INTRODUCTION OBJECTIVES

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1 INTRODUCTION The transport of particles in laminar and turbulent flows has numerous applications in engineering, biological and environmental systems. The deposition of aerosol particles in channels and pipes, as well as in bends and contractions, is important in many applications including dust inhalation and human respiratory systems, chip fabrication, particle size characterization, and sampling of radioactive aerosols. In coal combustion systems and coal liquefaction-gasification pipelines, the erosion of material by solid particle impact is an important problem. In order to gain a fundamental understanding of the particle impact and deposition phenomena, as well as to make meaningful design improvements, it is essential to develop reliable predictive models for the transient two -phase flows encountered in these systems. OBJECTIVES The objective of the present study is to numerically simulate and analyze particle transport and deposition in an unsteady particle laden flow over a square cylinder placed in a channel. The presence of the cylinder (bluff body) creates an unsteady or periodic flow field characterized by the presence of shedding vortices and recirculation in the wake region of the cylinder. In addition, the flow contains a stagnation region in front of the cylinder, as well as regions of flow acceleration and deceleration around it. The particles injected in such complex, transient flows are subjected to both steady and unsteady lift and drag forces, caused by the effects of flow non-uniformity (shear), pressure gradient and unsteady relative acceleration. In order to account for these effects, the particle dynamics is simulated by using the modified Basset-Bousinesq-Oseen (BBO) equation, which is further modified to include the large Reynolds number effects. A detailed two -phase simulation model and flow visualization are employed to analyze the transient flowfield and vortex dynamics, as well as the particle dynamics and dispersion behavior. Numerical experiments are performed to characterize particle deposition as a function of the Stokes number, Reynolds number, ratio of particle density to gas density, and other parameters. The relative contributions of various force terms in the BBO equation are also quantified.

2 THE PHSICAL-NUMERICAL MODEL Gas-Phase Equations The equations governing the unsteady incompressible gas flow are the continuity and Navier-Stokes equations, which can be written in the non-dimensional form as: u x + v y = 0 (1) u t v t + u u + x v u = p + 1 y x Re + u v x + v v y = p y + Fr u + 2 u x 2 y 2 Re 2 v + 2 v x 2 y 2 (2) (3) Here Re and Fr are the Reynolds and Froude numbers defined, respectively, as Re = BV 0 /ν and Fr = V 0 / gb, where g is the gravitational acceleration and V 0 the velocity at the channel entrance. The equations are normalized by using B (square length) as the length scale, V 0 as the velocity scale, and B/V 0 as the time scale. Particle Dynamics Equations Particles under consideration traverse a complex transient flow field containing regions of stagnation flow, separating flow, recirculation, and vortex structures. Consequently, a detailed particle dynamics model based on the modified BBO equation is developed to calculate their trajectories. The BBO equation is further modified to account for the high Reynolds number effects. The non-dimensional form of this equation can be written as: du pi dt = C Ds u i u pi St 1.2 C H ( εst ) 1/2 t t ε d ( u i Du i + C A Dt 2 u pi ) / dt t t * d ε dt ( u i u pi )+ dt * ( εst ) 1/2 d ij ( u i u pi ) ( d lk d kl ) 1/4 ε Fr 2 (4)

3 The equation is written in a vector form where u i and u pi represent, respectively, the gas and particle velocity vectors. The instantaneous particle location x i (t) is obtained from dx i dt =u pi (5) The Stokes number is defined as the ratio of particle response time (t p ) to a characteristic flow time (t f ), and can be written as: St = t p t f = d 2 p ε Re (6) 1 B 2 where d p is the particle diameter. C Ds in Eq. (4) is a correction factor that accounts for the high-reynolds number effect on the steady-state drag coefficient, and is given by: C Ds = 1 + Re 2/3 p 6 The particle Reynolds number is defined as Re p = v r d p ρ/µ, where v r is the magnitude of relative particle velocity. Further, C A and C H in Eq. (4) are the correction factors accounting for the high-reynolds number effects in the added mass and Basset history terms. Following Odar and Hamilton, C A and C H are expressed as: () C A = A c and C H = (1+ A c ) 3 () where A c is the relative acceleration factor defined as: A C = v 2 r / d p dv r / dt () Finally d ij in the Saffman lift term is the deformation rate tensor defined as: d ij =1 / 2 ( u ij + u ji ) u ij = u i x j

4 t = 0.0 t = 1.0 t = t = 3.0 t = 4.0 t = Flow evolution pattern for one vortex time (Re = 1000).

5 Viscous Pressure Basset Saffman Virtual Mass St = 0.5 Ac cel er ati on Time 10 0 St = Ac cel er ati on Time Temporal history of acceleration terms for different Stokes numbers, Re = 250, ε = 1000

6 ε = 1 ε = 5 ε = 10 (a) ε = 1 ε = 5 ε = 10 (b) Comparison of particle trajectories computed by using the Modified BBO equation (St =1.0, Re = 1000): (a) using all the terms in Eq (14), (b) using only the viscous drag force.

7 S t = 0 (F luid ) S t = S t = 0.1 S t = 0.3 Effect of Stokes number on particle distribution and deposition (Re = 250).

8 CONCLUSIONS 1. Results concerning particle dynamics in an unsteady vortical flow over a cylinder indicate that all the secondary terms in the modified BBO equation are negligible compared to the steady state viscous term for particle density ratios (e ) above 20. However, they become progressively important and comparable to the steady viscous term as e is reduced below 10. Amongst the secondary terms, the Basset history term has the largest amplitude, followed by the Saffman lift term. In addition, for a fixed Stokes number, the particle dynamics and deposition exhibit negligible dependence on particle density for e > The particle dispersion in the presence of vortical structures in the cylinder wake exhibits a typical non-monotonic variation with the Stokes number. Particles with St < 0.1 behave like fluid particles, whereas those with Stokes number in the range 0.1 < St < 0.5 exhibit intermediate-st behavior, i.e., they are dispersed more than the fluid particles. On the other hand, particles with St > 1.0 exhibit large-st behavior, i.e., they are essentially unaffected by the flow in the near wake region. In addition, while the majority of small-st particles are distributed in the vortex core, the majority of intermediate-st particles are distributed around the vortex periphery. 3. For large particle density ratios (e > 20), the particle deposition phenomena is essentially characterized by the Stokes number (St). The amount of deposition increases precipitously as the Stokes number is increased from low values (corresponding to fluid particles, i.e., St@0) to a value of unity. For 1.0 < St < 3.0, the particle deposition increases relatively slowly with the Stokes number. For St > 3.0, the deposition becomes essentially independent of the Stokes number. 4. For the range of Reynolds numbers investigated, which includes both the laminar and transitional regimes, the Reynolds number (Re) has very little effect on the deposition phenomena. However, Re has a more discernible influence on particle distribution and dispersion. 5. In the current research we are extending the simulations to (i) the two-phase turbulent flow over a fixed cylinder and (ii) two phase laminar flow over on a oscillating cylinder.

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