Finite volume method for unsteady flow

Introduction

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.

Governing Equation

The conservation equation for the transport of a scalar in unsteady flow has the general form as

${\frac {\partial \rho \phi }{\partial t}}+\operatorname {div} (\rho \phi \upsilon )=\operatorname {div} (\Gamma \operatorname {grad} \phi )+S_{\phi }$

$\rho$ is density and $\phi$ is conservative form of all fluid flow,
$\Gamma$ is the Diffusion coefficient and $S$ is the Source term. $\operatorname {div} (\rho \phi \upsilon )$ is Net rate of flow of $\phi$ out of fluid element(convection),
$\operatorname {div} (\Gamma \operatorname {grad} \phi )$ is Rate of increase of $\phi$ due to diffusion,
$S_{\phi }$ is Rate of increase of $\phi$ due to sources.

${\frac {\partial \rho \phi }{\partial t}}$ is Rate of increase of $\phi$ 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.

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