#### Name

Sajjad ur Rehman

#### Course

Ph.D Student (박사과정)

#### Research Area

##### Study of Isotropic turbulence

The term “turbolenze ” is first introduce by an Italian mathematician, engineer Leonardo da Vinci 500 years ago. Initially turbulence is considered as random process. The first attempt is done in 1930s by G. I. Taylor who introduced statistical methods involving correlation, Fourier transform and power spectra into turbulence literature. Then in 1941 Russian statistician A. N. Kolmogorov published three papers that provide some important and most-often quoted results of turbulence theory.

Statistically Homogenous Turbulence is that in which all statistics of fluctuating quantities are invariant under translation of the coordinate system. Scalar dissipation rate in statistically homogenous turbulence is look like as,

Statistically Isotropic turbulence is that in which all statistics are invariant under translation, rotation and reflection of coordinate system. In this case mean velocities are zero. Simplifications allow theoretical conclusions about turbulence. Turbulent motions on small scales are typically assumed to be isotropic (Kolmogorov hypotheses).

##### Numerical treatment for Isotropic turbulence:

In order to solve isotropic turbulence problem Direct Numerical Simulation (DNS) is used. This method is first introduced by Orszag and Patterson in 1970. In this method Naiver-Stokes equations are solved without any turbulence model. DNS is performed with and without using Pseudo-Spectral methods. Initial conditions can be controlled in such simulations. First convert Naiver-Stokes equation into rotational form by replacing a nonlinear term by a relationship. Pressure Possion equation is used to eliminate pressure. Then by applying Fourier series all derivatives are expanded. Viscous terms are solved analytically. For temporal discretization a low storage 3rd –order 3-stage Runge-Kutta scheme is applied. Nonlinear terms are solved by using Pseudo-spectral method. External forcing is necessary to maintain turbulence. Stochastic spectral scheme as proposed by Eswaren and Pope is used.

Eulerian Statistics are calculated other interesting results for example one dimensional spectra, structure functions and correlation function are calculated.

**Figure 1.** RMS(Root mean square) Velocity**Figure 2.** 1D Energy Spectrum

#### Email

sajjadqau2@gmail.com

#### Entrance date

2013/09