An introduction to topological field theory by Lawrence R.J.

By Lawrence R.J.

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17. 18 and Fig. 4a). However, if the discretisation produces an equation where several unknown values are related to several known values, for example in Fig. 19 with theta not zero, then the computation is known as an implicit scheme. To produce a solution with an explicit scheme each unknown value of U can be easily calculated, but to produce a solution with an implicit scheme a set of simultaneous equations must be solved to find the unknown values of U. '. The answer to this lies in the difference in the stability of the two schemes.

In the figure the variation is linear, but it could equally be a quadratic or cubic variation or a variation of even higher order. If the variation is linear we can describe the value of U at any point along the element as a function of the length along the element x and the values of U that are known at the endpoints of the element. These positions, which are used as reference positions on the element, are known as the nodes of the element. If the variation of the variable was assumed to be quadratic then we would need to know the value of U at three nodes placed at, for example, the end-points of the element and the middle of the element.

Another modelling approach is to try and find values for the Reynolds stresses themselves. Again, complex transport equations for these stresses have to be derived and solved. The advantage of doing this over the methods mentioned previously is that those methods give a single additional viscosity, whereas the direct 29 Chapter 2. Fluids in Motion modelling of the stress terms allows the effects of turbulence to vary in the three coordinate directions. It is this three-dimensional variation that is found when the stresses are measured experimentally.

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