HEAT TRANSFER IN PARTICLE-LADEN WALL-BOUNDED
TURBULENT FLOWS
Marek Jaszczur, Luis M. Portela Delft University of Technology Faculty of Applied Physics, MSP Prins Bernhardlaan 6,2628BW Delft,
The Netherlands e-mail: jaszczur@agh.edu.pl
web page: http://www.msp.tudelft.nl
Key words: Direct Numerical Simulation, Particle Laden Flow
Abstract. In present work heat transfer in particle-laden wall-bounded turbulent flows has
been study with the fluid-particle one way interaction approach. Direct Numerical Simulation of the flow, combined with Lagrangian particle tracking technique has been performed to study the problem. In presented configuration small solid and dense particles carrying by fluid forces are influence by turbulent non-isothermal flow. Numerical computation was performed for various grids for Reynolds number 180, 395 and the molecular Prandtl number 1.0. The effect of particle diameter and density as well as the effect of the Reynolds and Prandtl number on the statistical quantities was examined. The flow field, spatial pattern of particle concentration and particle temperature were presented. It has been shown that particles finally were distributed not uniformly nor hydrodynamic nor thermally.
1 INTRODUCTION
Particle-laden flow has recently received considerable attention due to its relevance to large number of engineering applications. However, the main focus of researchers has been concentrated on the particle-fluid interactions in isothermal flows. But it is well know that depending on the particle diameter and density the flow can be highly influence by the presence of the particles. This has been study in large numbers of works concerning mixing enhancement as well as drag reduction. But only few works has been concentrated on influence of the particles on the turbulent heat transfer. It is still not recognized how existence of particles will influence heat transfer. DNS can play an important role in understanding fundamental interaction between particles and turbulence. In the present work particle transport in fully developed turbulent wall-bounded non-isothermal flow has been investigated with the fluid-particle one way interaction approach. Direct Numerical Simulation of the incompressible flow, combined with Lagrangian particle tracking technique has been performed to study problem. For presented wall-bounded configuration and small solid and dense particles carrying by fluid forces Eulerian-Lagrangian point-particle approach
has been considerable good in study influence of particle on turbulent isothermal flow1,2.
L =6.4x δ L =2.0z δ L =3.2y δ Dp qw qw x z y Mean Flow particles
in number of fundamental studies3 performed in isotropic turbulence, and shown that particles
with time constant of order of smallest turbulence timescale are preferentially concentrated
into regions of low vorticity and high strain rate (Wang and Maxey3). In wall-bounded shear
flow (Rouson and Eaton4, Pedinotti5) have shown that particle are preferentially concentrated
in the low speed streaks and near-wall region. McLaughlin6 has shown that particles
accumulate in the near wall region. DNS is limited tools but especially attractive for computation of particle-laden flows not only for isothermal flows but also for no-isothermal flows and flow and thermal quantities can be predicted very detailed and accurately.
The principal objective of present study is application of direct numerical simulation for computation of turbulent shear flow in fully developed wall-bounded non-isothermal flow for which results are available in literature. This flow will be extended to particle laden non-isothermal flow for which any data can not be found in literature. Particle interaction with turbulence on hydrodynamically and thermally results in very complex statistically behaviour and complex structural features. Therefore primary objectives is to determine accurate flow and thermal field. Comparisons of numerical results with available data for isothermal and non-isothermal flows is presented for various grid resolutions. Then statistical properties for particles will be presented and analysed in detailed qualitatively and quantitatively.
2 MATHEMATICAL MODEL AND NUMERICAL METHODS
For computation of wall-bounded turbulent flow with particle driven by uniform
pressure gradient between the Eulerian-Lagrangian point-particle approach2 has been used.
The continuous-phase is solved using direct numerical simulation techniques for incompresible flow which is combined with the tracking of the individual particles. In symulation the transfer of momentum between the fluid and the point-particle is computed through a force located at the geometrical center of the particle, and is determined base on the velocities of the particle and of the surrounding fluid. The temperature is determined base on velocities of the particles and fluid and the temperatures of the particle and of the surrounding fluid. This approach is valid if the particles are significantly smaller than the smallest flow scales both for velocity and temperature. To consider only the particle-turbulence interactions, the simulations are performed without gravity force. The flow in the channel is assumed to be fully developed thermally and hydrodynamic. The presently computational domain and coordinate system are shown in Figure 1.
The streamwise wall normal and spanwise coordinate are denotes x,y,z. The wall heat flux qw
is assumed to be uniformly distributed on walls. The working fluid is assumed to be Newtonian fluid with constant properties. Temperature is treated as passive scalar. Under this assumption, the continuity, Navier-Stokes and energy equations are described, as follows:
∇ U⋅ =0 (1)
( )
U U P U t U 1 2 ∇ + ∇ − = ⋅ ∇ + ∂ ∂ ν ρ (2)(
U T)
T t T =−∇ ⋅ + ∇2 ∂ ∂ α (3)For small particles (diameter smaller then Komogorov scale) and heavy particles (density much larger than fluids) in the equation of particle motion the only significant force is the
drag3 (in present study gravity force is not consider) and the governing equations can be
written as: C Re
(
u v)
dt v d V p d r r r − = τ 1 24 (4)where ur is the velocity of the surrounding fluid interpolated at the center of the particle and
vris the velocity of the particle. The particle Reynolds number, Rep, and the hydrodynamic
particle-relaxation-time,τV, are defined as:
(
)
ν p p D v u Re r r − = , ν ρ ρ τ 18 2 p p V D = , p d Re C = 24whereρpandρare the particle and fluid densities, Dp is the diameter of the particles and Cd is
the drag coefficient for Stokes flow (valid for small Rep although some computations (Wang
Squires7) shows that particle Reynolds number does not necessary have small value). The
volume fraction of particles is assumed to be small enough such that particle-particle interactions are negligible. The equation for the temperature of particles, assuming a Biot number less than 0.1 (uniform temperature of particle) and thermal particle-relaxation-time,
T
τ , can be written in the form:
(
p)
T p T T Nu dt dT − = τ 1 2 , k D cp p p T 12 2 ρ τ = (or Pr c cp V T 2 3 τ τ = ) (5)where T is the temperature of the surrounding fluid interpolated at the center of the particle
and Tp is the temperature of the particle and τTis the thermal particle-relaxation-time. In order
to take into account effect of particles relative velocity and fluid number dependence on the Nusselt number the value of the Nusselt is calculated from the Ranz-Marshall correlation:
3 1 2 1 6 0 2 / / p Pr Re . Nu= + (6)
where the Prandtl number is defined as
2.1 Numerical method
The position, flow, and particle quantities are normalized by the channel half-width,δ , the
friction velocity, uτ, and the friction temperature, Tτ =qw/
(
ρcpuτ)
. The governing equations(1)-(3) are solved numerically using the fractional step method on a staggered grid. Second-order Adams-Bashforth was used for time advancement of all terms with the step determined by the Courant criterion. The Poisson equations for pressure is solved using Fast Fourier Solver using Fourier series expansion in the streamwise and spanwise direction and tri-diagonal matrix inversion in wall-normal direction. The equation of the particle motion (4) and the equation of the particle temperature (5) are solved use a second-order Adams-Bashforth scheme for the time-advancement, and a tri-linear interpolation for the velocity and temperature field at the center of the particle. The channel walls are considered to be perfectly smooth and a particle was assumed to contact wall when the center is one radius from the wall. When a particle hits the wall it is bounced-back using elastic specular-reflection (elastic collisions), and when it leaves the domain it is re-introduced at the opposite side with the same velocity and temperature. To force the fluid motion, a constant pressure-gradient is imposed along the streamwise direction. The flow is heated by a uniform heat-flux from both walls; hence there is no restriction on wall-temperature fluctuations. Periodic boundary conditions are imposed in both streamwise and spanwise directions for the hydro and thermal fields. In order to apply periodicity in the streamwise direction, the following transformation of the pressure and temperature is made:
x dx p d p pi= − x dx T d T Ti= − where ρ δ τ / u dx p d 2 2 − = , δ ρ p b w U c q dx T d = 3. NUMERICAL RESULTS
The DNS simulations were performed at Reynolds numbers based on friction velocity and
channel half-height of Reτ=180, 395 (corresponding to the Reynolds numbers of 5705 and
14147 base on mean centreline velocity and channel half-height) and Prandtl number Pr=1.0. For all computations the flow was resolved using (64x64x64), (128x128x66), (256x256x128) respectively (uniform in the streamwise and spanwise directions, and with an hyperbolic-tangent stretching in the normalwise direction). The channel domain for the calculation was (3.2x1.6x1.0), (6.4x3.2x1.0) of channel height. The grid spacing in wall coordinated in the x
and z direction was up to ∆x =9.0 and + ∆y =4.5 at Re+ τ=180, Pr=1.0 and ∆x =18.0 and +
+
∆y =8 at Reτ=395, Pr=1.0. The stretched grid was used in wall-normal direction and for all
computation first two computational points has been located at 1 <1
+
y and 2+ <4
y and last
around 6.0. Properties of the dispersed phase were obtained by tracking the trajectories of 64x64x64 particles uniformly distributed at initial time. Large number of particles are required in order to computed accurate statistics for the dispersed phase in all points of computational domain. The mean velocity and temperature profiles and it correlations for continuous phase obtained from the DNS computations are shown in Fig. 2 and 3 for different computational grid and computational domain results are compared with data available in
Fig.2a. Computed quantities for Reτ=180 in comparison with other DNS computations.
Fig.2b. Computed thermal quantities for Reτ=180 in comparison with other DNS computations.
Fig.3. Computed quantities for Reτ=395 in comparison with other DNS computations.
0 120 240 360 0 5 10 15 20 U m z* Kawamura 256 Moser Present 128 Present 256 1 10 100 0 5 10 15 20 U m Reτ Kawamura 256 Moser Present 128 Present 256 120 240 360 0 2 4 6 8 10 < U U > z* Kawamura 256 Moser Present 128 Present 256 120 240 360 0,0 0,4 0,8 1,2 < V V > z* Moser Present 128 Present 256 90 180 270 360 0 5 10 15 20 T m z* 256 short 256 long 128 short 0 60 120 180 0 4 8 12 16 20 Tm z+ 128 short 256 short 256 long
Fig.4. Influence of domain size on mean flow temperature for Reτ=180, 395 and Pr=1.0
90 180 270 360 0 5 10 15 20 T m z* 256 small 256 medium 128 large 0 60 120 180 0 4 8 12 16 20 T m z+ 128 small 256 medium 256 large
On Fig.4. and Fig.5. influence of domain size on mean flow temperature has been presented. In reference to standard channel size presented on Fig.1. (short) computational domain has
been increase by double channel length Lx(long) or double/decrease in each direction by
factor 2 large/small. Fig.6. presents instantaneous particle field for Reτ=395 and Stokes
number =2,7 and 29. Particles on this figure are coloured by particle temperature according to attached legend.
Fig.6. Instantaneous particle field for Reτ=395 and St=2, 7, 29 - colour by temperature
1 10 100 0,1 1 10 Re=180 St=2 St=7 St=29 Cm z+ 0 60 120 180 0 5 10 15 20 Fluid Particles St=2 Particles St=7 Particles St=29 V e lo c it y z+ 0 90 180 270 360 0 5 10 15 20 Fluid Particles St=2 Particles St=7 Particles St=29 V e lo c it y z+ 1 10 100 1 10 Re=395 St=2 St=7 St=29 Cm z+ 0 60 120 180 0 2 4 6 8 10 Re=180 Pr=1 St=2 UU flow UUp UpUP < u u > z+ 0 60 120 180 0 2 4 6 8 10 Re=180 Pr=1 St=29 UU flow UUp UpUP < u u > z+ 0 90 180 270 360 0 2 4 6 8 10 Re=395 Pr=1 St=2 UU flow UUp UpUP < u u > z+ 0 90 180 270 360 0 2 4 6 8 10 12 Re=395 Pr=1 St=29 UU flow UUp UpUP < u u > z+
Fig.7. Flow component, flow-particle and particle-particle correlations for Reτ=180,
Fig.7. shows particle concentration, flow and particle velocity and its correlation (flow-flow,
flow-particle and particle-particle) for two Reynolds number Reτ=180, 395 and molecular
Parndtl number Pr=1.0. On Fig.8. Thermal quantities for fluid and particles are presented. Temperature and fluid velocity–particle temperature correlations are presented for Stokes number St=2.0, 7.0, 29,0 and Pr=1.0. 0 60 120 180 0 5 10 15 20 z+ T m Fluid Temperature Particle Temperature St=2.0 Particle Temperature St=7.0 Particle Temperature St=29.0 0 60 120 180 0 -2 -4 -6 -8 -10 -12 Fluid Particle-fluid St=2.0 Particle-fluid St=7.0 Particle-fluid St=29.0 < u T > z+ 0 60 120 180 0,0 0,3 0,6 0,9 1,2 z* flow Particle-fluid St=2.0 Particle-fluid St=7.0 Particle-fluid St=29.0 < w T >
Fig.8. Particle and flow thermal quantities and velocity-temperature correlations for
4 CONCLUSIONS
We extended the Eulerian-Lagrangian point-particle DNS approach in order to include heat transfer. The methodology was applied to study a particle-laden turbulent channel flow with one-way coupling. The statistical results for the continuous-phase show a good agreement with previous work. The qualitative analysis of the turbulence and particle structures shows the well-known patterns for the hydrodynamics and indicates how these patterns are associated with the patterns for the temperature of the particles.
ACKNOWLEDGMENTS
This research was supported in part by European Commission (project Dev-CPPS, FP6-002968), and project AGH 10.10.180.22
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