Numerical Investigation of Nanofluid Turbulent Flow in a Wavy Channel With Different Wavelengths, Amplitudes & Phase Lag

Two dimensional incompressible turbulent nanofluid flow in a sinusoidal wavy channel is numerically investigated. Finite volume method and Rhie and Chow interpolation in a collocated grid arrangement are used for solving governing equations. The effects of the volume fraction of nanoparticles, Reynolds number, phase lag, frequency and amplitude of the wavy walls on the heat transfer rate are studied. The present work is shown good agreement with existing experimental and numerical results. Increasing the frequency and amplitude of the wave and nanoparticles volume fraction are the most important factor for changing heat transfer rate.

7 Conclusion 110 Changing the fluid properties and flow geometry are two common ways of making an improvement in heat transfer rate.Using porous media, micro scale channels, increasing the surface and put a shackle in the way of the fluid are some methods for increasing heat transfer with changing the geometry.Adding nanoparticles to base fluid and using fluids with better properties are the ways for increasing heat transfer.100 T. Tavangar, A. Ramiar, A. Arya, R. Mohammadyari, M. Rahimi-Esbo Although some researchers investigated the flow in a wavy channel, but none of them investigated the nanofluid flow in a wavy channel with phase lag.For the first time two dimensional steady turbulent flow of nanofluid in a sinusoidal curved wall channel have been numerically simulated.Suspension of water and nanoparticles of aluminum dioxide (AL 2 O 3 ) have been used for this purpose.Effect of Reynolds numbers, wave amplitudes, wave lengths and phase lags is studied.In this paper a wide range of Reynolds numbers, from 3 × 10 3 to 10 5 , wave amplitude changes from 0 m to 0.3 and wave length changes from 1 m to 24 has been investigated.The fluid enters the channel from left side and exits from the right side.The wavy walls in the top and bottom are considered isothermal and two direct sections of the channel at the top and bottom are considered as adiabatic walls.Amplitude, Reynolds number, phase lag and wavelength of the waves are parameters of the problem.The characteristic length for calculating Reynolds is h, which is average height of wavy walls.Function of wall is simple sinusoidal function of amplitude and wavelength.Maximum length for all cases is 20 m to guarantee the outflow boundary condition.The continuity, momentum and energy equation for the 2D flow problem are given by: 1. Continuity: 2. Momentum: 3. Energy: Turbulent Flow in a Wavy Channel

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Where Γ and Γ t are the molecular thermal diffusivity and turbulent thermal diffusivity, respectively and are given by: The Reynolds-averaged approach for turbulence modeling requires that the Reynolds stresses, i.e. ρ nf u ′ i u ′ j needs to be modeled.For this purpose k − ε model is chosen.A common method employs the Boussinesq hypothesis to relate the Reynolds stresses to the mean velocity gradients: The turbulent viscosity term is to be computed.It is defined as below equation: and G k is the rate of turbulent kinetic energy generation and ρε is its destruction rate, and is written as: For obtaining thermo physical properties, there are different approaches available.These approaches can be categorized by how they treat with nanoparticles and fluid interaction.Homogenous nanofluid is a common method.Homogenous nanofluid is a common method that nanoparticles and fluid are assumed to be in thermal 104 T. Tavangar, A. Ramiar, A. Arya, R. Mohammadyari, M. Rahimi-Esbo equilibrium and nanoparticles velocity is as the same as fluid velocity.So there is no slip between nanoparticles and the fluid.
The thermal conductivity of the nanofluid is calculated from Chon et al [8], which is expressed in the following form: where: where: The density and specific heat of the nanofluid are calculated by using the Pak and Cho [11] correlations, which are defined as follows: For inlet and outlet of the channel, and for isothermal and adiabatic walls, boundary conditions can be expressed like below: The Reynolds number is defined as: Local Nusselt number can be defined as: While ∂T ∂ n is the temperature gradient normal to the walls.
N u is the average Nusselt number.
Where Q is the total heat transfer rate through the channel and σ is the length of the curve and can be calculated as: (5.26) Q is the total heat transfer rate.
where T bulk is the mean temperature and can be calculated as defined below: (5.28)

Results and discussion
For mesh independency, the results of several meshes (250 × 70, 300 × 80, 350 × 90, 450 × 130, 450 × 150, 500 × 150, and 500 × 170) have been compared.It can be seen from Fig. 2 that 500 × 150 nodes is enough to achieve accurate results.The present results are compared with experimental data of Zhang et al. [1] and numerical results of Pham et al. [2].They have used Colburn factor for their analysis.They investigated the effect of Reynolds numbers from 750 to 4500.As it can be seen from Fig. 3 there is an acceptable agreement between their result and current study.  .The effect of nanoparticles on the temperature difference term is negligible.The effect of volume fraction of nanoparticles on the temperature gradient term and on the thermal conductivity ratio term is more pronounced.The equations show that any increase in volume fraction increases inertia forces because ρ nf will be increased and accordingly increases the temperature gradient.
Besides, the nanoparticles increase the thermal conductivity ratio term as it can be seen from Eq. .Therefore, both the temperature gradient term and thermal conductivity ratio term increase by increasing the volume fraction of nanoparticles.Accordingly, It can be seen that by increasing volume fraction the Nusselt number will be increased, because the heat transfer properties is improved.By increasing amplitude flow cannot follow the body so it will separates from it.This separations form recirculation zones in the dip of the wall.This recirculation zones prevent the cold fluid to contact with hot wall, so they decrease local Nusselt number.Figure 6 shows these recirculation zones using velocity vectors.On the other hand increasing the wave amplitudes will increase the chaotic behavior of the fluid flow which leads to be more impact between molecules of the fluid and the walls.These impacts cause more energy absorption by the fluid along the channel.These cause an average Nusselt number increase on both top and bottom walls.Another parameter which is important in heat transfer enhancement is the wavelength of the wavy walls.As it can be seen in these figure average Nusselt number takes a pick with reducing wavelength.The enhancement of average Nusselt number for greater wavelengths is a result of enhancement of the contact surface.But as it was debated before recirculation zones have a negative effect on heat transfer.The reduction of average Nusselt number in lower wavelengths can be justified with recirculation zone effect.In lower wavelengths, these recirculation zones will become stronger and will lead to heat transfer reduction.

Conclusion
Forced convection of a turbulent flow in a wavy wall channel has been numerically studied.The results show that by increasing the Reynolds number, aspect ratio and amplitude the average Nusselt number will increase.Also the effects of different phase and different wavelengths of the wavy walls have been investigated.The results reveal that by enhancement of aspect ratio, the penetration of the jet flow in the duct and the developing length increases simultaneously.Also, this effect can is seen by increasing the Reynolds number.Furthermore it was demonstrated in the investigations that by increasing the Reynolds number and aspect ratio, the length of the recirculation zones raises and as a result affects on the local Nusselt number and consequently the average Nusselt number.
102 T. Tavangar, A. Ramiar, A. Arya, R. Mohammadyari, M. Rahimi-Esbo5.Physics of the problem and governing equationsGeometry of the problem is a channel with two direct sections at the entrance and exit.Wavy walls are at the middle part as shown in Fig.1.

Figure 1 :
Figure 1: (a) Physical model, (b) Sample of grids used in present computing

. 9 )
s k and s ε are source term for k and ε and in this research are taken as zero.The boundary values for the turbulent quantities near the wall are specified with the standard wall-treatment method.The values of C µ = 0.009, C 1ε = 1.44,C 2ε = 1.92, σ k = 1, σ ε = 1.3 and Pr t = 0.9 are chosen to be the empirical constants in the turbulence transport equations [?].Below equation are used to calculate the turbulent intensity (I), turbulent kinetic energy (k) and turbulent dissipation rate (ε) at the inlet section of the channel.

Figure 3 :
Figure 3: Comparison of present work with available data Figure 4. shows the particle volume concentration and flow Reynolds number effect on the average Nusselt number.It can be seen that the average Nusselt number increases with an increase in the nanoparticles volume concentration as well as an increase in the flow Reynolds number.The figure reveals an enhancement in average Nusselt number by increasing the volume fraction of nanoparticles.This behavior can be inferred from Eq.

Figure 4 :Figure 5 :
Figure 4: Average Nusselt number for different Reynolds and different volume fractions λ = 2 and α = 0.2.The effect of amplitude on Nusselt number is depicted in Fig.5.
[4]y showed that Both C f and Nusslet number have the maximum value in the upper part of the wavy wall.Wall bending is found to effectively enhance the heat transfer.Another large eddy simulation (LES) is applied by Pham et al.[2]who studied turbulent flow in a wavy wall channel.They have used Colburn and Fanning factor for their analysis.Their work contains data for Reynolds numbers from 750 to 4500.Rostamiani et al.[3]numerically analyzed the turbulent nanofluid flow with different volume concentrations of nanoparticles through a two-dimensional duct under constant heat flux condition.They founded that by increasing the volume concentration, the wall shear stress and heat transfer rates increase.Also they reported for a constant volume concentration and Reynolds number, the effect of CuO nanoparticles yield to more enhancement in the Nusselt number and friction coefficient than Al 2 O 3 and TiO 2 nanoparticles.Nassan et al.[4]experimentally compared heat transfer characteristics of Al 2 O 3 -water and CuO-water nanofluids through a square crosssection cupric duct in laminar flow under uniform heat flux condition.