"Assignment 1:\r\n\r\nTwo parallel plates extended to infinity are a distance of h = 4cm apart. The fluid within the plates has a kinematic viscosity of v = 0.000217m2/s. The upper plate is stationary and the lower one is suddenly set in motion with a constant velocity of Uo = 40m/s. The governing equation is Navier-Stokes simplified as:\r\n\r\n∂u/∂t = v.∂2u/∂y2\r\n\r\nwhere y is the cross-stream direction and u(y, t) is the streamwise velocity component. Use a first-order forward-time and second-order central space (FTCS) scheme to discretize the PDE.\r\n\r\n1. What is the truncation error for FTCS? Is this scheme consistent?\r\n\r\n2. The stability condition for FTCS is given by: d = vΔt/Δy2 ≤ 0.5. Verify this condition numerically by examining the velocity profile for different Δt and Δy.\r\n\r\n3. Plot the velocity profile at various times till the flow reaches steady state.\r\n\r\n4. At steady state, obtain the analytical solution by solving ∂2u/∂y2 = 0. Validate your numerical results with the analytical solution.\r\n\r\nAssigment 2. Laminar Flow in a Pipe\r\n\r\nconsider the steady state flow in a. (2D) pipe with diameter U = 0.2M and length L = 8m. The inlet velocity is U = 1 m/s which is uniform acres the cross section. The fluid exhausts into the ambient atmosphere which is at a pressure of 1 atm. The fluid density is ρ = 1 kg/m3 and coefficient of viscosity μ = 2 x 10-3 kg/m.s. The Reynolds number Re based on the pipe. diameter is Re = ρUD/μ = 100. Use FLUENT to solve and analyze the flow inside this pipe.\r\n\r\n1. Plot the centerline axial velocity vs. axial coordinate,\r\n\r\n2. Plot the velocity vectors along the pipe.\r\n\r\n3. Plot the fully developed axial velocity vs. radial coordinate (r). Validate your results by comparing with the analytical solution (u = 3/2U[1 - (2r/D)2]). Note that this expression is valid for a 2D pipe; for a. circular pipe (3D or axisymmetric) the velocity profile is u = 2U[1 - (2r/D)2].\r\n\r\n4. Refine your mesh and repeat your calculations. Verify your results by comparing the fine- and coarse-mesh solutions.\r\n\r\nAssignment 3. Convective Heat transfer in a laminar pipe flow\r\n\r\nConsider the steady state flow in a (2D axisyrnmetric) pipe with diameter D = 0.2M and length L = 2m. The inlet velocity is U = 1 m/s and temperature T = 300K, which are uniform across the cross section. The fluid exhausts into the ambient atmosphere which is at a pressure of 1 atm. The fluid density is ρ = 1 kg/m3 and coefficient of viscosity μ = 2 x 10-3 kg/m.s. The Reynolds number Re based on the pipe diameter is Re = ρUD/μ = 100 and the Prandtl number of the fluid is Pr = 1. There is a constant heat flux at the wall q.\" = 1000 W/m2. The conductivity of the fluid is k = 0.5 W/m.K . Use FLUENT to solve and analyze the flow and heat transfer inside this pipe.\r\n\r\n1. Refine your mesh and repeat your calculations. Verify your results by comparing the fine-and coarse-mesh solutions.\r\n\r\n2. Plot the temperature at the centerline and at the wall vs. axial coordinate.\r\n\r\n3. Plot the contours of temperature within the pipe.\r\n\r\n4. Plot the fully developed temperature vs. radial coordinate (r). Validate your results by comparing with the analytical solution\r\n\r\nT - Tc = 1/α.(dTm/dx).(UcD2/16).[(2r/D)2 - 1/4(2r/D)4], where Tc and Uc, are the centerline temperature and axial velocity, respectively and Tm, is mean temperature. Mean temperature is defined by employing Newton's law of cooling: q·\" = h(Ts - Tm) where Ts, is surface temperature, Note that in thermally fully developed region, dTm/dx is constant. Also, α = μ/Pr.ρ is the thermal diffusivity.\r\n\r\n5. Find the hydrodynamic entrance length Lh by visual inspection of the velocity profile). How does your result compare with the theoretical formula Lh/D = 0.056Re?\r\n\r\n6. Compute the Nusselt number in the fully developed region. How does your result compare with the theoretical value Nu = hD/k = 4.364? (h is the convective heat transfer coefficient).\r\n\r\nAssignment 4. Turbulent Flow in a Pipe\r\n\r\nConsider the turbulent flow in a pipe with diameter D = 0.2 m and length = 4 m. The inlet velocity is U = 1 m/s which is uniform across the cross section. The fluid density p = 1 kg/m3 and viscosity μ = 2 x 10-3 kg /m.3. Therefore, the Reynolds number Re based on the pipe diameter is Re = ρUD/μ = 10,000. Use FLUENT to solve and analyze the flow inside this pipe. For turbulence modeling, use the standard k - ∈ model with the inlet turbulent intensity of %1. Using such Reynolds-averaged Navier-Stokes (RANS) description of turbulence we can consider a 2D a.xisymmetric domain at steady-state.\r\n\r\n1. Refine your mesh and repeat your calculations. Verify your results by comparing the fine- and coarse-mesh solutions.\r\n\r\n2. Plot the wall y+ vs. axial coordinate. For acceptable near wall resolution we must have y+ < 5 at the wall for all axial locations. Is this condition satisfied in your simulations?\r\n\r\n3. Find the hydrodynamic entrance length Lh (by visual inspection of the velocity profile). How does your result compare with the theoretical formula\r\n\r\nLh/D = 1.359Re1/4?\r\n\r\n4. Plot the skin friction coefficient Cf = τw/(1/2ρu‾2) vs. axial coordinate, where τw and u‾ denote the wall shear stress and the mean axial velocity across a cross section of the pipe, respectively. In the fully developed region, how does your result compare with the theoretical value Cf = 0.0085 for turbulent flow in a smooth pipe?\r\n\r\n5. Plot the velocity profile vs. radial coordinate. How does the peak velocity in the fully developed region compare with umax/u‾ = 1 + 1.33√f, where umax is the maximum velocity and f = 4Cf is the Darcy friction factor. From continuity we have u‾ = U = 1 m/s. Qualitatively compare this velocity profile with that of a laminar flow. What can you say about difference in mixing in laminar and turbulent cases?\r\n\r\nAssignment 5\r\n\r\nTwo parallel plates extended to infinity are a distance of h = 4 cm apart. The fluid within the plates has a kinematic viscosity of v = 0.000217m2/s. The upper plate is stationary and the lower one is suddenly set in motion with a. constant velocity of U0 = 40 m/s. The governing equation is Navier-Stokras simplified as:\r\n\r\nu/∂t = v.∂2u/∂y2\r\n\r\nwhere y is the cross-stream direction and u(y, t) is the streamwise velocity component. Use a first-order forward-time and second-order central space (FTCS) scheme to discretize the PDE. Write a computer program (MATLAB, C, Java, ...)\r\n\r\n1. The stability condition for FTCS is given by: d = vΔt/Δy2 ≤ 0.5. Verify this condition numerically by examining the velocity profile for different Δt and Δy. Show that if you violate this condition, numerical method becomes unstable.\r\n\r\n2. Plot the velocity profile at three different times till the flow readies steady state.\r\n\r\n3. At steady state, obtain the analytical solution by solving ∂2u/∂y2 = 0. Validate your numerical results with the analytical solution.\r\n\r\nAssignment 6. Laminar Flow past a circular cylinder\r\n\r\nConsider the steady state, two-dimensional laminar flow over a cylinder with diameter D = 1m. The free stream velocity is U∞ = 1 m./s. The density and viscosity of the fluid are ρ = 1 kg/m3 and μ= 0.025 kg/m.s. Therefore, the Reynolds number for this flow is Re = ρU∞D/μ = 40. Use FLUENT to obtain velocity and preure distribution on the cylinder.\r\n\r\n1. Plot the streamwise velocity on the cylinder and obtain the streamwi.se velocity at the edge of the boundary layer on top of the cylinder (θ = 90° where the angle θ is measured from the front stagnation point). How does your result compare with the theoretical value u = 2U∞?\r\n\r\n2. Obtain the pressure distribution on the cylinder and normalize it as Cp = (ps - p∞)/(1/2.ρU∞) where ps, and p∞ denote the surface and farfield pressure. respectively. Compare your result with the theoretical distribution Cp = 1 - 4sin2θ. How do you explain the discrepancy between the results?\r\n\r\n3. Obtain the drag coefficient CD = FD/(1/2.ρU2∞D) where FD is the drag force in the stremwise direction. How does you result compare with that from the literature [1]. CD = 1.498 (for Re = 40)?\r\n\r\n4. Obtain the separation angle and compare it with that from the literature [1] θ ≈123° (for Re = 40).\r\n\r\n5. Show verification of your numerical solution using at least three (fine, medium and coarse) mesh arrangements.\r\nCFD Report May 22, 2017 1 Assignment 1 The simpli?ed Navier-Stokes equation 2 ?u ? u =? 2 ?t ?y can be approximated by the numerical scheme (FTCS) du u (t)-2u (t)+u (t) i i+1 i i-1 = ? ,quadi= 1,2,...,N-1 2 dt h j+1 j du u -u i i i = , k =0,1,... dt k h = 1/N u (0) = 0,u (t) = 40,u (t) =0, i 0 N and N is the space discretization. This can be written as a linear system of equations du = Au(t)+b(t), dt where A is a (N-1)×(N-1) tridiagonal matrix ? ? -2 1 0 ... 0 ? ? 1 -2 1 ... 0 ? ? ? ? . . . . . . A = , ? ? . . . ? ? ? ? 1 -2 1 1 -2 1.1 Task 1 2 The truncation error for the FTCS sheme is O(?t) +O(?h ), whih follows from the Taylorexpansion. The FTCS scheme is clearly consistent in the stable region, since from the same expansion, the truncation error tends to zero as h? 0. 11.2 Task 2 The stability region for the FTCS scheme is 2 d =??t/?y =0.5, For d > 0.5, the round-o? errors do not decrease and the method fails to converge. 1.3 Task 3 The steady-state velocity distribution is shown in ?gure 1. Figure 1: Velocity pro?le for laminar ?ow between two in?nite plates 1.4 Task 4 The solution to the steady-state di?erential equation 2 ? u =0 2 ?y 2obtained by integrating twice, is u =C +C y. 1 2 Using the boundary conditions gives C = 40 and C = -1000, which is a 1 2 straight line which is in agreement with the numerical solution as t?8. 2 Assignment 2 The steady-state ?ow in a 2D pipe with diameter D =0.2m and length L =8m is determined usingFLUENT. Planar coordinates are used. The mesh was set to 100×10 elements in the x (axial) and y (radial) coordinates, respectively. -6 The grid is set to 100×10 elements. The monitor error limits were set to 10 . 2.1 Task 1 The axial velocity along the centerline is shown in ?gure 2. Figure 2: Centerline axial velocity along the pipe 32.2 Task 2 The velocity vectors for the beginning and the end of the pipe are in ?gures 3 and 4. Figure 3: Velocity vectors along the pipe (beginning) 2.3 Task 3 Thenumericalresultisingoodagreementwith analyticalresult, whichisshown in ?gure 5. 2.4 Task 4 Themesh isnowre?nedto200×20. The numericalresult ofthe fully developed axial velocity is closer to the theoretical result, as shown in the ?gure above. 3 Assignment 3 3.1 Task 1 Forthe convectiveheattransferinalaminarpipe?ow, thegridissetto200×30 elements. For comparison, the grid is set to 200×60 elements in the next run. -6 The monitor error limits were set to 10 . 4Figure 4: Velocity vectors along the pipe (end) 3.2 Task 2 The temperature at the centerline and the wall along the axis are shown in ?gure 6. 3.3 Task 3 The temperature contours are shown in ?gure 7. The temperature at the centerline and the wall along the axis are shown in ?gure 7. 3.4 Task 4 The fully developed temperature versus radial distance is shown in ?gure 8. 3.5 Task 5 The velocity pro?le is shown below. The theoretical value of the hydrodynamic entrance length is L /D = 0.056Re, h which gives (with Re = 100), L = 1.12. This corresponds reasonably well to h the point where the veocity increases linearly with axial distance in ?gure 9. 5Figure 5: Fully developed axial velocity with radial distance from the center line 3.6 Task 6 TheNusseltnumberidcalculatedto6.1622,whichishigherthanthetheoretical 4.364. 4 Assignment 4 4.1 Task 1 Forthe turbulent ?ow, an adaptivemesh isused which is ?ner closerto the pipe wall. Initially, the mesh is created with 200×30 elements. For comparison, the second (radial) dimension is increased to 60 elements. The monitor error limits -6 were set to 10 . 6Figure 6: Axial temperature at the centerline and the wall 4.2 Task 2 + The y -values along the axial dimension are shown in ?gure 10. It is clear from + this, that y < 5 for all x. The maximum value is slightly larger than 3. 4.3 Task 3 The hydrodymanic entrance length is theoretically 1/4 L /D = 1.359·Re , h so that L = 2.718. This corresponds roughly to the point in ?gure 11, at the h point just before the velocity starts to be a?ected by the back-turbulence at the outlet. 4.4 Task 4 The coe?cient of skin friction is show in ?gure 12. 7Figure 7: Temperature contours We can see that the value is in fairly good agreement with the theoretical value C =0.0085 for turbulent ?ow in a smooth pipe. f 4.5 Task 5 The velocity versus the radial coordinate is shown in ?gure 13. The theoretical value of u is given by max p u /u¯ =1+1/33 f, max where f = 4C , u is the maximum velocity and u¯ =1 equals the velocity at f max the inlet. This gives u =1.245. From the ?gure, u is larger with roughly max max 10%. The maximum velocity is lower than in the laminar case which indicates a larger degree of mixing in the turbulent case compared to the laminar case. 5 Assignment 5 The Matlab code is given the appendix. 8Figure 8: Temperature versus radial distance 5.1 Task 1 If we study the discretization with N = 10, the scheme becomes unstable at ?t = 0.038. This gives an increasingly oscilliating solution, as shown in the ?gure below. 5.2 Task 2 Figure 15 shows the convergence with space discretization N = 10. 5.3 Task 3 -13 For t = 1000 and N = 10, the truncation error is of order 10 . Thus, the max solutioniswithaverygoodapproximationastraigthlinethroughtheboundary values. 6 Assignment 6 The mesh for the laminar ?ow past a cylinder is based on two concentric cylin- ders, where the outer has a diameter of 64m. The resolution (medium mesh) is based on 192 circumferential divisions and 96 radial divisions with re?nement 9Figure 9: Velocity pro?le on the center line close to the cylinder. We note that steady solutions tend to be unstable for a -6 Reynolds number of Re˜ 40 [1]. The monitor error limits were set to 10 . 6.1 Task 1 The streamwise velocity on the cylinder is shown in ?gure 16. 6.2 Task 2 Figure 17 shows the pressure distribution compared to the theoretical value 2 C =1-4sin ?. p The theoretical value is deduced from idealizations, whereas the instability gives turbulence e?ects which change the pressure distribution. 6.3 Task 3 The drag coe?cient C is de?ned in the setup phase of the calculations. It D was found to be C = 1.525, which should be compared with the theoretical D 10"