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Unformatted text preview: Lecture Notes in Turbulence Steve Berg 2nd June 2004 Contents 1 Conservation Equations 2 1.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Conservation of Moment . . . . . . . . . . . . . . . . . . . . . . 4 2 Turbulent Theory 9 2.1 What is Turbulence ? . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Turbulence Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Correlation in Turbulence . . . . . . . . . . . . . . . . . . . . . 16 2.4 Turbulence Equations . . . . . . . . . . . . . . . . . . . . . . . 23 3 Turbulence Models 26 3.1 Zero Equation Model . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 One Equation Model . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 Two Equation Model . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4 Reynolds Stress Model (RSM) . . . . . . . . . . . . . . . . . . . 31 4 Summary 36 A Turbulence Pictures 38 B Examination Questions 41 1 Chapter 1 Conservation Equations Transport equations are the basics for all CFD simulations. Before we discuss turbulence models we will derive these equations for laminar flow. Turbulence will be added in chapter 2.4. This chapter is based on lectures notes from Mathiesen (2000) and as an alternative you can also read chapter 6 in Munson et al. (2002). 1.1 Conservation of Mass This equation is also known as equation of continuity. It is derived from the concept of control volumes, ie the mass entering a volume also has to leave it. Before we start discussing the details about mass flow, let us first define the basic equation for mass flow in one direction: m x = ρu x A [ kg s ] This equation is used for all directions for calculation of mass flow. In our case we will only derive the equation of mass for two direction ( x and y ). It is easier to understand the procedures for the derivation of the equations when there are a few variables as possible. The third flow direction part ( zdirection) can easily be added to the two direction equation. 2 6 6 ( ρu y )  y Δ x Δ z y + Δ y y ( ρu y )  y +Δ y Δ x Δ z ( ρu x )  x Δ y Δ z ( ρu x )  x +Δ x Δ y Δ z x + Δ x x Δ x Δ y Δ z = Δ V Incoming mass per time unit Outgoing mass per time unit = Accumulations of mass per time unit Accumulation of mass per time unit: ∂ρ ∂t Δ V = ∂ ∂t ( ρu x )Δ x Δ y Δ z Mass flow in and out of the control volume is then given by: ( ρu x )  x Δ y Δ z  {z } inflow x +( ρu y )  y Δ x Δ z  {z } inflow y ( ρu x )  x +Δ x Δ y Δ z  {z } outflow x ( ρu y )  y +Δ y Δ x Δ z  {z } outflow y We then put the inflow and outflow for x directed flow in the same equation, and do the same for the y directed flow: { ( ρu x )  x ( ρu x )  x +Δ x } Δ y Δ z { ( ρu y )  y ( ρu y )  y +Δ y } Δ x Δ z We want to express this variables with the mean value:{ ( ρu x )  x +Δ x ( ρu x )  x } Δ x Δ y Δ x Δ z{ ( ρu y )  y +Δ y ( ρu y )  y } Δ y Δ y Δ x Δ z The only thing we have done is multiplying with respectively Δ...
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This note was uploaded on 09/16/2011 for the course ME 563 taught by Professor Staff during the Spring '11 term at Auburn University.
 Spring '11
 Staff

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