Questions: Turbulent Kinetic Energy: Production and Dissipation

Question 1

In a high-Reynolds-number turbulent pipe flow, where does most of the irreversible conversion of mechanical energy to heat actually occur?

• A. At the pipe centerline, where the mean velocity and kinetic energy are highest
• B. At the large energy-containing eddies, which carry most of the turbulent kinetic energy
• C. At the Kolmogorov microscale eddies, where viscous forces dominate over inertial forces
• D. Uniformly distributed throughout the flow cross-section

Answer: (C)

Despite containing most of the turbulent kinetic energy, large-scale eddies do not dissipate it directly — they transfer it to progressively smaller scales through the energy cascade. Dissipation occurs at the Kolmogorov microscale η = (ν³/ε)^(1/4), where the local Reynolds number is of order unity and viscous forces finally dominate. It is at this tiny scale that kinetic energy is irreversibly converted to heat. This separation between where energy is produced (large scales near the mean shear) and where it is dissipated (Kolmogorov scales) is the defining feature of the turbulent energy cascade.

Question 2

Grid turbulence — created when flow passes through a mesh — decays rapidly as the flow moves downstream. The best explanation is:

• A. The mesh directly absorbs and dissipates the turbulent kinetic energy
• B. Without a mean velocity gradient downstream of the mesh to drive production, turbulence dissipates without being replenished
• C. The flow transitions back to laminar because the Reynolds number drops below the critical value
• D. Downstream boundary conditions absorb the turbulent fluctuations

Answer: (B)

Production of turbulent kinetic energy requires a mean velocity gradient (dŪ/dy), which acts on the Reynolds stresses to transfer energy from the mean flow into turbulent fluctuations. Once the flow passes the mesh and the mean velocity profile becomes uniform, there is no shear to drive production. Turbulence decays because dissipation continues at the Kolmogorov scales while production has ceased. This experiment isolates the decay process and validates the energy budget framework.

Question 3

In the turbulent energy cascade, energy is transferred from small eddies to large eddies before being dissipated at the largest scale.

• T. True
• F. False

Answer: False

The cascade runs in the opposite direction: from large to small scales. Large eddies (whose size is set by the flow geometry) are the first recipients of energy extracted from the mean flow. They are unstable and break up into smaller eddies through nonlinear inertial interactions, transferring their energy downward in scale. This process continues until eddies reach the Kolmogorov microscale, where viscosity dissipates the energy as heat. There is no inverse cascade in standard three-dimensional turbulence.

Question 4

As the Reynolds number of a turbulent flow increases, the ratio of the largest to smallest eddy scales grows, making Direct Numerical Simulation increasingly expensive.

• T. True
• F. False

Answer: True

The ratio L/η scales as Re^(3/4), where L is the integral scale and η is the Kolmogorov microscale. To resolve all scales from L down to η, a DNS grid must have spacing ~ η in each dimension, requiring grid points scaling as (L/η)³ ~ Re^(9/4). At Re = 10⁶, this is roughly 10^13.5 grid points — completely impractical with current computing. This is the fundamental reason turbulence modeling (k-ε, k-ω, LES) is necessary for engineering calculations.

Question 5

Explain why turbulent flows dissipate mechanical energy far more efficiently than laminar flows, using the energy cascade concept.

Model answer: In laminar flow, viscous dissipation acts directly on the smooth mean velocity gradients, which are limited in magnitude. In turbulent flow, the energy cascade creates intense small-scale velocity gradients throughout the entire flow volume: large eddies break into medium eddies, medium into small, continuing to the Kolmogorov scale where gradients are extremely steep. The total rate of dissipation is proportional to the total magnitude of velocity gradients across all scales. By manufacturing enormous small-scale velocity gradients everywhere in the flow, turbulence provides vastly more surface area for viscosity to act, converting kinetic energy to heat at rates orders of magnitude higher than laminar flow at the same bulk velocity.

This is why turbulent flows have much higher friction factors and require more pumping power: the cascade efficiently routes mean-flow energy through all scales to heat. Engineers designing low-drag systems (aircraft, pipelines) work to delay transition to turbulence precisely because turbulent dissipation is so much higher.