Partial Differential Equations of Fluid Dynamics

Transcription

1 Partial Differential Equations of Fluid Dynamics Ville Vuorinen,D.Sc.(Tech.) 1 1 Department of Energy Technology, Internal Combustion Engine Research Group Department of Energy Technology

2 Outline Introduction Modeling Transport Phenomena Using PDE s Discretization of Derivatives Numerical Solution of PDE s Matlab and PDE s 2/27

3 Role of CFD in Modern Engine Computations 1 Turbulent Intake, Exhaust and Incylinder Flow Using LES (Courtesy of Sandia) Matlab and PDE s 3/27

4 Role of CFD in Modern Engine Computations 2 Fuel Injection and Turbulence Using LES (Courtesy of Sandia) Matlab and PDE s 4/27

5 Role of CFD in Modern Engine Computations 3 Cross-Section of Turbulence in Exhaust Manifold Using LES Matlab and PDE s 5/27

6 Partial Differential Equations Can Be Used to Model and Simulate 1D, 2D and 3D Flows Diffusion, convection and radiation are basic trasport mechanisms of mass, momentum and energy. Matlab and PDE s 6/27

7 Partial Differential Equations Can Be Used to Model and Simulate 1D, 2D and 3D Flows Diffusion, convection and radiation are basic trasport mechanisms of mass, momentum and energy. Diffusion, convection and radiation are basic trasport mechanisms of mass, momentum and energy. Matlab and PDE s 6/27

8 Partial Differential Equations Can Be Used to Model and Simulate 1D, 2D and 3D Flows Diffusion, convection and radiation are basic trasport mechanisms of mass, momentum and energy. Diffusion, convection and radiation are basic trasport mechanisms of mass, momentum and energy. Example 1: heat conduction (diffusion) via walls of a hot engine cylinder. Matlab and PDE s 6/27

9 Partial Differential Equations Can Be Used to Model and Simulate 1D, 2D and 3D Flows Diffusion, convection and radiation are basic trasport mechanisms of mass, momentum and energy. Diffusion, convection and radiation are basic trasport mechanisms of mass, momentum and energy. Example 1: heat conduction (diffusion) via walls of a hot engine cylinder. Example 2: transport of concentration of a radioactive cloud in given wind conditions. Matlab and PDE s 6/27

10 Partial Differential Equations Can Be Used to Model and Simulate 1D, 2D and 3D Flows Diffusion, convection and radiation are basic trasport mechanisms of mass, momentum and energy. Diffusion, convection and radiation are basic trasport mechanisms of mass, momentum and energy. Example 1: heat conduction (diffusion) via walls of a hot engine cylinder. Example 2: transport of concentration of a radioactive cloud in given wind conditions. Example 3: convection of hot exhaust gases out of an engine. Matlab and PDE s 6/27

11 Partial Differential Equations Can Be Used to Model and Simulate 1D, 2D and 3D Flows Diffusion, convection and radiation are basic trasport mechanisms of mass, momentum and energy. Diffusion, convection and radiation are basic trasport mechanisms of mass, momentum and energy. Example 1: heat conduction (diffusion) via walls of a hot engine cylinder. Example 2: transport of concentration of a radioactive cloud in given wind conditions. Example 3: convection of hot exhaust gases out of an engine. The phenomena above can all be modeled with partial differential equations. Matlab and PDE s 6/27

12 Partial Differential Equations Can Be Used to Model and Simulate 1D, 2D and 3D Flows Diffusion, convection and radiation are basic trasport mechanisms of mass, momentum and energy. Diffusion, convection and radiation are basic trasport mechanisms of mass, momentum and energy. Example 1: heat conduction (diffusion) via walls of a hot engine cylinder. Example 2: transport of concentration of a radioactive cloud in given wind conditions. Example 3: convection of hot exhaust gases out of an engine. The phenomena above can all be modeled with partial differential equations. This lecture gives an overview of linear PDE s and the next lecture on the (non-linear) Navier-Stokes equations. Matlab and PDE s 6/27

13 Outline Introduction Modeling Transport Phenomena Using PDE s Discretization of Derivatives Numerical Solution of PDE s Matlab and PDE s 7/27

14 The General Form of Any Transport Phenomenon Also the Navier-Stokes Equation is of This Form The Convection-Diffusion Equation for the Quantity φ temporal derivative convection term diffusion term ρφ + (uρφ) (ρν φ) = t source term S φ (φ) φ convects and diffuses in a fluid (e.g. air,water...). φ could represent e.g. concentration of soot particles in cigarette smoke or water temperature in ocean currents etc. ρ is the density of the fluid and u = u(x,y,z,t) the velocity field, and ν the diffusivity (e.g. heat diffusivity) of φ. Matlab and PDE s 8/27

15 In 1D the Convection-Diffusion Equation for Temperature T Can Be Written as Follows 1D Form temporal derivative T + t convection term x (ut) diffusion term x (ν x T) = 0 Assumption Fluid is incompressible i.e. ρ= constant (e.g. water or low velocity <100m/s air). Matlab and PDE s 9/27

16 When u = 0 One Obtains the Heat Diffusion Equation 1D Heat Equation temporal derivative T t = heat diffusion term x (ν x T) Matlab and PDE s 10/27

17 When ν =Constant the Heat Diffusion Equation Further Simplifies 1D Heat Equation temporal derivative T t = ν heat diffusion term x ( x T) Matlab and PDE s 11/27

18 When ν = 0 One Obtains the Convection Equation 1D Convection Equation temporal derivative T + t convection term (ut) = 0. x Assumption Fluid is incompressible i.e. ρ= constant (e.g. water or low velocity <100m/s air). Matlab and PDE s 12/27

19 Again, When u =Constant the Convection Equation Further Simplifies 1D Convection Equation temporal derivative T + t convection term u (T) = 0. x Assumption Fluid is incompressible i.e. ρ= constant (e.g. water or low velocity <100m/s air). Matlab and PDE s 13/27

20 Outline Introduction Modeling Transport Phenomena Using PDE s Discretization of Derivatives Numerical Solution of PDE s Matlab and PDE s 14/27

21 How to Solve/Simulate the Basic PDE s Using Computers? The continuous equations need to be discretized. Matlab and PDE s 15/27

22 How to Solve/Simulate the Basic PDE s Using Computers? The continuous equations need to be discretized. This means that the derivatives have to be expressed using Taylor series which allows numerical evaluation of the derivative terms. Matlab and PDE s 15/27

23 How to Solve/Simulate the Basic PDE s Using Computers? The continuous equations need to be discretized. This means that the derivatives have to be expressed using Taylor series which allows numerical evaluation of the derivative terms. Space is divided into finite size elements by replacing dx x. Matlab and PDE s 15/27

24 How to Solve/Simulate the Basic PDE s Using Computers? The continuous equations need to be discretized. This means that the derivatives have to be expressed using Taylor series which allows numerical evaluation of the derivative terms. Space is divided into finite size elements by replacing dx x. Time is divided into finite size time steps by replacing dt t. Matlab and PDE s 15/27

25 How to Solve/Simulate the Basic PDE s Using Computers? The continuous equations need to be discretized. This means that the derivatives have to be expressed using Taylor series which allows numerical evaluation of the derivative terms. Space is divided into finite size elements by replacing dx x. Time is divided into finite size time steps by replacing dt t. The basic idea: timesteps are numbered from 0, 1, 2,..., n 1, n, n + 1,..., N corresponding to physical times total simulation period(s) 0, t, 2 t,..., N t, where N =. t Matlab and PDE s 15/27

26 How to Solve/Simulate the Basic PDE s Using Computers? The continuous equations need to be discretized. This means that the derivatives have to be expressed using Taylor series which allows numerical evaluation of the derivative terms. Space is divided into finite size elements by replacing dx x. Time is divided into finite size time steps by replacing dt t. The basic idea: timesteps are numbered from 0, 1, 2,..., n 1, n, n + 1,..., N corresponding to physical times total simulation period(s) 0, t, 2 t,..., N t, where N =. t At n th timestep we want to know the solution at the (n + 1) th timestep using solution T n. Matlab and PDE s 15/27

27 How to Solve/Simulate the Basic PDE s Using Computers? (cont.) Here T n = T n (k x) = Tn k, where k = 0, 1, 2,..., L/ x i.e. the solution at timestep is defined in discrete space points 0, x, 2 x,... Matlab and PDE s 16/27

28 How to Solve/Simulate the Basic PDE s Using Computers? (cont.) Here T n = T n (k x) = Tn k, where k = 0, 1, 2,..., L/ x i.e. the solution at timestep is defined in discrete space points 0, x, 2 x,... Simulation of e.g. heat or convection equation means that for every single discrete point (or cell ) we want to find a general update formula in order to advance the solution from timestep n to time n + 1. Matlab and PDE s 16/27

29 Discretization of Time A Simple First Order Taylor Series Approximation Gives temporal derivative T t numerical approximation T n+1 T n +O( t). t Above, = O( t) is the time discretization error. Matlab and PDE s 17/27

30 Discretization of Time A Simple First Order Taylor Series Approximation Gives temporal derivative T t numerical approximation T n+1 T n +O( t). t Above, = O( t) is the time discretization error. O( t) 0 when t 0. Matlab and PDE s 17/27

31 Discretization of First Derivative in Space A Simple Second Order Central Difference Taylor Series Approximation Gives spatial derivative T x numerical approximation T k+1 T k 1 2 x +O( x 2 ). Above, = O( x 2 ) is the space discretization error. Matlab and PDE s 18/27

32 Discretization of First Derivative in Space A Simple Second Order Central Difference Taylor Series Approximation Gives spatial derivative T x numerical approximation T k+1 T k 1 2 x +O( x 2 ). Above, = O( x 2 ) is the space discretization error. O( x 2 ) 0 when x 0. Matlab and PDE s 18/27

33 Discretization of Second Derivative in Space Central Difference Formula for T (x) second spatial derivative 2 T x 2 numerical approximation T k+1 2T k + T k 1 x 2 +O( x 2 ). Above, = O( x 2 ) is the space discretization error. Matlab and PDE s 19/27

34 Discretization of Second Derivative in Space Central Difference Formula for T (x) second spatial derivative 2 T x 2 numerical approximation T k+1 2T k + T k 1 x 2 +O( x 2 ). Above, = O( x 2 ) is the space discretization error. O( x 2 ) 0 when x 0. Matlab and PDE s 19/27

35 Outline Introduction Modeling Transport Phenomena Using PDE s Discretization of Derivatives Numerical Solution of PDE s Matlab and PDE s 20/27

36 Explicit Euler Method for the Convection Equation Update Formula T k n+1 Tk n t + u Tk+1 n T k 1 n 2 x = 0. Or Moving the Known Terms to Right Hand Side Tn+1 k = Tn k u t Tk+1 n T k 1 n 2 x = T k n Courant Number u t x T k+1 n T k 1 n 2 Matlab and PDE s 21/27

37 Explicit Euler Method for the Heat (Diffusion) Equation Update Formula T k n+1 Tk n t = ν Tk+1 n 2T k n + T k 1 n x 2 = 0. Or Moving the Known Terms to Right Hand Side T k n+1 = T k n + Courant-Friedrichs-Lewy Number ν t x 2 (Tn k+1 2Tn k + Tn k 1 ) Matlab and PDE s 22/27

38 Stability Limits Courant Number The Courant number Co = u t describes the propagation distance x of information during timestep t. Matlab and PDE s 23/27

39 Stability Limits Courant Number The Courant number Co = u t describes the propagation distance x of information during timestep t. It is necessary that Co < 1 for physically meaningfull solution. Matlab and PDE s 23/27

40 Stability Limits Courant Number The Courant number Co = u t describes the propagation distance x of information during timestep t. It is necessary that Co < 1 for physically meaningfull solution. For explicit Euler method Co < 1/2 sets the limit for timestep for the algorithm to be numerically stable. Matlab and PDE s 23/27

41 Stability Limits (cont.) Courant-Friedrichs-Lewy Number The Courant-Friedrichs-Lewy Number CFL = ν t describes the x2 diffusion distance of information during timestep t. Matlab and PDE s 24/27

42 Stability Limits (cont.) Courant-Friedrichs-Lewy Number The Courant-Friedrichs-Lewy Number CFL = ν t describes the x2 diffusion distance of information during timestep t. Again, it is necessary that CFL < 1 for physically meaningfull solution. Matlab and PDE s 24/27

43 Stability Limits (cont.) Courant-Friedrichs-Lewy Number The Courant-Friedrichs-Lewy Number CFL = ν t describes the x2 diffusion distance of information during timestep t. Again, it is necessary that CFL < 1 for physically meaningfull solution. Again, for explicit Euler method CFL < 1/2 sets the limit for timestep for the algorithm to be numerically stable. Matlab and PDE s 24/27

44 Matlab Implementation The folder /demos2012/ shows demo files in which the Explicit Euler method is applied for heat diffusion and convection-diffusion equations. ConvectionDiffusionEquation.m and HeatDiffusion.m demonstrate these examples. The files PlottingFigure.m, SurfaceAnimation.m and DrawingSurface.m show how graphs can be drawn, surface motion be animated etc. Matlab and PDE s 25/27

45 Summary An introduction to PDE s and numerical solution using the explicit Euler method was given. Matlab and PDE s 26/27

46 Summary An introduction to PDE s and numerical solution using the explicit Euler method was given. It is very easy to extend the present knowledge to e.g. the Runge-Kutta-methods which would provide more robustness and stability. Matlab and PDE s 26/27

47 Summary An introduction to PDE s and numerical solution using the explicit Euler method was given. It is very easy to extend the present knowledge to e.g. the Runge-Kutta-methods which would provide more robustness and stability. In the next lecture we will show that you can already directly apply the learned methods and PDE s to non-linear the non-linear realistic case of the Navier-Stokes equation for a 1D pipe flow. Matlab and PDE s 26/27

48 The End Thank you Questions? Matlab and PDE s 27/27