{ "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 engineering decision about the geometry of the CFD domain, choice of boundary conditions and numerical models for the CFD simulation. Furthermore, user will have a general big picture on the expected outcome from the CFD 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 a problem before developing a CFD simulation for it can be thought as turning a flash light on before taking any step in a completely dark room. This simple but wise action would avoid any potential accident and failure, while entering and discovering the detail of the room. Hence, let's review the fundamental physics and theory behind the problem of \"2D Laminar Flow in a Rectangular Pipe.\" as the first step for development and validation of CFD simulation for this problem of interest:**\n", "\n", "In order to discuss the theory of behind the flow field evolution as it moves inside a 2D rectangular pipe consider the general form of the Navier-Stokes equation as follows:\n", "\n", "$$\n", "\\frac{D\\mathbf{V}}{D t} = \n", "-\\frac{1}{\\rho} \\nabla{p} +\n", "\\mathbf{g} +\n", "\\nu \\nabla^2 \\mathbf{V}.\n", "$$\n", "\n", "This equation indicates that the material derivative of the velocity field (i.e. the summation of external forces per unit mass in a Lagrangian reference frame) is balanced with the summation of pressure, body and viscous forces per unit mass within the flow field (i.e. forces in Eulerian reference frame). For the case of laminar flow in a pipe, in most of the engineering applications the incoming flow into the pipe has an almost constant and uniform velocity. However, it is probable that the incoming flow velocity has some temporal fluctuations. In all cases these fluctuations should be estimated to declare the degree of steadiness of the flow. In generic cases the assumption of steady/uniform flow or flow with minor temporal fluctuations is a reasonable assumption. Furthermore, considering a horizontal pipe flow, the effect of body forces can become negligible. Applying these two assumptions the Navier-Stokes equation will be reduced to:\n", "\n", "$$\n", "0 = \n", "-\\frac{1}{\\rho} \\nabla{p} +\n", "\\mathbf{g} +\n", "\\nu \\nabla^2 \\mathbf{V}.\n", "$$\n", "\n", "Once the flow enters the pipe, due to the no slip boundary condition at the pipe's walls fluid velocity will instantly become zero at this boundary. This phenomena results into a strong velocity gradient across the width of the pipe and effects the velocity profile right at the entrance region of the pipe. This interaction between fluid element and solid walls forms a thin layer at the solid-fluid boundary called `Boundary Layer`. Inside the boundary layer the flow field evolution is complex and hard to theoretically model. However, moving further downstream the two boundary layers, formed at pipe's top and bottom walls, merge with each other and the velocity gradient becomes smoother. After a specific length the velocity profile gets a parabolic shape and the flow becomes `fully developed`. In this state the shape of the velocity profile becomes fix along the length of the pipe. This short length is referred to as the `entrance region`. The entrance length for laminar flows is experimentally related to the Reynolds number and hydraulic diameter of non-circular (i.e. rectangular) pipe as follows:\n", "\n", "$$ L_e = 0.05~.~Re_w~.~D_h~,$$\n", "\n", "where $ ~Re_w~ $ is the Reynolds number of the pipe based on the pipe's width. $ D_h $ is the hydraulic diameter of the pipe. Considering unit depth $D_h$ is defined as $ D_h=\\frac{4A}{P} $, where $ A $ and $ P $ are area and perimeter of the pipe's rectangular inlet respectively.\n", "\n", "For a fully developed flow, it can be assumed that the pressure field variation in any directions other than the streamwise direction (i.e. x-direction in this problem) will become negligible. Furthermore, the change in the streamwise velocity is only significant across the pipe's cross section (i.e. y-direction) due to the no slip boundary conditions at the bottom and top walls of the pipe. Applying these assumptions the Navier-Stokes equation can be furthermore reduced to:\n", "\n", "$$ \\frac{1}{\\rho} \\frac{dp}{dx} = \\nu \\frac{d^2 u}{dy^2}. $$\n", "\n", "In order to solve the above equation, to obtain the general velocity profile and pressure drop across the pipe we consider:\n", "\n", "$$ \\frac{dp}{dx} = -K'. $$\n", "\n", "$$ \\frac{\\nu}{r} \\frac{d}{dr} (r \\frac{du}{dr}) = -K'. $$\n", "\n", "In the above equations $K'$ is a constant and the negative sign implies the pressure drop across the pipe. Integration of these equation will result into the general velocity profiles with two constant values. Applying two following boundary conditions in the flow field would lead into a determined system equations and gives values for C1 and C2.\n", "\n", "$$ u=0,~~at~~r= \\pm \\frac{D}{2} ~~ (no~slip) $$\n", "$$ \\frac{du}{dr}=0,~~at~~r= 0 ~~ (symmetry) $$\n", "\n", "The general form of velocity profile will be as follows:\n", "\n", "$$ u = \\frac{K' D^2}{16 \\mu} [~1 - (\\frac{2r}{D})^2~], $$\n", "\n", "where K' is the only undefined constant. Considering the average velocity across the cross section of the duct one can write:\n", "\n", "$$ \n", "\\bar{V} = \\frac{1}{A} \\int u dA = \n", "\\frac{1}{\\frac{\\pi}{4} D^2} \\int_{0}^{\\frac{D}{2}} u~2\\pi~r dr = \n", "\\frac{K' D^2}{32 \\mu}.\n", "$$\n", "\n", "As a result of this algebraic manipulation the general form of velocity profile and pressure gradient across the pipe will be as follows:\n", "\n", "$$\n", "\\frac{u}{\\bar{V}} = 2~[1 - (\\frac{2r}{D})^2].\n", "$$\n", "\n", "$$\n", "K' = -\\frac{dp}{dx} = \\frac{32 \\mu \\bar{V}}{D^2}.\n", "$$\n", "\n", "It should be noted that this derivation shows that the velocity profile evolution is complex within the entrance length of the pipe and can not be modeled theoretically. The only approach to study flow field evolution in this region and determine this length is through experimental and numerical analyses. However, once the flow reaches the fully developed state, it will get a fixed parabolic shape with a defined equation that satisfies all physical boundary conditions within the flow field. With this review one has developed an in-depth physical and theoretical understanding of the laminar flow inside a 2D rectangular pipe and approximate the expected results from the CFD simulations. Now one can move forward to initiate developing the CFD domain and simulations for this problem of interest.\n", "\n", "> For more details on the physics, theory and equation derivation please see chapter 8 of \"A Physical Introduction to Fluid Mechanics by Alexander J. Smits\" 2nd edition. [Download Book Here!](http://www.efluids.com/efluids/books/efluids_books.htm)\n" ] }, { "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 }