1. Finite Volume Method for Unsteady Flows
Lecture 06

2. The conservation law for the transport of a scalar in an unsteady flow has the general form
(1)
The first term of the equation represents the rate of change term and is zero for steady flows.
To predict transient problems we must retain this term in the discretisation process. The finite volume integration of equation (1) over a control volume (CV) must be augmented with a further integration over a finite time step t.
By replacing the volume integrals of the convective and diffusive terms with surface integrals as before and changing the order of integration in the rate of change term we obtain
(2)

3. One-dimensional unsteady heat conduction
Unsteady one-dimensional heat conduction is governed by the equation
(3)
Where c is the specific heat of the material (J/kg/K).

4. One-dimensional unsteady heat conduction…
Integration of equation (3) over the control volume and over a time interval from t to t + t gives
(4)
Rewriting,
(5)
In equation (5), A is the face area of the control volume, V is its volume, which is equal to Ax where x is the width of the control volume, and S is the average source strength.

5. One-dimensional unsteady heat conduction…
If the temperature at a node is assumed to prevail over the whole control volume, the left hand side can be written as
(6)
In equation (6) superscript 'o' refers to temperatures at time t; temperatures at time level t + t are not superscripted.
The same result as (6) would be obtained by substituting (Tp - T°)/t for dT/dt so this term has been discretised using a first- order (backward) differencing scheme.
Higher order schemes, which may be used to discretise this term

6. One-dimensional unsteady heat conduction…
If we apply central differencing to the diffusion terms on the right hand side equation (5) may be written as
(7)
To evaluate the right hand side of this equation we need to make an assumption about the variation of TP, TE and Tw with time.
We could use temperatures at time t or at time t +t to calculate the time integral or, alternatively, a combination of temperatures at time t and t +  t.

7. We have highlighted the following values of integral IT:
We may generalise the approach by means of a weighting parameter  between 0 and 1 and write the integral IT of temperature Tp with respect to time as
(8)
Variation of temperature with time for three difference schemes
We have highlighted the following values of integral IT:
if  = 0 the temperature at (old) time level t is used;
if  = 1 the temperature at new time level t + t is used;
and finally if  = 1/2, the temperatures at t and t + t are equally weighted.

9. Using formula (8) for Tw and Te in equation (7), and dividing by At throughout, we have
(9)
which may be re-arranged to give
(10)
Now we identify the coefficients of Tw and Te as aW and aE and write equation (10) in the familiar standard form:

10. In standard form,
(11)
The exact form of the final discretised equation depends on the value of .
When  is zero, we only use temperatures T°p, T0w and T0E at the old time level t on the right hand side of equation (11) to evaluate TP at the new time; the resulting scheme is called explicit.
When 0 <  < 1 temperatures at the new time level are used on both sides of the equation; the resulting schemes are called implicit.
The extreme case of  = 1 is termed fully implicit and the case corresponding to  = 1/2 is called the Crank-Nicolson scheme (Crank and Nicolson, 1947).

11. Explicit scheme
In the explicit scheme the source term is linearised as b = Su + Sp T0p .
Substitution of  = 0 into (11) gives the explicit discretisation of the unsteady conductive heat transfer equation:
(12)

12. Explicit scheme…
The right hand side of equation (12) only contains values at the old time step so the left hand side can be calculated by forward marching in time.
The scheme is based on backward differencing and its Taylor series truncation error accuracy is first-order with respect to time.
All coefficients need to be positive in the discretised equation.
The coefficient of T0p may be viewed as the neighbour coefficient connecting the values at the old time level to those at the new time level.
For this coefficient to be positive we must have a0p - aw - aE > 0.
For constant k and uniform grid spacing, xPE = xWP = x, this condition may be written as
(13a)
(13b)

13. This inequality sets a stringent maximum limit to the time step size and represents a serious limitation for the explicit scheme.
It becomes very expensive to improve spatial accuracy because the maximum possible time step needs to be reduced as the square of x.
Consequently, this method is not recommended for general transient problems.
Explicit schemes with greater formal accuracy than the above one have been designed.
Nevertheless, provided that the time step size is chosen with care, the explicit scheme described above is efficient for simple conduction calculations.

14. Crank-Nicolson scheme
The Crank-Nicolson method results from setting  = 1/2 in equation (11).
The discretised unsteady heat conduction equation is
(14)

15. Crank-Nicolson scheme…
Since more than one unknown value of T at the new time level is present in equation (14) the method is implicit and simultaneous equations for all node points need to be solved at each time step.
Although schemes with 1/2 <  < 1, including the Crank-Nicolson scheme, are unconditionally stable for all values of the time step (Fletcher, 1991) it is more important to ensure that all coefficients are positive for physically realistic and bounded results.
This is the case if the coefficient of T°p satisfies the following condition:
which leads to

16. This time step limitation is only slightly less restrictive than (13) associated with the explicit method.
The Crank-Nicolson method is based on central differencing and hence it is second-order accurate in time.
With sufficiently small time steps it is possible to achieve considerably greater accuracy than with the explicit method.
The overall accuracy of a computation depends also on the spatial differencing practice, so the Crank-Nicolson scheme is normally used in conjunction with spatial central differencing.

17. The fully implicit scheme
When the value of  is set equal to 1 we obtain the fully implicit scheme. The discretised equation is
(16)
Both sides of the equation contain temperatures at the new time step, and a system of algebraic equations must be solved at each time level.
The time marching procedure starts with a given initial field of temperatures T°. The system of equations (16) is solved after selecting time step t. Next the solution T is assigned to T° and the procedure is repeated to progress the solution by a further time step.
It can be seen that all coefficients are positive, which makes the implicit scheme unconditionally stable for any size of time step. Since the accuracy of the scheme is only first-order in time, small time steps are needed to ensure the accuracy of results.
The implicit method is recommended for general purpose transient calculations because of its robustness and unconditional stability.

18. Solving transient heat conduction using
Finite Difference Method
Finite Volume Method (assignment 10)

19. Heat conduction equation: Explicit, implicit and Crank-Nicolson Method
Insulated
Hot
Cold
8 cm
Find the temperature at time t = 0.4 second
Assignment 10
Solve the above problem using Finite Volume Method using (a) Explicit method
(b) Implicit method ( = ½) and (c)  = 1

20. Heat conduction equation: Explicit, implicit and Crank-Nicolson Method
Insulated
Explicit
Hot
Cold
Fourier series expansion and using centered finite difference formulation, 
Grid point involved in time difference
Grid point involved in space difference
i-1,l
i,l+1
O(x2) t 
l+1
i+1,l 
Fourier series expansion and using backward finite difference formulation for time derivative, l
i,l
i-1 i
i+1
0,0
Put in heat conduction equation, x
Arranging,

21. Heat conduction equation: Explicit, implicit and Crank-Nicolson Method
Fourier series expansion and using centered finite difference formulation,
Grid point involved in time difference
Grid point involved in space difference
i,l+1
i-1,l+1
i+1,l+1
O(x2) t 
l+1  l
Fourier series expansion and using backward finite difference formulation for time derivative,
i,l
i-1 i
i+1
0,0 x
Put in heat conduction equation,
Arranging,

22. Comparison between explicit and implicit method
Explicit method
Implicit method

23. Assignment 11.
Solve above unsteady heat conduction equation for time step t = 10s, 5, 2, 1, 0.5 and 0.2. Compare the results with implicit and explicit method. Analytical solution at t = 10 s is T = Comment on the convergence and stability.

24. End