CFD Lecture 05 Finite Volume Discretization(2015)

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Finite-Volume Discretization 1 
Finite Volume (Control-Volume Finite Difference) Method 
 
Consider the 2-D diffusion equation of the form 
 
 
2 2
2 2
f f S
x y
∂ ∂
+ =
∂ ∂
, 0 , 0x a y b≤ ≤ ≤ ≤ (1) 
 
The procedure of the finite volume discretization technique for the PDE includes: 
 
(A) Domain discretization 
(B) Equation discretization 
 
Finite-Volume Discretization 2 
(A) Domain Discretization 
 
The solution domain is subdivided into a finite number of small control volumes 
(CVs) by a mesh which defines the control surfaces (CSs), not the computational 
grid points. For instance, a Cartesian 2-D grid system as illustrated below: 
 
 xi-1/2 xi+1/2 
yj+1/2 
yj-1/2 
 
Finite-Volume Discretization 3 
(B) Equation Discretization 
The finite volume discretization uses the integral form of the PDE over the 
control volume centered at the grid point P or (i, j) as the starting point: 
 
 
2 2
2 2[ ]
p p
p p p
V V
f f dV S dV
x y
∂ ∂
+ =
∂ ∂∫ ∫ (2) 
 
By means of the divergence theorem, the volume integral of LHS of Eq. (2) can be 
transformed into surface integral as 
 
 
4 42 2
2
2 2
1 1
[ ] ( )
p k
p p p k
p p A p A k k
k kV V A A
f f fdV f dV f n dA dA F
x y n
= =
∂ ∂ ∂
+ = ∇ = ∇ ⋅ = =
∂ ∂ ∂∑ ∑∫ ∫ ∫ ∫ (3) 
Finite-Volume Discretization 4 
 
where ( )
k
k
k A k
A
fF dA
n
∂
=
∂∫ , k = e, w, s, and n, which denotes the diffusion flux over 
the sub-control surface k around the grid point P. 
 
Finite-Volume Discretization 5 
Surface integrals 
 
The surface integral over each sub-control surface can be approximated by 
numerical integration schemes, such as 
 
(a) Mean-value theorem 
 
 ( ) ( ) ( )
k k
k
k A k A k k k
A
f f fF dA A A
n n n
∂ ∂ ∂
= = ≈
∂ ∂ ∂∫ (4) 
 
Here the mean diffusion flux over each sub-control surface is approximated by 
the local value at the surface point k. 
Finite-Volume Discretization 6 
(b) Trapezoidal integration 
 For instance, over the sub-control surface Ae, we have 
 
 ( ) [( ) ( ) ] / 2
e
e
e A e e ne se
A
f f fF dA A
n n n
∂ ∂ ∂
= ≈ +
∂ ∂ ∂∫ (5) 
 
(c) Simpson 1/3 integration rule 
 
 ( ) [( ) 4( ) ( ) ]/ 6
e
e
e A e e ne e se
A
f f f fF dA A
n n n n
∂ ∂ ∂ ∂
= ≈ + +
∂ ∂ ∂ ∂∫ (6) 
 
 Usually the simplest approximation using the mean-value theorem is used in 
FVD. 
Finite-Volume Discretization 7 
Volume Integrals 
 
 The simplest approximation for the volume integral is to assume that the 
integrand is either constant or varies linearly within the control volume so that 
 
 ( )
p
p p p p p
V
S dV S V S x y≈ ∆ = ∆ ∆∫ (7) 
 
Now, the finite volume approximation to the PDE can be expressed as 
 
 ( ) ( ) ( ) ( ) ( )e e w w s s n n p
f f f fA A A A S x y
n n n n
∂ ∂ ∂ ∂
+ + + = ∆ ∆
∂ ∂ ∂ ∂
 (8) 
Finite-Volume Discretization 8 
Now that 
 
 ( ) ( ) ( )e e e
f fA y
n x
∂ ∂
= ∆
∂ ∂
 ( ) ( ) ( )w e w
f fA y
n x
∂ ∂
= − ∆
∂ ∂
 (9a) 
 
 ( ) ( ) ( )n n n
f fA x
n y
∂ ∂
= ∆
∂ ∂
 ( ) ( ) ( )s s s
f fA x
n y
∂ ∂
= − ∆
∂ ∂
 (9b) 
 
Substituting into Eq. (8) yields 
 
 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )e w n s p
f f f fy y x x S x y
x x y y
∂ ∂ ∂ ∂
∆ − ∆ + ∆ − ∆ = ∆ ∆
∂ ∂ ∂ ∂
 
 
 [( ) ( ) ] /( ) [( ) ( ) ] /( )e w n s p
f f f fx y S
x x y y
∂ ∂ ∂ ∂
− ∆ + − ∆ =
∂ ∂ ∂ ∂
 (10) 
 
Finite-Volume Discretization 9 
[Interpolation at Control Surface Grids] 
(a) Piece-wise Linear Interpolation 
The assumption of a linear variation between the grid point P and its adjacent 
grid points E, W, N, and S. It follows that 
 
 ( ) ( )( )
( )
E P E P
e
E P
f f f ff
x x x x
− −∂
≈ =
∂ − ∆
 ( ) ( )( )
( )
P W P W
w
P W
f f f ff
x x x x
− −∂
≈ =
∂ − ∆
 (11a) 
 
 ( ) ( )( )
( )
N P N P
n
N p
f f f ff
y y y y
− −∂
≈ =
∂ − ∆
 ( ) ( )( )
( )
P S P S
s
P S
f f f ff
y y y y
− −∂
≈ =
∂ − ∆
 (11b) 
Finite-Volume Discretization 10 
Incorporating the above approximations, the finite volume equation analogue to 
the PDE can be formulated as 
 
2 2
( 2 ) ( 2 )S P N W P E
P
f f f f f f S
y x
− + − +
+ =
∆ ∆
 
or 
, -1 , , 1 -1, , 1,
,2 2
( 2 ) ( 2 )
=i j i j i j i j i j i j i j
f f f f f f
S
y x
+ +− + − ++
∆ ∆
 (12a) 
 
In compact form, 
 
 2 2 2 21, , 1 , 1, , 1 ,2(1 )i j i j i j i j i j i jf f f f f S xβ β β− − + ++ − + + + = ∆ (12b) 
 
where /x yβ = ∆ ∆ . 
Finite-Volume Discretization 11 
Note that the finite volume discretization thus developed gives an identical algebraic 
equation to that obtained using the Taylor series expansion. But this should be 
viewed as a special situation. 
Finite-Volume Discretization 12 
(b) Non-Linear Interpolation 
Consider 1-D advection-diffusion problem of 
 
 ( )( ) 0,d df d uf
dx dx dx
α − = 0 1x≤ ≤ (13a) 
 
subject to 
 
 ( 0) 0 and ( 1) 1f x f x= = = = (13b) 
 
where u and α are the flow velocity, which is assumed to be constant, and the 
diffusivity of the medium, respectively. 
 
user
螢光標示
Finite-Volume Discretization 13 
The analytical solution to this problem can be obtained as illustrated in Fig. 1: 
 
 
( / )
( / )
1( )
1
ux
u
ef x
e
α
α
−
=
−
 (14) 
 
Fig. 1 Solutions to the linear 1-D convection-diffusion problem. 
Finite-Volume Discretization 14 
Application of the finite volume discretization to Eq. (13a) for a control volume 
around an arbitrary interior grid xi of a uniform mesh over the solution domain 
gives 
 
xi xi-1 xi+1 
∆x 
a b 
 
 ( ) 0
b
a
d df uf dx
dx dx
α − =∫ ⇒ [( )( ) ] [( )( ) ] 0b b a adf dff fu dx u dxα α− − − = (15) 
 
Finite-Volume Discretization 15 
In view of the form of the analytical solution, we may assume the solution of the 
form 
 
 [ ( ) / ]0 1( ) i
u x xf x C C e α−= + , 1i ix x x +≤ ≤ (16) 
 
where the two coefficients C0 and C1 are determined by enforcing Eq. (16) to 
satisfy 
 
 
0 1
( / )
1 1 0 1
( )
( )
i i
u x
i i
f f x x C C
f f x x C C e α∆+ +
= = = +


 = = = +
 ⇒ 
( / )
1
0 ( / )
1
1 ( / )
1
1
u x
i i
u x
i i
u x
f e fC
e
f fC
e
α
α
α
∆
+
∆
+
∆
 −
= −

 − =
−
 (17) 
 
Finite-Volume Discretization 16 
Similarly, for the interval -1i ix x x≤ ≤ , we have 
 
 -1[ ( ) / ]0 1( ) i
u x xf x D D e α−= + (19) 
 
with 
( / )
1
0 ( / )1
u x
i i
u x
f e fD
e
α
α
∆
−
∆
−
=
−
 and 11 ( / )1
i i
u x
f fD
e α
−
∆
−
=
−
 
 
Then, the values and the gradient at the control surfaces can be evaluated as 
follows: 
 
 [ ( / 2) / ]1/ 2 0 1( )
u x
a if f x x D D e
α∆
−= = = + and 1/ 2
[ ( / 2) / ]
1( ) ( ) ( )i
u x
a x
df df uD e
dx dx
α
α−
∆= = (20) 
 
 [ ( / 2) / ]b +1/ 2 0 1( )
u x
if f x x C C e
α∆= = = + and 
1/ 2
[ ( / 2) / ]
1( ) ( ) ( )i
u x
b x
df df uC e
dx dx
α
α+
∆= = (21) 
Finite-Volume Discretization 17 
Substituting into Eq. (15) yields 
 
 ( / ) ( / )1 1(1 ) 0
u x u x
i i ie f e f f
α α∆ ∆
− +− + + = or 1 1(1 ) 0x x
Pe Pe
i i ie f e f f− +− + + = (22) 
 
where Pex(= u∆x/α) is called the grid Peclet number. 
 
Alternatively, incorporating the linear interpolation technique, Eq. (15) takes the 
form of 
 
 1 1[1 ( / 2)] 2 [1 ( / 2)] 0x i i x iPe f f Pe f− ++ − + − = (23) 
 
which is identical to that obtained using central-difference formula.Finite-Volume Discretization 18 
 
 
 
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