Bernoulli principle

A flow of air through a venturi meter. The kinetic energy increases at the expense of the fluid pressure, as shown by the difference in height of the two columns of water.

Video of a venturi meter used in a lab experiment

Bernoulli's principle is a key concept in fluid dynamics that relates pressure, speed and height. Bernoulli's principle states that an increase in the speed of a parcel of fluid occurs simultaneously with a decrease in either the pressure or the height above a datum.^[1]^: Ch.3^[2]^{: 156–164, § 3.5} The principle is named after the Swiss mathematician and physicist Daniel Bernoulli, who published it in his book Hydrodynamica in 1738.^[3] Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler in 1752 who derived Bernoulli's equation in its usual form.^[4]^[5]

Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a steady flow, the sum of all forms of energy in a fluid is the same at all points that are free of viscous forces. This requires that the sum of kinetic energy, potential energy and internal energy remains constant.^[2]^{: § 3.5} Thus an increase in the speed of the fluid—implying an increase in its kinetic energy—occurs with a simultaneous decrease in (the sum of) its potential energy (including the static pressure) and internal energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same because in a reservoir the energy per unit volume (the sum of pressure and gravitational potential $ρ g h$ ) is the same everywhere.^[6]^{: Example 3.5 and p.116}

Bernoulli's principle can also be derived directly from Isaac Newton's second Law of Motion. If a small volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then there is more pressure behind than in front. This gives a net force on the volume, accelerating it along the streamline.^[a]^[b]^[c]

Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.^[10]

Bernoulli's principle is only applicable for isentropic flows: when the effects of irreversible processes (like turbulence) and non-adiabatic processes (e.g. thermal radiation) are small and can be neglected. However, the principle can be applied to various types of flow within these bounds, resulting in various forms of Bernoulli's equation. The simple form of Bernoulli's equation is valid for incompressible flows (e.g. most liquid flows and gases moving at low Mach number). More advanced forms may be applied to compressible flows at higher Mach numbers.

Incompressible flow equation

In most flows of liquids, and of gases at low Mach number, the density of a fluid parcel can be considered to be constant, regardless of pressure variations in the flow. Therefore, the fluid can be considered to be incompressible, and these flows are called incompressible flows. Bernoulli performed his experiments on liquids, so his equation in its original form is valid only for incompressible flow.

A common form of Bernoulli's equation is:

{\frac {v^{2}}{2}}+gz+{\frac {p}{\rho }}={\text{constant}}

(A)

where:

$v$ is the fluid flow speed at a point,
$g$ is the acceleration due to gravity,
$z$ is the elevation of the point above a reference plane, with the positive $z$ -direction pointing upward—so in the direction opposite to the gravitational acceleration,
$p$ is the pressure at the chosen point, and
$\rho$ is the density of the fluid at all points in the fluid.

Bernoulli's equation and the Bernoulli constant are applicable throughout any region of flow where the energy per unit mass is uniform. Because the energy per unit mass of liquid in a well-mixed reservoir is uniform throughout, Bernoulli's equation can be used to analyze the fluid flow everywhere in that reservoir (including pipes or flow fields that the reservoir feeds) except where viscous forces dominate and erode the energy per unit mass.^[6]^{: Example 3.5 and p.116}

The following assumptions must be met for this Bernoulli equation to apply:^[2]^: 265

the flow must be steady, that is, the flow parameters (velocity, density, etc.) at any point cannot change with time,
the flow must be incompressible—even though pressure varies, the density must remain constant along a streamline;
friction by viscous forces must be negligible.

For conservative force fields (not limited to the gravitational field), Bernoulli's equation can be generalized as:^[2]^: 265

{\frac {v^{2}}{2}}+\Psi +{\frac {p}{\rho }}={\text{constant}}

where

Ψ

is the force potential at the point considered. For example, for the Earth's gravity

Ψ = gz

.

By multiplying with the fluid density $ρ$ , equation (A) can be rewritten as:

{\tfrac {1}{2}}\rho v^{2}+\rho gz+p={\text{constant}}

or:

q+\rho gh=p_{0}+\rho gz={\text{constant}}

where

$q = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1/2ρv2$ is dynamic pressure,
$h = z + p / ρg$ is the piezometric head or hydraulic head (the sum of the elevation $z$ and the pressure head)^[11]^[12] and
$p 0 = p + q$ is the stagnation pressure (the sum of the static pressure $p$ and dynamic pressure $q$ ).^[13]

The constant in the Bernoulli equation can be normalized. A common approach is in terms of total head or energy head $H$ :

H=z+{\frac {p}{\rho g}}+{\frac {v^{2}}{2g}}=h+{\frac {v^{2}}{2g}},

The above equations suggest there is a flow speed at which pressure is zero, and at even higher speeds the pressure is negative. Most often, gases and liquids are not capable of negative absolute pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be valid before zero pressure is reached. In liquids—when the pressure becomes too low—cavitation occurs. The above equations use a linear relationship between flow speed squared and pressure. At higher flow speeds in gases, or for sound waves in liquid, the changes in mass density become significant so that the assumption of constant density is invalid.

Simplified form

In many applications of Bernoulli's equation, the change in the $ρgz$ term is so small compared with the other terms that it can be ignored. For example, in the case of aircraft in flight, the change in height $z$ is so small the $ρgz$ term can be omitted. This allows the above equation to be presented in the following simplified form:

p+q=p_{0}

where

p 0

is called total pressure, and

q

is dynamic pressure.^[14] Many authors refer to the pressure

p

as static pressure to distinguish it from total pressure

p 0

and dynamic pressure

q

. In Aerodynamics, L.J. Clancy writes: "To distinguish it from the total and dynamic pressures, the actual pressure of the fluid, which is associated not with its motion but with its state, is often referred to as the static pressure, but where the term pressure alone is used it refers to this static pressure."^[1]^{: § 3.5}

The simplified form of Bernoulli's equation can be summarized in the following memorable word equation:^[1]^{: § 3.5}

static pressure + dynamic pressure = total pressure

Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own unique static pressure $p$ and dynamic pressure $q$ . Their sum $p + q$ is defined to be the total pressure $p 0$ . The significance of Bernoulli's principle can now be summarized as "total pressure is constant in any region free of viscous forces". If the fluid flow is brought to rest at some point, this point is called a stagnation point, and at this point the static pressure is equal to the stagnation pressure.

If the fluid flow is irrotational, the total pressure is uniform and Bernoulli's principle can be summarized as "total pressure is constant everywhere in the fluid flow".^[1]^{: Equation 3.12} It is reasonable to assume that irrotational flow exists in any situation where a large body of fluid is flowing past a solid body. Examples are aircraft in flight and ships moving in open bodies of water. However, Bernoulli's principle importantly does not apply in the boundary layer such as in flow through long pipes.

Unsteady potential flow

The Bernoulli equation for unsteady potential flow is used in the theory of ocean surface waves and acoustics. For an irrotational flow, the flow velocity can be described as the gradient $\nabla φ$ of a velocity potential $φ$ . In that case, and for a constant density $ρ$ , the momentum equations of the Euler equations can be integrated to:^[2]^: 383

{\frac {\partial \varphi }{\partial t}}+{\tfrac {1}{2}}v^{2}+{\frac {p}{\rho }}+gz=f(t),

which is a Bernoulli equation valid also for unsteady—or time dependent—flows. Here $\partial φ / \partial t$ denotes the partial derivative of the velocity potential $φ$ with respect to time $t$ , and $v = | \nabla φ |$ is the flow speed. The function $f (t)$ depends only on time and not on position in the fluid. As a result, the Bernoulli equation at some moment $t$ applies in the whole fluid domain. This is also true for the special case of a steady irrotational flow, in which case $f$ and $\partial φ / \partial t$ are constants so equation (A) can be applied in every point of the fluid domain.^[2]^: 383 Further $f (t)$ can be made equal to zero by incorporating it into the velocity potential using the transformation:

\Phi =\varphi -\int _{t_{0}}^{t}f(\tau )\,\mathrm {d} \tau ,

resulting in:

{\frac {\partial \Phi }{\partial t}}+{\tfrac {1}{2}}v^{2}+{\frac {p}{\rho }}+gz=0.

Note that the relation of the potential to the flow velocity is unaffected by this transformation: $\nablaΦ = \nabla φ$ .

The Bernoulli equation for unsteady potential flow also appears to play a central role in Luke's variational principle, a variational description of free-surface flows using the Lagrangian mechanics.

Compressible flow equation

Bernoulli developed his principle from observations on liquids, and Bernoulli's equation is valid for ideal fluids: those that are incompressible, irrotational, inviscid, and subjected to conservative forces. It is sometimes valid for the flow of gases: provided that there is no transfer of kinetic or potential energy from the gas flow to the compression or expansion of the gas. If both the gas pressure and volume change simultaneously, then work will be done on or by the gas. In this case, Bernoulli's equation—in its incompressible flow form—cannot be assumed to be valid. However, if the gas process is entirely isobaric, or isochoric, then no work is done on or by the gas (so the simple energy balance is not upset). According to the gas law, an isobaric or isochoric process is ordinarily the only way to ensure constant density in a gas. Also the gas density will be proportional to the ratio of pressure and absolute temperature; however, this ratio will vary upon compression or expansion, no matter what non-zero quantity of heat is added or removed. The only exception is if the net heat transfer is zero, as in a complete thermodynamic cycle or in an individual isentropic (frictionless adiabatic) process, and even then this reversible process must be reversed, to restore the gas to the original pressure and specific volume, and thus density. Only then is the original, unmodified Bernoulli equation applicable. In this case the equation can be used if the flow speed of the gas is sufficiently below the speed of sound, such that the variation in density of the gas (due to this effect) along each streamline can be ignored. Adiabatic flow at less than Mach 0.3 is generally considered to be slow enough.^[15]

It is possible to use the fundamental principles of physics to develop similar equations applicable to compressible fluids. There are numerous equations, each tailored for a particular application, but all are analogous to Bernoulli's equation and all rely on nothing more than the fundamental principles of physics such as Newton's laws of motion or the first law of thermodynamics.

Compressible flow in fluid dynamics

For a compressible fluid, with a barotropic equation of state, and under the action of conservative forces,^[16]

{\frac {v^{2}}{2}}+\int _{p_{1}}^{p}{\frac {\mathrm {d} {\tilde {p}}}{\rho \left({\tilde {p}}\right)}}+\Psi ={\text{constant (along a streamline)}}

where:

$p$ is the pressure
$ρ$ is the density and $ρ (p)$ indicates that it is a function of pressure
$v$ is the flow speed
$Ψ$ is the potential associated with the conservative force field, often the gravitational potential

In engineering situations, elevations are generally small compared to the size of the Earth, and the time scales of fluid flow are small enough to consider the equation of state as adiabatic. In this case, the above equation for an ideal gas becomes:^[1]^{: § 3.11}

{\frac {v^{2}}{2}}+gz+\left({\frac {\gamma }{\gamma -1}}\right){\frac {p}{\rho }}={\text{constant (along a streamline)}}

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