|
Kármán–Moore Theory
Kármán–Moore theory is a linearized theory for supersonic flows over a slender body, named after Theodore von Kármán and Norton B. Moore, who developed the theory in 1932. The theory, in particular, provides an explicit formula for the wave drag, which converts the kinetic energy of the moving body into outgoing sound waves behind the body. Mathematical description Consider a slender body with pointed edges at the front and back. The supersonic flow past this body will be nearly parallel to the x-axis everywhere since the shock waves formed (one at the leading edge and one at the trailing edge) will be weak; as a consequence, the flow will be potential everywhere, which can be described using the velocity potential \varphi = xv_1 + \phi, where v_1 is the incoming uniform velocity and \phi characterising the small deviation from the uniform flow. In the linearized theory, \phi satisfies :\frac + \frac - \beta^2 \frac =0, where \beta^2=(v_1^2-c_1^2)/c_1^2=M_1^2-1, c_1 is the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersonic
Supersonic speed is the speed of an object that exceeds the speed of sound (Mach 1). For objects traveling in dry air of a temperature of 20 °C (68 °F) at sea level, this speed is approximately . Speeds greater than five times the speed of sound (Mach 5) are often referred to as hypersonic. Flights during which only some parts of the air surrounding an object, such as the ends of rotor blades, reach supersonic speeds are called transonic. This occurs typically somewhere between Mach 0.8 and Mach 1.2. Sounds are traveling vibrations in 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, the speed of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theodore Von Kármán
Theodore von Kármán ( , May 11, 1881May 6, 1963) was a Hungarian-American mathematician, aerospace engineer, and physicist who worked in aeronautics and astronautics. He was responsible for crucial advances in aerodynamics characterizing supersonic and hypersonic airflow. The human-defined threshold of outer space is named the " Kármán line" in recognition of his work. Kármán is regarded as an outstanding aerodynamic theoretician of the 20th century. Early life Theodore von Kármán was born into a Jewish family in Budapest, then part of Austria-Hungary, as Kármán Tódor, the son of Helene (Konn or Kohn, ) and . Among his ancestors were Rabbi Judah Loew ben Bezalel, who was said to be the creator of the Golem of Prague, and Rabbi , who wrote about Zohar. His father, Mór, was a well-known educator, who reformed the Hungarian school system and founded Minta Gymnasium in Budapest. He became an influential figure and became a commissioner of the Ministry of Educa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Drag
In aeronautics, wave drag is a component of the aerodynamic drag In fluid dynamics, drag, sometimes referred to as fluid resistance, is a force acting opposite to the direction of motion of any object moving with respect to a surrounding fluid. This can exist between two fluid layers, two solid surfaces, or b ... on aircraft wings and fuselage, propeller blade tips and Shell (projectile), projectiles moving at transonic and supersonic speeds, due to the presence of shock waves. Wave drag is independent of viscous effects,Clancy, L.J. (1975), ''Aerodynamics'', Section 11.7 and tends to present itself as a sudden and dramatic increase in drag as the vehicle increases speed to the critical Mach number. It is the sudden and dramatic rise of wave drag that leads to the concept of a sound barrier. Overview Wave drag is a component of pressure drag due to compressibility effects. It is caused by the formation of shock waves around a body. Shock waves create a considerable amount ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Velocity Potential
A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788. It is used in continuum mechanics, when a continuum occupies a simply-connected region and is irrotational. In such a case, \nabla \times \mathbf =0 \,, where denotes the flow velocity. As a result, can be represented as the gradient of a scalar function : \mathbf = \nabla \varphi\ = \frac \mathbf + \frac \mathbf + \frac \mathbf \,. is known as a velocity potential for . A velocity potential is not unique. If is a velocity potential, then is also a velocity potential for , where is a scalar function of time and can be constant. Velocity potentials are unique up to a constant, or a function solely of the temporal variable. The Laplacian of a velocity potential is equal to the divergence of the corresponding flow. Hence if a velocity potential satisfies Laplace equation, the flow is incompressible. Unlike a stream function, a velocity potential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sound Speed
The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. More simply, the speed of sound is how fast vibrations travel. At , the speed of sound in air is about , or in or one mile in . It depends strongly on temperature as well as the medium through which a sound wave is propagating. At , the speed of sound in dry air (sea level 14.7 psi) is about . The speed of sound in an ideal gas depends only on its temperature and composition. The speed has a weak dependence on frequency and pressure in dry air, deviating slightly from ideal behavior. In colloquial speech, ''speed of sound'' refers to the speed of sound waves in air. However, the speed of sound varies from substance to substance: typically, sound travels most slowly in gases, faster in liquids, and fastest in solids. For example, while sound travels at in air, it travels at in water (almost 4.3 times as fast) and at in iron (almost 15 times as fast). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mach Number
The Mach number (M or Ma), often only Mach, (; ) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound. It is named after the Austrian physicist and philosopher Ernst Mach. \mathrm = \frac, where: * is the local Mach number, * is the local flow velocity with respect to the boundaries (either internal, such as an object immersed in the flow, or external, like a channel), and * is the speed of sound in the medium, which in air varies with the square root of the thermodynamic temperature. By definition, at Mach1, the local flow velocity is equal to the speed of sound. At Mach0.65, is 65% of the speed of sound (subsonic), and, at Mach1.35, is 35% faster than the speed of sound (supersonic). The local speed of sound, and hence the Mach number, depends on the temperature of the surrounding gas. The Mach number is primarily used to determine the approximation with which a flow can be treated as an i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Equation
The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. This article focuses on waves in classical physics. Quantum physics uses an operator-based wave equation often as a relativistic wave equation. Introduction The wave equation is a hyperbolic partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in scalars by scalar functions of a time variable (a variable representing time) and one or more spatial variables (variables representing a position in a space under discussion). At the same time, there a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mach Wave
In fluid dynamics, a Mach wave, also known as a weak discontinuity, is a pressure wave traveling with the speed of sound caused by a slight change of pressure added to a compressible flow. These weak waves can combine in supersonic flow to become a shock wave if sufficient Mach waves are present at any location. Such a shock wave is called a Mach stem or Mach front. Thus, it is possible to have shockless compression or expansion in a supersonic flow by having the production of Mach waves sufficiently spaced (''cf.'' isentropic compression in supersonic flows). A Mach wave is the weak limit of an oblique shock wave where time averages of flow quantities don't change (a normal shock is the other limit). If the size of the object moving at the speed of sound is near 0, then this domain of influence of the wave is called a Mach cone. Mach angle A Mach wave propagates across the flow at the Mach angle ''μ'', which is the angle formed between the Mach wave wavefront and a vector that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Drag (physics)
In fluid dynamics, drag, sometimes referred to as fluid resistance, is a force acting opposite to the direction of motion of any object moving with respect to a surrounding fluid. This can exist between two fluid layers, two solid surfaces, or between a fluid and a solid surface. Drag forces tend to decrease fluid velocity relative to the solid object in the fluid's path. Unlike other resistive forces, drag force depends on velocity. Drag force is proportional to the relative velocity for low-speed flow and is proportional to the velocity squared for high-speed flow. This distinction between low and high-speed flow is measured by the Reynolds number. Drag is instantaneously related to vorticity dynamics through the Josephson-Anderson relation. Examples Examples of drag include: * Net force, Net Aerodynamic force, aerodynamic or Fluid dynamics, hydrodynamic force: Drag acting opposite to the direction of movement of a solid object such as cars, aircraft, and boat hulls. * Viscou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sears–Haack Body
The Sears–Haack body is the shape with the lowest theoretical wave drag in supersonic flow, for a slender solid body of revolution with a given body length and volume. The mathematical derivation assumes small-disturbance (linearized) supersonic flow, which is governed by the Prandtl–Glauert equation. The derivation and shape were published independently by two separate researchers: Wolfgang Haack in 1941 and later by William Sears in 1947. The Kármán–Moore theory indicates that the wave drag scales as the square of the second derivative of the area distribution, D_\text \sim S''(x)2 (see full expression below), so for low wave drag it is necessary that S(x) be smooth. Thus, the Sears–Haack body is pointed at each end and grows smoothly to a maximum and then decreases smoothly toward the second point. Useful formulas The cross-sectional area of a Sears–Haack body is : S(x) = \frac x(1-x) = \pi R_\text^2 x(1-x), its volume is : V = \frac R_\text^2 L, its r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor–Maccoll Flow
Taylor–Maccoll flow refers to the steady flow behind a conical shock wave that is attached to a solid cone. The flow is named after G. I. Taylor and J. W. Maccoll, whom described the flow in 1933, guided by an earlier work of Theodore von Kármán.Von Karman, T., & Moore, N. B. (1932). Resistance of slender bodies moving with supersonic velocities, with special reference to projectiles. Transactions of the American Society of Mechanical Engineers, 54(2), 303-310.Maccoll, J. W. (1937). The conical shock wave formed by a cone moving at a high speed. Proceedings of the Royal Society of London. Series A-Mathematical and Physical Sciences, 159(898), 459-472. Mathematical description Consider a steady supersonic flow past a solid cone that has a semi-vertical angle \chi. A conical shock wave can form in this situation, with the vertex of the shock wave lying at the vertex of the solid cone. If it were a two-dimensional problem, i.e., for a supersonic flow past a wedge, then the incom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
