CFD-FVM-11: The finite volume method for unsteady flows
Introduction
The conservation law for the transport of a scalar in an unsteady flow has the general form
The finite volume integration with a further integration over a finite time step
One-dimensional unsteady heat conduction
Unsteady one-dimensional heat conduction is governed by the equation
Consider a control volume as shown in the figure below:
Integration over the control volume and over a time interval from
FVM discretization of the above equation gives
The left hand side can be written as
If we apply central differencing to the diffusion terms on the right hand side we may have
For right hand side we could use temperatures at time
Dividing the discretization equation by
which may be rearranged to give
In a general form we may write
where
When
Explicit scheme
Its Taylor series truncation error accuracy is first-order with respect to time. All coefficients need to be positive in the discretised equation. For constant k and uniform grid spacing, this condition may be written as
Crank-Nicolson scheme
The Crank–Nicolson method results from setting
Since more than one unknown value of
The fully implicit scheme
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. 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 generalpurpose transient calculations because of its robustness and unconditional stability.
Implicit method for two- and three-dimensional problems
The fully implicit method is recommended for general-purpose CFD computations on the grounds of its superior stability.
A three-dimensional control volume is considered for the discretisation. The resulting equation is
where
| 1D | - | - | - | - | ||
| 2D | - | - | ||||
| 3D |
| 1D | 2D | 3D | |
|---|---|---|---|
| 1 | |||
| - | |||
| - | - |
Discretisation of transient convection-diffusion equation
or
The fully implicit discretisation equation is
where
with
| One-dimensional | Two-dimensional | Three-dimensional | |
|---|---|---|---|
| 0 | |||
| 0 | |||
| 0 | 0 | ||
| 0 | 0 | ||
In the above expressions the values of
| Face | w | e | s | n | b | t |
|---|---|---|---|---|---|---|
Read the example in the book.
Solution procedures for unsteady flow calculations
Transient SIMPLE
The integrated form of this equation over a two-dimensional scalar control volume becomes
Note that compared to steady flow,
$$b_{I, J}^\prime = (\rho u^A)_{i, J} - (\rho u^A){i + 1, J} + (\rho v^*A){I, j} - (\rho v^*A)_{I, j + 1} + \frac{(\rho_P^0 - \rho_P)}{\Delta t}\Delta V$$
The transient PISO algorithm
The PISO method has yielded accurate results with sufficiently small time steps (see e.g. Issa et al, 1986; Kim and Benson, 1992). Since the PISO method does not require iterations within a time level it is less expensive than the implicit SIMPLE algorithm.
Steady state calculations using the pseudotransient approach
The under-relaxed form of the two-dimensional
Compare this with the transient (implicit)
We immediately note a clear analogy between transient calculations and underrelaxation in steady state calculations. It can be easily deduced that
This formula shows that it is possible to achieve the effects of under-relaxed iterative steady state calculations from a given initial field by means of a pseudo-transient computation starting from the same initial field by taking a step size that satisfies (8.48). Alternatively steady state calculations may be interpreted as pseudo-transient solutions with spatially varying time steps.
Summary
| Scheme | Stability | Accuracy | Positive coefficient criterion |
|---|---|---|---|
| Explicit | Conditionally stable | First-order | |
| Crank-Nicolson | Unconditionally stable | Second-order | |
| Implicit | Unconditionally stable | First-order | Always positive |