The increase of flow velocity and Reynolds number in coarse porous media and the subsequent violation of Darcy's law, force to analyze the flow based on nonlinear relations of hydraulic slope and flow velocity. So, it is necessary to study nonlinear relationships more accurately. The purpose of this study was to investigate the performance of fractional-order model and the effect of conformable derivatives on improving the relationship between flow velocity and hydraulic gradient. Therefore, by determining the acceptable range for the fractionalorder model, a nonlinear model based on conformable derivatives of the Izbash equation for the fully developed turbulent flow was presented and solved analytically and the parameters of the proposed model were determined using laboratory data analysis. The optimal values of the model parameters including coefficient a and the order of fractional derivative α , which can be varied in the range of (0-2), were calculated for each laboratory data set. The results were compared with the experimental data and the analytical solution of Izbash equation and a good agreement was found to the non-Darcian flow laboratory data. Moreover, using dimensional analysis method, Reynolds number was introduced as an effective factor on α coefficient and a suitable relationship was observed between the order of fractional derivative α and Reynolds number indicating the hydraulic concept of fractional-order model. According to the present study, the fractional order α is not only a fitting coefficient, but it represents a physical concept.

In this study, the fractional-order differential equations in range of (0, 1) were used to model the water surface profile under Darcy's law condition in porous medium for a fully developed turbulent flow. The developed equation is solved analytically. The laboratory model used in this study consists of a coarse-grained porous medium with 6. 4 m length, 0. 8 m width and 1 m height, including rounded corner materials, which are tested for different flow rates and three longitudinal slopes of 0, 4, 20. 3%. Then, parameters of model and porous media were calibrated based on laboratory data. In order to evaluate the proposed analytical solution, the obtained results from fractional-order differential model were compared with the laboratory data. The results showed a satisfactory agreement with experimental data of water surface profile (seepage line) in all three slopes. The maximum error of the proposed model is 3. 5% compared to the experimental data. It can be concluded that the proposed method can provide better description of water surface profile analysis under nonDarcy flow conditions as compared to Darcy model in porous media.

In this paper using one, two and three dimensional simulations, the accelerating flows in developing inlet pipe region are considered numerically. The developing length calculated based on different turbulence parameters is studied thoroughly. The SST turbulence model in comparison with recent reported experimental data have been used to achieve reliable predictions. The predictions obtained using all 1, 2, and 3 dimensional cases are generally the same and follow the experiments well.Thus, it seems that one dimensional simulation in fully developed region is sufficient. However, for the inlet region in developing state a two dimensional axisymmetric analysis is required. This research shows that calculating the developing length based on only mean averaged velocity is not adequate, but also turbulence kinetic energy and viscosity must be paid enough attention. Further, the comparison between steady and unsteady flows for the developing length shows that they are very different. This length also depends on both the value of flow acceleration and turbulence delay time in addition to Reynolds number and pipe diameter.

In the present paper, the turbulent reacting flow within a porous media is modeled by developing a computer code. Separate energy equations for fluid and solid phases and the k-ε turbulence equations have been applied by non-thermal equilibrium and double decomposition methods, respectively and the fuel consumption rate is obtained from one-step Arrhenius equation. Turbulence modeling helped to obtain closer results compared with experimental data. Turbulence caused an increase to the effect of diffusion and heat transfer in the preheat zone which resulted in a lower maximum temperature in the combustion zone. In the case of excess air combustion, no temperature difference is observed in the upstream zone. The results showed that with decreasing porosity in the combustion zone, the fluid temperature along the burner decreases. That is, for the porosities of 0. 95 and 0. 7 the maximum temperature is decreased by 16% and 18% respectively. The maximum temperature difference in the case of excess air of 67% is about 400K which occurs at x=0. 01m and for the case of no excess air is about 200K at x=0. 045m.

In recent years, extensive efforts have been made on the design and fabrication of surfaces having the ability to reduce viscous drag. In this article, the effect of hydrophobic surfaces on viscous drag has been investigated using large eddy simulation of a turbulent channel flow. Hydrophobic surfaces are known by their ability to trap an air layer and by the existence of slip boundary conditions on them. Using slip boundary condition, the viscous drag is reduced and the turbulence intensities and the near-wall eddies are weakened considerably. In this paper, the slip velocities and the shear stresses at the wall for different slip lengths of hydrophobic surfaces, at the viscous Reynolds number of Ret≅180, have been investigated. For slip lengths greater that 10-5m, an average slip velocity of more than 18% of the average velocity has been obtained and the wall shear stress has been reduced by more than 60%.The results show that, the slip length to have a tangible effect on turbulence should be greater than a certain amount.