Upwind Difference Solutions

Inserting equation (7.20) into equation (7.21) gives and for the i -I-1/2 interface  

In addition, we will still need to follow the stability criteria, Di 0.5. 

The prior discretization of equation (2.1) uses control volumes with exphcit differences. They are explicit because only the accumulation term contains a concentration at the n -k 1 time step, resulting in an exphcit equation for (equations (E7.1.4), (E7.2.5), (E7.3.4), and (7.25)). Another common option would be fully imphcit (Laasonen) discretization where flux rate terms in equations (7.24) and (7.23) are computed at the n -k 1 time increment, instead of the n increment. Fully implicit is generally preferred over Crank-Nicolson implicit UQ = U Q n + Q.n+i) /P) 

Because velocity will either be upwind or downwind, one of the two bracket terms will be zero, and equation (7.27) is solved as 

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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 ... 

Central differences are applied to diffusion problems, and upwind differences are applied to convective problems, but most cases have both diffusion and convection. This conundrum led Spaulding (1972) to develop exponential differences, which combines both central and upwind differences in an analytical solution of steady, one-dimensional convection and diffusion. Consider a control volume of length Ax, in a flow fleld of velocity U, and transporting a compound, C, at steady state with a diffusion coefficient, D. Then, the governing equation inside of the control volume is a simphflcation of Equation (2.14) ... 

First order hyperbolic differential equations transmit discontinuities without dispersion or dissipation. Unfortunately, as Carver (10) and Carver and Hinds (11) point out, the use of spatial finite difference formulas introduces unwanted dispersion and spurious oscillation problems into the numerical solution of the differential equations. They suggest the use of upwind difference formulas as a way to diminish the oscillation problem. This follows directly from the concept of domain of influence. For hyperbolic systems, the domain of influence of a given variable is downstream from the point of reference, and therefore, a natural consequence is to use upstream difference formulas to estimate downstream conditions. When necessary, the unwanted dispersion problem can be reduced by using low order upwind difference formulas. 

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. 

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. 

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. 

The almost certainly independent evolution of upwind orientation to chemicals in different phylogenetic groups (at least at the ordinal level) and the inherent variability in environmental (particularly wind) conditions in various habitats, ranging from open grassland to dense tropical forests, may have dictated disparate as well as multiple solutions to the problem of discovering the... 

While many different design solutions have been considered in the early stages, for commercial use the modem wind industry has now stabilized on horizontal axis wind turbines (HAWT). A typical example is shown in Fig. 1 a land-based tower with a nacelle mounted on the top, containing the generator, a gearbox, and the rotor. Typically, three-bladed upwind rotors are used. 

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