Turbulence modeling remains a continuously evolving discipline even today due to the highly fluctuating and complex nature of fluid flow conditions.
One of the complexities in studying turbulence is the scale or size range, which spans a wide spectrum, ranging from scales close to particle sizes to enormously large scales (kilometers) for flows in the Earth’s atmosphere or cosmic dust streams.
The diagram below illustrates the length scales of turbulent flows:
Figure 6.3. Turbulent eddies scale illustration
Let’s say we have a 100-meter building. The largest eddy size will be about a dozen meters, and the smallest are just in the order of millimeters, not to mention the eddy size near the walls.
This extreme size discrepancy makes equations (or mesh sizes) too expensive to capture all these phenomena if demanded.
The difference eddy diameter size also indicates the different characteristic of time-scale. Smaller the eddy, the “frequency” of fluid moving become larger. Which inturn makes the timestep needs to be extremily small to fully resolve it.
To date, there isn’t an exact definition of turbulent flow, but several characteristics define turbulent flow:
- Irregularity: Turbulent flow is highly irregular and chaotic, making it extremely difficult to calculate directly using the Navier-Stokes equations. Within this flow, localized vortices or eddies of varying sizes occur.
- Diffusivity: Flow containing these eddies results in internal frictions, increasing momentum exchange, such as within the boundary layer. This increased diffusivity also elevates wall drag forces and convective heat transfer near walls.
- High Reynolds Number: Turbulent flows generally occur at relatively high Reynolds numbers. For instance, in pipes, the transition from laminar to turbulent flow occurs at Re 2,300, and in the boundary layer at Re 500,000.
- Three dimensions: Turbulent flow always exists in a 3D domain due to eddies flowing in all directions, and it’s unsteady. In turbulence modeling, averaging over time and considering 2D simplifies calculations.
- Dissipation: Dissipative flow involves the conversion of kinetic energy within the flow into thermal energy (heat). This energy change occurs at small eddy scales. Small eddies gain energy from slightly larger eddies, continuing in a cascade until reaching the largest scale, while the largest eddies receive energy from the average flow. The process of transferring energy from the largest to the smallest eddies is known as the cascade process.
- Continuum: Despite eddies being very small in size, they remain significantly larger than the size of fluid molecules. Hence, mathematically, we still calculate this flow as a continuum.
Due to these characteristics, solving the Navier-Stokes equations for every detail of the flow is nearly impossible. Therefore, the simplest approach is to model it as simpler quantities and solve to obtain the required flow patterns, known as turbulent modeling.
This chapter will explain several commonly used methods in CFD for this purpose.
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