Prev Subsection 3.5.1: Weak Formulation Up Section 3.5: Computational Fluid Dynamics Subsection 3.5.3: Spatial Discretization Next
3.5.2 Temporal Discretization and Linearization
The time derivatives, which appears in the expression for in eq.(2.11.1-2), and which similarly appears in , may be discretized upon the choice of a time integration scheme, such as the generalized- method  (Section 3.9↓). In this scheme, is evaluated at an intermediate time step between the current time step and previous time step , though different values of are used for the primary variables and their time derivatives. The velocity and volume ratio are evaluated as and at the intermediate time step , whereas their time derivatives are evaluated as and at the intermediate time step . The parameters and are evaluated from the spectral radius for an infinite time step, , as described in Section 3.9↓. The solution of the nonlinear equation is obtained by linearizing this relation as where the operator represents the directional derivative of at along an increment of , or of . The aim of this analysis is to solve for the velocity and volume ratio at the current time step . Using the split form of between external and internal work contributions, this relation may be expanded as
In this framework the finite element mesh is defined on the spatial domain , which is fixed (time-invariant) in conventional CFD treatments. Thus, we can linearize along increments in the velocity and in the volume ratio , by simply bringing the directional derivative operator into the integrals of eqs.(3.5.1-2)-(3.5.1-3). The linearization of and is given by whereas that of and is given by Here, is the current time increment and is the Newmark integration parameter .
The linearization of along an increment is then where we have introduced the fourth-order tensor representing the tangent of the viscous stress with respect to the rate of deformation, Note that exhibits minor symmetries because of the symmetries of and ; in Cartesian components, we have and . In general, does not exhibit major symmetry ( ), though the common constitutive relations adopted in fluid mechanics produce such symmetry as shown below.
The linearization of along an increment is where we have used ; and respectively represent the first and second derivatives of . We have also defined as the tangent of the viscous stress with respect to ,
For the external work, when , and are prescribed, these linearizations simplify to and
We may define the fluid dilatation as an alternative essential variable, since initial and boundary conditions are more convenient to handle in a numerical scheme than . It follows that and . Therefore the changes to the above equations are minimal, simply requiring the substitution and . Steady-state analyses may be obtained by setting the terms involving to zero in eqs.(3.5.2-3)-(3.5.2-4), (3.5.2-5) and (3.5.2-7).