*********************************************************************
*********************************************************************
******** ********
******** "Establishment of ********
******** the Direct Numerical Simulation Data Bases of ********
******** Turbulent Transport Phenomena" ********
******** ********
******** Co-operative Research ********
******** No. 02302043 ********
******** ********
******** Supported by the Ministry of Education, ********
******** Science and Culture ********
******** ********
******** 1990 - 1992 ********
******** ********
******** <Research Collaborators> ********
******** ********
******** N. Kasagi (Organizer) ********
******** K. Horiuchi, Y. Miyake, T. Miyauchi, Y. Nagano ********
******** ********
*********************************************************************
*********************************************************************
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
++++++++ << CAUTION >> ++++++++
++++++++ ++++++++
++++++++ All rights are reserved by the computors of each ++++++++
++++++++ data base. No part of the data described herein ++++++++
++++++++ may be represented or otherwise used in any form ++++++++
++++++++ without fully referring to this data base and ++++++++
++++++++ the literature cited at the end of the data base. ++++++++
++++++++ The original data base will be revised without ++++++++
++++++++ notice, whenever necessary. Future revisions ++++++++
++++++++ will be notified to those who register by writing ++++++++
++++++++ to Research Organizer: ++++++++
++++++++ ++++++++
++++++++ N. Kasagi ++++++++
++++++++ Department of Mechanical Engineering ++++++++
++++++++ The University of Tokyo ++++++++
++++++++ Bunkyo-ku, Tokyo 113 ++++++++
++++++++ Japan. ++++++++
++++++++ ++++++++
++++++++ December 1990 ++++++++
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
============ this data base begins from this line ===================
Test case: Homogeneous Isotropic Flow with Passive Scalar Diffusion
Code number: IS____UT.HM1
Date of Release: March 20, 1991
Computors: T. Miyauchi, M. Tanahashi and T. Ishizu
Department of Mechanical Engineering Science
Tokyo Institute of Technology
Meguro-ku, Tokyo 152
Nomenclature:
E: spectral distribution of kinetic energy; cm**3/s**2,
E(k) = 2 x pai x k**2 x ui(k) x conjg(ui(k)) (i=1,2,3)
repeated summation is used for i
Ene: kinetic energy per unit mass (space averaged); cm**2/s**2,
Ens: enstrophy; 1/s**2
Et: spectral distribution of scalar fluctuation; cmK**2,
Et(k) = 2 x pai x k**2 x dT(k) x conjg(dT(k))
dT = fluctuation from mean temperature
<f>: spatialy averaged value of variable f
Fi: flatness factor of ui
Fdi: flatness factor of i-derivative of ui
Int: mean square value of temperature fluctuation; K**2
k: absolute value of wave number; 1/cm
L: reference length = 2 x pai(fundamental wave length); cm
Lu: integral scale of u1 in x1-derection; cm,
Lu = the integral of R11 from zero to pai
Lt: integral scale of T in x1-direction; cm,
Lt = the integral of Rtt from zero to pai
Ramdu: Taylor micro scale for u1; cm
Ramdt: Taylor micro scale for T; cm
Re_ramd:micro scale Reynolds number U*Ramdu/Ve
Rtt: two-point auto-correlation coefficient of scalar fluctuation
Rtt = <T(x1) x T(x1+tau)> / <T(x1)**2>
Rij: two-point correlation coefficient of ui and uj
Rij = <ui(x1) x uj(x1+tau)>
/sqrt(<ui(x1)**2>)/sqrt(<uj(x1)**2>)
Rut: two-point correlation coefficient of u1 and T
Rut = <u1(x1) x T(x1+tau)>
/sqrt(<u1(x1)**2>)/sqrt(<T(x1)**2>)
Si: skewness factor of ui
Sdi: skewness factor of i-th derivative of ui
tau: lag in x1-direction; cm
T: temprature; K
TS: time step; s,
1TS=0.005
ui: velocity in xi-direction; cm/s
U: reference velocity = root mean square velocity
fluctuation at initial stage; cm/s
Ve: kinematic viscosity; cm**2/s
1. Description of Flow Field
The flow field simulated is homogeneous isotropic turbulence
with temperature diffusion. Temperature fluctuation is not so
large so that temperature can be assumed to be a passive scalar.
We simulated two cases with Prandtl numbers of 0.20 and 0.71.
2. Numerical Method
Governing Equation: time dependent Navier Stokes equation
(rotational form) and energy conservation equation.
Discretization method:spectral method = 64 x 64 x 64 Fourier modes
in the x1-,x2-,x3-direction.
Aliasing treatment:nonlinear term is computed in physical space
by using padding method.
Time integration:second order Adams-Bashforth scheme for the
nonlinear term, Backward-Euler scheme for the pressure term,
and Crank-Nicolson scheme for the viscous and diffusion terms.
Size of computational box:(2 x pai) x (2 x pai) x (2 x pai)
Initial spectral distribution: E(k)=0.038 x k**4 x exp(-0.14 x k**2)
Et(k)=0.40 x k**2 x exp(-0.10 x k**2)
Error in continuity equation = Max{abs(div u)}: less than 1.0E-15
Computer:HITAC-S820/80 at the Computer Center of the University of Tokyo
3. Flow condition
Re = U*L/Ve =309
4. Numerical data of Velocity Field
4.1 Kinetic Energy per Unit Mass
TS Ene(TS)
0 1.22052E-1
100 1.05959E-1
200 9.10524E-2
300 7.75190E-2
400 6.58436E-2
500 5.60734E-2
600 4.80062E-2
4.2 3-D Energy Spectrum
TS=0
k E(k)
1.41421 1.00360E-1
2.00000 3.10754E-1
3.00000 8.31843E-1
4.00000 9.14860E-1
5.00000 6.44100E-1
6.00000 3.23011E-1
7.00000 9.38450E-2
8.06225 1.85709E-2
TS=200
k E(k)
1.00000 5.55352E-4
2.00000 2.40827E-1
3.00000 7.06399E-1
4.00000 4.35846E-1
5.00000 2.82298E-1
6.00000 1.74570E-1
7.00000 1.25323E-1
8.00000 5.20969E-3
9.00000 6.37578E-2
10.0000 1.13119E-2
15.0000 2.32024E-3
20.0000 2.74516E-2
25.0000 1.44430E-5
30.0000 1.20502E-6
35.0000 2.74168E-7
40.1249 2.04499E-8
45.0222 1.05743E-9
TS=400
k E(k)
1.00000 1.93256E-3
2.00000 1.78107E-1
3.00000 5.10856E-1
4.00000 7.11306E-2
5.00000 2.05873E-1
6.00000 1.01789E-1
7.00000 1.54639E-1
8.00000 3.47021E-3
9.00000 5.38501E-2
10.0000 9.67449E-3
15.0000 2.98650E-3
20.0000 4.92136E-5
25.0000 1.81891E-5
30.0000 1.54466E-6
35.0000 2.12039E-7
40.1249 2.19996E-8
45.0555 1.44693E-9
TS=600
k E(k)
1.00000 3.46109E-3
2.00000 1.29655E-1
3.00000 3.61931E-1
4.00000 1.71386E-2
5.00000 1.49187E-1
6.00000 6.01920E-2
7.00000 9.58069E-2
8.00000 3.81687E-3
9.00000 3.20410E-2
10.0000 5.81278E-3
15.0000 1.72995E-3
20.0000 1.88906E-5
25.0000 6.13948E-6
30.0000 3.91979E-7
35.0000 4.12451E-8
40.1249 3.64087E-9
42.7317 1.09891E-9
4.3 Two-point Velocity Correlations (x1-direction)
4.3.1 Two-point Auto-correlation of u1
TS=0
tau R11(tau)
0.00000 1.00000
3.92699E-1 8.01477E-1
7.85396E-1 4.16887E-1
1.17809 1.42585E-1
1.57079 2.34320E-2
1.96349 -1.39152E-2
2.35619 -1.12421E-2
2.74889 1.01116E-2
3.04341 2.15049E-2
TS=200
tau R11(tau)
0.00000 1.00000
3.92699E-1 7.82555E-1
7.85396E-1 3.98984E-1
1.17809 1.22451E-1
1.57079 -4.69710E-3
1.96349 -3.47308E-2
2.35619 -3.68531E-2
2.74889 -3.75797E-2
3.04341 -3.68706E-2
TS=400
tau R11(tau)
0.00000 1.00000
3.92699E-1 7.77175E-1
7.85396E-1 4.09594E-1
1.17809 1.53880E-1
1.57079 2.83990E-2
1.96349 -1.87648E-2
2.35619 -4.80518E-2
2.74889 -7.61841E-2
3.04341 -8.72665E-2
TS=600
tau R11(tau)
0.00000 1.00000
3.92699E-1 7.96694E-1
7.85396E-1 4.58991E-1
1.17809 2.21358E-1
1.57079 9.09038E-2
1.96349 1.67314E-2
2.35619 -3.34679E-2
2.74889 -6.46255E-2
3.04341 -7.46684E-2
4.3.2 Two-point Cross-correlation of u1 and u2
TS=0
tau R12(tau)
0.00000 -2.52921E-3
3.92699E-1 1.36380E-2
7.85396E-1 1.95040E-2
1.17809 -7.93034E-3
1.57079 -3.07035E-2
1.96349 -2.27267E-2
2.35619 -1.34261E-2
2.74889 -2.45913E-2
3.04341 -4.06192E-2
TS=200
tau R12(tau)
0.00000 -9.15311E-3
3.92699E-1 -1.66495E-3
7.85396E-1 -1.16247E-2
1.17809 -3.88549E-2
1.57079 -6.18614E-2
1.96349 -6.62977E-2
2.35619 -6.43712E-2
2.74889 -6.87116E-2
3.04341 -7.34356E-2
TS=400
tau R12(tau)
0.00000 4.86372E-3
3.92699E-1 1.88666E-3
7.85396E-1 -9.99898E-3
1.17809 -9.54547E-3
1.57079 -1.45949E-2
1.96349 -3.48256E-2
2.35619 -4.74712E-2
2.74889 -4.52225E-2
3.04341 -4.54916E-2
TS=600
tau R12(tau)
0.00000 1.57828E-2
3.92699E-1 -7.63564E-4
7.85396E-1 -3.18279E-3
1.17809 2.74346E-2
1.57079 3.79977E-2
1.96349 1.39818E-2
2.35619 -1.67513E-2
2.74889 -2.57774E-2
3.04341 -2.45575E-2
4.4 Enstrophy
TS Ens(TS)
0 1.72438
100 1.53886
200 1.43236
300 1.26384
400 1.06958
500 8.87075E-1
600 7.30283E-1
4.5 Flatness of u1
TS F1(TS)
0 3.22735
100 3.17413
200 3.09192
300 3.06572
400 3.07073
500 3.06446
600 3.03757
4.6 Flatness of x1 derivative of u1
TS Fd1(TS)
0 3.00407
100 3.21584
200 3.63375
300 3.86776
400 3.90112
500 3.81612
600 3.71646
4.7 Skewness of u1
TS S1(TS)
0 9.22152E-2
100 9.05295E-2
200 7.62247E-2
300 5.84252E-2
400 4.57007E-2
500 4.52819E-2
600 5.52007E-2
4.8 Skewness of x1 derivative of u1
TS Sd1(TS)
0 1.24398E-2
100 -3.71311E-1
200 -5.11329E-1
300 -5.42004E-1
400 -5.48522E-1
500 -5.45802E-1
600 -5.37426E-1
4.9 Taylor Micro Scale of u1
TS Ramdu
0 5.89143E-1
100 5.78069E-1
200 5.48347E-1
300 5.33189E-1
400 5.33179E-1
500 5.43749E-1
600 5.55990E-1
4.10 Micro Scale Reynolds Number
TS Re_ramda
0 29.1078
100 26.6111
200 23.4000
300 20.9942
400 19.3484
500 18.2092
600 17.3492
4.11 Integral Scale of u1
TS Lu
0 7.70470E-1
100 7.46703E-1
200 6.90412E-1
300 6.70576E-1
400 6.96117E-1
500 7.44976E-1
600 8.00514E-1
5. Numerical data of Scalar Field
5.1 Averaged Intensity of Scalar fluctuation
5.1.1 Prandtl Nnmber = 0.20
TS Int(TS)
0 3.02647E-1
100 1.88012E-1
200 1.25224E-1
300 8.70439E-2
400 6.23675E-2
500 4.57898E-2
600 3.43929E-2
5.1.2 Prandtl Number = 0.71
TS Int(TS)
0 3.02647E-1
100 2.59492E-1
200 2.16989E-1
300 1.77294E-1
400 1.43593E-1
500 1.16202E-1
600 9.44040E-2
5.2 Spectrum of fluctuation
5.2.1 Prandtl Nnmber = 0.20
TS=0
k Et(k)
1.41421 4.02415E-1
2.00000 6.70266E-1
3.00000 9.92101E-1
4.00000 7.48541E-1
5.00000 5.38204E-1
6.00000 2.72123E-1
7.00000 8.73092E-2
7.81024 3.59780E-2
TS=200
k Et(k)
1.00000 1.05466E-3
2.00000 3.36166E-1
3.00000 3.67973E-1
4.00000 6.25681E-2
5.00000 6.43270E-2
6.00000 2.94095E-2
7.00000 2.19399E-2
8.00000 5.00320E-4
9.00000 8.67487E-3
10.0000 9.70602E-4
15.0000 6.25349E-5
20.0000 1.95121E-7
25.0000 2.32717E-8
30.0000 1.23606E-9
31.0644 1.19609E-9
TS=400
k Et(k)
1.00000 2.55930E-3
2.00000 1.44344E-1
3.00000 1.44038E-1
4.00000 1.13994E-2
5.00000 1.51787E-2
6.00000 5.00846E-3
7.00000 8.01863E-3
8.00000 4.40039E-4
9.00000 2.17536E-3
10.0000 4.03067E-4
15.0000 1.64498E-5
20.0000 4.98806E-8
25.0000 4.28694E-9
28.4604 1.20757E-9
TS=600
k Et(k)
1.00000 3.38334E-3
2.00000 6.41323E-2
3.00000 5.92391E-2
4.00000 2.47813E-3
5.00000 9.64493E-3
6.00000 2.90652E-3
7.00000 2.23367E-3
8.00000 1.82921E-4
9.00000 7.40450E-4
10.0000 1.20997E-4
15.0000 3.92064E-6
20.0000 1.22954E-8
25.0000 1.06088E-9
26.2488 1.23265E-9
5.2.2 Prandtl Number = 0.71
TS=0
k Et(k)
1.41421 4.02415E-1
2.00000 6.70266E-1
3.00000 9.92101E-1
4.00000 7.48541E-1
5.00000 5.38204E-1
6.00000 2.72123E-1
7.00000 8.73092E-2
7.81024 3.59780E-2
TS=200
k Et(k)
1.00000 1.29962E-3
2.00000 4.39541E-1
3.00000 6.68757E-1
4.00000 1.78571E-1
5.00000 2.47688E-1
6.00000 1.62595E-1
7.00000 1.52707E-1
8.00000 2.84070E-3
9.00000 1.21958E-1
10.0000 2.15648E-2
15.0000 5.25002E-3
20.0249 6.66627E-4
25.0000 3.16783E-5
30.0000 1.91073E-6
35.0000 3.52409E-7
40.1249 2.62925E-8
45.0222 1.06691E-9
45.4972 1.07276E-9
TS=400
k Et(k)
1.00000 3.63503E-3
2.00000 2.30373E-1
3.00000 4.05406E-1
4.00000 5.89957E-2
5.00000 9.92578E-2
6.00000 4.18847E-2
7.00000 1.35679E-1
8.00000 1.25409E-2
9.00000 7.35297E-2
10.0000 1.78520E-2
15.0000 4.28378E-3
20.0249 6.60598E-4
25.0000 1.95621E-5
30.0000 1.92381E-6
35.0000 1.98541E-7
40.1249 1.09693E-8
43.7035 1.20398E-9
TS=600
k Et(k)
1.00000 5.30288E-3
2.00000 1.18290E-1
3.00000 2.31282E-1
4.00000 1.85658E-2
5.00000 8.56520E-2
6.00000 3.71559E-2
7.00000 6.90075E-2
8.00000 4.60320E-3
9.00000 4.70612E-2
10.0000 9.06811E-3
15.0000 2.33428E-3
20.0249 2.38316E-4
25.0000 9.34227E-6
30.0000 4.70629E-7
35.0000 3.57281E-8
40.1249 4.17494E-9
42.7317 1.11619E-9
5.3 Two-point Scalar Correlation (x1-direction)
5.3.1 Two-point Auto-correlation of T
5.3.1.1 Prandtl Nnmber = 0.20
TS=0
tau Rtt(tau)
0.00000 1.00000
3.92699E-1 7.65198E-1
7.85396E-1 3.80563E-1
1.17809 1.96080E-1
1.57079 1.29589E-1
1.96349 3.49275E-2
2.35619 -4.35902E-2
2.74889 -2.45681E-2
3.04341 7.40358E-3
TS=200
tau Rtt(tau)
0.00000 1.00000
3.92699E-1 8.42460E-1
7.85396E-1 5.47156E-1
1.17809 3.06791E-1
1.57079 1.36871E-1
1.96349 5.10942E-3
2.35619 -9.71935E-2
2.74889 -1.62158E-1
3.04341 -1.82194E-1
TS=400
tau Rtt(tau)
0.00000 1.00000
3.92699E-1 8.64442E-1
7.85396E-1 5.92639E-1
1.17809 3.41050E-1
1.57079 1.37506E-1
1.96349 -3.38250E-2
2.35619 -1.80986E-2
2.74889 -2.89195E-1
3.04341 -3.27239E-1
TS=600
tau Rtt(tau)
0.00000 1.00000
3.92699E-1 8.75289E-1
7.85396E-1 6.12474E-1
1.17809 3.49222E-1
1.57079 1.18035E-1
1.96349 -8.43321E-2
2.35619 -2.53471E-2
2.74889 -3.74504E-1
3.04341 -4.17435E-1
5.3.1.2 Prandtl Number = 0.71
TS=0
tau Rtt(tau)
0.00000 1.00000
3.92699E-1 7.65198E-1
7.85396E-1 3.80563E-1
1.17809 1.96080E-1
1.57079 1.29589E-1
1.96349 3.49275E-2
2.35619 -4.35902E-2
2.74889 -2.45681E-2
3.04341 7.40358E-3
TS=200
tau Rtt(tau)
0.00000 1.00000
3.92699E-1 7.41081E-1
7.85396E-1 4.01289E-1
1.17809 2.12377E-1
1.57079 1.02886E-1
1.96349 1.18153E-2
2.35619 -6.33214E-2
2.74889 -1.08048E-1
3.04341 -1.16941E-1
TS=400
tau Rtt(tau)
0.00000 1.00000
3.92699E-1 7.23875E-1
7.85396E-1 3.89504E-1
1.17809 2.03907E-1
1.57079 8.73791E-2
1.96349 -9.34931E-3
2.35619 -1.04056E-2
2.74889 -1.84025E-1
3.04341 -2.12603E-1
TS=600
tau Rtt(tau)
0.00000 1.00000
3.92699E-1 7.11879E-1
7.85396E-1 3.61104E-1
1.17809 1.74932E-1
1.57079 5.64651E-2
1.96349 -4.39034E-2
2.35619 -1.39069E-1
2.74889 -2.26159E-1
3.04341 -2.66649E-1
5.3.2 Two-point Cross-correlation of u1 and T
5.3.2.1 Prandtl Number = 0.20
TS=0
tau Rut(tau)
0.00000 -1.44586E-2
3.92699E-1 -6.11074E-3
7.85396E-1 -1.33807E-2
1.17809 -4.25926E-2
1.57079 -6.47857E-2
1.96349 -5.59928E-2
2.35619 -3.87882E-2
2.74889 -4.49668E-2
3.04341 -6.26223E-2
TS=200
tau Rut(tau)
0.00000 -1.04887E-3
3.92699E-1 8.77347E-4
7.85396E-1 -1.04764E-2
1.17809 -3.60806E-2
1.57079 -6.03131E-2
1.96349 -7.15026E-2
2.35619 -7.54783E-2
2.74889 -8.10526E-2
3.04341 -8.68597E-2
TS=400
tau Rut(tau)
0.00000 1.92512E-2
3.92699E-1 1.03815E-2
7.85396E-1 -2.50776E-3
1.17809 -8.33401E-3
1.57079 -1.85051E-2
1.96349 -4.12017E-2
2.35619 -6.19532E-2
2.74889 -7.13868E-2
3.04341 -7.65273E-2
TS=600
tau Rut(tau)
0.00000 2.80603E-2
3.92699E-1 1.27637E-2
7.85396E-1 7.49954E-3
1.17809 1.89814E-2
1.57079 1.81352E-2
1.96349 -6.40816E-3
2.35619 -3.68489E-2
2.74889 -5.48986E-2
3.04341 -6.30438E-2
5.3.2.2 Prandtl Number = 0.71
TS=0
tau Rut(tau)
0.00000 -1.44586E-2
3.92699E-1 -6.11074E-3
7.85396E-1 -1.33807E-2
1.17809 -4.25926E-2
1.57079 -6.47857E-2
1.96349 -5.59928E-2
2.35619 -3.87882E-2
2.74889 -4.49668E-2
3.04341 -6.26223E-2
TS=200
tau Rut(tau)
0.00000 -9.15311E-3
3.92699E-1 -1.66495E-3
7.85396E-1 -1.16247E-2
1.17809 -3.88549E-2
1.57079 -6.18614E-2
1.96349 -6.62977E-2
2.35619 -6.43712E-2
2.74889 -6.87116E-2
3.04341 -7.34356E-2
TS=400
tau Rut(tau)
0.00000 4.86372E-3
3.92699E-1 1.88666E-3
7.85396E-1 -9.99898E-3
1.17809 -9.54547E-3
1.57079 -1.45949E-2
1.96349 -3.48256E-2
2.35619 -4.74712E-2
2.74889 -4.52225E-2
3.04341 -4.54916E-2
TS=600
tau Rut(tau)
0.00000 1.57828E-2
3.92699E-1 -7.63564E-4
7.85396E-1 -3.18279E-3
1.17809 2.74346E-2
1.57079 3.79977E-2
1.96349 1.39818E-2
2.35619 -1.67513E-2
2.74889 -2.57774E-2
3.04341 -2.45575E-2
5.4 Taylor Micro Scale of T
5.4.1 Prandtl Number = 0.20
TS Ramdt
0 5.30994E-1
100 6.01636E-1
200 6.25611E-1
300 6.89005E-1
400 7.10113E-1
500 7.27146E-1
600 7.45882E-1
5.4.1 Prandtl Number = 0.71
TS Ramdt
0 5.30994E-1
100 5.09580E-1
200 4.71441E-1
300 4.56420E-1
400 4.49209E-1
500 4.44183E-1
600 4.43962E-1
5.5 Integral Scale of T
5.5.1 Prandtl Number = 0.20
TS Lt
0 7.63122E-1
100 8.24965E-1
200 8.19134E-1
300 7.85126E-1
400 7.37302E-1
500 6.89601E-1
600 6.48297E-1
5.5.1 Prandtl Number = 0.71
TS Lt
0 7.63122E-1
100 7.58729E-1
200 7.19570E-1
300 6.79317E-1
400 6.28680E-1
500 5.78482E-1
600 5.36436E-1
6. References
6.1 Miyauchi T. and Ishizu T., "Direct Numerical Simulation of
Homogeneous Isotropic Turbulence. - Decay of passive scalar fluctuation",
Prepr. of Jpn. Soc. Mech. Eng. No.914-2, pp. 166-168,
(Kansaishibu Dai66ki Teijisoukai), (1991).
(EOF)