======Homogeneous Isotropic Turbulence Decay of Passive Scalar Fluctuations======
=====DNS by Miyauchi and Ishizu=====

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====Description====

Decay of temperature fluctuations in isotropic decaying turbulence was investigated by
performing direct numerical simulations using a spectral method. Temperature was treated
as a passive scalar, thus, it has no effect on the velocity field. Two cases are reported;
for Prandtl numbers \(Pr=0.2\) and \(0.71\).


====Modelling Methods====

The governing equations are

\[ \nabla \cdot u = 0 \]
\[ \frac{\partial u}{\partial t} = u\times \omega - \nabla P + \frac{1}{Re} \nabla^2 u \]
\[ P=p/\rho + (1/2)|u^2| \]
\[ \frac{\partial \theta}{\partial t} + u\cdot \nabla \theta = \frac{1}{Re\cdot Pr} \nabla^2\theta \]

The Reynolds number was 309 based on the width of the calculation region and the 
root mean square value of the initial velocity fluctuations.  
Turbulent flows with two different Prandtl numbers, 0.2 and 0.71, were simulated. 

The integral length scale of the initial velocity and temperature fluctuations were 0.73 and
0.70, respectively (which should be compared with the width of the calculation region
which is \(2\pi\)). The Taylor micro scales of the initial velocity
and temperature fluctuations were 0.58 and 0.50, respectively. The Reynolds number based
on the Taylor length and velocity scales was 29.1.

The initial velocity field satisfied the divergence-free and isotropic conditions with
the energy spectrum given by
\[ E(k) = Ak^4 \exp(-Bk^2) \]
where \(A\) and \(B\) are constants whose values are: \(A = 0.038\), \(B = 0.140\).

The initial temperature field satisfied
\[ \hat{T}(-k) = \hat{T}(k)^* \]
This condition ensures that the calculated values of the temperature are real numbers
and not imaginary numbers. (This condition should be also satisfied in the velocity
field). The energy spectrum of the initial temperature fluctuations was given by:
\[ E_t(k) = Ak^2 \exp(-Bk^2) \]
with the constants \(A = 0.40\), \(B = 0.10\).


====Calculation Techniques====

The calculation domain was a \(2\pi\times 2\pi\times 2\pi\) cubic box, with periodic
boundary conditions applied in all three directions. 
A spectral methods with \(64\times 64\times 64\) Fourier modes was used to discretize
the problem. For time advancement, the
Adams-Bashforth method was used for the convective term, the Crank-Nicolson method for the
viscous and diffusion terms and the implicit Euler method for the pressure gradient term,
respectively. The numerical time step was 0.005. 


====Available Results====

Data available include:

  * Decay in time of turbulent kinetic energy and temperature variance
  * 3D energy and temperature fluctuation spectra at selected times
  * Two-point correlations for both dynamic and thermal field at selected times

[[case048-plots|Sample plots]] of selected quantities are available.

The data can be downloaded as compressed archives from the links below, or as the single uncompressed file.

  * {{cdata:case048:isdt-allfiles.zip|isdt-allfiles.zip}}
  * {{cdata:case048:isdt-allfiles.tar.gz|isdt-allfiles.tar.gz}}

|    |   Datafile  |
| Dynamic and thermal field data |  {{cdata:case048:is_ut_hm1.dat|is_ut_hm1.dat}}  |


====References====

  - Miyauchi, T., Ishizu, T. (1991). Direct numerical simulation of homogeneous isotropic turbulence. - Decay of passive scalar fluctuation. //Prepr. of JSME No.914-2//, pp. 166-168.

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Indexed data:
<data dbcase free_flow>
case       : 048
title      : Homogeneous Isotropic Decay of Passive Scalar Fluctuations
author*    : Miyauchi, Ishizu
year       : 1991
type       : DNS
flow_tag*  : homogeneous, scalar
</data>