A numerical method for solving the equations of a viscous shock layer

1995

The influence of the second-approximation effects of laminar boundary layer theory on the heat transfer in three-dimensional hypersonic flow over blunt cones with a large aspect ratio is investigated numerically .The influence of the second-approximation effects of boundary layer theory in the neighborhood of the stagnation point was first investigated in detail in [1] . In that study the principal second-approximation effects caused by the terms of order O(Re "2 ), which are not involved in the Prandtl equations, were identified . It was noted that the effects of entropy layer absorption by the boundary layer predominate over all the other second-order effects . In [2] it was shown that the contribution of the entropy layer absorption effect to the heat flux on the surface of an orbiter can reach 50% . The entropy layer absorption manifests itself at large distances from the bluntness . Various approximate approaches were developed for taking the entropy layer absorption into account within the framework of the classical Prandtl model for bodies with a large aspect ratio . These approaches involve both analytical studies of the problem and various engineering methods based on taking the vorticity on the outer edge of the boundary layer approximately into account (see, for example, [3][4][5][6][7]) .The system of equations of the complete viscous shock layer contains all the terms of the complete Navier-Stokes equations which contribute to the second approximation of boundary layer theory and makes it possible to determine with quantitative accuracy their total contribution . In study [8] the influence of the effects in question was determined for hypersonic axisymmetric flow over blunt cones with a large aspect ratio . It was found that the entropy layer absorption effect can lead to substantial changes in the heat transfer and friction coefficients (by up to 100%) in the zone of intense absorption, as compared with the results for the first-approximation boundary layer (classical theory) . It was noted that cooling the surface of the body leads to a decrease in the influence of the entropy layer absorption effect . The applicability of approximate methods of taking the entropy layer absorption into account was studied in [3][4][5][6][7] . In [9] the contribution of the second-approximation effects of boundary layer theory was investigated in the case of axisymmetric air flow over blunt bodies assuming equilibrium dissociation and ionization reactions in the shock layer .In the present paper the combined influence of the second-approximation effects in three-dimensional unseparated flow over long blunt cones is investigated . In contrast to [8], here we shall investigate a three-dimensional flow . The solution over the entire domain between the surface of the body and the surface of the shock wave, including subsonic flow regions, is found in a unified manner using a numerical method based on global iterations [10] . L FORMULATION OF THE PROBLEM AND METHOD OF SOLUTIONWe will consider an unseparated flow over l...

Section: Formulation Of the Problem And Methods Of Solutionmentioning

confidence: 99%

“…In contrast to [8], here we shall investigate a three-dimensional flow . The solution over the entire domain between the surface of the body and the surface of the shock wave, including subsonic flow regions, is found in a unified manner using a numerical method based on global iterations [10] . …”

mentioning

confidence: 99%

Effects of second-approximation of boundary layer theory in three-dimensional flow over bodies with a large aspect ratio at small angles of incidence

Lipchinskii¹,

Tirskii²,

1995

“…The set of viscous shock layer equations, described in [7,8], is used as the gasdynamic flow model. All the terms of the Euler equations are present in the equations of this set, together with all the second-order terms of asymptotic boundary layer theory.…”

Section: Problem Formulation and Methods Of Solutionmentioning

confidence: 99%

“…The two-layer Cebeci-Smith model [1] is used to describe the flow turbulence. The gas flow between the bow shock and the body surface is governed by the viscous shock layer equations for a perfect gas [7,8]: …”

Section: Problem Formulation and Methods Of Solutionmentioning

confidence: 99%

Study of turbulent hypersonic flow past elongated blunt-nosed cones

Паламарчук¹,

Tirskii²,

et al. 1994

Axisyrmnetdc turbulent hypersonic perfect-gas flow past slightly blunted cones is considered. The influence of the absorption of the entropy layer, developing on an elongated body, on friction and heat transfer to the body surface is studied using the viscous shock layer model. Correlations are established for the heat flux and the friction coefficient along the cone surface at large distances from the forward stagnation point. The results are presented together with their correlations in the framework of the viscous shock layer model and an approximate model of the boundary layer type, taking into account the entropy layer absorption. A generalization of the Reynolds analogy is obtained.There are many paper.s dealing with the calculation of the friction and heat transfer in flow past blunt bodies within the framework of the classical theory of the turbulent boundary layer (see [1]). The absorption of the entropy layer by the boundary layer may be regarded as the principal second-order effect in boundary layer theory. Approximate methods have been developed to take this effect into account (see [2--6]). It should be noted that the entropy layer is absorbed more rapidly by a turbulent than by a laminar boundary layer, due to the greater boundary layer thickness; in this case the absorption length is 5--50 bluntness radii. In [2--4] the effect of the absorption of the entropy layer by the boundary layer was taken into account by means of additional calculations for the flow parameters at the outer edge of the boundary layer, using the mass flow rate global balance method. The authors of [5] and [6] have proposed a technique based on the mass-average method. All these approaches enable the entropy layer absorption to be taken into account without solving a set of equations more intricate than that of boundary layer theory. All these approaches need theoretical or experimental verification.The similitude criteria for turbulent boundary layers on blunt-nosed cones were analyzed in [4,6]. In [4] the similitude parameters determining the heat transfer to the cone surface were established for the entropy layer absorption regime over the following ranges of the governing parameters: cone half-angle 0k= 10 deg, Reynolds number Re~ =1.0-105--3.1-105, Mach number Mo. =6--8, temperature factor tw=0.40--0.52 (here, tw=Tw/To, T w being the wall temperature and T O --the stagnation temperature). The similitude criteria governing hypersonic flow past slender blunt bodies in the turbulent boundary layer regime were determined in [6].Our aim is to study numerically the problem of hypersonic turbulent flow past blunt-nosed cones using the viscous shock layer model; to estimate the errors of the various approximate theories; and to establish the similitude criteria for the heat transfer at large distances from the bluntness and for the friction at both large and intermediate distances.

“…This makes it possible to calculate the distributions of all the gasdynamic parameters in the shock layer adjacent to a blunt cone with or without gas injection from the body surface, in uniform and nonuniform oncoming streams [10][11][12] . Comparison of the numerical solutions for flow past a sphere and a blunt cone obtained by this method with the experimental data and other numerical and asymptotic solutions [10][11][12] shows that the method possesses high accuracy and requires less computation time than time-dependent methods for the Navier-Stokes equations . On the basis of the approximate asymptotic solutions, a general similarity law was derived in [13] for convective heat transfer to the side surface of a slender blunt body in laminar hypersonic flow, as well as for other gasdynamic parameters .…”

mentioning

confidence: 99%

On the applicability of some approximate similitude laws in hypersonic aerodynamics

Pilyugin¹,

Talipov²,

1994

The range of applicability of some similitude laws for heat transfer, friction and drag coefficients is discussed on the basis of numerical solutions of the complete viscous shock layer equations describing hypersonic flow past blunt bodies .The method of matched asymptotic expansions [1] is used widely in the theory of hypersonic flow past bodies ; this method enables one to solve simplified gasdynamic equations in the first terms of the expansion of the unknown functions in asymptotic series in terms of a small parameter [2][3][4] . Similitude relations for the shock wave stand-off distance, the drag and friction coefficients, the convective and radiative heat transfer to impermeable bodies, which are of practical importance, have been obtained in the last few years on the basis of these solutions [2][3][4][5] . In the case of the intense subsonic injection of a foreign gas from the body surface, another small parameter appears, namely, the momentum ratio for the injected and oncoming gases [6,7] . In this case it is also possible to construct an asymptotic solution and to obtain, for example, the shape of the contact surface separating two flows [7] .The results of asymptotic and numerical studies of supersonic viscous nonuniform wake-type flows past blunt bodies with and without gas injection from the body surface were reviewed in [8] .Since the convergence of asymptotic solutions in the general case of nonlinear gasdynamic equations has not been proved strictly from the mathematical standpoint, the problem of the accuracy and the applicability range of the approximate similitude relations thus obtained arises . In order to arrive at an answer to this problem, it would be well to carry out a systematic comparison with the results of either specifically designed aerodynamical experiments or numerical solutions of the more accurate (non-simplified) gasdynamic equations .In recent years a numerical method of solving the complete viscous layer equations has been developed [9] . This makes it possible to calculate the distributions of all the gasdynamic parameters in the shock layer adjacent to a blunt cone with or without gas injection from the body surface, in uniform and nonuniform oncoming streams [10][11][12] . Comparison of the numerical solutions for flow past a sphere and a blunt cone obtained by this method with the experimental data and other numerical and asymptotic solutions [10][11][12] shows that the method possesses high accuracy and requires less computation time than time-dependent methods for the Navier-Stokes equations . On the basis of the approximate asymptotic solutions, a general similarity law was derived in [13] for convective heat transfer to the side surface of a slender blunt body in laminar hypersonic flow, as well as for other gasdynamic parameters . The similitude laws for inviscid flow past blunt slender bodies and viscous flow past sharp slender bodies follow from this law as special cases .