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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. + + 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

10. 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

11. 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.

12. Thanks End of Session #4 Next Session : Dr. Mohammad Jadidi CFD
Dr. Mohammad Jadidi
(Ph.D. in Mechanical Engineering)