# Turbulence Dissipation *Rate*

• K41
In summary, the conversation discusses the dissipation of energy in turbulent flows and the concept of an energy cascade. The main issue is understanding how the rate of dissipation of kinetic energy, denoted by ϵ, can be independent of viscosity at large scales (as shown in EQ1), but then be calculated from viscosity at small scales (as shown in EQ2). The book being read also uses different terminology for ϵ, referring to it as both "dissipation rate" and "dissipation of turbulent energy". The conversation also mentions the idea that the dissipation stage is the last process in the energy cascade and therefore the dissipation rate must be calculated from the first process, but the concept is not fully understood as it seems

#### K41

I've been reading about dissipation of energy in turbulent flows. I've read in various places about Richardson's idea of Energy cascade. But one of the biggest problems I have is understanding how the books (or texts online) all refer to ϵ, the rate of dissipation of kinetic energy, being independent of viscosity, that is:

ϵ ~ u³/l .........EQ1

where u and l denote the characteristic length and velocity scales of the large eddies.

My problem is as follows. The texts ALL say that the energy dissipation RATE is independent of viscosity. Okay. But then the majority of books then say, at the small scales:

ϵ = 2*v*Sij*Sij .............EQ2

So how can ϵ, the rate of dissipation of kinetic energy, be independent of viscosity, if at the small scales it can be calculated from viscosity?

The book I'm reading also says sometimes that EQ 2 is simply "dissipation" rather than "dissipation rate" or that other times ϵ represents "dissipation of turbulent energy". It seems to regularly use different terminology for the same equation and same symbol. So I need some clarity here.

Any help?

EDIT:

Some text alluded to the fact that the dissipation stage is last process in the entire energy cascade of processes and therefore the dissipation rate must be calculated from the first process in the cascade processes, hence ϵ ~ u³/l but I didn't understand this because, although I agree that the rate of dissipation may depend on how quickly energy cascades to the smaller scale, it must surely also depend on how quickly the energy is transformed to heat at the smallest Kolmogorov scale? I understand that the inertial forces dominate over viscous forces for large eddies, but I don't see how viscosity can be neglected from a rate calculation once you get down to the small scales.

djpailo said:
but I don't see how viscosity can be neglected from a rate calculation once you get down to the small scales.
Compare the magnitudes of the "dissipation" rates.

## 1. What is turbulence dissipation rate?

Turbulence dissipation rate is a measure of the rate at which energy is lost due to the dissipation of turbulence in a fluid. It is a fundamental quantity in the study of turbulent flows and is often denoted by the symbol ε.

## 2. How is turbulence dissipation rate calculated?

Turbulence dissipation rate can be calculated using various methods, such as direct numerical simulation or experimental measurements. It can also be estimated using turbulence models, which use mathematical equations to simulate the behavior of turbulence in a fluid.

## 3. What factors affect turbulence dissipation rate?

The turbulence dissipation rate is influenced by several factors, including the velocity and viscosity of the fluid, the size and shape of the turbulent structures, and the presence of external forces, such as gravity or pressure gradients.

## 4. Why is turbulence dissipation rate important?

Turbulence dissipation rate is important because it plays a crucial role in many natural and industrial processes. It affects mixing and transport of heat, mass, and momentum in fluids, as well as the development of turbulent structures and the overall stability of flows.

## 5. How can turbulence dissipation rate be controlled?

Controlling turbulence dissipation rate is a challenging task, as it is influenced by various factors. However, it can be altered by changing the properties of the fluid, such as its viscosity or density, or by manipulating the flow conditions, such as using flow control techniques or introducing obstacles to change the flow patterns.