Upwind cell

QUICK Differencing Scheme. The QUICK differencing scheme (Leonard and Mokhtari, 1990) is similar to the second-order upwind differencing scheme, with modifications that restrict its use to quadrilateral or hexahedral meshes. In addition to the value of the variable at the upwind cell center, the value from the next neighbor upwind is also used. Along with the value at the node P, a quadratic function is fitted to the variable at these three points and used to compute the face value. This scheme can offer improvements over the second-order upwind differencing scheme for some flows with high swirl. 

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. 

Figure 2.60 Pictorial representation of the SLIC scheme showing the updating scheme for an upwind and a downwind cell. Cells filled with fluid 1 are indicated in gray, those with fluid 2 in white. Cells containing a mixture of both fluids are represented by hatched areas. In the right column the configuration at the new time step is shown, with interface positions depicted explicitly.

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 flux terms are discretized at the cell interface. Because transport occurs in the direction of flow, we will use upwind differences. In addition, we will use an exphcit solution technique by discretizing our flux terms at the n (previous) time step ... 

One of the most popular VOF methods is that due to Hirt and Nichols (1981). This method uses an approximate interface reconstruction that forces the interface to align with one of the co-ordinate axis, depending on the prevailing direction of the interface normal. A schematic diagram of reconstruction of a two-dimensional interface is shown in Fig. 7.9. To compute fluxes in a direction parallel to the reconstructed interface, upwind fluxes are used. Fluxes in a direction perpendicular to the reconstructed interface are estimated using a donor-acceptor method. In a donor-acceptor method, a computational cell is identified as a donor of some amount of fluid from one phase and another neighbor cell is identified as the acceptor of that donated amount of fluid. The amount of fluid from one phase that can be convected (donated) across a cell boundary is limited by the minimum of the filled volume in the donor cell or the free volume in the acceptor cell. This minimizes numerical diffusion at the interface. 

If a first-order upwind spatial-differencing scheme is used, the reconstructed moments tW/t,i i/2 at the cell interfaces are equal to the upwind values and, in the case of positive u, the following equation is obtained ... 

With standard DQMOM, when the explicit Euler scheme in time and the first-order upwind differencing scheme for space are employed, the volume-average weights in the cell centered at X at time (n + l)Af are... 

A third problem plaguing numerical solution of advection problems can be seen if the upwind scheme is modified to include the updated information in the cell i — 1. Replacing c" with in (25.128), we get... 

The subscript i refers to gas-phase reactants and products the superscript c refers to gas-phase cell threads. The accumulation term in (3.43) is equalized to the sum of the convective and reactive fluxes. The reactive term is computed only when a control volume has a face adjacent to the catalytic geometrical surface Fext), while the other terms are considered for internal faces Pint). Face variables, denoted by the subscript f, are computed with a first-order upwind scheme. If Peclet number, defined as shown in (3.44), is significantly greater than 1 (as happens for the LNT case study described later), it is possible to neglect the diffusion term vL... 

The functional relationship of the HRIC scheme is also function of the angle 6 between the normal to the interface and the normal to the cell face (Fig. 8). For an interface aligned with the cell face (9 = 0), the bounded downwind scheme is used, while for an interface perpendicular to the cell face, the upwind scheme is used. For an interface with 6 between these two limits,/(0) is chosen to be V cos 6 and the blending formula is given by... 

As in the case of decretizing fluid flow equations. Equation (5.27) requires the appropriate "upwinding" and as a result Equation (5.28) represents the upstream donor cell difference whereby 7=1 gives a full upstream effect. For 7 = 0 the equation becomes numerically unstable (Anderson et al., 1984). 

A 3D solution for the film and pad temperature using the control cell technique, with upwinding of the film flows. 

The computational domain extended 170 m in the X direction (from 20 m upwind to 150 m downwind from the release point), 30 m in the Y direction (symmetric crosswind plan from the release point) and 10 m in the Z direction the cells were initially represented by 1 m x 1 m x 0.5 m cuboids (forming the macro grid). 

First-order upwind Good when convection dominates and the flow is aligned with the grid. Assumes that the face value for each variable is equal to the upstream cell center value. Stable, and a good way to start off a calculation. A switch to a higher-order scheme is usually recommended once the solution has partially converged. 

---