COMPUTATIONAL NUCLEAR THERMAL HYDRAULICS
Cho, Hyoung Kyu
Department of Nuclear Engineering Seoul National University
Cho, Hyoung Kyu
Department of Nuclear Engineering Seoul National University
Contents
Introduction
Conservation laws of fluid motion and boundary conditions
Turbulence and its modeling
The finite volume method for diffusion problems
The finite volume method for convection‐diffusion problems
Solution algorithms for pressure‐velocity coupling in steady flows
Solution of discretized equations
The finite volume method for unsteady flows
CHAPTER 8: FINITE VOLUME METHOD FOR
UNSTEADY FLOWS
Contents
Introduction
One‐dimensional unsteady heat conduction
Explicit scheme
Crank–Nicolson scheme
The fully implicit scheme
Illustrative examples
Implicit method for two‐ and three‐dimensional problems
Discretization of transient convection–diffusion equation
Worked example of transient convection–diffusion using QUICK differencing
Solution procedures for unsteady flow calculations
Transient SIMPLE
The transient PISO algorithm
Steady state calculations using the pseudo‐transient approach
A brief note on other transient schemes
Summary
Introduction
General transport equation
Applying FVM
Time integration
Contents
Introduction
One‐dimensional unsteady heat conduction
Explicit scheme
Crank–Nicolson scheme
The fully implicit scheme
Illustrative examples
Implicit method for two‐ and three‐dimensional problems
Discretization of transient convection–diffusion equation
Worked example of transient convection–diffusion using QUICK differencing
Solution procedures for unsteady flow calculations
Transient SIMPLE
The transient PISO algorithm
Steady state calculations using the pseudo‐transient approach
A brief note on other transient schemes
Summary
One‐dimensional unsteady heat conduction
Unsteady one‐dimensional heat conduction
One‐dimensional unsteady heat conduction
Unsteady one‐dimensional heat conduction
Superscript ‘o’ refers to temperatures at time t.
Temperatures at time level t+Δt are not superscripted.
First order (backward) differencing scheme.
Higher order schemes: will be discussed later in this chapter
t T T
t
T p p
0
One‐dimensional unsteady heat conduction
Variation of TP, TE and TW with time
Integral of temperature with respect to time
One‐dimensional unsteady heat conduction
Standard form
θ = a weighting parameter between zero and one
θ = 0 explicit scheme
0 < θ < 1 : implicit scheme, for θ = 0.5 Crank‐Nicolson scheme
θ = 1.0 fully implicit scheme
Contents
Introduction
One‐dimensional unsteady heat conduction
Explicit scheme
Crank–Nicolson scheme
The fully implicit scheme
Illustrative examples
Implicit method for two‐ and three‐dimensional problems
Discretization of transient convection–diffusion equation
Worked example of transient convection–diffusion using QUICK differencing
Solution procedures for unsteady flow calculations
Transient SIMPLE
The transient PISO algorithm
Steady state calculations using the pseudo‐transient approach
A brief note on other transient schemes
Summary
Explicit scheme
Standard form of the unsteady conductive heat transfer equation
θ = 0 explicit scheme
Right‐hand side: only contains values at the old time step
Left‐hand side: can be calculated by forward marching in time
First‐order accurate
Explicit scheme
Time step restriction
All coefficients need to be positive in the discretized equation.
The coefficient of may be viewed
as the neighbor coefficient connecting the values at the old time level to those at the new time level.
This inequality sets a stringent maximum limit to the time step size and represents a serious limitation for the explicit scheme.
Not recommended for general transient problems.
k c x
t x or
k t
c x
2
2 2
Explicit scheme
Explicit scheme with higher order accuracy
DuFort‐Frankel methods
Use temperatures at more than two time levels
Contents
Introduction
One‐dimensional unsteady heat conduction
Explicit scheme
Crank–Nicolson scheme
The fully implicit scheme
Illustrative examples
Implicit method for two‐ and three‐dimensional problems
Discretization of transient convection–diffusion equation
Worked example of transient convection–diffusion using QUICK differencing
Solution procedures for unsteady flow calculations
Transient SIMPLE
The transient PISO algorithm
Steady state calculations using the pseudo‐transient approach
A brief note on other transient schemes
Summary
Crank–Nicolson scheme
Crank‐Nicolson scheme
θ = 1/2
Linearization of the source term
Discretized unsteady heat conduction equation
Crank–Nicolson scheme
Assessment of Crank‐Nicolson scheme
Unconditionally stable for all values of time step, t
Schemes with 1/2 θ 1
Boundedness issue
All coefficients are required to be positive for physically realistic and bounded results.
Coefficient for
Time step restriction for Crank‐Nicolson
k c x
t 2
2
Time step restriction for explicit method
Crank–Nicolson scheme
Assessment of Crank‐Nicolson scheme
Accuracy
Second‐order accurate in time
Contents
Introduction
One‐dimensional unsteady heat conduction
Explicit scheme
Crank–Nicolson scheme
The fully implicit scheme
Illustrative examples
Implicit method for two‐ and three‐dimensional problems
Discretization of transient convection–diffusion equation
Worked example of transient convection–diffusion using QUICK differencing
Solution procedures for unsteady flow calculations
Transient SIMPLE
The transient PISO algorithm
Steady state calculations using the pseudo‐transient approach
A brief note on other transient schemes
Summary
The fully implicit scheme
The fully implicit scheme
θ = 1
Discretized unsteady heat conduction equation
System of algebraic equations at each time level
Time marching procedure
Start with given temperature field,
The system of equations is solved after selecting time step t
is assigned to
The fully implicit scheme
Assessment of the fully implicit scheme
All coefficients are positive.
Unconditionally stable for any size of time step
Accuracy
First order accurate in time
Small time steps are needed
to ensure the accuracy of results.
Recommended for general purpose transient calculations because of its robustness and unconditional stability.
Contents
Introduction
One‐dimensional unsteady heat conduction
Explicit scheme
Crank–Nicolson scheme
The fully implicit scheme
Illustrative examples
Implicit method for two‐ and three‐dimensional problems
Discretization of transient convection–diffusion equation
Worked example of transient convection–diffusion using QUICK differencing
Solution procedures for unsteady flow calculations
Transient SIMPLE
The transient PISO algorithm
Steady state calculations using the pseudo‐transient approach
A brief note on other transient schemes
Summary
Illustrative examples
1D unsteady conduction example with analytical solutions
• Initially, uniform temperature of 200C
• At t=0, Teast becomes 0C
Illustrative examples
1D unsteady conduction example with analytical solutions
Governing equation
Initial condition
Boundary conditions
Analytical solution
Illustrative examples
1D unsteady conduction example with analytical solutions
Discretized with 5 control volumes
x=0.004 m
At node 1
At node 5
Illustrative examples
1D unsteady conduction example with analytical solutions
Standard form
Time step restriction
Illustrative examples
1D unsteady conduction example with analytical solutions
With a suitable time step: = 2 s.
Discretized equation at each cell
Illustrative examples
1D unsteady conduction example with analytical solutions
With a suitable time step: = 2 s.
Illustrative examples
1D unsteady conduction example with analytical solutions
With a suitable time step: = 2 s.
With large time step: = 8 s.
Illustrative examples
1D unsteady conduction example with analytical solutions
At node 1 & 5
Illustrative examples
1D unsteady conduction example with analytical solutions
General form
Calculation with reasonably small time step: 2s
Illustrative examples
1D unsteady conduction example with analytical solutions
With implicit scheme: 2s vs. 8s
Whereas the explicit method gives unrealistic oscillations at this step size, the implicit method gives results that are in reasonable agreement with the exact solution.
This clearly illustrates a key advantage of the implicit method, which tolerates much larger time steps. However, we stress that good solution accuracy can, of course, only be achieved with small time steps.
Contents
Introduction
One‐dimensional unsteady heat conduction
Explicit scheme
Crank–Nicolson scheme
The fully implicit scheme
Illustrative examples
Implicit method for two‐ and three‐dimensional problems
Discretization of transient convection–diffusion equation
Worked example of transient convection–diffusion using QUICK differencing
Solution procedures for unsteady flow calculations
Transient SIMPLE
The transient PISO algorithm
Steady state calculations using the pseudo‐transient approach
A brief note on other transient schemes
Summary
Implicit method for 2‐D and 3‐D problems
Extension of the 1D calculation to 2D and 3D problems
3D transient diffusion problem
General form: fully implicit method
Contents
Introduction
One‐dimensional unsteady heat conduction
Explicit scheme
Crank–Nicolson scheme
The fully implicit scheme
Illustrative examples
Implicit method for two‐ and three‐dimensional problems
Discretization of transient convection–diffusion equation
Worked example of transient convection–diffusion using QUICK differencing
Solution procedures for unsteady flow calculations
Transient SIMPLE
The transient PISO algorithm
Steady state calculations using the pseudo‐transient approach
A brief note on other transient schemes
Summary
Discretization of transient convection–diffusion eq.
Transient problem
Central coefficient
Additional source term
Transport equation
3D equation
Discretization of transient convection–diffusion eq.
Coefficients with the hybrid scheme
Discretization of transient convection–diffusion eq.
Coefficients with the hybrid scheme
