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The classic linear Heaslet–Lomax problem, which is concerned with the indicial response of a supersonic airfoil experiencing a sudden small step motion, was recently extended to nonlinear cases whe...

The aim of this work is to develop a new numerical calculation program to determine the effect of the stagnation temperature on the calculation of the supersonic flow around a pointed airfoils using the equations for oblique shock wave and the Prandtl Meyer expansion, under the model at high temperature, calorically imperfect and thermally perfect gas, lower than the dissociation threshold of the molecules. The specific heat at constant pressure does not remain constant and varies with the temperature. The new model allows making corrections to the perfect gas model designed for low stagnation temperature, low Mach number, low incidence angle and low airfoil thickness. The stagnation temperature is an important parameter in our model. The airfoil should be pointed at the leading edge to allow an attached shock solution to be seen. The airfoil is discretized into several panels on the extrados and the intrados, placed one adjacent to the other. The distribution of the flow on the panel in question gives a compression or an expansion according to the deviation of the flow with respect to the old adjacent panel. The program determines all the aerodynamic characteristics of the flow and in particular the aerodynamic coefficients. The calculation accuracy depends on the number of panels considered on the airfoil. The application is made for high values of stagnation temperature, Mach number and airfoil thickness. A comparison between our high temperature model and the perfect gas model is presented, in order to determine an application limit of the latter. The application is for air.

This paper examines the steady interaction of a shear layer separating two uniform supersonic streams of Mach numbers M1 and M2 with an oblique shock approaching from the faster stream at an incident angle sigmai sufficiently small for the postshock flow to remain supersonic everywhere. The development begins by considering the related problem of oblique-shock impingement on a supersonic vortex sheet of infinitesimal thickness, for which the region of existence of regular shock refractions with downstream supersonic flow is delineated in the parametric space (M1,M2,sigmai). The interaction region located about the impingement point, scaling with the shear-layer thickness, is described next by integrating the Euler equations in the postshock region, formulated in characteristic form, subject to the Rankine-Hugoniot jump conditions at the shock front. The results are used to investigate the accuracy and limitations of a simplified treatment, the so-called Moeckel-Chisnell approach, commonly employed for determining the shape of the shock wave in these scenarios, which does not account for the influence of the postshock flow. It is found that, although the Moeckel-Chisnell method predicts accurately the shape of the shock front as it evolves across the shear layer, it is unable to predict the final transition to the transmitted-shock solution, which occurs beyond the edge of the shear layer. The structure of the shear layer in the far field also is addressed here for the first time, with the objective being to lay the groundwork for future studies of shock-induced ignition in supersonic fuel-air mixing layers. The work of A. L. Sánchez and F. A. Williams was supported by the U.S. Air Force Office of Scientific Research grant FA9550-16-1-0443, and that of D. Martínez-Ruiz and C. Huete was supported by the Spanish Ministry of Science (ENE2015-65852-C2-2-R) and the Fundación Iberdrola España (BYNV-ua37crdy).

A method for modeling the internal flowfield in a rotating detonation engine is developed using shock-expansion theory combined with the steady two-dimensional isentropic method of characteristics. An analytical model using the oblique shock relations, the Prandtl–Meyer function, and the detonation jump conditions is used to determine the basic shock structure. Once the structure is known, a shock-fitted method of characteristics solution is marched out to generate the rest of the flowfield. Reactant injection is handled analytically by solving the conservation equations for a flow undergoing a sudden expansion along with the method of characteristics compatibility relations to provide a new boundary condition. A new solution is then initialized using information from the previous solution to calculate the new shock structure. This process is repeated until the solutions converge. The converged solution is the ideal steady-state solution of a rotating detonation engine in the wave-fixed reference frame. T...

The aim of present study is to obtain mathematical expressions for surface pressure and its derivative with respect to Piston Mach number for an oscillating cone. The effects of Mach number and it’s geometry on Pressure and its derivative with respect to inertia level and incidence angle, and the results have been obtained. The non-dimensional surface pressure distribution on the cone remain linear for lower angle of incidence and Mach number, however, for higher Mach numbers non-linearity creeps in. For higher Mach number due to viscous-inviscid interaction the flow behaves differently. Also, it is observed that there is interaction between the boundary layer and entropy layer and this interaction makes the flow on the surface of the cone very complex. At large Mach numbers the aerodynamic heating will be a serious concern. At high angle theta and low Mach number shock wave will be detached. The surface pressure derivative with Mach number on the cone exhibits linear behaviour for lower Mach number and angle theta and non-linearity for large Mach number and angle theta.

When the stagnation pressure of a perfect gas increases, the specific heat and their ratio do not remain constant anymore and start to vary with this pressure. The gas does not stay perfect. Its state equation change and it becomes for a real gas. In this case, the effects of molecular size and intermolecular attraction forces intervene to correct the state equation, the thermodynamic parameters and the value of Prandtl Meyer function. The aim of this work is developing a new form of Prandtl Meyer function based on those assumptions; and determining the effect of stagnation pressure on this function. With the assumptions that Berthelot’s state equation accounts for molecular size and intermolecular force effects, expressions are developed for analysing the supersonic flow for thermally and calorically imperfect gas lower than the dissociation molecules threshold. The supersonic parameters depend directly on the stagnation parameters of the combustion chamber. The application is for air. A computation of error was made in this case to give a limit of the perfect gas and the high temperature models compared to the real gas model.

New numerical algorithm for low Mach number droplet-laden turbulent channel flow with phase transitions.The new algorithm eliminates the dependence of the time step on the Mach number.The results of the new algorithm show close agreement with those of an explicit solver.The new algorithm is 200 times faster for Ma = 0.005 . In this study we propose a new numerical algorithm for droplet-laden turbulent channel flow with phase transitions at low Mach numbers. The carrier gas is treated as compressible flow. In order to avoid very small time steps at low Mach numbers that would arise from stability requirements associated with explicit time-stepping we propose a new semi-explicit time integration method, applied to the low Mach number compressible flow equations. We perform a perturbation analysis in powers of the Mach number of the system of governing equations. The obtained decomposition of pressure into a space-independent part and a hydrodynamic part permits to apply a special pressure-based time integration algorithm for compressible flows at low Mach numbers. An important feature of the new numerical approach is the independence of the maximum allowed time step on the Mach number. In this study we validate the new method by comparing it with a fully explicit code for compressible flow at general Mach numbers showing a good agreement in all quantities of interest. The differences between the results of the two codes are on the order of the square of the Mach number caused by the disregard of high-order terms in the Mach number in the new algorithm. The relative difference found for a specific low value of the Mach number of 0.05 is on the order of 1 % for instantaneous and mean quantities of the two phases. We also quantify the efficiency of the new algorithm by comparing the computational time it takes to simulate one time unit with both codes.

When the stagnation temperature of the combustion chamber or ambient air increases, the specific heats and their ratio do not remain constant any more, and start to vary with this temperature. The gas remains perfect, except, it will be calorically imperfect and thermally perfect. A new generalized form of the Prandtl Meyer function is developed, by adding the effect of variation of this temperature, lower than the threshold of dissociation. The new relation is presented in the form of integral of a complex analytical function, having an infinite derivative at the critical temperature. A robust numerical integration quadrature is presented in this context. The classical form of the Prandtl Meyer function of a perfect gas becomes a particular case of the developed form. The comparison is made with the perfect gas model for aim to present a limit of its application. The application is for air.

When the stagnation temperature of a perfect gas increases, the specific heats and their ratio do not remain constant any more and start to vary with this temperature. The gas remains perfect; its state equation remains always valid, except, it is named in more by calorically imperfect gas. The aim of this work is to trace the profiles of the supersonic axisymmetric minimum length nozzle to have a uniform and parallel flow at the exit section, when the stagnation temperature is taken into account, lower than the dissociation threshold of the molecules, and to have for each exit Mach number and stagnation temperature shape of nozzle. The method of characteristics is used with the algorithm of the second order finite differences method. The form of the nozzle has a point of deflection and an initial angle of expansion. The comparison is made with the calorically perfect gas.

When the stagnation temperature of the combustion chamber or ambient air increases, the specific heats and their ratio do not remain constant any more, and start to vary with this temperature. The gas remains perfect, except, it will be calorically imperfect and thermally perfect. A new generalized form of the Prandtl Meyer function is developed, by adding the effect of variation of this temperature, lower than the threshold of dissociation. The new relation is presented in the form of integral of a complex analytical function, having an infinite derivative at the critical temperature. A robust numerical integration quadrature is presented in this context. The classical form of the Prandtl Meyer function of a perfect gas becomes a particular case of the developed form. The comparison is made with the perfect gas model for aim to present a limit of its application. The application is for air.

When the stagnation temperature of a perfect gas increases, the specific heats and their ratio do not remain constant any more and start to vary with this temperature. The gas remains perfect; its state equation remains always valid, except, it is named in more by calorically imperfect gas. The aim of this work is to trace the profiles of the supersonic axisymmetric Minimum Length Nozzle to have a uniform and parallel flow at the exit section, when the stagnation temperature is taken into account, lower than the dissociation threshold of the molecules, and to have for each exit Mach number and stagnation temperature shape of nozzle. The method of characteristics is used with the algorithm of the second order finite differences method. The form of the nozzle has a point of deflection and an initial angle of expansion. The comparison is made with the calorically perfect gas. The application is for air.

When the stagnation temperature of a perfect gas increases, the specific heats and their ratio do not remain constant any more and start to vary with this temperature. The gas remains perfect, its state equation remains always valid, except it will name in more calorically imperfect gas or gas at High Temperature. The goal of this research is to trace the profiles of the supersonic plug nozzle when this stagnation temperature is taken into account, lower than the threshold of dissociation of the molecules, by using the new formula of the Prandtl Meyer function, and to have for each exit Mach number, several nozzles shapes by changing the value of this temperature. A study on the error given by the PG (perfect gas) model compared to our model at high temperature is presented. The comparison is made with the case of a calorically perfect gas aiming to give a limit of application of this model. The application is for the air.

When the stagnation temperature of a perfect gas increases, the specific heat ratio does not remain constant any more, and start to vary with this temperature. The gas remains perfect, its state equation remains always valid, except it will name in more calorically imperfect gas or gas at High Temperature. The goal of this work is to trace the profiles of the supersonic Minimum Length Nozzle with centered expansion when the stagnation temperature is taken into account, lower than the threshold of dissociation of the molecules and to have for each exit Mach number several nozzles shapes by changing the value of the temperature. The method of characteristics is used with a new form of the Prandtl Meyer function at high temperature. The resolution of the obtained equations is done by the second order of finite differences method by using the predictor corrector algorithm. A study on the error given by the perfect gas model compared to our model is presented. The comparison is made with a calorically perfect gas for goal to give a limit of application of this model. The application is for the air.

A method to design the contour and conception of a plug nozzle of arbitrary shape, but specified exit flow conditions is presented. Severals shapes can be obtained for exit Mach number by changing the specific heats ratio. The characteristics of the nozzle in terms of length, weight and pressure force exerted on the wall are compared to the Minimum Length Nozzle and found to be better. Our field of study is limited to the supersonic mode to not to have the dissociation of the molecules. The design method is based on the use of the Prandtl Meyer function of a perfect gas. The flow is not axial at the throat, which may be advantageous for many propulsion applications. The performance benefits of the plug nozzle compared to the Minimum Length Nozzle are also presented.

著者人数: 5名, 資料番号: AC1000031000

Nomenclature A = area, m D = diameter, m f M = relationship between area ratio and Mach number at a given section g M = relationship between area ratio and pressure at a given section L = length of nozzle measured from throat, m M = Mach number p = pressure, Pa p0 = stagnation pressure, Pa x = axial distance, m = semicone angle, deg = Mach angle, deg = Prandtl–Meyer function, rad = ratio of specific heats

This work is to develop a new computational program to determine the effect of using gas propellant of the combustion chamber at high temperature on the calculation of the value of the Prandtl Meyer function and application to supersonic nozzles design. The selected gases are the H2 , O2 , N2 , CO, CO2 , H2 O, NH3 , CH4 and air. Prandtl Meyer function depends on the stagnation temperature, Mach number and the gas used. The specific heat at constant pressure and the enthalpy of the gases vary with the temperature and the selected gas. The gas is still considered as perfect and it will be calorically imperfect and thermally perfect below the threshold of dissociation of molecules. A calculation of the difference between the Prandtl Meyer function for different gases with the air is made for the purpose of comparison. The application is made for designing the supersonic MLN nozzle at high temperature.

For a plasma, flowing through a cascaded arc channel with a varying cross-section, and flowing from a subsonic to a supersonic state, the sonic condition moves downstream and the plasma Mach number at the smallest cross section is less than one, although in case of a transonic isentropic gas flow the sonic condition is found at the smallest cross section. This shift in sonic condition is due to the lack of isentropic behavior of the plasma flow. Sources causing the anisentropy are viscosity, heat and ionization, of which ionization is vital for a plasma. It is found that the plasma Mach number is always lower than the corresponding gas Mach number. A quasi one-dimensional analysis and simulations with a two-dimensional plasma model, which support the analysis, are presented. © 1999 American Institute of Physics. © 1999 American Institute of Physics

The problem of constructing the leading part of a semi-infinite solid of revolution with an axially symmetric flow of an ideal (inviscid and non-heat-conducting) gas around it which has the maximum critical Mach number is considered. A numerical-analytic method is proposed for solving this problem. Results are presented for the case of a perfect gas with an adiabatic index γ = 1.4.

The aim of this work is to calculate the flow parameters and to trace suitably the profiles of the supersonic nozzle, so as to obtain an uniform and parallel flow at the exit section, and to have for the same exit Mach number, several shapes by changing the repport of specific heats γ. The nozzle chosen in this study is the plug nozzle with a central body. The performances obtained like the length and the weight are better compared to the other types in particular the Minimum Length Nozzle. Our field of study is limited in the supersonic regim not to have the dissociation ofthe molecules. The design method is based on the use of the Prandtl Meyer function of a gas thermally and calorifiquement perfect. The flow is not horizontal at the col, and consequently is inclined comparaed to the horizontal. The comparaison will be made with the minimum length nozzle type.

Introduction I N two-dimensional supersonic isentropic flow, an infinitesimal change in flow angle is equal to an infinitesimal change in the Prandtl-Meyer function.Anderson states that this function was first presented in the doctoral dissertation of Theodor Meyer who was a student of Ludwig Prandtl. The Prandtl-Meyer function v, which is an explicit function of the Mach number M, is defined by

'Ozcan, O., Edis, F. O., Asian, A. R., and Pinar, L, "Inverse Solutions of the Prandtl-Meyer Function,' Journal of Aircraft, Vol. 31, No. 6, 1994, pp. 1422-1424. Hayes, W. D., and Probstein, R. F., Hypersonic Flow Theory, Academic, New York, 1959, p. 264. 'Probstein, R. F., "Inversion of the Prandtl-Meyer Relation for Specific Heat Ratios of 5/3 and 5/4," Journal of the Aeronautical Sciences, Vol. 24, 1957, pp. 316, 317, 632.

The pressure distribution due to the expansion of a boundary layer at a sharp trailing edge is investigated analytically. The entire expansion is assumed to be inviscid, and the subsonic and supersonic regions of the boundary layer are treated with the streamtube approximation and Prandtl-Meyer relation, respectively. Matching of streamfunction and pressure on the initially sonic streamline determines the expansion process upstream of the trailing edge, whose location is uniquely determined. The extent of the upstream expansion increases linearly with the Mach number at the edge of the boundary layer for Mach numbers greater than five. At the trailing edge the wall pressure is less than its initial value, and the final expansion to the base pressure requires a smaller turning angle of the dividing streamline than was previously calculated. The effects of rotationality in the supersonic region of the boundary layer and the influence of viscosity are also considered.

L is considerably less than 4 x 107, a significant Reynolds-number effect is expected on the basis of the Whitfield and Potter conclusions. High-Reynolds-number predictions1'3 for the M^ = 10.5 data are also included in Fig. 3. Reasons for the discrepancy between these two semiempirical predictions have been discussed by Cassanto and Men del son;11 hence, it suffices only to point out that both methods indicate a base pressure level considerably lower than what was in fact measured. The rather severe variation (~60%) in pb/pe seen for Ree L ^ 6.3 x 106 illustrates the Reynolds-number influence on a full-scale hypersonic flight vehicle. Conclusions The influence of Reynolds number on turbulent base pressure has been discussed relative to hypersonic re-entry. Flight data from slightly blunt slender cones, for which Ree L ^> 4 x 107 (Ree L « 2Re00tL), reveal no Reynolds-numb er effect, whereas an appreciable Reynolds-number dependence (~60% variation) is shown for a very blunt slender cone for which Ree L

In gas dynamics it is frequently necessary to solve for Mach number for a given Prandtl-Meyer angle from the well known Prandtl-Meyer function. Analytical methods are possible only for special cases o f the ratio o f specific heats. In this note a hybrid computer method for solving this problem is presented. The accuracy is comparable to the pure digital solution.

ing of hydrogen lines in plasma," National Research Lab Rept 5455 (March 1962) 4 Unsold, A , ' Continuous spectrum of high-pressure Hg lamp and similar gas discharges," Ann Physik33, 607-616 (1938) 5 Biberman, L M and Norman, G E , "On the calculation of photoionization absorption," Opt i Spektroskopiy 8, 433 (1960); also Opt Spectr (USSR)(English Transl ) 8, 230 (1960) 6 Fay, J A and Kemp, N H , "Theory of stagnation point heat transfer in a partially ionized diatomic gas," Avco-Everett Res Lab Res Rept 144 (April 1963) Fig 1 The test model showing angular measurements

This chapter discusses the plane irrotational flow of an ideal gas. The chapter discusses subsonic flow and supersonic flow past a thin aerofoil. The Mach number of the flow in a supersonic wind tunnel can be determined by introducing a fine needle or probe into the stream and measuring the inclination of the Mach waves relative to the direction of the undisturbed flow. The Mach waves can be made visible by virtue of the fact that they involve changes of density and, therefore, of the refractive index of the gas. In the case of a subsonic stream, just as in the flow of an incompressible fluid, the pressure perturbation at any point of the flow depends on the shape of the contour as a whole. The equations of gas dynamics in the hodograph plane are discussed. The Prandtl-Glauert rule applies only to thin aerofoils of low camber and at a small angle of attack in a stream of relatively low Mach number.