Strömningslära Fluid Dynamics. Computer laboratories using COMSOL v4.4

Transcription

1 UMEÅ UNIVERSITY Department of Physics Claude Dion Olexii Iukhymenko May 15, 2015 Strömningslära Fluid Dynamics (5FY144) Computer laboratories using COMSOL v4.4!!

2 Report requirements Computer labs must be done by each student individually, not in groups. In order to pass the computer labs every student has to either show all results of each of the 4 labs to a supervisor during the computer lab sessions (see the schedule); send by a brief report including all necessary plots and the answers to all questions. A report is the be sent only if you haven t passed any or some of the labs after the last scheduled lab session. The reports should be sent to by to oleksii.iukhymenko@physics.umu.se before , 23:59. It is more convenient if you put your name in the report filename, like Eric_Andersson_fd_2015 or similar. 0 Introduction 0.1 What is COMSOL? COMSOL Multiphysics is a multipurpose software platform for simulating physicsbased problems. It is used in computer labs in Vector Analysis, Electrostatics, Electrodynamics, Fluid Dynamics and in other related courses. COMSOL is a powerful software package based on the Finite Elements Method (FEM) for solving Partial Differential Equations (PDE), and includes a number of pre-built modules and submodules. COMSOL has an interactive Graphical User Interface (GUI) and can be run together with MATLAB (using the appropriate module). COMSOL provides extremely useful and representative tools for preliminary studies as well as for thorough research in many different areas of physics. In addition, it is perfectly suited to educational purposes, producing results after just a few mouse clicks. 0.2 What are we going to study? The main purpose of the following labs is to study some examples of incompressible flows: flows inside channels and pipes, flows outside cylinders and spheres and flows outside other objects with more complex shape, like an aerofoil. With the help of COMSOL you will see the flow fields distributions, like velocity and pressure in a chosen geometry; integration procedure to calculate drag and lift forces is also rather simple, as you will see below. 1

3 The flow motion is governed by the set of the Navier-Stokes equations together with the continuity equations ρ v t + ρ (v ) v = P + µ 2 v, (1) v = 0, (2) where v is the flow velocity, ρ = const. is the density, P is the pressure, and µ is the dynamic viscosity. It is very often convenient to study a process using dimensionless variables. In a steady state, the stationary momentum equation then takes the form (u ) u = P + Re 2 u, (3) where u v/u is the scaled velocity, P P/(ρU 2 ), Re ρul/ν is the Reynolds number, and ν = µ/ρ is the kinematic viscosity. In the first two labs, you are going to study dependence of the entry length and the drag force as a function of the Reynolds number, so it is convenient to use this number as the main parameter to specify viscosity. 0.3 Getting started with COMSOL v4.4 Launch COMSOL Multiphysics 4.4 from the desktop or under Window s Start button menu. Start the Model Wizard to setup the general settings of the model. In all the labs, we will work with a two-dimensional geometry, so you must choose 2D. In the next window, Select Physics, you select the physics involved, in our case fluid flow. In the Fluid Flows module, select Single-Phase Flow Laminar Flow, press Add and proceed to Study. We will study only stationary problems in these labs, so in the following window, Select Study, select Stationary and then press the Done button. You will now see the Model Builder, figure 1, which consists of a model tree with several items and sub-items. You can also use the ribbon for specifying your model. In the first item, Global Definitions, you can define important parameters and constants that you will use in the labs, like flow density, length sizes, and velocity value. Although not necessary, it is much more convenient to have all the parameters in one place. Here you can also determine the value of the Reynolds number, which we will later substitute into the viscosity term. In the next item, Component 1, you will define the geometry and set the boundary and initial conditions. In addition, you can take advantage of the material database to use the characteristics of certain materials, although in these labs we will use just the simplest model flow of unity density and viscosity defined through the Reynolds number. At the end of the Component 1 section, you can specify the mesh settings; the default is a Physicscontrolled mesh with Normal element size. The geometries used in these labs are quite simple, so you can leave the setting to Physics-controlled mesh. After having created some object, you can check that the normal meshing provides a rather coarse mesh, hence it is better it change it to the Finer or Extra fine mesh. While it is better and more accurate to use 2

4 IntroductionToCOMSOLMultiphysics.book Page 2 Tuesday, January 14, :03 PM COMSOL Desktop QUICK ACCESS TOOLBAR Use these buttons for access to functionality such as file open/save, undo/redo, copy/paste, and delete. RIBBON The ribbon tabs have buttons and drop-down lists for controlling all steps of the modeling process. MODEL BUILDER TOOLBAR MODEL TREE The model tree gives an overview of the model and all the functionality and operations needed for building and solving a model as well as processing the results. MODEL BUILDER WINDOW The Model Builder window with its model tree and the associated toolbar buttons gives you an overview of the model. The modeling process can be controlled from context-sensitive menus accessed by right-clicking a node. SETTINGS WINDOW Click any node in the model tree to see its associated settings window displayed next to the Model Builder. 2 COMSOL Desktop Figure 1: Appearance of the COMSOL 4.4 interface. 3

5 a finer mesh resolution, convergence problem may arise if you make it too fine. If you obtain such an error, change the mesh resolution and see how it influences the results. The above settings are common to all these labs, with the exception of lab 2 which uses a 3D geometry. More details will be given in the individual labs. 0.4 Comment on the Reynolds number In all the labs, we restrict ourselves to laminar flow, so that the Reynolds number is limited to some critical value, about , depending on particular geometry. If your solution doesn t converge, then you probably should reduce the Reynolds number. In the first two labs you are to study the dependence with respect to the Reynolds number, which implies that you should take sufficiently different values, starting from a very viscous flow to an almost turbulent one. The typical range of the Reynolds number is from 0.01 to 100, and you can also try higher values if the solution converges. Within this interval, you should have about 10 points to draw the plot asked in a lab. 4

6 1 Lab 1 Entry length in the Poiseuille flow In this lab, we consider a stationary incompressible flow in a tube in two geometries: first in a 2D channel, and then in a cylindrical pipe. 1.1 Two-dimensional channel flow We start with the simplest geometry of a channel of width d formed by two parallel plates, as shown in Fig. 2. The plates are assumed to be infinitely long in the z-direction. Then the velocity depends on the x and y coordinates only. The velocity at the inlet (x = 0) is u 0 in the x-direction. Downstream from the inlet, the velocity profile resembles that of a fully developed Poiseuille flow in a channel, ( ) ] 2y 2 u(y) = U max [1. (4) d The flow in the configuration of Fig. 2 is characterized by the entry length and the Reynolds number of the flow. The entry length may be defined as the distance from the inlet to the point on the x-axis where the flow becomes plane parallel and the velocity saturates to its maximum value. In order to determine entry length from COMSOL simulations, you can fix some velocity value close to the maximal one, of the maximal value U max. This value can be determined by means of a cross-section along the central line of the channel. Main task: Determine the dependence of the entry length on the Reynolds number. Figure 2: Flow between two parallel plates.! 5

7 You should plot this dependence together with the second task of this lab. In addition, you should check that the velocity profile corresponds to a fully-developed Poiseuille flow, see Eq. (4). Draw the appropriate cross-section of the channel, to check the parabolic velocity profile. Below are step-by-step instructions on how to build the geometry and complete the lab tasks. In further labs these instructions won t be repeated, so you are to get used to working with COMSOL in this lab. Under the Geometry item in Component 1, choose Rectangle from the geometry ribbon above. Draw an elongated rectangle with width to height ratio of about 10. In the Rectangle window, check that Corner is selected for the Base. It is more convenient if 0 is used as the x-coordinate. Next step is to define important parameters of the flow. Right click on Global Definitions and add a Parameters section. Here you can specify the density, inlet velocity, Reynolds number, and rectangle sizes as well. For example, to define the density, enter rho under Name, and in the Expression field enter its value, for instance 1. In addition, you can set the viscosity here as proportional to the inverse of the Reynolds number. Under Model Laminar Flow, specify the initial and boundary conditions. In the Fluid Properties item you can see what equation is solved. In the Fluid Properties Settings, you must specify density and viscosity as User defined and put your values or parameter names there. Right click on Laminar Flow and you will see different kinds of initial and boundary conditions, choose Inlet and Outlet conditions. By default, all walls are set as non-slip. Select Inlet and in the Settings choose the Boundary condition as Velocity (default option) and specify the inlet velocity. In the Graphics window select the left boundary as the inlet. (The Boundary Selection should be set to Manual). In a similar manner, select the Outlet boundary condition, leaving the settings to the default. Check the mesh element size under Mesh. You can see the present grid by pressing the Build All button,. The Zoom In, Zoom Out, Zoom Box, and Zoom Extents buttons, found on the toolbar of the Graphics window, are very convenient. Solve the problem, by pressing the Compute button,, under the Study item. After a few seconds you will see the velocity profile as the default result plot. In order to see the velocity profile in details along some line, you should add a cross-section. In the Model Builder, under Results Velocity or Pressure, you can add a Cut line. From the top toolbar choose Select First Point for Cut Line and Select Second Point for Cut Line. You can adjust these points under the Results Data Sets Cut Line 2D settings. Also in the Results section, a 1D Plot Group must appear, where you will see the velocity profile along the particular cross-section. 6

8 You should draw two cross-sections, one along the central line and one across the channel. The first cross-section will allow you to set the entry length, by fixing some appropriate velocity value. Then change the Reynolds number and solve the problem again. How does the entry length change? Pipe flow Now you will study the same problem in a pipe, which stimulates the 3D case, however using a two-dimensional cylindrical coordinate system. For this, in the Model Wizard, you need to choose the 2D Axisymmetric geometry. Follow the same procedure as above to create a rectangle, set the boundary conditions, and produce the results. Note that the cylinder axis is vertical by default, such that one of the long sides of your rectangle must coincide with the axis. Consequently, the inlet and outlet should be set to the bottom and top sides, respectively. Calculate the entry length for the same Reynolds numbers and draw the two curves on the same plot. In addition, compare the velocity profile across the channel and the pipe as well. Discuss the difference between the two cases. 1 You can also do a Study Parametric Sweep, where you choose the Reynolds number as a parameter and specify a range of values. 7

9 2 Lab 2 Drag on a cylinder and on a sphere In this lab, you will study the flow outside a cylinder and outside a sphere and how it depends on the Reynolds number. 2.1 Cylinder 2D case Lets consider the flow around a circular cylinder of radius R. We are interested in the drag force and the drag coefficient C D, which is given by C D = 2D (5) R, where D is the drag per unit length. It may be calculated in COMSOL by means of a line integration over the cylinder surface. In this exercise, you should draw one large elongated rectangle the computational domain (width-to-height ratio around 5) and then a small circle inside it. Right-click on Geometry to select Boolean Operations Difference to subtract the circle from the rectangle (i.e., the rectangle appears under Objects to add and the circle under Objects to subtract). For the rectangle, specify the boundary and initial conditions as in Lab 1, except that it is better to set the outer walls as Slip, to eliminate an additional influence of the walls on the drag force upon the cylinder. To do that, you need to add a Wall by right-clicking on Laminar Flow. Check that the boundaries of the cylinder are No slip and Compute the solution. Under Results, right-click on Derived Values and choose Line Integration and add all the boundaries of the cylinder to the Selection window. Under Expression, choose (using ) Laminar Flow Total stress, and its appropriate component to calculate the drag force. Repeat this calculation for other values of the Reynolds number, similar to Lab 1. ρu Sphere 2D axisymmetrical case Next, you should do the same calculations for a sphere instead of a cylinder. As in Lab 1, you have to select 2D Axisymmetric in the Model Wizard, and then proceed as in the previous exercise above. Present the drag for both cases (cylinder and sphere) as a function of the Reynolds number. 2.3 Sphere 3D case (optional) If you are fast and/or interested by this problem, you can repeat the previous study for the full 3D case. It can take a longer time to do all the calculations, so you can check the drag force only for one or a few specific values of the Reynolds number. 8

10 3 Lab 3 Flow around airfoil and elongated body In this lab, you will investigate the flow around objects with more complex shapes than a cylinder or a sphere. The first part of the lab is devoted to drag and lift forces acting on an aerofoil. In the second part of the lab you are to study the drag force acting on a streamlined body of variable shape. 3.1 Flow around an aerofoil In this exercise, we are interested how the drag and lift on an aerofoil depends on the angle of attack, α, see Fig. 3. For simplicity and reasonable computation time, we will use a two-dimensional airfoil with a characteristic length l, with the flow velocity u 0 and flow density equal to unity. Calculations should be performed at a relatively large Reynolds number, say 200 (may be varied from 100 to 1000), although it is limited due to problems with the numerical convergence. You should either set the viscosity to obtain such a Reynolds number or simply set a certain value of the Reynolds number and calculate the viscosity, as in previous laboratories. The drag C D and the lift coefficients C L of the aerofoil are defined in a way similar to Eq. (5), with D and L the drag and lift per unit length, respectively. C D = 2D ρu 2 0 l, (6) C L = 2L ρu 2 0 l, (7) Main task: Determine how the drag and lift coefficients for an aerofoil depend on the angle of attack, α, at Re = 500 (use negative angles as well as positive). Find the optimal angle for the lift to drag ratio C L /C D. In order to rotate the airfoil, right-click on Geometry to choose Transform Rotate and select the body. After that, do a Boolean subtraction, as in Lab 2. To calculate the lift you are to integrate the y-component of the total stress force. Hint: you can save time if you start with the maximal angle and then reduce it. Otherwise, if you start with the smaller angle and increase it, you can end up with convergence Figure 3: Flow around an aerofoil.! 9

11 problems at some angle; you would then have to decrease the Reynolds number and redo all the calculations. 3.2 Flow around a streamlined body In this part of the lab, you are to study the dependence of the drag on a streamlined body on its shape. As usual, we consider two-dimensional flow, assuming the body to be infinitely long in the direction perpendicular to the flow. The maximum width of the body, d, is kept constant while the length, L, of the tail is varied, see Fig. 4. You can use similar Reynolds number as in the previous part, Re 200. Main task: Determine how the drag varies as a function of the ratio of the length of the body tail to the body width d of the tail of the body. Figure 4: Flow around a streamlined body.! 10