Boundary Layer Analysis with Navier-stokes Equation in 2d Channel Flow

The work on this theme will comprise a boundary layer analysis in channel flow. Here we will be looking at both the laminar and turbulent case of incompressible flow within the presence of shear stress and vorticity. This study for both cases is a very important concept to understand for boundary layers in channel flows. To accomplish this study, we used the Finite Element Method and Finite Volume Method, and compared with Direct Numerical Simulation data for channel flow. Boundary layer simulations of fully developed laminar and turbulent channel flow at two Reynolds numbers up to Reτ = 590 are reported. INTRODUCTION The plane channel, which is also called plane Poiseuille flow or duct flow, is a canonical configuration for studying internal flows. Understanding the structure of channel flow is obviously of great engineering interest since this can be applied in many applications. This flow is obviously a Newtonian fluid, so that the important boundary problems are raised. To study the plane channel flow, we need to understand the behavior of flow in boundary layers for both laminar and turbulence flows. For the laminar channel flow, we know the solutions since we could calculate it analytically, but in turbulent case, we can not get an analytical turbulent solution since turbulence is more complex, high Reynolds number is applied, and becomes unstable. Moreover, the reason why turbulence is more complex is the boundary layer starts off laminar, but at some critical Reynolds number, it becomes unstable to disturbance, e.g. noise, vibration, surface, and so on. Therefore, we will discuss the boundary layer in detail in this study. However, we could also approach the turbulent channel flow solution diffusion term, and understand the phenomena of boundary layer in turbulent channel flow with modeling. The Finite Element Method is widely used for numerical analysis because it is not only accurate but also provides complex mesh grids. Therefore, we could solve many other applications of channel flows with Finite Element Method. In this study, we solve two cases of channel flow. First one is the laminar case (Reτ = 90) of channel flow. We can obtain analytical solutions and compare with results from Finite Element Method (Ansys) and Finite Volume Method (Fluent). For the next step, we solve the turbulence case (Reτ = 590), and compare with results from the FVM and MKM1 DNS (Direct Numerical Simulation) data. We are interested in understanding the phenomena of the differences between a laminar and a turbulent flow, and also how the finite element method can predict well the channel flow with comparing to other data. For turbulent channel flow, the k− ε model has been employed. Formulation We begin with the equations for two dimensional steady continuity and Navier-Stokes equations. ∂u ∂x + ∂v ∂y = 0 (1) 1Moser,R.D.,Kim,J.and Mansour,N,N(1999) 1 Copyright c © by ASME u ∂u ∂x + v ∂u ∂y =− ρ ∂p ∂x +ν( ∂2u ∂x2 + ∂2u ∂y2 ) (2) u ∂v ∂x + v ∂v ∂y =− ρ ∂p ∂y +ν( ∂2v ∂x2 + ∂2v ∂y2 ) (3) Laminar channel flow For laminar channel flow, the no-slip boundary condition has been employed and we can apply the conservation of mass and momentum, then we can get the solution for the horizontal velocity, average velocity, vorticity, and the shear stress at the bottom wall; u(y) =−umax[1−4( y h )],umax = h2 8μ d p dx (4)