{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# CFD Problem's Physics & Theory\n", "\n", "**In developing CFD simulation for a fluid mechanic problem understanding the general physics and fundamental theory of the problem is extremely important. This knowledge would provide users the ability to make logical decision about the geometry of CFD domain, choice of proper boundary conditions and numerical models for the CFD simulation. Furthermore, user will have a general big picture on the expected outcome of simulation. These knowledges and abilities form the foundation for a successful implementation and validation of the CFD simulation.**\n", "\n", "**In other words, understanding physics and theory of problem before developing a CFD simulation can be thought of turning on a flash light before taking any step in a completely dark room. Hence, let's review the fundamental physics and theory behind the problem of 2D laminar flow over a cylinder as the first step for development and validation of CFD simulation for this problem of interest:**\n", "\n", "One of the most fundamental and interesting case studies in the field of fluid mechanics is the evolution of the flow field as the fluid interacts with a blunt object. Among different shapes for a blunt object, 2D cylinder (simple version of 3D sphere) is known to be one of the most applicable and popular geometries that many scientists investigate the flow field around it both numerically and experimentally. The behavior and physics of the flow field around a 2D cylinder is strongly dependent on the ratio of inertial to viscous forces (a.k.a Reynolds number) within the flow, which will be proved theoretically later in this section. As this force ratio increases the physics of the flow around and in the wake of the cylinder would change and becomes more complicated. Fig. 1 visualizes this flow field evolution via simple sketch.\n", "\n", "\n", "Fig 1. Sketch of the flow field evolution around a 2D cylinder according to the ratio of inertial to viscous forces (i.e. Reynolds number) within the flow.\n", "\n", "In order to understand the physics of this flow field and perform a general analysis on it, first the Control Volume (CV) analysis is used. By definition, a control volume is a limited inertial frame of reference within the continuum of a flow that encloses the blunt object and flow field around it, shown in Fig. 2. According to the control volume analysis the two major physical variables that will be conserved within this control volume are **mass** and **momentum**. Writing down and balancing these conservation equations for each of these variables would provide important information about the behavior of the flow around 2D cylinder.\n", "\n", "\n", "Fig 2. Sketch of the defined Control Volume for the flow around a 2D cylinder ([source](http://www.disasterzone.net/projects/docs/mae171a/water_tunnel_experiment.pdf)).\n", "\n", "### Conservation of mass ###\n", "\n", "The general form of conservation of mass equation applied to the defined control volume, visualized in Fig. 2, has the following form:\n", "\n", "$$\\frac{\\partial}{\\partial t} \\int_{CV} \\rho~dV + \\int_{CS} \\mathbf{n}~.~\\rho \\mathbf{V}~dA = 0 .$$\n", "\n", "In this equation $\\rho$ is the density of the fluid, $\\mathbf{V}$ is the flow velocity on the control surfaces of the control volume boundaries, which has an area of $dA$ and a normal vector of $\\mathbf{n}$. In this case study since the flow is steady state the first term of the equation will be equal to zero. The second term, however gives the incoming and outgoing mass flux within the control volume surfaces. Considering that the most important component of the flow velocity is the streamwise component, this equation can be simplified further and knowing the variation of incoming velocity into the control volume will give the variation of the outgoing velocity field, after interacting with the cylinder. Therefore, conservation of mass would provide the velocity field variation within the flow.\n", "\n", "### Conservation of momentum ###\n", "\n", "To obtain a general understanding about the interaction between the flow and cylinder the equation for conservation of momentum is used. Upon interaction between the flow and cylinder there are two force components acting on the body of cylinder. One force component is in the streamwise direction and is called the **Drag Force**. The other component is perpendicular to the streamwise flow direction and is called the **Lift Force** (i.e. flow tries to lift-up the cylinder). For a stationary cylinder due to the symmetry of the flow field on the top and bottom of the cylinder, the lift forces acting on these regions cancel each other out. Therefore, the net lift force on the cylinder will always be equal to zero. However, there will be a significant drag force on the cylinder wall as a result of it's interaction with the flow. To obtain an estimate for this drag force acting on the cylinder, one can consider the conservation of momentum in streamwise direction. The general form of conservation of momentum equation applied to the defined control volume showed in Fig.2 is as follows:\n", "\n", "$$\n", "\\mathbf{R_{ext}} + \\mathbf{F_v} -\n", "\\int \\mathbf{n}~p~dA +\n", "\\int \\rho \\mathbf{g}~dV = \n", "\\frac{\\partial}{\\partial t} \\int \\rho \\mathbf{V} dV +\n", "\\int (\\mathbf{n}~.~\\rho \\mathbf{V}) \\mathbf{V} dA .\n", "$$\n", "\n", "In the above equation assuming that the effect of viscous forces within the flow ($F_v$) are negligible within the control volume around the cylinder, neglecting body forces and assuming a steady state flow, terms one, four and five from right in the above equation will become zero respectively. Furthermore, since the surfaces of the inflow and outflow within the control volume are far from the cylinder one can conclude the pressure would be constant and in the opposite directions on the surfaces of control volume. Application of all these assumptions to the conservation of momentum equation will result in:\n", "\n", "$$\n", "\\mathbf{R_{ext}}= \n", "\\int (\\mathbf{n}~.~\\rho \\mathbf{V}) \\mathbf{V} dA .\n", "$$\n", "\n", "This simple equation implies that the total drag force on the cylinder is equal to the change of momentum between the incoming and outgoing flow form the boundaries of the defined control volume. It should be noted that this equation is based on the assumption that the boundaries of the defined control volume are far from the cylinder body. Therefore, the effect of pressure forces on the drag force are indirectly reflected through changes in velocity. \n", "\n", "### Dimensional Analysis ###\n", "\n", "After performing the control volume analysis on this flow field, the conclusion is that the drag force on the cylinder is a function of fluid properties (i.e. density), area of interaction and the velocity evolution within the flow. Although this conclusion is important, it reflect a complex dependency including multiple variables. Performing dimensional analysis on the flow over a cylinder immersed in a flow of a constant velocity $V$ one can reduce this complex dependency in a simpler format. Starting the dimensional analysis and using the previous developed understandings it can be showed that the drag force acting on the cylinder is a function of following flow variables:\n", "\n", "$$ \\mathbf{F_D} = f~(d,\\mu,\\rho,V) ,$$\n", "\n", "where $d$ is the dimension of the cylinder, $\\mu$ and $\\rho$ are the viscosity and density of the fluid. As there are 5 variables and 3 reference dimensions (i.e., Mass, Length, Time), two non-dimensional groups (i.e. $\\Pi$ groups) are needed to fully define the flow field around a cylinder defined. Performing dimensional analysis result in tothe two following non-dimensional groups:\n", "\n", "$$\n", "\\Pi_1 = \\frac{\\rho V d}{\\mu}\n", "$$\n", "\n", "$$\n", "\\Pi_2 = \\frac{\\mathbf{F_D}}{d^2 \\rho V^2 }\n", "$$\n", "\n", "Therefore, as shown previously using control volume analysis, it can be concluded the drag force on the cylinder is a function of velocity field through an unknown function $\\phi$ as follows:\n", "\n", "$$ \\frac{\\mathbf{F_D}}{d^2 \\rho V^2 } = \\phi( \\frac{\\rho V d}{\\mu}) $$\n", "\n", "The left hand side of this equation is the non-dimensional drag force and referred to as the drag force coefficient. The right hand side of this equation is the ratio of inertial forces to the viscous forces within the flow, which is known as Reynolds number. This outcome of the dimensional analysis confirms that the drag force on the cylinder in a uniform flow is a direct function of Reynolds number. Therefore, majority of scientists investigating on this flow field look at the variation of these two variables to find the function $\\phi$.\n", "\n", "$$ C_D = \\phi(Re) $$\n", "\n", "In this equation $C_D$, the total coefficient of the drag force due to pressure and viscous forces within the flow acting on the cylinder's surface in the streamwise direction. This coefficient is the normalized drag force component with the available momentum in the undisturbed incoming flow that could be exerted on the cylinder. The value of this momentum is calculated using $\\rho$ as the reference density of the flow, which is the density of the flow. $V$ as the undisturbed streamwise flow velocity in the inlet and $d$ is the diameter of the cylinder surface. In some formulation and books the term $d^2$ is replaced with $A$ as the projected area of the cylinder. For a two-dimensional cylinder the projected area is equal to the diameter of the cylinder. This is the area where pressure and viscous forces are acting on the cylinder in the streamwise direction. It should be reminded that the pressure and viscous forces in the perpendicular direction to streamwise direction cancel each other. Hence, the forces and areas that they act on does not have any contribution to the drag force and coefficient.\n", "\n", "As shown in the above discussion the velocity field and drag force are the most important variables of interest in the case study of flow over cylinder. Although this analysis provide detail understanding about the flow field, however is limited in providing the detail of evolution of flow field variables. Furthermore, it includes multiple simplified assumptions in it. This is particularly true if one want to reach a high accuracy, let along dealing with some more complex flow field condition. This is when Computational Fluid Dynamics (CFD) comes into play. With the help of CFD, one can divide the area of interest into multiple small *control volumes*, called meshes elements, and obtain exact flow field variables. With this general and detail theoretical knowledge about the flow field in the next sections it will be discuss how one can move forward to initiate developing the CFD domain and simulations for this problem of this flow field of interest interest." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.5.1" } }, "nbformat": 4, "nbformat_minor": 0 }