ME/AE 339 Computational Fluid Dynamics K. M. Isaac Topic1_ODE 11/9/2018 ODE

Please email me ( isaac@umr.edu ) the following Information: Computational Fluid Dynamics (AE/ME 339) K. M. Isaac Course Outline 1 MAEEM Dept., UMR Please email me ( isaac@umr.edu ) the following Information: Mailing address Phone number Fax number Any other information you want me to know 11/9/2018 ODE

Password to access files: will be emailed to you Course Information Computational Fluid Dynamics (AE/ME 339) K. M. Isaac Course Outline 1 MAEEM Dept., UMR Password to access files: will be emailed to you Course Information 11/9/2018 ODE

Ordinary differential equations (ODE) Computational Fluid Dynamics (AE/ME 339) K. M. Isaac Course Outline 1 MAEEM Dept., UMR Course Outline Ordinary differential equations (ODE) Numerical techniques for solving ODEs Example: Flow in constant area pipe with heat addition and friction Partial differential equations, classification Discretization of derivatives Errors and analysis of stability Example: Unsteady heat conduction in a rod Example: Natural convection at a heated vertical plate Discretization techniques 11/9/2018 ODE

Course Outline (continued) Couette flow Computational Fluid Dynamics (AE/ME 339) K. M. Isaac Course Outline 2 MAEEM Dept., UMR Course Outline (continued) Couette flow The shock tube problem Introduction to packaged codes: Grid generation Problem setup Solution Turbulence modeling 11/9/2018 ODE

ODEs and PDEs may be discretized-approximated- Computational Fluid Dynamics (AE/ME 339) K. M. Isaac Introduction 1 MAEEM Dept., UMR ODEs and PDEs may be discretized-approximated- as a set of algebraic equations and solved Discretization methods for ODEs are well known e.g., Runge-Kutta methods for initial value problems and shooting methods for BV problems PDEs involve more than 1 independent variable e.g., x, y, z, t in Cartesian coordinates for time-dependent Problems PDEs can be discretized using finite difference Methods 11/9/2018 ODE

PDEs can also be discretized in integral form, Computational Fluid Dynamics (AE/ME 339) K. M. Isaac Introduction 2 MAEEM Dept., UMR PDEs can also be discretized in integral form, known as finite volume methods Sometimes coordinate transformation is necessary before discretization 11/9/2018 ODE

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 0a MAEEM Dept., UMR Flow with heat addition and friction Ref: Hill & Peterson, Mechanics and Thermodynamics of Propulsion, Addison-Wesley 11/9/2018 ODE

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 0b MAEEM Dept., UMR Flow with heat addition and friction Ref: Hill & Peterson, Mechanics and Thermodynamics of Propulsion, Addison-Wesley 11/9/2018 ODE

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 0c MAEEM Dept., UMR Flow with heat addition and friction Ref: Hill & Peterson, Mechanics and Thermodynamics of Propulsion, Addison-Wesley 11/9/2018 ODE

Flow with heat addition and friction Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 1 MAEEM Dept., UMR Flow with heat addition and friction Ref: Hill & Peterson, Mechanics and Thermodynamics of Propulsion, Addison-Wesley CV 11/9/2018 ODE

Flow with heat addition and friction Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 2 MAEEM Dept., UMR Flow with heat addition and friction CV Perfect gas flows from left to right in a constant area duct Heat addition and/or friction may be present Flow properties will change during the process Equations can be solved analytically when either heat addition or friction is present, but not both 11/9/2018 ODE

Flow with heat addition Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 3 MAEEM Dept., UMR Flow with heat addition Stagnation enthalpy change (Conservation of Energy/First Law) (1) Conservation of mass (2) 11/9/2018 ODE

Flow with heat addition Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 4 MAEEM Dept., UMR Flow with heat addition Momentum (3) Stagnation Enthalpy (4) 11/9/2018 ODE

Flow with heat addition Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 5 MAEEM Dept., UMR Flow with heat addition Integration yields (5) Equation of state Speed of sound 11/9/2018 ODE

Flow with heat addition Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 6 MAEEM Dept., UMR Flow with heat addition Above equations can be combined to yield the following 11/9/2018 ODE

Flow with heat addition Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 7 MAEEM Dept., UMR Flow with heat addition The above equation + the following adiabatic flow relation can be used to get stagnation temperature ratio 11/9/2018 ODE

Flow with heat addition Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 8 MAEEM Dept., UMR Flow with heat addition Note: Final Mach number depends on initial Mach number and final stagnation temperature. 11/9/2018 ODE

Flow with heat addition Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 9 MAEEM Dept., UMR Flow with heat addition Reference conditions Note that the stagnation conditions change due to heat addition For given initial conditions, Mach 1 conditions, denoted by (*) can be used for reference Thus 11/9/2018 ODE

Flow with heat addition Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 10 MAEEM Dept., UMR Flow with heat addition The above equation shows that, for given initial conditions, fluid properties are only a function of the local Mach number 11/9/2018 ODE

Flow with heat addition Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 11 MAEEM Dept., UMR Flow with heat addition 11/9/2018 ODE

Flow with heat addition Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 12 MAEEM Dept., UMR Flow with heat addition Calculation procedure, given p01, T01, M1, and q Determine T0* using T01 and M1 Determine T02/T0* using Calculate M2, p2, p02 Flow with heat addition Observe in figure, for subsonic and supersonic cases Heat addition drives M towards 1. Results in “thermal choking.” There is a loss of stagnation pressure. 11/9/2018 ODE

Flow with heat addition and friction Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 13 MAEEM Dept., UMR Flow with heat addition and friction Subscript h is for heat addition and subscript f is for friction Three processes to attain choking are shown: 1) constant entropy process (vertical direction), 2) heating process, and 3) friction process Note stagnation pressure falls for both heating and friction cases Travel in either direction is possible for isentropic and and heat addition cases Friction process is possible only in one direction 11/9/2018 ODE

Flow with heat addition and friction Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 14 MAEEM Dept., UMR Flow with heat addition and friction 11/9/2018 ODE

where c is the circumference and A is the cross-section Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 15 MAEEM Dept., UMR Flow with friction Momentum equation where c is the circumference and A is the cross-section area. cdx is the curved surface area of the tube of length dx. t0 is the wall shear stress (N/m2) 11/9/2018 ODE

The energy equation in this case is h0 = h + u2/2 = constant Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 16 MAEEM Dept., UMR Flow with friction The energy equation in this case is h0 = h + u2/2 = constant or dh + udu = 0 Shear stress correlation for fully developed pipe flow 11/9/2018 ODE

where e is the rms roughness of the pipe wall Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 17 MAEEM Dept., UMR Flow with friction where e is the rms roughness of the pipe wall cf is the skin coefficient The equations of continuity, momentum and energy can now be combined with the perfect gas equation of state to get the equations for flow with friction 11/9/2018 ODE

Note the behavior of the flow for subsonic and supersonic Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 18 MAEEM Dept., UMR Flow with friction See Hill & Peterson for detailed derivation of the following equation Note the behavior of the flow for subsonic and supersonic cases. In both cases, Mach number tends towards 1. Condition is called friction choking 11/9/2018 ODE

Integrating and applying the limit between M = M and M = 1 yields the Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 19 MAEEM Dept., UMR Flow with friction Integrating and applying the limit between M = M and M = 1 yields the following result for “length to choke,” L* 11/9/2018 ODE

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 20 MAEEM Dept., UMR Flow with friction 11/9/2018 ODE

Flow with heat addition and friction Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 21 MAEEM Dept., UMR Flow with heat addition and friction The continuity equation is the same as before. The momentum equation has the shear stress term. The energy equation has the heat addition (q) term. These equations can now be combined with the perfect gas equation to get the differential equation which does not have a closed-form solution. Solution can be obtained by numerical integration. 11/9/2018 ODE

(1) (2) (3) Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 22 MAEEM Dept., UMR (1) (2) (3) 11/9/2018 ODE

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 23 MAEEM Dept., UMR (4) (5) Momentum (6) 11/9/2018 ODE

From (1) (7) Substitute in ( 4 ) (8) Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 24 MAEEM Dept., UMR From (1) (7) Substitute in ( 4 ) (8) 11/9/2018 ODE

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 25 MAEEM Dept., UMR Combine (3) and (6) (9) 11/9/2018 ODE

(10) Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 26 MAEEM Dept., UMR (10) 11/9/2018 ODE

(11) Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 27 MAEEM Dept., UMR (11) 11/9/2018 ODE

Substitute (9) for dp/p in (10) Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 28 MAEEM Dept., UMR Substitute (9) for dp/p in (10) 2nd RHS term: 11/9/2018 ODE

(12) Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 29 MAEEM Dept., UMR (12) 11/9/2018 ODE

Substitute in (8) using (9), (10), (11) and (12) Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 30 MAEEM Dept., UMR (11) Substitute in (8) using (9), (10), (11) and (12) 11/9/2018 ODE

LHS factor: Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 31 MAEEM Dept., UMR LHS factor: 11/9/2018 ODE

or Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 32 MAEEM Dept., UMR or 11/9/2018 ODE

Flow with heat addition and friction Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 33 MAEEM Dept., UMR Flow with heat addition and friction The final form of the equation is as follows 11/9/2018 ODE

Flow with heat addition and friction Computational Fluid Dynamics (AE/ME 339) K. M. Isaac ODEs 34 MAEEM Dept., UMR Flow with heat addition and friction The above ODE can be integrated by using methods such as Runge-Kutta or using software packages such as Matlab which has routines for solving ODEs 11/9/2018 ODE

Runge-Kutta Method 4th order method: Computational Fluid Dynamics (AE/ME 339) K. M. Isaac Runge-Kutta MAEEM Dept., UMR Runge-Kutta Method 4th order method: Let the first order ODE be represented as 11/9/2018 ODE

The 4th order RK-Gill algorithm is then given by Computational Fluid Dynamics (AE/ME 339) K. M. Isaac Runge-Kutta MAEEM Dept., UMR The 4th order RK-Gill algorithm is then given by 11/9/2018 ODE

The method can be used for several simultaneous first-order equations Computational Fluid Dynamics (AE/ME 339) K. M. Isaac Runge-Kutta MAEEM Dept., UMR The method can be used for several simultaneous first-order equations as well as a single higher-order equation. See Applied Numerical Methods, Carnahan, Luther and Wilkes, or some other numerical methods book for details. 11/9/2018 ODE

Program Completed University of Missouri-Rolla Copyright 2002 Curators of University of Missouri 11/9/2018 ODE