Numerical Investigation of Turbulent Flows Using k-epsilon

Numerical Investigation of Turbulent Flows Using k-epsilon
In The Name of God Numerical Investigation of Turbulent Flows Using k-epsilon By Reza Barati Under Guidance of Prof. G. Heidarinejad Large Eddy Simulation Tarbiat Modares University July,

Overview The computer program k-epsilon procedure Wall function
6/11/ :16 PM Overview The computer program k-epsilon procedure Wall function Lid Driven Cavity Backward Facing Step Numerical Investigation of Turbulent Flows k-epsilon © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

The computer program Numerical Investigation of Turbulent Flows k-epsilon 3

Brief description of the Computer Program
6/11/ :16 PM Brief description of the Computer Program A program for simulation of two-dimensional (2D) incompressible Navier-stokes (N-S) equations using Semi-Implicit Method for Pressure-Linked Equations (SIMPLE method) through a finite difference scheme with staggered grid was developed. The large eddy simulation (LES) and k-epsilon were used as turbulent models that were approximated in the finite difference scheme. The system of linear equations for pressure corrections can be solve by the following procedures in the computer program: (1) The Gauss-Seidel method. (2) Inverse of the coefficient matrix. (3) Alternating Direction Implicit (ADI) scheme. The program can simulate the laminar and turbulent flow for the following geometries: (1) Lid Driven Cavity (2) Backward Facing Step Numerical Investigation of Turbulent Flows Using k-epsilon 4 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

k-epsilon procedure Numerical Investigation of Turbulent Flows Using k-epsilon 5

6/11/ :16 PM k-epsilon procedure The governing equations for laminar, transitional, and turbulent flows that are the Navier-Stokes equations for the velocity components ui (i = 1, 2, 3) and the pressure p, given here in non-dimensional conservation form for an incompressible flow, can be written as: Complemented with the incompressibility constraint, Numerical Investigation of Turbulent Flows Using k-epsilon 6 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

k-epsilon procedure (2)
6/11/ :16 PM k-epsilon procedure (2) Any flow can be divided into steady and fluctuating parts. For our purposes, we will define turbulence as the fluctuating part of that flow. The underlying average velocities over which these turbulent fluctuations exist will be called the mean flow. The k-epsilon model will not resolve the turbulent fluctuations themselves but rather the turbulent kinetic energy per unit mass, the amount of kinetic energy per unit mass present in the turbulent fluctuations. This two-dimensional array will be defined at the cell centers and designated by a K. A variable, ε, will also be calculated over the mesh to represent the rate of dissipation of turbulent kinetic energy in different subregions in the fluid. Numerical Investigation of Turbulent Flows Using k-epsilon 7 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

6/11/ :16 PM k-epsilon procedure (3) Each velocity and pressure in the momentum equation is made up of a mean value and a fluctuating value: Substituting these definitions into the momentum equation Or Numerical Investigation of Turbulent Flows Using k-epsilon 8 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

6/11/ :16 PM k-epsilon procedure (4) After time averaging and simplifying, Or, Rij is called the Reynolds stress tensor. This second-order tensor represents the effect of turbulence on the mean flow. Computationally, this tensor is approximated by calculating a turbulent viscosity that is added to the molecular viscosity to represent the total viscous forces on the fluid. Numerical Investigation of Turbulent Flows Using k-epsilon 9 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

6/11/ :16 PM k-epsilon procedure (5) The Boussinesq approximation is used for the Reynolds stress tensor as We calculate νt by using the variables K and ε as Where Numerical Investigation of Turbulent Flows Using k-epsilon 10 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

6/11/ :16 PM k-epsilon procedure (6) Having related K and ε to the momentum equation, we must also derive transport equations for these two quantities. K is calculated by relating it to the Reynolds stress tensor, and using the classical equation for kinetic energy Or Numerical Investigation of Turbulent Flows Using k-epsilon 11 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

6/11/ :16 PM k-epsilon procedure (7) Using this equation for K in terms of the Reynolds tensor, we calculate K by first deriving an equation for Rij as In this equation the term denoted by 1 represents the time rate of change of K, 2 represents advection, 3 represents diffusion of turbulent energy, 4 represents the generation of turbulence by shear forces (forces similar to friction that are caused by flows at different velocities rubbing against each other), and 5 (ε) represents the dissipation of turbulence. Where Numerical Investigation of Turbulent Flows Using k-epsilon 12 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

6/11/ :16 PM k-epsilon procedure (8) A transport equation for ε is “derived” by modeling an equation after the transport equation for K. The ε transport equation is In the transport equation for epsilon, σε, Cε1, Cε2 are constants which have been determined as a result of experimentation. The k and ε transport equations and the turbulence viscosity equation, make up the K – ε turbulence model. Numerical Investigation of Turbulent Flows Using k-epsilon 13 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

6/11/ :16 PM k-epsilon procedure (9) The turbulence transport equations are calculated at every point on the mesh. For this calculation local arrays of variables are employed to represent each term in the equations. The turbulence transport equations are then written as Where Numerical Investigation of Turbulent Flows Using k-epsilon 14 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

6/11/ :16 PM k-epsilon procedure (10) k and ε are calculated using the finite-difference versions of the transport equations In finite difference form of the transport equation for k is Numerical Investigation of Turbulent Flows Using k-epsilon 15 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

6/11/ :16 PM k-epsilon procedure (11) Or And finally Numerical Investigation of Turbulent Flows Using k-epsilon 16 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

6/11/ :16 PM k-epsilon procedure (12) In finite difference form of the transport equation for ε is Numerical Investigation of Turbulent Flows Using k-epsilon 17 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Wall function Numerical Investigation of Turbulent Flows Using k-epsilon 18

6/11/ :16 PM Wall function The classical wall models were developed based on the analysis of boundary layer flows. In the motion of a fluid above a wall surface, the influence of viscosity is mainly confined to a boundary layer close to the wall surface. Within the wall boundary layer, the velocity changes rap-idly from zero at the wall (no-slip condition) to the free stream velocity. Experimental studies of a turbulent wall boundary layer suggest that it may be divided into two regions (or layers): the inner (wall) layer and an outer layer. Numerical Investigation of Turbulent Flows Using k-epsilon 19 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

6/11/ :16 PM Wall function (2) The velocity distribution in the wall layer can be analyzed by using dimensional analysis, which led to the law of the wall. The law of the wall states that u+ is a function of y+ only. As determined by the wall distance, a corresponding law for the viscous sublayer (region I of the inner layer), the logarithmic buffer layer (region II of the inner layer), and the logarithmic outer layer (region III of the inner layer) is assumed: Where a2, a3, b2, and b3 are empirical constants, which may be given as a2 = 5.0, a3 = 2.5, b2 = 3.05, and b3 = 5.0–5.2. Numerical Investigation of Turbulent Flows Using k-epsilon 20 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Lid Driven Cavity Numerical Investigation of Turbulent Flows Using k-epsilon 21

6/11/ :16 PM Lid Driven Cavity The fluid flow in a rectangular container driven which moves tangentially to itself and parallel to one of the side walls is called the lid-driven cavity. Because of its simple geometry and boundary conditions, the lid-driven-cavity flow is a fundamental model for the vortex dynamics in closed systems. It is notable that the ﬂow in a driven cavity is neither two-dimensional nor steady, most probably, even at Re = 1000, physically. Numerical Investigation of Turbulent Flows Using k-epsilon 22 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Streamlines RE =100 (Laminar) RE =400 (Laminar)
6/11/ :16 PM Streamlines RE =100 (Laminar) RE =400 (Laminar) Numerical Investigation of Turbulent Flows Using k-epsilon 23 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Streamlines (2) RE =1000 (Laminar) RE =1000 (LES) RE =1000 (k-epsilon)
6/11/ :16 PM Streamlines (2) RE =1000 (Laminar) RE =1000 (LES) RE =1000 (k-epsilon) Numerical Investigation of Turbulent Flows Using k-epsilon 24 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Streamlines (3) RE =3200 (LES) RE =5000 (LES) RE =10000 (LES)
6/11/ :16 PM Streamlines (3) RE =3200 (LES) RE =5000 (LES) RE =10000 (LES) RE =3200 (k-epsilon) RE =5000 (k-epsilon) RE =10000 (k-epsilon) Numerical Investigation of Turbulent Flows Using k-epsilon 25 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Pressure contours RE =100 (Laminar) RE =400 (Laminar)
6/11/ :16 PM Pressure contours RE =100 (Laminar) RE =400 (Laminar) Numerical Investigation of Turbulent Flows Using k-epsilon 26 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Pressure contours (2) RE =1000 (Laminar) RE =1000 (LES)
6/11/ :16 PM Pressure contours (2) RE =1000 (Laminar) RE =1000 (LES) RE =1000 (k-epsilon) Numerical Investigation of Turbulent Flows Using k-epsilon 27 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Pressure contours (3) RE =3200 (LES) RE =5000 (LES) RE =10000 (LES)
6/11/ :16 PM Pressure contours (3) RE =3200 (LES) RE =5000 (LES) RE =10000 (LES) RE =3200 (k-epsilon) RE =5000 (k-epsilon) RE =10000 (k-epsilon) Numerical Investigation of Turbulent Flows Using k-epsilon 28 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Vorticity contours RE =100 (Laminar) RE =400 (Laminar)
6/11/ :16 PM Vorticity contours RE =100 (Laminar) RE =400 (Laminar) Numerical Investigation of Turbulent Flows Using k-epsilon 29 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Vorticity contours (2) RE =1000 (Laminar) RE =1000 (LES)
6/11/ :16 PM Vorticity contours (2) RE =1000 (Laminar) RE =1000 (LES) RE =1000 (k-epsilon) Numerical Investigation of Turbulent Flows Using k-epsilon 30 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Vorticity contours (3) RE =3200 (LES) RE =5000 (LES) RE =10000 (LES)
6/11/ :16 PM Vorticity contours (3) RE =3200 (LES) RE =5000 (LES) RE =10000 (LES) RE =3200 (k-epsilon) RE =5000 (k-epsilon) RE =10000 (k-epsilon) Numerical Investigation of Turbulent Flows Using k-epsilon 31 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Contours of the kinetic energy (k)
6/11/ :16 PM Contours of the kinetic energy (k) RE = 1000 RE = 3200 RE = 10000 RE = 5000 Numerical Investigation of Turbulent Flows Using k-epsilon 32 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Contours of the dissipation (epsilon)
6/11/ :16 PM Contours of the dissipation (epsilon) RE = 1000 RE = 3200 RE = 10000 RE = 5000 Numerical Investigation of Turbulent Flows Using k-epsilon 33 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Convergence RE = 3200 RE = 1000 RE = 5000 RE = 10000
6/11/ :16 PM Convergence RE = 3200 RE = 1000 RE = 5000 RE = 10000 Numerical Investigation of Turbulent Flows Using k-epsilon 34 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

CPU Time Numerical Investigation of Turbulent Flows Using k-epsilon 35
6/11/ :16 PM CPU Time Numerical Investigation of Turbulent Flows Using k-epsilon 35 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Comparison of the Results
6/11/ :16 PM Comparison of the Results RE =100 RE =400 Numerical Investigation of Turbulent Flows Using k-epsilon 36 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Comparison of the Results (2)
6/11/ :16 PM Comparison of the Results (2) RE =1000 Numerical Investigation of Turbulent Flows Using k-epsilon 37 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Backward Facing Step Numerical Investigation of Turbulent Flows Using k-epsilon 41

6/11/ :16 PM Backward Facing Step The flow over a backward-facing step in a channel is a good test case for assessment of the numerical method. A dissipative scheme can not predict the correct reattachment length of the recirculation zone downstream of the step. The regime of fluid flow on backward facing step, for Re<400, 400<Re<3400 and Re>3400 is laminar, transient and turbulent, respectively. For Re<1000 there is an eddy in the downstream, but as the Reynolds number is increased, secondary or roof eddy is created in the upstream of the step and, therefore, the reattachment length is decreased. Numerical Investigation of Turbulent Flows Using k-epsilon 42 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

6/11/ :16 PM Streamlines RE =100 RE =200 Numerical Investigation of Turbulent Flows Using k-epsilon 43 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Streamlines (2) RE =400 LES RE =400 k-epsilon
6/11/ :16 PM Streamlines (2) RE =400 LES RE =400 k-epsilon Numerical Investigation of Turbulent Flows Using k-epsilon 44 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Vorticity contours RE =400 LES RE =400 k-epsilon
6/11/ :16 PM Vorticity contours RE =400 LES RE =400 k-epsilon Numerical Investigation of Turbulent Flows Using k-epsilon 48 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Vorticity contours (2) RE =1000 LES RE =1000 k-epsilon
6/11/ :16 PM Vorticity contours (2) RE =1000 LES RE =1000 k-epsilon Numerical Investigation of Turbulent Flows Using k-epsilon 49 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Contours of the kinetic energy (k)
6/11/ :16 PM Contours of the kinetic energy (k) RE =400 RE =600 Numerical Investigation of Turbulent Flows Using k-epsilon 50 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Contours of the kinetic energy (k) (2)
6/11/ :16 PM Contours of the kinetic energy (k) (2) RE =800 RE =1000 Numerical Investigation of Turbulent Flows Using k-epsilon 51 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Contours of the dissipation (epsilon)
6/11/ :16 PM Contours of the dissipation (epsilon) RE =400 RE =600 Numerical Investigation of Turbulent Flows Using k-epsilon 52 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Contours of the dissipation (epsilon) (2)
6/11/ :16 PM Contours of the dissipation (epsilon) (2) RE =800 RE =1000 Numerical Investigation of Turbulent Flows Using k-epsilon 53 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Convergence RE =400 RE =600 RE =1000 RE =800
6/11/ :16 PM Convergence RE =400 RE =600 RE =1000 RE =800 Numerical Investigation of Turbulent Flows Using k-epsilon 54 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

CPU Time Numerical Investigation of Turbulent Flows Using k-epsilon 55
6/11/ :16 PM CPU Time Numerical Investigation of Turbulent Flows Using k-epsilon 55 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Comparison with Numerical (Erturk ) & Experimental (Armaly) results
6/11/ :16 PM Comparison with Numerical (Erturk ) & Experimental (Armaly) results Numerical Investigation of Turbulent Flows Using k-epsilon 56 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Difference with Erturk's results
6/11/ :16 PM Comparison with Numerical (Erturk ) & Experimental (Armaly) results (2) It is notable that several works reporting that the k-epsilon model underestimates Xr by 10% to 25%. Re 75% Erturk's results k-epsilon 90% Erturk's results Erturk's results 400 6.11 6.88 7.34 8.15 600 7.82 9.18 9.38 10.42 800 8.97 10.63 10.77 11.96 1000 9.94 11.88 11.93 13.25 Re Difference with Erturk's results LES k-epsilon 400 0.029 1.279 600 0.089 1.249 800 0.090 1.340 1000 0.129 1.378 Numerical Investigation of Turbulent Flows Using k-epsilon 56 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

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Numerical Investigation of Turbulent Flows Using k-epsilon

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Numerical Investigation of Turbulent Flows Using k-epsilon

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