Convection upwinding

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. 

The first order derivative in Equation (2.80) corresponds to the convection in a field problem and the examples shown in Figure 2.26 illustraTes the ina bility of the standard Galerkin method to produce meaningful results for convection-dominated equations. As described in the previous section to resolve this difficulty, in the solution of hyperbolic (convection-dominated) equations, upwind-ing or Petrov-Galerkin methods are employed. To demonstrate the application of upwinding we consider the case where only the weight function applied to the first-order derivative in the weak variational statement of the problem, represented by Equation (2.82), is modified. 

The finite element results obtained for various values of (3 are compared with the analytical solution in Figure 2.27. As can be seen using a value of /3 = 0.5 a stable numerical solution is obtained. However, this solution is over-damped and inaccurate. Therefore the main problem is to find a value of upwinding parameter that eliminates oscillations without generating over-damped results. To illustrate this concept let us consider the following convection-diffusion equation... 

Brooks, A. N, and Hughes, T. J.R., 1982. Streamline-upwind/Petrov Galerldn formulations for convection dominated hows with particular emphasis on the incompressible Navier -Stokes equations. Cornput. Methods Appl Meek Eng. 32, 199-259. 

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. 

The integrals in Equation (3.32) are found using a quadrature over the element domain The viscoelastic constitutive equations used in the described model are hyperbolic equations and to obtain numerically stable solutions the convection terms in Equation (3.32) are weighted using streamline upwinding as (inconsistent upwinding)... 

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 origin of atmospheric turbulence is diurnal heating of the Earth s surface, which gives rise to the convection currents that ultimately drive weather. Differential velocities caused perhaps when the wind encounters an obstacle such as a mountain, result in turbulent flow. The strength of the turbulence depends on a number of factors, including geography it is noted that the best observation sites tend to be the most windward mountaintops of a range— downwind sites experience more severe turbulence caused by the disturbance of those mountains upwind. 

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. 

Streamline upwind Petrov/Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations,... 

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. 

Upwind differences are typical for convective flux, where the upstream concentration is important to determine the convective flux at the upstream interface. Upwind differences have a lower numerical diffusion than central differences... 

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) ... 

An upwind difference of the convective term presumes that v is always positive, that is, that vertical flow from the lower toward the upper plate. The discrete boundary conditions are given as... 

Care must be taken with the convective terms in the transport equations to account for the axial flow direction. In the stagnation-flow problems for flow against a surface, the axial velocity is always negative (i.e., flowing toward the surface). The convective term in the radial-momentum equation uses the following upwind difference approximation ... 

The convective terms are approximated using an upwind difference formula. For example,... 

There are many ways of solving the energy equation with convection effects. One that will be presented here is the widely accepted streamline upwind Petrov-Galerkin method... 

SUPG) developed by Brooks and Hughes [3]. Essentially, the finite element equations remain the same however, as shown here, modified shape functions are introduced on the upwind side of a nodal point. Hence, we have two interpolation, or shape, functions that define the temperature, or convected variable, distribution. One definition uses the conventional shape functions given by... 

A.N. Brooks and T.J.R. Hughes. Streamline upwind-Petrov-Garlerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comp. Meth. Appl. Mech. Eng., 32 199, 1982. 

The newer method of Bortels et al., called multidimensional upwinding method (MDUM) should also be mentioned [127]. It was applied to a problem involving diffusion, convection and migration, both steady state and timemarching. 

Because of the success encountered by finite elements in the solution of elliptic problems, it was extended (in the 80s) to the advection or transport equation which is a hyperbolic equation with only one real characteristic. This equation can be solved naturally for an analytical velocity field by solving a time differential equation. It appeared important, when the velocity field was numerically obtained, to be able to solve simultaneously propagation and diffusion equations at low cost. By introducing upwinding in test functions or in the discretization scheme, the particular nature of the transport equation was considered. In this case, a particular direction is given at each point (the direction of the convecting flow) and boundary conditions are only considered on the part of the boundary where the flow is entrant. 

An element for the stress components composed of 16 sub-elements (4x4) on which bilinear (continuous) polynomials are used, was introduced by Marchal and Crochet in [28]. This leads to a continuous C° approximation of the three variables. The velocity is approximated by biquadratic polynomials while the pressure is linear. Fortin and Pierre ([17]) made a mathematical analysis of the Stokes problem for this three-field formulation. They conclude that the polynomial approximations of the different variables should satisfy the generalized inf-sup (Brezzi-Babuska) condition introduced by Marchal and Crochet and they proved it was the case for the Marchal and Crochet element. In order to take into account the hyperbolic character of the constitutive equation, Marchal and Crochet have implemented and compared two different methods. The first is the Streamline-Upwind/Petrov-Galerkin (SUPG). Thus a so-called non-consistent Streamline-Upwind (SU) is also considered (already used in [13]). As a test problem, they selected the "stick-slip" flow. With SUPG method applied to this problem, wiggles in the stress and the velocity field were obtained. In the SU method, the modified weighting function only applies to the convective terms in the constitutive equations. 

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