Accuracy of present data

1
Accuracy of present data
2
Grid dependency (I)
urms at r/D0.5
urms at r/D1
U at r/D1
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
urms
urms
U

• 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.

Re20000 LES ratio141 ? ?0.007D (?
average eddy size) DNS ratio1700
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/D2.

U at r/D1
k at y/D0.15
H/D4
H/D4
y/D
No. cells on M-G level 1 (?) 194 x 194 x 194
(?) 170 x 194 x 170
k
U
4
Grid stretching in y (H/D2)
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
Physical
Signal U within the wall jet at r/D1.5
(?) Mean velocity (?) RMS velocity
Numerical
?
Re20000 LES steps1400 DNS
steps17000 for 10 cycles
t

• Convergence error below 10-3, fallen approx.
  three magnitudes ? accuracy of approx 0.1.
• Doubling of the total sampling time
• ?
• Negligible change of 1st order statistics.
• Second order slightly higher change, not
  influencial.

6
Model errorsand errors due to the boundary
conditions
(?) LES (pipe) (?) LES (top-hat) (?)
Cooper et al. (?) Geers et al.
y/D
urms
U

• 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
Velocity spectrum at r/D0.5, y/D1 for H/D4
PSD
St

• 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/D4.
• 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 1st 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 R0
• ? 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
• Due to the non-linear term (also for uniform
  filters, meshes).
• 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 kk

• 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 kc.
• Im(k) dissipation (CDO6FilterO4 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).

kckmax
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?
  kc, 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.