Week 8: Literature review RANS derivation and analysis

Let's consider the incompressible Navier Stokes Equation in differential form:

∂u/∂t +(u⋅∇) u=−1/ ρ(∇p) + ν ∇2 u

On Applying Reynold’s Decomposition Velocity & Pressure Fields:

u = U + u′

P = P + p′

Next,

On Substituting these decompositions into Navier Stokes Equations,

∂U/∂t +(U⋅∇)U=−1 /ρ ∇P + ν∇2 U−1/ ρ ∇⋅R

∇⋅U=0

Where R is Reynold’s Stress sensor given by:

Rij =u’i u′j

The above equation can be further decomposed as:

Rij =−ui′ uj′ +2/3 kδij

Where k is Turbulent Kinetic Energy given by :

K = 1/2 ui′ ui′

Thus, above equation shows Reynold’s Stress is related to Turbulent Fluctuations of the flow & also plays crucial roles in their prediction.

Reynolds stress is a term that appears in the equations describing the behavior of fluid flow. It arises from the fluctuating nature of turbulence in a fluid and is responsible for transferring momentum between different regions of the flow It measures the correlation of fluctuating velocity components in different directions. It represents the effect of turbulent fluctuations on the mean flow and is responsible for transferring momentum between different regions of the flow It is a tensor quantity that describes the stress due to turbulent uplift in the fluid. It can be decomposed into two parts: an isotropic pressure-like term and an anisotropic shear-like term. This is an important concept in the study of fluid dynamics and plays an important role in the prediction and modeling of turbulent flows.