transient, source terms and relaxation · transient, source terms and relaxation chapter 13....
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FMIA
ISBN 978-3-319-16873-9
Fluid Mechanics and Its ApplicationsFluid Mechanics and Its ApplicationsSeries Editor: A. Thess
F. MoukalledL. ManganiM. Darwish
The Finite Volume Method in Computational Fluid DynamicsAn Advanced Introduction with OpenFOAM® and Matlab®
The Finite Volume Method in Computational Fluid Dynamics
Moukalled · Mangani · Darwish
113
F. Moukalled · L. Mangani · M. Darwish The Finite Volume Method in Computational Fluid Dynamics An Advanced Introduction with OpenFOAM® and Matlab ®
This textbook explores both the theoretical foundation of the Finite Volume Method (FVM) and its applications in Computational Fluid Dynamics (CFD). Readers will discover a thorough explanation of the FVM numerics and algorithms used in the simulation of incompressible and compressible fluid flows, along with a detailed examination of the components needed for the development of a collocated unstructured pressure-based CFD solver. Two particular CFD codes are explored. The first is uFVM, a three-dimensional unstructured pressure-based finite volume academic CFD code, implemented within Matlab®. The second is OpenFOAM®, an open source framework used in the development of a range of CFD programs for the simulation of industrial scale flow problems.
With over 220 figures, numerous examples and more than one hundred exercises on FVM numerics, programming, and applications, this textbook is suitable for use in an introductory course on the FVM, in an advanced course on CFD algorithms, and as a reference for CFD programmers and researchers.
Engineering
9 783319 168739
Transient, Source Terms and Relaxation
Chapter 13
Transient Term Discretization
Transient coordinate
C
C
C
trans
ient o
pera
tor
∂ ρφ( )∂t
t
L φ t( )spatial operator
t
t+∆t
t-∆t
Transient Problems
integrate over spatial control volume
integrate over temporal control volume
transient term spatial terms
time
t
t-1
t-2
t+1
L(φ(t))
φ(t+1/2)
φ(t-1/2)
∂ ρφ( )∂t
= L φ( )
∂ ρφ( )∂t
dVΩ∫ = L φ( )dV
Ω∫ ⇒
∂ ρφ( )∂t
V = L φ( )V
∂ ρφ( )∂t
V⎛⎝⎜
⎞⎠⎟ dt
t− 12 ∆ t
t+ 12 ∆ t
∫ = L φ( )V( )dtt− 12 ∆ t
t+ 12 ∆ t
∫
ρφ( )t+ 12 ∆ t − ρφ( )t− 12 ∆ t⎡⎣ ⎤⎦V = L φ( )V⎡⎣ ⎤⎦∆ t ⇒ρφ( )t+ 12 ∆ t − ρφ( )t− 12 ∆ t
∆ tV = L φ( )V
ρφ( )t+ 12 ∆ t − ρφ( )t− 12 ∆ t∆ t
V = L φ( )V
Order of Discretization
time
t
t-1
t-2
t+1
L(φ(t))
φ(t+1/2)
φ(t-1/2)L φ( )V t− 12 ∆ t→t+ 12 ∆ t = L φ( )V
ρφ( )t+ 12 ∆ t = f ρφ t+1,ρφ t ,ρφ t−1,ρφ t−2 ,...( )
Implicit vs Explicit
C
C
backward Euler
C
C
Forward Euler
φ t+Δt
φ t
φ t−Δt /2
φ t+Δt /2
L φ t( )
φ t−Δt
∂ ρφ( )∂t
= L φ( )V⎡⎣ ⎤⎦t +∆ t2
∂2 ρφ( )∂t 2
+ ...
neglected terms! "## $##
⇒ρφ( )t − ρφ( )t−∆ t
∆ tV = L φ( )V⎡⎣ ⎤⎦
t
Backward Euler
Implicit Scheme
Solve system of equations
Iterate
first order accuracy Numerical diffusion
ρφ( )t+ 12 ∆ t − ρφ( )t− 12 ∆ t∆ t
V = L φ( )V
ρφ( )t+ 12 ∆ t ← ρφ( )t
ρφ( )t− 12 ∆ t ← ρφ( )t−∆ t
ρφ( )t−∆ t = ρφ( )t − ∂ ρφ( )∂t
∆ t +∂2 ρφ( )∂t 2
∆ t 2
2−∂3 ρφ( )∂t 4
∆ t 3
6+ ....
Accuracy
⇒ρφ( )t − ρφ( )t−∆ t
∆ t=∂ ρφ( )∂t
−∂2 ρφ( )∂t 2
∆ t2
+∂3 ρφ( )∂t 4
∆ t 2
6+ ....
Stable for all ∆t
⇒ρφ( )t+∆ t − ρφ( )t
∆ tV = L φ( )V⎡⎣ ⎤⎦
t
Explicit Scheme
Point EvaluationNo Iterations
ρφ( )t+ 12 ∆ t − ρφ( )t− 12 ∆ t∆ t
V = L φ( )V
ρφ( )t+ 12 ∆ t ← ρφ( )t+∆ t
ρφ( )t− 12 ∆ t ← ρφ( )t
Forward Euler
Unstable for Courant>1
ρφ( )t+∆ t = ρφ( )t + ∂ ρφ( )∂t
∆ t +∂2 ρφ( )∂t 2
∆ t 2
2+∂3 ρφ( )∂t 4
∆ t 3
6+ ....
Accuracy
⇒ρφ( )t+∆ t − ρφ( )t
∆ t=∂ ρφ( )∂t
+∂2 ρφ( )∂t 2
∆ t2
+∂3 ρφ( )∂t 4
∆ t 2
6+ ....
∂ ρφ( )∂t
= L φ( )V⎡⎣ ⎤⎦t −∆ t2
∂2 ρφ( )∂t 2
+ ...
neglected terms! "## $##
first order accuracy
Numerical anti-diffusion
ρφ( )t+∆ t = ρφ( )t + L φ( )∆ t
φ t+Δt
φ t
φ t+Δt /2
L φ t( )φ t−Δt /2
φ t−Δt
φ t
φ t+Δt /2
φ t−Δt /2
L φ t( )
φ t−Δt
φ t−2Δt
⇒3 ρφ( )t − 4 ρφ( )t−∆ t + 2 ρφ( )t−2∆ t
2∆ tV = L φ( )V⎡⎣ ⎤⎦
t
Adam-Bashforth
2 old time-steps needed
Second Order Accuracy
Implicit Scheme
ρφ( )t+ 12 ∆ t − ρφ( )t− 12 ∆ t∆ t
V = L φ( )V
ρφ( )t+ 12 ∆ t ← 32
ρφ( )t − 12
ρφ( )t−∆ t
ρφ( )t− 12 ∆ t ← 32
ρφ( )t−∆ t − 12
ρφ( )t−2∆ t
Crank-Nicholson
L(φ(t))
φ(t+1/2)
φ(t-1/2)
Crank-Nicholson
⇒ρφ( )t+∆ t + 2 ρφ( )t−∆ t
2∆ tV = L φ( )V⎡⎣ ⎤⎦
t
ρφ( )t+ 12 ∆ t − ρφ( )t− 12 ∆ t∆ t
V = L φ( )V
ρφ( )t+ 12 ∆ t ← 12
ρφ( )t+∆ t + 12
ρφ( )t
ρφ( )t− 12 ∆ t ← 12
ρφ( )t + 12
ρφ( )t−∆ t
2 old time-steps needed
Second Order Accuracy
Explicit Scheme
Unstable for Courant>2
Crank-Nicholson
C
C
C
φ t+Δt
φ t
φ t+Δt /2
φ t−Δt /2
φ t−Δt
L φ t( )
⇒ρφ( )t − ρφ( )t−∆ t
∆ tV = L φ( )V⎡⎣ ⎤⎦
t
⇒ρφ( )t+∆ t − ρφ( )t
∆ tV = L φ( )V⎡⎣ ⎤⎦
t
Backward Euler
Forward Euler
Crank-Nicholson-Implementation
L(φ(t+1/2))
φ(t+1)
φ(t)
L(φ (t-1/2))
φ(t-1)
Crank-Nicholsonρφ( )t+∆ t − ρφ( )t−∆ t
∆ tV = 2 L φ( )V⎡⎣ ⎤⎦
t
ρφ( )t − ρφ( )t−∆ t∆ t
V = L φ( )V⎡⎣ ⎤⎦t1
2
ρφ( )t+∆ t − ρφ( )t∆ t
V = L φ( )V⎡⎣ ⎤⎦t =
ρφ( )t − ρφ( )t−∆ t∆ t
V
ρφ( )t+∆ t = 2 ρφ( )t − ρφ( )t−∆ t
Note that
Thus step 2 can be written as
Stability of the Explicit Scheme
�
2D : ρΔxΔyΔt
≥ ΓeΔyδxe
+ ΓwΔyδxw
+ ΓnΔxδyn
+ ΓsΔxδys
⎛
⎝ ⎜
⎞
⎠ ⎟ ⇒Δt ≤
ρ Δx( )24Γ�
1D : ρΔxΔt
≥ ΓeΔyδxe
+ ΓwΔyδxw
⎛
⎝ ⎜
⎞
⎠ ⎟ ⇒Δt ≤
ρ Δx( )22Γ
Von Neumann Stability Criterion
∂ ρφ( )∂t
= ∇ ⋅Γ∇φ +Q
aPt φP
t = aNt−∆ tφN
t−∆ t
NB∑ + aP
t−∆ t − aNt−∆ t
NB∑( )φPt−∆ t + bP
⇒ρφ( )t − ρφ( )t−∆ t
∆ tV = Γ∇φ( ) f ⋅Sf
nb∑ +QV
aPt =
ρt
∆ tV aP
t−∆ t =ρt−∆ t
∆ tV
bP = QV
aEt−∆ t =
Γet−∆ t ∆ yδx
aWt−∆ t =
Γwt−∆ t ∆ yδx
aNt−∆ t =
Γnt−∆ t ∆ xδy
aSt−∆ t =
Γ st−∆ t ∆ xδy
if aPt−∆ t − aNB
t−∆ t
NB∑( ) < 0⇒Unphysical feedback
Initial Condition
Δt
tinitial
Δt / 2
Δt
tinitial
Δt
Δt / 2
Conclusion
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