Many numerical approaches to CFD can be extended to arbitrarily high order (for instance, discontinuous Galerkin methods, WENO methods, spectral differencing, etc.). How should I choose an appropriate order of accuracy for a given problem?
In practice, most people stick to relatively low orders, usually first or second order. This view is often challenged by more theoretical researchers that believe in more accurate answers . The rate of convergence for simple smooth problems is well documented, for example see Bill Mitchell's comparison of hp adaptivity.
While for theoretical works it is nice to see what the convergence rate are, for more application oriented among us this concern is balanced with constitutive laws, necessary precision, and code complexity. It doesn't make much since in many porous media problems that solve over a highly discontinuous media to have high order methods, the numerical error will dominate the discretization errors. The same concern applies for problems that include a large number of degrees of freedom. Since low-order implicit methods have a smaller bandwidth and often a better conditioning, the high order method becomes too costly to solve. Finally the code complexity of switching orders and types of polynomials is usually too much for the graduate students running the application codes.
Guidelines: High-order methods for problem where solution is expected to be smooth and otherwise low-order methods and/or methods which can handle discontinuities in solutions. In cases where high-order methods can be exploited there can be significant saving in computational effort measured in terms CPU time as a result of high convergence rate. For elliptic problems which require solution of linear systems, high-order methods leads to less sparse operators and this have to be compensated by faster convergence rate. For time-dependent problems, if high-order methods can be exploited faster convergence rate and more accuracy can be achieved and for long integration times high-order methods are superior in terms of both accuracy and computational effort due to low numerical dispersion and dissipation errors.
Higher order methods can be used e.g. to solve the level set equation when using it to describe a two phase fluid flow within a Finite Volume Method framework. In this case, the WENO and ENO schemes are used to advect the level set function and a re-initialisation step is used to maintain it as a distance function from the fluid interface.
Check this out: http://ftp.cc.ac.cn/lcfd/WENO_mem.html
Basically, they are used in CFD simulations when dealing with discontinuities in the flow.
Always implement at least two distinct orders. On a representative problem, solve once using each order. Compare the two on a grid fine enough to be converged at the lower order. Ensure your two answers are reasonably close which gives some indication that the numerical behavior of the lower order scheme hasn't overwhelmingly damaged the solution. If it has, toss the lower order scheme and start over.
Assuming you didn't have to start over, coarsen the grid for the higher order as much as possible while still maintaining a reasonably accurate solution as measured by the specific quantity of interest you want. Compare the computational cost for the lower order on the finer grid to that of the higher order on the coarser grid.
Choose whichever is more operationally advantageous. Document the process for naysayers and so that you can repeat it when the representative problem or quantity of interest changes.