Upwinding scheme

In the earlier versions of the streamline upwinding scheme the modified weight function was only applied to the convection tenns (i.e. first-order derivatives in the hyperbolic equations) while all other terms were weighted in the usual manner. This is called selective or inconsistent upwinding. Selective upwinding can be interpreted as the introduction of an artificial diffusion in addition to the physical diffusion to the weighted residual statement of the differential equation. This improves the stability of the scheme but the accuracy of the solution declines. 

Hughes, T. J.R. and Brooks, A.N., 1979, A multidimensional upwind scheme with no cross-wind diffusion. In Hughes, I . J. R. (ed.), Finite Element Methods for Convection Dominated Flows, AMD Vol. 34, ASME, New York. 

Derivation of the working equations of upwinded schemes for heat transport in a polymeric flow is similar to the previously described weighted residual Petrov-Galerkm finite element method. In this section a basic outline of this derivation is given using a steady-state heat balance equation as an example. 

Extension of the streamline Petrov -Galerkin method to transient heat transport problems by a space-time least-squares procedure is reported by Nguen and Reynen (1984). The close relationship between SUPG and the least-squares finite element discretizations is discussed in Chapter 4. An analogous transient upwinding scheme, based on the previously described 0 time-stepping technique, can also be developed (Zienkiewicz and Taylor, 1994). 

Luo, X. L, and Tanner, R. L, 1989. A decoupled finite element streamline-upwind scheme for viscoelastic flow problems. J. Non-Newtonian Fluid Mech. 31, 143-162. 

The inconsistent streamline upwind scheme described in the last section is fonuulated in an ad hoc manner and does not correspond to a weighted residual statement in a strict sense. In tins seetion we consider the development of weighted residual schemes for the finite element solution of the energy equation. Using vector notation for simplicity the energy equation is written as... 

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. 

It can easily be shown that for the upwind scheme all coefficients a appearing in Eq. (37) are positive [81]. Thus, no unphysical oscillatory solutions are foimd and stability problems with iterative equation solvers are usually avoided. The disadvantage of the upwind scheme is its low approximation order. The convective fluxes at the cell faces are only approximated up to corrections of order h, which leaves room for large errors on course grids. 

Solve the convection equation of high order (3rd order) essentially non-oscillatory (ENO) upwind scheme (Sussman et al., 1994) is used to calculate the convective term V V

velocity field P". The time advancement is accomplished using the second-order total variation diminishing (TVD) Runge-Kutta method (Chen and Fan, 2004). 

The advection—diffusion equation with a source term can be solved by CFD algorithms in general. Patankar provided an excellent introduction to numerical fluid flow and heat transfer. Oran and Boris discussed numerical solutions of diffusion—convection problems with chemical reactions. Since fuel cells feature an aspect ratio of the order of 100, 0(100), the upwind scheme for the flow-field solution is applicable and proves to be very effective. Unstructured meshes are commonly employed in commercial CFD codes. 

The finite volume methods have been used to discretised the partial differential equations of the model using the Simple method for pressure-velocity coupling and the second order upwind scheme to interpolate the variables on the surface of the control volume. The segregated solution algorithm was selected. The Reynolds stress turbulence model was used in this model due to the anisotropic nature of the turbulence in cyclones. Standard fluent wall functions were applied and high order discretisation schemes were also used. 

P 77] Simulation was done using CFD-ACE+ [55,163], The physical properties of water were assumed. Discretization with structured grids of 15 pm length was used. The first-order upwind scheme and conjugate gradient with preconditioning solver were applied. 

Many CFD codes also add an artificial viscosity / . This viscosity can be explicit or it can be hidden, for example in the case of upwind schemes. An important dissipation is also introduced by large time steps and implicit schemes which are commonly used in RANS. 

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. 

Several schemes and algorithms for solving the fluid dynamic part of the model have been published. This work has been concentrated on several items. Most important, one avoids using the very diffusive first order upwind schemes discretizing the convective terms in the multi-fluid transport equations. Instead higher order schemes that are more accurate have been implemented into the codes [62, 139, 140, 65, 105, 66[. The numerical truncation errors induced by the discretization scheme employed for the convective terms may severely alter the numerical solution and this can destroy the physics reflected... 

The first order explicit upwind scheme was introduced by Courant, Isaacson and Reeves [31], and later on several extensions to second order accuracy and implicit time integrations have been developed. 

In the upwind scheme tpe is approximated by the tp value at the node upstream of the surface location e ... 

MUSCL (Monotonic Upwind Scheme for Convective Laws) limiter of van Leer [194, 195] ... 

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. 

To illustrate the principles of the finite volume method, as a first approach, the implicit upwind differencing scheme is used for a multi-dimensional problem. Although the upwind differencing scheme is very diffusive, this scheme is frequently recommended on the grounds of its stability as the preferred method for treatment of convection terms in multiphase flow and determines the basis for the implementation of many higher order upwinding schemes. 

The upwind scheme is obtained by substituting the UDS approximation of the convective terms and central difference approximation of the diffusive terms into the equation. 

For unsteady problems the discretized algebraic equation, for the upwind scheme example, is generally written ... 

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. 

By use of the upwind scheme for the convective terms, the generalized transport equation becomes ... 

MUSCL Monotone Upwind Scheme for Conservative Laws MUSIC MUltiple-SIze-Group MWR Method of Weighted Residuals NG Number of Groups... 

Then, if the first-order upwind scheme is employed, the final equation is... 

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-... 

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