1.5 Conservation of Scalar Quantities
9
The pressure gradient is then regarded as a body force; this amounts to
non-conservative treatment of the pressure term. The non-conservative form
of equations is often used in finite difference methods, since it is somewhat
simpler. In the limit of a very fine grid, all equation forms and numerical
solution methods give the same solution; however, on coarse grids the nonconservative form introduces additional errors which may become important.
If the expression for the viscous part of the stress tensor, Eq. (1.13),
is substituted into Eq. (1.16) written in index notation and for Cartesian
coordinates, and if gravity is the only body force, one has:
where g, is the component of the gravitational acceleration g in the direction
of the Cartesian coordinate xi. For the case of constant density and gravity,
the term pg can be written as grad (pg . r ) , where r is the position vector,
r = x,ii (usually, gravity is assumed to act in the negative t-direction, i.e.
g = g t k , g, being negative; in this case g . r = g,z). Then -pg,t is the
hydrostatic pressure, and it is convenient - and for numerical solution more
efficient - to define 5 = p-pg,z as the head and use it in place of the pressure.
The term pgi then disappears from the above equation. If the actual pressure
is needed, one has only t o add pg,z to @.
Since only the gradient of the pressure appears in the equation, the absolute value of the pressure is not important except in compressible flows.
In variable density flows (the variation of gravity can be neglected in all
flows considered in this book), one can split the pgi term into two parts:
pogi + (p - po)gi, where po is a reference density. The first part can then be
included with pressure and if the density variation is retained only in the
gravitational term, we have the Boussinesq approximation, see Sect. 1.7.
1.5 Conservation of Scalar Quantities
The integral form of the equation describing conservation of a scalar quantity,
4, is analogous to the previous equations and reads:
where fm represents transport of 4 by mechanisms other than convection and
any sources or sinks of the scalar. Diffusive transport is always present (even
in stagnant fluids), and it is usually described by a gradient approximation,
e.g. Fourier's law for heat diffusion and Fick's law for mass diffusion:
9
The pressure gradient is then regarded as a body force; this amounts to
non-conservative treatment of the pressure term. The non-conservative form
of equations is often used in finite difference methods, since it is somewhat
simpler. In the limit of a very fine grid, all equation forms and numerical
solution methods give the same solution; however, on coarse grids the nonconservative form introduces additional errors which may become important.
If the expression for the viscous part of the stress tensor, Eq. (1.13),
is substituted into Eq. (1.16) written in index notation and for Cartesian
coordinates, and if gravity is the only body force, one has:
where g, is the component of the gravitational acceleration g in the direction
of the Cartesian coordinate xi. For the case of constant density and gravity,
the term pg can be written as grad (pg . r ) , where r is the position vector,
r = x,ii (usually, gravity is assumed to act in the negative t-direction, i.e.
g = g t k , g, being negative; in this case g . r = g,z). Then -pg,t is the
hydrostatic pressure, and it is convenient - and for numerical solution more
efficient - to define 5 = p-pg,z as the head and use it in place of the pressure.
The term pgi then disappears from the above equation. If the actual pressure
is needed, one has only t o add pg,z to @.
Since only the gradient of the pressure appears in the equation, the absolute value of the pressure is not important except in compressible flows.
In variable density flows (the variation of gravity can be neglected in all
flows considered in this book), one can split the pgi term into two parts:
pogi + (p - po)gi, where po is a reference density. The first part can then be
included with pressure and if the density variation is retained only in the
gravitational term, we have the Boussinesq approximation, see Sect. 1.7.
1.5 Conservation of Scalar Quantities
The integral form of the equation describing conservation of a scalar quantity,
4, is analogous to the previous equations and reads:
where fm represents transport of 4 by mechanisms other than convection and
any sources or sinks of the scalar. Diffusive transport is always present (even
in stagnant fluids), and it is usually described by a gradient approximation,
e.g. Fourier's law for heat diffusion and Fick's law for mass diffusion: