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
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
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 convec ...
),
is Rate of increase of
due to
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
,
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
In systems theory, a system or a Process theory, process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those p ...
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
Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its ''thermal conductivity'', and is denoted .
Heat spontaneously flows along a te ...
equation.
Now, holding the assumption of the
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
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