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Finite Volume Method
Published in D. G. Roychowdhury, Computational Fluid Dynamics for Incompressible Flows, 2020
One major problem associated with the central differencing scheme is that it gives equal weightage of both upstream and downstream nodes and does not take care of the flow direction. For a strong convective flow from west-to-east, west cell face, w, should receive much stronger influence from node W than from node P. In upwind differencing scheme which is also known as “donor-cell” differencing scheme, face value ∅f is taken as the value at whichever is the upwind node i.e. ∅f=∅U. Here f stands for the face, while U and D is upstream and downstream node respectively as shown in Figure 5.11.
Numerical Methods for Convection Heat Transfer
Published in Yogesh Jaluria, Kenneth E. Torrance, Computational Heat Transfer, 2017
Yogesh Jaluria, Kenneth E. Torrance
The equation may also be solved numerically by using central and upwind differencing schemes. Figure 6.14 shows the results from Fletcher (1991) for central differencing at various mesh lengths, as well as comparison with upwind differencing results at grid Peclet number Pe =|u| Δx/α or |v |Δy/α = 4. Clearly, physically unrealistic results are obained by using central differencing at large Pe and not very accurate results, though non-oscillatory, are obtained by using upwind differencing. Therefore, the central differencing scheme cannot be taken as an accurate standard approach to evaluate other schemes. As discussed in detail by Patankar (1980), false diffusion can be reduced by grid re nement and by employing schemes that are based on the multi-dimensional nature of the ow. At relatively low grid Peclet numbers, the variable-order schemes perform satisfactorily. However, for high grid Peclet numbers, higher-order differencing schemes are needed to obtain stable and accurate results. Such schemes are considered next.
Longitudinal dispersion
Published in A. W. Jayawardena, Environmental and Hydrological Systems Modelling, 2013
Thus, the accuracy of all finite difference schemes depends on the step size, Δx. The forward and backward differencing schemes have trailing first-order terms and are therefore called first-order approximations. The central differencing scheme has trailing second-order terms and is therefore called second-order approximation. In a first-order scheme, the error would decrease by a factor of 2 if the mesh size (Δx) is halved. In a second-order scheme, the error would be reduced by a factor of 4 when the mesh size is halved. It does not, however, say anything about the magnitude of the error. A second-order scheme may not necessarily be capable of modelling a process more accurately than a first-order approximation. Formulation can be done to obtain the unknown value expressed explicitly in terms of a combination of known values, in which case solutions can be obtained in a recursive manner. However, explicit methods have the inherent problem resulting from the accumulation of round-off errors that leads to numerical instability. Small time steps, governed by certain stability conditions, are required to avoid this problem. In implicit methods, the unknown value is related to other unknown values as well as known values. Implicit methods are unconditionally stable. Solutions cannot be obtained in a recursive manner. Simultaneous solutions are therefore needed.
A new higher-order RBF-FD scheme with optimal variable shape parameter for partial differential equation
Published in Numerical Heat Transfer, Part B: Fundamentals, 2019
Y. L. Ng, K. C. Ng, T. W. H. Sheu
The calculation begins with the initial value of thus simplified the Eq. (28) to conventional second order central differencing scheme. The optimal value of is computed from the initial solution obtained after solving the PDE in Eq. (31). The coefficients of the PDE matrix are updated with the new value of optimal leading to the new refined solution The iterative solution process continues until the solution becomes converged, that is, when
An a priori study of different tabulation methods for turbulent pulverised coal combustion
Published in Combustion Theory and Modelling, 2018
Yujuan Luo, Xu Wen, Haiou Wang, Kun Luo, Hanhui Jin, Jianren Fan
The detailed chemistry simulations are performed with a flow solver called pccFRFoam while the flamelet chemtable extractions are performed with a solver called pccFPVFoam. Both solvers are developed based on the open-source finite-volume CFD code OpenFOAM-v2.3.0 [53] using the same discretisation schemes, which have been extensively validated in our previous works [2,5,6] by comparing with the well-documented experimental data. The gaseous phase governing equations are solved with a finite volume method (FVM) using the PIMPLE algorithm [53]. The discretisation schemes for the conservation equations are listed as follows. For the convection terms, a second-order central differencing scheme is used for momentum and total variation diminishing for the bounded scalars. For the diffusion terms, the central differencing scheme is used. The backward differencing scheme is used for time integration. For purpose of interphase two-way coupling, a second-order linear interpolation scheme is used to interpolate the particle source terms onto the mesh, and to interpolate the gas phase quantities to the particle location. To solve the chemistry, an Euler-implicit solver [53] is adopted, in which the chemistry time-scale is set to be 1.0. The total run time of the present simulation is about 15 days on 256 processors (Intel Xeon E5-2660, Intel, US), resulting in a physical time of 0.1 s using a time step of 1.0 × 10−8 s.
Exponential Time Differencing Schemes for Fuel Depletion and Transport in Molten Salt Reactors: Theory and Implementation
Published in Nuclear Science and Engineering, 2022
Zack Taylor, Benjamin S. Collins, G. Ivan Maldonado
The convective transport term is more difficult to deal with than diffusion. Diffusion has no primary direction of flow; it simply causes a species to evenly distribute through a medium through a concentration gradient. Convection, on the other hand, has a primary flow direction that is driven by a pressure gradient. One of the major drawbacks of using a second-order central differencing scheme is the inability to identify flow direction. In addition to the neglect in identifying the flow direction, the central differencing scheme will cause numerical instability problems for flows with high Péclet numbers.20 To combat these numerical problems and to handle flow direction, a second-order upwind differencing scheme is used.