Unsteady flows are characterized as flows in which the properties of the fluid are time dependent. It gets reflected in the governing equations as the time derivative of the properties are absent.
For Studying
Finite-volume method for unsteady flow there is some governing equations
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Governing Equation
The conservation equation for the transport of a scalar in unsteady flow has the general form as
is
density and
is conservative form of all fluid flow,
is the Diffusion coefficient and
is the Source term.
is Net rate of flow of
out of fluid element(
convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convecti ...
),
is Rate of increase of
due to
diffusion,
is Rate of increase of
due to sources.
is Rate of increase of
of fluid element(transient),
The first term of the equation reflects the unsteadiness of the flow and is absent in case of steady flows. The finite volume integration of the governing equation is carried out over a control volume and also over a finite time step ∆t.
The
control volume
In continuum mechanics and thermodynamics, a control volume (CV) is a mathematical abstraction employed in the process of creating mathematical models of physical processes. In an inertial frame of reference, it is a fictitious region of a given v ...
integration of the
steady part of the equation is similar to the
steady state governing equation's integration. We need to focus on the integration of the unsteady component of the equation. To get a feel of the integration technique, we refer to the one-dimensional unsteady
heat conduction equation.
Now, holding the assumption of the
temperature at the node being prevalent in the entire control volume, the left side of the equation can be written as
By using a
first order backward differencing scheme, we can write the right hand side of the equation as
Now to evaluate the right hand side of the equation we use a weighting parameter
between 0 and 1, and we write the integration of
Now, the exact form of the final discretised equation depends on the value of
. As the variance of
is 0<
<1, the scheme to be used to calculate
depends on the value of the
Different Schemes
1. Explicit Scheme in the explicit scheme the source term is linearised as
. We substitute
to get the explicit discretisation i.e.:
where
. One thing worth noting is that the right side contains values at the old time step and hence the left side can be calculated by forward matching in time. The scheme is based on backward differencing and its Taylor series truncation error is first order with respect to time. All coefficients need to be positive. For constant k and uniform grid spacing,
this condition may be written as
This inequality sets a stringent condition on the maximum time step that can be used and represents a serious limitation on the scheme. It becomes very expensive to improve the spatial accuracy because the maximum possible time step needs to be reduced as the square of
2. Crank-Nicolson scheme : the
Crank-Nicolson method results from setting
. The discretised unsteady heat conduction equation becomes
Where
Since more than one unknown value of T at the new time level is present in equation the method is implicit and simultaneous equations for all node points need to be solved at each time step. Although schemes with
including the Crank-Nicolson scheme, are unconditionally stable for all values of the time step 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
satisfies the following condition