Week 8: Literature review - RANS derivation and analysis

To derive the Reynolds stress terms and explain the concepts of turbulent and molecular viscosity, I'll go through each question one by one:

1. Applying Reynolds Decomposition to Navier-Stokes Equations

Reynolds decomposition is used to separate the instantaneous variables in turbulent flow into their mean and fluctuating components. Given any flow variable $\phi$ϕ (like velocity, pressure), we decompose it as:

$$\phi = \overline{\phi} + \phi'$$

where:

• $\overline{\phi}$​ is the time-averaged (mean) component, and
• $\phi'$ is the fluctuating (deviation) component.

For velocity $u_i$ui​, this decomposition becomes:

$$u_i = \overline{u_i} + u_i'$$​

When this decomposition is applied to the Navier-Stokes (NS) equations, it introduces terms that represent the interaction between fluctuating components. The NS equations are:

$$\frac{\partial u_i}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + \nu \frac{\partial^2 u_i}{\partial x_j^2}$$​​

Expanding this with Reynolds decomposition and averaging, we obtain an additional term, the Reynolds stress tensor $-\rho \overline{u_i' u_j'}$​​.

2. Expression for Reynolds Stress

The Reynolds stress tensor arises from averaging the momentum equation with the fluctuating components. It is expressed as:

$$\tau_{ij} = -\rho \overline{u_i' u_j'}$$​​

This term represents the additional stresses in turbulent flow due to velocity fluctuations. These fluctuations cause apparent stresses that affect the momentum exchange and energy dissipation in turbulent flow. Reynolds stresses are critical in modeling turbulent flows, as they encapsulate the effect of turbulence without tracking every small fluctuation in the flow.

3. Understanding Reynolds Stress

Reynolds stress represents the rate at which turbulent eddies transfer momentum across layers in a fluid, effectively capturing the impact of turbulence on the mean flow. Since turbulence involves chaotic fluctuations, Reynolds stresses are not easily predictable and usually require modeling approaches like turbulent viscosity models or Reynolds-averaged Navier-Stokes (RANS) equations.

4. Turbulent Viscosity and Molecular Viscosity

• Molecular Viscosity $(\mu)$: Refers to the inherent resistance of a fluid to flow, caused by molecular interactions. It is constant for a given fluid at a fixed temperature and is responsible for viscous dissipation at small scales.

• Turbulent Viscosity $(\mu_t)$: A conceptual model used in turbulence modeling that accounts for the momentum transfer due to turbulent eddies. Unlike molecular viscosity, turbulent viscosity is not a physical property of the fluid but rather an effective viscosity that varies with flow conditions. It is generally much larger than molecular viscosity in turbulent flows and depends on the scale and intensity of turbulence.