NUMERICAL STUDY OF PARTICLE DISTRIBUTION IN THE WAKE OF PARTICLE-LADEN AIR FLOWS PAST A RECTANGULAR BUILDING USING DISCRETE VORTEX METHOD

A numerical investigation on particle distribution in the wake of particle-laden airflows past a rectangular building at Reynolds number of 10 is presented. The Discrete Vortex Method is employed to evaluate the unsteady airflow fields and a Lagrangian approach is applied for tracking individual solid particles. A dispersion function is defined to represent the dispersion scale of the particle. The wake vortex patterns and the distributions of particles with Stokes numbers, St, of 0.01, 0.1, 1.0 and 10 are obtained. The results show that the particle distribution in the wake of the rectangular building is closely related to the particle’s Stokes number and the structure of wake vortices: (1) very small sized particles with St=0.01 behave like gas elements and can distribute both in the vortex core and around the vortex periphery, whereas the intermediate sized particles with St=1.0, 10 can not enter the vortex cores and concentrate near the peripheries of the vortex structures due to the centrifugal forces acting on the particles by the wake vortices; (2) the dispersion intensity of particle decreases as St increases from 0.01 to 10. This work will be helpful in predicting and analyzing particle concentrations around rectangular buildings. INDEX TERMS Particle-laden airflows, Rectangular building, Discrete Vortex Method, Particle distribution INTRODUCTION Unsteady particle-laden airflows (such as dust-polluted airflows from process operations in industries, vehicle-induced dust-laden airflows, and dust storms etc.) past rectangular buildings (rectangular cylinders) at high Reynolds numbers are a common occurrence. Because gas flows past a rectangular cylinder are characterized by the development of wake vortices, understanding the mechanism of particle distribution in the wake of a rectangular cylinder is crucial for predicting particle pollution around rectangular buildings. Up to now, no numerical or experimental studies on particle dispersion in gas-particle flows past a rectangular cylinder at high Reynolds number have been reported. D.J. Brandon and S.K. Aggarwal (2001) conducted a numerical investigation on particle dispersion in the wake of a rectangular cylinder and found that the particle dispersion pattern was determined by the particle’s stokes number. However, D.J. Brandon and S.K. Aggarwal in their work only investigated the laminar and transitional regimes (for the range of Reynolds numbers) and did not quantitatively study the dispersion intensity of particles. The present work aims to numerically investigate the distributions of different sized particles in the wake of a rectangular building at high Reynolds number. Here, we study the two-dimensional, dilute gas-particle flows past a rectangular cylinder at Reynolds number of 10. In the numerical procedure, the Discrete Vortex Method is used to compute the unsteady airflow fields and a Lagrangian approach is applied for tracking individual particles; In the simulation, the change in the particle distribution pattern and in the intensity of particle dispersion with the Stokes number are explored. BASIC EQUATIONS AND NUMERICAL METHOD Unsteady gas flow field We use the Discrete Vortex Method (Wu wenquan and F sisto 1987, A.J.Chorin 1973) to evaluate the unsteady airflow fields numerically. * Corresponding author email: huangyuandong@tsinghua.org. cn Proceedings: Indoor Air 2005 2735 For two-dimensional, viscous, homogeneous incompressible gas flows past a rectangular cylinder at high Reynolds number, the vorticity transport equation is given as ω ν ω 2 ∇ = Dt D (1) where Dt D / is the constitutive derivative, t is the time, ν is the kinematic viscosity of the gas, the vorticity ω is expressed by y u x v t y x ∂ ∂ − ∂ ∂ = ) , , ( ω (2) where u and v are the gas velocity components in xand ydirections, respectively. The stream function ψ is related to the velocity components through y u ∂ ∂ = ψ and x v ∂ ∂ − = ψ (3) which satisfies the continuity equation. Combing Eqs. (2) and (3) yields ω ψ − = ∇ (4) The Poisson’s equation (4) enables one to determine ψ from ω . In the Discrete Vortex Method, the continuous vorticity field is represented by the sum of a large number of discrete vortex blobs. During the simulation, the discrete vortex blobs are tracked using a grid-free vortex-tracing algorithm, once the position and strength of the vortex blobs are determined, the whole velocity field can be obtained by using the Biot-Savart Law. The Lagrangian equation of motion for a particle In the present simulation, the following assumptions are made: (1) The gas-particle two-phase flow is dilute, i.e., the particle-to-particle collisions can be neglected (R.Ishill et al. 1989), and the gas flow is not affected by the particles; (2) The particles are spherical and not rotating; (3) The flow is two dimensional, and the flow plane is perpendicular to the gravity. Then, the Lagrangian equation of motion for a particle we used is ( ) ( ) ∫ −      − + − + − = t p p p p p p P p p t d d V d d V d d V V dt d d V V f d dt V d d 0 2 3 3 2 3 12 3 6 τ τ τ τ πν ρ ρ π πμ ρ π v v v v v v v + ( ) ( ) ( )           ∂ ∂ − + ∂ ∂ − j y u u u i x v v v d p p p v v 2 1 2 1 2 1 2 615 . 1 μρ Dt V D d p v ρ π 3 6 + (5) where p ρ and ρ are the particle and gas density, respectively, p d is the particle diameter, μ is the dynamic viscosity of the gas, V v and p V v are the instantaneous velocity of the gas and the particle, respectively, f is the modification factor for the Stokes drag coefficient, dt d / is the temporal derivative along the discrete particle trajectory and Dt D / is the temporal derivative along the gas motion. The terms in the right-hand side of Equation (5) are referred to as drag, added-mass, Basset, saffman, and fluid (due to the fluid pressure gradient and viscous stresses) forces. The particle Reynolds number is defined by Proceedings: Indoor Air 2005 2736 ν p p p d V V v v − = Re (6) f related to the particle Reynolds number is given below 1 = f ( 1 Re < p ),      + = 6 / Re 1 3 2 p f ( 1000 Re 1 ≤ ≤ p ), 24 / Re 44 . 0 p f = ( 1000 Re > p ) Let L be a characteristic length, U be a characteristic velocity, and U L T / = be the characteristic time, we set L x x ′ = , L y y ′ = , L / ∇′ = ∇ , U V V v v ′ = , U V V p p ′ = v v , U L t t ′ = . After substituting and dropping primes, Equation (5) takes the following dimensionless form ( ) τ τ τ τ α π α α d t d V d d V d St dt V d V V t V V V St f dt V d t p p p ∫ −         − +       + ∇ ⋅ + ∂ ∂ + − =       + 0 2 3 2 1 2 1 1 v v v v v v v v v