] a P U P + a N U N = b P p g x ρ calculate the velocity without ρ and p U P = a 1 P (b P a N U N ) interpolation of gradients is bad! (Rhie-Chow). Face flux using starred velocity φ = U f S f now the flux with the density gradient: φ = φ g x f ρ n a 1 P,f S f Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 13 / 22

14 Pressure correction, cont. use the continuity equation to find pressure that makes the velocity discretely divergence free. U = U f S f = φ = 0 φ will not satisfy continuity because it is a numerical approximation, and it does not contain the pressure gradient term. Return to the momentum equation for a single cell, and note the use of the starred velocity. U P = U a 1 p a 1 g x ρ P insert into continuity a 1 P p = φ after solving for p, then update the face flux and velocity φ = φ a 1 P,f p f U = U g x f p P Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 14 / 22

15 Time-step size Courant number in simple terms: C o = U t x For arbitrary polyhedral finite volume: C o = U S f d S f t d is the vector from pole center to neighbor center if we solve implicit equations, what is an acceptable time step size based on the Courant number? Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 15 / 22

16 PISO-settings momentumpredictor: relatively small additional expense recommended ncorrectors: this is to loop over pressure system, also known as PISO loops. For strict time accuracy, minimum of 2. Calm-water resistance, 1 should do. nnonorthogonalcorrectors: due to small time step, and use of ncorrectors, this may be set to 0 in most cases. Perhaps for initial time steps on bad grids a few may help. nalphacorr: loop over α equation. For time-dependent flows 1-2. For steady flow like calm-water resistance, 0. nalphasubcycles: this reduces the time-step size for the explicit integration of the α transport equation. As you increase the time-step size for the total system of equations, and if you need time accuracy (maybe retain stability), increase the number of sub-cycles according. calpha: the compression term in the advection of α is scaled by this parameter. Set to zero to deactivate compression. Set to 1 as a nominal value. Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 16 / 22

17 Discretization Settings time: Euler, Courant number restriction leads to small time steps, first order accuracy is fine for calm-water resistance. gradient: linear divergence: upwind to aid in convergence. vanleer is second-order away from extrema. limitedlinearv may be less diffusive than vanleer Laplacian: Gauss linear corrected. Second-order, with correction for non-orthogonal part. Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 17 / 22

18 Wigley Hull Experiments Well used test data. Body fixed and free to sink and trim. SRI 0.08 < F < < R < Item Symbol Value Unit Length L 4.0 m Beam B 0.4 m Draft T 0.25 m Wetted Surface S m 2 Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 18 / 22

19 +08$ * $ 10-97$ :$ 7066*$ Wigley Hull Computations: Table 5: Resistance coefficients Time results for experimental Integration values As it can be seen from Table 4 and 5, the numerical approximations using the CFD solver are in agreement with the experimental results. Especially the Fine grid computations for Fr# = caught 100% convergence with experimental value. 144K cell coarse grid (Pointwise) Figure 2: Comparison of Total Resistance courtesy Coefficient of Mert Results Türkol with Experimental Value for Fr# = Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 19 / 22

20 Wigley Hull Computations: Convergence courtesy of Ensign William Garland Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 20 / 22

21 Wigley Hull Computations: Full Scale 400 m Ship courtesy of Ensign William Garland Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 21 / 22

22 Hull Force Library control over quantity calculated and the write syntax F p = M p = F v = M v = pnds S (x f x o ) pnds S τ nds S (x f x o ) τ nds S column 1:time, 2-4: F p, 5-7: F p, 8-10: M v, 11-13: M v Maki (UofM) Training Session: Ship Resistance 6th OpenFOAM Workshop 22 / 22