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52-497: Transonic (or transsonic ) flow is air flowing around an object at a speed that generates regions of both subsonic and supersonic airflow around that object. The exact range of speeds depends on the object's critical Mach number , but transonic flow is seen at flight speeds close to the speed of sound (343 m/s at sea level), typically between Mach 0.8 and 1.2. The issue of transonic speed (or transonic region) first appeared during World War II. Pilots found as they approached

104-411: A ( x ) {\displaystyle L_{a}(x)} becomes y = f ( a ) + M ( x − a ) {\displaystyle y=f(a)+M(x-a)} . Because differentiable functions are locally linear , the best slope to substitute in would be the slope of the line tangent to f ( x ) {\displaystyle f(x)} at x =

156-441: A {\displaystyle a} is f ′ ( a ) {\displaystyle f'(a)} . To find 4.001 {\displaystyle {\sqrt {4.001}}} , we can use the fact that 4 = 2 {\displaystyle {\sqrt {4}}=2} . The linearization of f ( x ) = x {\displaystyle f(x)={\sqrt {x}}} at x =

208-671: A {\displaystyle a} is close to b {\displaystyle b} . In short, linearization approximates the output of a function near x = a {\displaystyle x=a} . For example, 4 = 2 {\displaystyle {\sqrt {4}}=2} . However, what would be a good approximation of 4.001 = 4 + .001 {\displaystyle {\sqrt {4.001}}={\sqrt {4+.001}}} ? For any given function y = f ( x ) {\displaystyle y=f(x)} , f ( x ) {\displaystyle f(x)} can be approximated if it

260-395: A {\displaystyle x=a} based on the value and slope of the function at x = b {\displaystyle x=b} , given that f ( x ) {\displaystyle f(x)} is differentiable on [ a , b ] {\displaystyle [a,b]} (or [ b , a ] {\displaystyle [b,a]} ) and that

312-409: A {\displaystyle x=a} is y = a + 1 2 a ( x − a ) {\displaystyle y={\sqrt {a}}+{\frac {1}{2{\sqrt {a}}}}(x-a)} , because the function f ′ ( x ) = 1 2 x {\displaystyle f'(x)={\frac {1}{2{\sqrt {x}}}}} defines the slope of

364-444: A {\displaystyle x=a} . While the concept of local linearity applies the most to points arbitrarily close to x = a {\displaystyle x=a} , those relatively close work relatively well for linear approximations. The slope M {\displaystyle M} should be, most accurately, the slope of the tangent line at x = a {\displaystyle x=a} . Visually,

416-475: A ) ) {\displaystyle y=(f(a)+f'(a)(x-a))} For x = a {\displaystyle x=a} , f ( a ) = f ( x ) {\displaystyle f(a)=f(x)} . The derivative of f ( x ) {\displaystyle f(x)} is f ′ ( x ) {\displaystyle f'(x)} , and the slope of f ( x ) {\displaystyle f(x)} at

468-539: A global optimum . In multiphysics systems—systems involving multiple physical fields that interact with one another—linearization with respect to each of the physical fields may be performed. This linearization of the system with respect to each of the fields results in a linearized monolithic equation system that can be solved using monolithic iterative solution procedures such as the Newton–Raphson method . Examples of this include MRI scanner systems which results in

520-461: A system of nonlinear differential equations or discrete dynamical systems . This method is used in fields such as engineering , physics , economics , and ecology . Linearizations of a function are lines —usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function y = f ( x ) {\displaystyle y=f(x)} at any x =

572-458: A German mathematician and engineer at Braunschweig , discovered Tricomi's work in the process of applying the hodograph method to transonic flow near the end of World War II. He focused on the nonlinear thin-airfoil compressible flow equations, the same as what Tricomi derived, though his goal of using these equations to solve flow over an airfoil presented unique challenges. Guderley and Hideo Yoshihara, along with some input from Busemann, later used

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624-452: A far distance they are invariably supersonic. Supernovae explosions are accompanied by supersonic flows and shock waves. Bow shocks formed in solar winds are a direct result of transonic winds from a star. It had been long thought that a bow shock was present around the heliosphere of our solar system, but this was found not to be the case according to IBEX data published in 2012. Supersonic speed Sounds are traveling vibrations in

676-449: A fatal plane accident. He lost control of the plane when a shock wave caused by supersonic airflow developed over the wing, causing it to stall. Virden flew well below the speed of sound at Mach 0.675, which brought forth the idea of different airflows forming around the plane. In the 40s, Kelly Johnson became one of the first engineers to investigate the effect of compressibility on aircraft. However, contemporary wind tunnels did not have

728-483: A flow speed close to or at Mach 1 does not allow the streamtubes (3D flow paths) to contract enough around the object to minimize the disturbance, and thus the disturbance propagates. Aerodynamicists struggled during the earlier studies of transonic flow because the then-current theory implied that these disturbances– and thus drag– approached infinity as local Mach number approached 1, an obviously unrealistic result which could not be remedied using known methods. One of

780-425: A line, given a point ( H , K ) {\displaystyle (H,K)} and slope M {\displaystyle M} . The general form of this equation is: y − K = M ( x − H ) {\displaystyle y-K=M(x-H)} . Using the point ( a , f ( a ) ) {\displaystyle (a,f(a))} , L

832-476: A point p ( a , b ) {\displaystyle p(a,b)} is: The general equation for the linearization of a multivariable function f ( x ) {\displaystyle f(\mathbf {x} )} at a point p {\displaystyle \mathbf {p} } is: where x {\displaystyle \mathbf {x} } is the vector of variables, ∇ f {\displaystyle {\nabla f}}

884-510: A rapid increase in drag from about Mach 0.8, and it is the fuel costs of the drag that typically limits the airspeed. Attempts to reduce wave drag can be seen on all high-speed aircraft. Most notable is the use of swept wings , but another common form is a wasp-waist fuselage as a side effect of the Whitcomb area rule . Transonic speeds can also occur at the tips of rotor blades of helicopters and aircraft. This puts severe, unequal stresses on

936-496: A singular solution of Tricomi's equations to analytically solve the behavior of transonic flow over a double wedge airfoil , the first to do so with only the assumptions of thin-airfoil theory. Although successful, Guderley's work was still focused on the theoretical, and only resulted in a single solution for a double wedge airfoil at Mach 1. Walter Vincenti , an American engineer at Ames Laboratory , aimed to supplement Guderley's Mach 1 work with numerical solutions that would cover

988-520: Is Mach 1 and the Prandtl–Glauert singularity . In astrophysics, wherever there is evidence of shocks (standing, propagating or oscillating), the flow close by must be transonic, as only supersonic flows form shocks. All black hole accretions are transonic. Many such flows also have shocks very close to the black holes. The outflows or jets from young stellar objects or disks around black holes can also be transonic since they start subsonically and at

1040-515: Is actually just a sonic boom . The first human-made supersonic boom was likely caused by a piece of common cloth, leading to the whip's eventual development. It's the wave motion travelling through the bullwhip that makes it capable of achieving supersonic speeds. Most modern firearm bullets are supersonic, with rifle projectiles often travelling at speeds approaching and in some cases well exceeding Mach 3 . Most spacecraft are supersonic at least during portions of their reentry, though

1092-421: Is approximately 2 + 4.001 − 4 4 = 2.00025 {\displaystyle 2+{\frac {4.001-4}{4}}=2.00025} . The true value is close to 2.00024998, so the linearization approximation has a relative error of less than 1 millionth of a percent. The equation for the linearization of a function f ( x , y ) {\displaystyle f(x,y)} at

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1144-444: Is more complex. The main key to having low supersonic drag is to properly shape the overall aircraft to be long and thin, and close to a "perfect" shape, the von Karman ogive or Sears-Haack body . This has led to almost every supersonic cruising aircraft looking very similar to every other, with a very long and slender fuselage and large delta wings, cf. SR-71 , Concorde , etc. Although not ideal for passenger aircraft, this shaping

1196-478: Is near a known differentiable point. The most basic requisite is that L a ( a ) = f ( a ) {\displaystyle L_{a}(a)=f(a)} , where L a ( x ) {\displaystyle L_{a}(x)} is the linearization of f ( x ) {\displaystyle f(x)} at x = a {\displaystyle x=a} . The point-slope form of an equation forms an equation of

1248-400: Is quite adaptable for bomber use. Linearization In mathematics , linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems , linearization is a method for assessing the local stability of an equilibrium point of

1300-418: Is that disturbances within the flow are relatively small, which allows mathematicians and engineers to linearize the compressible flow equations into a relatively easily solvable set of differential equations for either wholly subsonic or supersonic flows. This assumption is fundamentally untrue for transonic flows because the disturbance caused by an object is much larger than in subsonic or supersonic flows;

1352-408: Is the gradient , and p {\displaystyle \mathbf {p} } is the linearization point of interest . Linearization makes it possible to use tools for studying linear systems to analyze the behavior of a nonlinear function near a given point. The linearization of a function is the first order term of its Taylor expansion around the point of interest. For a system defined by

1404-397: Is very nearly the value of the tangent line at the point ( x + h , L ( x + h ) ) {\displaystyle (x+h,L(x+h))} . The final equation for the linearization of a function at x = a {\displaystyle x=a} is: y = ( f ( a ) + f ′ ( a ) ( x −

1456-558: The Black Rock Desert on 15 October 1997. The Bloodhound LSR project planned an attempt on the record in 2020 at Hakskeenpan in South Africa with a combination jet and hybrid rocket propelled car. The aim was to break the existing record, then make further attempts during which (the members of) the team hoped to reach speeds of up to 1,600 km/h (1,000 mph). The effort was originally run by Richard Noble who

1508-636: The Tupolev Tu-144 . Both of these passenger aircraft and some modern fighters are also capable of supercruise , a condition of sustained supersonic flight without the use of an afterburner . Due to its ability to supercruise for several hours and the relatively high frequency of flight over several decades, Concorde spent more time flying supersonically than all other aircraft combined by a considerable margin. Since Concorde's final retirement flight on November 26, 2003, there are no supersonic passenger aircraft left in service. Some large bombers , such as

1560-562: The Tupolev Tu-160 and Rockwell B-1 Lancer are also supersonic-capable. The aerodynamics of supersonic aircraft is simpler than subsonic aerodynamics because the airsheets at different points along the plane often cannot affect each other. Supersonic jets and rocket vehicles require several times greater thrust to push through the extra aerodynamic drag experienced within the transonic region (around Mach 0.85–1.2). At these speeds aerospace engineers can gently guide air around

1612-527: The eigenvalues of the Jacobian matrix evaluated at a hyperbolic equilibrium point to determine the nature of that equilibrium. This is the content of the linearization theorem . For time-varying systems, the linearization requires additional justification. In microeconomics , decision rules may be approximated under the state-space approach to linearization. Under this approach, the Euler equations of

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1664-469: The fuselage of the aircraft without producing new shock waves , but any change in cross area farther down the vehicle leads to shock waves along the body. Designers use the Supersonic area rule and the Whitcomb area rule to minimize sudden changes in size. However, in practical applications, a supersonic aircraft must operate stably in both subsonic and supersonic profiles, hence aerodynamic design

1716-472: The utility maximization problem are linearized around the stationary steady state. A unique solution to the resulting system of dynamic equations then is found. In mathematical optimization , cost functions and non-linear components within can be linearized in order to apply a linear solving method such as the Simplex algorithm . The optimized result is reached much more efficiently and is deterministic as

1768-405: The accompanying diagram shows the tangent line of f ( x ) {\displaystyle f(x)} at x {\displaystyle x} . At f ( x + h ) {\displaystyle f(x+h)} , where h {\displaystyle h} is any small positive or negative value, f ( x + h ) {\displaystyle f(x+h)}

1820-446: The beginning of the 20th century, the term "supersonic" was used as an adjective to describe sound whose frequency is above the range of normal human hearing. The modern term for this meaning is " ultrasonic ", but the older meaning sometimes still lives on, as in the word superheterodyne The tip of a bullwhip is generally seen as the first object designed to reach the speed of sound. This action results in its telltale "crack", which

1872-406: The best wingtip shape for sonic speeds. After World War II , major changes in aircraft design were seen to improve transonic flight. The main way to stabilize an aircraft was to reduce the speed of the airflow around the wings by changing the chord of the plane wings, and one solution to prevent transonic waves was swept wings. Since the airflow would hit the wings at an angle, this would decrease

1924-491: The capability to create wind speeds close to Mach 1 to test the effects of transonic speeds. Not long after, the term "transonic" was defined to mean "across the speed of sound" and was invented by NACA director Hugh Dryden and Theodore von Kármán of the California Institute of Technology. Initially, NACA designed "dive flaps" to help stabilize the plane when reaching transonic flight. This small flap on

1976-414: The effects on the spacecraft are reduced by low air densities. During ascent, launch vehicles generally avoid going supersonic below 30 km (~98,400 feet) to reduce air drag. Note that the speed of sound decreases somewhat with altitude, due to lower temperatures found there (typically up to 25 km). At even higher altitudes the temperature starts increasing, with the corresponding increase in

2028-641: The equation the linearized system can be written as where x 0 {\displaystyle \mathbf {x_{0}} } is the point of interest and D F ( x 0 , t ) {\displaystyle D\mathbf {F} (\mathbf {x_{0}} ,t)} is the x {\displaystyle \mathbf {x} } - Jacobian of F ( x , t ) {\displaystyle \mathbf {F} (\mathbf {x} ,t)} evaluated at x 0 {\displaystyle \mathbf {x_{0}} } . In stability analysis of autonomous systems , one can use

2080-559: The first methods used to circumvent the nonlinearity of transonic flow models was the hodograph transformation. This concept was originally explored in 1923 by an Italian mathematician named Francesco Tricomi , who used the transformation to simplify the compressible flow equations and prove that they were solvable. The hodograph transformation itself was also explored by both Ludwig Prandtl and O.G. Tietjen's textbooks in 1929 and by Adolf Busemann in 1937, though neither applied this method specifically to transonic flow. Gottfried Guderley,

2132-410: The form of pressure waves in an elastic medium. Objects move at supersonic speed when the objects move faster than the speed at which sound propagates through the medium. In gases, sound travels longitudinally at different speeds, mostly depending on the molecular mass and temperature of the gas, and pressure has little effect. Since air temperature and composition varies significantly with altitude,

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2184-525: The function f ( x ) = x {\displaystyle f(x)={\sqrt {x}}} at x {\displaystyle x} . Substituting in a = 4 {\displaystyle a=4} , the linearization at 4 is y = 2 + x − 4 4 {\displaystyle y=2+{\frac {x-4}{4}}} . In this case x = 4.001 {\displaystyle x=4.001} , so 4.001 {\displaystyle {\sqrt {4.001}}}

2236-410: The range of transonic speeds between Mach 1 and wholly supersonic flow. Vincenti and his assistants drew upon the work of Howard Emmons as well as Tricomi's original equations to complete a set of four numerical solutions for the drag over a double wedge airfoil in transonic flow above Mach 1. The gap between subsonic and Mach 1 flow was later covered by both Julian Cole and Leon Trilling , completing

2288-405: The rotor blade and may lead to accidents if it occurs. It is one of the limiting factors of the size of rotors and the forward speeds of helicopters (as this speed is added to the forward-sweeping [leading] side of the rotor, possibly causing localized transonics). Issues with aircraft flight relating to speed first appeared during the supersonic era in 1941. Ralph Virden, a test pilot, crashed in

2340-479: The sound barrier the airflow caused aircraft to become unsteady. Experts found that shock waves can cause large-scale separation downstream, increasing drag, adding asymmetry and unsteadiness to the flow around the vehicle. Research has been done into weakening shock waves in transonic flight through the use of anti-shock bodies and supercritical airfoils . Most modern jet powered aircraft are engineered to operate at transonic air speeds. Transonic airspeeds see

2392-549: The speed of sound, and Mach numbers for a steadily moving object may change. In water at room temperature supersonic speed means any speed greater than 1,440 m/s (4,724 ft/s). In solids, sound waves can be polarized longitudinally or transversely and have higher velocities. Supersonic fracture is crack formation faster than the speed of sound in a brittle material. The word supersonic comes from two Latin derived words ; 1) super : above and 2) sonus : sound, which together mean above sound, or faster than sound. At

2444-490: The speed of sound. When an inflated balloon is burst, the torn pieces of latex contract at supersonic speed, which contributes to the sharp and loud popping noise. To date, only one land vehicle has officially travelled at supersonic speed, the ThrustSSC . The vehicle, driven by Andy Green , holds the world land speed record, having achieved an average speed on its bi-directional run of 1,228 km/h (763 mph) in

2496-401: The tail of the aircraft will reach supersonic flight while the nose of the aircraft is still in subsonic flight. A bubble of supersonic expansion fans terminating by a wake shockwave surround the tail. As the aircraft continues to accelerate, the supersonic expansion fans will intensify and the wake shockwave will grow in size until infinity is reached, at which point the bow shockwave forms. This

2548-436: The transonic behavior of the airfoil by the early 1950s. At transonic speeds supersonic expansion fans form intense low-pressure, low-temperature areas at various points around an aircraft. If the temperature drops below the dew point a visible cloud will form. These clouds remain with the aircraft as it travels. It is not necessary for the aircraft as a whole to reach supersonic speeds for these clouds to form. Typically,

2600-417: The underside of the plane slowed the plane to prevent shock waves, but this design only delayed finding a solution to aircraft flying at supersonic speed. Newer wind tunnels were designed, so researchers could test newer wing designs without risking test pilots' lives. The slotted-wall transonic tunnel was designed by NASA and allowed researchers to test wings and different airfoils in transonic airflow to find

2652-474: The wing thickness and chord ratio. Airfoils wing shapes were designed flatter at the top to prevent shock waves and reduce the distance of airflow over the wing. Later on, Richard Whitcomb designed the first supercritical airfoil using similar principles. Prior to the advent of powerful computers, even the simplest forms of the compressible flow equations were difficult to solve due to their nonlinearity . A common assumption used to circumvent this nonlinearity

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2704-570: Was the leader of the ThrustSSC project, however following funding issues in 2018, the team was bought by Ian Warhurst and renamed Bloodhound LSR. Later the project was indefinitely delayed due to the COVID-19 pandemic and the vehicle was put up for sale. Most modern fighter aircraft are supersonic aircraft. No modern-day passenger aircraft are capable of supersonic speed, but there have been supersonic passenger aircraft , namely Concorde and

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