Theory Manual Version 3.4
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Subsection 3.5.1: Weak Formulation Up Section 3.5: Computational Fluid Dynamics Subsection 3.5.3: Spatial Discretization

### 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 [42] (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 [23]. 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 [42].
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).
Subsection 3.5.1: Weak Formulation Up Section 3.5: Computational Fluid Dynamics Subsection 3.5.3: Spatial Discretization