This article is about complex Dynamics Systems Turbulence Flow in Fluid Mechanics mathematical model and analysis method.

**Mathematical Model of **Fluid Mechanics

There are three types of flow in fluid mechanics. 1-Laminar flow, 2-Transition flow, 3-Turbulence flow. The reason why flow is examined under these three headings is due to change in their structure. 1-Laminar flow is a situation that particle of liquid and therefor the flow follow straight and smooth path (pattern). This flow can be easily analysed and result can be interpreted. Since the laminar flow characteristic understood, it can be used easily n applied engineering systems. Because, we know how to get the results we want to obtain on the flow by changing which parameters in

the flow. 2-Transition flow represents the transformation from laminar flow to turbulence flow. 3-Turbulence flow is a situation that particle of liquid and therefor the flow not follow straight and smooth path, on the contrary flow follows unpredictable, complex pattern format. Also, this flow is 3D that is another reason it is difficult to analyse, and consequently explain this flow. That is why, it is tries to escape this kind of flow in engineering practice. To recognize and distinguish the flow type it use Reynolds equation.

**Analysis Method**

Andrei N. Kolmogorov studied about flow of fluids. In 1941, he published a paper shows mathematical analysis to formula for he energy spectrum of turbulence flow.[4] However, Kolmogorov’s results are not empirically correct. On the other hand, it is a splendid part of applied mathematical analysis. He stated that it is caused by an energy cascade that does not dissipate from large scales (large eddy) to small scales (small eddy). The Navier-Stokes equations describing the dynamics of the flow of viscous fluids can take a spectral form where the variables are wave numbers for eddies of various sizes.

Kolmogorov’s studies shows that the energy density per unit wave number of E should depend solely on the number of waves k and ψ, which is the rate of energy loss per unit volume. İf equation 4.2 for some constant C dimensional compatibility requires

E(k,ψ) = Ckαψβ L3 = L-αL2β

T-2 = T-3β

-α+2β = 3 -3β = -2

Thus Kolmogorov’s energy spectrum is;

E(k,ψ) = Ck-5/3ψ2/3

Joseph Smagorinsky proposed a simulation model for atmospheric flow in 1963. This mathematical model is feasible to simulate for turbulence flow. It is called Large Eddy Simulation.[5] LES is applied for complex engineering flow, including combustion, acoustic, and simulations of the atmospheric boundary layer. However, significant challenges remain for the LES. The equations used in LES derived from the laws of conservation of mass, energy and momentum. Finite volume method used to solve these equations.

Equations expressing the conservation of mass and Momentum in Newtonian incompressible flow can be written as below;

S(ij)=1(?? +??) ?? 2 ? ? ? ?

??? = ?(???? − ????)

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