2.3 Water in Motion: Hydrodynamics
27
2.3.3 Rotational and Irrotational Flows
A classification of water flow as steady or unsteady, laminar or turbulent (see
Sect. 2.4), is based on the physical properties of flow. However, there is another
very important division in hydrodynamics which distinguishes rotational and
irrotational flows. The irrotationality of water flow is an abstract, essentially
mathematical concept which results in many simple and powerful methods used
in the solution of hydrodynamic problems. As will be shown in Appendix C.3.1,
for irrotational flow, unknown quantities such as velocity, pressure and acceleration can be expressed through one function called the velocity potential.
A brief analysis of fluid rotationality (or irrotationality) is given in Appendixes
C.3 and C.4, but more information can be found in many text books (for example, Batchelor, 1967; Le Mehaute, 1976; Pedlosky, 1979; Massel, 1989). It
is difficult to establish simple practical rules for assessing the validity of the
irrotationalityassumption. Some of these rules, are given below, without proof;
• A fluid flow which initially is irrotational remains irrotational in absence
of viscous and friction forces. A good example is that of still water onto
which waves propagate. The initial flow is irrotational and after the
waves arrive it will tend to remain irrotational, unless they move into a
region where viscous forces are significant. Such forces are induced by
jets, wakes, or solid boundaries. Near solid surfaces a boundary layer is
created through which the stream velocity drops to zero.
• Flow becomes rotational when density gradients are caused by stratification, rather than by pressure gradients. This is a typical situation for
large scale ocean circulation.
• Flow becomes rotational when there is significant forcing, apart from
gravity. One example of such an effect, is the so called Coriolis acceleration due to the Earth's rotation (for more details, see Chaps. 5 and 7).
• Water motion is rotational in the neighbourhood of boundaries (sea bottom and sea surface), and motion may be considered irrotational only if
the boundary layer is of little importance, i. e. relatively thin.
In this book, the property of irrotationality or rotationality will be used in
many places.
2.3.4 Mass-Conservation Equation
In general, when studying the motion of ocean water, the following quantities
must be known: three components of velocity (u, v, w), pressure p, sea surface
elevation ((x, y, t) relative to the still water level, and water density Pw' As
was shown in Chap. 1, water density, Pw, is dependent on water salinity, S, and
temperature, T, and therefore can be considered as a known. The fundamental
relationships for velocity and pressure result from the physical principles of
conservation of mass and momentum.
27
2.3.3 Rotational and Irrotational Flows
A classification of water flow as steady or unsteady, laminar or turbulent (see
Sect. 2.4), is based on the physical properties of flow. However, there is another
very important division in hydrodynamics which distinguishes rotational and
irrotational flows. The irrotationality of water flow is an abstract, essentially
mathematical concept which results in many simple and powerful methods used
in the solution of hydrodynamic problems. As will be shown in Appendix C.3.1,
for irrotational flow, unknown quantities such as velocity, pressure and acceleration can be expressed through one function called the velocity potential.
A brief analysis of fluid rotationality (or irrotationality) is given in Appendixes
C.3 and C.4, but more information can be found in many text books (for example, Batchelor, 1967; Le Mehaute, 1976; Pedlosky, 1979; Massel, 1989). It
is difficult to establish simple practical rules for assessing the validity of the
irrotationalityassumption. Some of these rules, are given below, without proof;
• A fluid flow which initially is irrotational remains irrotational in absence
of viscous and friction forces. A good example is that of still water onto
which waves propagate. The initial flow is irrotational and after the
waves arrive it will tend to remain irrotational, unless they move into a
region where viscous forces are significant. Such forces are induced by
jets, wakes, or solid boundaries. Near solid surfaces a boundary layer is
created through which the stream velocity drops to zero.
• Flow becomes rotational when density gradients are caused by stratification, rather than by pressure gradients. This is a typical situation for
large scale ocean circulation.
• Flow becomes rotational when there is significant forcing, apart from
gravity. One example of such an effect, is the so called Coriolis acceleration due to the Earth's rotation (for more details, see Chaps. 5 and 7).
• Water motion is rotational in the neighbourhood of boundaries (sea bottom and sea surface), and motion may be considered irrotational only if
the boundary layer is of little importance, i. e. relatively thin.
In this book, the property of irrotationality or rotationality will be used in
many places.
2.3.4 Mass-Conservation Equation
In general, when studying the motion of ocean water, the following quantities
must be known: three components of velocity (u, v, w), pressure p, sea surface
elevation ((x, y, t) relative to the still water level, and water density Pw' As
was shown in Chap. 1, water density, Pw, is dependent on water salinity, S, and
temperature, T, and therefore can be considered as a known. The fundamental
relationships for velocity and pressure result from the physical principles of
conservation of mass and momentum.