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65-513: The Compact Kinetic Energy Missile ( CKEM ) was a developmental program to produce a hypersonic anti-tank guided missile for the U.S. Army . Lockheed Martin was the primary contractor. The program was the third in a series of projects based on kinetic energy missiles that stretches back to 1981's Vought HVM through the 1990s to 2000s LOSAT and finally to the CKEM. The Army Aviation and Missile Command (AMCOM) developed this program as part of

130-466: A {\displaystyle \rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=-\nabla p+\nabla \cdot {\boldsymbol {\tau }}+\rho \,\mathbf {a} } where In this form, it is apparent that in the assumption of an inviscid fluid – no deviatoric stress – Cauchy equations reduce to the Euler equations . Assuming conservation of mass , with the known properties of divergence and gradient we can use

195-644: A . {\displaystyle \left({\frac {\partial }{\partial t}}+\mathbf {u} \cdot \nabla -\nu \,\nabla ^{2}-({\tfrac {1}{3}}\nu +\xi )\,\nabla (\nabla \cdot )\right)\mathbf {u} =-{\frac {1}{\rho }}\nabla p+\mathbf {a} .} The convective acceleration term can also be written as u ⋅ ∇ u = ( ∇ × u ) × u + 1 2 ∇ u 2 , {\displaystyle \mathbf {u} \cdot \nabla \mathbf {u} =(\nabla \times \mathbf {u} )\times \mathbf {u} +{\tfrac {1}{2}}\nabla \mathbf {u} ^{2},} where

260-446: A . {\displaystyle \rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} \right)=-\nabla p+\nabla \cdot \left\{\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]\right\}+\nabla [\zeta (\nabla \cdot \mathbf {u} )]+\rho \mathbf {a} .} in index notation,

325-520: A . {\displaystyle \rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} \right)=-\nabla p+\nabla \cdot \left\{\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]\right\}+\rho \mathbf {a} .} If the dynamic μ and bulk ζ {\displaystyle \zeta } viscosities are assumed to be uniform in space,

390-459: A . {\displaystyle {\frac {D\mathbf {u} }{Dt}}=-{\frac {1}{\rho }}\nabla p+\nu \,\nabla ^{2}\mathbf {u} +({\tfrac {1}{3}}\nu +\xi )\,\nabla (\nabla \cdot \mathbf {u} )+\mathbf {a} .} where D D t {\textstyle {\frac {\mathrm {D} }{\mathrm {D} t}}} is the material derivative . ν = μ ρ {\displaystyle \nu ={\frac {\mu }{\rho }}}

455-409: A . {\displaystyle {\frac {\partial }{\partial t}}(\rho \mathbf {u} )+\nabla \cdot \left(\rho \mathbf {u} \otimes \mathbf {u} +[p-\zeta (\nabla \cdot \mathbf {u} )]\mathbf {I} -\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]\right)=\rho \mathbf {a} .} Apart from its dependence of pressure and temperature,

520-601: A i . {\displaystyle \rho \left({\frac {\partial u_{i}}{\partial t}}+u_{k}{\frac {\partial u_{i}}{\partial x_{k}}}\right)=-{\frac {\partial p}{\partial x_{i}}}+{\frac {\partial }{\partial x_{k}}}\left[\mu \left({\frac {\partial u_{i}}{\partial x_{k}}}+{\frac {\partial u_{k}}{\partial x_{i}}}-{\frac {2}{3}}\delta _{ik}{\frac {\partial u_{l}}{\partial x_{l}}}\right)\right]+{\frac {\partial }{\partial x_{i}}}\left(\zeta {\frac {\partial u_{l}}{\partial x_{l}}}\right)+\rho a_{i}.} The corresponding equation in conservation form can be obtained by considering that, given

585-525: A hypersonic speed is one that exceeds five times the speed of sound , often stated as starting at speeds of Mach 5 and above. The precise Mach number at which a craft can be said to be flying at hypersonic speed varies, since individual physical changes in the airflow (like molecular dissociation and ionization ) occur at different speeds; these effects collectively become important around Mach 5–10. The hypersonic regime can also be alternatively defined as speeds where specific heat capacity changes with

650-408: A parabolic equation and therefore have better analytic properties, at the expense of having less mathematical structure (e.g. they are never completely integrable ). The Navier–Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents , water flow in a pipe and air flow around

715-637: A wing . The Navier–Stokes equations, in their full and simplified forms, help with the design of aircraft and cars, the study of blood flow , the design of power stations , the analysis of pollution , and many other problems. Coupled with Maxwell's equations , they can be used to model and study magnetohydrodynamics . The Navier–Stokes equations are also of great interest in a purely mathematical sense. Despite their wide range of practical uses, it has not yet been proven whether smooth solutions always exist in three dimensions—i.e., whether they are infinitely differentiable (or even just bounded) at all points in

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780-459: A body's Mach number increases, the density behind a bow shock generated by the body also increases, which corresponds to a decrease in volume behind the shock due to conservation of mass . Consequently, the distance between the bow shock and the body decreases at higher Mach numbers. As Mach numbers increase, the entropy change across the shock also increases, which results in a strong entropy gradient and highly vortical flow that mixes with

845-410: A flow with respect to space. While individual fluid particles indeed experience time-dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle. Remark: here, the deviatoric stress tensor is denoted τ {\textstyle {\boldsymbol {\tau }}} as it was in the general continuum equations and in

910-521: A number of similarity parameters , which allow the simplification of a nearly infinite number of test cases into groups of similarity. For transonic and compressible flow , the Mach and Reynolds numbers alone allow good categorization of many flow cases. Hypersonic flows, however, require other similarity parameters. First, the analytic equations for the oblique shock angle become nearly independent of Mach number at high (~>10) Mach numbers. Second,

975-419: A number of regimes. The selection of these regimes is rough, due to the blurring of the boundaries where a particular effect can be found. In this regime, the gas can be regarded as an ideal gas . Flow in this regime is still Mach number dependent. Simulations start to depend on the use of a constant-temperature wall, rather than the adiabatic wall typically used at lower speeds. The lower border of this region

1040-579: A similarity parameter, similar to the Whitcomb area rule , which allowed similar configurations to be compared. In the study of hypersonic flow over slender bodies, the product of the freestream Mach number M ∞ {\displaystyle M_{\infty }} and the flow deflection angle θ {\displaystyle \theta } , known as the hypersonic similarity parameter: K = M ∞ θ {\displaystyle K=M_{\infty }\theta }

1105-532: A vector whose direction and magnitude are those of the velocity of the fluid at that point in space and at that moment in time. It is usually studied in three spatial dimensions and one time dimension, although two (spatial) dimensional and steady-state cases are often used as models, and higher-dimensional analogues are studied in both pure and applied mathematics. Once the velocity field is calculated, other quantities of interest such as pressure or temperature may be found using dynamical equations and relations. This

1170-525: Is ∇ ( ∇ ⋅ u ) {\textstyle \nabla \left(\nabla \cdot \mathbf {u} \right)} , one finally arrives to the compressible Navier–Stokes momentum equation: D u D t = − 1 ρ ∇ p + ν ∇ 2 u + ( 1 3 ν + ξ ) ∇ ( ∇ ⋅ u ) +

1235-555: Is around Mach 5, where ramjets become inefficient, and the upper border around Mach 10–12. This is a subset of the perfect gas regime, where the gas can be considered chemically perfect, but the rotational and vibrational temperatures of the gas must be considered separately, leading to two temperature models. See particularly the modeling of supersonic nozzles, where vibrational freezing becomes important. In this regime, diatomic or polyatomic gases (the gases found in most atmospheres) begin to dissociate as they come into contact with

1300-491: Is called the transonic range. Aircraft designed to fly at supersonic speeds show large differences in their aerodynamic design because of the radical differences in the behavior of flows above Mach 1. Sharp edges, thin aerofoil -sections, and all-moving tailplane / canards are common. Modern combat aircraft must compromise in order to maintain low-speed handling; "true" supersonic designs, generally incorporating delta wings, are rarer. The categorization of airflow relies on

1365-449: Is considered to be an important governing parameter. The slenderness ratio of a vehicle τ = d / l {\displaystyle \tau =d/l} , where d {\displaystyle d} is the diameter and l {\displaystyle l} is the length, is often substituted for θ {\displaystyle \theta } . Hypersonic flow can be approximately separated into

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1430-399: Is different from what one normally sees in classical mechanics , where solutions are typically trajectories of position of a particle or deflection of a continuum . Studying velocity instead of position makes more sense for a fluid, although for visualization purposes one can compute various trajectories . In particular, the streamlines of a vector field, interpreted as flow velocity, are

1495-431: Is less than Mach 1. The critical Mach number (Mcrit) is lowest free stream Mach number at which airflow over any part of the aircraft first reaches Mach 1. So the subsonic speed range includes all speeds that are less than Mcrit. The transonic speed range is that range of speeds within which the airflow over different parts of an aircraft is between subsonic and supersonic. So the regime of flight from Mcrit up to Mach 1.3

1560-748: Is proportional to the shear viscosity: σ ′ = τ = μ [ ∇ u + ( ∇ u ) T − 2 3 ( ∇ ⋅ u ) I ] {\displaystyle {\boldsymbol {\sigma }}'={\boldsymbol {\tau }}=\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]} Both bulk viscosity ζ {\textstyle \zeta } and dynamic viscosity μ {\textstyle \mu } need not be constant – in general, they depend on two thermodynamics variables if

1625-704: Is the identity tensor , and tr ⁡ ( ε ) {\textstyle \operatorname {tr} ({\boldsymbol {\varepsilon }})} is the trace of the rate-of-strain tensor. So this decomposition can be explicitly defined as: σ = − p I + λ ( ∇ ⋅ u ) I + μ ( ∇ u + ( ∇ u ) T ) . {\displaystyle {\boldsymbol {\sigma }}=-p\mathbf {I} +\lambda (\nabla \cdot \mathbf {u} )\mathbf {I} +\mu \left(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }\right).} Since

1690-415: Is the outer product of the flow velocity ( u {\displaystyle \mathbf {u} } ): u ⊗ u = u u T {\displaystyle \mathbf {u} \otimes \mathbf {u} =\mathbf {u} \mathbf {u} ^{\mathrm {T} }} The left side of the equation describes acceleration, and may be composed of time-dependent and convective components (also

1755-784: Is the shear kinematic viscosity and ξ = ζ ρ {\displaystyle \xi ={\frac {\zeta }{\rho }}} is the bulk kinematic viscosity. The left-hand side changes in the conservation form of the Navier–Stokes momentum equation. By bringing the operator on the flow velocity on the left side, on also has: ( ∂ ∂ t + u ⋅ ∇ − ν ∇ 2 − ( 1 3 ν + ξ ) ∇ ( ∇ ⋅ ) ) u = − 1 ρ ∇ p +

1820-535: Is three: tr ⁡ ( I ) = 3. {\displaystyle \operatorname {tr} ({\boldsymbol {I}})=3.} the trace of the stress tensor in three dimensions becomes: tr ⁡ ( σ ) = − 3 p + ( 3 λ + 2 μ ) ∇ ⋅ u . {\displaystyle \operatorname {tr} ({\boldsymbol {\sigma }})=-3p+(3\lambda +2\mu )\nabla \cdot \mathbf {u} .} So by alternatively decomposing

1885-400: Is usually neglected most of the time (that is whenever we are not dealing with processes such as sound absorption and attenuation of shock waves, where second viscosity coefficient becomes important) by explicitly assuming ζ = 0 {\textstyle \zeta =0} . The assumption of setting ζ = 0 {\textstyle \zeta =0} is called as

1950-595: The Cauchy stress tensor σ {\textstyle {\boldsymbol {\sigma }}} to be the sum of a viscosity term τ {\textstyle {\boldsymbol {\tau }}} (the deviatoric stress ) and a pressure term − p I {\textstyle -p\mathbf {I} } (volumetric stress), we arrive at: ρ D u D t = − ∇ p + ∇ ⋅ τ + ρ

2015-907: The Stokes hypothesis . The validity of Stokes hypothesis can be demonstrated for monoatomic gas both experimentally and from the kinetic theory; for other gases and liquids, Stokes hypothesis is generally incorrect. With the Stokes hypothesis, the Navier–Stokes equations become ρ D u D t = ρ ( ∂ u ∂ t + ( u ⋅ ∇ ) u ) = − ∇ p + ∇ ⋅ { μ [ ∇ u + ( ∇ u ) T − 2 3 ( ∇ ⋅ u ) I ] } + ρ

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2080-425: The boundary layer . A portion of the large kinetic energy associated with flow at high Mach numbers transforms into internal energy in the fluid due to viscous effects. The increase in internal energy is realized as an increase in temperature. Since the pressure gradient normal to the flow within a boundary layer is approximately zero for low to moderate hypersonic Mach numbers, the increase of temperature through

2145-447: The bow shock generated by the body. Surface catalysis plays a role in the calculation of surface heating, meaning that the type of surface material also has an effect on the flow. The lower border of this regime is where any component of a gas mixture first begins to dissociate in the stagnation point of a flow (which for nitrogen is around 2000 K). At the upper border of this regime, the effects of ionization start to have an effect on

2210-955: The bulk viscosity ζ {\textstyle \zeta } , ζ ≡ λ + 2 3 μ , {\displaystyle \zeta \equiv \lambda +{\tfrac {2}{3}}\mu ,} we arrive to the linear constitutive equation in the form usually employed in thermal hydraulics : σ = − [ p − ζ ( ∇ ⋅ u ) ] I + μ [ ∇ u + ( ∇ u ) T − 2 3 ( ∇ ⋅ u ) I ] {\displaystyle {\boldsymbol {\sigma }}=-[p-\zeta (\nabla \cdot \mathbf {u} )]\mathbf {I} +\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]} which can also be arranged in

2275-417: The deviatoric stress tensor σ ′ {\displaystyle {\boldsymbol {\sigma }}'} is still coincident with the shear stress tensor τ {\displaystyle {\boldsymbol {\tau }}} (i.e. the deviatoric stress in a Newtonian fluid has no normal stress components), and it has a compressibility term in addition to the incompressible case, which

2340-478: The domain . This is called the Navier–Stokes existence and smoothness problem. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$ 1 million prize for a solution or a counterexample. The solution of the equations is a flow velocity . It is a vector field —to every point in a fluid, at any moment in a time interval, it gives

2405-668: The incompressible flow section . The compressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor: σ ( ε ) = − p I + λ tr ⁡ ( ε ) I + 2 μ ε {\displaystyle {\boldsymbol {\sigma }}({\boldsymbol {\varepsilon }})=-p\mathbf {I} +\lambda \operatorname {tr} ({\boldsymbol {\varepsilon }})\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}} where I {\textstyle \mathbf {I} }

2470-410: The trace of the rate-of-strain tensor in three dimensions is the divergence (i.e. rate of expansion) of the flow: tr ⁡ ( ε ) = ∇ ⋅ u . {\displaystyle \operatorname {tr} ({\boldsymbol {\varepsilon }})=\nabla \cdot \mathbf {u} .} Given this relation, and since the trace of the identity tensor in three dimensions

2535-520: The Army's Future Combat Systems . This missile was primarily an anti-tank weapon, and could be mounted on land vehicles and low-altitude aircraft. The goal of these weapons was to demonstrate a state-of-the-art system for the next generation. The program was cancelled in 2009 with the rest of the future combat systems program. This article relating to missiles is a stub . You can help Misplaced Pages by expanding it . Hypersonic In aerodynamics ,

2600-461: The assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing viscous flow . The difference between them and the closely related Euler equations is that Navier–Stokes equations take viscosity into account while the Euler equations model only inviscid flow . As a result, the Navier–Stokes are

2665-562: The boundary layer coincides with a decrease in density. This causes the bottom of the boundary layer to expand, so that the boundary layer over the body grows thicker and can often merge with the shock wave near the body leading edge. High temperatures due to a manifestation of viscous dissipation cause non-equilibrium chemical flow properties such as vibrational excitation and dissociation and ionization of molecules resulting in convective and radiative heat-flux . Although "subsonic" and "supersonic" usually refer to speeds below and above

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2730-431: The compressible case the pressure is no more proportional to the isotropic stress term, since there is the additional bulk viscosity term: p = − 1 3 tr ⁡ ( σ ) + ζ ( ∇ ⋅ u ) {\displaystyle p=-{\frac {1}{3}}\operatorname {tr} ({\boldsymbol {\sigma }})+\zeta (\nabla \cdot \mathbf {u} )} and

2795-622: The conservation form of the equations of motion. This is often written: ∂ ∂ t ( ρ u ) + ∇ ⋅ ( ρ u ⊗ u ) = − ∇ p + ∇ ⋅ τ + ρ a {\displaystyle {\frac {\partial }{\partial t}}(\rho \,\mathbf {u} )+\nabla \cdot (\rho \,\mathbf {u} \otimes \mathbf {u} )=-\nabla p+\nabla \cdot {\boldsymbol {\tau }}+\rho \,\mathbf {a} } where ⊗ {\textstyle \otimes }

2860-407: The deviatoric (shear) stress tensor in terms of viscosity and the fluid velocity gradient, and assuming constant viscosity, the above Cauchy equations will lead to the Navier–Stokes equations below. A significant feature of the Cauchy equation and consequently all other continuum equations (including Euler and Navier–Stokes) is the presence of convective acceleration: the effect of acceleration of

2925-705: The effect of the volume viscosity ζ {\textstyle \zeta } is that the mechanical pressure is not equivalent to the thermodynamic pressure : as demonstrated below. ∇ ⋅ ( ∇ ⋅ u ) I = ∇ ( ∇ ⋅ u ) , {\displaystyle \nabla \cdot (\nabla \cdot \mathbf {u} )\mathbf {I} =\nabla (\nabla \cdot \mathbf {u} ),} p ¯ ≡ p − ζ ∇ ⋅ u , {\displaystyle {\bar {p}}\equiv p-\zeta \,\nabla \cdot \mathbf {u} ,} However, this difference

2990-418: The effects of non-inertial coordinates if present). The right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces (such as gravity). All non-relativistic balance equations, such as the Navier–Stokes equations, can be derived by beginning with the Cauchy equations and specifying the stress tensor through a constitutive relation . By expressing

3055-919: The equation can be written as ρ ( ∂ u i ∂ t + u k ∂ u i ∂ x k ) = − ∂ p ∂ x i + ∂ ∂ x k [ μ ( ∂ u i ∂ x k + ∂ u k ∂ x i − 2 3 δ i k ∂ u l ∂ x l ) ] + ∂ ∂ x i ( ζ ∂ u l ∂ x l ) + ρ

3120-478: The equations in convective form can be simplified further. By computing the divergence of the stress tensor, since the divergence of tensor ∇ u {\textstyle \nabla \mathbf {u} } is ∇ 2 u {\textstyle \nabla ^{2}\mathbf {u} } and the divergence of tensor ( ∇ u ) T {\textstyle \left(\nabla \mathbf {u} \right)^{\mathrm {T} }}

3185-420: The flow. In this regime the ionized electron population of the stagnated flow becomes significant, and the electrons must be modeled separately. Often the electron temperature is handled separately from the temperature of the remaining gas components. This region occurs for freestream flow velocities around 3–4 km/s. Gases in this region are modeled as non-radiating plasmas . Above around 12 km/s,

3250-888: The fluid contains a single chemical species, say for example, pressure and temperature. Any equation that makes explicit one of these transport coefficient in the conservation variables is called an equation of state . The most general of the Navier–Stokes equations become ρ D u D t = ρ ( ∂ u ∂ t + ( u ⋅ ∇ ) u ) = − ∇ p + ∇ ⋅ { μ [ ∇ u + ( ∇ u ) T − 2 3 ( ∇ ⋅ u ) I ] } + ∇ [ ζ ( ∇ ⋅ u ) ] + ρ

3315-508: The formation of strong shocks around aerodynamic bodies means that the freestream Reynolds number is less useful as an estimate of the behavior of the boundary layer over a body (although it is still important). Finally, the increased temperature of hypersonic flow mean that real gas effects become important. Research in hypersonics is therefore often called aerothermodynamics , rather than aerodynamics . The introduction of real gas effects means that more variables are required to describe

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3380-479: The full state of a gas. Whereas a stationary gas can be described by three variables ( pressure , temperature , adiabatic index ), and a moving gas by four ( flow velocity ), a hot gas in chemical equilibrium also requires state equations for the chemical components of the gas, and a gas in nonequilibrium solves those state equations using time as an extra variable. This means that for nonequilibrium flow, something between 10 and 100 variables may be required to describe

3445-768: The heat transfer to a vehicle changes from being conductively dominated to radiatively dominated. The modeling of gases in this regime is split into two classes: The modeling of optically thick gases is extremely difficult, since, due to the calculation of the radiation at each point, the computation load theoretically expands exponentially as the number of points considered increases. Navier%E2%80%93Stokes equations The Navier–Stokes equations mathematically express momentum balance for Newtonian fluids and make use of conservation of mass . They are sometimes accompanied by an equation of state relating pressure , temperature and density . They arise from applying Isaac Newton's second law to fluid motion , together with

3510-540: The local speed of sound respectively, aerodynamicists often use these terms to refer to particular ranges of Mach values. When an aircraft approaches transonic speeds (around Mach 1), it enters a special regime. The usual approximations based on the Navier–Stokes equations , which work well for subsonic designs, start to break down because, even in the freestream, some parts of the flow locally exceed Mach 1. So, more sophisticated methods are needed to handle this complex behavior. The "supersonic regime" usually refers to

3575-1012: The mass continuity equation , the left side is equivalent to: ρ D u D t = ∂ ∂ t ( ρ u ) + ∇ ⋅ ( ρ u ⊗ u ) {\displaystyle \rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}={\frac {\partial }{\partial t}}(\rho \mathbf {u} )+\nabla \cdot (\rho \mathbf {u} \otimes \mathbf {u} )} To give finally: ∂ ∂ t ( ρ u ) + ∇ ⋅ ( ρ u ⊗ u + [ p − ζ ( ∇ ⋅ u ) ] I − μ [ ∇ u + ( ∇ u ) T − 2 3 ( ∇ ⋅ u ) I ] ) = ρ

3640-1574: The mass continuity equation, which represents the mass per unit volume of a homogenous fluid with respect to space and time (i.e., material derivative D D t {\displaystyle {\frac {\mathbf {D} }{\mathbf {Dt} }}} ) of any finite volume ( V ) to represent the change of velocity in fluid media: D m D t = ∭ V ( D ρ D t + ρ ( ∇ ⋅ u ) ) d V D ρ D t + ρ ( ∇ ⋅ u ) = ∂ ρ ∂ t + ( ∇ ρ ) ⋅ u + ρ ( ∇ ⋅ u ) = ∂ ρ ∂ t + ∇ ⋅ ( ρ u ) = 0 {\displaystyle {\begin{aligned}{\frac {\mathbf {D} m}{\mathbf {Dt} }}&={\iiint \limits _{V}}\left({{\frac {\mathbf {D} \rho }{\mathbf {Dt} }}+\rho (\nabla \cdot \mathbf {u} )}\right)dV\\{\frac {\mathbf {D} \rho }{\mathbf {Dt} }}+\rho (\nabla \cdot {\mathbf {u} })&={\frac {\partial \rho }{\partial t}}+({\nabla \rho })\cdot {\mathbf {u} }+{\rho }(\nabla \cdot \mathbf {u} )={\frac {\partial \rho }{\partial t}}+\nabla \cdot ({\rho \mathbf {u} })=0\end{aligned}}} where to arrive at

3705-567: The other usual form: σ = − p I + μ ( ∇ u + ( ∇ u ) T ) + ( ζ − 2 3 μ ) ( ∇ ⋅ u ) I . {\displaystyle {\boldsymbol {\sigma }}=-p\mathbf {I} +\mu \left(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }\right)+\left(\zeta -{\frac {2}{3}}\mu \right)(\nabla \cdot \mathbf {u} )\mathbf {I} .} Note that in

3770-697: The paths along which a massless fluid particle would travel. These paths are the integral curves whose derivative at each point is equal to the vector field, and they can represent visually the behavior of the vector field at a point in time. The Navier–Stokes momentum equation can be derived as a particular form of the Cauchy momentum equation , whose general convective form is: D u D t = 1 ρ ∇ ⋅ σ + f . {\displaystyle {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}={\frac {1}{\rho }}\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {f} .} By setting

3835-420: The previously-operated Space Shuttle ; various reusable spacecraft in development such as SpaceX Starship and Rocket Lab Electron ; and (theoretical) spaceplanes . In the following table, the "regimes" or "ranges of Mach values" are referenced instead of the usual meanings of "subsonic" and "supersonic". The subsonic speed range is that range of speeds within which, all of the airflow over an aircraft

3900-521: The second viscosity coefficient also depends on the process, that is to say, the second viscosity coefficient is not just a material property. Example: in the case of a sound wave with a definitive frequency that alternatively compresses and expands a fluid element, the second viscosity coefficient depends on the frequency of the wave. This dependence is called the dispersion . In some cases, the second viscosity ζ {\textstyle \zeta } can be assumed to be constant in which case,

3965-455: The set of Mach numbers for which linearised theory may be used; for example, where the ( air ) flow is not chemically reacting and where heat transfer between air and vehicle may be reasonably neglected in calculations. Generally, NASA defines "high" hypersonic as any Mach number from 10 to 25, and re-entry speeds as anything greater than Mach 25. Among the spacecraft operating in these regimes are returning Soyuz and Dragon space capsules ;

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4030-437: The state of the gas at any given time. Additionally, rarefied hypersonic flows (usually defined as those with a Knudsen number above 0.1) do not follow the Navier–Stokes equations . Hypersonic flows are typically categorized by their total energy, expressed as total enthalpy (MJ/kg), total pressure (kPa-MPa), stagnation pressure (kPa-MPa), stagnation temperature (K), or flow velocity (km/s). Wallace D. Hayes developed

4095-843: The stress tensor into isotropic and deviatoric parts, as usual in fluid dynamics: σ = − [ p − ( λ + 2 3 μ ) ( ∇ ⋅ u ) ] I + μ ( ∇ u + ( ∇ u ) T − 2 3 ( ∇ ⋅ u ) I ) {\displaystyle {\boldsymbol {\sigma }}=-\left[p-\left(\lambda +{\tfrac {2}{3}}\mu \right)\left(\nabla \cdot \mathbf {u} \right)\right]\mathbf {I} +\mu \left(\nabla \mathbf {u} +\left(\nabla \mathbf {u} \right)^{\mathrm {T} }-{\tfrac {2}{3}}\left(\nabla \cdot \mathbf {u} \right)\mathbf {I} \right)} Introducing

4160-470: The temperature of the flow as kinetic energy of the moving object is converted into heat. While the definition of hypersonic flow can be quite vague and is generally debatable (especially due to the absence of discontinuity between supersonic and hypersonic flows), a hypersonic flow may be characterized by certain physical phenomena that can no longer be analytically discounted as in supersonic flow. The peculiarities in hypersonic flows are as follows: As

4225-621: The vector ( ∇ × u ) × u {\textstyle (\nabla \times \mathbf {u} )\times \mathbf {u} } is known as the Lamb vector . For the special case of an incompressible flow , the pressure constrains the flow so that the volume of fluid elements is constant: isochoric flow resulting in a solenoidal velocity field with ∇ ⋅ u = 0 {\textstyle \nabla \cdot \mathbf {u} =0} . The incompressible momentum Navier–Stokes equation results from

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