TANGENTIAL INJECTION TO A SUPERSONIC FLOW ON A BLUNTED NOSE

The §ow pattern and the heat §ux to a body surface at a tangential gas injecting have been investigated. The cooling air was injected to a §ow through the tangential axisymmetric slot on the spherically blunted cylinder. The experiments were conducted at M∞ = 6, Re∞,Rw = 0.76 · 10, angle of attack α = 030, and the slot width hk/Rw = 0 0.021. The mass rate of the injecting gas was G = gj/(πρ∞u 2 ∞w) = 0 0.16. It has been shown that maximum of the heat §ux toward the sphere surface can be su©ciently decreased. Numerical investigations have been carried out using the solution of the NavierStokes equations for axisymmetric two-dimensional (2D) viscous compressible unsteady §ows at α = 0.


INTRODUCTION
To gain high a lift-to-drag ratio at a high-speed aircraft, a small bluntness radius is needed for fuselage nosetips, wing leading edges, and other projected elements.Small bluntness is also needed for operating e©ciency of inlet of a supersonic air-breathing jet engine.In some cases, low temperature of the vehicle surface is required (e. g., for mounting optical window used by photo devices).
However, with high levels of stagnation temperature and pressure behind the normal shock along with small bluntness radius of aircraft elements, the heat §ux is so high that reliable reusable thermal protection cannot be implemented by traditional methods, even using best materials.
Decreasing of heat loads towards the vehicle surface is possible by means of coolant injecting through the permeable surface or using special devices.The detailed analysis showed that the most e¨ective coolants (providing minimal weight and volume of the coolant system) are water, glycerol, ammonia, etc. [14].
However, in many cases where gas is available aboard (hydrogen, nitrogen, etc.), it is reasonable to implement the vehicle surface cooling via gas injecting.
In the absence of local zones of increased pressure on the wall (e. g., shock wave incidence on the protected surface), the lowest weight cost of the coolant matter injecting is attained by injecting coolant through a microporous material [5,6].But hydraulic properties of microporous material are not stable in operating time and they vary from sample to sample.This is caused, in particular, by the presence of the condensed particles in a coolant and its supply system.
In real-time use of cooling via gas injecting through perforation, injected trickles penetrate into free §ow across the boundary layer, even at the smallest size of holes to be manufactured at present time.For nose elements of streamlined bodies (where the boundary layer is generally thin), the penetration of trickles into the outer §ow is observed most of the time.This requires very high mass §ow rate of the injecting matter [7].
An alteration of hydraulic properties of injecting devices during operation is possible to prevent by injecting through a slot, including the cases with dust and moisture in the coolant and its supply system.This is conditioned by a large width of the slot (tenths of millimeter) in comparison with micropore diameter (microns and tenths of micron), and also because slot channels are almost straight.In [8,9], it was shown that it is possible to cool the lateral surface of a streamlined body using low mass rates of coolant.Tangential injecting through a slot could be more e¨ective in comparison with porous injecting in the case of §at shock wave incidence on a vehicle surface or in the dusted §ow [10,11].The present authors do not know any works on §ow and heat transfer study in the case of gas injecting through axisymmetric slot placed on the front surface of blunted nose.
This work represents the results of experimental and numerical study of §ow and heat transfer on the spherical bluntness surface of a longitudinally streamlined cylinder when air is injecting through a tangential axisymmetric slot.The outlet section of the slot nozzle was located at a small distance S * = S/R w = 0.23 (the central angle ψ = 13 • ) from the model critical point.It is shown that even at the substantial angle of attack of 30 • , it is possible to essentially reduce the maximum of the heat §ux towards the model surface using low mass rates of the injecting matter.
The study was carried out in the TsAGI wind tunnel operating in Ludwig scheme at the free §ow Mach number í ∞ = 6, total pressure p 0 = 30 bar, and stagnation temperature T 0 = 710 K.In the experiments, pressure of the injecting gas, the slot width, and the angle of attack were varied.
The case of zero angle of attack was investigated numerically.The 2D axisymmetric NavierStokes equations were solved using the implicit second-order total variation diminishing (TVD) scheme.It is shown that the computed §ow ¦elds are in qualitative agreement with the experimental data.
The similar investigations were performed on the sharp wedge model [1214].

Wind Tunnel
The study was carried out in the TsAGI wind tunnel operating in Ludwig scheme at the free §ow Mach number M ∞ = 6, total pressure p 0 = 30 bar, and stagnation temperature T 0 = 710 ë.Pressure of the injecting gas, the slot width and the angle of attack have been varied in the experiments.The high-pressure channel of the wind tunnel with the inner diameter of 70 mm and length 6 or 12 m is equipped by an external ohmic heater which provides gas heating to T 0 = 800 K (Fig. 1 [15]).The test section diameter is 2R n = 0.5 m.Duration of the steady-state §ow is τ = 23 or 46 ms.

Model
The model is a spherically blunted cylinder (Fig. 2).The cooling gas (air) is supplied through a central tube 4. The tangential axisymmetric slot is implemented between a §at ¤cap¥ and the sphere surface (A).The slot width h can be changed by rotating of the stem ¤cap¥ with very small thread in the central tube.
Because of the sphere curvature, the slot width in the exit section h e is larger than in the minimal entrance section h k .Thus, the ratio of the areas in critical and exit sections of the slot is F k /F e = (d k h k )/(d e h e ).When the minimal slot width is h k = 0.2 mm, the rated Mach number at the slot exit is M = 2.56 and at the slot width h k = 0.8 mm, the Mach number is M = 2.04.In all cases, the injecting §ow is supersonic that guarantees a uniform distribution of coolant in the azimuthal direction.
In the experiments, gas was supplied to the model by means of the fastoperating system.The cross section of each element of the system essentially exceeds the minimal cross section of the slot.Therefore, the gas velocity in the Figure 2 The model with the tangential slot on the spherical nosetip: 1 ¡ nose; 2 ¡ cup of slot; 3 ¡ thin wall; 4 ¡ coolant channel; T ¡ thermocouples; K ¡ calorimeters; and h k ¡ slot system is much lower than the sound speed even in the case of maximum slot width h k = 0.8 mm.The tests of the supply system together with the model showed that hydraulic resistance of the system is less than 1% of the model resistance.
Duration of the hydraulic experiments essentially exceeded duration of tests in the wind tunnel.That is why, the error in injecting gas rate measurement did not exceed 3%.
During experiments in the wind tunnel, the cooling gas supply was started prior to the diaphragm rupture with the advancing τ = 0.10.5 ms and it was stopped after the end of heat §ux 12 where ρ ∞ and u ∞ are the free stream density and velocity.Prior to the wind-tunnel experiments, careful measurements of injecting gas rate G j were carried out (Fig. 3) at various pressures P b in the vessel 18 where T b is the coolant temperature ahead the model and c(h k ) is the experimentally determined dependence of the hydraulic resistance of the slot having the width h k (see Fig. 2).
The thin wall technique and calorimeters were used for measuring the heat §ux to the body surface.A stainless steel foil band of 21-millimeter width and 0.2-millimeter thickness was welded to the model surface.From the inside, 0.1millimeter diameter kopel alloy wires were welded to the foil.Near the welding point, the wires were expanded to the thickness of 0.030.04mm.The dimensions of the thermocouple contact zone were 0.2 × 0.2 mm.
In accord with the thin wall technique, the heat §ux was obtained using the one-dimensional heat conduction relation: q = k dT /dτ where k ≈ ρcδ is the calibration coe©cient with ρ and c being the density and the heat capacity of the wall material and δ its thickness; T is the temperature; and τ is the time.The coe©cient k is determined using the calibration device.
The e¨ect of 2D heat conduction was investigated by numerical calculations and experimentally for the wall of di¨erent thicknesses (δ = 0.1 and 0.2 mm).It has been found that the lateral heat conduction along the wall weakly a¨ects the measurement results.

Supply System for Gas Injecting
The coolant supply system is shown in Fig. 4.
The basic elements of this system are: the vessel 18 of about 40-liter capacity; the valve 16 for ¦lling the vessel; the valve 15 for releasing pressure to atmosphere; the main pneumoelectric high-speed valve 11.
The maximum gas pressure in the supply system was 150 bar.When main valve 10 is opened, the pressure in the model sets in approximately 0.2 ms.Pressure P j and temperature T j in the gas supply system were registered at the vessel exit (gages 13 and 14, respectively) and at the model inlet (gages 4 and 5).
The pressure gage 4 has small inertia (t ∼ 2 ms).Inertia of the temperature gage 5 is approximately the same as of the pressure gage.

NUMERICAL PROBLEM FORMULATION AND SOLUTION METHOD
The NavierStokes equations for axisymmetric 2D viscous compressible unsteady §ows are solved numerically.The dimensionless conservative form of these equations is where (ξ, η) is the curvilinear coordinate system; x = x(ξ, η) and r = r(ξ, η) are the Cartesian coordinates in a plane section of the cylindrical coordinate system; Q(ξ, η) is the vector of conservative variables; E(ξ, η) and G(ξ, η) are the §ux vectors; and S(ξ, η) is the source vector.These vectors are expressed in terms of the corresponding vectors Q c (x, r), E c (x, r), G c (x, r), and S c (x, r) in the Cartesian coordinate system as where J = r det [∂(x, y)/∂(ξ, η)] is the transformation Jacobian.Cartesian vector components for 2D NavierStokes equations are: Here, e = p/(γ − 1) + ρ(u 2 + v 2 )/2 is the total energy; H = T /((γ − 1)M 2 ∞ ) +(u 2 + v 2 )/2 is the total speci¦c enthalpy; div V = ∂u/∂x + ∂v/∂r + v/r; and t is the stress tensor with components: The §uid is a perfect gas with the speci¦c heat ratio γ = const and Prandtl number Pr = const.The system of equations is closed by the state equation The dynamic viscosity µ is calculated using Sutherland£s formula where T µ = 110 K/T * ∞ .The second viscosity is assumed to be zero.The dependent variables are normalized to the corresponding free-stream parameters: pressure ¡ to the doubled dynamic pressure ρ * ∞ U * 2 ∞ ; the coordinates ¡ to the reference length L * = 1 mm; and time t ¡ to L * /U * ∞ .The computations were carried out at the free-stream Mach number M ∞ = 6, the Reynolds number Re 4, Pr = 0.72, T * ∞ = 86.33K, and T * w = 300 K.These §ow parameters correspond to the experimental conditions.
The NavierStokes equations are integrated numerically using an implicit ¦nite-volume method with the second-order approximation in space and time.A quasi-monotonic Godunov-type scheme (TVD scheme) with Van-Leer limiter [16] are used.This gives a system of nonlinear algebraic equations, which is solved using the Newton iteration method.At each iteration step, the corre-sponding linear system is solved by  the GMRes (generalized minimal residual) method.Note that this approach is most e©cient if the computational domain contains shock waves and other strong spatial inhomogeneities of a §ow such as boundary-layer separation.
The boundary conditions are: no-slip conditions on the wall (u = v = 0); symmetry conditions on the axis in front of the body; free-stream conditions on the upper boundary (u = 1; v = 0; p = 1/(γM 2 ∞ ); and T = 1); and the linear extrapolation from the interior for the dependent variables u, v, p, and T on the right (out §ow) boundary (f i − 2f i−1 + f i−2 = 0 ¡ ¤soft¥ boundary conditions).The body surface is isothermal with temperature T w = 3.47.Numerical simulations require an additional condition on the wall pressure.This condition is obtained by extrapolation of the near-wall pressure to the surface assuming that ∂p w /∂n = 0.
The computational domain is shown in Fig. 5.The generatrix of the upper boundary is parabola with the front point coordinates: (−12 mm, 0 mm) and the rear point coordinates: (55 mm, 79.5 mm).The body (and, therefore, the wall boundary) is composed of a hemisphere of radius R w = 37.5 mm followed by a longitudinal cylinder.In front of the hemisphere, there is a ¤cap¥ with radius r = 27.3 mm and height of 25 mm stepped out by h = 0.4 mm from the main body.
Computations are performed on a curved orthogonal grid.It is generated using numerical conformal mapping of a rectangle onto the computational domain [17].The grid is clustered near the surface so that 55% of the nodes are within the boundary layer or in the separation region with the mixing layer.The grid segment near the injecting region is shown in Fig. 6.

Numerical Results
The §ow is computed in the two Mach number 2.53; pressure 15.2 bar; and temperature 273 K.These parameters are set via the boundary conditions at the step between ¤cap¥ and hemisphere (Fig. 7).
The calculated §ow temperature ¦eld is shown in Fig. 8a.It is seen that the separation zone is formed immediately downstream from the step between sphere and ¤cap.¥The separation bubble sizes agree well with the experimental results.When gas is injected into the free stream, the §ow ¦eld near the step is completely di¨erent (Fig. 8b).
The separation zone immediately downstream from the step is not observed, the injecting jet width notably increases with a distance from the slot.At a distance of 3 jet widths, the injecting jet separates from the sphere surface.Further downstream, the jet reattaches and then the cooling gas veil §ows further without separation.Temperature of the cooling gas veil is lower than that of the near-wall main §ow everywhere in the computational domain.
The density ¦eld of the main §ow and the injecting jet is shown in Fig. 9.The zones of higher density formed in the shock waves are clearly seen in the injecting jet.The velocity vectors in Fig. 10 show the presence of the reverse §ow inside the separation zone.
The calculated heat §ux is in qualitative agreement with the experimental data (Fig. 11a).However, the predicted heat §ux is substantially lower than that measured in the experiment.The simpli¦cation (uniform speed at the output jet and the use of larger boundaries of jet than in an experiment) is an important reason for the di¨erences between experimental and numerical data [18].In the future work, the present authors will try to include the e¨ects of nonuniform and nonstationary §ow.11a shows also that the numerical solution does not depend on the grid ¦neness except for the heat- §ux peak region where the coolant jet reattaches the surface.Finer grids are needed to simulate the §ow in this region correctly.
The heat- §ux distribution with injecting essentially di¨ers from that of without injecting.The substantial decrease of the heat §ux is observed (Fig. 11b) on the whole model surface (up to S/R w = 1.9).

Experimental Results
The heat §uxes obtained in the experiments are nondimensionalized by q 0 ratabled using the Fay£s and Riddell£s formula for the critical point on sphere [19].Under the conditions of UT-1 wind tunnel (in presence of moisture, dust in free stream, and other reasons), the actual heat §ux on the model can exceed this value.The excess level increase can reach 60%.Therefore, at the further analysis, the relation of a thermal stream in the given point of a surface q to the experimentally received maximum value of a thermal stream on a model surface in the critical point q 0e will be used.
The ¤cap¥ disturbs the spherical shape of the blunted nose.For the experiments with sealed slot, this can be seen on shadow patterns as deformation of the bow shock in front of the slot.On the other hand, the coolant injecting a¨ects the e©cient shape of bluntness.For moderate injecting rates, the shape is approximately spherical.
Therefore, the mentioned shock deformation can be appreciably smoothed (Fig. 12a).Note that with the §ow rates used in the experiments, the e¨ect of injecting on the shock-layer thickness is practically negligible.However, as it can be seen in shadow pattern, the cooling air jet introduces the disturbances into the §ow over the model.
At the angle of attack of the model and the small expense of injected gas P j0 , di¨erent §ow pattern on windward generatrix of the model is possible.Momentum of injected gas can be insu©cient for penetration into area of a stagnation point.Then, the basic §ow forces the injected jet to change its direction toward the base of the model (Fig. 12b).Before the jet, there is a local shock wave (A).Thus, the neighborhood of a critical point is not cooled.
In some runs without cooling, the slot was sealed by the rubber pad.Without sealing, the heat §uxes near the slot can be higher (approximately by 10%) due to slight suction of hot air through the slot into the model.The results of the experiments with the sealed slot are shown in Fig. 13.The heat §uxes at some distance form the slot appreciably increase with the growth of the angle of attack.It is associated with a number of causes, in particular, with the displacement of critical point relative to the slot and with forming a separation zone behind it.) and injecting on the relative heat §ux distribution q = q/q0,e along the s-coordinate; h k = 0.2 mm; windward: 1 ¡ G = 0; 2 ¡ 0.005; 3 ¡ 0.010; 4 ¡ 0.020; 5 ¡ 0.078; and 6 ¡ G = 0.155 Figure 15 E¨ect of the angle of attack (α = 20 • ) and injecting on the relative heat §ux distribution q = q/q0,c along the s-coordinate (see Fig. 2); h k = 0.2 mm (h * = 0.0053): 1 ¡ G = 0; 1 ¡ 0.0100; 3 ¡ 0.0195; 4 ¡ 0.0398; 5 ¡ 0.0789; and 6 ¡ 0.157 Downstream from this zone, the separated §ow reattaches to the model surface that is accompanied by the heat §ux growth.
The experiments with injecting showed that with the coolant mass rate being ¦xed, the relative heat §ux is negligibly a¨ected by the slot width.Therefore, in further consideration, only the e¨ect of coolant mass rate G and angle of attack α on heat transfer rates are discussed (Figs. 14 and 15).
The injecting reduces the heat §uxes most in the case of zero angle of attack (see Fig. 14), when even for small mass rates (of order of G = 0.005), the peak value q m = q m /q 0 is decreased by a factor of 2 and the peak moves away from the slot to the point s = s m /R ≈ 0.25.
Qualitatively, the dependence q(G) remains the same for nonzero angles of attack (see Fig. 15).It is seen that when G is constant, the heat- §ux peak location with respect to the slot slightly depends upon the angle of attack.However, the peak value increases that is an evidence of decreasing ef-¦ciency of the injecting.For instance, at G = 0.157, the maximum heat §ux is reduced by a factor of 2.5 rather than by a factor of 10 in the case of α = 0 • .According with the foregoing discussion, the heat- §ux peak location is driven primarily by the coolant mass rate.
This conclusion is con¦rmed by the experimental data shown in Fig. 16 in terms of s m (G) for all angles of attack considered.The curve shows the approximate dependence.
The dependence of maximal heat §ux q m on the injecting rate is shown in Fig. 17 for di¨erent angles of attack.It is normalized by the maximal heat §ux q mg on the model with the sealed slot.For G > 0, the e¨ect of the angle of attack on the relative maximal heat §ux can be estimated using the relation q m q mg ≈ q m q mg n + 0.013α where (q m /q mg ) n is the relative heat §ux at zero angle of attack.

CONCLUDING REMARKS
The tangential gas blowing through an axisymmetric slot located near the critical point of a spherically blunted cylinder leads to decreasing of the heat §ux on the blunted nose and the adjacent cylindrical part.The heat §ux reduction is attained in the range of angles of attack from 0 • to 30 • .However, at high angles of attack, essentially greater coolant §ow rate is needed compared to the case of zero angle of attack.
The main parameter governing the heat §ux decrease appears to be the mass rates ratio G * of the blowing matter and free §ow.
The shape of the bow shock in front of the blunted nose does not practically depend on G * .
Numerical simulations provide insight into the §ow-¦eld structures of the main §ow and the blowing gas.The predicted heat- §ux distribution qualitatively agrees with the experimental results.However, turbulence should be accounted for to get a good quantitative agreement.

Figure 3
Figure 3The hydraulic resistance of the slot c(h k ) vs. the slot width h k

Figure 5
Figure 5 Computational domain

Figure 6
Figure 6 Grid near the injecting region (every 10th line is shown)

Figure 7
Figure 7 Injecting scheme: 1 ¡ sphere; 2 ¡ slot; 3 ¡ ¤cap¥ edge; 4 ¡ cap front surface; 5 and 6 ¡ boundaries of the injecting jet for numerical simulations; 7 ¡ injecting jet; and 8 ¡ turned main §ow regimes: with and without coolant injecting.The injecting gas parameters are:Mach number 2.53; pressure 15.2 bar; and temperature 273 K.These parameters are set via the boundary conditions at the step between ¤cap¥ and hemisphere (Fig.7).The calculated §ow temperature ¦eld is shown in Fig.8a.It is seen that the separation zone is formed immediately downstream from the step between sphere and ¤cap.¥The separation bubble sizes agree well with the experimental results.When gas is injected into the free stream, the §ow ¦eld near the step is completely di¨erent (Fig.8b).

Figure 10
Figure 10 The Mach number ¦eld and the velocity vectors at the injecting mass rate G * = 0.072: 1 ¡ mixing injected and near-wall §ows; 2 ¡ second separation zone; 3 ¡ separation zone; 4 ¡ shock wave in injected jet; 5 ¡ boundary points of the slot outlet section; and 6 ¡ critical point

Figure
Figure11ashows also that the numerical solution does not depend on the grid ¦neness except for the heat- §ux peak region where the coolant jet reattaches the surface.Finer grids are needed to simulate the §ow in this region correctly.The heat- §ux distribution with injecting essentially di¨ers from that of without injecting.The substantial decrease of the heat §ux is observed (Fig.11b) on the whole model surface (up to S/R w = 1.9).