Applied Computational Fluid Dynamics Techniques - Wiley Episode 2 Part 8 docx

422 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES where d w (x) is given by the Hermitian polynomial (Löhner (2001)) d w = 1 −3ξ 2 + 2ξ 3 , (19.13) and ξ is defined in (19.11). The final semi-discrete scheme takes the form M l β ,t = r =r a (u, v, β) + r s (d w ,w)+ r d (d h ,β), (19.14) where the subscripts a, s and d stand for advection, source and damping. This system of ODE’s is integrated in time using explicit time-marching schemes, e.g. a standard five-stage Runge–Kutta scheme. 19.1.2. OVERALL SCHEME One complete timestep consists of the following steps: - given the boundary conditions for the pressure , update the solution in the 3-D fluid mesh (velocities, pressures, turbulence variables, etc.); - extract the velocity vector v =(u, v, w) at the free surface and transfer it to the 2-D free surface module; - given the velocity field, update the free surface β; - transfer back the new free surface β to the 3-D fluid mesh, and impose new boundary conditions for the pressure . For steady-state applications, the fluid and free surface domains are updated using local timesteps. This allows some room for variants that may converge faster to the final solution, e.g. n steps of the fluid followed by m steps of the free surface, complete convergence of the free surface between fluid updates, etc. Empirical evidence (Löhner et al. (1998, 1999a,c)) indicates that most of these variants prove unstable, or do not accelerate convergence measurably. For steady-state applications it was found that an equivalent ‘time-interval’ ratio between the fluid and the free surface of 1:8 yielded the fastest convergence (e.g. a Courant number of C f = 0.25 for the fluid and C s = 2.0 for the free surface). 19.1.3. MESH UPDATE Schemes that work with structured grids (e.g. Hino (1989,1997), Hino et al. (1993), Farmer et al. (1993), Martinelli and Farmer (1994), Cowles and Martinelli (1996)) march the solution in time until a steady state is reached. At each timestep, a volume update is followed by a free surface update. The repositioning of points at each timestep implies a complete recalculation of geometrical parameters, as well as interrogation of the CAD information defining the surface. For general unstructured grids, this can lead to a doubling of CPU requirements.For this reason, when solving steady-state problems,it is advisable not to move the grid at each timestep, but only change the pressure boundary condition after each update of the free surface β. The mesh is updated every 100 to 250 timesteps, thereby minimizing the costs associated with geometry recalculations and grid repositioning along surfaces. One can also observe that this strategy has the advantage of not moving the mesh unduly at the beginning of a run, where large wave amplitudes may be present. One mesh update consists of the following steps. TREATMENT OF FREE SURFACES 423 - Obtain the new elevation for the points on the free surface from β. This only results in a vertical (z-direction) displacement field d  for the boundary points. - Apply the proper boundary conditions for the points on the waterline. This results in an additional horizontal (x,y-direction) displacement field for the points on the water line. - Smooth the displacement field in order to avoid mesh distortion. This may be accom- plished with any of the techniques described in Chapter 12. - Interrogate the CAD data to reposition the points on the hull. Denoting by d ∗ , n and t the predicted displacement of each point, surface normals and tangential directions, the boundary conditions for the mesh movement are as follows (see Figure 19.2). x z y a d b b b e c c f g a-f Hull, On Surface Patch c-f Hull, End-Point e-f Hull, Water Line End-Point In Plane of Symmetry g -f Water Surface Point in Plane of S y mmetr y b-f Hull, Line Point d-f Hull, Water Line Point f-f Water Surface Point Figure 19.2. Boundary conditions for mesh movement (a) Hull, on surface patch. The movement of these points has to be along the surface, i.e. the normal component of d ∗ is removed: d = d ∗ − (d ∗ · n) n. (19.15) (b) Hull, line point. The movement of these points has to be along the lines, resulting in a tangential boundary displacement of the form d =(d ∗ · t)t. (19.16) 424 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES (c) Hull, endpoint. No displacement is allowed for these points, i.e. d =0. (d) Hull/water line point, water line endpoint. The displacement of these points is fixed, given by the change in elevation z and the surface normal of the hull. Defining d 0 = (0, 0,z),wehave d = d 0 − (d 0 · n) n 1 − n 2 z . (19.17) (e) Hull/water line endpoint in plane of symmetry. The displacement of these points is fixed, and dictated by the tangential vector to the hull line in the symmetry plane: d = (d 0 · t)t 1 − n 2 t . (19.18) (f) Water surface points. These points start with an initial displacement d 0 , but may glide along the water surface, allowing the mesh to accommodate the displacements in the x,y-directions due to points on the hull. The normal to the water line is taken, and (19.17) is used to correct any further displacements. (g) Water surface points in plane of symmetry. As before, these points start with an initial displacement d 0 , but may glide along the water surface, remaining in the plane of symmetry, thus allowing the mesh to accommodate the displacements in the x-direction due to points on the hull. The tangential direction is obtained from the sides lying on the water surface in the plane of symmetry, and (19.18) is used to correct any further displacements. An option to restrict the movement of points completely in ‘difficult’ regions of the mesh is often employed. Regions where such an option is required are transom sterns, as well as the points lying in the half-plane given by the minimum z-value of the hull. Should negative elements arise due to surface point repositioning, they are removed and a local remeshing takes place. Naturally, these situations should be avoided as much as possible. 19.1.4. EXAMPLES FOR SURFACE FITTING We include some examples that were computed using the algorithm outlined in the preceeding sections. All of these consider the prediction of steady wave patterns for hulls over a wide range of Froude numbers. For all of these cases, local timestepping was employed for the 3-D incompressible flow solvers as well as the free surface solver. At the start of a run, the 3-D flowfield was updated for 10 timesteps without any free surface update. Thereafter, the free surface was updated after every 3-D flowfield timestep. The mesh was moved every 100 to 250 timesteps. 19.1.4.1. Submerged NACA0012 The first case considered is a submerged NACA0012 at α =5 ◦ angle of attack. This same configuration was tested experimentally by Duncan (1983) and modelled numerically by Hino et al. (1993), Hino (1997). Although the case is 2-D, it was modelled as 3-D, with two parallel walls in the y-direction. The mesh consisted of 2 409 720 tetrahedral elements, TREATMENT OF FREE SURFACES 425 (a) (b) (c) (d) Figure 19.3. Submerged NACA0012: (a), (b) surface grids; (c), (d) pressure and velocity fields; (e), (f) velocity field (zoom); (g) wave profiles 465752 points and 11 093 boundary points, of which 6929 were on the free surface. The Froude number was set to Fr = 0.5672. This case was run in Euler mode and using the Baldwin–Lomax model with a Reynolds number of Re = 10 6 . Figures 19.3(a)–(f) show the surface grid, pressure and velocity fields, as well as a zoom of the velocity field close to 426 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES (e) (f) -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 -2 -1 0 1 2 3 4 5 Wave Elevation (Fr=0.5672) X-Coordinates RANS Euler Hino 93 Experiment (g) Figure 19.3. Continued the airfoil. Note the boundary layer from the velocity fields. Figure 19.3(g) compares the wave profiles for the Euler, Baldwin–Lomax, Hino et al.’s (1993) Euler and Duncan’s (1983) experiment data. The wave amplitudes are noticeably lower for the RANS case. Interestingly, this was also observed by Hino (1997). TREATMENT OF FREE SURFACES 427 19.1.4.2. Wigley hull The next case is the well-known Wigley hull, given by the analytical formula y = 0.5 ·B ·[1 −4x 2 ]·  1 −  z D  2  , (19.19) where B and D are the beam and the draft of the ship at still water. For the case considered here, D = 0.0625 and B =0.1. This same configuration was tested experimentally at the University of Tokyo (ITTC (1983a,b)) and modelled numerically by Farmer et al. (1993), Raven (1996) and others. At first, a fine triangulation for the surface given by (19.19) was generated. This triangulation was subsequently used to define, in a discrete manner, the hull. The surface definition of the complete computational domain consisted of discrete (hull) and analytical surface patches. The mesh consisted of 1119703 tetrahedral elements, 204155 points and 30358 boundary points, of which 15 515 were on the free surface. The parameters for this simulations were as follows: Fr = 0.25, Re = 10 6 and the k −  model with the law of the wall approximation. Figures 19.4(a) and (b) show the surface grids employed, and the extent of the region with high-aspect-ratio elements. In Figures 19.4(c) and (d) the wave profiles and surface velocities obtained from Euler and RANS calculations are compared. As expected, the effect of viscosity becomes noticeable in the stern region. Figure 19.4(e) shows the comparison to the experiments conducted at the University of Tokyo. It can be noticed that the first wave is accurately reproduced, but that the second wave is not well reproduced by the calculations. 19.1.4.3. Double Wigley hull The third case considered is an inviscid case comprising two Wigley hulls positioned close to each other. Figures 19.5(a)–(f) show the resulting wave pattern for a Froude number of Fr = 0.316 for different relative spacings in the x-andy-directions. As one can see, the effect on the resulting wave pattern is considerable. 19.1.4.4. Wigley carrier group The next case considered is again inviscid. The configuration is composed of a Wigley hull in the centre that has been enlarged by a factor of 1:3, surrounded by normal Wigley hulls. The mesh consisted of approximately 4.2 million tetrahedra. Figures 19.6(a) and (b) show the resulting wave pattern for a Froude number of Fr = 0.316. The CPU time for this problem was approximately 24 hours using eight processors on an SGI Origin2000. 19.1.5. PRACTICAL LIMITATIONS OF FREE SURFACE FITTING While very accurate and extremely competitive in terms of storage and CPU requirements as compared to other methods, free surface fitting also has limitations. The key limitation stems from the free surface description given by (19.4). Any free surface described in this manner can only be single-valued in the z-direction. Therefore, it will be impossible to describe a breaking wave. Another case where this method fails is the situation where transom sterns have vertical walls. Here, as before, the free surface becomes multi-valued. Summarizing, free surface fitting methods cannot be used if the interface topology changes significantly. 428 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES (a) (b) Euler RANS (kŦH) Euler RANS (kŦH) (c) (d) -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Wave Elevation (Fr=0.25) X-Coordinates RANS Euler Experiment (e) Figure 19.4. Wigley hull: (a), (b) surface of mesh; (c) wave elevation; and (d) surface velocity; (e) wave elevation at the hull TREATMENT OF FREE SURFACES 429 dx=0.10, dy=0.50 dx=0.25, dy=0.50 dx=0.50, dy=0.50 (a) (b) (c) dx=0.10, dy=0.25 dx=0.25, dy=0.25 dx=0.50, dy=0.25 (d) (e) (f) Figure 19.5. (a)–(c) Wave elevation for two Wigley hulls (Fr = 0.316); (d)–(f) wave elevation for two Wigley hulls (Fr =0.316) 19.2. Interface capturing methods As stated before, the third possible approach to treat free surfaces is given by the so- called interface-capturing methods (Nichols and Hirt (1975), Hirt and Nichols (1981), Yabe and Aoki (1991), Unverdi and Tryggvason (1992), Sussman et al. (1994), Yabe (1997), 430 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES (a) (b) Figure 19.6. (a) Wigley carrier group; (b) wave elevation (Fr =0.316) TREATMENT OF FREE SURFACES 431 Scardovelli and Zaleski (1999), Chen and Kharif (1999), Fekken et al. (1999), Enright et al. (2003), Biausser et al. (2004), Huijsmans and van Grosen (2004),Coppola-Owen and Codina (2005)). These consider both fluids as a single effective fluid with variable properties; the interface is captured as a region of sudden change in fluid properties. The main problem of complex free surface flows is that the density ρ jumps by three orders of magnitude between the gaseous and liquid phases. Moreover, this surface can move, bend and reconnect in arbitrary ways. In order to illustrate the difficulties that can arise if one treats the complete system, consider a hydrostatic flow, where the exact solution is v = 0,p=−ρg · (x − x 0 ), and x 0 denotes the position of the free surface. Unless the free surface coincides with the faces of elements, there is no way for typical finite element shape functions to capture the discontinuity in the gradient of the pressure. This implies that one has to either increase the number of Gauss points (Codina and Soto (2002)) or modify (e.g. enrich) the shape function space (Coppola-Owen and Codina (2005), Kölke (2005)). Using the standard linear element procedure leads to spurious velocity jumps at the interface, as any small pressure gradient that ‘pollutes over’ from the water to the air region will accelerate the air considerably. This in turn will lead to loss of divergence, causing more spurious pressures. The whole cycle may, in fact, lead to a complete divergence of the solution. Faced with this dilemma, most flows with free surfaces have been solved neglecting the air. This approach does not account for the pressure buildup due to volumes of gas enclosed by liquid, and therefore is not universal. The liquid–gas interface is described by a scalar equation of the form  ,t + v a ·∇ = 0. (19.20) For the classic volume of fluid (VOF) technique,  represents the percentage of liquid in a cell/element or control volume (see Nichols and Hirt (1975), Hirt and Nichols (1981), Scardovelli and Zaleski (1999), Chen and Kharif (1999), Fekken et al. (1999), Biausser et al. (2004), Huijsmans and van Grosen (2004)). For pseudo-concentration (PC) techniques,  represents the total density of the material in a cell/element or control volume. For the level set (LS) approach  represents the signed distance to the interface (Enright et al. (2003)). One complete timestep for a projection-based incompressible flow solver as described in Chapter 11 then comprises of the following substeps: - predict velocity (advective-diffusive predictor, equations (11.32a), (11.41) and (11.42)); - extrapolate the pressure (imposition of boundary conditions); - update the pressure (equation (11.32b)); - correct the velocity field (equation (11.32c)); - extrapolate the velocity field; and - update the scalar interface indicator. The extension of a solver for the incompressible Navier–Stokes equations to handle free surface flows via the VOF or LS techniques requires a series of extensions which are the subject of the present section. At this point, we remark that the implementation of the VOF and LS approaches is very similar. Moreover, experience indicates that both work well. [...]... time 446 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES 120 35 x_c 20 z-Position 25 60 x_c vs z_c 30 80 Position 100 40 20 15 10 5 0 0 -2 0 -5 -4 0 -4 0 0 20 40 60 80 100 120 140 160 180 20 0 22 0 Time -2 0 (c) 0 20 40 60 x-Position 80 100 120 (d) 0 .2 a_x 0.15 Angle 0.1 0.05 0 -0 .05 -0 .1 -0 .15 0 20 40 60 80 100 120 140 160 180 20 0 22 0 Time (e) Figure 19. 12 Continued almost no 3-D effects for the first 25 oscillating... wave elevation (ζ /L) 0.6 0.4 wave elevation (ζ /L) 0.6 0 .2 0 -0 .2 -0 .4 0 .2 0 -0 .2 -0 .4 -0 .6 -0 .6 0 5 10 15 20 t/T 25 30 35 40 0 5 10 15 20 t/T 25 VOF, T 1.3, A/L 0. 025 35 40 VOF, T 1.3, A/L 0.05 0.6 0.4 0.4 wave elevation (ζ /L) 0.6 wave elevation (ζ /L) 30 0 .2 0 -0 .2 -0 .4 0 .2 0 -0 .2 -0 .4 -0 .6 -0 .6 0 5 10 15 20 t/T 25 30 35 40 0 5 10 15 20 t/T 25 30 35 40 (c) Figure 19.10 Continued surface Note also... gL2b 50 3 0 -5 0 -1 00 0 -5 0 -1 00 -1 50 -1 50 0 5 10 15 20 t/T 25 150 30 35 40 0 10 15 20 t/T 25 35 40 VOF, T 1.3, A/L 0.05 50 Fx 103 /ρ gL2b 50 30 100 2 3 5 150 VOF, T 1.3, A/L 0. 025 100 Fx 10 /ρ gL b VOF, T 1 .2, A/L 0.05 100 2 Fx 10 /ρ gL b 100 0 -5 0 -1 00 0 -5 0 -1 00 -1 50 -1 50 0 5 10 15 20 t/T 25 30 35 40 0 5 10 15 20 t/T 25 30 35 40 (b) VOF, T 1 .2, A/L 0. 025 VOF, T 1 .2, A/L 0.05 0.4 wave elevation (ζ /L)... (b) a 2- D tank and (c) a 3-D tank at A/L = 0. 025 , T /T1 = 1; (d) time history of force Fz for a 3-D tank at A/L = 0. 025 , T /T1 = 1; (e), (f), (g) snap shots of the free surface wave elevation for a 3-D tank 441 TREATMENT OF FREE SURFACES 150 VOF, A/L 0. 025 , 2D Tank Fx 103 /ρ gL2b 100 50 0 -5 0 -1 00 -1 50 0 10 20 30 40 50 60 70 80 t/T (b) 150 VOF, A/L 0. 025 , 3D Tank Fx 103 /ρ gL2b 100 50 0 -5 0 -1 00 -1 50... 1.1 T/T1 1 .2 1.3 1.4 1.5 0.7 0 .8 0.9 1 1.1 T/T1 1 .2 1.3 1.4 1.5 (e) VOF, A/L=0.05 Exp, A/L=0.05 SPH, A/L=0.05 180 160 160 140 140 Fx 103 / gL2b Fx 103 / gL2b VOF, A/L=0.050 VOF, A/L=0. 025 180 120 100 80 60 40 120 100 80 60 40 20 20 0 0 0.7 0 .8 0.9 1 1.1 T/T1 1 .2 1.3 1.4 1.5 0.7 0 .8 (f) 0.9 1 1.1 T/T1 1 .2 1.3 1.4 1.5 (g) Figure 19.10 Continued Y B 1m A X H 1m Z h 0.35m L 1m (a) Figure 19.11 3-D tank:... 0. 025 , 0.05 A = 0. 025 , 0.05 are shown in Figure 19.10(b) The corresponding time history of the wave elevation at the wave probe A1 (see Figure 19.10(a)) are shown in Figure 19.10(c) Some free surface snapshots are shown in Figure 19.10(d) The dark line represents the free 4 38 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES 150 150 VOF, T 1 .2, A/L 0. 025 50 Fx 103 /ρ gL2b 50 3 0 -5 0 -1 00 0 -5 0 -1 00 -1 50... nelem=54, 124 elements The numerical simulations are carried out for both 3-D and 2- D tanks, where both tanks are undergoing the same prescribed sway motion 440 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES VOF, A/L 0. 025 Exp, A/L 0. 025 SPH, A/L 0. 025 0.7 VOF, A/L 0.05 Exp, A/L 0.05 SPH, A/L 0.05 0.7 0.6 wave height (ζ /L) wave height (ζ /L) 0.6 0.5 0.4 0.3 0.5 0.4 0.3 0 .2 0 .2 0.1 0.1 0 0 0.7 0 .8 0.9... from Figures 19.11(c) and (d) that there are 4 42 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES (e) Figure 19.11 Continued 443 TREATMENT OF FREE SURFACES (f) Figure 19.11 Continued 444 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES (g) Figure 19.11 Continued 445 TREATMENT OF FREE SURFACES 310m y x z 24 5.7m 120 m 700m x=x0+a sin( t) (a) (b) Figure 19. 12 Ship adrift: (a) problem definition; (b) evolution... 19 .2. 8. 3 Sloshing of a 3-D tank due to sway excitation In order to study the 3-D effects, the sloshing of a partially filled 3-D tank is considered The main tank dimensions are L = H = 1 m, with tank width b = 1 m The problem definition is shown in Figure 19.11(a) The 3-D tank has the same filling level h/L = 0.35 as the 2- D tank The 3-D tank case is run on a mesh with nelem=561 ,80 8 elements, and the 2- D... 80 t/T (b) 150 VOF, A/L 0. 025 , 3D Tank Fx 103 /ρ gL2b 100 50 0 -5 0 -1 00 -1 50 0 10 20 30 40 50 60 70 80 t/T (c) 150 VOF, A/L 0. 025 , 3D Tank Fz 103 /ρ gL2b 100 50 0 -5 0 -1 00 -1 50 0 10 20 30 40 t/T 50 60 70 80 (d) Figure 19.11 Continued given by x = A sin (2 t/T ) The simulations were carried out for A = 0. 025 and T = 1 .27 (i.e T /T1 = 1) The forced oscillation amplitude increases smoothly in time and . field close to 426 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES (e) (f) -0 .1 -0 . 08 -0 .06 -0 .04 -0 . 02 0 0. 02 0.04 0.06 0. 08 0.1 -2 -1 0 1 2 3 4 5 Wave Elevation (Fr=0.56 72) X-Coordinates RANS Euler Hino. free 4 38 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES -1 50 -1 00 -5 0 0 50 100 150 0 5 10 15 20 25 30 35 40 F x 10 3 /ρ gL 2 b t/T VOF, T 1 .2, A/L 0. 025 -1 50 -1 00 -5 0 0 50 100 150 0 5 10 15 20 25 . (kŦH) Euler RANS (kŦH) (c) (d) -0 .006 -0 .004 -0 .0 02 0 0.0 02 0.004 0.006 0.0 08 0.01 0.0 12 0.014 -1 -0 .8 -0 .6 -0 .4 -0 .2 0 0 .2 0.4 0.6 0 .8 1 Wave Elevation (Fr=0 .25 ) X-Coordinates RANS Euler Experiment (e) Figure