Proceedings of,, BOUNDARY LAYER ANALYSIS WITH NAVIER-STOKES EQUATION IN 2D CHANNEL FLOW Yunho Jang Department of Mechanical and Industrial Engineering University of Massachusetts Amherst, MA 01002 Email: yujang@ecs.umass.edu ABSTRACT 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 Re τ = 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 (Re τ = 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 (Re τ = 590), and compare with results from the FVM and MKM 1 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. 1 Moser,R.D.,Kim,J.and Mansour,N,N(1999) u x + v y = 0 (1) 1 Copyright c by ASME
u u x + v u y = 1 p ρ x + u ( 2 x 2 + 2 u y 2 ) (2) u v x + v v y = 1 p ρ y + v ( 2 x 2 + 2 v y 2 ) (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) = u max [1 4( y h )],u max = h2 d p 8µ dx (4) u m = 2 3 u max (5) τ yx = µ du dy = 8µu y max h 2 (6) From equation (5) and (6), we know the maximum horizontal velocity is 45m/s, and the mean velocity is 30m/s. Moreover, we can calculate following quantities with this analytical solutions. Re m = u m 2 δ Re c = u cδ (10) (11) where u c is centerline velocity, and C f is skin fraction based on τ w and u m. C f = τ w 1 2 ρu (12) m 2 C f is skin fraction based on τ w and u m We also use equations for displacement thickness (δ ) and momentum thickness (θ). δ δ = (1 u )dy (13) 0 u c ω yx = du dy = 8u y max h 2 (7) where h, u m, and u max are height of channel, mean velocity, and maximum of horizontal velocity, respectively. Re τ = u τδ (8) δ θ = [(1 u ) u ]dy (14) 0 u c u c Turbulent channel flow For turbulent channel flow, the k ε model has been used, and Re τ is 590, and we introduce the standard k ε model which is modeling the turbulent viscosity from the transport equation. The turbulent (or eddy) viscosity, µ t, is computed by combining k and ε as follows, τw u τ = ρ (9) µ t = ρc µ k 2 ε (15) where δ and u τ are boundary layer thickness and wall shear velocity, respectively; We use constant dynamic viscosity and density (µ= 0.01111 and ρ = 1) for incompressible flow so that we have constant kinematic viscosity ( = µ/ρ = 0.01111). For this laminar case, the height of channel is 2m, length of channel is 100m, the grid of domain is 60*200, and Reynolds number based on u τ is 90. where C µ is a constant. In this study we follow default values of the model constants, C 1ε,C 2ε,C µ, and σ ε, which have been determined from experiments with air and water for fundamental turbulent flows and they have been found to work fairly well for a wide range of wall-bounded and free shear flows. The model constants are, C 1ε = 1.44, C 2ε = 1.92, C µ = 0.09, σ ε = 1.3, σ k = 1.0 2 Copyright c by ASME
For using k ε model, it is assumed that the flow is fully turbulent, and the effects of molecular viscosity are negligible. The standard k ε model is therefore valid only for fully turbulent flows. The standard wall function has been used for wall boundary treatment, resulting in, where u + = 1 κ ln(ey+ ) (16) Table 1. COMMONLY COMPUTED QUANTITIES Method u max (m/s) τ w (pascal) ω w (s 1 ) Analytic solution 45 1.0-90 FVM 44.9313 0.987-88.8 FEA 44.248 0.935-84.144 Method C f C f o δ Analytic solution 0.00222 0.000987 0.333 FVM 0.00219 0.000974 0.356 FEA 0.00207 0.000923 0.338 u + U pcµ 1/4 k 1/2 p τ w /ρ y + ρc1/4 µ µ k 1/2 p y p (17) (18) Method θ Re m Re c Analytic solution 0.1333 5400 4050 FVM 0.1343 5400 4043 FEA 0.1338 5400 3982 and κ = Von Karman constant (= 0.42) E = empirical constant (= 9.81) U p = mean velocity of the fluid at point p k p = turbulence kinetic energy at point p y p = distance from point p to the wall µ = dynamic viscosity of the fluid In turbulent channel flow, the kinematic viscosity and mean velocity are reduced to get Re = 590 ( = 0.001695, u m = 18.4539). Results Laminar channel flow As mentioned before in this chapter, the data from Finite Element Method will be compared with an analytical solution and the data from Finite Volume Method. From the analytical solution for laminar channel flow, we have predictable maximum velocity, shear stress, vorticity, and so on. As shown in Table 1, the quantities from the FEA and FVM simulations are very similar with those from analytical solution. Figure 1 shows velocity profiles from FVM and FEA. The mean velocity is 30 m/s in this case, and as shown this is the maximum velocity which is centerline velocity is almost 45 m/s (analytical solution) in both cases. Figure 2 and Figure 3 show the behaviors of shear stress and vorticity in channel flow. The shear stress and vorticity are at a maximum at the wall, then gradually decrease with distance. Those values are also very similar with those of analytical solution. Figure 1. VELOCITY PROFILES IN FULLY DEVELOPED LAMINAR CHANNEL FLOW AT X/L = 1, Re τ = 90 Turbulent channel flow In this part, we will discuss the results of turbulent channel flow. As shown before, in this case Re τ is 590, and viscosity ( = 0.001695) has been reduced to match this Re τ. The velocity profiles (Figure 4) are very different from the laminar channel flow. As shown, the velocities in middle of channel level-off due to the behavior of turbulence viscosity. For FEA data, the velocities in middle of channel are smaller than those of FVM and Direct Numerical Simulation (DNS) data, however, as 3 Copyright c by ASME
Figure 2. SHEAR STRESS PROFILES IN FULLY DEVELOPED LAMI- NAR CHANNEL FLOW AT X/L = 1, Re τ = 90 Figure 4. VELOCITY PROFILES IN FULLY DEVELOPED TURBULENT CHANNEL FLOW AT X/L = 1, Re τ = 590 Figure 3. VORTICITY PROFILES IN FULLY DEVELOPED LAMINAR CHANNEL FLOW AT X/L = 1, Re τ = 90 we approach the wall, they are much similar to DNS data. Figure 5 and Figure 6 show the shear stress and vorticity profiles in turbulent channel flow. In this picture, turbulent channel flow is not dominated by the shear stress and vorticity. At the wall, vorticity is increasing compared with laminar channel flow. Figure 7 shows the wall law plot for turbulent boundary layers with three sets of data. The logarithmic law for mean velocity is known to be valid for y + > about 30 to 60. In our case, the loglaw is employed when y + > 11.225. When the mesh is such that Figure 5. SHEAR STRESS PROFILES IN FULLY DEVELOPED TUR- BULENT CHANNEL FLOW AT X/L = 1, Re τ = 590 y + < 11.225 at the wall-adjacent cell, FVM and FEA apply the laminar stress-strain relationship that can be written as u + = y +. In this picture, the velocities of FEA near wall are similar to DNS data until y + = 10 which is the viscous sublayer. In part of buffer layer (5 < y + < 30), the velocities of FVM are predicted with creater accuracy. 4 Copyright c by ASME
ble to apply different simulations to the boundary layer problems. In the turbulent channel flow case, there are some errors since we are using turbulent model (k ε) which is not an exact solution. However, the data from FEA and FVM are believable and still going on right track. In order to understand about the turbulent flow, we need to develop more exact model for this flow. Figure 6. VORTICITY PROFILES IN FULLY DEVELOPED TURBU- LENT CHANNEL FLOW AT X/L = 1, Re τ = 590 REFERENCES [1] Moser,R.D., Kim,J. and Mansour,N.N., Direct Numerical Simulation of Turbulent Channel Flow up to Re τ = 590, Phys.Fluid, Vol 11,No4, pp 943-945. 1999. [2] Pope, Stephen B., Book: Turbulent Flows, Cambridge University Press, New York, 2000. [3] Schetz, Joseph A., Book: Boundary Layer Analysis, Prentice-Hall,Inc., New Jersey, 1993. [4] Wilcox, David C., Book: Basic Fluid Mechanics, DCW Industries,Inc., California, 2000. [5] Fluent 6.0 Manual. [6] Ansys 5.7 Manual. Figure 7. at X/L = 1, Re τ = 590 Mean velocity profiles in fully developed turbulent channel flow CONCLUSION We simulated both laminar and turbulent channel flow using FEA, and compared with data from FVM and DNS. As shown in results, in the laminar case the data from FEA and FVM are quite similar to the analytical solution; and velocity, shear stress, and vorticity profiles are close to each other. Therefore, we can say that the simulations of laminar flow which is dominated by viscosity are well predictable in both Finite Volume Method and Finite Element Method, and it is possi- 5 Copyright c by ASME