AerE 546 ' Lecture 17
Finite volume method
Natural for unstructured meshed. Also, working with fluxes leads into hyperbolic equations.
A. Recall origin of p.d.e.ʼs
� � �
∂t qdV = − F · ndS + Sources dV
CV CS CV
� �
∂q/∂t F =− ∇ · F dV + Sources dV
CV CV
Source ⇒ ∂t q = −∇ · F + Sources
Donʼt take second step: apply discretization to the integral balance. Control volume is polyline
(polyhedron in 3-D).
Divergence theorem (Gaussʼ) is used in f.v. method. Rationale:
f = ∫df/dx dx but now f ! ∫F·dS and df/dx dx !
∫ ∇·F dV =∫ ∇·F dS dxn =∫ dF/dxn dxn dS = ∫n·FdS
Divergence theorem <-> fundamental theorem of calculus:
Sometimes useful to start with p.d.e. (see below)
1. Control volumes are defined by mesh:
� �
∂t qdV = − F · ndS
CV CS
(without sources). The integrals are
S2 S3
∂t ∫ q dV = -Σ ∫ F·n dS S1 V S4
S6 S5
This is exact. In 2-D, V = area and S = length: ∂t ∫ q dA= -Σ ∫ F·n dℓ
The integrals must be discretized.
What is F? Convective uT ; diffusive -κ∇T --- How does one compute the gradient?
This study source was downloaded by 100000899610689 from CourseHero.com on 09-06-2025 01:33:36 GMT -05:00
1
https://www.coursehero.com/file/33324883/Lecture17pdf/
, AerE 546 ' Lecture 17
2. Sometimes have to think of starting with diff eq. E.g., down gradient heat flux:
∫-∇T dV = - ∫∇·(I T) dV = - ∫T (n·I) dS = - ∫T ndS n = outward normal
Temperature gradient is computed from face temperatures: ∫-∇T dV = -Σ ∫ n T dS
B. Control volumes
1. Order of accuracy determined by `reconstructionʼ.
2. Natural approach for unstructured grids (vs. finite difference)
3. Can derive by integrating diff. eqs. over control volume = mesh cell
4. Centers and vertices define dual grids
' ' ' ' ' ' Data can be stored on grid (vertex)
' ' ' ' ' ' ' ' or on dual (cell center)
g rid
al
Du
c.v.
Grid
Mesh = set of control volumes. ∂t ∫ q dV = -Σ ∫ F·n dS
Note flux out of one cell = flux into neighbor.
Xʼs can be computational nodes of triangular c.v.s. Or vertices can be nodes and
dual defines c.v.ʼs
control
surface
x dual created
x
x from cell centers
x control
x volume
x
This study source was downloaded by 100000899610689 from CourseHero.com on 09-06-2025 01:33:36 GMT -05:00
2
https://www.coursehero.com/file/33324883/Lecture17pdf/
Finite volume method
Natural for unstructured meshed. Also, working with fluxes leads into hyperbolic equations.
A. Recall origin of p.d.e.ʼs
� � �
∂t qdV = − F · ndS + Sources dV
CV CS CV
� �
∂q/∂t F =− ∇ · F dV + Sources dV
CV CV
Source ⇒ ∂t q = −∇ · F + Sources
Donʼt take second step: apply discretization to the integral balance. Control volume is polyline
(polyhedron in 3-D).
Divergence theorem (Gaussʼ) is used in f.v. method. Rationale:
f = ∫df/dx dx but now f ! ∫F·dS and df/dx dx !
∫ ∇·F dV =∫ ∇·F dS dxn =∫ dF/dxn dxn dS = ∫n·FdS
Divergence theorem <-> fundamental theorem of calculus:
Sometimes useful to start with p.d.e. (see below)
1. Control volumes are defined by mesh:
� �
∂t qdV = − F · ndS
CV CS
(without sources). The integrals are
S2 S3
∂t ∫ q dV = -Σ ∫ F·n dS S1 V S4
S6 S5
This is exact. In 2-D, V = area and S = length: ∂t ∫ q dA= -Σ ∫ F·n dℓ
The integrals must be discretized.
What is F? Convective uT ; diffusive -κ∇T --- How does one compute the gradient?
This study source was downloaded by 100000899610689 from CourseHero.com on 09-06-2025 01:33:36 GMT -05:00
1
https://www.coursehero.com/file/33324883/Lecture17pdf/
, AerE 546 ' Lecture 17
2. Sometimes have to think of starting with diff eq. E.g., down gradient heat flux:
∫-∇T dV = - ∫∇·(I T) dV = - ∫T (n·I) dS = - ∫T ndS n = outward normal
Temperature gradient is computed from face temperatures: ∫-∇T dV = -Σ ∫ n T dS
B. Control volumes
1. Order of accuracy determined by `reconstructionʼ.
2. Natural approach for unstructured grids (vs. finite difference)
3. Can derive by integrating diff. eqs. over control volume = mesh cell
4. Centers and vertices define dual grids
' ' ' ' ' ' Data can be stored on grid (vertex)
' ' ' ' ' ' ' ' or on dual (cell center)
g rid
al
Du
c.v.
Grid
Mesh = set of control volumes. ∂t ∫ q dV = -Σ ∫ F·n dS
Note flux out of one cell = flux into neighbor.
Xʼs can be computational nodes of triangular c.v.s. Or vertices can be nodes and
dual defines c.v.ʼs
control
surface
x dual created
x
x from cell centers
x control
x volume
x
This study source was downloaded by 100000899610689 from CourseHero.com on 09-06-2025 01:33:36 GMT -05:00
2
https://www.coursehero.com/file/33324883/Lecture17pdf/
