In physics and engineering, fluid dynamics
is a subdiscipline of fluid mechanics that
describes the flow of fluids—liquids and
gases. It has several subdisciplines, including
aerodynamics (the study of air and other gases
in motion) and hydrodynamics (the study of
liquids in motion). Fluid dynamics has a wide
range of applications, including calculating
forces and moments on aircraft, determining
the mass flow rate of petroleum through pipelines,
predicting weather patterns, understanding
nebulae in interstellar space and modelling
fission weapon detonation,
Fluid dynamics offers a systematic structure—which
underlies these practical disciplines—that
embraces empirical and semi-empirical laws
derived from flow measurement and used to
solve practical problems. The solution to
a fluid dynamics problem typically involves
the calculation of various properties of the
fluid, such as flow velocity, pressure, density,
and temperature, as functions of space and
time.
Before the twentieth century, hydrodynamics
was synonymous with fluid dynamics. This is
still reflected in names of some fluid dynamics
topics, like magnetohydrodynamics and hydrodynamic
stability, both of which can also be applied
to gases.
== Equations of fluid dynamics ==
The foundational axioms of fluid dynamics
are the conservation laws, specifically, conservation
of mass, conservation of linear momentum (also
known as Newton's Second Law of Motion), and
conservation of energy (also known as First
Law of Thermodynamics). These are based on
classical mechanics and are modified in quantum
mechanics and general relativity. They are
expressed using the Reynolds transport theorem.
In addition to the above, fluids are assumed
to obey the continuum assumption. Fluids are
composed of molecules that collide with one
another and solid objects. However, the continuum
assumption assumes that fluids are continuous,
rather than discrete. Consequently, it is
assumed that properties such as density, pressure,
temperature, and flow velocity are well-defined
at infinitesimally small points in space and
vary continuously from one point to another.
The fact that the fluid is made up of discrete
molecules is ignored.
For fluids that are sufficiently dense to
be a continuum, do not contain ionized species,
and have flow velocities small in relation
to the speed of light, the momentum equations
for Newtonian fluids are the Navier–Stokes
equations—which is a non-linear set of differential
equations that describes the flow of a fluid
whose stress depends linearly on flow velocity
gradients and pressure. The unsimplified equations
do not have a general closed-form solution,
so they are primarily of use in Computational
Fluid Dynamics. The equations can be simplified
in a number of ways, all of which make them
easier to solve. Some of the simplifications
allow some simple fluid dynamics problems
to be solved in closed form.In addition to
the mass, momentum, and energy conservation
equations, a thermodynamic equation of state
that gives the pressure as a function of other
thermodynamic variables is required to completely
describe the problem. An example of this would
be the perfect gas equation of state:
p
=
ρ
R
u
T
M
{\displaystyle p={\frac {\rho R_{u}T}{M}}}
where p is pressure, ρ is density, T the
absolute temperature, while Ru is the gas
constant and M is molar mass for a particular
gas.
=== Conservation laws ===
Three conservation laws are used to solve
fluid dynamics problems, and may be written
in integral or differential form. The conservation
laws may be applied to a region of the flow
called a control volume. A control volume
is a discrete volume in space through which
fluid is assumed to flow. The integral formulations
of the conservation laws are used to describe
the change of mass, momentum, or energy within
the control volume. Differential formulations
of the conservation laws apply Stokes' theorem
to yield an expression which may be interpreted
as the integral form of the law applied to
an infinitesimally small volume (at a point)
within the flow.
Mass continuity (conservation of mass): The
rate of change of fluid mass inside a control
volume must be equal to the net rate of fluid
flow into the volume. Physically, this statement
requires that mass is neither created nor
destroyed in the control volume, and can be
translated into the integral form of the continuity
equation:
∂
∂
t
∭
V
ρ
d
V
=
−
{\displaystyle {\partial \over \partial t}\iiint
_{V}\rho \,dV=-\,{}}
S
{\displaystyle {\scriptstyle S}}
ρ
u
⋅
d
S
{\displaystyle {}\,\rho \mathbf {u} \cdot
d\mathbf {S} }
Above,
ρ
{\displaystyle \rho }
is the fluid density, u is the flow velocity
vector, and t is time. The left-hand side
of the above expression is the rate of increase
of mass within the volume and contains a triple
integral over the control volume, whereas
the right-hand side contains an integration
over the surface of the control volume of
mass convected into the system. Mass flow
into the system is accounted as positive,
and since the normal vector to the surface
is opposite the sense of flow into the system
the term is negated. The differential form
of the continuity equation is, by the divergence
theorem:
∂
ρ
∂
t
+
∇
⋅
(
ρ
u
)
=
0
{\displaystyle \ {\partial \rho \over \partial
t}+\nabla \cdot (\rho \mathbf {u} )=0}
Conservation of momentum: Newton's second
law of motion applied to a control volume,
is a statement that any change in momentum
of the fluid within that control volume will
be due to the net flow of momentum into the
volume and the action of external forces acting
on the fluid within the volume.
∂
∂
t
∭
V
ρ
u
d
V
=
−
{\displaystyle {\frac {\partial }{\partial
t}}\iiint _{\scriptstyle V}\rho \mathbf {u}
\,dV=-\,{}}
S
{\displaystyle _{\scriptstyle S}}
(
ρ
u
⋅
d
S
)
u
−
{\displaystyle (\rho \mathbf {u} \cdot d\mathbf
{S} )\mathbf {u} -{}}
S
{\displaystyle {\scriptstyle S}}
p
d
S
{\displaystyle {}\,p\,d\mathbf {S} }
+
∭
V
ρ
f
body
d
V
+
F
surf
{\displaystyle \displaystyle {}+\iiint _{\scriptstyle
V}\rho \mathbf {f} _{\text{body}}\,dV+\mathbf
{F} _{\text{surf}}}
In the above integral formulation of this
equation, the term on the left is the net
change of momentum within the volume. The
first term on the right is the net rate at
which momentum is convected into the volume.
The second term on the right is the force
due to pressure on the volume's surfaces.
The first two terms on the right are negated
since momentum entering the system is accounted
as positive, and the normal is opposite the
direction of the velocity
u
{\displaystyle \mathbf {u} }
and pressure forces. The third term on the
right is the net acceleration of the mass
within the volume due to any body forces (here
represented by fbody). Surface forces, such
as viscous forces, are represented by
F
surf
{\displaystyle \mathbf {F} _{\text{surf}}}
, the net force due to shear forces acting
on the volume surface. The momentum balance
can also be written for a moving control volume.The
following is the differential form of the
momentum conservation equation. Here, the
volume is reduced to an infinitesimally small
point, and both surface and body forces are
accounted for in one total force, F. For example,
F may be expanded into an expression for the
frictional and gravitational forces acting
at a point in a flow.
D
u
D
t
=
F
−
∇
p
ρ
{\displaystyle \ {D\mathbf {u} \over Dt}=\mathbf
{F} -{\nabla p \over \rho }}
In aerodynamics, air is assumed to be a Newtonian
fluid, which posits a linear relationship
between the shear stress (due to internal
friction forces) and the rate of strain of
the fluid. The equation above is a vector
equation in a three-dimensional flow, but
it can be expressed as three scalar equations
in three coordinate directions. The conservation
of momentum equations for the compressible,
viscous flow case are called the Navier–Stokes
equations.Conservation of energy: Although
energy can be converted from one form to another,
the total energy in a closed system remains
constant.
ρ
D
h
D
t
=
D
p
D
t
+
∇
⋅
(
k
∇
T
)
+
Φ
{\displaystyle \ \rho {Dh \over Dt}={Dp \over
Dt}+\nabla \cdot \left(k\nabla T\right)+\Phi
}
Above, h is enthalpy, k is the thermal conductivity
of the fluid, T is temperature, and
Φ
{\displaystyle \Phi }
is the viscous dissipation function. The viscous
dissipation function governs the rate at which
mechanical energy of the flow is converted
to heat. The second law of thermodynamics
requires that the dissipation term is always
positive: viscosity cannot create energy within
the control volume. The expression on the
left side is a material derivative.
=== Compressible vs incompressible flow ===
All fluids are compressible to some extent;
that is, changes in pressure or temperature
cause changes in density. However, in many
situations the changes in pressure and temperature
are sufficiently small that the changes in
density are negligible. In this case the flow
can be modelled as an incompressible flow.
Otherwise the more general compressible flow
equations must be used.
Mathematically, incompressibility is expressed
by saying that the density ρ of a fluid parcel
does not change as it moves in the flow field,
i.e.,
D
ρ
D
t
=
0
,
{\displaystyle {\frac {\mathrm {D} \rho }{\mathrm
{D} t}}=0\,,}
where D/Dt is the material derivative, which
is the sum of local and convective derivatives.
This additional constraint simplifies the
governing equations, especially in the case
when the fluid has a uniform density.
For flow of gases, to determine whether to
use compressible or incompressible fluid dynamics,
the Mach number of the flow is evaluated.
As a rough guide, compressible effects can
be ignored at Mach numbers below approximately
0.3. For liquids, whether the incompressible
assumption is valid depends on the fluid properties
(specifically the critical pressure and temperature
of the fluid) and the flow conditions (how
close to the critical pressure the actual
flow pressure becomes). Acoustic problems
always require allowing compressibility, since
sound waves are compression waves involving
changes in pressure and density of the medium
through which they propagate.
=== Newtonian vs non-Newtonian fluids ===
All fluids are viscous, meaning that they
exert some resistance to deformation: neighbouring
parcels of fluid moving at different velocities
exert viscous forces on each other. The velocity
gradient is referred to as a strain rate;
it has dimensions
T
−
1
{\displaystyle T^{-1}}
. Isaac Newton showed that for many familiar
fluids such as water and air, the stress due
to these viscous forces is linearly related
to the strain rate. Such fluids are called
Newtonian fluids. The coefficient of proportionality
is called the fluid's viscosity; for Newtonian
fluids, it is a fluid property that is independent
of the strain rate.
Non-Newtonian fluids have a more complicated,
non-linear stress-strain behaviour. The sub-discipline
of rheology describes the stress-strain behaviours
of such fluids, which include emulsions and
slurries, some viscoelastic materials such
as blood and some polymers, and sticky liquids
such as latex, honey and lubricants.
=== Inviscid vs viscous vs Stokes flow ===
The dynamic of fluid parcels is described
with the help of Newton's second law. An accelerating
parcel of fluid is subject to inertial effects.
The Reynolds number is a dimensionless quantity
which characterises the magnitude of inertial
effects compared to the magnitude of viscous
effects. A low Reynolds number (Re<<1) indicates
that viscous forces are very strong compared
to inertial forces. In such cases, inertial
forces are sometimes neglected; this flow
regime is called Stokes or creeping flow.
In contrast, high Reynolds numbers (Re>>1)
indicate that the inertial effects have more
effect on the velocity field than the viscous
(friction) effects. In high Reynolds number
flows, the flow is often modeled as an inviscid
flow, an approximation in which viscosity
is completely neglected. Eliminating viscosity
allows the Navier–Stokes equations to be
simplified into the Euler equations. The integration
of the Euler equations along a streamline
in an inviscid flow yields Bernoulli's equation.
When, in addition to being inviscid, the flow
is irrotational everywhere, Bernoulli's equation
can completely describe the flow everywhere.
Such flows are called potential flows, because
the velocity field may be expressed as the
gradient of a potential energy expression.
This idea can work fairly well when the Reynolds
number is high. However, problems such as
those involving solid boundaries may require
that the viscosity be included. Viscosity
cannot be neglected near solid boundaries
because the no-slip condition generates a
thin region of large strain rate, the boundary
layer, in which viscosity effects dominate
and which thus generates vorticity. Therefore,
to calculate net forces on bodies (such as
wings), viscous flow equations must be used:
inviscid flow theory fails to predict drag
forces, a limitation known as the d'Alembert's
paradox.
A commonly used model, especially in computational
fluid dynamics, is to use two flow models:
the Euler equations away from the body, and
boundary layer equations in a region close
to the body. The two solutions can then be
matched with each other, using the method
of matched asymptotic expansions.
=== Steady vs unsteady flow ===
A flow that is not a function of time is called
steady flow. Steady-state flow refers to the
condition where the fluid properties at a
point in the system do not change over time.
Time dependent flow is known as unsteady (also
called transient). Whether a particular flow
is steady or unsteady, can depend on the chosen
frame of reference. For instance, laminar
flow over a sphere is steady in the frame
of reference that is stationary with respect
to the sphere. In a frame of reference that
is stationary with respect to a background
flow, the flow is unsteady.
Turbulent flows are unsteady by definition.
A turbulent flow can, however, be statistically
stationary. According to Pope:
The random field U(x,t) is statistically stationary
if all statistics are invariant under a shift
in time.
This roughly means that all statistical properties
are constant in time. Often, the mean field
is the object of interest, and this is constant
too in a statistically stationary flow.
Steady flows are often more tractable than
otherwise similar unsteady flows. The governing
equations of a steady problem have one dimension
fewer (time) than the governing equations
of the same problem without taking advantage
of the steadiness of the flow field.
=== Laminar vs turbulent flow ===
Turbulence is flow characterized by recirculation,
eddies, and apparent randomness. Flow in which
turbulence is not exhibited is called laminar.
The presence of eddies or recirculation alone
does not necessarily indicate turbulent flow—these
phenomena may be present in laminar flow as
well. Mathematically, turbulent flow is often
represented via a Reynolds decomposition,
in which the flow is broken down into the
sum of an average component and a perturbation
component.
It is believed that turbulent flows can be
described well through the use of the Navier–Stokes
equations. Direct numerical simulation (DNS),
based on the Navier–Stokes equations, makes
it possible to simulate turbulent flows at
moderate Reynolds numbers. Restrictions depend
on the power of the computer used and the
efficiency of the solution algorithm. The
results of DNS have been found to agree well
with experimental data for some flows.Most
flows of interest have Reynolds numbers much
too high for DNS to be a viable option, given
the state of computational power for the next
few decades. Any flight vehicle large enough
to carry a human (L > 3 m), moving faster
than 20 m/s (72 km/h) is well beyond the limit
of DNS simulation (Re = 4 million). Transport
aircraft wings (such as on an Airbus A300
or Boeing 747) have Reynolds numbers of 40
million (based on the wing chord dimension).
Solving these real-life flow problems requires
turbulence models for the foreseeable future.
Reynolds-averaged Navier–Stokes equations
(RANS) combined with turbulence modelling
provides a model of the effects of the turbulent
flow. Such a modelling mainly provides the
additional momentum transfer by the Reynolds
stresses, although the turbulence also enhances
the heat and mass transfer. Another promising
methodology is large eddy simulation (LES),
especially in the guise of detached eddy simulation
(DES)—which is a combination of RANS turbulence
modelling and large eddy simulation.
=== Subsonic vs transonic, supersonic and
hypersonic flows ===
While many flows (e.g. flow of water through
a pipe) occur at low Mach numbers, many flows
of practical interest in aerodynamics or in
turbomachines occur at high fractions of M=1
(transonic flows) or in excess of it (supersonic
or even hypersonic flows). New phenomena occur
at these regimes such as instabilities in
transonic flow, shock waves for supersonic
flow, or non-equilibrium chemical behaviour
due to ionization in hypersonic flows. In
practice, each of those flow regimes is treated
separately.
=== Reactive vs non-reactive flows ===
Reactive flows are flows that are chemically
reactive, which finds its applications in
many areas such as combustion(IC engine),
propulsion devices (Rockets, jet engines etc.),
detonations, fire and safety hazards, astrophysics
etc. In addition to conservation of mass,
momentum and energy, conservation of individual
species (for example, mass fraction of methane
in methane combustion) need to be derived,
where the production/depletion rate of any
species are obtained by simultaneously solving
the equations of chemical kinetics.
=== Magnetohydrodynamics ===
Magnetohydrodynamics is the multi-disciplinary
study of the flow of electrically conducting
fluids in electromagnetic fields. Examples
of such fluids include plasmas, liquid metals,
and salt water. The fluid flow equations are
solved simultaneously with Maxwell's equations
of electromagnetism.
=== Relativistic fluid dynamics ===
Relativistic fluid dynamics studies the macroscopic
and microscopic fluid motion at large velocities
comparable to the velocity of light. This
branch of fluid dynamics accounts the relativistic
effects both from the special theory of relativity
and the general theory of relativity. The
governing equations are derived in Riemannian
geometry for Minkowski spacetime.
=== Other approximations ===
There are a large number of other possible
approximations to fluid dynamic problems.
Some of the more commonly used are listed
below.
The Boussinesq approximation neglects variations
in density except to calculate buoyancy forces.
It is often used in free convection problems
where density changes are small.
Lubrication theory and Hele–Shaw flow exploits
the large aspect ratio of the domain to show
that certain terms in the equations are small
and so can be neglected.
Slender-body theory is a methodology used
in Stokes flow problems to estimate the force
on, or flow field around, a long slender object
in a viscous fluid.
The shallow-water equations can be used to
describe a layer of relatively inviscid fluid
with a free surface, in which surface gradients
are small.
Darcy's law is used for flow in porous media,
and works with variables averaged over several
pore-widths.
In rotating systems, the quasi-geostrophic
equations assume an almost perfect balance
between pressure gradients and the Coriolis
force. It is useful in the study of atmospheric
dynamics.
== Terminology in fluid dynamics ==
The concept of pressure is central to the
study of both fluid statics and fluid dynamics.
A pressure can be identified for every point
in a body of fluid, regardless of whether
the fluid is in motion or not. Pressure can
be measured using an aneroid, Bourdon tube,
mercury column, or various other methods.
Some of the terminology that is necessary
in the study of fluid dynamics is not found
in other similar areas of study. In particular,
some of the terminology used in fluid dynamics
is not used in fluid statics.
=== Terminology in incompressible fluid dynamics
===
The concepts of total pressure and dynamic
pressure arise from Bernoulli's equation and
are significant in the study of all fluid
flows. (These two pressures are not pressures
in the usual sense—they cannot be measured
using an aneroid, Bourdon tube or mercury
column.) To avoid potential ambiguity when
referring to pressure in fluid dynamics, many
authors use the term static pressure to distinguish
it from total pressure and dynamic pressure.
Static pressure is identical to pressure and
can be identified for every point in a fluid
flow field.
A point in a fluid flow where the flow has
come to rest (i.e. speed is equal to zero
adjacent to some solid body immersed in the
fluid flow) is of special significance. It
is of such importance that it is given a special
name—a stagnation point. The static pressure
at the stagnation point is of special significance
and is given its own name—stagnation pressure.
In incompressible flows, the stagnation pressure
at a stagnation point is equal to the total
pressure throughout the flow field.
=== Terminology in compressible fluid dynamics
===
In a compressible fluid, it is convenient
to define the total conditions (also called
stagnation conditions) for all thermodynamic
state properties (e.g. total temperature,
total enthalpy, total speed of sound). These
total flow conditions are a function of the
fluid velocity and have different values in
frames of reference with different motion.
To avoid potential ambiguity when referring
to the properties of the fluid associated
with the state of the fluid rather than its
motion, the prefix "static" is commonly used
(e.g. static temperature, static enthalpy).
Where there is no prefix, the fluid property
is the static condition (i.e. "density" and
"static density" mean the same thing). The
static conditions are independent of the frame
of reference.
Because the total flow conditions are defined
by isentropically bringing the fluid to rest,
there is no need to distinguish between total
entropy and static entropy as they are always
equal by definition. As such, entropy is most
commonly referred to as simply "entropy".
== See also ==
=== 
Fields of study ===
=== 
Mathematical equations and concepts ===
=== 
Types of fluid flow ===
=== 
Fluid properties ===
=== 
Fluid phenomena ===
=== 
Applications ===
=== 
Fluid dynamics journals ===
=== 
Miscellaneous ===
=== See also
