However, the proposed IMLE method takes advantage of the Eulerian method so that the spatial derivative terms can be approximated to give an even higher order of accuracy, e.g. sixth- and fifth-order accuracy for the pressure gradient and velocity Laplacian terms, respectively. Also, with the help of the interpolation procedure, there is no longer any need to take care of particle. It is noted that the source term for the PPE is calculated from the interpolated velocities.
Therefore, at the beginning of the next time step, the velocity Laplacian term on Eulerian grids can be approximated using a higher accurate numerical scheme without adopting any interpolation procedure that would degrade the order of accuracy. Using the moving least squares (MLS) interpolation method [44], velocity components can be interpolated back to the Eulerian grid to calculate the source. Moreover, the order of accuracy in the calculation of diffusion terms is as high as that of the numerical schemes adopted on Eulerian grids and is not limited by the interpolation scheme presented in the original MLE method.
In the original MLE method, the continuity equation is solved on Eulerian grids aiming to preserve the elliptic property of the PPE. In solving the periodic boundary value problem, the same procedures can be used with only a slight change of the matricesLandU. 3.3 The newly proposed improved mixed Lagrangian-Eulerian method. 2 Schematic of the interpolation process (circles indicate Eulerian grids; triangles indicate Lagrangian particles; solid symbols indicate the points participating in the interpolation process; hollow symbols indicate the points not participating in the interpolation process).
Therefore, the precision order of the diffusion terms on Lagrangian particles is limited by the precision order of the assumed interpolation scheme.
Boundary treatments for intermediate velocity
Boundary treatment for Neumann boundary condition
For completeness, the numerical procedure of the IMLE method is summarized below. After the interpolation step (step 9 of Algorithm1), the MLE particles continue to move from the intermediate locations r∗ to the next locations rn+1 (step 12 of Algorithm 1) until they are deleted. This means that MLE particles are not always located at the Eulerian grid points.
Thus, another interpolation (step 5 in Algorithm 1) is required to interpolate the diffusion terms from the Eulerian grids to the Lagrangian particles after these terms have been computed on the Eulerian grids with the VC-CCD scheme. In each advection step (step 7 in Algorithm 1 and step 6 in Algorithm 5), the departure points of the MLE particles may not be located at the Euler grid points, while the IMLE particles are precisely offset from the Euler grid points (shown in Fig. 3). In other words, the distance between a Lagrangian particle and its corresponding Euler grid point can be longer for the MLE method, i.e. |rn−rn−2|MLE>|r∗−rn|IMLE.
Consequently, we can expect IMLE particles to be more uniformly distributed than MLE particles (small/large deviation from uniform Euler grids in IMLE/MLE methods). In this situation, a particle distribution with a smaller bias can reduce the interpolation error of MLS [63] in the IMLE method. The interpolation process (step 7 in Algorithm 5) can be thought of as a re-meshing step that reconnects the slightly non-uniform Lagrangian particles to uniform Eulerian grids.
IMLE adopts lumped grid, and there is no need for interpolating pressure gradient term; therefore IMLE is more efficient. With respect to the conventional finite difference method, which directly discretizes the convection terms, the previous process (steps 6–7 in Algorithm5) can be seen as an alternative way to calculate the addition of particle derivative and convection terms at grid points. As shown above, the particles move along the streamlines so that no spurious diffusion error is generated using any local one-dimensional excitation scheme.
Furthermore, since no convection terms are used in the IMLE method, there is no dispersion error resulting from the discretization of the convection terms. Red circles indicate cell centers, blue diamonds indicate MLE particles at different time steps, and blue triangle indicates the intermediate location of IMLE particles. 4 Schematic of determining the location and velocity of particles of the boundary surface (green solid diamond) for the Neumann boundary condition in the x-direction.
Boundary treatment for interpolation procedure When using MLS interpolation, there are six unknowns in
However, most of the kernel functions are developed in a continuous sense, i.e., satisfy Eq. 27) only, they are not guaranteed to satisfy the discrete unit condition as stated. In order to enforce the unsatisfied discrete unitary condition, reproduce kernel particle method (RKPM) [71]. or improved SPH methods [69,70] have been developed. On the other hand, MLS interpolation method is based on the concept of surface reconstruction. 19), a smooth function can be accurately reconstructed under the nine-particle framework.
4 Verification and validation studies 4.1 Taylor–Green vortex flow
Backward-facing step flow
In this subsection, a backward step flow problem is considered to demonstrate the ability to apply IMLE to simulate inflow-outflow problem. Reynolds number for this problem is defined as Re=ρU L/ν, whereρ is the density which is 1,Uthe average inflow velocity which is 1 andL the hydraulic length which is equal to 2Hi. Uniform grid sizex =y =0.1 and time step =0.01 are assumed so that Courant number is approximately equal to 0.15.
Simulation times of T =500.0 and 1000.0 are taken for the cases with low and high Reynolds numbers, respectively. To compare the numerical results with the experimental data [74], normalized velocity profiles (u/U) at different cross-sections (x/Hs) are compared. Second, fig. 11 that velocity profiles obtained from the IMLE method are in good agreement with reference experimental data.
The degree of computational improvement is defined as the fraction of the computation time of the MLE method compared to that of the IMLE method. The numerical data of the velocity u at different cross-sections are shown in Tables 3 and 4 for different Reynolds numbers.
Lid-driven cavity flow
13 Comparison of the velocity profiles between IMLE, MLE and Ghia [75] for different Reynolds numbers for the lid driven cavity flow problem. more particles) with the same number of grid points show more accurate velocity profiles (compare cases (1)–(2) and (4)–(5)). However, comparing the middle and bottom rows of Fig. 14 and 15, the IMLE method shows even smoother contours both foru, v and p under the same number of grids/particles. In comparison with the conventional particle method, the IMLE method shows accurate results with a smaller number of grids and particles (nc2=802and 2002), while the SPH method reported in [55] used much larger number of particles (N = 2002and 4002).
Therefore, the IMLE method is said to be more accurate than the previously proposed MLE method. For solving the MLS matrix equation, only the backward elimination procedure is taken in the MLS interpolation in the IMLE method. It is not easy for the MLE method to reach steady-state solutions, while the application of the IMLE method can obtain nominally steady-state solutions.
Therefore, it is clearly shown that IMLE method is more stable than the previously proposed MLE method. After illustrating the advantages of the proposed IMLE method in terms of the accuracy, efficiency and stability, the effects of numerical schemes on the accuracy of the. 16 Convergence profiles of the MLE (top row) and IMLE (bottom row) methods of the lid-driven cavity flow problem.
17 Number of CG iterations of the MLE (top row) and IMLE (bottom row) methods of the lid-driven cavity flow problem. Apparently, the accuracy of the numerical solution is more sensitive to the MLS interpolation scheme. Finally, the proposed IMLE method is compared with the colocated finite volume solutions obtained from Open-Foam [76].
Figure 19 shows that the predicted velocity profiles using the proposed IMLE method approach those of the third-order QUICK scheme, while the FUD scheme (from OpenFoam) shows significant numerical scatter. Meanwhile, Figure 20 shows that the contours of the u and v velocity components predicted by the proposed IMLE method are significantly smoother than those obtained using the QUICK scheme. Figure 21 shows the streamlines and vorticity contours obtained by the proposed IMLE method and OpenFoam with the QUICK convective scheme.
It is clearly seen that non-oscillatory numerical solutions can be obtained using the proposed IMLE method. However, the proposed IMLE method is more physically reasonable as the moving particles are merely redirected along the streamlines at their respective velocities.
![Fig. 13 Comparison of the velocity profiles between IMLE, MLE and Ghia [75] for different Reynolds numbers of the lid-driven cavity flow problem](https://thumb-ap.123doks.com/thumbv2/azpdforg/10239164.0/19.892.115.775.79.387/comparison-velocity-profiles-different-reynolds-numbers-driven-problem.webp)
5 Concluding remarks
The icoFoam module in OpenFoam is used, solving the unsteady incompressible laminar Navier–Stokes equations with the PISO scheme. 20 Contours of velocity components obtained from the proposed IMLE method and OpenFoamicoFoammodule with FUD and QUICK convective schemes. 21 Streamlines and vorticity contours obtained from IMLE and OpenFoam solutions with the QUICK scheme.a Streamlines obtained from the calculated IMLE solutions;bstreamlines obtained from the OpenFoam solutions with the QUICK scheme;cvor-.
In the last section, the IMLE method shows good numerical results for various types of problems including periodic flows, input-output and bounded flows. Finally, it is reiterated that for the fully periodic problem, our proposed IMLE method can provide fourth- and second-order accuracy for velocity and pressure, respectively. Also, IMLE outperforms MLE in terms of numerical accuracy, computational efficiency, and scheme stability.
It is also worth noting that the proposed method can be easily extended to solve the three-dimensional. Since the proposed IMLE method uses Eulerian grids to calculate the velocity Laplacian term and solve the PPE, one has to make extra efforts to confront the problems of complex geometries within the flow domain, irregular physical domain and distorted free surface . For these issues, the coupled immersed boundary (IB) [77] and IMLE method, and the coupled level-set [10,29] and IMLE method are under our current development.
