Velocity vorticity vector behavior in supersonic flows behind discontinuity surfaces

Левин, В. А.; Skopina, G. A.

doi:10.1134/s0869864307030067

Thermophys. Aeromech.

2007

DOI: 10.1134/s0869864307030067

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Velocity vorticity vector behavior in supersonic flows behind discontinuity surfaces

В. А. Левин

G. A. Skopina

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Cited by 3 publications

(7 citation statements)

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“…For D = 0, this formula coincides with the expression for vorticity behind a curvilinear steady shock or detonation wave with constant values of the initial gas parameters, which was obtained in [1][2][3][4][5][6][7].…”

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confidence: 68%

“…Thus, the normal component of vorticity remains a continuous function across the discontinuity surface, as in the case of steady flows [6,7]. For unsteady axisymmetric flows, the law of conservation of the quantity ω τ /ρ across the discontinuity surface is valid for all distributions of free-stream gas parameters, as in the case of steady axisymmetric flows [6].…”

mentioning

confidence: 93%

“…In this case, the normal component of vorticity is demonstrated to remain a continuous function across the discontinuity surface. For axisymmetric flows, the law of conservation of the quantity ω τ /ρ is satisfied for all distributions of the free-stream parameters, regardless of the wave type (shock or detonation wave), as in the case of steady flows [6,7]. This work was supported by the Far-East Division of the Russian Academy of Sciences (Grant No.…”

mentioning

confidence: 95%

“…Levin and Skopina [6] studied the behavior of the vorticity vector in supersonic axisymmetric swirl flows behind a steady curvilinear shock or detonation wave. In the general three-dimensional case, expressions for the components of the vorticity vector behind a discontinuity surface generated by a steady supersonic nonuniform flow of a combustible gas around a solid were derived in [7]. For unsteady flows, Levin and Skopina [8] obtained formulas for vorticity on a cylin-1 Institute of Automation and Control Processes, Far-East Division, Russian Academy of Sciences, Vladivostok 690041; levin@imec.msu.ru.…”

mentioning

confidence: 99%

“…If ∂ ∂t = 0, and s is a natural coordinate introduced on the discontinuity surface, then the formulas for the vorticity components (29) completely coincide with the formulas for vorticity behind a steady detonation wave in a supersonic nonuniform vortex flow of a combustible gas, which were derived in [6,7]:…”

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confidence: 99%

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Behavior of the vorticity vector in a supersonic nonuniform swirl flow of a combustible gas behind a moving curvilinear shock or detonation wave

Левин

Skopina

2009

Combust Explos Shock Waves

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Expressions for components of the vorticity vector behind a curvilinear shock or detonation wave propagating in a supersonic nonuniform flow of a combustible gas are derived. Plane and axisymmetric gas flows are considered. The free stream in the general case is a vortex flow with a specified distribution of parameters. Formulas for the vorticity components in the plane of the flow for axisymmetric flows are found to have the same form as formulas for steady axisymmetric flows. As in the case of steady flows, the normal-to-wave component of vorticity is demonstrated to remain continuous across the discontinuity surface; in the case of axisymmetric flows, the ratio of the tangential component of vorticity aligned in the plane of the flow to density also remains continuous, though the quantities themselves become discontinuous.In the case of steady flows, the expression for vorticity behind a curved shock wave for a flow with constant parameters was first derived by Truesdell [1] and later by other authors. Lighthill [2] obtained formulas for components of vorticity behind an arbitrary curved wave in the general case, under the assumption of an infinite shock-wave intensity. A generalized formula was given by Hayes [3]. Maikapar [4] and Rusanov [5] obtained formulas for vorticity components behind a shock wave of an arbitrary intensity with constant free-stream parameters. Levin and Skopina [6] studied the behavior of the vorticity vector in supersonic axisymmetric swirl flows behind a steady curvilinear shock or detonation wave. In the general three-dimensional case, expressions for the components of the vorticity vector behind a discontinuity surface generated by a steady supersonic nonuniform flow of a combustible gas around a solid were derived in [7]. For unsteady flows, Levin and Skopina [8] obtained formulas for vorticity on a cylin-1 Institute of Automation and Control Processes, Far-East Division, Russian Academy of Sciences, Vladivostok 690041; levin@imec.msu.ru.drical discontinuity surface propagating in an axisymmetric swirl flow of an ideal gas away from the axis of symmetry. In the present activities, we determine the vorticity directly behind a curvilinear shock or detonation wave propagating over a nonuniform swirl flow of a combustible gas. We study plane-parallel and axisymmetric unsteady motions of the gas, which depend only on two coordinates: x and y. The plane of the flow is the plane (x, y). The axis of symmetry coincides with the straight line x = 0. The main examined quantities (velocity vector V , pressure p, density ρ, and vorticity 2ω = rotV ) are considered as functions of the Cartesian coordinates (x, y) and time t.The velocity vector V in the coordinate system (x 1 , x 2 , x 3 ) has the components u, υ, and w. For planeparallel motion, we have x 1 = x, x 2 = y, and x 3 = z; for axisymmetric motion, we have x 1 = x, x 2 = y, and x 3 = ϕ. For a plane-parallel flow, we have w = 0.

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confidence: 68%

mentioning

confidence: 93%

mentioning

confidence: 95%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Behavior of the vorticity vector in a supersonic nonuniform swirl flow of a combustible gas behind a moving curvilinear shock or detonation wave

Левин

Skopina

2009

Combust Explos Shock Waves

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Gradient relations at the front of shock and detonation waves

Prokhorov

2009

Combust Explos Shock Waves

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A natural assumption on the form of the calorific equations of state (internal energy) for one-dimensional motion was used to obtain the so-called gradient relations that give a one-to-one correspondence between the first partial spatial derivatives of the pressure, density, mass velocity (gradients of parameters) at shock and detonation fronts and the time derivative (acceleration) of the front. The assumption is based on the fact that, taking into account the thermal equation of state, the total internal energy, including both the thermodynamic part and potential chemical energy, can be represented as a function of pressure and density. This holds for both inert media and reaction products in the state of chemical equilibrium.Key words: shock wave, detonation wave, gradients of parameters, acceleration of the front.The strong-discontinuity relations, which are often called the laws of conservation of mass, momentum, and energy at the shock front, are well-known [1,2]. With the thermal effect of chemical reactions taken into account, these laws of conservation are also applicable to a detonation front (a strong discontinuity with heat release) [3,4]. If the motion of the medium behind the front is described by a smooth one-dimensional solution and the parameters ahead of the front are constant, it is possible to give a one-to-one correspondence between the partial spatial derivative (gradient) of any parameter and the time derivative of the velocity (acceleration) of the front. For one-dimensional adiabatic flow of a perfect gas, such gradient relations at the shockwave front are given in [5,6]. These relations can be used to develop numerical and analytical approximate methods for solving gas-dynamic problems and establishing asymptotic laws of attenuation of shock waves [7,8]. Here it is also pertinent to note a paper on a related topic [9], in which spatial derivatives were obtained for gas-dynamic functions behind a curved stationary shock wave subjected to an oncoming uniform supersonic flow. These results were further developed in a study [10] of the behavior of the vortex velocity 1 Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090; prokh@hydro.nsc.ru.vector behind strong-discontinuity surfaces. However, in the cited papers [9, 10], the effect of acceleration of the front on parameter gradients was not considered. The range of application of the gradient relations at the shock front presented in [5, 6] is appreciably limited by the conditions of the perfect gas model [11]; for polyatomic gas and mixtures of various chemically inert gases, this model is approximately valid only in a narrow range of temperatures. Therefore, the previously obtained gradient relations cannot be used for practically important cases such as: 1) gas motion behind strong shock waves, where excitation of additional degrees of freedom and dissociation of molecules are possible; 2) equilibrium flow of reacting gases behind a detonation front propagating in a chemically active m...

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Vorticity on the Surface of an Axially Symmetric Body behind a Detached Shock Wave

2018

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Velocity vorticity vector behavior in supersonic flows behind discontinuity surfaces

Cited by 3 publications

References 4 publications

Behavior of the vorticity vector in a supersonic nonuniform swirl flow of a combustible gas behind a moving curvilinear shock or detonation wave

Behavior of the vorticity vector in a supersonic nonuniform swirl flow of a combustible gas behind a moving curvilinear shock or detonation wave

Gradient relations at the front of shock and detonation waves

Vorticity on the Surface of an Axially Symmetric Body behind a Detached Shock Wave

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