Turbulence modeling

From Wikipedia, the free encyclopedia

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

A simulation of a physical wind tunnel airplane model

In fluid dynamics, turbulence modeling is the construction and use of a mathematical model to predict the effects of turbulence. Turbulent flows are commonplace in most real-life scenarios. In spite of decades of research, there is no analytical theory to predict the evolution of these turbulent flows. The equations governing turbulent flows can only be solved directly for simple cases of flow. For most real-life turbulent flows, CFD simulations use turbulent models to predict the evolution of turbulence. These turbulence models are simplified constitutive equations that predict the statistical evolution of turbulent flows.

Closure problem

The Navier–Stokes equations govern the velocity and pressure of a fluid flow. In a turbulent flow, each of these quantities may be decomposed into a mean part and a fluctuating part. Averaging the equations gives the Reynolds-averaged Navier–Stokes (RANS) equations, which govern the mean flow. However, the nonlinearity of the Navier–Stokes equations means that the velocity fluctuations still appear in the RANS equations, in the nonlinear term ${\displaystyle -\rho \overline{v_i^{\mathrm{\prime }}{v}_j^{\mathrm{\prime }}}}$ from the convective acceleration. This term is known as the Reynolds stress, ${\displaystyle R_{ij}}$. Its effect on the mean flow is like that of a stress term, such as from pressure or viscosity.

To obtain equations containing only the mean velocity and pressure, we need to close the RANS equations by modelling the Reynolds stress term ${\displaystyle R_{ij}}$ as a function of the mean flow, removing any reference to the fluctuating part of the velocity. This is the closure problem.

Eddy viscosity

Joseph Valentin Boussinesq was the first to attack the closure problem, by introducing the concept of eddy viscosity. In 1877 Boussinesq proposed relating the turbulence stresses to the mean flow to close the system of equations. Here the Boussinesq hypothesis is applied to model the Reynolds stress term. Note that a new proportionality constant ${\displaystyle {\nu }_t>0}$, the (kinematic) turbulence eddy viscosity, has been introduced. Models of this type are known as eddy viscosity models (EVMs).

$${\displaystyle -\overline{v_i^{\mathrm{\prime }}{v}_j^{\mathrm{\prime }}}={\nu }_t\left(\frac{\mathrm{\partial }\overline{v_i}}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }\overline{v_j}}{\mathrm{\partial }{x}_i}\right)-\frac{2}{3}k{\delta }_{ij}}$$ which can be written in shorthand as $${\displaystyle -\overline{v_i^{\mathrm{\prime }}{v}_j^{\mathrm{\prime }}}=2{\nu }_t{S}_{ij}-\frac{2}{3}k{\delta }_{ij}}$$ where

• ${\displaystyle S_{ij}}$ is the mean rate of strain tensor
• ${\displaystyle {\nu }_t}$ is the (kinematic) turbulence eddy viscosity
• ${\displaystyle k=\frac{1}{2}\overline{v_i^{\prime }{v}_i^{\prime }}}$ is the turbulence kinetic energy
• and ${\displaystyle {\delta }_{ij}}$ is the Kronecker delta.

In this model, the additional turbulence stresses are given by augmenting the molecular viscosity with an eddy viscosity. This can be a simple constant eddy viscosity (which works well for some free shear flows such as axisymmetric jets, 2-D jets, and mixing layers).

The Boussinesq hypothesis – although not explicitly stated by Boussinesq at the time – effectively consists of the assumption that the Reynolds stress tensor is aligned with the strain tensor of the mean flow (i.e.: that the shear stresses due to turbulence act in the same direction as the shear stresses produced by the averaged flow). It has since been found to be significantly less accurate than most practitioners would assume. Still, turbulence models which employ the Boussinesq hypothesis have demonstrated significant practical value. In cases with well-defined shear layers, this is likely due the dominance of streamwise shear components, so that considerable relative errors in flow-normal components are still negligible in absolute terms. Beyond this, most eddy viscosity turbulence models contain coefficients which are calibrated against measurements, and thus produce reasonably accurate overall outcomes for flow fields of similar type as used for calibration.

Prandtl's mixing-length concept

Later, Ludwig Prandtl introduced the additional concept of the mixing length, along with the idea of a boundary layer. For wall-bounded turbulent flows, the eddy viscosity must vary with distance from the wall, hence the addition of the concept of a 'mixing length'. In the simplest wall-bounded flow model, the eddy viscosity is given by the equation: $${\displaystyle {\nu }_t=\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right|l_m^2}$$ where

• ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}$ is the partial derivative of the streamwise velocity (u) with respect to the wall normal direction (y)
• ${\displaystyle l_m}$ is the mixing length.

This simple model is the basis for the "law of the wall", which is a surprisingly accurate model for wall-bounded, attached (not separated) flow fields with small pressure gradients.

More general turbulence models have evolved over time, with most modern turbulence models given by field equations similar to the Navier–Stokes equations.

Smagorinsky model for the sub-grid scale eddy viscosity

Joseph Smagorinsky was the first who proposed a formula for the eddy viscosity in Large Eddy Simulation models, based on the local derivatives of the velocity field and the local grid size:

${\displaystyle {\nu }_t=\mathrm{\Delta }x\mathrm{\Delta }y\sqrt{{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\right)}^2+{\left(\frac{\mathrm{\partial }v}{\mathrm{\partial }y}\right)}^2+\frac{1}{2}{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+\frac{\mathrm{\partial }v}{\mathrm{\partial }x}\right)}^2}}$

In the context of Large Eddy Simulation, turbulence modeling refers to the need to parameterize the subgrid scale stress in terms of features of the filtered velocity field. This field is called subgrid-scale modeling.

Spalart–Allmaras, k–ε and k–ω models

The Boussinesq hypothesis is employed in the Spalart–Allmaras (S–A), k–ε (k–epsilon), and k–ω (k–omega) models and offers a relatively low cost computation for the turbulence viscosity ${\displaystyle {\nu }_t}$. The S–A model uses only one additional equation to model turbulence viscosity transport, while the k–ε and k–ω models use two.

Common models

The following is a brief overview of commonly employed models in modern engineering applications.

• Spalart–Allmaras (S–A)
  The Spalart–Allmaras model is a one-equation model that solves a modelled transport equation for the kinematic eddy turbulent viscosity. The Spalart–Allmaras model was designed specifically for aerospace applications involving wall-bounded flows and has been shown to give good results for boundary layers subjected to adverse pressure gradients. It is also gaining popularity in turbomachinery applications.
• k–ε (k–epsilon)
  K-epsilon (k-ε) turbulence model is the most common model used in computational fluid dynamics (CFD) to simulate mean flow characteristics for turbulent flow conditions. It is a two-equation model which gives a general description of turbulence by means of two transport equations (PDEs). The original impetus for the K-epsilon model was to improve the mixing-length model, as well as to find an alternative to algebraically prescribing turbulent length scales in moderate to high complexity flows.
• k–ω (k–omega)
  In computational fluid dynamics, the k–omega (k–ω) turbulence model is a common two-equation turbulence model that is used as a closure for the Reynolds-averaged Navier–Stokes equations (RANS equations). The model attempts to predict turbulence by two partial differential equations for two variables, k and ω, with the first variable being the turbulence kinetic energy (k) while the second (ω) is the specific rate of dissipation (of the turbulence kinetic energy k into internal thermal energy).
• SST (Menter’s Shear Stress Transport)
  SST (Menter's shear stress transport) turbulence model is a widely used and robust two-equation eddy-viscosity turbulence model used in computational fluid dynamics. The model combines the k-omega turbulence model and K-epsilon turbulence model such that the k-omega is used in the inner region of the boundary layer and switches to the k-epsilon in the free shear flow.
• Reynolds stress equation model

  The Reynolds stress equation model (RSM), also referred to as second moment closure model, is the most complete classical turbulence modelling approach. Popular eddy-viscosity based models like the k–ε (k–epsilon) model and the k–ω (k–omega) models have significant shortcomings in complex engineering flows. This arises due to the use of the eddy-viscosity hypothesis in their formulation. For instance, in flows with high degrees of anisotropy, significant streamline curvature, flow separation, zones of recirculating flow or flows influenced by rotational effects, the performance of such models is unsatisfactory. In such flows, Reynolds stress equation models offer much better accuracy.

  Eddy viscosity based closures cannot account for the return to isotropy of turbulence, observed in decaying turbulent flows. Eddy-viscosity based models cannot replicate the behaviour of turbulent flows in the Rapid Distortion limit, where the turbulent flow essentially behaves like an elastic medium.

Radio propagation

From Wikipedia, the free encyclopedia

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

Radio propagation is the behavior of radio waves as they travel, or are propagated, from one point to another in vacuum, or into various parts of the atmosphere. As a form of electromagnetic radiation, like light waves, radio waves are affected by the phenomena of reflection, refraction, diffraction, absorption, polarization, and scattering. Understanding the effects of varying conditions on radio propagation has many practical applications, from choosing frequencies for amateur radio communications, international shortwave broadcasters, to designing reliable mobile telephone systems, to radio navigation, to operation of radar systems.

Several different types of propagation are used in practical radio transmission systems. Line-of-sight propagation means radio waves which travel in a straight line from the transmitting antenna to the receiving antenna. Line of sight transmission is used for medium-distance radio transmission, such as cell phones, cordless phones, walkie-talkies, wireless networks, FM radio, television broadcasting, radar, and satellite communication (such as satellite television). Line-of-sight transmission on the surface of the Earth is limited to the distance to the visual horizon, which depends on the height of transmitting and receiving antennas. It is the only propagation method possible at microwave frequencies and above.

At lower frequencies in the MF, LF, and VLF bands, diffraction allows radio waves to bend over hills and other obstacles, and travel beyond the horizon, following the contour of the Earth. These are called surface waves or ground wave propagation. AM broadcast and amateur radio stations use ground waves to cover their listening areas. As the frequency gets lower, the attenuation with distance decreases, so very low frequency (VLF) to extremely low frequency (ELF) ground waves can be used to communicate worldwide. VLF to ELF waves can penetrate significant distances through water and earth, and these frequencies are used for mine communication and military communication with submerged submarines.

At medium wave and shortwave frequencies (MF and HF bands), radio waves can refract from the ionosphere, a layer of charged particles (ions) high in the atmosphere. This means that medium and short radio waves transmitted at an angle into the sky can be refracted back to Earth at great distances beyond the horizon – even transcontinental distances. This is called skywave propagation. It is used by amateur radio operators to communicate with operators in distant countries, and by shortwave broadcast stations to transmit internationally.

In addition, there are several less common radio propagation mechanisms, such as tropospheric scattering (troposcatter), tropospheric ducting (ducting) at VHF frequencies and near vertical incidence skywave (NVIS) which are used when HF communications are desired within a few hundred miles.

Frequency dependence

At different frequencies, radio waves travel through the atmosphere by different mechanisms or modes:

Radio frequencies and their primary mode of propagation

Band		Frequency	Wavelength	Propagation via
ELF	Extremely Low Frequency	3–30 Hz	100,000–10,000 km	Guided between the Earth and the D layer of the ionosphere.
SLF	Super Low Frequency	30–300 Hz	10,000–1,000 km	Guided between the Earth and the ionosphere.
ULF	Ultra Low Frequency	0.3–3 kHz (300–3,000 Hz)	1,000–100 km	Guided between the Earth and the ionosphere.
VLF	Very Low Frequency	3–30 kHz (3,000–30,000 Hz)	100–10 km	Guided between the Earth and the ionosphere. Ground waves.
LF	Low Frequency	30–300 kHz (30,000–300,000 Hz)	10–1 km	Guided between the Earth and the ionosphere. Ground waves.
MF	Medium Frequency	300–3,000 kHz (300,000–3,000,000 Hz)	1000–100 m	Ground waves. E, F layer ionospheric refraction at night, when D layer absorption weakens.
HF	High Frequency (Short Wave)	3–30 MHz (3,000,000–30,000,000 Hz)	100–10 m	E layer ionospheric refraction. F1, F2 layer ionospheric refraction.
VHF	Very High Frequency	30–300 MHz (30,000,000–     300,000,000 Hz)	10–1 m	Line-of-sight propagation. Infrequent E ionospheric (Es) refraction. Uncommonly F2 layer ionospheric refraction during high sunspot activity up to 50 MHz and rarely to 80 MHz. Sometimes tropospheric ducting or meteor scatter
UHF	Ultra High Frequency	300–3,000 MHz (300,000,000–     3,000,000,000 Hz)	100–10 cm	Line-of-sight propagation. Sometimes tropospheric ducting.
SHF	Super High Frequency	3–30 GHz (3,000,000,000–     30,000,000,000 Hz)	10–1 cm	Line-of-sight propagation. Sometimes rain scatter.
EHF	Extremely High Frequency	30–300 GHz (30,000,000,000–     300,000,000,000 Hz)	10–1 mm	Line-of-sight propagation, limited by atmospheric absorption to a few kilometers (miles)
THF	Tremendously High frequency	0.3–3 THz (300,000,000,000–     3,000,000,000,000 Hz)	1–0.1 mm	Line-of-sight propagation, limited by atmospheric absorption to a few meters.
FIR	Far infrared light (overlaps radio)	0.3–20 THz (300,000,000,000–     20,000,000,000,000 Hz)	1,000–150 μm	Line-of-sight propagation, mostly limited by atmospheric absorption to a few meters.

Free space propagation

Further information: Free-space path loss

In free space, all electromagnetic waves (radio, light, X-rays, etc.) obey the inverse-square law which states that the power density ${\displaystyle \rho }$ of an electromagnetic wave is proportional to the inverse of the square of the distance ${\displaystyle r}$ from a point source or:

${\displaystyle \rho \propto \frac{1}{r^2}.}$

At typical communication distances from a transmitter, the transmitting antenna usually can be approximated by a point source. Doubling the distance of a receiver from a transmitter means that the power density of the radiated wave at that new location is reduced to one-quarter of its previous value.

The power density per surface unit is proportional to the product of the electric and magnetic field strengths. Thus, doubling the propagation path distance from the transmitter reduces each of these received field strengths over a free-space path by one-half.

Radio waves in vacuum travel at the speed of light. The Earth's atmosphere is thin enough that radio waves in the atmosphere travel very close to the speed of light, but variations in density and temperature can cause some slight refraction (bending) of waves over distances.

Direct modes (line-of-sight)

Main article: Line-of-sight propagation

Line-of-sight refers to radio waves which travel directly in a line from the transmitting antenna to the receiving antenna, often also called direct-wave. It does not necessarily require a cleared sight path; at lower frequencies radio waves can pass through buildings, foliage and other obstructions. This is the most common propagation mode at VHF and above, and the only possible mode at microwave frequencies and above. On the surface of the Earth, line of sight propagation is limited by the visual horizon to about 40 miles (64 km). This is the method used by cell phones, cordless phones, walkie-talkies, wireless networks, point-to-point microwave radio relay links, FM and television broadcasting and radar. Satellite communication uses longer line-of-sight paths; for example home satellite dishes receive signals from communication satellites 22,000 miles (35,000 km) above the Earth, and ground stations can communicate with spacecraft billions of miles from Earth.

Ground plane reflection effects are an important factor in VHF line-of-sight propagation. The interference between the direct beam line-of-sight and the ground reflected beam often leads to an effective inverse-fourth-power (1⁄distance⁴) law for ground-plane limited radiation.

Surface modes (groundwave)

Main article: Ground wave

Ground wave propagation

Lower frequency (between 30 and 3,000 kHz) vertically polarized radio waves can travel as surface waves following the contour of the Earth; this is called ground wave propagation.

In this mode the radio wave propagates by interacting with the conductive surface of the Earth. The wave "clings" to the surface and thus follows the curvature of the Earth, so ground waves can travel over mountains and beyond the horizon. Ground waves propagate in vertical polarization so vertical antennas (monopoles) are required. Since the ground is not a perfect electrical conductor, ground waves are attenuated as they follow the Earth's surface. Attenuation is proportional to frequency, so ground waves are the main mode of propagation at lower frequencies, in the MF, LF and VLF bands. Ground waves are used by radio broadcasting stations in the MF and LF bands, and for time signals and radio navigation systems.

At even lower frequencies, in the VLF to ELF bands, an Earth-ionosphere waveguide mechanism allows even longer range transmission. These frequencies are used for secure military communications. They can also penetrate to a significant depth into seawater, and so are used for one-way military communication to submerged submarines.

Early long-distance radio communication (wireless telegraphy) before the mid-1920s used low frequencies in the longwave bands and relied exclusively on ground-wave propagation. Frequencies above 3 MHz were regarded as useless and were given to hobbyists (radio amateurs). The discovery around 1920 of the ionospheric reflection or skywave mechanism made the medium wave and short wave frequencies useful for long-distance communication and they were allocated to commercial and military users.

Non-line-of-sight modes

Non-line-of-sight (NLOS) radio propagation occurs outside of the typical line-of-sight (LOS) between the transmitter and receiver, such as in ground reflections. Near-line-of-sight (also NLOS) conditions refer to partial obstruction by a physical object present in the innermost Fresnel zone.

Obstacles that commonly cause NLOS propagation include buildings, trees, hills, mountains, and, in some cases, high voltage electric power lines. Some of these obstructions reflect certain radio frequencies, while some simply absorb or garble the signals; but, in either case, they limit the use of many types of radio transmissions, especially when low on power budget.

Lower power levels at a receiver reduce the chance of successfully receiving a transmission. Low levels can be caused by at least three basic reasons: low transmit level, for example Wi-Fi power levels; far-away transmitter, such as 3G more than 5 miles (8.0 km) away or TV more than 31 miles (50 km) away; and obstruction between the transmitter and the receiver, leaving no clear path.

NLOS lowers the effective received power. Near Line Of Sight can usually be dealt with using better antennas, but Non Line Of Sight usually requires alternative paths or multipath propagation methods.

How to achieve effective NLOS networking has become one of the major questions of modern computer networking. Currently, the most common method for dealing with NLOS conditions on wireless computer networks is simply to circumvent the NLOS condition and place relays at additional locations, sending the content of the radio transmission around the obstructions. Some more advanced NLOS transmission schemes now use multipath signal propagation, bouncing the radio signal off other nearby objects to get to the receiver.

Non-Line-of-Sight (NLOS) is a term often used in radio communications to describe a radio channel or link where there is no visual line of sight (LOS) between the transmitting antenna and the receiving antenna. In this context LOS is taken

• Either as a straight line free of any form of visual obstruction, even if it is actually too distant to see with the unaided human eye
• As a virtual LOS i.e., as a straight line through visually obstructing material, thus leaving sufficient transmission for radio waves to be detected

There are many electrical characteristics of the transmission media that affect the radio wave propagation and therefore the quality of operation of a radio channel, if it is possible at all, over an NLOS path.

The acronym NLOS has become more popular in the context of wireless local area networks (WLANs) and wireless metropolitan area networks such as WiMAX because the capability of such links to provide a reasonable level of NLOS coverage greatly improves their marketability and versatility in the typical urban environments where they are most frequently used. However, NLOS contains many other subsets of radio communications.

The influence of a visual obstruction on a NLOS link may be anything from negligible to complete suppression. An example might apply to a LOS path between a television broadcast antenna and a roof mounted receiving antenna. If a cloud passed between the antennas the link could actually become NLOS but the quality of the radio channel could be virtually unaffected. If, instead, a large building was constructed in the path making it NLOS, the channel may be impossible to receive.

Beyond line-of-sight (BLOS) is a related term often used in the military to describe radio communications capabilities that link personnel or systems too distant or too fully obscured by terrain for LOS communications. These radios utilize active repeaters, groundwave propagation, tropospheric scatter links, and ionospheric propagation to extend communication ranges from a few kilometers to a few thousand kilometers.

Measuring HF propagation

HF propagation conditions can be simulated using radio propagation models, such as the Voice of America Coverage Analysis Program, and realtime measurements can be done using chirp transmitters. For radio amateurs the WSPR mode provides maps with real time propagation conditions between a network of transmitters and receivers. Even without special beacons the realtime propagation conditions can be measured: A worldwide network of receivers decodes morse code signals on amateur radio frequencies in realtime and provides sophisticated search functions and propagation maps for every station received.

Practical effects

The average person can notice the effects of changes in radio propagation in several ways.

In AM broadcasting, the dramatic ionospheric changes that occur overnight in the mediumwave band drive a unique broadcast license scheme in the United States, with entirely different transmitter power output levels and directional antenna patterns to cope with skywave propagation at night. Very few stations are allowed to run without modifications during dark hours, typically only those on clear channels in North America. Many stations have no authorization to run at all outside of daylight hours.

For FM broadcasting (and the few remaining low-band TV stations), weather is the primary cause for changes in VHF propagation, along with some diurnal changes when the sky is mostly without cloud cover. These changes are most obvious during temperature inversions, such as in the late-night and early-morning hours when it is clear, allowing the ground and the air near it to cool more rapidly. This not only causes dew, frost, or fog, but also causes a slight "drag" on the bottom of the radio waves, bending the signals down such that they can follow the Earth's curvature over the normal radio horizon. The result is typically several stations being heard from another media market – usually a neighboring one, but sometimes ones from a few hundred kilometers (miles) away. Ice storms are also the result of inversions, but these normally cause more scattered omnidirection propagation, resulting mainly in interference, often among weather radio stations. In late spring and early summer, a combination of other atmospheric factors can occasionally cause skips that duct high-power signals to places well over 1000 km (600 miles) away.

Non-broadcast signals are also affected. Mobile phone signals are in the UHF band, ranging from 700 to over 2600 MHz, a range which makes them even more prone to weather-induced propagation changes. In urban (and to some extent suburban) areas with a high population density, this is partly offset by the use of smaller cells, which use lower effective radiated power and beam tilt to reduce interference, and therefore increase frequency reuse and user capacity. However, since this would not be very cost-effective in more rural areas, these cells are larger and so more likely to cause interference over longer distances when propagation conditions allow.

While this is generally transparent to the user thanks to the way that cellular networks handle cell-to-cell handoffs, when cross-border signals are involved, unexpected charges for international roaming may occur despite not having left the country at all. This often occurs between southern San Diego and northern Tijuana at the western end of the U.S./Mexico border, and between eastern Detroit and western Windsor along the U.S./Canada border. Since signals can travel unobstructed over a body of water far larger than the Detroit River, and cool water temperatures also cause inversions in surface air, this "fringe roaming" sometimes occurs across the Great Lakes, and between islands in the Caribbean. Signals can skip from the Dominican Republic to a mountainside in Puerto Rico and vice versa, or between the U.S. and British Virgin Islands, among others. While unintended cross-border roaming is often automatically removed by mobile phone company billing systems, inter-island roaming is typically not.

Empirical models

A radio propagation model, also known as the radio wave propagation model or the radio frequency propagation model, is an empirical mathematical formulation for the characterization of radio wave propagation as a function of frequency, distance and other conditions. A single model is usually developed to predict the behavior of propagation for all similar links under similar constraints. Created with the goal of formalizing the way radio waves are propagated from one place to another, such models typically predict the path loss along a link or the effective coverage area of a transmitter.

The inventor of radio communication, Guglielmo Marconi, before 1900 formulated the first crude empirical rule of radio propagation: the maximum transmission distance varied as the square of the height of the antenna.

As the path loss encountered along any radio link serves as the dominant factor for characterization of propagation for the link, radio propagation models typically focus on realization of the path loss with the auxiliary task of predicting the area of coverage for a transmitter or modeling the distribution of signals over different regions.

Because each individual telecommunication link has to encounter different terrain, path, obstructions, atmospheric conditions and other phenomena, it is intractable to formulate the exact loss for all telecommunication systems in a single mathematical equation. As a result, different models exist for different types of radio links under different conditions. The models rely on computing the median path loss for a link under a certain probability that the considered conditions will occur.

Radio propagation models are empirical in nature, which means, they are developed based on large collections of data collected for the specific scenario. For any model, the collection of data has to be sufficiently large to provide enough likeliness (or enough scope) to all kind of situations that can happen in that specific scenario. Like all empirical models, radio propagation models do not point out the exact behavior of a link, rather, they predict the most likely behavior the link may exhibit under the specified conditions.

Different models have been developed to meet the needs of realizing the propagation behavior in different conditions. Types of models for radio propagation include:

Models for free space attenuation

• Free-space path loss
• Dipole field strength in free space
• Friis transmission equation

Models for outdoor attenuation

• Terrain models
  • ITU terrain model
  • Egli model
  • Longley–Rice irregular terrain model (ITM)
  • Two-ray ground-reflection model
• City models
  • Okumura model
  • Hata model
  • COST Hata model

Models for indoor attenuation

• ITU model for indoor attenuation
• Log-distance path loss model

Recoil

From Wikipedia, the free encyclopedia

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

An early naval cannon, which is allowed to roll backwards slightly when fired, and therefore must be tethered with strong ropes

Recoil (often called knockback, kickback or simply kick) is the rearward thrust generated when a gun is being discharged. In technical terms, the recoil is a result of conservation of momentum, for according to Newton's third law the force required to accelerate something will evoke an equal but opposite reactional force, which means the forward momentum gained by the projectile and exhaust gases (ejectae) will be mathematically balanced out by an equal and opposite momentum exerted back upon the gun.

Basics

Any launching system (weapon or not) generates recoil. However recoil only constitutes a problem in the field of artillery and firearms due to the magnitude of the forces at play. Gun chamber pressures and projectile acceleration forces are tremendous, on the order of tens to hundreds megapascal and tens of thousands of times the acceleration of gravity (g's), both necessary to launch the projectile at useful velocity during the very short time (typically only a few milliseconds) it is travelling inside the barrel. Meanwhile, the same pressures acting on the base of the projectile are acting on the rear face of the gun chamber, accelerating the gun rearward during firing with just the same force it is accelerating the projectile forward.

This moves the gun rearward and generates the recoil momentum. This recoil momentum is the product of the mass and the acceleration of the projectile and propellant gasses combined, reversed: the projectile moves forward, the recoil is rearward. The heavier and the faster the projectile, the more recoil will be generated. The gun acquires a rearward velocity that is ratio of this momentum by the mass of the gun: the heavier the gun, the slower the rearward velocity. As an example, a 8 g (124 gr) bullet of 9×19mm Parabellum flying forward at 350 m/s muzzle speed generates a momentum to push a 0.8 kg pistol firing it at 3.5 m/s rearward, if unopposed by the shooter.

Countering recoil

In order to bring the rearward moving gun to a halt, the momentum acquired by the gun is dissipated by a forward-acting counter-recoil force applied to the gun over a period of time during and after the projectile exits the muzzle. In hand-held small arms, the shooter will apply this force using their own body, resulting in a noticeable impulse commonly referred to as a "kick". In heavier mounted guns, such as heavy machine guns or artillery pieces, recoil momentum is transferred through the platform on which the weapon is mounted. Practical weight gun mounts are typically not strong enough to withstand the maximum forces accelerating the gun during the short time the projectile is in the barrel. To mitigate these large recoil forces, recoil buffering mechanisms spread out the counter-recoiling force over a longer time, typically ten to a hundred times longer than the duration of the forces accelerating the projectile. This results in the required counter-recoiling force being proportionally lower, and easily absorbed by the gun mount.

To apply this counter-recoiling force, modern mounted guns may employ recoil buffering comprising springs and hydraulic recoil mechanisms, similar to shock-absorbing suspension on automobiles. Early cannons used systems of ropes along with rolling or sliding friction to provide forces to slow the recoiling cannon to a stop. Recoil buffering allows the maximum counter-recoil force to be lowered so that strength limitations of the gun mount are not exceeded.

Contribution of propellant gasses

Modern cannons also employ muzzle brakes very effectively to redirect some of the propellant gasses rearward after projectile exit. This provides a counter-recoiling force to the barrel, allowing the buffering system and gun mount to be more efficiently designed at even lower weight.

Propellant gases are even more tapped in recoilless guns, where much of the high pressure gas remaining in the barrel after projectile exit is vented rearward though a nozzle at the back of the chamber, creating a large counter-recoiling force sufficient to eliminate the need for heavy recoil mitigating buffers on the mount (although at the cost of a reduced muzzle velocity of the projectile).

Hand-held guns

The same physics principles affecting recoil in mounted guns also applies to hand-held guns. However, the shooter's body assumes the role of gun mount, and must similarly dissipate the gun's recoiling momentum over a longer period of time than the bullet travel-time in the barrel, in order not to injure the shooter. Hands, arms and shoulders have considerable strength and elasticity for this purpose, up to certain practical limits. Nevertheless, "perceived" recoil limits vary from shooter to shooter, depending on body size, the use of recoil padding, individual pain tolerance, the weight of the firearm, and whether recoil buffering systems and muzzle devices (muzzle brake or suppressor) are employed. For this reason, establishing recoil safety standards for small arms remains challenging, in spite of the straightforward physics involved.

Physics: momentum, energy and impulse

There are two conservation laws at work when a gun is fired: conservation of momentum and conservation of energy. Recoil is explained by the law of conservation of momentum, and so it is easier to discuss it separately from energy.

Momentum is simply mass multiplied by velocity. Velocity is speed in a particular direction (not just speed). In a very technical sense, speed is a scalar (mathematics): a magnitude; while velocity is a vector (physics): magnitude and direction. Momentum is conservative: any change in momentum of an object requires an equal and opposite change of some other objects. Hence the recoil: imparting momentum to the projectile requires imparting opposite momentum to the gun.

A change in the momentum of a mass requires the application of a force (see Newton's laws of motion). Forces within a firearm wildly change, so what matters is impulse: the change of momentum is equal to the impulse. The rapid change of velocity (acceleration) of the gun is a shock and will be countered as if by a shock absorber.

Energy in firing a firearm comes in many forms (thermal, pressure) but for understanding recoil what matters is kinetic energy, which is half mass multiplied by squared speed. For the recoiling gun, this means that for a given rearward momentum, doubling the mass halves the speed and also halves the kinetic energy of the gun, making it easier to dissipate.

Momentum

If all the masses and velocities involved are accounted for, the vector sum, magnitude and direction, of the momentum of all the bodies involved does not change; that is, momentum of the system is conserved. This conservation of momentum is why gun recoil occurs in the opposite direction of bullet projection—the mass times velocity of the projectile (gas included) in the positive direction equals the mass times velocity of the gun in the negative direction. In summation, the total momentum of the system (ammunition, gun and shooter/shooting platform)) equals zero just as it did before the trigger was pulled.

From a practical engineering perspective, therefore, through the mathematical application of conservation of momentum, it is possible to calculate a first approximation of a gun's recoil momentum and kinetic energy simply based on estimates of the projectile speed (and mass) coming out the barrel. And then to properly design recoil buffering systems to safely dissipate that momentum and energy. To confirm analytical calculations and estimates, once a prototype gun is manufactured, the projectile and gun recoil energy and momentum can be directly measured using a ballistic pendulum and ballistic chronograph.

The nature of the recoil process is determined by the force of the expanding gases in the barrel upon the gun (recoil force), which is equal and opposite to the force upon the ejecta. It is also determined by the counter-recoil force applied to the gun (e.g. an operator's hand or shoulder, or a mount). The recoil force only acts during the time that the ejecta are still in the barrel of the gun. The counter-recoil force is generally applied over a longer time period and adds forward momentum to the gun equal to the backward momentum supplied by the recoil force, in order to bring the gun to a halt. There are two special cases of counter recoil force: Free-recoil, in which the time duration of the counter-recoil force is very much larger than the duration of the recoil force, and zero-recoil, in which the counter-recoil force matches the recoil force in magnitude and duration. Except for the case of zero-recoil, the counter-recoil force is smaller than the recoil force but lasts for a longer time. Since the recoil force and the counter-recoil force are not matched, the gun will move rearward, slowing down until it comes to rest. In the zero-recoil case, the two forces are matched and the gun will not move when fired. In most cases, a gun is very close to a free-recoil condition, since the recoil process generally lasts much longer than the time needed to move the ejecta down the barrel. An example of near zero-recoil would be a gun securely clamped to a massive or well-anchored table, or supported from behind by a massive wall. However, employing zero-recoil systems is often neither practical nor safe for the structure of the gun, as the recoil momentum must be absorbed directly through the very small distance of elastic deformation of the materials the gun and mount are made from, perhaps exceeding their strength limits. For example, placing the butt of a large caliber gun directly against a wall and pulling the trigger risks cracking both the gun stock and the surface of the wall.

The recoil of a firearm, whether large or small, is a result of the law of conservation of momentum. Assuming that the firearm and projectile are both at rest before firing, then their total momentum is zero. Assuming a near free-recoil condition, and neglecting the gases ejected from the barrel, (an acceptable first estimate), then immediately after firing, conservation of momentum requires that the total momentum of the firearm and projectile is the same as before, namely zero. Stating this mathematically: $${\displaystyle p_{\text{f}}+p_{\text{p}}=0}$$ where ${\displaystyle p_{\text{f}}}$ is the momentum of the firearm and ${\displaystyle p_{\text{p}}}$ is the momentum of the projectile. In other words, immediately after firing, the momentum of the firearm is equal and opposite to the momentum of the projectile.

Since momentum of a body is defined as its mass multiplied by its velocity, we can rewrite the above equation as: $${\displaystyle m_{\text{f}}{v}_{\text{f}}+m_{\text{p}}{v}_{\text{p}}=0}$$ where:

• ${\displaystyle m_{\text{f}}}$ is the mass of the firearm
• ${\displaystyle v_{\text{f}}}$ is the velocity of the firearm immediately after firing
• ${\displaystyle m_{\text{p}}}$ is the mass of the projectile
• ${\displaystyle v_{\text{p}}}$ is the velocity of the projectile immediately after firing

A force integrated over the time period during which it acts will yield the momentum supplied by that force. The counter-recoil force must supply enough momentum to the firearm to bring it to a halt. This means that:

$${\displaystyle {\int }_0^{t_{\text{cr}}}{F}_{\text{cr}}(t)dt=-m_{\text{f}}{v}_{\text{f}}=m_{\text{p}}{v}_{\text{p}}}$$

where:

• ${\displaystyle F_{\text{cr}}(t)}$ is the counter-recoil force as a function of time (t)
• ${\displaystyle t_{\text{cr}}}$ is duration of the counter-recoil force

A similar equation can be written for the recoil force on the firearm:

$${\displaystyle {\int }_0^{t_{\text{r}}}{F}_{\text{r}}(t)dt=m_{\text{f}}{v}_{\text{f}}=-m_{\text{p}}{v}_{\text{p}}}$$

where:

• ${\displaystyle F_{\text{r}}(t)}$ is the recoil force as a function of time (t)
• ${\displaystyle t_{\text{r}}}$ is duration of the recoil force

Assuming the forces are somewhat evenly spread out over their respective durations, the condition for free-recoil is ${\displaystyle t_{\text{r}}\ll t_{\text{cr}}}$, while for zero-recoil, ${\displaystyle F_{\text{r}}(t)+F_{\text{cr}}(t)=0}$.

Angular momentum

For a gun firing under free-recoil conditions, the force on the gun may not only force the gun backwards, but may also cause it to rotate about its center of mass or recoil mount. This is particularly true of older firearms, such as the classic Kentucky rifle, where the butt stock angles down significantly lower than the barrel, providing a pivot point about which the muzzle may rise during recoil. Modern firearms, such as the M16 rifle, employ stock designs that are in direct line with the barrel, in order to minimize any rotational effects. If there is an angle for the recoil parts to rotate about, the torque (${\displaystyle \tau }$) on the gun is given by:

$${\displaystyle \tau =I\frac{d^2\theta }{d{t}^2}=hF(t)}$$

where $h$ is the perpendicular distance of the center of mass of the gun below the barrel axis, $F(t)$ is the force on the gun due to the expanding gases, equal and opposite to the force on the bullet, $I$ is the moment of inertia of the gun about its center of mass, or its pivot point, and ${\displaystyle \theta }$ is the angle of rotation of the barrel axis "up" from its orientation at ignition (aim angle). The angular momentum of the gun is found by integrating this equation to obtain: $${\displaystyle I\frac{d\theta }{dt}=h{\int }_0^{t}F(t)dt=h{m}_{\text{g}}{V}_{\text{g}}(t)=h{m}_{\text{b}}{V}_{\text{b}}(t)}$$ where the equality of the momenta of the gun and bullet have been used. The angular rotation of the gun as the bullet exits the barrel is then found by integrating again: $${\displaystyle I{\theta }_f=h{\int }_0^{t_f}{m}_{\text{b}}{V}_{\text{b}}dt=2h{m}_{\text{b}}L}$$

where ${\displaystyle {\theta }_f}$ is the angle above the aim angle at which the bullet leaves the barrel, ${\displaystyle t_f}$ is the time of travel of the bullet in the barrel (because of the acceleration ${\displaystyle a=2x/t^2}$ the time is longer than ${\displaystyle L/V_{\text{b}}}$ : ${\displaystyle t_f=2L/V_{\text{b}}}$) and L is the distance the bullet travels from its rest position to the tip of the barrel. The angle at which the bullet leaves the barrel above the aim angle is then given by: $${\displaystyle {\theta }_f=\frac{2h{m}_{\text{b}}L}{I}}$$

Including the ejected gas

Before the projectile leaves the gun barrel, it obturates the bore and "plugs up" the expanding gas generated by the propellant combustion behind it. This means that the gas is essentially contained within a closed system and acts as a neutral element in the overall momentum of the system's physics. However, when the projectile exits the barrel, this functional seal is removed and the highly energetic bore gas is suddenly free to exit the muzzle and expand in the form of a supersonic shockwave (which can be often fast enough to momentarily overtake the projectile and affect its flight dynamics), creating a phenomenon known as the muzzle blast. The forward vector of this blast creates a jet propulsion effect that exerts back upon the barrel, and creates an additional momentum on top of the backward momentum generated by the projectile before it exits the gun.

The overall recoil applied to the firearm is equal and opposite to the total forward momentum of not only the projectile, but also the ejected gas. Likewise, the recoil energy given to the firearm is affected by the ejected gas. By conservation of mass, the mass of the ejected gas will be equal to the original mass of the propellant (assuming complete burning). As a rough approximation, the ejected gas can be considered to have an effective exit velocity of ${\displaystyle \alpha V_0}$ where ${\displaystyle V_0}$ is the muzzle velocity of the projectile and ${\displaystyle \alpha }$ is approximately constant. The total momentum ${\displaystyle p_e}$ of the propellant and projectile will then be: $${\displaystyle p_e=m_p{V}_0+m_{\text{g}}\alpha V_0}$$ where ${\displaystyle m_{\text{g}}}$ is the mass of the propellant charge, equal to the mass of the ejected gas.

This expression should be substituted into the expression for projectile momentum in order to obtain a more accurate description of the recoil process. The effective velocity may be used in the energy equation as well, but since the value of α used is generally specified for the momentum equation, the energy values obtained may be less accurate. The value of the constant α is generally taken to lie between 1.25 and 1.75. It is mostly dependent upon the type of propellant used, but may depend slightly on other things such as the ratio of the length of the barrel to its radius.

Muzzle devices can reduce the recoil impulse by altering the pattern of gas expansion. For instance, muzzle brakes primarily works by diverting some of the gas ejecta towards the sides, increasing the lateral blast intensity (hence louder to the sides) but reducing the thrust from the forward-projection (thus less recoil). Similarly, recoil compensators divert the gas ejecta mostly upwards to counteract the muzzle rise. However, suppressors work on a different principle, not by vectoring the gas expansion laterally but instead by modulating the forward speed of the gas expansion. By using internal baffles, the gas is made to travel through a convoluted path before eventually released outside at the front of the suppressor, thus dissipating its energy over a larger area and a longer time. This reduces both the intensity of the blast (thus lower loudness) and the recoil generated (as for the same impulse, force is inversely proportional to time).

Perception of recoil

Recoil while firing Smith & Wesson Model 500 revolver

For small arms, the way in which the shooter perceives the recoil, or kick, can have a significant impact on the shooter's experience and performance. For example, a gun that is said to "kick like a mule" is going to be approached with trepidation, and the shooter may anticipate the recoil and flinch in anticipation as the shot is released. This leads to the shooter jerking the trigger, rather than pulling it smoothly, and the jerking motion is almost certain to disturb the alignment of the gun and may result in a miss. The shooter may also be physically injured by firing a weapon generating recoil in excess of what the body can safely absorb or restrain; perhaps getting hit in the eye by the rifle scope, hit in the forehead by a handgun as the elbow bends under the force, or soft tissue damage to the shoulder, wrist and hand; and these results vary for individuals. In addition, as pictured in the image, excessive recoil can create serious range safety concerns, if the shooter cannot adequately restrain the firearm in a down-range direction.

Perception of recoil is related to the deceleration the body provides against a recoiling gun, deceleration being a force that slows the velocity of the recoiling mass. Force applied over a distance is energy. The force that the body feels, therefore, is dissipating the kinetic energy of the recoiling gun mass. A heavier gun, that is a gun with more mass, will manifest lower recoil kinetic energy, and, generally, result in a lessened perception of recoil. Therefore, although determining the recoiling energy that must be dissipated through a counter-recoiling force is arrived at by conservation of momentum, kinetic energy of recoil is what is actually being restrained and dissipated. The ballistics analyst discovers this recoil kinetic energy through analysis of projectile momentum.

One of the common ways of describing the felt recoil of a particular gun-cartridge combination is as "soft" or "sharp" recoiling; soft recoil is recoil spread over a longer period of time, that is at a lower deceleration, and sharp recoil is spread over a shorter period of time, that is with a higher deceleration. Like pushing softer or harder on the brakes of a car, the driver feels less or more deceleration force being applied, over a longer or shorter distance to bring the car to a stop. However, for the human body to mechanically adjust recoil time, and hence length, to lessen felt recoil force is perhaps an impossible task. Other than employing less safe and less accurate practices, such as shooting from the hip, shoulder padding is a safe and effective mechanism that allows sharp recoiling to be lengthened into soft recoiling, as lower decelerating force is transmitted into the body over a slightly greater distance and time, and spread out over a slightly larger surface.

Keeping the above in mind, one can generally base the relative recoil of firearms by factoring in a small number of parameters: bullet momentum and the mass of the firearm. Lowering momentum lowers recoil, all else being the same. Increasing the firearm's mass also lowers recoil, again all else being the same. The following are base examples calculated through the Handloads.com free online calculator, and bullet and firearm data from respective reloading manuals (of medium/common loads) and manufacturer specifications:

• In a Glock 22 frame, using the empty weight of 1.43 lb (0.65 kg), the following was obtained:
  • 9 mm Luger: Recoil impulse of 0.78 lb_f·s (3.5 N·s); Recoil velocity of 17.55 ft/s (5.3 m/s); Recoil energy of 6.84 ft⋅lb_f (9.3 J)
  • .357 SIG: Recoil impulse of 1.06 lb_f·s (4.7 N·s); Recoil velocity of 23.78 ft/s (7.2 m/s); Recoil energy of 12.56 ft⋅lb_f (17.0 J)
  • .40 S&W: Recoil impulse of 0.88 lb_f·s (3.9 N·s); Recoil velocity of 19.73 ft/s (6.0 m/s); Recoil energy of 8.64 ft⋅lb_f (11.7 J)
• In a Smith & Wesson .44 Magnum with 7.5-inch barrel, with an empty weight of 3.125 lb (1.417 kg), the following was obtained:
  • .44 Remington Magnum: Recoil impulse of 1.91 lb_f·s (8.5 N·s); Recoil velocity of 19.69 ft/s (6.0 m/s); Recoil energy of 18.81 ft⋅lb_f (25.5 J)
• In a Smith & Wesson 460 7.5-inch barrel, with an empty weight of 3.5 lb (1.6 kg), the following was obtained:
  • .460 S&W Magnum: Recoil impulse of 3.14 lb_f·s (14.0 N·s); Recoil velocity of 28.91 ft/s (8.8 m/s); Recoil energy of 45.43 ft⋅lb_f (61.6 J)
• In a Smith & Wesson 500 4.5-inch barrel, with an empty weight of 3.5 lb (1.6 kg), the following was obtained:
  • .500 S&W Magnum: Recoil impulse of 3.76 lb_f·s (16.7 N·s); Recoil velocity of 34.63 ft/s (10.6 m/s); Recoil energy of 65.17 ft⋅lb_f (88.4 J)

In addition to the overall mass of the gun, reciprocating parts of the gun will affect how the shooter perceives recoil. While these parts are not part of the ejecta, and do not alter the overall momentum of the system, they do involve moving masses during the operation of firing. For example, gas-operated shotguns are widely held to have a "softer" recoil than fixed breech or recoil-operated guns (although many semi-automatic recoil and gas-operated guns incorporate recoil buffer systems into the stock that effectively spread out peak felt recoil forces). In a gas-operated gun, the bolt is accelerated rearwards by propellant gases during firing, which results in a forward force on the body of the gun. This is countered by a rearward force as the bolt reaches the limit of travel and moves forwards, resulting in a zero sum, but to the shooter, the recoil has been spread out over a longer period of time, resulting in the "softer" feel.

Mounted guns

Photograph of the kickback of a cannon, taken in Morges Castle, Switzerland

Recoilless designs allow larger and faster projectiles to be shoulder-launched.

A recoil system absorbs recoil energy, reducing the peak force that is conveyed to whatever the gun is mounted on. Old-fashioned cannons without a recoil system roll several meters backwards when fired; systems were used to somewhat limit this movement (ropes, friction including brakes on wheels, slopes so that the recoil would force the gun uphill,...), but utterly preventing any movement would just have resulted in the mount breaking. As a result, guns had to be put back into firing position and carefully aimed again after each shot, dramatically slowing the firing rate. The modern quick-firing guns was made possible by the invention of a much more efficient device: the hydro-pneumatic recoil system. First developed by Wladimir Baranovsky in 1872–5 and adopted by the Russian army, then later in France, in the 75mm field gun of 1897, it is still the main device used by big guns nowadays.

Main article: Hydraulic recoil mechanism

In this system, the barrel is mounted on rails on which it can recoil to the rear, and the recoil is taken up by a cylinder which is similar in operation to an automotive gas-charged shock absorber, and is commonly visible as a cylinder shorter and smaller than the barrel mounted parallel to it. The cylinder contains a charge of compressed air that will act as a spring, as well as hydraulic oil; in operation, the barrel's energy is taken up in compressing the air as the barrel recoils backward, then is dissipated via hydraulic damping as the barrel is returned forward to the firing position under the pressure of the compressed air. The recoil impulse is thus spread out over the time in which the barrel is compressing the air, rather than over the much narrower interval of time when the projectile is being fired. This greatly reduces the peak force conveyed to the mount (or to the ground on which the gun has been placed).

Soft-recoil

In a soft-recoil system, the spring (or air cylinder) that returns the barrel to the forward position starts out in a nearly fully compressed state, then the gun's barrel is released free to fly forward in the moment before firing; the charge is then ignited just as the barrel reaches the fully forward position. Since the barrel is still moving forward when the charge is ignited, about half of the recoil impulse is applied to stopping the forward motion of the barrel, while the other half is, as in the usual system, taken up in recompressing the spring. A latch then catches the barrel and holds it in the starting position. This roughly halves the energy that the spring needs to absorb, and also roughly halves the peak force conveyed to the mount, as compared to the usual system. However, the need to reliably achieve ignition at a single precise instant is a major practical difficulty with this system; and unlike the usual hydro-pneumatic system, soft-recoil systems do not easily deal with hangfires or misfires. One of the early guns to use this system was the French 65 mm mle.1906; it was also used by the World War II British PIAT man-portable anti-tank weapon.

Other devices

Recoilless rifles and rocket launchers exhaust gas to the rear, balancing the recoil. They are used often as light anti-tank weapons. The Swedish-made Carl Gustav 84mm recoilless gun is such a weapon.

In machine guns following Hiram Maxim's design – e.g. the Vickers machine gun – the recoil of the barrel is used to drive the feed mechanism.