Q2. What contributions have the authors mentioned in the paper "Linear dispersion-diffusion analysis and its application to under-resolved turbulence simulations using discontinuous galerkin spectral/hp methods" ?
The authors investigate the potential of linear dispersion-diffusion analysis in providing direct guidelines for turbulence simulations through the under-resolved DNS ( sometimes called implicit LES ) approach via spectral/hp methods. The authors revisit the eigensolutions technique as applied to linear advection and suggest a new perspective on the role of multiple numerical modes, peculiar to spectral/hp methods.
Q3. What is the efficient way to convert the available resources into resolution power?
In particular, the use of higher-order discretizations along with coarser meshes is expected to be the most efficient way to translate available resources into resolution power.
Q4. What is the hp discretization space’s nature?
If a single Fourier component is prescribed as initial condition through a projection, the hp discretization space will perceive it as a variety of polynomial components which will correspond to a series of numerical eigenfunctions, instead of just one.
Q5. What is the length of a wave in a hp setting?
Since h is the length measure of one degree of freedom in an hp setting, Δt is the time it takes for a signal to cross a single DOF.
Q6. What is the significance of diffusion errors?
And since diffusion errors have been verified to be more significant than dispersion errors [23, 27], the authors now focus on defining the extent of the plateau region of diffusioncurves.
Q7. What makes the iLES and uDNS approaches attractive?
Bypassing the need for explicit SGS models makes both iLES and uDNS approaches attractive since most of the theoretical and implementation complexities of traditional LES are avoided.
Q8. What is the effect of numerical diffusion on the energy spectrum?
Fig. 7 also helps to clarify why the DG formulation can be suitable for under-resolved simulations of turbulence: the numerical discretization is capable of resolving scales up to k1% with good accuracy while dissipation is provided at the end of the energy spectrum in the form of numerical diffusion.
Q9. What is the significance of the eigenmodes?
The authors conclude that the numerical eigenmodes related to well-resolved wavenumbers, i.e. those within the linear regions of dispersion/diffusion curves, will propagate correctly the components of the numerical solution associated to their wavenumbers.
Q10. Why is the comparison in Fig. 9 appropriate?
The comparison in Fig. 9 is appropriate because the random number generator employed for the forcing variable σk(t) is deterministic and therefore the same forcing function was used in both test cases through the whole integration period.
Q11. Why is the hump in Fig. 8 peculiar?
Case P = 1 is peculiar because dispersion errors are only significant for the second wavenumbers’ portion, by which energy should accumulate almost exclusively before k1%.
Q12. What is the range of wavenumbers for which a DG simulation can be considered?
As shown in Sec. 3.2, the range of wavenumbers for which wave propagation can be considered accurate corresponds to the linear regions of dispersion/diffusion curves.
Q13. What is the difference between the DG-uDNS approach and LES approach?
In this sense, the DG-uDNS approach seems closer to hyper-viscosity approaches than to LES approaches based on (explicit or implicit) subgrid-scale modelling.
Q14. Why is the resolution gain given by log10(2) 0.3?
This is because, for each P,log(k′1%) − log(k1%) = log ( |kh|1%h′) − log ( |kh|1% h ) = log(h/h′) = log( f ) . (37)In the test cases considered here, where a mesh refinement factor of f = 2 separates cases a-b or b-c for each value of P, the expected resolution gain is given by log10(2) ≈ 0.3 which is also visible in Fig. 7.