12
1. Basic Concepts of Fluid Flow
The choice of the normalization quantities is obvious in simple flows; vo
is the mean velocity and Lo is a geometric length scale; To and TI are the
cold and hot wall temperatures. If the geometry is complicated, the fluid
properties are not constant, or the boundary conditions are unsteady, the
number of dimensionless parameters needed to describe a flow can become
very large and the use of dimensionless equations may no longer be useful.
The dimensionless equations are useful for analytical studies and for determining the relative importance of various terms in the equations. They
show, for example, that steady flow in a channel or pipe depends only on
the Reynolds number; however, if the geometry changes, the flow will also be
influenced by the shape of boundary. Since we are interested in computing
flows in complex geometries, we shall use the dimensional form of transport
equations throughout this book.
1.7 Simplified Mathematical Models
The conservation equations for mass and momentum are more complex than
they appear. They are non-linear, coupled, and difficult to solve. It is difficult to prove by the existing mathematical tools that a unique solution exists
for particular boundary conditions. Experience shows that the Navier-Stokes
equations describe the flow of a Newtonian fluid accurately. Only in a small
number of cases - mostly fully developed flows in simple geometries, e.g. in
pipes, between parallel plates etc. - is it possible to obtain an analytical solution of the Navier-Stokes equations. These flows are important for studying
the fundamentals of fluid dynamics, but their practical relevance is limited.
In all cases in which such a solution is possible, many terms in the equations are zero. For other flows some terms are unimportant and we may
neglect them; this simplification introduces an error. In most cases, even the
simplified equations cannot be solved analytically; one has to use numerical methods. The computing effort may be much smaller than for the full
equations, which is a justification for simplifications. We list below some flow
types for which the equations of motion can be simplified.
1.7.1 Incompressible Flow
The conservation equations for mass and momentum presented in Sects. 1.3
and 1.4 are the most general ones; they assume that all fluid and flow properties vary in space and time. In many applications the fluid density may be
assumed constant. This is true not only for flows of liquids, whose compressibility may indeed be neglected, but also for gases if the Mach number is
below 0.3. Such flows are said to be incompressible. If the flow is also isothermal, the viscosity is also constant. In that case the mass and momentum
conservation equations (1.6) and (1.16) reduce to:
1. Basic Concepts of Fluid Flow
The choice of the normalization quantities is obvious in simple flows; vo
is the mean velocity and Lo is a geometric length scale; To and TI are the
cold and hot wall temperatures. If the geometry is complicated, the fluid
properties are not constant, or the boundary conditions are unsteady, the
number of dimensionless parameters needed to describe a flow can become
very large and the use of dimensionless equations may no longer be useful.
The dimensionless equations are useful for analytical studies and for determining the relative importance of various terms in the equations. They
show, for example, that steady flow in a channel or pipe depends only on
the Reynolds number; however, if the geometry changes, the flow will also be
influenced by the shape of boundary. Since we are interested in computing
flows in complex geometries, we shall use the dimensional form of transport
equations throughout this book.
1.7 Simplified Mathematical Models
The conservation equations for mass and momentum are more complex than
they appear. They are non-linear, coupled, and difficult to solve. It is difficult to prove by the existing mathematical tools that a unique solution exists
for particular boundary conditions. Experience shows that the Navier-Stokes
equations describe the flow of a Newtonian fluid accurately. Only in a small
number of cases - mostly fully developed flows in simple geometries, e.g. in
pipes, between parallel plates etc. - is it possible to obtain an analytical solution of the Navier-Stokes equations. These flows are important for studying
the fundamentals of fluid dynamics, but their practical relevance is limited.
In all cases in which such a solution is possible, many terms in the equations are zero. For other flows some terms are unimportant and we may
neglect them; this simplification introduces an error. In most cases, even the
simplified equations cannot be solved analytically; one has to use numerical methods. The computing effort may be much smaller than for the full
equations, which is a justification for simplifications. We list below some flow
types for which the equations of motion can be simplified.
1.7.1 Incompressible Flow
The conservation equations for mass and momentum presented in Sects. 1.3
and 1.4 are the most general ones; they assume that all fluid and flow properties vary in space and time. In many applications the fluid density may be
assumed constant. This is true not only for flows of liquids, whose compressibility may indeed be neglected, but also for gases if the Mach number is
below 0.3. Such flows are said to be incompressible. If the flow is also isothermal, the viscosity is also constant. In that case the mass and momentum
conservation equations (1.6) and (1.16) reduce to: