1 Accuracy of present data

2 Grid dependency (I) u rms y/D u rms at r/D=0.5 u rms at r/D=1 U at r/D=1 u rms U No. cells on M-G level 1 (): 194 x 98 x 194 (A) (): 194 x 146 x 194 (B) (): 194 x 226 x 194 (C) The grid refinement study shows a monotonic behavior. Doubling the number of cells in y does not affect the general flow character. The character of the wall jet is only weakly influenced by four times higher wall- normal resolution. Additional studies have shown that the growth of instabilities at the nozzle outlet is very little influenced by the axial and radial grid resolution.  The present grid (B) provides a grid independent solution, within appropriate error limit, with respect to the wall-normal grid resolution. The important large-scale dynamics is not affected by the finite grid size. Re=20000 LES: ratio≈141  ≈0.007D (: average eddy size) DNS: ratio≈1700

3 Grid dependency (II) As the grid spacing increases with the radius, the flow becomes successively less resolved. This has a negative influence on the transition process in the wall jet.  Accurate results within this work is only obtained within a certain limit on r/D. This limit has not exactly been determined (significant cost to refine the grid). May contribute to the lack of a second maximum in Nu for H/D=2. U y/D U at r/D=1 k at y/D=0.15 k No. cells on M-G level 1 (): 194 x 194 x 194 (): 170 x 194 x 170 H/D=4

4 Grid stretching in y (H/D=2) No. cells on M-G level 1 (): 194 x 98 x 194 (A) (): 194 x 146 x 194 (B) (): 194 x 226 x 194 (C) y/D y*/D

5 Convergence error (): Mean velocity (): RMS velocity t  Convergence error below 10 -3, fallen approx. three magnitudes  accuracy of approx 0.1. Doubling of the total sampling time  Negligible change of 1 st order statistics. Second order slightly higher change, not influencial. Signal: U within the wall jet at r/D=1.5 Physical Numerical Re=20000 LES: steps≈1400 DNS: steps≈17000 for 10 cycles

6 Model errors and errors due to the boundary conditions U y/D (  ): LES (pipe) (  ): LES (top-hat) (О): Cooper et al. (  ): Geers et al. u rms With correct boundary conditions the results correlate well with experimental data ("same" boundary conditions). The model/scheme provides a physically relevant solution.

7 Model error St PSD Velocity spectrum at r/D=0.5, y/D=1 for H/D=4 The flow is not fully developed but fairly close to that state. The spectrum shows a behavior close to the -5/3 law within approx. one magnitude of frequencies.

8 Additional observations Additional to the figures shown: The frequency of the natural mode agrees with theory and experiments. The frequency of the jet-column mode agrees with experimental data for H/D=4. Growth of instabilities exhibit physically correct behavior, i.e. the model/scheme provides appropriate amount of dissipation. From visual observations the dynamical behavior of the jet agrees with expermental observations. The near-wall resolution is appropriate in terms of describing turbulent mechanisms (see e.g. near-wall peak of 1 st order statistics).

9 SGS-models

10 Smagorinsky model Boussinesq hypothesis: The SGS-scales can be described by a length- and a time-scale  The second important parameter at cut-off is the dissipation: Assuming the equilibrium hypothesis  Production = Dissipation: Considers the energetic action and neglects the structural information.

11 SSM Idea: The difference between the once and twice filtered field is related to the unresolved scales. Simplest division: 1) the largest resolved scales, 2) the smallest resolved scales, 3) the unresolved scales. Approx SSM:  if filter is a Reyn. Op.  terms cancels out. L: large-scale interactions, C: cross-scale interactions, R: SGS interaction In Bardina model: L is computed exactly, C is modeled and R=0  non-dissipative  combine linearly with the Smagorinsky: approx.

12 Dynamic approach In order to adapt the model to the structure of the flow the model constant is dynamically calculated. Applicable to any model that explicitly uses an arbitrary constant. Assumption: the SGS-stresses have the same asymptotic behavior at different filter widths. Germano identity: Assume that the two SGS tensors can be modeled by the same constant  the SGS and STS stress are related through the resolved stress

13 Dynamic approach Using the same SGS model: assumption of scale-similarity. Smagorinsky: In order to proceed: approx.,i.e. C is a slowly varying function in space. C is computed such as the introduced error becomes minimal. C can take negative values  account for backscatter. Averaging or clipping of C to avoid numerical instability since the denominator may become zero.

14 LES

15 Filtering Properties Linear: Conservation of constant: Cummutation with derivatives:

16 LES-errors

17 Errors A second order filter introduces commutation error of second order. A fourth-order scheme using fourth-order commutative explicit filters with a filter width of at least twice the grid-spacing adds a numerically clean environment, suitable for SGS model evaluation. Due to the effect from the modified wave-number, aliasing is less important at high wave-numbers for FD than for spectral methods. However, the representation at high wave-numbers for FD methods is less accurate because of the large truncation errors.

18 Commutation errors Commutation errors are introduced 1.Due to the non-linear term (also for uniform filters, meshes). 2.Due to non-uniform filtering, meshes (also the linear terms). The first commutation error is accounted for by the SGS model the second group can also be accounted for but this adds significant complexity to the model. The commutation errors can be made smaller by using high order schemes and by having a smooth varying grid.

19 Galilean invariance

20 Galilean invariance If the method is not GI additional terms (errors) are introduced under translation, that must be accounted for. The N-S are GI so also the filtered N-S with the tripple decomposition of the SGS stress.

21 Numerical Errors

22 Modified wavenumber The formal order of accuracy of FD scheme describes the behavior for fairly well solved features. For scales that are described by only a few grid points the wave-# is replaced by a modified wave-#. With FD the derivative become: For spectral methods k*=k Any UW-scheme is equiv. to the next higher CD-scheme. Third order UW (Kawamura et al.): k*: the actual wave-# that is resolved  large influence on the smallest resolved scales. The energetic scales must be smaller than k c. Im(k*): dissipation (CDO6+FilterO4 = UWO3) The Im-part  change the amplitude of u. Due to the modified wave-# the effect from aliasing is of less importance for FD than for spectral. However, the high wave-# for FD is influenced by large amount of T.E.  less accurate (Chow & Moin, JCP, No.184, 203). k c =k * max

23 Modified wavenumber CDO2: For low wave-# (smooth functions) (Ferziger & Peric) : For UW-schemes the modified wave-# is complex, and is an indication of the dissipative nature of this type of approximation (Ferziger & Peric).

24 Numerical dissipation Sagaut: The numerical and the subgrid dissipation is correlated in space. A SGS viscosity model: second order dissipation, associated with a spectrum: A nth-order numerical dissipation is associated with a spectrum:  Third order dissipative scheme: the numerical dissipation will be largest for high wave-#. SGS-dissipation be largest for small wave-#. The numerical dissipation increases exp. as k  k c, i.e. as k  Numerical dissipative schemes may give to steep spectrum at large k: se pipe simulation. The dissipative role of the SGS terms can be replaced by a numerical one if it does not affect the energy level of the larger scales. For fine enough resolution the effects of the SGS scales (backscatter etc) does not affect the large scales. With implicit modeling the effects from the SGS scales cannot be accounted for in a physical manner. ‘Non-physical’ related backscatter may occur due to the dispersive behavior of the scheme, i.e. not monotone.

25 Truncation error For linear problems the truncation error acts as a source of the discretization error, which is convected and diffused by the operator, in this case the NS-operator. The T.E. cannot be computed exactly since the exact solution is not known. Exact analysis is not possible for non-linear problems. Small enough errors  linearization. An approximation to the T.E. can be found from successive refinement of the grid. Guide to where grid refinement is needed. For sufficiently fine grids the T.E. is prop. to the leading term in the Taylor series. The error can be estimated as, where F is the grid coarsening ratio and p the order of the scheme. When solutions on several grids are available, an the convergence is monotonic, one can use Richardson extrapolation to find a better approximation to the solution on the finest grid h:

26 Truncation error For the present UWO3 scheme: The fourth order derivatives are dissipative in nature.