Abaqus/CFD Sample Problems. Abaqus 6.10

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1 Abaqus/CFD Sample Problems Abaqus 6.10

2 Contents 1. Oscillatory Laminar Plane Poiseuille Flow 2. Flow in Shear Driven Cavities 3. Buoyancy Driven Flow in Cavities 4. Turbulent Flow in a Rectangular Channel 5. Von Karman Vortex Street Behind a Circular Cylinder 6. Flow Over a Backward Facing Step This document provides a set of sample problems that can be used as a starting point to perform rigorous verification and validation studies. The associated Python scripts that can be used to create the Abaqus/CAE database and associated input files are provided. 2

3 1. Oscillatory Laminar Plane Poiseuille Flow

4 Oscillatory Laminar Plane Poiseuille Flow Overview This example compares the prediction of the time-dependent velocity profile in a channel subjected to an oscillatory pressure gradient to the analytical solution. Problem description A rectangular 2-dimensional channel of width = 1m and length = 2m is considered. An oscillatory pressure gradient (with zero mean) is imposed at the inlet. The analysis is carried out in two steps. In the first analysis step, a constant pressure gradient is prescribed for the first 5 seconds of the simulation to initialize the velocity field to match that of the analytical steady-state solution. In the second analysis step, the flow is subjected to an oscillatory pressure gradient. A 40x20 uniform mesh is used for this problem. Two dimensional geometry is modeled as three dimensional with one element in thickness direction. Wall boundary condition 1 m Time-dependent inlet pressure z x Wall boundary condition 2 m Schematic of the geometry used Outlet dp P 0 cos( ( t to )) dx Inlet pressure profile

5 Oscillatory Laminar Plane Poiseuille Flow Features Laminar flow Time-dependent pressure inlet Multi-step analysis Boundary conditions Pressure inlet t < t o : p = t > t o : p = 10*Cos( (t-t o )) ; t o = 5, = p/5 Pressure outlet (p = 0) No-slip wall boundary condition on top and bottom (V = 0) Analytical solution dp i t P0 e dx 2 Po s u( y, t) Re e 2i i s i t cos( z h) 1 cos( h) 2 h is the half-channel width P o is the amplitude of pressure gradient oscillation is the circular frequency References Fluid Mechanics, Second Edition: Volume 6 (Course of Theoretical Physics), Authors: L. D. Landau, E.M. Lifshitz 5

6 Oscillatory Laminar Plane Poiseuille Flow Results T= 5 sec Velocity profile: Comparison with analytical solution Files ex1_oscillatory_planeflow.py ex1_oscillatory_planeflow_mesh.inp T= 10 sec 6

7 2. Flow in Shear Driven Cavities 7

8 Flow in Shear Driven Cavities Overview This sample problem compares the prediction of velocity profiles in shear-driven cavities of different shapes. 2-dimensional 90 0, 45 0 and 15 0 cavities are considered. Shear driven flow in a cubical box is also presented. Velocity profiles in each of these cases are compared against numerical results available in literature. Problem description The following cavity configurations and Reynolds numbers are presented 1 m 1 m 1 m 1 m 1 m 1 m 1 m 1 m 1 m (a) 90 0 Cavity (b) 45 0 Cavity (c) 15 0 Cavity (d) Cubical box 90 0 Cavity: Reynolds number = 100, 3200; Mesh: 129x129x Cavity: Reynolds number = 100; Mesh: 512x512x Cavity: Reynolds number = 100; Mesh: 256x256x1 Cubical box: Reynolds number = 400; Mesh: 32x32x32

9 Flow in Shear Driven Cavities Features Laminar shear driven flow Boundary conditions Specified velocity at top plane to match flow Reynolds number No-slip at all other planes (V = 0) Hydrostatic mode is eliminated by setting reference pressure to zero at a single node References 1. High-Re Solutions for Incompressible Flow Using the Navier-Stokes Equations and a Multigrid Method U. Ghia, K. N. Ghia, and C. T. Shin Journal of Computational Physics 48, (1982) 2. Numerical solutions of 2-D steady incompressible flow in a driven skewed cavity Ercan Erturk and Bahtiyar Dursun ZAMM Z. Angew. Math. Mech. 87, No. 5, (2007) 3. "Flow topology in a steady three-dimensional lid-driven cavity" T.W.H. Sheu, S.F. Tsai, Computers & Fluids, 31, (2002) 9

10 Flow in Shear Driven Cavities 90 0 Cavity Results Velocities along horizontal and vertical centerlines Re = 100 Re =

11 Flow in Shear Driven Cavities Skew Cavity (45 0 and 15 0 ) Results Velocities along horizontal and vertical centerlines

12 Flow in Shear Driven Cavities Cubical Box Results Velocity vectors Velocities along horizontal and vertical centerlines 12

13 Flow in Shear Driven Cavities Files ex2_sheardriven_cavity.py ex2_cavity15deg_mesh.inp ex2_cavity45deg_mesh.inp ex2_cubicalbox_mesh.inp ex2_sqcavity_mesh.inp 13

14 3. Buoyancy Driven Flow in Cavities 14

15 Buoyancy Driven Flow in Cavities Overview This sample problem compares the prediction of velocity profiles due to buoyancy driven flow in square and cubical cavities. The cavities are differentially heated to obtain a temperature gradient. Velocity profiles in each of these cases are compared against numerical results available in the literature. Problem description The material properties are chosen to match the desired Rayleigh number, Ra q = 0.0 T = 1.0 T = g L T Ra,Kinematic viscosity,thermal diffusivity,thermal expansion coefficient g, Acceleration due to gravity x = 0, T = 1.0 x = 1.0, T = 1.0 q = 0.0 All other faces are adiabatic Square Cavity: Rayleigh number = 1e3, 1e6 Cubical Cavity: Rayleigh number = 1e4

16 Buoyancy Driven Flow in Cavities Features Buoyancy driven flow Boussinesq body forces Boundary conditions No-slip velocity boundary condition on all the planes (V = 0) Specified temperatures Hydrostatic mode is eliminated by setting reference pressure to zero at a single node References 1. Natural Convection in a Square Cavity: A Comparison Exercise G. de Vahl Davis and I. P. Jones International Journal for Numerical Methods in Fluids, 3, , (1983) 2. "Benchmark solutions for natural convection in a cubic cavity using the high-order time space method" Shinichiro Wakashima, Takeo S. Saitoh International Journal of Heat and Mass Transfer, 47, , (2004) 16

17 Buoyancy Driven Flow in Cavities 2D Square Cavity Results Present Benchmark (de Vahl Davis) Present Benchmark (de Vahl Davis) Temperature q = 0.0 Velocity u x u x T = 1.0 T = 0.0 q = 0.0 Ra = 1000, Pr = 0.71 Temperature Velocity Present Benchmark (de Vahl Davis) u x Present Benchmark (de Vahl Davis) u x Ra = 1.0e6, Pr =

18 Buoyancy Driven Flow in Cavities Cubical Box Results Ra = 1.0e4, Pr = 0.71 Benchmark (Wakashima & Saitoh (2004)) 2(0,0,0) U 1 (0.5,0.5, z) U (,0.5,0.5) max 3 x (z = ) max (x = ) Benchmark Present Temperature contours at the mid plane (y=0.5) Abaqus/CFD Error % 0.1% 0.2% Benchmark Present Vorticity contours at the mid plane (y=0.5) 18

19 Buoyancy Driven Flow in Cavities Files ex3_buoyancydriven_flow.py ex3_sqcavity_mesh.inp ex3_cubicbox_mesh.inp 19

20 4. Turbulent Flow in a Rectangular Channel 20

21 Turbulent Flow in a Rectangular Channel Overview This example models the turbulent flow in a rectangular channel at a friction Reynolds number = 180 and Reynolds number (based on mean velocity) = The one equation Spalart-Allmaras turbulence model is used. The results are compared with direct numerical simulation (DNS) results available in the literature as well as experimental results. Abaqus/CFD results at a friction Reynolds number = 395 and Reynolds number (based on mean velocity) = are also presented. Problem description A rectangular 2-dimensional channel of width = 1 unit and length = 10 units is considered. A pressure gradient is imposed along the length of the channel by means of specified pressure at the inlet and zero pressure at the outlet. The pressure gradient is chosen to impose the desired friction Reynolds number for the flow. Wall boundary condition Inlet Outlet H = 1 Wall boundary condition L = 10

22 Turbulent Flow in a Rectangular Channel Problem description (cont d) Features The Reynolds number is defined as Re = ru av H/m The friction Reynolds number is defined as Re t = ru t (H/2)/m, where u t is the friction velocity defined as Turbulent flow Spalart-Allmaras turbulence model Boundary conditions Mesh No-slip velocity boundary condition on channel walls(v = 0) Set through thickness velocity components to zero Distance function = 0, Kinematic turbulent viscosity = 0 at No-slip velocity boundary condition Mesh 50 (Streamwise) X 91 (Normal) X 1 (Through thickness) y + at first grid point ~ t u w t r t w = wall shear stress 22

23 Turbulent Flow in a Rectangular Channel References 1. Turbulence statistics in fully developed channel flow at low Reynolds number J. Kim, P. Moin and R. Moser Journal of Fluid Mechanics, 177, , (1987) 2. The structure of the viscous sublayer and the adjacent wall region in a turbulent channel flow H. Eckelmann Journal of Fluid Mechanics, 65, 439, (1974) 23

24 Turbulent Flow in a Rectangular Channel Results Law of the wall u U, y u t uy t Results are presented at Re t 180 Abaqus/CFD results at Re t 395 is also presented 24

25 Turbulent Flow in a Rectangular Channel Results Velocity profile 25

26 Turbulent Flow in a Rectangular Channel Files ex4_turb_channelflow.py ex4_turb_channelflow_mesh.inp Note Abaqus/CFD results at a friction Reynolds number ~ 395 and Reynolds number (based on mean velocity) ~ can be obtained by setting a pressure gradient of

27 5. Von Karman Vortex Street Behind a Circular Cylinder 27

28 Von Karman Vortex Street Behind a Circular Cylinder Overview This example simulates the Von Karman vortex street behind a circular cylinder at a flow Reynolds number of 100 based on the cylinder diameter. The frequency of the vortex shedding is compared with results available in the literature. Problem description The computational domain for the vortex shedding calculations consists of an interior square region (-4 < x < 4; -4 < y < 4) surrounding a cylinder of unit diameter. The domain is extended in the wake of cylinder up to x = 20 units Inlet D = 1 Outlet H = 8 Tow-tank condition H = 20

29 Von Karman Vortex Street Behind a Circular Cylinder Features Unsteady laminar flow Reynolds number = ru inlet D/m Fluid Properties Density = 1 unit Viscosity = 0.01 units Boundary conditions Tow tank condition at top and bottom walls: U = 1, V = 0 Inlet velocity: U inlet = 1, V = 0 Set through thickness velocity components to zero Outlet: P = 0 Cylinder surface: No-slip velocity boundary condition (V = 0) 29

30 Von Karman Vortex Street Behind a Circular Cylinder References 1. Transient flow past a circular cylinder: a benchmark solution, M. S. Engelman and M. A. Jamnia International Journal for Numerical Methods in Fluids, 11, , (1990) 30

31 Von Karman Vortex Street Behind a Circular Cylinder Results Total number of element Strouhal Number * t Coarse Fine * St = fd/u inlet, f is the frequency of vortex shedding The results are consistent with the Strouhal numbers reported in the reference ( ) The frequency of the vortex shedding is obtained as the half of the peak frequency obtained by a FFT of kinetic energy plot (from t = 200 sec to 300 sec) 31

32 Von Karman Vortex Street Behind a Circular Cylinder Results Kinetic energy 32

33 Von Karman Vortex Street Behind a Circular Cylinder Results Velocity contour plot at t = 300 sec Pressure line plot at t = 300 sec 33

34 6. Flow Over a Backward Facing Step 34

35 Flow Over a Backward Facing Step Overview This example simulates the laminar flow over a backward facing step at a Reynolds number of 800 based on channel height. The results are compared with numerical as well as experimental results available in literature. Problem description The computational domain for the flow calculations consists of an rectangular region (0 < x < 30; -0.5 < y < 0.5). The flow enters the solution domain from 0< y < 0.5 while -0.5 < y < 0.0 represents the step. A parabolic velocity profile is specified at the inlet. V x (y) = 24y(0.5 y) V av = 1.0 No-slip H = 1 No-slip No-slip Outlet P = 0 L = 30 Re rvavh m

36 Flow Over a Backward Facing Step Features Steady laminar flow Fluid Properties Density = 1 unit Viscosity = units (the viscosity is chosen so as to set the flow Reynolds number to 800) Boundary conditions Set through thickness velocity components to zero No-slip velocity boundary condition at top and bottom walls V x = 0, V y = 0 Inlet velocity: Parabolic velocity profile - V x = f(y) V y = 0 Outlet: P = 0 No-slip velocity boundary condition at the step boundary V x = 0, V y = 0 36

37 Flow Over a Backward Facing Step References A test problem for outflow boundary conditions Flow over a backward-facing step D. K. Gartling, International Journal for Numerical Methods in Fluids Vol 11, , (1990) 37

38 Flow Over a Backward Facing Step Results L 2 L 3 L 1 Zone 1 (0 < x < 15) Uniform mesh x, y Zone 2 (15 < x < 30) Graded mesh x 2 x, y = constant Mesh (Across the channel x along the channel length) L 1 Length from the step face to the lower re-attachment point L 2 Length from the step face to upper separation point L 3 Length of the upper separation bubble Gartling (1990) 40x Abaqus/CFD Abaqus/CFD Abaqus/CFD Fine 80x1200x1 (Zone 1) 80x832x1 (Zone 2) Medium 40x600x1 (Zone 1) 40x416x1 (Zone 2) Coarse 20x300x1 (Zone 1) 20x208x1 (Zone 2) Elements used by Gartling (1990) were biquadratic in velocity and linear discontinuous pressure elements. In contrast, the fluid elements in Abaqus/CFD use linear discontinuous in velocity and linear continuous in pressure. 38

39 Flow Over a Backward Facing Step Results Pressure line plot (0 < x < 30) Velocity line plot (0 < x < 15) Out of plane vorticity (0 < x < 30) 39

40 Flow Over Backward Facing Step Results Horizontal velocity at x = 7 & x = 15 40

41 Flow Over Backward Facing Step Results Vertical velocity at x = 7 & x = 15 41

42 Flow Over Backward Facing Step Files ex6_backwardfacingstep.py ex6_backwardfacingstep_coarse.inp ex6_backwardfacingstep_medium.inp ex6_backwardfacingstep_fine.inp coarse_parabolic_inlet_velocity.inp medium_parabolic_inlet_velocity.inp fine_parabolic_inlet_velocity.inp Note The models require a parabolic velocity profile at the inlet. This needs to be manually included as boundary condition in the generated input file. The parabolic velocity profile required is provided in files coarse_parabolic_inlet_velocity.inp, medium_parabolic_inlet_velocity.inp and fine_parabolic_inlet_velocity.inp for coarse, medium and fine meshes, respectively. 42

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