Navier-Stokes Equations

Question 1

What do the Navier-Stokes Equations Describe?

Answer 1

How the Velocity, Pressure, Temperature, and the density of a moving fluid are related.

Question 2

What is a number that is an indicator of continuum?

Answer 2

Knudsen number: Kn = lambda / l = mean free path length / characteristic length

Question 3

For what Knood’s number is Navier-Stokes Applicable?

Answer 3

Kn < 0.01

Question 4

What are the four independent variables in Navier-Stokes?

Answer 4

The spatial coordinates and time.

Question 5

What are the six dependent variables for Navier-Stokes?

Answer 5

Pressure
Density
Temperature
Velocity components

Question 6

What is the formula for Stokes hypothesis?

Answer 6

Lambda = -2/3 micro

Question 7

What are the Euler Equations?

Answer 7

Set of partial Differential Equations governing adiabatic and inviscid flow. Momentum equation with simplifications of no viscosity and heat loss.

Question 8

What do Navier-Stokes Equations Consist of?

Answer 8

Conservation of Mass, Momentum and Energy

Question 9

What is Time Averaging?

Answer 9

Unsteady signal is decomposed in a mean and a fluctuating part.

Question 10

What is the characteristic length of ambient pressurized air?

Answer 10

70nm

Question 11

What dimensionless number is important in viscous / compressible flow?

Answer 11

Re / Ma

Question 12

What is the focus of time averaging?

Answer 12

Stationary Turbulance

Question 13

What is the closure problem in Reynolds Average Navier Stokes Equations (incompressible)?

Answer 13

Because of the introduction of the mean flow properties, we have ten unknowns but only four equations so the system is not closed and cannot be solved.

Question 14

Which property do we need to calculate all the mean flow properties of turbulent flow?

Answer 14

Reynolds Stress Tensor

Question 15

What else has to be considered in compressible flows (wrt averaging)?

Answer 15

We have significant density and temperature fluctuations. Needs to be decomposed into mean and fluctuating parts.

Question 16

Why do we introduce mass averaged velocities in compressible turbulent flow?

Answer 16

Because of the double and triple correlations between the quantities (fluctuating density(not for RA), fluctuating velocity components)

Question 17

What are possible forces for momentum equation?

Answer 17

• Volume Forces
• Body Forces
• Surface Forces

Question 18

What are the four important components of the momentum equation?

Answer 18

• Variation
• Divergence
• Internal Source
• Diffusion

Question 19

Where is Stokes hypothesis used?

Answer 19

To find the constant lambda in the Momentum equation for Newtonian fluids

Question 20

What law is used to derive the energy equation in NS?

Answer 20

First law of thermodynamics

Question 21

What is the formula for time averaging a function?

Answer 21

\bar f(x) = limt_{T->infty} (1/T) \int_(t)^(t+T) f(x, t) dt

Question 22

What is the Reynolds stress tensor?

Answer 22

It is the additional term that appears in the RANS equation (incompressible), consists of the time averaged terms.

Question 23

How many independent components does the Reynolds stress tensor have?

Answer 23

6

Question 24

What is Favre averaging?

Answer 24

It is the averaging method for compressible flows where we also have to average the density distribution and because of this have to introduce mass averaged velocities.