Bernoulli’s equation

The Bernoulli equation, which is also called Bernoulli’s law or (unambiguously) Bernoulli’s theorem, is a statement about currents after Bernoulli and Venturi. The theory of these essentially one-dimensional currents along a filament was created in the 18th century by Daniel Bernoulli and Giovanni Battista Venturi and represents the basis for important aerodynamic and hydrodynamic calculations.

The Bernoulli equation for the stationary flow of viscosity-free incompressible fluids (liquids and gases) states that the specific energy of the fluid elements along a flow line is constant:

 

 

Here is u the velocity, p the operating pressure, ρ the density, g the gravity acceleration and z the height above a reference plane in the gravity field of the earth. The operating pressure is the proportion of the static pressure that does not result from the dead weight of the fluid. The first summand represents the specific kinetic energy, the second the specific pressure energy and the third the specific position energy. The specific total specific energy e constant on the streamline is determined by appropriate reference values on the streamline.

The multiplication of the equation with the (constant) density ρ results in the Bernoull pressure equation and division by the (constant) gravity acceleration g produces an elevation equation. It contains the sum of the speed altitude , the pressure height  and the geodetic height z (in a loss-free case) constant and equal to the energy height .

Unsteadiness of flow, compressibility and viscosity of the fluid can be taken into account by extending the Bernoulli equation. It can be used for a wide range of applications in the design of technical pipe flows, in turbine and wind energy plant construction as well as for measuring devices (pitot tube, boundary probe).

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Today, the Bernoulli equation can be derived from the Navier-Stokes equations or the energy conservation theorem for the fluid elements along a streamline. However, since these connections were only found in the 19th century, Daniel Bernoulli was not able to fall back on it in his derivation of 1738. Instead, he used the preparatory work of Evangelista Torricelli, Christiaan Huygens and Gottfried Wilhelm Leibniz.

In 1640 Torricelli transferred the Galilean laws of gravity to flowing liquids, which led to Torricelli’s outflow law. In 1669, Huygens recognized that the equations of elastic shock established by René Descartes are correct when counting the velocities, taking into account their sign. In 1678, Leibniz concluded from Huygen’s law of elastic impact that the product of mass and the speed square, double the kinetic energy, are identical before and after the impact.

Daniel Bernoulli published his Hydrodynamica in 1738, see pictures, where he combined the results of Torricelli and Huygens on a fluid ball (abdc in his Fig. 72) in Sectio 12. Thus, he succeeded in determining the pressure of flowing fluids on walls and uncovering the role of the loss of kinetic energy, which he called vis viva, in the event of sudden changes in the flow cross-section.

In 1797, the Italian physicist Giovanni Battista Venturi published his discovery that the flow velocity of a liquid flowing through a pipe is inversely proportional to a changing pipe cross-section. Venturi could also prove experimentally, that static pressure at bottlenecks is lower than at other parts, see illustration for Bernoulli-effect below.

Bernoulli and Venturi looked at the quasi one-dimensional flow with plane cross-sections, which today is the subject of hydraulics and not hydrodynamics.

Characteristics of the currents according to Bernoulli and Venturi

venturi effect

Giovanni Battista Venturi discovered the continuity law for incompressible fluids: At a given volume flow A · v, the flow velocity v of an incompressible pipe flow is inversely proportional to the pipe cross-section A, so that the volume flow is constant over each cross-section, see figure. There is Δx1,2 = v1,2 Δt and follows with the constant volume V = A1 Δx1 = A2 Δx2:

 

 

This means that the speed of the fluid is greatest where the cross-section of the pipe is smallest. This effect is colloquially called jet effect. However, the above connection is only valid as long as density changes are insignificant, which is given in good approximation at flow velocities far below the speed of sound, see picture at the derivation below. In the case of a supersonic flow in a nozzle, the effect is reversed: A decreasing cross-section leads to a decrease in velocity and vice versa, which is explained in the last two articles and exploited in the lava nozzle.

The Venturi effect is noticeable in everyday life, for example, when the strength of wind increases between houses.

Bernoulli effect and hydrodynamic paradox

Venturi could also prove experimentally what the Bernoulli equation predicted, namely that the static pressure decreases at the constrictions in the Venturi nozzle, see Fig. 11 in the picture. The force to accelerate the fluid particles into the constriction is the pressure gradient force and its work p · V (specifically p / ρ) leads to an increase in the kinetic energy of the fluid particles.

The fact that where the flow is faster, the pressure is lower is called Bernoulli-effect. This effect can be shown in a simple experiment, see test sketch: Air is blown (light blue) between two sheets of paper (grey) hanging over rods (brown). Due to the higher ambient pressure compared to the reduced pressure in the air flow, the blades are compressed (black).

This fact is the hydrodynamic paradox: instead of the air stream blown in pushing the blades apart, they move together. Objects adjacent to flow zones of gases or liquids are drawn into them. Also, a water-emitting hose that is held vertically to a wall under water is not repelled from the wall, but drawn towards the wall.

The Bernoulli effect causes the human vocal lips to vibrate, which leads to vocal formation. The Bernoulli effect is technically exploited in jet pumps, chimneys and when flying, see also the Bernoulli equation below.

However, the Bernoulli effect also has negative effects: If two ships are on a parallel course, the effect can distract the ships in such a way that they collide. Likewise, a ship can go aground under the keel at high speed and with little water, because the Bernoulli effect sucks it towards the bottom. The same principle of action can lead to shortness of breath in strong winds if the wind sucks out the air resting in the airways as a result of the Bernoulli effect. Strong winds sweeping over houses also have a lower pressure than the rooms under the roof tiles, which leads to a wind load that can cover the roofs of houses.

Further conclusions from the Bernoulli equation

The Bernoulli equation explains the following facts in a steady, lossless and incompressible flow along a streamline:

  • Pascalsche’s law: At constant flow speed – especially at rest – the pressure decreases with height (or increases with depth): .
  • Torricellisches outflow law: With constant external pressure the speed square increases with decreasing height (or increasing depth): .
  • Venturi effect: With horizontal flow, the velocity square decreases in the direction of a pressure increase: .

The delta Δ stands for the difference at locations 1 and 2 on the streamline. In addition, when comparing the physical states at two points on the streamline:

  • At the same speed and pressure, the height at the points must also be the same.
  • At the same speed and same height, the pressure difference is zero.
  • With the same pressure and the same height, the speed at the points corresponds.

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This principle can be found in many things in everyday life. Here are some examples:

  • The airflow around a wing is sufficient to comply with the Bernoulli-law in case of incompressibility, see picture. The equation describes sufficiently well lift of airplane wings up to speeds of about 300 km/h (see graph below). If the air flows faster along the upper surface of the wing than on the lower surface, as shown in the picture, the Bernoulli equation implies that the static pressure on the upper surface is lower than on the underside, resulting in lift. The Bernoulli equation, however, does not explain why the flow on the upper side is faster than on the lower side.
  • the Prandtl’s stowage tube, which is used to measure the speed of an aircraft, among other things. Because of the assumed incompressibility, it delivers reliable results with the same restriction only in subsonic flight (e. g. sports aircraft).
  • the water jet pump.
  • the lip brake, a breathing technique for bronchial asthma and COPD.
  • the intake funnel of a carburettor. The Venturi flow meter and the Venturi nozzle are also named after their inventor.
  • Venting of ships through wind deflectors and Dorade fans
  • Ecological energy supply through vertical wind turbines in the Pearl River Tower (a high-rise building in Guangzhou).
  • The inductor in the fire brigade is used to produce foam in firefighting operations.

The following applies to these applications:

  • Since Bernoulli’s law only applies along a streamline, vortex-freedom is to be demanded in the considered area of the current.
  • Bernoulli’s law does not establish causality, but a relation between speed and pressure field.

Classic formulation for incompressible fluids

energy equation

The Bernoulli equation consists of three terms and can be explained graphically using a pipe system for a dam and the principle of energy conservation. Energy is given as a specific quantity, i. e. in relation to mass of fluid (, per kg in SI units).

The first partial term is formed by the velocity energy ek (kinetic energy) of the fluid resulting from the flow velocity u. If the pipe narrows, the flow velocity increases in accordance with the continuity law and thus the kinetic energy.

If the pipeline now runs over a difference in height z then gravitational acceleration g produces the position energy eto be valid. At the lower end of the dam, the positional energy of the flowing fluid is lower.

In order for the energy to remain constant in accordance with the principle of conservation of energy, a third partial term, the pressure energy wp, is required in addition to the positional energy and the kinetic energy. It is the work required to push a particle from a lower pressure area into an area of higher pressure, as is the case at the lower end of the dam. The smaller the density ρ of the particle (large volume) and the larger the pressure difference p, the more work is required. In the case of funnel-shaped constriction, the flow rate increases due to the constant volume flow, with which the fluid must be accelerated by a force. This force results from the pressure prior to constriction and reflects the pressure energy that each mass unit of fluid flowing through the pipe carries with it. This results in the energy equation given above for incompressible media

 

 

 

 

Elevation equation

Division of the Bernoulli equation by gravity acceleration: At the stationary (temporarily not changing) motion of an ideal (viscosity-free), incompressible liquid, which is only subject to gravity, the sum of velocity height is for all points of a streamline , pressure height  and geodetic height  constant:

The velocity height can be understood as the back pressure of the flow, the pressure height as a measure of the pressure of the liquid. This constant sum h along the streamline is called energy height and is given in meters.

Pressure equation

Multiplying the Bernoullian energy equation with density ρ produces the Bernoullian pressure equation:

The total pressure pt is therefore the sum of

  • the operating pressure p, which is that portion of the static pressure which does not result from the dead weight of the fluid,
  • the hydrostatic pressure ρ g z, which is the product of density ρ, gravity g and height z, and
  • the dynamic pressure ½ ρ u², which is half the product of density and velocity square,

along a streamline. From the pressure equation, for example, it can be seen that in a pipeline, an increase in velocity due to the constriction of the cross-section due to the constant mass flow in the course of the flow must lead to a reduction in pressure if the geodetic height remains constant.

Enhancements to the classical formulation

The derivation of the Bernoulli equation from the Navier-Stokes equations leads to the general Bernoulli equation in the form

They’re in there:

  • u the speed
  •  specific enthalpy in the case of isentropic flow or p / ρ in the case of incompressibility,
  • V is the specific positional energy, which assumes the shape V = g z in the gravity field of the earth,
  • η a loss term, which in the case of an incompressible flow leads to a pressure loss pV = ρ η and in the gravity field of the earth to a loss height HV = η / g, and
  •  a contribution which occurs only at unsteady flow.

The individual terms of this general Bernoulli equation are the subject of the following sections.

Extended bernoullic pressure equation of viscosity-free, ideal gases

The Bernoulli equation given above applies only to fluids with negligible density changes with sufficient accuracy. In the case of gases and major changes in velocity, the density changes associated with the pressure change must be taken into account in the energy approach:

The following formulations are available for the dependence of density on pressure

  • In the case of isothermal or isthalonic state changes, the following applies  with the specific gas constant Rs and the absolute temperature T. This is relative to a datum 0:

In there, it forms the natural logarithm.

  • In the case of an isentropic or adiabatic change of state, the following applies  with the isentropic exponent κ relative to a reference point 0 and thus 

The differential of the specific enthalpy h is dh = T ds + v dp.. In it T is the absolute temperature, s the specific entropy and v = 1/ρ the specific volume. If isentropic flow (ds = 0) is thus dh = dp/ρ and the integrand in the Bernoulli equation above corresponds to the specific enthalpy. Thus, the Bernoulli equation for real gases with isentropic flow is as follows:

Inside, is the specific enthalpy.

The following connections apply to pressure-driven equalizing currents through converging nozzles. The specific enthalpy for an ideal gas is h = cp T and with the interrelationships to be found in ideal gases.

 follows:

In it, cp,v are the specific heat capacities of the gas at constant pressure or volume.

The picture shows the enthalpy contributions h / h0 with h0 = Rs T of air according to the given formulas and isentropic state change (except for isothermal state change) relative to the reference point 0 under normal conditions.

For the orange curve “h = cv T“, the following values are used   and as with the red curve “h = cp T” was used  is used.

Extended Bernoullic energy equation of viscous liquids

The extended Bernoullian energy equation deals with viscous liquids. The friction losses are taken into account. The so-called loss level HV is usually empirically determined by a pressure loss coefficient  (Zeta) with the following function:

with

ζ: Coefficient of pressure loss
u: Speed
g: Gravity acceleration (i. e. position energy V = g z)

This assumption is based on the empirical observation that the pressure losses in pipelines increase with the square of the flow velocity in turbulent flows. The loss coefficients or the sum of the loss coefficients in an overall system are composed of:

  • Individual losses such as inlet and outlet losses, installation losses (headers, restriction, valves) and
  • Losses from pipe friction

Therefore, the pressure equation extended by the pressure loss ρ g HV is therefore:

With this equation, the usual questions of the design of piping systems with turbulent flow can be solved by knowing the loss coefficients.

For the calculation of energy losses, a distinction should be made between individual losses and losses in straight pipes.

individual losses

These are calculated according to the formula

is calculated. The pressure loss coefficients ζ are as follows

  • for inlets in pipelines:

ζ = 0.50 (vertical inlet, sharp-edged),
ζ = 0.06 to 0.005 (vertical, rounded inlet),

  • in case of sudden cross section expansion F1 → 2

or

  • Gradual constriction (angle of constriction < 20°)

ζ = 0,04.

The parameter ζ is determined according to empirical formulas that depend on the roughness of the pipeline and the flow behaviour of the medium.

Losses in straight pipelines

These are calculated according to the Darcy-Weisbach equation to

: Energy line gradient, i. e. loss height per length unit of the pipeline.

: Pipe friction coefficient (loss coefficient)

: tube diameter

is calculated.

Extended bernoullic energy equation for transient flows

The contribution of speed changes over time is suppressed in the Bernoulli equation, but can be taken into account:

The integral of local acceleration  along the streamline between points 1 and 2 is evaluated at a fixed point in time, see the example below.

The equation is considerably simplified if the flow is free of loss and rotation or – equivalent to a potential flow. Then there is a speed potential φ whose gradient is the speed. In such a flow, the extended Bernoulli equation applies

even global, i. e. for any points within flow field. The constant quantity C, which is constant at one moment within the whole flow field, could still depend on time but this time-dependency can be added to potential φ without affecting its physical meaning   would change.

Derivation

Derivation from the energy law

The Bernoulli equation can be derived from the energy balance, which requires work to be done in a stationary flow to change the energy of a fluid element. The work is that of pressure and the energies are the positional energy and the kinetic energy. It is then shown that the sum of pressure work (some inaccurate pressure energy), kinetic energy and positional energy along a streamline is constant.

Derivation from the Navier-Stokes equations

Today, the Bernoulli equation can be derived from the Navier-Stokes equations in a conservative gravitational field in a barotropic Newtonian fluid. The preconditions met allow the preliminary integration of the gradients occurring in the Navier-Stokes equations along a streamline, which leads to the Bernoulli equation. Since the pressure-density ratio of gases is temperature-dependent – gases are generally not barotropic – and liquids are often incompressible in good approximation, it is usually assumed that they are incompressible. At flows, this is well below wave propagation speed in fluid, see picture.

Example

A container as shown in the picture is located in the homogeneous gravitational field of the earth with gravity acceleration g and ambient pressure p0 and is filled with an ideal, incompressible liquid with density ρ. The difference in height between the surface AB and the discharge o between F and D is h and the diameter FD is negligibly small compared to the surface AB and the height h. The difference in height h is negligible. At time t0 = 0, the discharge is opened so that the vessel flows out in an unsteady flow, whereby the level of the vessel is kept constant by an inflow. Wanted is the outflow velocity in the discharge pipe as a function of time.

At a time t > t0, a current filament connects the surface AB (point 1) and the discharge o (point 2). The speed in the tank ACFTU can be neglected compared to AB because of the small diameter FD and the speed on the cross-sections in the drainpipe between EG and FD (over distance L) is the same everywhere u and also its change ∂u/∂t in the drainpipe is constant. The extended Bernoulli equation for transient currents thus delivers:

This non-linear ordinary first-order differential equation can be solved by separating the variables:

In ihm ist artanh der acacanthus hyperbolicus und tanh seine inverse Tangente hyperbolicus. Die Geschwindigkeit hat zur Zeit t = 0 den Wert Null und erreicht den Grenzwert für t → ∞ asymptotisch , which is Torricelli’s effluent law.

This law is derived from the Bernoulli equation faster with the assumption of a steady state flow:

 

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