Nikiya Anton Bettey 2021: 09-22-2021-2047

Thursday, September 23, 2021

09-22-2021-2047 - dynamic pressure

In incompressible fluid dynamics dynamic pressure (indicated with $q$ , or Q, and sometimes called velocity pressure) is the quantity defined by:^[1]

q={\frac {1}{2}}\rho \,u^{2}

where (using SI units):

$q,\;$	dynamic pressure in pascals (i.e., kg/m⋅s²),
$\rho ,\;$	fluid mass density (e.g. in kg/m³, in SI units),
$u,\;$	flow speed in m/s.

It can be thought of as the fluid's kinetic energy per unit volume.

For incompressible flow, the dynamic pressure of a fluid is the difference between its total pressure and static pressure. From Bernoulli's law, dynamic pressure is given by

p_{0}-p_{\text{s}}={\frac {1}{2}}\rho \,u^{2}

where $p_{0}$ and $p_{\text{s}}$ are the total and static pressures, respectively.

https://en.wikipedia.org/wiki/Dynamic_pressure

In fluid mechanics the term static pressure has several uses:

In the design and operation of aircraft, static pressure is the air pressure in the aircraft's static pressure system.
In fluid dynamics, many authors use the term static pressure in preference to just pressure to avoid ambiguity. Often however, the word ‘static’ may be dropped and in that usage pressure is the same as static pressure at a nominated point in a fluid.
The term static pressure is also used by some authors in fluid statics.

https://en.wikipedia.org/wiki/Static_pressure

A pitot-static system is a system of pressure-sensitive instruments that is most often used in aviation to determine an aircraft's airspeed, Mach number, altitude, and altitude trend. A pitot-static system generally consists of a pitot tube, a static port, and the pitot-static instruments.^[1] Other instruments that might be connected are air data computers, flight data recorders, altitude encoders, cabin pressurization controllers, and various airspeed switches. Errors in pitot-static system readings can be extremely dangerous as the information obtained from the pitot static system, such as altitude, is potentially safety-critical. Several commercial airline disasters have been traced to a failure of the pitot-static system.^[2]

Pitot-static pressure[edit]

Examples of pitot tube, static tube, and pitot-static tube.

Static ports fitted to an Airbus A330 passenger airliner.

The pitot-static system of instruments uses the principle of air pressure gradient. It works by measuring pressures or pressure differences and using these values to assess the speed and altitude.^[1] These pressures can be measured either from the static port (static pressure) or the pitot tube (pitot pressure). The static pressure is used in all measurements, while the pitot pressure is used only to determine airspeed.

Pitot pressure[edit]

The pitot pressure is obtained from the pitot tube. The pitot pressure is a measure of ram air pressure (the air pressure created by vehicle motion or the air ramming into the tube), which, under ideal conditions, is equal to stagnation pressure, also called total pressure. The pitot tube is most often located on the wing or front section of an aircraft, facing forward, where its opening is exposed to the relative wind.^[1] By situating the pitot tube in such a location, the ram air pressure is more accurately measured since it will be less distorted by the aircraft's structure. When airspeed increases, the ram air pressure is increased, which can be translated by the airspeed indicator.^[1]

Static pressure[edit]

The static pressure is obtained through a static port. The static port is most often a flush-mounted hole on the fuselage of an aircraft, and is located where it can access the air flow in a relatively undisturbed area.^[1] Some aircraft may have a single static port, while others may have more than one. In situations where an aircraft has more than one static port, there is usually one located on each side of the fuselage. With this positioning, an average pressure can be taken, which allows for more accurate readings in specific flight situations.^[1] An alternative static port may be located inside the cabin of the aircraft as a backup for when the external static port(s) are blocked. A pitot-static tube effectively integrates the static ports into the pitot probe. It incorporates a second coaxial tube (or tubes) with pressure sampling holes on the sides of the probe, outside the direct airflow, to measure the static pressure. When the aircraft climbs, static pressure will decrease.

Multiple pressure[edit]

Some pitot-static systems incorporate single probes that contain multiple pressure-transmitting ports that allow for the sensing of air pressure, angle of attack, and angle of sideslip data. Depending on the design, such air data probes may be referred to as 5-hole or 7-hole air data probes. Differential pressure sensing techniques can be used to produce angle of attack and angle of sideslip indications.

Pitot-static instrument[edit]

Airspeed indicator diagram showing pressure sources from both the pitot tube and static port

The pitot-static system obtains pressures for interpretation by the pitot-static instruments. While the explanations below explain traditional, mechanical instruments, many modern aircraft use an air data computer (ADC) to calculate airspeed, rate of climb, altitude and Mach number. In some aircraft, two ADCs receive total and static pressure from independent pitot tubes and static ports, and the aircraft's flight data computer compares the information from both computers and checks one against the other. There are also "standby instruments", which are back-up pneumatic instruments employed in the case of problems with the primary instruments.

Airspeed indicator[edit]

The airspeed indicator is connected to both the pitot and static pressure sources. The difference between the pitot pressure and the static pressure is called dynamic pressure. The greater the dynamic pressure, the higher the airspeed reported. A traditional mechanical airspeed indicator contains a pressure diaphragm that is connected to the pitot tube. The case around the diaphragm is airtight and is vented to the static port. The higher the speed, the higher the ram pressure, the more pressure exerted on the diaphragm, and the larger the needle movement through the mechanical linkage.^[3]

Aneroid Wafer of an altimeter

Altimeter[edit]

The pressure altimeter, also known as the barometric altimeter, is used to determine changes in air pressure that occur as the aircraft's altitude changes.^[3] Pressure altimeters must be calibrated prior to flight to register the pressure as an altitude above sea level. The instrument case of the altimeter is airtight and has a vent to the static port. Inside the instrument, there is a sealed aneroid barometer. As pressure in the case decreases, the internal barometer expands, which is mechanically translated into a determination of altitude. The reverse is true when descending from higher to lower altitudes.^[3]

Machmeter[edit]

Aircraft designed to operate at transonic or supersonic speeds will incorporate a machmeter. The machmeter is used to show the ratio of true airspeed in relation to the speed of sound. Most supersonic aircraft are limited as to the maximum Mach number they can fly, which is known as the "Mach limit". The Mach number is displayed on a machmeter as a decimal fraction.^[3]

A vertical speed indicator

Vertical speed indicator[edit]

The variometer, also known as the vertical speed indicator (VSI) or the vertical velocity indicator (VVI), is the pitot-static instrument used to determine whether or not an aircraft is flying in level flight.^[4] The vertical speed specifically shows the rate of climb or the rate of descent, which is measured in feet per minute or meters per second.^[4] The vertical speed is measured through a mechanical linkage to a diaphragm located within the instrument. The area surrounding the diaphragm is vented to the static port through a calibrated leak (which also may be known as a "restricted diffuser").^[3] When the aircraft begins to increase altitude, the diaphragm will begin to contract at a rate faster than that of the calibrated leak, causing the needle to show a positive vertical speed. The reverse of this situation is true when an aircraft is descending.^[3] The calibrated leak varies from model to model, but the average time for the diaphragm to equalize pressure is between 6 and 9 seconds.^[3]

https://en.wikipedia.org/wiki/Pitot-static_system#Static_pressure

Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body "fluids at hydrostatic equilibrium^[1] and the pressure in a fluid, or exerted by a fluid, on an immersed body".^[2]

It encompasses the study of the conditions under which fluids are at rest in stable equilibrium as opposed to fluid dynamics, the study of fluids in motion. Hydrostatics is a subcategory of fluid statics, which is the study of all fluids, both compressible or incompressible, at rest.

Hydrostatics is fundamental to hydraulics, the engineering of equipment for storing, transporting and using fluids. It is also relevant to geophysics and astrophysics (for example, in understanding plate tectonics and the anomalies of the Earth's gravitational field), to meteorology, to medicine (in the context of blood pressure), and many other fields.

Hydrostatics offers physical explanations for many phenomena of everyday life, such as why atmospheric pressurechanges with altitude, why wood and oil float on water, and why the surface of still water is always level.

History[edit]

Some principles of hydrostatics have been known in an empirical and intuitive sense since antiquity, by the builders of boats, cisterns, aqueducts and fountains. Archimedes is credited with the discovery of Archimedes' Principle, which relates the buoyancy force on an object that is submerged in a fluid to the weight of fluid displaced by the object. The Roman engineer Vitruvius warned readers about lead pipes bursting under hydrostatic pressure.^[3]

The concept of pressure and the way it is transmitted by fluids was formulated by the French mathematician and philosopher Blaise Pascal in 1647.

Hydrostatics in ancient Greece and Rome[edit]

Pythagorean Cup[edit]

The "fair cup" or Pythagorean cup, which dates from about the 6th century BC, is a hydraulic technology whose invention is credited to the Greek mathematician and geometer Pythagoras. It was used as a learning tool.

The cup consists of a line carved into the interior of the cup, and a small vertical pipe in the center of the cup that leads to the bottom. The height of this pipe is the same as the line carved into the interior of the cup. The cup may be filled to the line without any fluid passing into the pipe in the center of the cup. However, when the amount of fluid exceeds this fill line, fluid will overflow into the pipe in the center of the cup. Due to the drag that molecules exert on one another, the cup will be emptied.

Heron's fountain[edit]

Heron's fountain is a device invented by Heron of Alexandria that consists of a jet of fluid being fed by a reservoir of fluid. The fountain is constructed in such a way that the height of the jet exceeds the height of the fluid in the reservoir, apparently in violation of principles of hydrostatic pressure. The device consisted of an opening and two containers arranged one above the other. The intermediate pot, which was sealed, was filled with fluid, and several cannula (a small tube for transferring fluid between vessels) connecting the various vessels. Trapped air inside the vessels induces a jet of water out of a nozzle, emptying all water from the intermediate reservoir.

Pascal's contribution in hydrostatics[edit]

Pascal made contributions to developments in both hydrostatics and hydrodynamics. Pascal's Law is a fundamental principle of fluid mechanics that states that any pressure applied to the surface of a fluid is transmitted uniformly throughout the fluid in all directions, in such a way that initial variations in pressure are not changed.

Pressure in fluids at rest[edit]

Due to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a shear stress. However, fluids can exert pressure normal to any contacting surface. If a point in the fluid is thought of as an infinitesimally small cube, then it follows from the principles of equilibrium that the pressure on every side of this unit of fluid must be equal. If this were not the case, the fluid would move in the direction of the resulting force. Thus, the pressure on a fluid at rest is isotropic; i.e., it acts with equal magnitude in all directions. This characteristic allows fluids to transmit force through the length of pipes or tubes; i.e., a force applied to a fluid in a pipe is transmitted, via the fluid, to the other end of the pipe. This principle was first formulated, in a slightly extended form, by Blaise Pascal, and is now called Pascal's law.

Hydrostatic pressure[edit]

In a fluid at rest, all frictional and inertial stresses vanish and the state of stress of the system is called hydrostatic. When this condition of $V = 0$ is applied to the Navier–Stokes equations, the gradient of pressure becomes a function of body forces only. For a barotropic fluid in a conservative force field like a gravitational force field, the pressure exerted by a fluid at equilibrium becomes a function of force exerted by gravity.

The hydrostatic pressure can be determined from a control volume analysis of an infinitesimally small cube of fluid. Since pressure is defined as the force exerted on a test area ( $p = F / A$ , with $p$ : pressure, $F$ : force normal to area $A$ , $A$ : area), and the only force acting on any such small cube of fluid is the weight of the fluid column above it, hydrostatic pressure can be calculated according to the following formula:

p(z)-p(z_{0})={\frac {1}{A}}\int _{z_{0}}^{z}dz'\iint _{A}dx'dy'\,\rho (z')g(z')=\int _{z_{0}}^{z}dz'\,\rho (z')g(z'),

where:

$p$ is the hydrostatic pressure (Pa),
$ρ$ is the fluid density (kg/m³),
$g$ is gravitational acceleration (m/s²),
$A$ is the test area (m²),
$z$ is the height (parallel to the direction of gravity) of the test area (m),
$z 0$ is the height of the zero reference point of the pressure (m).

For water and other liquids, this integral can be simplified significantly for many practical applications, based on the following two assumptions: Since many liquids can be considered incompressible, a reasonable good estimation can be made from assuming a constant density throughout the liquid. (The same assumption cannot be made within a gaseous environment.) Also, since the height $h$ of the fluid column between $z$ and $z 0$ is often reasonably small compared to the radius of the Earth, one can neglect the variation of $g$ . Under these circumstances, the integral is simplified into the formula:

p-p_{0}=\rho gh,

where $h$ is the height $z - z 0$ of the liquid column between the test volume and the zero reference point of the pressure. This formula is often called Stevin's law.^[4]^[5]Note that this reference point should lie at or below the surface of the liquid. Otherwise, one has to split the integral into two (or more) terms with the constant $ρ liquid$ and $ρ (z') above$ . For example, the absolute pressure compared to vacuum is:

p=\rho gH+p_{\mathrm {atm} },

where $H$ is the total height of the liquid column above the test area to the surface, and $p atm$ is the atmospheric pressure, i.e., the pressure calculated from the remaining integral over the air column from the liquid surface to infinity. This can easily be visualized using a pressure prism.

Hydrostatic pressure has been used in the preservation of foods in a process called pascalization.^[6]

Medicine[edit]

In medicine, hydrostatic pressure in blood vessels is the pressure of the blood against the wall. It is the opposing force to oncotic pressure.

Atmospheric pressure[edit]

Statistical mechanics shows that, for a pure ideal gas of constant temperature in a gravitational field, T, its pressure, p will vary with height, h, as:

p(h)=p(0)e^{-{\frac {Mgh}{kT}}}

where:

$g$ is the acceleration due to gravity
$T$ is the absolute temperature
$k$ is Boltzmann constant
$M$ is the mass of a single molecule of gas
$p$ is the pressure
$h$ is the height

This is known as the barometric formula, and maybe derived from assuming the pressure is hydrostatic.

If there are multiple types of molecules in the gas, the partial pressure of each type will be given by this equation. Under most conditions, the distribution of each species of gas is independent of the other species.

Buoyancy[edit]

Any body of arbitrary shape which is immersed, partly or fully, in a fluid will experience the action of a net force in the opposite direction of the local pressure gradient. If this pressure gradient arises from gravity, the net force is in the vertical direction opposite that of the gravitational force. This vertical force is termed buoyancy or buoyant force and is equal in magnitude, but opposite in direction, to the weight of the displaced fluid. Mathematically,

F = \rho g V

where $ρ$ is the density of the fluid, $g$ is the acceleration due to gravity, and $V$ is the volume of fluid directly above the curved surface.^[7] In the case of a ship, for instance, its weight is balanced by pressure forces from the surrounding water, allowing it to float. If more cargo is loaded onto the ship, it would sink more into the water – displacing more water and thus receive a higher buoyant force to balance the increased weight.

Discovery of the principle of buoyancy is attributed to Archimedes.

Hydrostatic force on submerged surfaces[edit]

The horizontal and vertical components of the hydrostatic force acting on a submerged surface are given by the following:^[7]

{\begin{aligned}F_{\mathrm {h} }&=p_{\mathrm {c} }A\\F_{\mathrm {v} }&=\rho gV\end{aligned}}

where:

$p c$ is the pressure at the centroid of the vertical projection of the submerged surface
$A$ is the area of the same vertical projection of the surface
$ρ$ is the density of the fluid
$g$ is the acceleration due to gravity
$V$ is the volume of fluid directly above the curved surface

Liquids (fluids with free surfaces)[edit]

Liquids can have free surfaces at which they interface with gases, or with a vacuum. In general, the lack of the ability to sustain a shear stress entails that free surfaces rapidly adjust towards an equilibrium. However, on small length scales, there is an important balancing force from surface tension.

Capillary action[edit]

When liquids are constrained in vessels whose dimensions are small, compared to the relevant length scales, surface tension effects become important leading to the formation of a meniscus through capillary action. This capillary action has profound consequences for biological systems as it is part of one of the two driving mechanisms of the flow of water in plant xylem, the transpirational pull.

Hanging drops[edit]

Without surface tension, drops would not be able to form. The dimensions and stability of drops are determined by surface tension. The drop's surface tension is directly proportional to the cohesion property of the fluid.

Static pressure in design and operation of aircraft[edit]

An aircraft's altimeter is operated by the static pressure system. An aircraft's airspeed indicator is operated by the static pressure system and the pitot pressure system.^[1]

The static pressure system is open to the exterior of the aircraft to sense the pressure of the atmosphere at the altitude at which the aircraft is flying. This small opening is called the static port. In flight the air pressure is slightly different at different positions around the exterior of the aircraft. The aircraft designer must select the position of the static port carefully. There is no position on the exterior of an aircraft at which the air pressure, for all angles of attack, is identical to the atmospheric pressure at the altitude at which the aircraft is flying.^[2] The difference in pressure causes a small error in the altitude indicated on the altimeter, and the airspeedindicated on the airspeed indicator. This error in indicated altitude and airspeed is called position error.^[3]^[4]

When selecting the position for the static port, the aircraft designer's objective is to ensure the pressure in the aircraft's static pressure system is as close as possible to the atmospheric pressure at the altitude at which the aircraft is flying, across the operating range of weight and airspeed. Many authors describe the atmospheric pressure at the altitude at which the aircraft is flying as the freestream static pressure. At least one author takes a different approach in order to avoid a need for the expression freestream static pressure. Gracey has written "The static pressure is the atmospheric pressure at the flight level of the aircraft".^[5]^[6] Gracey then refers to the air pressure at any point close to the aircraft as the local static pressure.

Static pressure in fluid dynamics[edit]

The concept of pressure is central to the study of fluids. A pressure can be identified for every point in a body of fluid, regardless of whether the fluid is in motion. Pressure can be measured using an aneroid, Bourdon tube, mercury column, or various other methods.

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 pressurein fluid dynamics, many authors use the term static pressure to distinguish it from total pressure and dynamic pressure; the term static pressure is identical to the term pressure, and can be identified for every point in a fluid flow field.

In Aerodynamics, L.J. Clancy^[7] 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."

Bernoulli's equation is fundamental to the dynamics of incompressible fluids. In many fluid flow situations of interest, changes in elevation are insignificant and can be ignored. With this simplification, Bernoulli's equation for incompressible flows can be expressed as^[8]^[9]^[10]

P+{\tfrac 12}\rho v^{2}=P_{0},

where:

$P\;$ is static pressure,
${\tfrac 12}\rho v^{2}$ is dynamic pressure, usually denoted by $q\;$ ,
$\rho \,$ is the density of the fluid,
$v\,$ is the flow velocity, and
$P_{0}\;$ is total pressure which is constant along any streamline. It is also known as the stagnation pressure.

Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own static pressure $P$ , dynamic pressure $q$ , and total pressure $P_{0}$ . Static pressure and dynamic pressure are likely to vary significantly throughout the fluid but total pressure is constant along each streamline. In irrotational flow, total pressure is the same on all streamlines and is therefore constant throughout the flow.^[11]

The simplified form of Bernoulli's equation can be summarised in the following memorable word equation:^[12]^[13]^[14]

static pressure + dynamic pressure = total pressure.

This simplified form of Bernoulli's equation is fundamental to an understanding of the design and operation of ships, low speed aircraft, and airspeed indicators for low speed aircraft – that is aircraft whose maximum speed will be less than about 30% of the speed of sound.

As a consequence of the widespread understanding of the term static pressure in relation to Bernoulli's equation, many authors^[15] in the field of fluid dynamics also use static pressure rather than pressure in applications not directly related to Bernoulli's equation.

The British Standards Institution, in its Standard^[16] Glossary of Aeronautical Terms, gives the following definition:

4412 Static pressure The pressure at a point on a body moving with the fluid.

Static pressure in fluid statics[edit]

The term (hydro)static pressure is sometimes used in fluid statics to refer to the pressure of a fluid at a nominated depth in the fluid. In fluid statics the fluid is stationary everywhere and the concepts of dynamic pressure and total pressure are not applicable. Consequently, there is little risk of ambiguity in using the term pressure, but some authors^[17] choose to use static pressure in some situations.

Nikiya Anton Bettey 2021

Blog Archive

Thursday, September 23, 2021

09-22-2021-2047 - dynamic pressure

Pitot-static pressure[edit]

Pitot pressure[edit]

Static pressure[edit]

Multiple pressure[edit]

Pitot-static instrument[edit]

Airspeed indicator[edit]

Altimeter[edit]

Machmeter[edit]

Vertical speed indicator[edit]

History[edit]

Hydrostatics in ancient Greece and Rome[edit]

Pythagorean Cup[edit]

Heron's fountain[edit]

Pascal's contribution in hydrostatics[edit]

Pressure in fluids at rest[edit]

Hydrostatic pressure[edit]

Medicine[edit]

Atmospheric pressure[edit]

Buoyancy[edit]

Hydrostatic force on submerged surfaces[edit]

Liquids (fluids with free surfaces)[edit]

Capillary action[edit]

Hanging drops[edit]

See also[edit]

Static pressure in design and operation of aircraft[edit]

Static pressure in fluid dynamics[edit]

Static pressure in fluid statics[edit]

See also[edit]

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