DOI 10.1007/s10891-018-1734-y

Journal of Engineering Physics and Thermophysics, Vol. 91, No. 1, January, 2018

HEAT AND MASS TRANSFER IN DISPERSED AND POROUS MEDIA CONCENTRATION DISTRIBUTION OF SOLID PARTICLES IN THE COMPLETELY DEVELOPED TURBULENT FLOW IN A CHANNEL K. N. Volkov and V. N. Emel′yanov

UDC 532.529

The formation of regions with a higher concentration of solid particles in the completely developed turbulent flow in a channel with impermeable walls was investigated. A gas flow was simulated on the basis of nonstationary three-dimensional Navier–Stokes equations. The discrete-trajectory approach was used for simulation of the movement of particles. On the basis of the data of a direct numerical simulation, the distributions of the average and pulsating characteristics of the flow in the channel and the concentration distribution of the dispersed phase in it were determined. It was established that the formation of regions with a higher concentration of solid particles in the channel is due to the instantaneous redistribution of vorticity near its walls. The numerical-simulation data obtained are in qualitative and quantitative agreement with the corresponding results of physical and computational experiments. Keywords: completely developed turbulent flow, solid particles, channel with impermeable walls, direct numerical simulation. Introduction. When solid particles are introduced into liquid or gas flows, the pattern of a flow is complicated because the dispersed inclusions different in properties (the inertia of particles and their concentration distribution) give rise to a number of different flow regimes [1]. In the case where the inertia of the particles in a flow is infinitely low, the velocity fields of the dispersed and carrying phases coincide, and, if the initial concentration of particles in the flow is uniform, the concentration of the dispersed phase in it remains constant everywhere over its space. When the small but finite inertia of particles in a flow is taken into account, the pattern of the density distribution of the dispersed phase in the flow changes; in it there appear local concentration nonuniformities of the dispersed phase which manifest themselves in the neighborhood of kinematic features of the velocity field of the carrying flow (critical points, discontinuities, local-vorticity zones). The formation of clusters (compact regions with a higher concentration of the dispersed phase, surrounded by the flow regions with a low dispersed-phase concentration) in a flow is due to the interaction of particles with turbulent vortices. Clusterization of particles takes place in many physical processes and leads, among other things, to an increase in the rate of deposition of particles and in the frequency of their coagulation [2–4]. The clusterization of heavy particles in an inhomogeneous turbulent flow in a channel is explained by their turbulent migration (turbophoresis) from the flow regions with a high turbulence intensity to the flow zones with a low degree of turbulence (in particular, to the viscous sublayer at the channel walls) [5]. However, clusterization of inertial particles also can take place in a homogeneous turbulent flow with no gradients of the pulsating velocity of the gas phase. The spatial distribution of particles in a statistically homogeneous turbulent flow is local and accidental in character, and it changes with time. The concentration of heavy particles in the regions with a low vorticity in such a flow can be increased locally because of the centrifugal-force action and the interaction of particles with the small-scale vortex structures [6, 7]. The compressibility of the velocity field of the dispersed phase, taking place, even in an incompressible flow, can cause concentration fluctuations of inertial particles in a flow and their clusterization manifesting themselves as the interrelation between the divergence of the flow velocity and the formation of zones with a higher concentration of particles. The clusterization of particles in a flow is most marked in the case where the time of their dynamic relaxation is coincident with the Kolmogorov turbulence scale of the flow (Stk ~ 1). At definite Stokes numbers, clusters with a fractal dimension smaller than the dimension of the physical space are formed [8]. D. F. Ustinov Baltic State Technical University "VOENMEKH," 1st Krasnoarmeiskaya Str., 1, St. Petersburg, 190005, Russia; email: [email protected]. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 91, No. 1, pp. 198–207, January– February, 2018. Original article submitted May 10, 2017. 0062-0125/18/9101-0185 ©2018 Springer Science+Business Media, LLC

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Various effects occurring in turbulent flows of a gas suspension can be estimated by the probability density function method [3]. In [8], on the basis of the kinetic equation for the two-point probability density function of the relative velocity of two particles, an analytical solution has been obtained for the dispersion and clusterization of low-inertia particles in a homogeneous isotropic turbulent gas flow at a large Reynolds number without regard for the influence of the particles on the gas. The theoretical model of a gas-suspension flow in a vertical channel, developed in [9], shows that the behavior of the volume concentration of the dispersed phase near the channel walls is singular in character for particles that are not very inertial, which points to the fact that the particles are accumulated in the region of the viscous sublayer of the flow. The clusterization of particles near the channel walls decreases with increasing their inertia because the profile of the average longitudinal velocity of the carrying flow becomes more flat and the turbophoresis effect decreases in this case. The particles found in a flow in a channel influence the intensity of the vortices formed near the channel walls but practically do not change their size and shape [10]. The clusterization of particles in this flow enhances with increase in the ratio between the correlation time scale of their rotational velocity and the corresponding deformation rate scale of the vortices [11]. The concentration of particles near the channel walls is sensitive to the parameters determining the interaction between the particles and the walls in the transverse direction. Investigations of the gas-suspension flow in a vertical channel have shown that, in the case where the particles colliding with the walls of the channel lose their momentum, the concentration of high-inertia particles near the channel walls increases compared to that in the case of elastic interaction of particles with these walls [9]. The clusterization of high-inertia particles in a flow in a channel as a result of the inelastic collisions of them with the channel walls is larger compared to that in the case of elastic collisions, which is explained by the accumulation of particles in the near-wall region of the flow because of the loss of their momentum [12, 13]. At Stk = 0(1), particles concentrate at the boundaries of two-dimensional vortices. In the case where in a flow in a channel there arise vortices extended in its transverse direction, the distribution of particles in the flow becomes mushroom-shaped. The dynamics of inertial particles in a two-dimensional flow defined by a periodic stream function and their clusterization were investigated in [15] with the use of the Lyapunov method of local characteristics. As investigation parameters, the Stokes number and the ratio between the density of the particle material and the density of the carrying flow were used. It was shown that the particles in the indicated flow have a tendency to accumulate in the flow regions with a low vorticity and a large deformation rate of vortices. Data on the completely developed turbulent flow of a pure gas in a channel are presented in [16, 17] (physical experiment) and in [18–21] (direct numerical simulation and simulation of the formation of large vortices). Direct numerical simulation of the movement of particles in a gas flow in a channel and investigation of the formation of regions with a higher concentration of particles in a flow in a channel have been performed in [22, 23] without regard for the influence of the particles on the gas flow. In [10, 24], large vortices in a turbulent flow with particles in a channel were simulated with account of their interaction with a small-scale turbulence. The formation and evolution of large-scale vortices in the boundary layer of such a flow and their influence on the dynamics of particles in it were investigated in [25]. In the present work, on the basis of the data of a direct numerical simulation of the completely developed turbulent flow of a gas suspension in a channel with impermeable walls, the distribution of characteristics of the carrying flow and the concentration distribution of the dispersed phase in it were investigated. The formation of regions with a higher concentration of solid particles in the indicated flow and its connection to the instantaneous vorticity distribution in the channel were considered. Results of the numerical simulation were compared with the data of physical and computational experiments. Mathematical Model. A completely developed turbulent flow of a gas suspension in a channel with impermeable walls was simulated on the basis of the Euler–Lagrange approach without regard for the influence of the particles on the characteristics of the carrying gas flow. Main equations. A nonstationary flow of a viscous compressible gas was defined in the Cartesian coordinate system x, y, z by the following equation written in conservative variables:

∂Q ∂Fx ∂Fy ∂Fz + + + = 0. ∂t ∂x ∂y ∂z The relation between the pressure in the gas flow and its total energy was determined as 1 ⎡ ⎤ p = ( γ − 1) ρ ⎢e − (v x2 + v 2y + v z2 )⎥ . 2 ⎣ ⎦ 186

The conservative-variable vector Q and the flow-vector components Fx, Fy, and Fz were represented in the form ⎛ ρ ⎞ ⎜ ⎟ ⎜ ρv x ⎟ Q = ⎜ ρv y ⎟ , ⎜ ⎟ ⎜ ρv z ⎟ ⎜ ρe ⎟ ⎝ ⎠ ρv x ⎛ ⎞ ⎜ ⎟ ρv x v x + p − τ xx ⎜ ⎟ ⎟, ρv x v y − τ xy Fx = ⎜ ⎜ ⎟ ρv x v z − τ xz ⎜ ⎟ ⎜ (ρe + p )v x − v x τ xx − v y τ xy − v z τ xz + q x ⎟ ⎝ ⎠ ρv y ⎛ ⎞ ⎜ ⎟ ρ v v − τ y x yx ⎜ ⎟ ⎟, ρv y v y + p − τ yy Fy = ⎜ ⎜ ⎟ ρv y v z − τ yz ⎜ ⎟ ⎜ (ρe + p )v y − v x τ yx − v y τ yy − v z τ yz + q y ⎟ ⎝ ⎠ ρv z ⎛ ⎞ ⎜ ⎟ ρ − τ v v z x zx ⎜ ⎟ ⎟. ρv z v y − τ zy Fz = ⎜ ⎜ ⎟ ρv z v z + p − τ zz ⎜ ⎟ ⎜ (ρe + p )v z − v x τ zx − v y τ zy − v z τ zz + q z ⎟ ⎝ ⎠

Components of the viscous-stress tensor and the heat-flow vector were determined from the relations ⎛ ∂v ⎞ ∂v j 2 ∂vk τij = μ ⎜ i + − δ ij ⎟ , ⎜ ∂x j ⎟ 3 ∂xk ∂xi ⎝ ⎠

qi = −λ

∂T . ∂xi

The viscosity of the gas was represented as a function of its temperature in accordance with the Sutherland law: ⎛T ⎞ μ = ⎜ ⎟ μ∗ ⎝ T∗ ⎠

3/ 2

T∗ + S 0 , T + S0

where μ∗ = 1.68·10–5 kg /(m·s), T∗ = 273 K, and S0 = 110.5 K for air. The heat-conduction coefficient of the gas was expressed in terms of its viscosity and the Prandtl number: λ = cpμ /Pr. The Prandtl number was assigned a constant value (Pr = 0.72 for air). Movement of a particle. The movement of particles in the flow being considered was defined using the trajectory approach, in which the equations of movement of the impurity in the gas flow were written in Lagrange variables and were integrated over the trajectories of individual particles. The translational movement of a test spherical particle in the gas flow was defined by the equation. mp

dv p dt

=

1 C D ρS m ( v − v p ) . 2

The drag coefficient of the particle was determined as

CD =

24 f D (Re p ) . Re p

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Here, the function fD makes a correction for the inertia of the particle. The Reynolds number of the relative movement of the particle and the gas was determined by the formula 2rp ρ | v − v p | Re p = . μ To the equation of movement of the particle was added the following kinematic relation, allowing one to calculate the radius-vector of the center of mass of the particle: drp dt

= vp .

The influence of the temperature of the particle on its movement was determined by the correction to its drag coefficient. In many flow regimes, this correction was small and, therefore, was disregarded. The equations for the movement of the particle were integrated over its mechanical trajectory at definite initial conditions determined by the coordinates of the particle and the velocity of its movement at an instant of time t = 0. The concentration of the dispersed phase was determined on the basis of processing the results of trajectory calculations of test particles [1]. Numerical Method. A numerical model was constructed on the basis of solution of the nonstationary Navier– Stokes equations for a three-dimensional flow of a compressible viscous gas with the use of a structured mesh [26]. For the digitization of the main equations, a high-accuracy explicit quasi-monotone scheme constructed by the modified Godunov method was used. The convective heat flows in each coordinate direction were calculated independently using an approximate solution of the Riemann problem (the HLLC method). The order of spatial accuracy of the digitization scheme was increased without loss of its monotonicity with the use of the minmod limiter. The diffusion heat flows were digitized bythe centered finite-difference schemes of the second order of accuracy. For integration of equations with respect to the time, the third-order Runge–Kutta method was used. The velocity of the gas was calculated at points lying on the mechanical trajectory of a particle: v = v(rp ). This velocity was determined from the solution of the Navier–Stokes equations for the control volumes of an Euler mesh whose nodes were not coincident with the position of the particle. This generated a need for the completion (interpolation) of the gasdynamic parameters of the carrying flow and, therefore, for the localization of the particle and determination of the computational-mesh control volume in which the particle is found at a definite instant of time [1]. The Cauchy problem was solved by methods allowing one to separate the rapidly- and slowly-decaying components of a solution [1]. The difference schemes have been developed on the basis of linearization of the initial system of equations by the principle of freezing of their individual terms, representation of these equations in the form of simplified functional dependences, and subsequent analytical interpretation of the approximate equation in each time step. Geometry of the Problem and Its Boundary Conditions. A coordinate system was selected so that its x axis is directed along the flow and the y and z are related to a cross section of the channel. The channel had a halfwidth h. The length of the channel was assumed to be equal to 4πh, and its extent in the transverse direction was 2πh. At the upper and lower walls of the channel, the adhesion and impermeability conditions were set. The walls of the channel were assumed to be heatinsulated. Periodic boundary conditions were prescribed for the other coordinate directions. It was assumed that the density of the air is ρ = 1.2 kg /m3 and its viscosity is ν = 1.6·10–5 m2/s. The formulation of the problem is described in detail in [21]. The gas flow was characterized by the Reynolds number Reτ = uτh/ν, where uτ = (τw /ρ)1/2, and the dimensionless time t = uτ y/h was introduced into consideration. The quality of the computational mesh was determined by the dimensionless near-wall coordinate y+ = yuτ /h. In the main calculations, we used the Reynolds number Reτ = 160 corresponding to the Reynolds number Re = 2450 determined by the average velocity of the flow and the halfwidth of the channel (Re = uh/ν). The pure gas flow was calculated for different Reynolds numbers, and the data obtained were compared with available literature data. Particles were introduced into the channel at an instant of time t = 0, and their position at the input cross section of the channel was chosen randomly. The initial velocity of a particle was assumed to be equal to the instantaneous velocity of the flow at the point of introduction of particles. The ratio between the density of the gas and the density of the particle material was ρ/ρp = 10–3. The inertia of the particle was characterized by the Stokes number Stk = τpuτ /ν, where τp is the relaxation time of the particle. The calculations were performed on a mesh containing 256 × 128 × 128 nodes. The mesh nodes were bunched near the channel walls so that the condition y+ = 0.05 was fulfilled, and 12 nodes were in the region y+ < 8. The calculations were 188

Fig. 1. Visualization of the vortex flow in the channel with the use of the isosurface criterion Δ at Reτ = 590.

Fig. 2. Distribution of the longitudinal flow velocity over the cross section of the channel: full line) calculation; points) data of [21]; dotted line) velocity profile of the laminar flow. conducted with a time step Δt = 4.8·10–4 as long as tf = 10. The size of the channel and the number of nodes in the mesh corresponded to those used in [18] where the Reynolds number was Reτ = 178 (Re = 2792). The time step selected was smaller than the step necessary for the resolution of the flow structure changing with time [27]. The number of particles used in the calculation was 105. The equations defining the movement of a particle were integrated as long as tf = 10 or until the particle went out of the computational region. Results of Calculations. Direct numerical simulation of the pure gas flow in the channel being considered was performed for different Reynolds numbers, and the results obtained were compared with the corresponding data from [18, 10]. The vortex structure of the flow visualized with the use of the criterion Δ, which was determined by invariants of the velocity-gradient tensor, is shown in Fig. 1. A vortex was represented as a flow region in which the inequality Δ > 0 is fulfilled (in this region, the velocity-gradient tensor has complex eigenvalues) [28] and the rotary motion (the antisymmetric part of the velocity-gradient tensor) prevails over the tension or compression (the symmetric part of the velocity-gradient tensor). The profiles of the longitudinal velocity of the flow at a cross section of the channel are shown in Fig. 2. The velocity of the flow in the boundary layer was normalized to its dynamic velocity (u+ = u/uτ). The velocity distribution of the flow near the channel walls correlates fairly well with Reichard′s law [29] obtained on the basis of the experimental data for a viscous sublayer and for the buffer and logarithmic regions of a boundary layer (Fig. 3). The calculation data on the kinetic energy of turbulence of the flow at Reτ = 395 are compared with the corresponding data of the direct numerical simulation in Fig. 4. The calculated distributions of the pulsating velocity components of the flow at a cross section of the channel and those 189

Fig. 3. Distribution of the velocity of the flow in the boundary layer near the wall of the channel: points) Reichard′s law; dotted line) linear distribution of the flow velocity in the viscous sublayer; dash-dot line) logarithmic distribution of the flow velocity in the boundary layer of the turbulent region.

Fig. 4. Distribution of the kinetic turbulence energy near the walls of the channel at Reτ = 395: full line) calculation; points) data of the direct numerical simulation [18]. obtained as a result of the direct numerical simulation in [18] are shown in Fig. 5. The numerical-simulation data agree fairly well with the corresponding literature data in a wide range of Reynolds numbers. The movement of particles in the completely developed turbulent flow in the channel was investigated for a definite Reynolds number equal to Reτ = 160. The distribution of particles over a cross section of the channel at Stk = 25 at a definite instant of time is shown in Fig. 6. It is seen that particles are distributed over this cross section nonuniformly and have a tendency to accumulate in the boundary layer at the lower and upper walls of the channel. The accumulation of particles in the viscous sublayer is due to the near-wall turbulence and the formation of vortices in the turbulent boundary layer [25]. In such vortices, the tangential components of the tensor of instantaneous Reynolds stress is negative (ν′ < 0 and w′ > 0), and their level is fairly high, which leads to the generation of turbulence. The indicated vortices are oriented along the direction of the 190

Fig. 5. Distributions of the pulsating components of the flow velocity u ′ (1), v ′ (2), and w′ (3) over the cross section of the channel at Reτ = 180 (a), 360 (b), and 590 (c): points) data of [18, 20]; full lines) calculation.

Fig. 6. Distribution of particles over the cross section of the channel at the instant of time t = 10 at Stk = 25. Fig. 7. Formaiton of vortices and diagram of accumulation of particles in the near-wall region of the channel. 191

Fig. 8. Concentration distribution of particles in the boundary layer of the channel at Stk = 0.2 (1), 1 (2), 5 (3), and 25 (4).

Fig. 9. Concentration distribution of particles in the boundary layer of the channel at Stk = 0.2 (a), 1 (b), 5 (c), and 25 (d): full lines) calculation; points) data from [23]. 192

flow [30]. The transfer of particles to the wall of the channel and their accumulation near these walls takes place in the regions of the channel in which the longitudinal velocity of the flow is lower than its average velocity (Fig. 7), which is supported by the data of [23, 31, 32]. The concentration distribution of particles different in sizes in the boundary layer of the flow, constructed in a logarithmic scale, is shown in Fig. 8. The results of calculations were normalized to the volume concentration of particles αp0 corresponding to their uniform distribution over the cross section of the channel at the initial instant of time. The formation of regions with a higher concentration of particles near the walls of the channel becomes marked when the inertia of particles increases. Small particles follow the small-scale disturbances of the flow and enter the low-velocity flow zones in the turbulent boundary layer. The results of our calculations are compared with the corresponding data of [33] in Fig. 9 (for simplicity, not all the calculation points are presented in this figure). Despite the fact that the indicated data are in fairly good agreement, between them there is quantitative disagreement, reaching a maximum value near the walls of the channel. Conclusions. A direct numerical simulation of the completely developed turbulent flow of a gas suspension in a channel with impermeable walls has been performed. The data obtained point to the formation of regions with a higher concentration of dispersed-phase particles near the walls of the channel, which was explained by the instantaneous distribution of vorticity. For visualization of the numerical-simulation data, the vorticity and invariants of the velocitygradient tensor were used. The calculation data obtained are in qualitative and quantitative agreement with the corresponded literature data. It was established that the clusterization of particles in their random velocity field in a flow is a kinematic phenomenon occurring in the absence of interactions between the particles. In the case where the random velocity field of the particles in a flow is averaged over an ensemble of its realizations, the features of the dynamics of the particles disappear. The typical methods used for the statistical averaging of the movement of particles in a flow level the qualitative features of realizations of their trajectories. The statistical averaging of the concentration field of particles in the random field of their velocities over all its realizations makes the concentration field smooth, while each individual realization of the indicated field tends, because of the mixing of the regions having very different concentrations, to be cut to pieces in the space. A statistical average usually characterizes the global space–time scales of the region in which stochastical processes occur, and it does not say anything about details of the development of these processes.

NOTATION CD, drag coefficient of a particle; d, diameter of the channel, m; e, total energy of a unit mass of the gas, J/ kg; h, halfwidth of the channel, m; k, kinetic energy of turbulence of the gas flow, m2/s2; mp, mass of a particle, kg; p, pressure, Pa; q, heat flow, W/m2; rp, radius of a particle, m; Sm, area of the midsection, m2; t, time, s; T, temperature, K; uτ, dynamic velocity of the gas flow, m /s; u, v, and w, velocity components of the flow in the Cartesian coordinate system, m /s; vp, velocity of movement of a particle, m /s; x, y, and z, Cartesian coordinates, m; α, volume concentration of the gas; γ, ratio between the specific heats of the gas and the particles; δij, Kroneker symbol; Δ, invariant of the velocity-gradient tensor; λ, heat conductivity of the gas, W/(m·K); μ, dynamic viscosity of the gas, kg /(m·s); ν, kinematic viscosity of the gas, m2/s; ρ, density of the gas, kg/m3; τ, viscous stress tensor; τp, relaxation time of a particle, s; τw, shear stress at the channel wall, 1/s. Subscripts: f, final; m, midsection; p, particle; w, wall; 0, initial instant of time; +, dimensionless near-wall coordinate.

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