AEME 339 Computational Fluid Dynamics CFD K M

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AE/ME 339 Computational Fluid Dynamics (CFD) K. M. Isaac 12/19/2021 topic 20_Burger's_Equation 1

AE/ME 339 Computational Fluid Dynamics (CFD) K. M. Isaac 12/19/2021 topic 20_Burger's_Equation 1

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Burger’s Equation

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Burger’s Equation (Ref. Tannehill, Anderson and Pletcher, Computational Fluid Mechanics and Heat Transfer, 1997) It is a model equation used to test finite difference techniques Inviscid and viscous forms can be used Has a time dependent term, non-linear term similar to the convection term, and a viscous dissipation term 12/19/2021 topic 20_Burger's_Equation 2

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Equation (1)

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Equation (1) is parabolic when the viscous dissipation term is included. With only the terms on the LHS, the equation is hyperbolic. Equation 2 can be thought of as the non-linear wave equation, where each point on the wave can propagate with a different speed. 12/19/2021 topic 20_Burger's_Equation 3

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. ,

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. , UMR 4

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. ,

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. , UMR 5

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Tranveling Discontinuity

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Tranveling Discontinuity Problem for Burger’s Equation 12/19/2021 topic 20_Burger's_Equation 6

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. ,

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. , UMR 7

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. ,

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. , UMR 8

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Figure 2.

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Figure 2. Characteristics for Centered Expansion 12/19/2021 topic 20_Burger's_Equation 9

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. ,

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. , UMR 10

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR The initial

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR The initial distribution of u results in a centered expansion where the width of the expansion grows linearly with time. The above solutions can now be used to evaluate finite difference algorithms. 12/19/2021 topic 20_Burger's_Equation 11

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. ,

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. , UMR 12

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. ,

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. , UMR 13

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. ,

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. , UMR 14

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. ,

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. , UMR 15

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR The method

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR The method is 2 nd order accurate (e = O[(Dt)2, (Dx)2]) and unconditionally stable for all time steps. A tridiagonal system must be solved for each time step. 12/19/2021 topic 20_Burger's_Equation 16

Beam and Warming method Summary 12/19/2021 topic 20_Burger's_Equation 17

Beam and Warming method Summary 12/19/2021 topic 20_Burger's_Equation 17

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR The Beam-Warming

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR The Beam-Warming method can now be applied to the inviscid Burger’s equation 12/19/2021 topic 20_Burger's_Equation 18

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. ,

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. , UMR 19

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. ,

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. , UMR 20

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. ,

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. , UMR 21

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. ,

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. , UMR 22

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. ,

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. , UMR 23

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Some Examples

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Some Examples 12/19/2021 topic 20_Burger's_Equation 24

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Solution of

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Solution of Burger’s equation Use Mac. Cormack’s method to solve inviscid Burger’s equation using a mesh with 51 points in the x-direction. Solve the equation for a right propagating discontinuity with u = 1 at the first 11 nodes and u = 0 at the rest of the nodes. Use Courant number = 1. Solution 12/19/2021 topic 20_Burger's_Equation 25

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. ,

Computational Fluid Dynamics (AE/ME 339) 12/19/2021 topic 20_Burger's_Equation K. M. Isaac MAEEM Dept. , UMR 26

Program Completed University of Missouri-Rolla Copyright 2002 Curators of University of Missouri 12/19/2021 topic

Program Completed University of Missouri-Rolla Copyright 2002 Curators of University of Missouri 12/19/2021 topic 20_Burger's_Equation 27

AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
382021 AEME 339 Computational Fluid Dynamics CFD K382021 AEME 339 Computational Fluid Dynamics CFD K
AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
AEME 339 Computational Fluid Dynamics CFD K MAEME 339 Computational Fluid Dynamics CFD K M
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