Although GCI estimation, the accuracy of the SLS - GCI method was found to be better than that by Mod. The rod bundle simulation takes advantage of the symmetries of the domain which can be divided into twelve homologous sections.
Introduction
Overview
The statistical analysis of the probability density functions (PDFs) of the Nusselt number for different Prandtl numbers is also presented. The fully developed turbulent channel flow by direct numerical simulation using the finite difference method and the three Reynolds numbers (𝑅𝑒𝜏 =180,395 and 640) by Hiroyuki Abe et al.
Research object
To reduce the computational cost, one can take advantage of the symmetries of the domain. The SLS – GCI is chosen for GCI estimation in this case with three belt levels arrangement because of the GCI estimation results of the pipe flow simulation show that the accuracy of the estimation by SLS – GCI is greater than that by Mod.
Thesis Outline
ASME and SLS - GCI used to measure the percentage difference between the numerical value and the asymptotic value, and to indicate changes in the solution as the grid is further refined and the polynomial order is increased or decreased in the spectral element method. The three different grid levels are considered with the fixed 7th order polynomial order.
Governing equation and Numerical Methods
Governing equations
- Boundary conditions
- Time discretization
- Interpolating Polynomial
The mathematical formulation and physical interpretation of the boundary conditions for the velocity and temperature fields are used in this dissertation as shown below. The wall boundary condition ("W") is known as the no-slip condition, mathematically equivalent to the homogeneous Dirichlet condition, 𝒖 = 0. The symmetric boundary condition ("SYM") is defined by the formula with a normal vector, 𝒏 , and a tangent vector, 𝒕 , as below,.
In the Navier–Stokes discretization, it is difficult to solve if the treatment implicitly creates a non-symmetric, non-linear system. While the explicit method uses a high-order extrapolation to the nonlinear terms, including the advection part. The two families most important in the context of the spectral element method are the Legendre polynomial and the Lobatt polynomial.
In this thesis, the elementary mesh is generated by python code and gmsh, the msh2nek tool converts this to a .re2 file and then the GLL nodes are distributed at initialization in Nek5000.
Large eddy simulation models
Grid convergence index
- Roache’s GCI method
- Modified ASME V&V 20 method
- Simplified least square version GCI
- Combination of Uncertainties
Roache's GCI method can be used to compute a higher-order estimate of the flow field from a range of lower-order discrete values. The GCI provides an estimate of the discretization error in the finest grid solution relative to the converged numerical solution. However, in the original method, the order of accuracy is calculated by the logarithmic function of the differences of the 𝑓𝑘 value.
However, convergence would be problematic if the mesh refinement factors did not have a constant rate, and the iteration step must be chosen to determine the observed order of convergence. As a result, the GCI value 𝑢𝐺𝐶𝐼 and the estimation error between the numerical and estimated results 𝑢𝑒 can be calculated according to the convergence order 𝑝:. 2.3.17). The index of 𝜀𝑐 estimated from the RMS of the differences between the reference result 𝑓0 (eg, experimental and analytical results) and the extrapolation result 𝑓𝑐 is also an uncertainty factor.
The index of 𝜀𝑐 estimated is estimated as follows:. 2.3.19) In addition, for the SLS-GCI, the local estimation error 𝑢𝑒 is counted as an uncertainty factor as follows:. 2.3.20) Therefore, the combined standard uncertainty 𝑈𝑆 of the numerical discretization as the overall value in the domain can be estimated by the root sum square (RSS) function with respect to the global values of the GCI of 𝑈GCI with a coefficient of change of 1, 15 to adjust the value for the standard deviation, the estimation error 𝑈𝑒 and the difference 𝜀𝑐 between the extrapolation and the reference:.
Deterministic Simulation for Pipe Flow
Simulation domain and numerical scheme
Three mesh levels were created in coarse, medium and fine mesh with three different polynomial orders for each mesh. The GLL score distribution density of each element is based on the number of element orders. This is in contrast to increasing the mesh refinement by dividing the computational domain multiple times to reduce the element size.
Thus, the total number of elements and nodes was promoted by increases in polynomial order.
Statistical properties of velocity and temperature
- Mean velocity profile
- Root mean square velocity
- Mean temperature profile
- Heat flux
- Instantaneous flow fields
- Computing memory and runtime
The results of each order agree well with the DNS data and the law of the wall in the viscous sub-layer region. However, the 4th order polynomial shows higher values than the results of the other orders and the reference data in the log law region are the results even though the finest mesh was used. The effect of the mesh type, coarse, medium and fine, under the same polynomial order 𝑁 = 8 is shown in Figure 4(b).
In the case of the 8th polynomial order (Figure 7(b)), the predicted average temperature profile does not depend on the grid level, and all the predicted values agree with each other and the reference data (Ould – Rouiss, Bousbai, and Mazouz 2013) [8]. Figure 8 shows the streamwise/axial turbulent heat flux as a function of the distance from the wall 𝑦+, at various polynomial orders (Figure 8(a)) and grid levels (Figure 8(b)). The instantaneous temperature fields in the pipe at the middle cross-section of the computational domain (Figure 3) for the different polynomial orders are shown in Figure 10(a), (b) and (c).
The effect of grid number on the instantaneous temperature field is shown in Figure 10(c), (d) and (e). Consideration of mesh size and polynomial order is important for optimal computation time and memory capacity while still ensuring accuracy of results. The total running time and required memory capacity based on the 6th order polynomial are much less than those of the 8th order polynomial on the same grid.
Grid convergence index
Among the three available GCI estimation methods (the original Roache method, Mod. – ASME, and SLS – GCI), we used the latter two in the present study since the original Roache method requires a monotonic solution (increase or decrease of results simultaneously between different grid systems). In this simulation, the calculated Nusselt number by the Gnielinski correlation [21] is 𝑁𝑢 = 49.753 when the Reynolds number Re = 19,000 and the Prandtl number 𝑃𝑟 = 0.71 as the reference result. The extrapolation value of the SLS–GCI method appears to be better in terms of difference from the reference solution: 𝜀 = 0.2248% for the polynomial order effect case and 𝜀 = 0.2365% for the mesh number effect case.
ASME method, the difference from the reference solution, 𝜀, is more than two times greater than that of the SLS–GCI, as reported in Table 5. The GCI values, 𝑢𝐺𝐶𝐼, by SLS–GCI are 0.0635 and 0, 0179 for the polynomial order and the mesh cases, respectively, while the Mod. The ASME method was estimated by the difference between the extrapolated value and the numerical value at the fine mesh, while the SLS-GCI method estimated the GCI value by the difference between the two values by a power law.
The insufficient GCI value results from ASME were due to its poor extrapolation compared to SLS - GCI, although the difference between Mod.
Deterministic Simulation for Rod Bundle flow
Computational domain and numerical scheme
The 𝑃𝑁− 𝑃𝑁−2 SEM formulation is considered for solution with 𝑁 = 7 which means that the velocity space is approximated by typical 7th order Lagrangian polynomial interpolations on the GLL points, and pressure space uses 5th order Lagrange interpolants on the Gaussdra-Legendre marks. The time integration scheme with the viscous terms treated implicitly by a second-order backward differentiation (BDF2), and the non-linear terms explicitly by a second-order extrapolation (EXT2) treatment, is BDF2/EXT2 one. Structured meshes were considered for three different grid levels (coarse, medium and fine) at the fixed order of 7th order, which is shown in figure 14.
Evaluation results of velocity and temperature
- Velocity profile
- Instantaneous flow fields
In Figure 15 , the profiles of the axial velocity ( Figure 15(a) ) and radial velocity ( Figure 15(b ) ) in the present simulation are compared with the results of single rod bundles by Javier Martínez et al. The axial velocity profiles approached the values of the single-bar LES results of Javier Martínez et al. This shows that the resolution of fine mesh is higher than that of coarse mesh.
Grid convergence index
For the radial velocity (Figure 15(b)), the finer grid profile matches that of the reference results (Javier Martínez et al. Simulation results for the axial/radial velocity are performed in Section 4.2 to provide data for GCI estimation. The error of the exact and the numerical results of 𝜀0 at axial and radial velocity are 13.24% and 6.38%, respectively, as shown in Table 7.
The US of 16.46% and 8.47% of axial velocity and radial velocity are shown by the error bar on the numerical results, respectively, as shown by the error bar on the numerical results in Figure 16. Where the index 𝜀0 in Table 6 is the error between the reference results and the numerical result in case – F, 𝑓1.
Conclusion
Conclusion
Although GCI estimation of polynomial order and grid levels affects the case in the pipe results, the accuracy of the SLS – GCI method was found to be greater than that by Mod. Thus, the SLS – GCI was chosen to study uncertainty quantification by GCI estimation for bar – beam results effect of grid level in bar beam simulation.
Future work
Miyake, Numerical simulation of thermal streak phenomena in a T-junction piping system for fundamental validation and uncertainty quantification by GCI estimation. Mazouz, Large eddy simulation of turbulent heat transfer in pipe flow with respect to Reynolds and Prandtl number effects. Matsuo, Direct numerical simulation of a fully developed turbulent channel flow with respect to the Reynolds number dependence.
El Khoury, G., Schlatter, P., Noorani, A., Fischer, P., Brethouwer, G., Johansson, A., Direct numerical simulation of turbulent pipe flow at moderately high Reynolds numbers. Antoranz, A., Gonzalo, A., Flores, O., García-Villalba, M., Numerical simulation of heat transfer in tubes with inhomogeneous thermal boundary conditions. Ryzhenkov, V., Ivashchenko, V., Vinuesa, R., Mullyadzhanov, R., Simulation of heat and mass transfer in turbulent channel flow using the spectral element method: Effect of spatial resolution.
Monji, Numerical simulation of heat streak phenomena in a T-junction piping system using large eddy simulation.
