4
1. Basic Concepts of Fluid Flow
where flcv is the CV volume, Scv is the surface enclosing CV, n is the unit
vector orthogonal to Scv and directed outwards, v is the fluid velocity and vb
is the velocity with which the CV surface is moving. For a fixed CV, which
we shall be considering most of the time, vb = 0 and the first derivative
on the right hand side becomes a local (partial) derivative. This equation
states that the rate of change of the amount of the property in the control
mass, @, is the rate of change of the property within the control volume plus
the net flux of it through the CV boundary due to fluid motion relative to
CV boundary. The last term is usually called the convective (or sometimes,
advective) flux of q5 through the CV boundary. If the CV moves so that its
boundary coincides with the boundary of a control mass, then v = vb and
this term will be zero as required.
A detailed derivation of this equation is given in in many textbooks on
fluid dynamics (e.g. in Bird et al., 1962; Fox and McDonald, 1982) and will not
be repeated here. The mass, momentum and scalar conservation equations
will be presented in the next three sections. For convenience, a fixed CV will
be considered; fl represents the CV volume and S its surface.
1.3 Mass Conservation
The integral form of the mass conservation (continuity) equation follows directly from the control volume equation, by setting 4 = 1:
By applying the Gauss' divergence theorem to the convection term, we can
transform the surface integral into a volume integral. Allowing the control
volume to become infinitesimally small leads to a differential coordinate-free
form of the continuity equation:
ap
- + div (pv) = 0
at
This form can be transformed into a form specific to a given coordinate
system by providing the expression for the divergence operator in that system.
Expressions for common coordinate systems such as the Cartesian, cylindrical
and spherical systems can be found in many textbooks (e.g. Bird et al., 1962);
expressions applicable to general non-orthogonal coordinate systems are given
e.g. in Truesdell (1977), Aris (1989), Sedov (1971). We present below the
Cartesian form in both tensor and expanded notation. Here and throughout
this book we shall adopt the Einstein convention that whenever the same
1. Basic Concepts of Fluid Flow
where flcv is the CV volume, Scv is the surface enclosing CV, n is the unit
vector orthogonal to Scv and directed outwards, v is the fluid velocity and vb
is the velocity with which the CV surface is moving. For a fixed CV, which
we shall be considering most of the time, vb = 0 and the first derivative
on the right hand side becomes a local (partial) derivative. This equation
states that the rate of change of the amount of the property in the control
mass, @, is the rate of change of the property within the control volume plus
the net flux of it through the CV boundary due to fluid motion relative to
CV boundary. The last term is usually called the convective (or sometimes,
advective) flux of q5 through the CV boundary. If the CV moves so that its
boundary coincides with the boundary of a control mass, then v = vb and
this term will be zero as required.
A detailed derivation of this equation is given in in many textbooks on
fluid dynamics (e.g. in Bird et al., 1962; Fox and McDonald, 1982) and will not
be repeated here. The mass, momentum and scalar conservation equations
will be presented in the next three sections. For convenience, a fixed CV will
be considered; fl represents the CV volume and S its surface.
1.3 Mass Conservation
The integral form of the mass conservation (continuity) equation follows directly from the control volume equation, by setting 4 = 1:
By applying the Gauss' divergence theorem to the convection term, we can
transform the surface integral into a volume integral. Allowing the control
volume to become infinitesimally small leads to a differential coordinate-free
form of the continuity equation:
ap
- + div (pv) = 0
at
This form can be transformed into a form specific to a given coordinate
system by providing the expression for the divergence operator in that system.
Expressions for common coordinate systems such as the Cartesian, cylindrical
and spherical systems can be found in many textbooks (e.g. Bird et al., 1962);
expressions applicable to general non-orthogonal coordinate systems are given
e.g. in Truesdell (1977), Aris (1989), Sedov (1971). We present below the
Cartesian form in both tensor and expanded notation. Here and throughout
this book we shall adopt the Einstein convention that whenever the same