RECIRCULATING FLOW [1] Lars Davidson, www.tfd.chalmers.se/lada QLES - - PowerPoint PPT Presentation

HOW TO ESTIMATE THE RESOLUTION OF AN LES OF

RECIRCULATING FLOW [1]

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept

HOW TO ESTIMATE RESOLUTION OF AN LES?

In boundary layers there are guidelines ` a priori. The cells size in the streamwise and spanwise direction should be approximately 100 and 30 respectively. First wall-adjacent node at y+ ≃ 1. No guidelines in free-flow region (shear layers, re-circulation region . . . ) Worse: even after having carried out an LES, it is difficult to know if the resolution is good! I have recently made a similar study for channel flow [2]

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 2 / 34

ENERGY SPECTRUM

10 10

1

10

• 4

Energy spectrum wavenumber

ww z

− 5 / 3

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 3 / 34

ENERGY SPECTRUM AND TWO-POINT CORRELATION

10 10

1

10

• 4

Energy spectrum wavenumber

ww z

− 5 / 3 s e e m s O K . . .

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 3 / 34

ENERGY SPECTRUM AND TWO-POINT CORRELATION

10 10

1

10

• 4

Energy spectrum wavenumber

ww z

0.1 0.2 0.3 0.4 0.2 0.4 0.6 0.8 1

Two-point correlation Separation distance in z − 5 / 3 s e e m s O K . . .

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 3 / 34

ENERGY SPECTRUM AND TWO-POINT CORRELATION

10 10

1

10

• 4

Energy spectrum wavenumber

ww z

0.1 0.2 0.3 0.4 0.2 0.4 0.6 0.8 1

Two-point correlation Separation distance in z − 5 / 3 s e e m s O K . . . b u t i s B A D !

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 3 / 34

ENERGY SPECTRUM VS TIME AND TWO-POINT CORRELATION

10

• 2

10

• 1

10 10

• 8

10

• 7

10

• 6

10

• 5

Energy spectrum frequency, f Eww(f)

0.1 0.2 0.3 0.4 0.2 0.4 0.6 0.8 1

Two-point correlation Separation distance in z Nx = 256, Nz = 32

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 4 / 34

ENERGY SPECTRUM VS TIME AND TWO-POINT CORRELATION

10

• 2

10

• 1

10 10

• 8

10

• 7

10

• 6

10

• 5

Energy spectrum frequency, f Eww(f)

0.1 0.2 0.3 0.4 0.2 0.4 0.6 0.8 1

Two-point correlation Separation distance in z Nx = 256, Nz = 32; Nx = 512, Nz = 128

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 4 / 34

ENERGY SPECTRUM VS TIME AND TWO-POINT CORRELATION

10

• 2

10

• 1

10 10

• 8

10

• 7

10

• 6

10

• 5

Energy spectrum frequency, f Eww(f)

0.1 0.2 0.3 0.4 0.2 0.4 0.6 0.8 1

Two-point correlation Separation distance in z d

• n

’ t t r u s t e n e r g y s p e c t r a Nx = 256, Nz = 32; Nx = 512, Nz = 128

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 4 / 34

PLANE ASYMMETRIC DIFFUSER (NOT TO SCALE)

L H 4.7H L1 L2 Inlet x y L1 = 7.9H, L = 21H, L2 = 28H. The spanwise width is zmax = 4H.

• Mesh (x × y × z)

258 × 64 × 32, 258 × 64 × 64, 258 × 64 × 128 · 512 × 64 × 32, 512 × 64 × 64, 512 × 64 × 128

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 5 / 34

COMPUTATIONAL METHOD

Finite volume with central differencing in space and time (Crank-Nicolson) Fractional step Dynamic Smagorinsky model Inlet fluctuating boundary conditions: synthetic isotropic turbulence [3] All simulations run on a single CPU. Averaging during one week (the finest mesh: two weeks)

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 6 / 34

¯ u/Ub,in PROFILES

x = 3 6 14 17 20 24H x = 3 6 14 17 20 24H Nz = 32; Nz = 64; Nz = 128; ◦ exp. [4]. Nx = 256 Nx = 512

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 7 / 34

u′v′/U2

b,in PROFILES

x = 3 6 13 16 19 23H x = 3 6 13 16 19 23H Nz = 32; Nz = 64; Nz = 128; ◦ exp. [4]. Nx = 256 Nx = 512

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 8 / 34

u′v′/U2

b,in PROFILES AT x = −H

• 2
• 1

1 2 x 10

• 3

0.2 0.4 0.6 0.8 1

y/H Nx = 256

• 2
• 1

1 2 x 10

• 3

0.2 0.4 0.6 0.8 1

y/H Nx = 512 Nz = 32; Nz = 64; Nz = 128

x = −H

Attached flow ∆x/∆z = 0.6, 1.2, 2.4 ∆x/∆z = 0.3, 0.6, 1.2

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 9 / 34

u′v′/U2

b,in PROFILES AT x = 20H

• 4
• 3
• 2
• 1

1 x 10

• 3
• 4
• 3
• 2
• 1

1

y/H Nx = 256

• 4
• 3
• 2
• 1

1 2 x 10

• 3
• 4
• 3
• 2
• 1

1

y/H Nx = 512 Nz = 32; Nz = 64; Nz = 128; ◦ exp. [4].

x = 20H

Incipient separation ∆x/∆z = 2.2, 4.4, 8.8 ∆x/∆z = 1.1, 2.2, 4.4

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 10 / 34

DIFFERENT WAYS TO ESTIMATE RESOLUTION

Energy spectra (both in spanwise direction and time) Two-point correlations Ratio of SGS shear stress τsgs,12 to resolved u′v′ Ratio of SGS viscosity, νsgs to molecular, ν Energy spectra of SGS dissipation Comparison of SGS dissipation due to ∂u′

i/∂xj and ∂¯

ui/∂xj

• Below we will only analyze results from the Nx = 256 meshes

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 11 / 34

ENERGY SPECTRA, TWO-POINT CORR. AT x = −H

10 10

1

10

2

10

• 6

10

• 5

10

• 4

Energy spectrum wavenumber, κz

ww z

0.1 0.2 0.3 0.4 0.2 0.4 0.6 0.8 1

Two-point correlation Separation distance in z Nz = 32; Nz = 64; Nz = 128.

x = −H

Attached flow −5/3

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 12 / 34

ENERGY SPECTRA, TWO-POINT CORR. AT x = 20H

10 10

1

10

2

10

• 8

10

• 6

10

• 4

Energy spectrum wavenumber, κz

ww z

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1

Two-point correlation Separation distance in z Nz = 32; Nz = 64; Nz = 128.

x = 20H

Incipient separation − 5 / 3

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 13 / 34

ENERGY SPECTRA IN TIME. x = −1.

10

• 2

10

• 1

10 10

• 8

10

• 7

10

• 6

10

• 5

ww( )

frequency, f Energy spectra in time

0.1 0.2 0.3 0.4 0.2 0.4 0.6 0.8 1

Two-point correlation Separation distance in z Nz = 32; Nz = 64; Nz = 128.

x = −H

Attached flow −5/3

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 14 / 34

ENERGY SPECTRA IN TIME. x = 20.

10

• 2

10

• 1

10 10

• 10

10

• 8

10

• 6

10

• 4

Eww(f) frequency, f Energy spectra in time

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1

Two-point correlation Separation distance in z Nz = 32; Nz = 64; Nz = 128.

x = 20H

Incipient separation − 5 / 3

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 15 / 34

SGS VS. RESOLVED SHEAR STRESSES

0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

y/H τsgs,12/u′v′

0.01 0.02 0.03 0.04 0.05

• 3.5
• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

y/H τsgs,12/u′v′ Nz = 32; Nz = 64; Nz = 128.

x = −H x = 20H

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 16 / 34

SGS VS. MOLECULAR VISCOSITY

1 2 3 4 0.2 0.4 0.6 0.8 1

y/H νsgs/ν

2 4 6 8 10 12

• 4
• 3
• 2
• 1

1

y/H νsgs/ν Nz = 32; Nz = 64; Nz = 128.

x = −H x = 20H

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 17 / 34

SGS VS. MOLECULAR VISCOSITY, Nx = 512

1 1.5 2 2.5 3 3.5 0.2 0.4 0.6 0.8 1

y/H νsgs/ν

2 4 6 8 10

• 4
• 3
• 2
• 1

1

y/H νsgs/ν Nz = 32; Nz = 64; Nz = 128.

x = −H x = 20H

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 18 / 34

DISSIPATION ENERGY SPECTRA: THEORY VS. REALITY

Theory κc κ E(κ) εsgs Reality κ E(κ) κc εsgs,κ εsgs = κc εsgs,κ(κ)dκ

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 19 / 34

APPROXIMATED DISSIPATION ENERGY SPECTRA

At which wavenumber is the SGS dissipation largest? In the homogeneous direction, z, the SGS dissipation can be analyzed in the wavenumber space εwz, can — in theory — be obtained from the two-point correlation [5] as εwz = 2ν ∂w′ ∂z 2 = 2ν ∂2Bww(ˆ z) ∂ˆ z2

• ˆ

z=0

= 2ν

Nz

• kz=1

κ2

zEww(kz)

When the equations are discretized, the left side = the right side The right side gives εwz ∝ κ2

zEww = κ2 zκ−5/3 = κ1/3

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 20 / 34

EXACT DISSIPATION ENERGY SPECTRA

A discrete Fourier transform of ∂w′/∂z is formed as ˆ Dz(kz) = 1 Nz

Nz

• n=1

∂w′(n) ∂z

• cos

2π(n − 1)(kz − 1) Nz

• − ı sin

2π(n − 1)(kz − 1) Nz

• (1)

where n is node number in z direction. Power Spectral Density (PSD) ∂w′ ∂z 2 =

Nz

• kz=1

ˆ Dz ∗ ˆ D∗

z = Nz

• kz=1

PSD ∂w′ ∂z

• Lars Davidson, www.tfd.chalmers.se/˜lada

QLES 2009, Pisa, 9-11 Sept 21 / 34

PREDICTED DISSIPATION ENERGY SPECTRA

20 40 60 80 100 1 2 3 4 5 6x 10

• 7

2ν · PSD(∂w′/∂z) κz = 2π(kz − 1)/zmax Exact

20 40 60 80 100 0.5 1 1.5 2 2.5x 10

• 6

2νk2

z Eww(kz)

κz = 2π(kz − 1)/zmax Approximated Nz = 32; Nz = 64; Nz = 128.

x = −H, y = 0.15H

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 22 / 34

PREDICTED DISSIPATION ENERGY SPECTRA

20 40 60 80 100 1 2 3 4 5 6x 10

• 7

2ν · PSD(∂w′/∂z) κz = 2π(kz − 1)/zmax Exact

20 40 60 80 100 0.5 1 1.5 2 2.5x 10

• 6

2νk2

z Eww(kz)

κz = 2π(kz − 1)/zmax Approximated Nz = 32; Nz = 64; Nz = 128.

x = −H, y = 0.15H

εwz=2ν κc PSDdκ εwz=2ν κc

0 κ2 zEwwdκ

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 22 / 34

PREDICTED DISSIPATION ENERGY SPECTRA

20 40 60 80 100 0.2 0.4 0.6 0.8 1 1.2 1.4x 10

• 7

2ν · PSD(∂w′/∂z) κz = 2π(kz − 1)/zmax Exact

20 40 60 80 100 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6x 10

• 7

2νk2

z Eww(kz)

κz = 2π(kz − 1)/zmax Approximated Nz = 32; Nz = 64; Nz = 128.

x = 20H, y = 2.9H

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 23 / 34

SGS DISSIPATION ENERGY SPECTRA

Above, energy spectra for ∂w′/∂z have been presented which is part of the viscous dissipation What about energy spectra for the SGS dissipation εsgs =

• νsgs

∂¯ ui ∂xj ∂¯ ui ∂xj ? Form a discrete Fourier transform of ε1/2

• sgs. Replace ∂w′/∂z in
• Eq. 1 on Slide 26 by ε1/2

sgs.

Strange unphysical Fourier coefficients! but the energy spectra εsgs =

Nz

• kz=1

ˆ Dz ∗ ˆ D∗

z = Nz

• kz=1

PSD

• ε1/2

sgs

• are OK

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 24 / 34

SGS DISSIPATION ENERGY SPECTRA

20 40 60 80 100 2 4 6 8 x 10

• 6

PSD(ε1/2

sgs,inst)

κz = 2π(kz − 1)/zmax

20 40 60 80 100 1 2 3 4 x 10

• 6

PSD(ε1/2

sgs,inst)

κz = 2π(kz − 1)/zmax Nz = 32; Nz = 64; Nz = 128.

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 25 / 34

SNAPSHOTS OF w′ VS. z

1 2 3 4

• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15 0.2

z/H

1 2 3 4

• 0.15
• 0.1
• 0.05

0.05 0.1 0.15 0.2

z/H Nz = 32; Nz = 64, w′ − 0.1; Nz = 128, w′ + 0.14.

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 26 / 34

TRANSFER OF KINETIC TURBULENT ENERGY

• τ ′

ij,sgs

∂u′

i

∂xj

• νsgs

∂¯ ui ∂xj ∂¯ ui ∂xj K kres ksgs εsgs,mean ε′

sgs

time-averaged K = 1

2¯

ui¯ ui (RANS) resolved kres = 1

2u′ iu′ i (RANS and LES)

SGS kinetic energy, ksgs.

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 27 / 34

RATIO OF SGS DISSIPATION

0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

y/H ε′

sgs/(εsgs,mean + ε′ sgs)

0.9 0.92 0.94 0.96 0.98 1

• 3.5
• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

y/H ε′

sgs/(εsgs,mean + ε′ sgs)

Nz = 32; Nz = 64; Nz = 128.

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 28 / 34

DECAYING GRID TURBULENCE

Diffuser flow: peaks in ∂w′/∂z at surprisingly low wavenumbers. The “decaying grid turbulence” is presented below in order to find at which wavenumbers the dissipation attain its peak The domain is a cubic box of side 2π. Three computations have been carried out.

• 1. Fine LES using a Smagorinsky model (CS = 0.1) on a 1283 grid.
• 2. DNS on a 1283 grid.
• 3. Coarse LES using a Smagorinsky model (CS = 0.1) on a 643 grid.

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 29 / 34

DECAYING GRID TURBULENCE: RESULTS

10 10

1

10

• 4

10

• 3

10

• 2

Eww z Energy spectra. t = 2

10 20 30 40 50 60 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

2ν · PSD(∂w′/∂z) κz = 2π(kz − 1)/zmax Exact dissipation spectra. t = 2 Fine LES; DNS; coarse LES; + exp. [6].

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 30 / 34

CONCLUSIONS

Two-point correlation best. They show by how many cells the largest scales are resolved. The energy spectra do not give any reliable information on the resolution. The νt/ν is not a good measure. It compares LES with DNS. τsgs,12/u′v′ is a good measure. However, it is difficult to give any quantitative guidelines. Ratio of ε′

sgs/εsgs,mean useful but difficult to give any quantitative

guidelines.

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 31 / 34

CONCLUSIONS CONT’D

• Energy spectra of the SGS dissipation show that the peak takes

place at surprisingly low wavenumber (length scale corresponding to 10 cells or more). κc κ E(κ) εsgs

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 32 / 34

CONCLUSIONS CONT’D

• Energy spectra of the SGS dissipation show that the peak takes

place at surprisingly low wavenumber (length scale corresponding to 10 cells or more). κc κ E(κ) εsgs εsgs,κ

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 32 / 34

REFERENCES I

• L. Davidson.

How to estimate the resolution of an LES of recirculating flow. In M. V. Salvetti, B. Geurts, J. Meyers, and P . Sagaut, editors, ERCOFTAC, volume 16 of Quality and Reliability of Large-Eddy Simulations II, pages 269–286. Springer, 2010.

• L. Davidson.

Large eddy simulations: how to evaluate resolution. International Journal of Heat and Fluid Flow, 30(5):1016–1025, 2009.

• L. Davidson.

Using isotropic synthetic fluctuations as inlet boundary conditions for unsteady simulations. Advances and Applications in Fluid Mechanics, 1(1):1–35, 2007.

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 33 / 34

REFERENCES II

C.U. Buice and J.K. Eaton. Experimental investigation of flow through an asymmetric plane diffuser. Report No. TSD-107, Thermosciences Division, Department of Mechanical Engineering, Stanford University, Stanford, California 94305, 1997. J.O. Hinze. Turbulence. McGraw-Hill, New York, 2nd edition, 1975.

• G. Comte-Bellot and S. Corrsin.

Simple Eularian time correlation of full- and narrow-band velocity signals in grid-generated “isotropic” turbulence. Journal of Fluid Mechanics, 48(2):273–337, 1971.

Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 34 / 34