Upwind difference scheme

Equations (4.2), (4.3), and (4.4) are for the fuel channel, water rod channel, and outside core, respectively. The governing equations are discretized using the upwind difference scheme and the full implicit scheme. The boundary conditions are the feedwater flow rate, the feedwater temperature, and the mrbine inlet flow rate. The characteristic of the turbine control valve, expressed as the change of steam flow rate, is shown in Fig. 4.4 [6]. The feedwater flow rate changes with the core pressure as shown in Fig. 4.5 [6]. 

Write finite-difference approximations to the governing equations. In deciding on the difference approximations, be sure to consider the boundary-condition information and the need for upwind differences. Sketch the difference scheme using a stencil. 

The simplest TVD schemes are constructed combining the first-order (and diffusive) upwind scheme and the second order dispersive central difference scheme. These TVD schemes are globally second order accurate, but reduce to first order accuracy at local extrema of the solution. 

The convective terms were solved using a second order TVD scheme in space, and a first order explicit Euler scheme in time. The TVD scheme applied was constructed by combining the central difference scheme and the classical upwind scheme by adopting the smoothness monitor of van Leer [193] and the monotonic centered limiter [194]. The diffusive terms were discretized with a second order central difference scheme. The time-splitting scheme employed is of first order. 

The upwind scheme described here is first-order accurate in space while the central difference scheme is second-order accurate. Hence a central-difference scheme is preferred whenever possible. Since it is the grid Peclet number that decides the behavior of the numerical schemes, it is, in principle, possible to refine the grids until the grid Peclet is smaller than 2. This strategy, however, is often limited by the required computing time. With sufficiently fine meshes, the two schemes should give essen-... 

Both point-by-point and line-by-line overrelaxation methods were used to resolve the algebraic equations. ° An overrelaxation parameter of 1.5-1.8 was typically used. The two methods required similar computational times. An upwind scheme was used for all variables for high-Pe problems, while a central-difference scheme was used for low Pe. For some high-Pe cases, a central-difference scheme was used for the potential, but no appreciable differences in the results were observed. 

FIGURE 25.12 Solution of the square-wave problem using Ax = 0.1 m, At = 0.01 s, and u = lms-1 with the upwind finite difference scheme. Numerical results and true solution after one timestep. 

R. Mittal, P. Moin, Stability of upwind-biased finite difference schemes for large-eddy simulation of turbulent flows, AIAA Journal 35 (8) (1997) 1415-1417. 

Based on the finite volume method, the control equation can be converted to a numerical method for solving algebraic equations. Convection of equation use second-order upwind difference during the discrete process, the solver is based on the pressure, the pressure-velocity coupling adopt the SIMPLE algorithm, pressure interpolation scheme use PRESTO Format. 

Here, central difference scheme and the upwind scheme are used to treat the diffusion and convection terms, respectively. 

The hybrid LBM-FDM method is used for the simulation, the convection term is discretized by upwind weighted scheme, and the diffusion term is discretized by central difference scheme. Runge-Kutta scheme is employed for time stepping. 

Finite difference schemes based on the principles above are called central differencing . Central differencing schemes have stability problems in situations where convective transport is significant compared to the diffusive transport. Various methods have been devised to overcome this problem. The upwind scheme assigns the value of at a cell boundary to be the value at the node point from which the fluid is flowing, rather than the mean as in Eqs. (7.1.5) and (7.1.6). 

In addition to the central differencing and upwind differencing schemes, which are first-order schemes, another popular finite difference scheme is the QUICK scheme, a second-order upwind differencing scheme. Higher order means that more node points are involved when estimating the values of the dependent variables and their derivatives for formulating the finite difference equations. 

All governing equations are all solved using a finite volume discretization, see [7]. All vectors quantities, e.g. position vector, velocity and moment of momentum, are expressed in Cartesian coordinates. Non-staggered variable arrangement is used to define dependent variables all physical quantities are stored and computed at cell centers. An interpolation practice of second order accuracy is adopted to calculate the physical quantities at cell-face center [8]. The deferred correction approach [9] is used to compute the convection term appearing in the governing equations by blending the upwind difference and the centi difference scheme. 

It should also be remembered that the discretization scheme influences the accuracy of the results. In most CFD codes, different discretization schemes can be chosen for the convective terms. Usually, one can choose between first-order schemes (e.g., the first-order upwind scheme or the hybrid scheme) or second-order schemes (e.g., a second-order upwind scheme or some modified QUICK scheme). Second-order schemes are, as the name implies, more accurate than first-order schemes. However, it should also be remembered that the second-order schemes are numerically more unstable than the first-order schemes. Usually, it is a good idea to start the computations using a first-order scheme. Then, when a converged solution has been obtained, the user can continue the calculations with a second-order scheme. 

When the transport equation for c is solved with a discretization scheme such as upwind, artificial diffusive fluxes are induced, effecting a smearing of the interface. When these diffusive fluxes are significant on the time-scale of the simulation, the information on the location of different fluid volumes is lost. The use of higher order discretization schemes is usually not sufficient to reduce the artificial smearing of the interface to a tolerable level. Hence special methods are used to guarantee that a physically reasonable distribution of the volume fraction field is maintained. 

Second method consists of a straightforward discretization method first order (Euler) explicit in time and finite differences in space. Both the time step and the grid size are kept constant and satisfying the Courant Friedrichs Lewy (CFL) condition to ensure the stability of the calculations. To deal with the transport part we have considered the minmod slope limiting method based on the first order upwind flux and the higher order Richtmyer scheme (see, e.g. Quarteroni and Valli, 1994, Chapter 14). We call this method SlopeLimit. 

P (1, 1), are oscillatory in two dimensions (second-order upwinding for details of second-order upwinding schemes, see Shyy et ai, 1992). Characteristics passing through point Q (0.5,0.75) have second-order accuracy (third order, if the slope at Q is 3/4). Thus, NVD can be used to evaluate different discretization schemes as well as devise new ones. 

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