1.7 Simplified Mathematical Models
15
fluid motion. If the density variation is not large, one may treat the density
as constant in the unsteady and convection terms, and treat it as variable
only in the gravitational term. This is called the Boussinesq approximation.
One usually assumes that the density varies linearly with temperature. If one
includes the effect of the body force on the mean density in the pressure term
as described in Sect. 1.4, the remaining term can be expressed as:
where p is the coefficient of volumetric expansion. This approximation introduces errors of the order of 1% if the temperature differences are below e.g.
2" for water and 15" for air. The error may be more substantial when temperature differences are larger; the solution may even be qualitatively wrong
(for an example, see Biickle and PeriC, 1992).
1.7.6 Boundary Layer Approximation
When the flow has a predominant direction (i.e. there is no reversed flow or
recirculation) and the variation of the geometry is gradual, the flow is mainly
influenced by what happened upstream. Examples are flows in channels and
pipes and flows over plane or mildly curved solid walls. Such flows are called
thin shear layer or boundary layer flows. The Navier-Stokes equations can be
simplified for such flows as follows:
0 diffusive transport of momentum in the principal flow direction is much
smaller than convection and can be neglected;
0 the velocity component in the main flow direction is much larger than the
components in other directions;
0 the pressure gradient across the flow is much smaller than in the principal
flow direction.
The two-dimensional boundary layer equations reduce to:
which must be solved together with the continuity equation; the equation for
the momentum normal to the principal flow direction reduces to dpldx2 = 0.
The pressure as a function of X I must be supplied by a calculation of the flow
exterior t o the boundary layer - which is usually assumed to be potential
flow, so the boundary layer equations themselves are not a complete description of the flow. The simplified equations can be solved by using marching
techniques similar t o those used to solve ordinary differential equations with
initial conditions. These techniques see considerable use in aerodynamics.
The methods are very efficient but can be applied only to problems without
separation.
15
fluid motion. If the density variation is not large, one may treat the density
as constant in the unsteady and convection terms, and treat it as variable
only in the gravitational term. This is called the Boussinesq approximation.
One usually assumes that the density varies linearly with temperature. If one
includes the effect of the body force on the mean density in the pressure term
as described in Sect. 1.4, the remaining term can be expressed as:
where p is the coefficient of volumetric expansion. This approximation introduces errors of the order of 1% if the temperature differences are below e.g.
2" for water and 15" for air. The error may be more substantial when temperature differences are larger; the solution may even be qualitatively wrong
(for an example, see Biickle and PeriC, 1992).
1.7.6 Boundary Layer Approximation
When the flow has a predominant direction (i.e. there is no reversed flow or
recirculation) and the variation of the geometry is gradual, the flow is mainly
influenced by what happened upstream. Examples are flows in channels and
pipes and flows over plane or mildly curved solid walls. Such flows are called
thin shear layer or boundary layer flows. The Navier-Stokes equations can be
simplified for such flows as follows:
0 diffusive transport of momentum in the principal flow direction is much
smaller than convection and can be neglected;
0 the velocity component in the main flow direction is much larger than the
components in other directions;
0 the pressure gradient across the flow is much smaller than in the principal
flow direction.
The two-dimensional boundary layer equations reduce to:
which must be solved together with the continuity equation; the equation for
the momentum normal to the principal flow direction reduces to dpldx2 = 0.
The pressure as a function of X I must be supplied by a calculation of the flow
exterior t o the boundary layer - which is usually assumed to be potential
flow, so the boundary layer equations themselves are not a complete description of the flow. The simplified equations can be solved by using marching
techniques similar t o those used to solve ordinary differential equations with
initial conditions. These techniques see considerable use in aerodynamics.
The methods are very efficient but can be applied only to problems without
separation.