Parabolic Flow in Parallel Plate Channel ME 412 Project 4 Jingwei Zhu April 12, 2014 Instructor: Surya Pratap Vanka 1 Project Description The objective of this project is to develop and apply a computer program to compute developing parabolic flow in a straight channel. Such a flow is of boundary layer type, and can be computed by solving the boundary layer form of the Navier-Stokes equations shown below: u v + v y = 0 (1) ρ(u u x + v u y ) = dp dx + µ 2 u y 2 (2) where u and v are the two Cartesian velocities, x and y are spatial coordinates, ρ is the density, P is the static pressure, and µ is the dynamic viscosity. Numerical results for different Reynolds numbers of 50, 100 and 200 will be shown. The channel Reynolds number is based on the channel height, mean velocity and the kinematic viscosity of the fluid, as is shown in Equation 3: Re = ρ V H µ where V is the mean velocity and H is the channel height. (3) The flow development and development lengths as a function of the Reynolds number will be studied. Fully-developed length will be taken as the distance when the centerline velocity reaches 0.99 of the fully-developed value. Note that here numerical fully-developed value (i.e. one that has the grid errors) instead of the analytical solution will be used. In this project, we consider a parallel plate channel with a height H of 0.1m and a length of 3m, as is shown in Figure 1. Note that the fully-developed length usually can be approximated as 0.1 Re H. Therefore we set the length to 3m so that the flow has enough space to be fully 1
Figure 1: Flow in parallel plate channel developed. µ, the dynamic viscosity, is set to 1 10 3 kg/m s and density ρ is set to 1 10 3 kg/m 3. The spatial intervals x and y of the grid yield to the following limits: µ ρ x 0.5 (4) u i,j y2 v i,j y µ/ρ < 2 (5) In our project, the spatial intervals in horizontal direction will be generated dynamically to be 0.4ρu i,j y 2 /µ as the explicit scheme marching along the horizontal direction while those in vertical direction are set to 0.002m. 2 Differencing schemes The code uses an explicit form of the marching algorithm, which is shown as follows: u i,j u i+1,j u i,j x + v i,j u i,j+1 u i,j 1 2 y = 1 ρ (dp dx ) i+1 + µ ρ (u i,j+1 2u i,j + u i,j 1 y 2 ) (6) u i+1,j + u i+1,j 1 u i,j u i,j 1 2 x where i and j are the node numbers in x and y direction. + v i+1,j v i+1,j 1 y = 0 (7) Since the flow is internal, ( dp dx ) i+1 is unknown ahead of time. So small adjustments need to be done to the previous algorithm to find ( dp dx ) i+1. We first guess ( dp dx ) i+1 to be equal to ( dp dx ) i and then correct it if needed. Solve the x-direction momentum equation at (i + 1) using ( dp dx ) i. Here we use symbol û i+1,j for this solution. u i,j û i+1,j u i,j x + v i,j u i,j+1 u i,j 1 2 y = 1 ρ (dp dx ) i + µ ρ (u i,j+1 2u i,j + u i,j 1 y 2 ) (8) 2
Then we have ny ˆṁ i+1 = ρ (û i+1,j + û i+1,j 1 ) y 2 j=2 (9) Inflow mass ṁ in = ρv in H. ˆṁ i+1 will not equal ṁ in until the dp dx becomes constant in x. Hence ṁ i+1 = ṁ in ˆṁ i+1. We now need to update ( dp dx ) i+1 with ( dp dx ) i + ( dp dx ) i+1. Since we use an explicit scheme, ny ρ ( u i+1,j + u i+1,j 1 ) y 2 = ṁ i+1 (10) j=2 u i+1,j = x ρu i,j ( dp dx ) i+1, j [2, ny 1] (11) u i+1,j = 0, j {1, ny} (12) Therefore we find the relationship between ( dp dx ) i+1 and ṁ i+1 to be x( dp dx ) 1 i+1[ + u i,2 3 Results and Discussion ny 1 1 + u i,ny 1 j=3 ( 1 u i,j + 1 u i,j 1 )] y 2 = ṁ i+1 (13) Reynolds Number 50 100 200 Fully Developed Length (m) 0.2116 0.4231 0.8462 Table 1: Fully developed length in the parallel plate channel Table 1 shows the fully developed length in the parallel plate channel for different Reynolds numbers. It can be observed that the computed fully developed length has a linear relationship with the Reynolds number. The larger the Reynolds number is, the longer the fully developed distance is. Figure 2 shows a comparison of contour of horizontal velocity in the parallel plate channel for different Reynolds numbers. We can observe the flow becomes uniform in the horizontal direction after going through the entrance length. Moreover the centerline speed when the flow is fully developed is larger than the initial inlet speed. Figure 3 displays plots of horizontal velocity profile in the parallel plate channel for different Reynolds numbers at 30%, 60%, 90% and 120% of the corresponding fully developed length. At the entrance of the parallel channel, horizontal velocity profile is flat around the centerline. When the flow is fully developed, horizontal velocity profile is parabolic. Figure 4 shows plots of vertical velocity profile in the parallel plate channel for different Reynolds numbers at 30%, 60%, 90% and 120% of the corresponding fully developed length. The vertical velocity profile has a shape similar to a sine function. It is flattened out as the flow develops and 3
(a) Re=50 (b) Re=100 (c) Re=200 Figure 2: (a)(b)(c): Contours of horizontal velocity in the parallel plate channel for different Reynolds numbers 4
(a) Re=50 (b) Re=100 (c) Re=200 Figure 3: (a)(b)(c): Plots of horizontal velocity profile in the parallel plate channel for different Reynolds numbers at four corresponding locations 5
eventually vertical velocity becomes zero. Figure 5 shows profiles of dp/dx as a function of distance from the channel inlet for different Reynolds numbers. There is a huge pressure differential drop where the free stream enters the channel as flow velocity at the boundary is zero while velocity elsewhere is not zero. Smaller grid mesh size will result in larger drop in pressure differential. After entering the channel, pressure differential gradually increases and converges to a constant value when the flow is fully developed. 6
(a) Re=50 (b) Re=100 (c) Re=200 Figure 4: (a)(b)(c): Plots of vertical velocity profile in the parallel plate channel for different Reynolds numbers at four corresponding locations 7
(a) Re=50 (b) Re=100 (c) Re=200 Figure 5: (a)(b)(c): Profiles of dp/dx as a function of distance from the channel inlet for different Reynolds numbers 8
4 Summary In this project, we successfully developed a program to solve the boundary layer form of Navier- Stokes equations in a parallel plate channel. Fully developed length as a function of Reynolds number is shown in a table and numerical results for the parabolic flow in the parallel plate channel are displayed with contours and plots. It can be observed that there is a positive linear relationship between fully developed length and Reynolds number. Horizontal velocity profile of the flow has a parabolic shape when the flow is fully developed. Pressure differential will converge to a negative constant value when the flow is fully developed and becomes uniform in the horizontal direction. References [1] Joel H. Ferziger and Milovan Peric, Computational Methods for Fluid Dynamics, 3rd edition. 9