D DAVID PUBLISHING. Uncertainty Analysis in CFD for Resistance. 1. Introduction

Transcription

1 Journal of Shipping and Ocean Engineering 7 (2017) doi / / D DAVID PUBLISHING WANG Zhongcheng 1, LIU Xiaoyu 1, ZHANG Shenglong 1, XU Leping 1 and ZHOU Peilin 2 1. MMC of Shanghai Maritime University, Shanghai , China 2. Strathclyde University, Glasgow G4 0LZ,UK Abstract: Based on the Latin square design of statistics, the thickness of first boundary layer, the turbulence model and the cell number were taken as the three main factors of uncertainty in CFD (computational fluid dynamics). Total resistance of hull was calculated and the flow field around the hull was simulated by CFD method. Then, the influence of uncertainty factors on the hull resistance was discussed by regression analysis with trimmed mesh and overset mesh. Through a series of calculation and analysis, the optimal calculation method was put forward, and the relevant parameters of the calculation were determined. Thirdly, the total resistance of different speed was calculated by using these two kinds of grids, which were in good agreement with the experimental results. Finally, according to the ITTC recommended procedures, uncertainty analysis in CFD was carried out with the numerical results of the total resistance by three sets of grids with uniform refinement ratio r G = 2. Then the modified resistance was compared with the experimental result, which improved the accuracy of the resistance prediction. Key words: Resistance prediction, Latin square design, regression analysis, trimmed mesh, overset mesh. 1. Introduction With the rapid development of computer technology, CFD (computational fluid dynamics) has been widely used in the field of ship resistance prediction, optimization design of ship hull, propeller performance research and so on. It can not only simulate the motion response of the hull in viscous fluid, but also can capture the strong nonlinear phenomenon [1]. In recent years, scholars have done a series of research work in this field, and have made gratifying achievements. Ahmed [2] used CFX tool to calculate the resistance of DTMB5415 and simulate the flow field around the hull at different speeds in calm water. SHI et al. [3] used the hybrid grids, fewer structured grids far away from the hull, and refinement of unstructured grids near the hull, to calculate the resistance of INSEAN2340. Sadat-Hosseini et al. [4] used the RANS method and free surface tracking method to predict the resistance of KVLCC2 and discussed the pitch and heave of hull in waves. The results of calculation were in good Corresponding author: LIU Xiaoyu, doctor, lecturer, research fields: naval architecture and ocean engineering. agreement with the experimental data. Although the satisfactory results have been achieved in numerical simulation, there are more or less errors compared with the experimental results, which are caused by various uncertain factors. Therefore, the study of the affecting factors in CFD has become a problem that we pay close attention to. DENG et al. [5] used the CFD method to calculate the resistance of catamaran, and appropriate method was presented by analyzing mesh generation, numerical algorithm and turbulence model one by one. Taking SFS as an example, the mesh generation method for complex ship was studied by WANG et al. [6]. The results show that the grids form did not significantly affect the calculation results and the hybrid grids were better than structural grids in resistance prediction. Although a large number of scholars have discussed the factors that affect the results of CFD, it is bound to increase the computation time and decrease efficiency. In this paper, Latin square design was introduced in analysis of CFD. First of all, the thickness of first boundary layer, the turbulence model and the cell number were taken as the three main factors of uncertainty in CFD. Secondly, for 3 factors 4//3 levels

2 193 problem, 16//9 experiments were designed by Latin square design. Then the main and secondary factors affecting CFD in the prediction of hull resistance were found. Thirdly, according to the most accurate calculation method, the resistance of DTMB5512 at different speed was calculated by trimmed mesh and overset mesh. Finally, in accordance with the ITTC recommended procedures, uncertainty analysis in CFD was carried out with the numerical results of the total resistance by three sets of grids with refinement ratio r G = 2. Then the modified resistance was compared with the experimental result, which improved the accuracy of the resistance prediction. 2. The Calculation Method of Resistance 2.1 Governing Equations The governing equations of the whole flow field are composed of continuous equations and motion equations [7]: U = 0 (1) U t i + U U i X i pˆ = + 1 Re 2 U 2.2 The Method of Free Surface Capturing i ' uiu ' + f * b i (2) VOF [8] is the most common method of free surface capturing, and now it has been widely applied to the research of ships. It is a surface tracking method fixed under the Euler grid and simulates the multiphase flow model by solving the momentum equation and the volume fraction of one or more fluids. Within each control volume, the sum of the volume fractions of all the phases is 1. As to Phase q, its equation is: a q t ( ua + q ) ( va + y q ) ( wa + z q ) = 0 (3) where, a 1 and a 2 respectively are the volume fractions of air and water, and a q = 0.5 is the interface of water and air, q = 0 means the unit is filled with water, q = 1 means the unit is filled with air. 3. Mesh Generation 3.1 Trimmed Mesh Trimmed mesh is a new technology of mesh generation. It contains not only the advantages of structured and unstructured grids, but also has better solution computing power for the large amplitude motion. So far, it has been widely used. Trimmed mesh is a method that the grids are firstly generated around the whole computational domain. Then according to user-defined cuboid, the whole computational domain is cut repeatedly with mesh refinement which is combined with neighbored grids. When dividing the grids, firstly, information of the adacent grids in the same layer needs to be recorded, then comparing the parent layer grids in its six directions. Grid resolution is done and sub-grids should be inserted in the back of the parent layer in order to calculate the neighbor information and boundary information, in the end, delete the parent layer. Fig. 1 shows the grid number in its two-dimension. And 40 parent grids are generated in the whole area, then parent grids with number 11-14, 19-22, 27-30, and are divided into 4 sub-grids. After re-numbering those grids, 92 grids are generated. In this paper, the trimmed mesh is divided into the whole calculation domain as shown in Fig Overset Mesh If CFD method is selected to solve the model, the transformation of geometric model into grids is an important step to seperate the governing equations. With further research in the field of shipbuilding, ship motion problem must be taken into account in the calculation of the ship resistance. Meanwhile, for the problem of two-phase flow, the grids between the water and the air interface only fine enough can accurately capture the nonlinear phenomena. This may

3 194 Fig. 1 Mesh refinement. Fig. 2 Trimmed mesh. cause great difficulties in the simulation of motion by the traditionally structured and unstructured grids. However, the emergence of overset mesh technique can solve such complex motion problem easily, and the calculation process is much easier and more convenient. It can generate high-quality grids, and may be more accurate to solve the large amplitude motion of ship [9]. Overset mesh is a method that grids are divided in each part of model and then nested into the background grids. First, according to the model execution point hole to dig command to remove the obect plane inside the unit and redundant overlapping unit and then interpolating in the overlap region by the difference between grid points, so that between the grids and the grid can be carried out in the border area data exchange, the final completion of the whole flow field solver calculation. Two acceptor cells are shown using dashed lines, one in the background mesh and one in the overset mesh, as shown in Fig. 3. The fluxes through the cell face between the last active cell and the acceptor cell are approximated in the same way as between two active cells. However, whenever the variable value at the acceptor cell centroid is referenced, the weighted variable values at the donor cells are substituted: φ φ (4) acceptor = αi i where, α i is the interpolation weighting factors, Ø is the values of the dependent variable at donor cells N i and subscript I runs over all donor nodes of an interpolation element (denoted by the green triangles in the figure). This way, the algebraic equation for the cell C in the above figure involves three neighbor cells from the same mesh (N 1 to N 3 ) and three cells from the overlapping mesh (N 4 to N 6 ). The coefficient matrix of each equation solved (both for the segregated and coupled solution method) is updated accordingly to ensure that equations can be solved up to the round-off level of residuals. There are many kinds of interpolation methods, in which the linear interpolation uses shape function to connect the center of the variable grids. And the interpolation unit is transferred from one interpolation

4 195 Fig. 3 Schematic diagram of overset mesh. Fig. 4 Overset mesh. elements to another from the center of mesh. Although this method may face with the low computational efficiency, it is more accurate on resistance prediction. In this paper, overset mesh is used to divide the computing domain and hull with linear interpolation. The mesh is shown in Fig Resistance Calculation (1) Boundary conditions: (a) Trimmed mesh: front, top and bottom boundaries are selected as velocity inlet, and ship speed is added in the inlet boundary. Back boundary is selected as pressure outlet. The hull is selected as wall, and both sides of tank are selected as symmetry plane. (b) Overset mesh: according to the requirement of overset mesh, the whole model needs two regions, that is, the background region and the overlapping region which are obtained by Boolean operation between the cuboid and the hull. In background region: the boundary condition is the same as the trimmed mesh. In overlapping region: the left side of the cuboid is set as the symmetry plane and the rest of surface is set as overset mesh. The hull is set as wall. (2) Calculation process: Taking DTMB5512 as an example, the total resistance of hull is calculated in calm water by CFD method. (a) Pre-calculation: establish geometric model, divide grids, check mesh quality and set the boundary conditions. (b) Calculation process: hull is fixed and the speed is offered along the x-direction at the inlet boundary. The continuity Eq. (1) and RANS Eq. (2) are selected as control equation of the whole computational domain. The appropriate turbulence model is used to close the equations. VOF model is selected for tracking free surface. SIMPLE is selected for solving equation, and the ship speed is taken as the initial velocity to start the iteration. The specific calculation process is shown in Fig. 5.

5 196 Fig. 5 Calculation flow of resistance. 5. Analysis of Main Factors Undoubtedly, there are some errors between the results of CFD and the experimental data, and according to different calculation conditions, errors are different which are mainly caused by a series of uncertain factors, such as in the process of the establishment of modeling, the use of calculation method, and the iterative calculation. It mainly includes five aspects: mathematical model error, iteration error, rounding error, truncation error and calculation error. In order to improve the accuracy of the resistance prediction, it is necessary to analyze which factor plays a dominant role and which plays a secondary role in the calculation of resistance. It will waste a lot of computing time and decrease efficiency if analyzing them one by one. Therefore, in this paper, experimental design is introduced to analyze the influence factors in CFD. Experimental design can find the close relationship in large amounts of data with the minimal number of tests and test cycle. It is a kind of high efficiency, fast and economical calculation method. There are many kinds of experimental design methods, among which the Latin square design is invented by the famous mathematician and physicist Euler. It is written to n different design elements in the n squares, each row and each column becomes a complete unit group, and each processing appears only once in each row and each column. (1) Latin square design The main factors that affect the results of CFD include three factors: the thickness of first boundary layer, the turbulence model and the cell number. Therefore, the design and analysis of these factors are carried out by the Latin square matrix. The design model of 3 factors of 4 levels is constructed on trimmed mesh and 3 factors of 3 levels are constructed on overset mesh (due to computing error of RST model, resistance cannot be obtained), as shown in Table 1. The 16//9 experiments are designed by Latin square design, as shown in Table 2. The total resistance coefficients at Fr = are also listed in this table which is obtained by the CFD method. Fig. 6 shows the comparison of errors between CFD results and the experimental results [10, 11]. (2) Results and discussion The hydrodynamic performance of DTMB5512 in calm water is simulated using 16 methods by trimmed mesh and 9 methods by overset mesh. It can be seen from Fig. 6: on the trimmed mesh, for the total resistance coefficients, the deviations between CFD and experimental results are in the range of 2.5% to 18.5%, the calculating result of case 9 is considerably

6 197 Table 1 The 3 factors of 4//3 levels. Factor Level Trimmed mesh Overset mesh Thickness of first boundary layer A Turbulence model B Grid number (relative ratio) C Table 2 Design matrix and total resistance coefficients C QT. No. A B C A A A A Null B 1 k ε k ε B 2 SST k ω SST k ω B 3 SA SA B 4 RST Null C C C C Null Trimmed mesh C QT A B C Overset mesh 1 A 1 B 1 C A 1 B 1 C A 1 B 2 C A 1 B 2 C A 1 B 3 C A 1 B 3 C A 1 B 4 C A 2 B 1 C A 2 B 1 C A 2 B 2 C A 2 B 2 C A 2 B 3 C A 2 B 3 C A 3 B 1 C A 2 B 4 C A 3 B 2 C A 3 B 1 C A 3 B 3 C A 3 B 2 C A 3 B 3 C A 3 B 4 C A 4 B 1 C Null 14 A 4 B 2 C A 4 B 3 C A 4 B 4 C C CT Fig. 6 Deviations between CFD and experimental results.

7 198 close to experimental result, and case 15 has maximal error; on the overset mesh, the deviations between CFD and experimental results are in the range of 1.25% to 14.5%, the calculating result of case 2 is considerably close to experimental result, and case 1 has maximal error. It shows that different calculation methods have great influence on the resistance and it is necessary to evaluate these three factors. Based on the calculation results of Table 2, the data are calculated by regression analysis, as shown in Table 3. From the table, it can be sorted by regression coefficient in descending order: B > C > A on the trimmed mesh and C > B > A on the overset mesh. Sum of squares of deviations and variance also have the same order with regression coefficient. It shows that turbulence model and the cell number have the greatest influence on resistance with trimmed mesh and overset mesh respectively, thickness of first boundary layer has minimal impact on resistance. The data in Table 2 are classified by turbulence model, as shown in Fig. 7. As seen in the figure, the minimum and maximum of the overall-average deviation were 3.25% by k ε and 9.55% by SA on trimmed mesh, and were 5.51% by SST k ω and 10.03% by SA on overset mesh. Therefore, with appropriate number of grids and thickness of first boundary layer, resistance can be predicted accurately by k ε on trimmed mesh and SST k ω on overset mesh. 6. Multi-speed Resistance Calculation In this section, the numerical simulation of DTMB5512 is carried out by using the trimmed mesh and overset mesh. In order to improve the accuracy of the calculation, the same grid division method and calculation method of cases 9 and 2 are used to predict the hull resistance. Then the results are compared with the experimental results, as shown in Fig. 8. The results are consistent with the experimental results, and the errors of two kinds of grids are very close which is 2.5% by trimmed mesh and 2.78% by overset mesh. At design speed Fr = 0.281, errors of resistance are 1.85% and 11.88% respectively. It can Table 3 Variance analysis. Grids Trimmed mesh Overset mesh Factor Sum of squares of deviations Freedom ss df Variance MS Regression analysis A 3.39E E E-04 B 7.10E E E-04 C 1.25E E E-04 A 3.71E E E-04 B 1.14E E E-04 C 2.17E E E-04 Fig. 7 Deviations of different turbulence models.

8 199 Fig. 8 Trimmed mesh Resistance changes with Fr. Overset mesh Table 4 Flow field distribution. Grids Flow field Bow pressure distribution Trimmed mesh Overset mesh be seen that the calculation method in this paper can precisely predict the hull resistance, and have more accuracy in the design speed and its nearby speed by overset mesh. Although the error is smaller at Fr = 0.28 by overset mesh, the average error is slightly higher, the cell number is more and the calculation time is longer than trimmed mesh. It is found that trimmed mesh has the advantage in predicting the resistance at multi-speed, and it can improve the accuracy of resistance prediction as long as increasing the cell number. Table 4 shows flow field and bow pressure distribution of DTMB5512 at Fr = From the table, the red area is the background area, and the blue area is the overset area. 7. Verification and Validation The uncertainty of CFD determines whether the data are useful. It is also difficult to compare the results from different researchers by different evaluation methods [12]. Therefore, uncertainty analysis in CFD has become an important part in CFD

9 200 research and application. According to the ITTC recommended procedures, uncertainty analysis in CFD was carried out with the numerical results of the total resistance, namely, verification and validation. (1) Verification In this paper, the results of total resistance in calm water are analyzed and discussed by trimmed mesh at Fr = The calculation model is analyzed by 3 sets of grids with uniform refinement ratio r G = 2. The meshes of free surface and wave contours are shown in Table 5. The wave has not been captured well with coarse grids, especially in the bow and stern. According to mesh refinement, the wave has been captured very well, as shown in Table 5. The total resistance coefficients of three sets of grids are shown in Table 6. As can be seen from the table, with the increasing of grids, more accurate resistance can be obtained. Table 7 is the verification of the total resistance coefficient C QT, including convergence rate R G, accuracy order P G, correction factor C G, grid uncertainty U G, error with correction factor σ * G1, the corrected uncertainties U GC, corrected simulation value S C. From Table 7, it can be seen that R G < 1, which shows that the grid is monotonic convergence. S C is equal to 4.67E-3 which is close to the experimental result and the error between them was only 1.3%. For detailed information on how to calculate these parameters, reference can be made from Stern et al. [13]. (2) Validation Confirmation means to use test data to evaluate the numerical simulation uncertainty U SM and the model error σ SN under the conditions permit. It determines Table 5 Mesh and wave contours. Grids Mesh of free surface Wave contours Coarse Medium Fine

10 201 Table 6 C QT with three sets of grids. Scalar Coarse Medium Fine EFD C QT Table 7 Verification of C QT. Grids R G P G C G U G σ * G1 U GC S C Trimmed mesh E E E E-3 Table 8 Validation. Comparison error Result Validation uncertainty Result Relation E E-04 U V1 2.07E-04 E 1 < U V1 E C1-5.68E-05 U V1C 1.55E-04 E C1 < U V1C whether the confirmation is realized by comparing comparison error and the uncertainty. If the absolute value of the comparison error is smaller than the validation uncertainty, the uncertainty of this level can be achieved. The validation of total resistance is shown in Table 8. It can be seen from Table 8, E is less than U V, which means uncertainty can be confirmed. 8. Conclusions (1) The Latin square design was used to establish 3 factors of 4//3 levels experimental matrix in which the thickness of first boundary layer, the turbulence model and the cell number were taken as the three main factors. Then the influencing factors on the resistance prediction in CFD were discussed with trimmed mesh and overset mesh. According to the calculation and discussion, the appropriate calculation method and mesh method were proposed. Results show that turbulence model and the cell number have the greatest influence on resistance with trimmed mesh and overset mesh respectively, and thickness of first boundary layer has minimal impact on resistance. The RNG k ε model is more suitable for the resistance prediction based on trimmed mesh, and SST k ω model is more suitable for the resistance prediction based on overset mesh. (2) The total resistance of the hull at different speed was calculated by using trimmed mesh and overset mesh which were in accordance with experiment results. The resistance prediction with trimmed mesh in multi-speed was more accurate than overset mesh except the design speed and its nearby speed. (3) According to the ITTC recommended procedures, uncertainty analysis in CFD was carried out with three sets of grids by trimmed mesh. The numerical solution of the model based on CFD was monotonic convergent with the mesh refinement, uncertainty of total resistance can be confirmed, and modified resistance was closer to the experimental result. Acknowledgements This paper is funded by Shanghai Science and Technology Committee ( ). References [1] YE, H. X., SHEN, Z. R., and WAN, D. C Numerical Simulation Calculation of Ship Wave Drag Resistance and Movement Response of DTMB5512 Ship. The 17th China International Boat and Its Technical Equipment Exhibition and High Performance Ship Academic Report. [2] Ahmed, Y. M Numerical Simulation for the Surface Flow around a Complex Ship Hull Form at Different Froude Numbers. Alexandria Engineering Journal 50: [3] SHIA, A. G., WU, M., YANG, B., WANG. X., and WANG, Z. C Resistance Calculation and Motions Simulation for Free Surface Ship Based on CFD. Procedia Engineering 31: [4] Sadat-Hosseini, H., WU, P. C., Carrica, P. M., Kim, H., Toda, Y., and Stern, F CFD Verification and Validation of Added Resistance and Motions of KVLCC2 with Fixed and Free Surge in Short and Long Head Waves. Ocean Engineering 59:

11 202 [5] DENG, R., HUANG, D. B., LI, J., CHENG, X. K., and YU, L Discussion of Grid Generation for Catamaran Resistance Calculation. Journal of Marine Science and Application 9 (2): [6] WANG, J. L., GAO, Y., and LIU, C. M Research on Mesh Type for Complex Ship. Journal of Huazhong University of Sci. & Tech. 43: [7] WANG, F. J Computational Fluid Dynamics Analysis-CFD Software Principles and Applications. Beiing: Tsinghua University Press. [8] Hirt, C. W., and Nichols, B. D Volume of Fluid Method for the Dynamics of Free Boundaries. Comp. Phys. 39 (1): [9] ZHAO, F. M., GAO, C. J., and XIA, Q Overlap Grid Research on the Application of Ship CFD. Journal of Ship Mechanics 15 (4): [10] Longo, J., and Stern, F Uncertainty Assessment for Towing Tank Tests with Example for Surface Combatant DTMB Model Journal of Ship Research 49 (1): [11] GUI, L., Longo, J., and Stern, F Biases of PIV Measurement of Turbulent Flow and the Masked Correlation-Based Interrogation. Experiments in Fluids 30 (1): [12] YANG, C. L., ZHU, R. C., MIAO, G. P., and FAN, J Uncertainty Analysis in CFD for Flow Simulation around Ship Using RANS and DES. Journal of Shanghai University 46 (3): [13] Stern, F., Wilson, R., and SHAO, J Quantitative V&V of CFD Simulations and Certification of CFD Codes. Int. J. Numer. Methods Fluids 50: