CFD Lecture 03 Partial Differential Equations(2015)

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Partial Differential Equations
General Quasi-Linear Second-Order PDE
The general quasi-linear second-order non-homogeneous 
partial differential equation in two independent variables is:
The characteristic equation corresponding to the PDE is
which can be solved to yield the differential equation of the 
characteristic curves:
GFf
y
fE
x
fD
y
fC
yx
fB
x
fA =+
∂
∂
+
∂
∂
+
∂
∂
+
∂∂
∂
+
∂
∂
2
22
2
2
0)())(()( 22 =+− dxCdydxBdyA
A
ACBB
dx
dy
2
)4( 2 −±
=
Mathematical Classification of PDEs
The classification of the PDE depends on the sign of the 
discriminant (B2−4AC) as follows: 
(B2−4AC) < 0 → Elliptic PDE (complex characteristic 
curves)
(B2−4AC) = 0 → Parabolic PDE (real and repeated 
characteristic curves)
(B2−4AC) > 0 → Hyperbolic PDE (real and distinct 
characteristic curves)
Mathematical Behavior of PDEs
The mathematical behavior of PDEs is closely 
related to their characteristics curves in terms 
of the concepts:
Domain of Dependence
Zone of Influence
Domain of Dependence
Consider a point P in the solution domain R with boundary B. 
The domain of dependence of point P is defined as the region of 
the solution domain upon which the solution at point P depends. 
The domain of dependence of the point P for (a) elliptic, (b) 
parabolic, and (c) hyperbolic PDE are shown as follows:
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影響P點的solution範圍有多大
Zone of Influence
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P點影響周遭的範圍有多大
Physical Classification of PDEs
Equilibrium or Jury Problems
Eigen-Problems
Marching or Propagation Problems
Equilibrium Problems
Equilibrium or jury problem are steady state problem in closed
domain D, in which the solution is governed by an elliptic 
PDE subjected to boundary conditions specified at each point 
on the boundary B of the domain.
Examples of Equilibrium Problems
(a) Steady state conduction problem:
(b) Consider x-component of the dimensionless 
momentum equation for a laminar incompressible 
Stokes flow:
02 =∇ T
u
x
p 2
Re
10 ∇+
∂
∂
−=
Eigen Problems
Eigen problems are special equilibrium 
problems in which the solution exists only for 
special values (i.e., eigenvalues) of a parameter 
of the problem.
Example: 
Hydrodynamic instability phenomena
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求流動狀態的轉變點
Propagation Problems
Propagation or marching problems are initial-value problems in 
open domain (with respect to one of the independent variables) in 
which the solution in the domain is marched forward from the 
initial state, guided and modified by boundary conditions.
Propagation problems are governed by parabolic or hyperbolic
PDEs.
The Mach Cone
Examples of Propagation Problems
(a) Transient Conduction/Diffusion Equation:
(b) Transient Stokes Equation:
(c) 1-D Euler Equation:
T
t
T 2∇=
∂
∂ α
u
x
pu 2
Re
1
∇+
∂
∂
−=
∂
∂
τ
x
p
x
uuu
∂
∂
−=
∂
∂
+
∂
∂
τ
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General Features of PDEs
Mathematical 
classification
Elliptic Parabolic Hyperbolic
Physical 
classification
Equilibrium Propagation Propagation
Characteristics Complex Real repeated Real distinct
Signal 
propagation speed
Undefined Infinite Finite
Domain of 
dependence
Entire solution 
domain
Present and 
entire past 
solution domain
Past solution 
domain between 
characteristics
Zone of influence Entire solution 
domain
Present and 
entire future 
solution domain
Future solution 
domain between 
characteristics
Type of numerical 
method
Iterative Marching Marching
PDEs in CFD
The PDEs in CFD can be generally written in 
Cartesian tensor of the form:
Convective or Diffusion terms
advective transport terms
( )( ) ( )j
j j j
u
S
t x x x φ
ρ φρφ φ
µ
∂∂ ∂ ∂
+ = +
∂ ∂ ∂ ∂
 
PDEs of Interest
a. First-order linear wave equation:
b. Inviscid Burger equation (non-linear wave equation):
c. Diffusion equation:
0=
∂
∂
+
∂
∂
x
uc
t
u
0=
∂
∂
+
∂
∂
x
uu
t
u
)(
x
u
xt
u
∂
∂
∂
∂
=
∂
∂ α
PDEs of Interest (continued)
d. Poisson/Laplace equation:
e. Linear convection-diffusion equation:
f. Burger equation (nonlinear convection-diffusion equation):
2
2
x
u
x
uu
t
u
∂
∂
=
∂
∂
+
∂
∂ ν
),(2
2
2
2
yxS
y
u
x
u
=
∂
∂
+
∂
∂
2
2
x
f
x
fu
t
f
∂
∂
=
∂
∂
+
∂
∂ ν
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PDEs of Interest (continued)
g. Tricomi equation:
h. 2-D velocity potential equation:
i. Helmholtz equation for acoustic waves:
02
2
2
2
=
∂
∂
+
∂
∂
y
u
x
uy
0)1( 2
2
2
2
2 =
∂
∂
+
∂
∂
−
yx
Ma φφ
022
2
2
2
=+
∂
∂
+
∂
∂ ku
y
u
x
u
Well-Posed Problems
Recall that a mathematical problem is said to be well-
posed if it satisfies the following requirements:
1. Existence:
There is at least one solution.
2. Uniqueness: 
There is at most one solution.
3. Stability: 
The solution depends continuously on the 
initial/boundary conditions given in the problem.
Solution Possibility of PDEs
For a linear equation, one of three possibilities holds:
a) There is no solution.
b) There is one and only one solution.
c) There is an infinite continuum of solutions.
While for a nonlinear equation, there is an additional 
possibility:
d) There are more than one, but count ably many 
solutions. (multiple solutions)
Can CFD problems always be well-posed?
There is usually no proof of the well-posedness of the 
complete set of equations in CFD.
Existence of numerical solution is somewhat less a 
problem
The question of uniqueness of an attained numerical 
solution is even more worrisome, simply because of 
the non-linear PDEs (Navier-Stokes equations) in 
CFD
END
	Partial Differential Equations
	General Quasi-Linear Second-Order PDE
	Mathematical Classification of PDEs
	Mathematical Behavior of PDEs
	Domain of Dependence
	Zone of Influence
	Physical Classification of PDEs
	Equilibrium Problems
	Examples of Equilibrium Problems
	Eigen Problems
	Propagation Problems
	The Mach Cone
	Examples of Propagation Problems
	General Features of PDEs
	PDEs in CFD
	PDEs of Interest
	PDEs of Interest (continued)
	PDEs of Interest (continued)
	Well-Posed Problems
	Solution Possibility of PDEs
	Can CFD problems always be well-posed?
	END