1 Chapter 8: The Finite Volume Method for Transient Problems Presented by: Prof. Ir. Dr. Shahrir Abdullah Dr. Wan Mohd Faizal Wan Mahmood Dept. of Mechanical & Materials Engineering Universiti Kebangsaan Malaysia KKKJ4164 COMPUTATIONAL FLUID DYNAMICS
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Chapter 8: The Finite Volume Method for Transient Problems Presented by: Prof. Ir. Dr. Shahrir Abdullah Dr. Wan Mohd Faizal Wan Mahmood Dept. of Mechanical & Materials Engineering Universiti Kebangsaan Malaysia
KKKJ4164 COMPUTATIONAL FLUID DYNAMICS
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FVM for Transient Problems Transient 1-D Diffusion Problems Implicit Method for 2-D and 3-D Problems Transient Convective-Diffusion Problems
Contents
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FVM for Transient Problems • The equation for the unsteady convection-diffusion problems is:
( ) ( ) ( ) φφφρρφ St
+∇Γ⋅∇=⋅∇+∂∂ u (1)
• Integrating throughout a finite volume produces:
( ) ( )
( ) dtdVSdtdA
dtdAdtdVt
tt
t
tt
t
tt
t
tt
t
∫ ∫∫ ∫
∫ ∫∫ ∫∆+∆+
∆+∆+
+
∇Γ⋅=
⋅+
∂∂
CVCS
CSCV
φφ
φρρφ
n
un
(2)
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FVM for Transient Problems or, in alternative form as
( ) ( )
( ) dtdVSdtdA
dtdAdVdtt
tt
t
tt
t
tt
t
tt
t
∫ ∫∫ ∫
∫ ∫∫ ∫∆+∆+
∆+∆+
+
∇Γ⋅=
⋅+
∂∂
CVCS
CSCV
φφ
φρρφ
n
un
• For 1D cases without generation/source term, the equation reduces to:
( ) ( )
∂∂
Γ∂∂
=∂∂
+∂∂
xxu
xtφφρρφ
In addition, a 1D flow also has to follow the flow continuity principle:
( ) 0=∂∂
+∂∂ u
xtρρ
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Transient 1-D Diffusion Problems
• An unsteady 1-D diffusion problem, e.g. heat conduction, may be modelled using the following equation:
SxTk
xtTc +
∂∂
∂∂
=∂∂ρ (3)
Control Volume around Node P
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Transient 1-D Diffusion Problems • Integration throughout a finite volume produces:
A thin plate having a thickness of L = 2 cm and a uniform initial temperature of 200°C. At time t = 0, the temperature at its left side drops to 0°C instantly, whereas the other surface is insulated. By using a grid of 5 nodes and appropriate timestep, use an explicit scheme to obtain the following times:
(a) t = 40 s, (b) t = 80 s, (c) t = 120 s.
and compare each case with the analytical solutions. Repeat the question using a timestep sufficient to fulfil the requirement for stability at t = 40 s. Given that the coefficient for heat conductance k = 10 W/m⋅K and ρc = 10 × 106 J/m3⋅K.
F ( ) ww Auρ ( ) ee Auρ ( ) ss Avρ ( ) nn Avρ ( ) bb Awρ ( ) tt Awρ
D WP
ww
xA
δΓ
PE
ee
xA
δΓ
SP
ss
yA
δΓ
PN
nn
yA
δΓ
BP
bb
zA
δΓ
PT
tt
zA
δΓ
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Transient Convective-Diffusion Problems Case Study
Consider a 1-D convection-diffusion problem with the boundary condition as followed:
Other data include L = 1.5 m, u = 2 m/s, ρ = 1.0 kg/m3, and Γ = 0.03 kg/m⋅s. Given that the source term is a time-dependent function as shown below for t > 0:
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Transient Convective-Diffusion Problems
where a = −200, b = 100, x1 = 0.6, x2 = 0.2. Obtain the temperature distribution until it reaches the steady state condition using the explicit and hybrid schemes. Use appropriate time step and number of node.