PARTICLE-LADEN FLOW - ERCOFTAC SERIES Phần 5 pptx

DNS of particle-laden flow over a backward

facing step at a moderate Reynolds number

A. Kubik and L. Kleiser

Institute of Fluid Dynamics, ETH Z¨urich, Switzerland

[email protected]

Summary. The present study investigates turbulence modification by particles in

a backward-facing step flow with fully developed channel flow at the inlet. This

flow configuration provides a range of flow regimes, such as wall turbulence, free

shear layer and separation, in which to compare turbulence modification. Fluid-phase

velocities in the presence of different mass loadings of particles with a Stokes number

of St = 3.0 are studied. Local enhancement and attenuation of the streamwise

component of the fluid turbulence of up to 27% is observed in the channel extension

region for a mass loading of φ = 0.2. The amount of modification decreases with

decreasing mass loading. No modification of the turbulence is found in the separated

shear layer or in the re-development region behind the re-attachment, although there

were significant particle loadings in these regions.

1 Background

The use of Direct Numerical Simulations (DNS) to predict particle-laden flows

is appealing as it promises to provide accurate results and a detailed insight

into flow and particle characteristics that are not always, or not easily, access￾ible to experimental investigations. In the present study, a vertical turbulent

flow over a backward-facing step (with gravity pointing in the mean flow dir￾ection) at moderate Reynolds number Reτ ≈ 210 (based on friction velocity

uτ and inflow channel half-width h) is investigated by means of DNS. The

main focus is directed towards particle statistics and turbulence modification.

Fessler and Eaton [9] reported the results of experiments on particle-laden

flows in a backward-facing step configuration. Like in our simulations, in this

work the bulk flow rate was fixed. The corresponding Reynolds number was

approximately Reτ ≈ 644. The experiments were performed with glass and

copper particles of different diameters in downward air flows. The particles

used in our simulations were chosen to match those in experiments and our

previous studies. (Due to limited space results for only one particle species

are presented here.)

Bernard J. Geurts et al. (eds), Particle Laden Flow: From Geophysical to Kolmogorov Scales, 165–177.

© 2007 Springer. Printed in the Netherlands.

166 A. Kubik and L. Kleiser

Our previous studies, e.g. [13], concentrated on particle-laden flows in a

vertical channel down-flow at the above-mentioned Reynolds number. It was

confirmed that particle feedback causes the turbulence intensities to become

more non-isotropic as the particle loading is increased. The particles tended to

increase the characteristic length scales of the fluctuations in the streamwise

velocity, which reduces the transfer of energy between the streamwise and the

transverse velocity components.

2 Methodology, numerical approach and parameters

The Eulerian-Lagrangian approach is adopted for the calculations of the fluid

flow and the particle trajectories. The two phases are coupled, as the fluid

phase exerts forces on the particles and experiences a feedback force from the

dispersed phase.

Fluid phase

The fluid phase is described by the 3D time-dependent modified Navier-Stokes

equations in which the feedback force on the fluid is added as an effective body

force. Additionally, the incompressibility constraint must be satisfied.

Duf

Dt = −∇p +

1

Re∆uf + f g + f r , ∇ · uf = 0 (1)

The symbols uf , t, p, Re, f g and f r denote the fluid velocity, time, pressure,

Reynolds number, gravity force and feedback force per unit mass, respectively.

To compute the feedback force the sum of the drag and lift forces acting on a

particle is redistributed to the nearest grid points, summed up with feedback

forces from other particles and divided by the mass of fluid contained in the

volume surrounding the grid point. [11]

The geometry and dimensions of the backward-facing step domain are

shown in fig. 1. The Reynolds number of the inlet channel flow in the present

simulation is chosen to be Reτ ≈ 210, as in our previous work [12],[13]. This

is a moderate number, still manageable in terms of computational costs but

securely located in the range of flows considered turbulent. Based on the bulk

velocity of the fluid, the Reynolds number is around 3333. This translates to

a Reynolds number of the back-step, based on bulk velocity and step height

H of ReH ≈ 6666.

The equations are solved using a spectral–spectral-element Fourier–Le￾gendre code [20] with no-slip boundary conditions on the walls and peri￾odic boundary conditions in the spanwise direction. Fully developed turbu￾lent channel flow from a separate calculation is applied at the inlet, whereas

a convective boundary condition is imposed at the outlet.

DNS of particle-laden flow over BFS at moderate Re 167

H z x

y

2h

u

Fig. 1. The geometry of the backward-facing step domain.

Inlet channel

Channel half-width h

Channel span 3.2h

Channel length 5H

Backward-facing step domain

Step height H = 2h

Expansion ratio H : 2h = 1

Domain behind step 52H

Dispersed phase

The particles are tracked individually. Their trajectories are calculated simul￾taneously in time with the fluid phase equations by integrating the equation of

motion for each particle. This is done by solving the equations for the particle

velocity and position vectors as given by Maxey and Riley [15]. Several modi￾fications were necessary: A lift force was supplemented ad hoc as described in

[16],[18]. Empirical and analytical corrections for the drag and lift were neces￾sary to accommodate moderate Reynolds numbers [5],[17] and the proximity

of walls [1],[8],[19]. Only the effects of drag, gravity and lift are taken into

account [11]. The equations for the particle velocity and position are thus

dup(k)

dt = Fdrag + Fg + Flif t , dxp(k)

dt = up(k) (2)

where up(k) denotes the velocity and xp(k) the position of the particle k.

Fdrag, Fg and Flif t represent the drag, gravity and lift force per particle

mass, respectively. Equations (2) were discretized in time and solved using

a third-order Adams-Bashforth scheme. Particle-wall collisions are modeled

taking into account the elasticity of the impact and particle deposition for

low-velocity particles in regions of low shear [12],[11]. Particle-particle colli￾sions are omitted in this study and the parameter range for the calculations

is restricted such as to keep this assumption valid. At the initial time the

backward-facing step computational domain contains no particles. They are

168 A. Kubik and L. Kleiser

introduced via the inlet channel flow, in which they have reached a statistically

s spatial distribution (starting with a random field) in a separate calculation.

Monodisperse particles with a particle-to-fluid density ratio of ρp/ρf =

7458 are used. The particle Reynolds number Rep characterizing the flow

around the particle is defined as

Rep = dp |uf − up|

ν (3)

where dp is the particle diameter, |uf − up| the velocity slip between the

particle and the fluid at the particle position and ν the kinematic viscosity of

the fluid. The particles response time τp for small particles with high particle￾to-fluid density ratios can be derived from the expression by Stokes τp,Stokes

corrected for non-negligible Reynolds numbers by the relation [5]

τp = τp,Stokes

[1 + 0.15Re0.687 p ]

≈ ρpd2

p

18µ[1 + 0.15Re0.687 p ] (4)

where µ the fluid dynamic viscosity. The Stokes number is the ratio of the

particle response time to a representative time scale of the flow, St = τp/τf .

There are several fluid time scales appropriate for analyzing the backward￾facing step flow, such as the approximate large-eddy passing frequency in the

separated shear layer 5H/ucl [9] or the local turbulence time-scale k/. (ucl is

the fluid velocity at the centerline.) In the present study the nominal Stokes

number was chosen to be St = 3.66, based on τp,Stokes and the large-eddy

time scale. This corresponds to a Stokes number of St = 3.0 based on τp and

turbulence time-scale k/ at the inlet channel centerline.

3 Results and Discussion

Figure 2 shows a contour plot of the mean fluid velocity with superimposed

streamlines. The flow topology includes the recirculation region behind the

step, an enlarged boundary layer at the step-opposite wall (due to the pressure

gradient), the re-attachment point at x/H = 7.4, a deceleration of the flow

behind the step, and a re-development toward an symmetric channel flow at

approximately x/H = 20. It should be noted that the mean velocity profile

is unchanged by the presence of particles, as constant fluid mass flow was

enforced in the simulation. (Additionally, relatively low mass loadings of the

particles combined with the high particle-to-fluid density ratio result in very

low volume loadings of the particles.)

Figure 3 shows a contour plot of the mean particle number density c di￾vided by the particle number density averaged over the inlet c. Very few

particles are found in the recirculation region directly behind the step. After

the re-attachment point an increasing number of particles can be found below