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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.
Convection and Diffusion
Published in Suhas V. Patankar, Numerical Heat Transfer and Fluid Flow, 2018
where U is the resultant velocity, and θ is the angle (between 0 and 90°) made by the velocity vector with the x direction. It is easy to see from this equation that no false diffusion is present when the resultant flow is along one of the sets of grid lines; on the other hand, the false diffusion is most serious when the flow direction makes an angle of 45° with the grid lines. (3) The amount of false diffusion can be reduced by using smaller Δx and Δy and, whenever possible, by orienting the grid such that the grid lines more or less align with the flow direction. (4) Since real diffusion is present in many problems, it is then sufficient to make the false diffusion small in comparison with the real diffusion. (5) The use of the central-difference scheme is no remedy for false diffusion. As mentioned earlier, the central-difference scheme gives highly unrealistic solutions when large Peclet numbers are involved. (6) The basic cause of false diffusion is the practice of treating the flow across each control-volume face as locally one-dimensional. For the situation shown in the inset of Fig. 5.15, the value of ϕ convected by the oblique flow to the grid point P actually comes from the corner grid point SW. However, this convection is represented as the effect of two separate streams coming from the grid points W and S. (7) Schemes that would give less false diffusion should take account of the multidimensional nature of the flow. It would also be necessary to involve more neighbors in the discretization equation. Although a few such schemes have been worked out [for example, Raithby (1976b)] and have shown an impressive reduction in false diffusion, they are significantly more complicated and so far insufficiently tested. For these reasons, we shall not discuss them here. (8) A more detailed discussion of false diffusion has been given by Raithby (1976a).
Synthetic jet application in the wind turbine concentrator design
Published in Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 2023
Teoman Oktay Kutluca, Emre Koç, Tahir Yavuz
In the computational fluid dynamics (CFD) analyses, the URANS solver is utilized to solve the turbulent flow in a time-dependent manner. The k-ω SST turbulence model is chosen for the modeling of turbulent flow since it provides high accuracy in determining the separation points and slowing down flows. The flow is assumed to be incompressible with a Mach number of 0.15, and thus, the Pressure-Based Solver is utilized for the analysis. The COUPLED scheme, which obtains more precise results and does not experience convergence problems, is preferred as the solution method. For the calculation of gradients in the CFD analysis, the Green-Gaussian Node-Based Approach is used since it gives more accurate computationally intensive results, minimizes false diffusion, and is recommended for unstructured mesh. To calculate cell surface pressures when using a pressure-based solver in FLUENT, the PRESTO interpolation scheme is selected. The Second-Order Upwind method is utilized for the discretization of momentum and turbulent viscosity equations (Ansys Inc 2010).
A meshless local Petrov–Galerkin approach for solving the convection-dominated problems based on the streamline upwind idea and the variational multiscale concept
Published in Numerical Heat Transfer, Part B: Fundamentals, 2018
Zheng-Ji Chen, Zeng-Yao Li, Xue-Hong Wu, Wen-Quan Tao
The numerical errors from different numerical methods are exhibited in Figure 10. It is illustrated that SUMLPG method produces the largest maximum error over 5.80 × 10−1. The stability term changes the discretized equation and produces apparent false diffusion. So the maximum error of VMS-SUMPLG method close to VMS-MLPG method reaches about 4.26 × 10−1 which is larger than that of FVM with QUICK. The relative error of VMS-SUMLPG method is about 3.83 × 10−2 which approaches that of FVM with QUICK, whereas the SUMPLG has the error over 5.25 × 10−2. Therefore, the present VMS-SUMLPG method has the global numerical stability and acceptable numerical accuracy at large Pe.
Two dimensional bed deformation model in turbulent streams
Published in Australian Journal of Civil Engineering, 2019
Amin Gharehbaghi, Birol Kaya, Gökmen Tayfur
In recent years, due to its advantages, TVD scheme has become more popular in open channel flows and rivers. The basic upwind differencing scheme introduces a high level of false diffusion due to its low order of accuracy (first-order) (Versteeg and Malalasekera 2007). ‘Higher-order schemes such as central differencing and Quadratic upwind differencing (QUICK) can give spurious oscillations or ‘wiggles’ when the Peclet number is high. When such higher-order schemes are used to solve turbulent quantities, turbulence energy and rate of dissipation, wiggles can give physically unrealistic negative values and instability. TVD schemes are designed to address this undesirable oscillatory behaviour of higher-order schemes. In TVD schemes, the tendency towards oscillation is counteracted by adding an artificial diffusion fragment or by adding a weighting towards upstream contribution’ (Versteeg and Malalasekera 2007). García‐Navarro, Alcrudo, and Savirón (1992) introduced an addition of a dissipation step to the McCormack numerical scheme for solving 1D open‐channel flow equations. This extra step is devised according to the theory of TVD schemes. They presented results from several computations and compared the results with the analytical solutions for some test problems. As presented in Castro Diaz et al. (2009), TVD scheme was applied in 2D model to determine water surface profile and sediment transport phenomenon. However, they did not consider any turbulence models. Liu, Landry, and García (2008) used the extrapolation method of the TVD scheme for the upwind ratio of consecutive gradients. Researchers have preferred to apply TVD scheme by Riemann solver. In this study however it is preferred to use TVD approximation directly. The codes are developed in MATLAB. One of the novel contributions of this paper is that TVD scheme has been directly employed in the simulation of two dimensional bed deformation and suspended sediment load by two simple but useful turbulence models.