3. Conservation equations and turbulence

For mass, momentum and energy conservation

Balancing equation

Rate of change of ϕ

= flow transport in

- flow transport out

+ molecular transport in

- molecular transport out

-+sources/sinks in CV

-+effects on CV-surfaces (2D,3D)

Example: Heat flow

Flow transport: convection

Molecular transport: conduction

Mass = Density * Volume

Mass flow rate = Density * Volume flow rate

= Density * Velocity * Area

Mass conservation equation ; Continuity equation

• Momentum = mass x velocity

• Momentum is a vector because velocity is a vector.

• One momentum equation per coordinate direction (x, y, z)

  Conservation equation:

• Temporal change of momentum = mass * acceleration

• Newton: mass * acceleration * force

• Momentum balance = force balance

CV Balancing

2D-conservation equations for x-, y- momentum:

3D -conservation equations for x-, y-, z-momentum:

• Set of partial, coupled, nonlinear differential equations with variable coefficients.

Stokes: Momentum flows contrary to the velocity gradient, from “fast” to “slow”

Terms on the right side of the normal stresses are zero for incompressible fluids

Stresses (e.g. molecular momentum fluxes) are symmetric tensor of second order (3 x 3-Matrix)

Conservation equations in tensor-notation

Turbulence Characteristics

Turbulence Effects

• Negative

Increased friction & resistance

Increased losses

Increased energy (heat) transfer

Noise

Vibrations

• Positive

Increased energy (heat) transfer

Increased mass (species concentration) transfer

Mixing

Mean Values has many industrial applications

• interest in mean (averaged) quantities:

• Friction coefficients

• Lift coefficients

• Heat transfer rates

  Alternatives:

• Averaging of the results of a Direct(3D, time-dependent) Numerical Simulation

• Averaging of the equations

• averaged quantities as result

DNS = Direct Numerical Simulation

• solves Navier-Stokes equations without averaging or approximation (only discretization)

• 3D , time dependent

• all motions contained in the flow are resolved by using a very fine grid

• limited by processing speed and memory of the machine

• most exact and „simplest“ approach

• possible only at low Reynolds numbers

• Resolves all scales and is used for none of the models.

LES = Large Eddy Simulation

• 3D , time dependent

• treats only large eddies exactly (more energetic,mainly responsible for turbulent transport), uses models for small scales

• prefered method for flows in which the Reynolds number is too high or the geometry is too complex for DNS

• Resolves all scales and works for small models.

  
  

RANS (=Reynolds Averaged Navier Stokes) Model