V.I. Lapygin et al.

About Configuration of a Lifting Body of Minimal Heat Flux

Lapygin V.I., Gorshkov A.B., Mikhalin V.A., Fofonov D.M., Sazonova T.V.

Central Research Institute of Machine Building (TSNIIMASH), Korolev, Moscow region,Russia

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Received **** 2015

Copyright © 2015 byLapygin V.I., Gorshkov A.B., Mikhalin V.A., Fofonov D.M., Sazonova T.V. and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

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Abstract

Approximate dependency for estimating the friction coefficient and Stanton number is derived basing on calculations of viscous gas flow around blunted delta wing. Configuration of lifting body with minimal convective heat flux is found. Hypersonic flows around delta wings with blunted leading edges and nose at v=5000 m/s and Re = 106 are calculated for the angles of attack less 14. It is shown that values of heat fluxes at surfaces of the wings with minimal heat flux, maximal lift-drag ratio and minimal axial force are close at similar value of lift-drag ratio.

Keywords

Viscous gas flow; Heat flux; Lift-drag ratio; Reynolds number;Friction coefficient.

1. Introduction

Gas flow at high supersonic velocity is characterized by high enthalpy and high heat flux at a body surface that is necessary to take into consideration when optimal configurations are constructed. Commonly used assumption on constant value of friction coefficient Сf and, consequently, Stanton number St brings to erroneous predictions when the problem on configuration with minimal heat flux at side surface is solved. At the same time,some formulas for estimatingСfandStvalues against Reynolds number and pressure coefficient for definite bodiesare known [1-3]. Validity of such formulas for three-dimensional bodies is analyzed and configuration of bodies with minimal convective heat flux is discussedbelow.

2.Estimation of friction coefficient and Stanton number

Flow parameters and heat transfer of discussed delta wings and space vehicles were calculated through numerical integration of Navier-Stokes equations in common with conservation equations for mass of chemical components. The conservative Navier-Stokes equations at arbitrary system of coordinates were solved using Gauss-Seidel iteration. Nonequilibrium dissociated air was described by five-component model. Description of numerical algorithm and physical-chemical air model is presented in [4].

This mathematical model was used for calculation of nonequilibrium air flow around delta wing with wedge-like profile and flat lower surface (Figure 1) at V=5000 m/s and Re = 106 for the angles of attack from 0 to 14. Radii of bluntness of the nose and leading edges were equal to r=0.01l, where l – length of the wing chord, sweep angle of the leading edges = 70, volume W=0.0195l3. With given Mach number, surface temperature, and specific heat ratio, the values of Сf and St () at a blunted plate are defined by the following relations [1]:

,(1)

.(2)

Here n0.5; рwandp – pressureonbodysurfaceandinoncomingflow; l – referencelength;– distancefromtheleadingedgetowardsX-axis; Аs, Аf –proportionality coefficients: Аs/Аf = 0.5-0.7.

Figure 2. Accurate and approximate values of Cf(x,z) over surface of delta wing
at nonequilibrium flow regime

Comparison of Сfvalues found through solution of Navier-Stokes equations and calculated by the formula (1) (Figure 1) illustrates satisfactory accuracy of the approximate relation on the wing surface except for the vicinity of the plane of symmetry. Analogous comparison for distribution of St(х,z) in the plane of symmetry of the winged space vehicle [5] is shown in Figure 2 that also demonstrates good correlation of accurate and approximate values. Here X-axis is directed along the wing axis of symmetry, and Z-axis is directed spanwise.

Figure 2. Accurate and approximate values of St(x,z) over surface of winged spacevehicle at nonequilibrium flow regime

3. Optimalconfigurations

Let’s consider supersonic flow over a body with blunted leading edges in horizontal plane XOZ (right-hand rectangular coordinates). X-axis is directed downwards and makes the angle  withvelocity vector of oncoming flow. Y-axis is directed upwards at sin>0. The body side surface is prescribed by the equation y+f(x,z)=0. Total heat flux is written in form

,,

S body projection at XOZ plane,

Coefficients of aerodynamic forces Cy, Cx and lift-drag ratio K are written in the following way:

,

The problem is to find the function f(x,z) such that the functional Q, Cx, K possess maximal/minimal values at given planform area, volume,bluntness radii of the nose and leading edges. The body dimensions are taken in ratio to its maximal length.

Optimal configurations (Figure 3) maximal lift-drag ratio at *=12 (Form 1) and *=10 (Form 4), minimal area of side surface at *=0 (Form 3), minimal axial force coefficient at *=0 (Form 5), minimal heat flux at *=0 (Form 6) were found with the help of the numerical method of local variations [6]. According to this method, a body surface is partitioned into small triangular meshes, where the values Cf , St, Q, Cy, Cx, and K are calculated. Optimization procedure consists in varying the coordinates yi,jand yk,l of mesh point on the body surface: yi,ji,j and yk,l k,l and choosing such variation (i,j, k,l) which decreases/increases the functional to be optimized keeping the body volume.

The Form 2 was used as initial configuration for the optimization procedure. It was assumed that leading edges of the wing are situated in horizontal plane XOZ. That is why the optimal configurations depend on given angle of attack *. Bluntness radii of the nose and leading edges of all examined configurations are equal to r =0.01l, wherel – length of wing root chord, sweep angle of the leading edge volume W =0.0195l3. The friction coefficient and Stanton number are calculated by the formulas (1-2). The pressure coefficient is determined through the formulas of tangent wedge [7]:

,

hereM – oncoming flow Mach number,

 – specific heat ratio,

– vector of local internal normal to the body surface.

Figure 3. Wings with various shapes of side surfaces

4. Heatfluxtooptimal wings

Aerodynamic characteristics Cy, Cx, K of optimal wings shown in Figure 3 and heat fluxes to their surfaces were found by calculations of viscous nonequilibrium air flow at velocity v=5000 m/s and Re = 106in the range of the angles of attack from 0 to 14. Cross-section shapes of the optimal wings are different therefore with equal values of the angle of attack the heat fluxes to their surfaces are distinct. It is advantageous to compare heat fluxes with similar value of lift-drag ratio. Corresponding dependencies of heat fluxes for windward surface Q1(K) and for the whole side surfaceQ2(K) are illustrated in Figures 4, 5. It is seen in these plots that atK=constthe values of heat fluxQ1(K) are similar for all examined wings (Figure 4).

Figure 4. Heat flux Q1(K) to windward surface

Heat flux to the side surface of the wing, which is optimal with respect to Q, is 3% less as compared with the wing, which is optimal with respect to the parameter K. On the whole the distinction of total heat flux to side surfaces of examined wings does not exceed 7%. ThedependenciesQ2(x) arealmostequidistantat K2, and distinction of maximal lift-drag ratio of examined optimal wings does not exceed 1% (Figure 5).

Figure 5. Heat flux Q2(K) to side surface

5. Conclusion

The relations are suggested for satisfactory approximationsof the friction coefficient and Stanton number during numerical calculation of hypersonic laminar viscous gas flow around three-dimensional bodies. Hypersonic nonequilibrium laminar flows around optimal delta wings are calculated with Mach number M15. It is shown that with similar volume, planform and lift-drag ratio the total convective heat flux to surfaces of these bodies does not depend on the shape of their side surface. Optimality criteria were the following: lift-drag ratio, axial force coefficient, total convective heat flux to the surface.

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