I- Computational Fluid Dynamics (CFD-I) Session #4 GOVERNING EQUATIONS Part 3

Classification of governing PDEs in fluid flow and heat transfer The information in the solutions tends to propagate along the characteristics if they exist. B2−4AC>0. There are two real characteristics intersecting at this point. The equation is called hyperbolic. 2. B2−4AC=0. There is one real characteristic. The equation is Called parabolic. 3. B2−4AC<0. No real characteristics exist at this point. The equation is called elliptic

Classification of governing PDEs in fluid flow and heat transfer 1-Hyperbolic Equations initial conditions: Characteristics: Example: The wave equation The initial perturbation of u(x,0)=f(x) around the point x0 (e.g., a localized deformation of a string) is split into halves, which propagate without changing their shape along the characteristics x+at= x0 and x−at=x0

An important feature of the hyperbolic systems : represents the response to the initial perturbation of ‘velocity’ . If, for example, the initial velocity is a delta function g(x)=δ(x−x0), the solution is a constant equal to 1/2a within the cone between the left-running and right-running characteristics and zero outside this cone. An important feature of the hyperbolic systems : The perturbations propagate in space with a finite speed

The perturbations propagate in space with a finite speed Two more comments: 1- When a source of perturbations suddenly appears at the time moment t0 and space location x0 (point P in Figure 3.5) An observer located at the distance L from the source will not notice the perturbations 2- The state of the solution at the point P only affects the solution within a cone between the left-running and right-running characteristics intersecting at P. The cone is called the domain of influence. Similarly, the solution at P itself is affected only by the solution within the domain of dependence. The behavior described by hyperbolic equations and, thus, determined by characteristics appears in many physical systems. All these processes involve wavelike motions along the characteristics or discontinuities across them (e.g., shock waves in supersonic flows). The Navier-Stokes equations have some features of a hyperbolic system due to the presence of the nonlinear terms. The perturbations propagate in space with a finite speed

The characteristics are lines t =constant Classification of governing PDEs in fluid flow and heat transfer 2- Parabolic Equations Example: 1D heat equation Characteristics: The characteristics are lines t =constant The perturbation that occurs at the space location x0 and time moment t0 (point P) affects the solution in the entire space domain 0<x<L, although the effect becomes weaker with the distance to P. 1- The domain of influence of the point P includes the domain 0 <x<L and times t>t0. 2- The domain of dependence of P includes all points 0<x<L and all moments of time prior to the time of P. Characteristics and domains of influence and dependence of the parabolic equation

The perturbations propagate in space with a in finite speed 1- In the physical systems described by parabolic equations, the perturbations are usually propagated by diffusion. 2- The interaction occurs at infinite speed but relaxes with distance 3-viscous terms in the Navier-Stokes equations lead to diffusion of gradients of the velocity field, thus giving the equations parabolic properties. another example : the parabolic behavior is the steady-state viscous boundary layers. The reduced Navier-stokes equations that describe the flow within a boundary layer are of the parabolic type. The characteristics are perpendicular to the wall, and the flow evolves along the boundary layer similarly to the time evolution of solutions of other parabolic systems (see Figure 3.6b). The perturbations propagate in space with a in finite speed

Classification of governing PDEs in fluid flow and heat transfer 3- Elliptic Equations The elliptic equations do not have real characteristics at all. Effect of any perturbation is felt immediately and to full degree in the entire domain of solution There are no limited domains of influence or dependence As opposite to the hyperbolic and parabolic systems that involve time evolution (or evolution along a spatial direction as in the case of a boundary layer flow) and have to be treated as marching problems in CFD, the elliptic PDE problems are always of equilibrium type. pressure field in incompressible flows is a solution of an elliptic Poisson equation

+ + DIFFERENT KINDS OF CFD discretization discretization exact solution of PDEs + in a continuum domain approximate numerical solution + in a discrete domain discretization A complete PDE problem consisting of: Governing Equation, Domain of solution, Boundary Conditions Initial Conditions. discretization continuum Solution domain discretization of the solution domain

Finite Difference/Finite volume Partial differential equation Exact analytical solution Domain and time-interval [t0,t1] Exact solution—function u(x,y,z,t) discretization System of algebraic discretization equations Approximate solution Computational grid—set of Points (x,y,z)i and time layers tn Approximate solution—set of Values uin approximating u at (x,y,z)i and tn Mathematical PDE Finite Difference/Finite volume Approximation of PDE PDE discretization BC

A complete PDE problem consisting of: The discretization can be implemented in different ways: finite difference/finite volume finite element Spectral method A complete PDE problem consisting of: Governing Equation, Domain of solution, Boundary Conditions Initial Conditions.

Thanks End of Session #4 Next Session : Dr. Mohammad Jadidi CFD https://ir.linkedin.com/in/moammad-jadidi-03ab8399 Jadidi.cfd@gmail.com Dr. Mohammad Jadidi (Ph.D. in Mechanical Engineering) https://www.researchgate.net/profile/Mohammad_Jadidi https://www.slideshare.net/MohammadJadidi