FD生成的PMM数据在初步设计阶段进行评估英文文献和中文翻译(10)
The uncertainty assessment procedure is very extensive and is not described here. It has only been done for a few conditions as spot checks of the uncertainty level. It is described in details in Larsen (2012). Table 3 and Table 4 show the results of the uncertainty assessment for the “static drift” case with ߚൌ8, and “static rudder” case with ߜൌ10, respectively at model speed of 1.318m/s, i.e. 0.775U0. Table 4. Uncertainties for “static rudder”, δ=10°, Fr=0.201. X’ Y’ N’ ܤ´0.000724 0.000705 0.000228 ܲ´0.000160 0.000408 0.000218 ܷ´ 0.000742 0.000815 0.000316 ܦ´ 0.0165 0.0150 -0.0077 ܷ݅݊%ܦ´ 4.50 5.45 4.11 29th Symposium on Naval Hydrodynamics Gothenburg, Sweden, 26-31 August 2012 The level of the uncertainties is higher than for a resistance test, but this is to be expected since the PMM test is conducted with a two-gauge system, which typically gives higher uncertainties than a single gauge system. The uncertainty level seems reasonable and the level is comparable to other to PMM tests in Benedetti et al. (2006), Yoon (2009) and Simonsen (2004). With respect to the measured data, examples are shown in Figure 8 to Figure 19 in the section below, in connection with the discussion of the comparison between computed and measured PMM data. Data generation based on CFD computations The CFD computations in the present work are supposed to replace both dynamic and static PMM conditions. However, at present time of writing the dynamic conditions are not completed, so only results for static cases are covered. Computational method The computations are performed with the Reynolds Averaged Navier-Stokes (RANS) solver STAR-CCM+ from CD-adapco. The code solves the RANS and continuity equations on integral form on an unstructured mesh by means of the finite volume technique. Both steady state and transient calculations are considered depending on the considered test condition. For the steady state calculations the temporal discretization is based on a first order Euler difference, while a second order difference is used for transient calculations. Spatial discretization is performed with second order schemes for both convective and viscous terms. The pressure and the velocities are coupled by means of the SIMPLE method. Closure of the Reynolds stress problem is achieved by means of the isotropic blended k-ε/k-ω SST turbulence model with an all Y+ wall treatment, which depending on the Y+ value selects the near wall treatment. The rotating propeller is not modelled directly, as this requires very long computational times that are not realistic when computing a large number of cases. Therefore, a prescribed body force propeller is applied instead as is commonly done for this type of simulations. The propeller model is based on a radial varying body force field, which follows the variation of a theoretically derived circulation distribution. It is prescribed by means of the ship speed based advance coefficient ܬ௩ and the propeller coefficients ܥ் and ܭொ. The model is integrated in the CFD-code, where the axial and tangential force components are given by h*h* *θ θ* *X Xr r rr rA fb , r r A fb ) 1 (11 (17) Where the non-dimensional propeller and hub radii are defined by ) 1 ( ) ( h h*r / r r r and p h h /R R r . With ∆ݔ being the longitudinal extension of the disk pided by the ship length, the coefficients ܣ௫ and ܣఏ are defined by h hThxr r x ΔC A 1 3 4 16105 (18) h h VQθr r π J x ΔKA 1 3 41052
(19) The free surface is modelled with the two phase volume of fluid technique (VOF). Further details about the code can be found in STAR-CCM+ user guide (2012). Finally, it must be noted that all computations are done in model scale with a model similar to the one used in the model test. The motions of the ship are described in terms of an earth-fixed inertial reference coordinate system and a ship fixed coordinate system. The motion of the ship, i.e. translations and rotations, are described with respect to the inertial frame. The linear and angular velocities plus the forces and moments are described with respect to the ship-fixed frame. In the present calculation, three degrees of freedom are considered and the ship motions in surge, sway and yaw are all prescribed. The motions are applied in the earth-fixed frame by mowing the complete computational domain and hereby introduce grid velocities according to the considered motion. This applies to both static conditions like static drift where the ship is moved with constant speed to obtain the desired drift angle and to dynamic tests like pure yaw motion, which are not covered in this paper. Heave and pitch motions in terms of dynamic sinkage and trim are taken from the model test, so they are not predicted in the CFD solver. The thrust and torque values applied in the prescribed body-force model are also taken from the model test. Computational domain The computational mesh is based on the trimmed mesh approach, which means that the mesh is dominated by hexahedral cells in most regions except close to the hull where the hull surface is used to trim the hexahedral cells leaving a set of polyhedral cells. Further, prism layer mesh is used near the hull surface to resolve the boundary layer flow. Concerning boundary conditions on the domain, an inlet condition 29th Symposium on Naval Hydrodynamics Gothenburg, Sweden, 26-31 August 2012 prescribing the inlet velocity and the still water level for the volume fraction is given up-stream of the model and on the sides of the domain. On the outlet boundary a pressure condition is applied in order to have zero gradients for the velocity and volume fraction and prescribed hydrostatic pressure. On the top and bottom of the domain, slip walls are applied so there is no flow through the boundary. The inlet boundary is located 4.0 ship lengths from the ship, Outlet and sides are located 6.4 and 5.0 ship lengths from the ship, respectively. Figure 5. Illustration of domain boundaries. Figure 6 and Figure 7 show the computational mesh in the bow and stern regions, respectively. Refinement zones are applied in the free surface region and on the hull where the wave formation is expected to take place. Figure 6. Mesh in bow region. Figure 7. Mesh in bow region. Verification and validation Concerning verification focus is on mesh sensitivity. To study the sensitivity of the mesh size on the results one condition has been picked to be investigated. Three different grid sizes have been used and the corresponding forces have been evaluated for a static drift condition with 20 degrees drift angle at model speed of 1.318m/s, i.e. 0.775U0. Further, no running body-force propeller is included in this case. With the unstructured mesh, it is not possible to make a completely systematic grid refinement, since it is not possible to directly control the size of the inpidual cells. However, all mesh quantities are given as a percentage of a base size, so in order to change the grid as systematically as possible the coarse (No. 3), medium (No. 2) and fine (No. 1) grids were made by changing the base size with a constant factor. Based on the three grids, the CFD simulations were conducted and the longitudinal X-force, X, the transverse Y-force, Y, and the yaw moments, N, taken around the Z-axis at mid-ship position were computed. It is difficult to estimate the grid uncertainty for unstructured meshes, so in lieu of an uncertainty method for unstructured meshes and since the present grid is hexahedral dominated, it has been attempted to estimate the grid uncertainty based on Richardson extrapolation. Solutions were obtained on 3 systematically refined grids by means of the refinement ratio, ݎீ ൌ √2. This gave mesh sizes ranging from approximately 1.5 to 7.0 million cells. Exact grid numbers and computed force and moment coefficients can be seen in Table 5. Forces and moments are non-dimensionalized with ሺ½ߩܷଶܮܶሻ and ሺ½ߩܷଶܮଶ ܶሻ respectively. 29th Symposium on Naval Hydrodynamics Gothenburg, Sweden, 26-31 August 2012 Table 5: Non-dim forces and moments for different grids. No. of cells X’ Y’ N’ 1,491,946 -0.0208 0.1372 0.0451 3,068,474 -0.0214 0.1442 0.0446 6,969,575 -0.0211 0.1473 0.0451 The grid uncertainty ܷீ is estimated according to the procedure given in Stern et al. 2001 and Wilson et al. 2004. Briefly, the procedure is as follows. The changes in solutions between coarse and medium grids, 2 3 32S S G , and between medium and fine grids, 1 2 21S S G , are used to calculate the convergence ratio 32 21/ G G G R . Depending on the value of ܴீ three conditions can occur: i) 0 ൏ ܴீ ൏1, grid convergence, ii) ܴீ ൏0, oscillatory convergence and iii) 1൏ܴீ, grid pergence. In condition iii) no uncertainty can be estimated. In condition ii) the uncertainty is estimated by ) ½( L U G S S U (20) where U S and L S are the maximum and minimum of the solutions from the considered grids. In condition i) grid convergence occurs and generalized Richardson extrapolation (RE) is used to estimate the grid error ) 1 *(1 G RE and the order of accuracy G p , which are given as 1211) 1 ( *G G pGGREr (21) and ) ln(/ ln 21 32GG GGrp (22) 2 3 1 2 / / G G G G G x x x x r is the refinement factor, which defines the relation between the considered grids. When ) 1 *(1 G RE and G p are know it is possible to estimate the grid uncertainty. There are two ways to do this depending on whether the solutions are close to the asymptotic range or not. If the correction factor defined by 11GestGpGpGGrrC (23) where Gestp is the limiting or theoretical accuracy of the applied numerical method, is close to unity, the solutions are close to the asymptotic range. In this case the sign of the error is known, so the numerical error,*SN , benchmark, C S and uncertainty, GC U can be calculated from ) 1 *( *1 G RE G SN C (24) *SN C S S (25) 125 . 0 1 , 1125 . 0 1 , ) 1 . 0 ) 1 ( 4 . 2 (** 211G RE GG RE GGCC CC CUGG (26) If the correction factor is away from unity only the numerical uncertainty is calculated 125 . 0 1 , ) 1 1 2 (125 . 0 1 , ) 1 . 1 ) 1 ( 6 . 9 (** 211G RE GG RE GGC CC CUGG (27) Table 6: % change in non-dim forces and moments between grids. % Change X’ Y’ N’ 3 to 2 -2.74 -4.74 1.14 2 to 1 1.28 -2.12 -0.97 Table 7: Estimated order of accuracy and grid uncertainty. Quantity G p G U G U % 1 S X’ Oscill. 0.00029 1.4 Y’ 2.3 0.00372 2.5 N’ Oscill. 0.00026 0.6 Table 6 shows the change in quantities between grids. It is seen that all quantities are converging as the grid is refined, but both X’ and N’ show oscillatory convergence, i.e. the order of accuracy cannot be estimated. For Y’ Table 7 shows that an order of accuracy of 2.3 is achieved. This is close to the theoretical order of accuracy of 2. Finally, Table 7 also shows the estimated grid uncertainty, which ranges from 0.6 to 1.4 % of the fine grid solution. It should be noted that if the solution for Y’ is corrected with the numerical error, the grid uncertainty can be reduced to 2.3%. In order to validate the computations against measurements, the measured data shown in Table 8 are used. No specific uncertainty assessment was made for the un-propelled 20 degrees static drift case, so the uncertainty estimates from the propelled 8 degrees drift case is applied, by assuming that the percentage in Table 3 is representative for the un-propelled case. Table 8: Measured values and data uncertainty Quantity D D U D U % D X’ -0.0220 0.00094 4.26 Y’ 0.1551 0.00326 2.10 N’ 0.0434 0.00099 2.28 Table 9: Validation investigation. Quantity E=D-S V U E % D X’ 0.00086 0.00098 3.90 Y’ -0.00779 0.00494 5.03 N’ 0.00167 0.00102 3.84 To finally check if Validation has been obtained it is necessary to check if the absolute comparison error E is smaller than the validation uncertainty calculated as2 2D G V U U U , i.e. if the comparison error is within the combined noise from the CFD simulation and the experiments. From Table 9 it is seen that the computation agrees with the measurement within approximately maximum 5%, so the agreement is pretty good taking into account that it is a pretty complex flow field at 20 degrees drift angle. However, in spite of this it is only X´ that is validated. Taking into account that going from medium to fine grids will only change the results with approximately 2% while the computational time will be more than doubled, it was judged that the medium mesh was sufficient for the present study, which has much focus on computation of large data sets for practical application. It should also be mentioned that later in the project it was necessary to increase the mesh size a little in order to have a mesh that work for non-zero rudder angles.