Swirling flows

“When I meet God, I am going to ask him two questions: why

relativity and why turbulence? I really believe he will have an

answer for the first” Werner Heisenberg

OutLineI. Swirling flows

II. Turbulence modeling of swirling flows

III. Application of swirling flows

Rotation

An essential ingredient in many industrial processes:

● mixing, ● separation● stabilisation

However, in some cases is an inevitable by product causing damage and financial loss:

● Temperature-differences between ocean and atmosphere leading to thunderstorms and tornadoes

● vortices generated by wings of large airplanes leading to delay during landing-procedures.

Swirling flows:

Many Engineering applications involve swirling or rotating flow:

● In combustion chambers of Jet engines

● Turbomachinery ● Mixing tanks

2D Swirling or Rotating flowsAlso known as the axisymmetric flow with swirl or rotation:

Assumption: no circumferential gradients in the flow.

Solving this problem includes the prediction of the circumferential or swirl velocity.

The momentum conservation equation for swirl velocity is given by :

Where x: is the axial coordinate

r: the radial coordinate

u: the axial velocity

v: the radial velocity

w: the swirl velocity

Physics of the swirling flows:

In swirling flow, conservation of angular momentum results in the creation of a free vortex flow in which circumferential velocity w increases as the radius decreases and then decays to zero at r=0 due to the action of viscosity.

For an ideal free vortex: The circumferential forces are in equilibrium with radial pressure gradient

For non-ideal vortices the radial pressure gradient also changes affecting the radial and axial flows.

Turbulence modeling in swirling flow:

Turbulent flows with significant amount of swirl: swirling jets or cyclone flows.

The strength of the swirl is gauged by the swirl number S, defined as the ratio of the axial flux of angular momentum to the axial flux of the axial momentum.

Where is R is the hydraulic radius*

Turbulence modeling in swirling flow:

Realizable k-epsilon model ● Realizable: model meets certain

mathematical constraints on the Reynold’s stresses that correspond to the physics of the turbulent flows.

● Considered as an improvement of the k-epsilon standard model.

● Characterized by both the new formulation of the turbulent viscosity and a new equation for the dissipation rate epsilon that is derived from the transport of the mean square vorticity fluctuation.

● Provides improved predictions for the spreading rate of both planar and round jets.

● Exhibits superior performance for flows involving rotation, boundary layers under strong adverse pressure gradients, separation, and recirculation.

RNG k-epsilon model

● Developed using Renormalisation Group (RNG) methods to renormalize the Navier-Stokes equations to account for small scales of motion.

● This is different from the original k-epsilon model where the eddy viscosity is determined from a single turbulence length scale.

● Mathematically similar to the k-epsilon model, but it has a different epsilon equation that accounts for different scales contribution to the production term.

Reynolds Stress Models:

● Also known as the Reynolds stress Transport (RST)

● Usually used for high level turbulence models

● The method of closure used is called Second Order Closure

● The Eddy viscosity approach is not used and the Reynold stresses are directly calculated using differential transport equation.

● The calculated Reynold’s stresses are then used to obtain closure for the Reynolds’ averaged momentum equation.

Application of the swirling flow: Simulating the internal flow of a pressure swirl fuel

injector

Fuel injectors

High velocity liquid fuel⇒ atomization and oxidation with air⇒ evaporation ⇒ combustion

Swirl injectors:

Hollow cone spray ⇒ more fuel droplets exposed to the hot air in the combustion chamber⇒ shorter evaporation time

The Physics of the atomization inside a PSI:

1. Film formation: - Liquid fuel is introduced through the tangential

ports into the swirl chamber.- The swirling motion pushes the liquid to the walls

of the injector which constitutes the origin of the thin film

2. Free Sheet : - At the exit of the nozzle, the free sheet is formed

in the shape of a cone.3. Atomization:

- The liquid free sheet is an unstable structure. As it interacts with air, it starts to break down into ligaments, these ligament disintegrate into small droplets [1].

Mathematical formulation

Favre averaging:

Favre averaging is a time averaging method that takes into account a changing density:

Applying the Favre averaging on Y, the gas mass fraction :

Mathematical formulation

Conservation of mass:

Conservation of momentum:

Conservation of energy

Closure: Homogeneous relaxation model

● Used to study thermal non equilibrium two phase flow

● Assumes adiabatic conditions ● Provides an equation for the return or the

relaxation of the quality to the equilibrium value

Computational methods

The geometry

Creating the mesh

Boundary conditions:

Fuel Inlets: -zero pressure gradient

Outlets: -atmospheric pressure -zero velocity

Walls: -zero velocity -zero pressure gradient

Post-processing:

A swirling velocity field

Simulation results:

Density field Pressure field

Volume fraction Temperature field

Spray angle predictions:

Problems and challenges:

Schmidt Number consideration:

● Swirling flows have a higher critical Re to transition from laminar flow to turbulent flow: relatively stable

● The Schmidt number is the ratio of the momentum diffusion rate over the mass diffusion rate.

● If there is no mass diffusion between the liquid phase and the gas phase (no mixing) then the Schmidt number goes to infinity

● A Schmidt number of 1: means that both types of diffusion are occurring at the same rate.