2.5. Governing Equations

The governing equations for turbulence include the continuity equation and the momentum equations. We express this as:

• $D D t \tau_{ij} = -\frac{1}{\rho} \frac{\partial p}{\partial x_{i}} + \frac{\partial x_{j}}{2\nu} + \nu_{t} \delta_{ij}$

In fluid dynamics, the governing equations describe how fluid flows under various conditions. The continuity equation ensures mass conservation, while the momentum equations dictate how momentum changes in a fluid due to forces acting on it. The equation provided expresses how the Reynolds shear stress evolves, linking it with pressure gradients, viscosity, and turbulence-induced stresses.

Think of a crowd at a concert. The continuity equation is like ensuring that the number of people entering and leaving the concert is balanced, so the crowd doesn't overflow or empty. The momentum equation describes how people move through the crowd when they push or are pushed.

The term $\nu_t$ represents the turbulent eddy viscosity, which links the turbulent kinetic energy (k) with the energy dissipation rate ($\epsilon$). It is expressed as:

• $\nu_t = c_\mu \frac{k^2}{\epsilon}$

Turbulent eddy viscosity is a model parameter that quantifies the effective viscosity of a turbulent fluid. It indicates how much momentum is transported by turbulent eddies. The equation shows that this viscosity depends on the turbulent kinetic energy and how quickly that energy is dissipated, allowing for a better understanding of turbulent flows.

Consider a large spoon stirring a pot of soup. The spoon represents the mean flow, while the soup’s motion becomes chaotic and swirling around the spoon, like turbulence. The turbulent eddies can be likened to the swirling movements in the soup, which transport heat and flavors throughout, similar to how eddy viscosity works in a turbulent flow.

To compute $\tau_{ij}$, formulas from the turbulent kinetic energy (k) and dissipation rate ($\epsilon$) are utilized, expressed with constants:

• $C_\mu = 0.09$
• $\sigma_k = 1$ and others for turbulent models.

The equations for turbulent kinetic energy and dissipation rate provide a way to model turbulence quantitatively. Constants like $C_\mu$ and $\sigma_k$ are empirically derived values that must be known to perform calculations accurately. These constants help relate the turbulence characteristics to the governing equations, allowing for better predictions and simulations of fluid behavior.

Think of baking. To achieve the best cake, you need the right ingredients in specific proportions. The constants in turbulence modeling are like the ingredients in your recipe; if you don’t get them right, your 'cake' (or flow model) won't turn out as expected!

Direct numerical simulation (DNS) resolves the Navier-Stokes equations without turbulence models, requiring high spatial resolution and computational power, especially for high Re flows.

DNS is a numerical method that calculates fluid dynamics equations directly, capturing all scales of turbulence without approximations. This method requires an enormous amount of computational resources, especially in high Reynolds number flows, where the complexity of turbulence increases significantly.

Imagine trying to capture every detail in a crowded market scene painting. If you want to portray every person, movement, and interaction accurately, you’ll need a very large canvas and an immense amount of paint, just like how DNS requires vast computational resources to capture the intricacies of fluid motion.

In turbulent flows, the balance of kinetic energy supplied and dissipated is crucial. The concept of Kolmogorov length scale ($\eta$) is used to describe the smallest scales of turbulence where energy is dissipated.

Understanding the energy balance is vital in turbulent flows as it explains how energy flows from large-scale movements to smaller scales until it's dissipated. The Kolmogorov length scale indicates where this dissipation occurs, highlighting the scale differences that exist within turbulent flows.

Consider a waterfall. The large gushing water (larger vortices) crashes down and breaks into smaller streams (smaller vortices). At some point, the energy in the smaller streams dissipates as they hit rocks and the ground, similar to turbulence dissipating at the Kolmogorov scale.

--