5.4. TEMPORAL RESOLUTION 47
Figure 5.2: Spatial mesh and numerical convergence analysis. (a) Triangular mesh with a gradually in-
creasing element size from 0.5 m at the boundaries to 20 m in the bulk, consisting of 30246 elements. (b)
Rectangular mesh with thin elongated 0.1-m-by-10-m elements at the boundaries and gradually changing
to nearly square 10-m-by-10-m elements in the bulk, consisting of 3308 elements. (c) The convergence
parameter C (semilog scale) of Eq. (5.19) for all rst-order steady harmonic elds and second-order time-
averaged elds versus the numerical resolution dened by s=dbd, where s is the viscous boundary layer
thickness Eq. (2.64) and dbd is the mesh-element size at the boundary. The elds are solved on triangular
meshes with dierent boundary element sizes, but all with xed bulk element size dbk = 20 µm and growth
rate = 1:3, while the reference solution is calculated for dbd = 0:15 µm, dbk = 2 µm, and = 1:3. The
vertical dotted line indicates the solution for dbd = 0:5 µm which was chosen as the default value for the
simulations in Ref. 29 Appendix D. Figure adapted from Ref. 29 Appendix D.
where C(g) is the relative convergence of a solution g with respect to a reference solution
gref . The reference solution should be calculated for a spatial resolution that is better than
what is expected to be necessary. Figure 5.2(c) shows an example of the numerical convergence
with respect to the spatial resolution s=dbd, where s is the viscous boundary layer
thickness Eq. (2.64) and dbd is the mesh-element size at the boundary. The convergence
parameter C is shown on a logarithmic scale and the graphs show how C initially decays
exponentially, until a point where no further improvement is seen for the second-order velocity
and temperature. This demonstrates internal consistency for the numerical results
and provides insight to choose the most ecient mesh sizes.
5.4 Temporal resolution
When solving the governing equations in time domain, compared to solving the periodic
state, the solution is obtained by a time-marching scheme that estimates the timederivatives
and steps forward in time with a given time step t. The necessary temporal
resolution for the time-marching scheme can be determined by the CourantFriedrichs
Lewy (CFL) condition, also referred to as just the Courant number
CFL =
cs t
r CFLmax; (5.20)
where t is the temporal discretization and r is the spatial discretization. This means
that the length over which a disturbance travels within a time step t should be some
fraction of the mesh element size, ultimately ensuring that disturbances do not travel
through a mesh element in one time step. A more accurate interpretation of the CFL