1 Introduction

High-speed flying vehicles often face the problem of high surface-temperature induced by aero-thermal heating. In terms of long-distance flights,measures for protecting them from being damaged by the heating are necessary. For instance,heat resisting tiles are used for space shuttles^[1]. However,they add much weight to the vehicles,and a large amount of maintenance work is needed after each flight. Also,this method is obviously inappropriate for the case of heat protection of optical windows on the flying vehicles^[2],because tiles are not optically transparent.

The film-cooling technique has already been successfully applied for the protection of turbine blades from being damaged by the oncoming flow with a high temperature^[3-4]. Naturally,one would consider the possibility of its application for the protection of the surfaces of high-speed flying vehicles from the aero-thermal heating.

In fact,the cause of the high temperature is fundamentally different between the two cases. For turbine blades,the high temperature is induced by the combustion of the fuel in the combustion chamber^[5]. In order to cool the gas,the liquid,which needs a large amount of heat for its evaporation,is injected from inside of the blade through small holes. Through its evaporation,a large amount of heat is consumed,therefore,the gas film around the blades has a temperature appreciably lower than that of the oncoming flow. On the other hand,for high-speed flying vehicles,the original temperature of the oncoming flow is very low,e.g., -40 ℃,and the high temperature is induced by the conversion of kinetic energy of the oncoming fluid into the thermal energy,when the relative speed of the oncoming fluid drastically reduces to be nearly zero in approaching the surface of the vehicle. Therefore,if an injected gas film forms successfully,the oncoming flow will not reach the close neighborhood of the surface of the vehicle,and correspondingly,its relative velocity will not become so small,leading to a lower temperature than the cases without injection. Moreover,the injected gas can always carry away a certain amount of heat,which further enhances the heat protection effect. In addition,because the oncoming flow is prevented from reaching the surface of the vehicle,the drag reduction is also expected.

So far,it seems that no one has ever offered such an interpretation for the gas-film-cooling effect applied to high-speed flying vehicles. We believe that the above interpretation would encourage people to put more efforts on the realization of the gas-film-cooling technique for high-speed flying vehicles. Although there were investigations on the use of the gas-film-cooling technique for hypersonic vehicles,such as Ref. ^[6],they were merely for the protection on the stagnation region of the vehicles,and the working principle is different.

In order to show that,in principle,the gas-film-cooling technique for protecting the surface of high-speed flying vehicles could be as effective as discussed above,two model problems are studied via the numerical simulation. Being different from the application for turbine blades,in which many factors,such as pressure gradient and acceleration of the mainstream,would affect the film cooling performance and the heat transfer under the film^[7],the models to be studied in this paper are quite simple due to the simplicity of the hypersonic flying vehicles in geometry. The first model is a hypersonic boundary layer on a flat plate,and the second one is a hypersonic boundary layer over an optical window on a sphere cone.

2 Problem I: boundary layer on flat plate 2.1 Physical model

The model to be studied is a two-dimensional (2D) boundary layer on a flat plate,with a slot at a distance $x_\mathrm s^\ast$ downstream of the leading edge for the gas injection,as shown in Fig. 1. By assuming that the plate is sufficiently thin and at zero angle of attack,we ignore the effect of the very weak shock at the leading edge,and consequently,the flow quantities at the edge of the boundary layer are considered to be the same with those of the oncoming flow. Denote the velocity,pressure,temperature,density,sound speed,and dynamic viscosity of the oncoming flow by $u_\mathrm e,p_\mathrm e,T_\mathrm e,\rho_\mathrm e,a_\mathrm e$,and $\mu_\mathrm e$,respectively. The Mach number is defined as $Ma=u_\mathrm e/a_\mathrm e$. The displacement thickness $\delta^\ast$ of the boundary layer for the case without injection at $x=x_\mathrm s^\ast$,is chosen as the reference length,and the Reynolds number is defined as $Re={{\rho }_{\text{e}}}{{\delta }^{*}}{{u}_{\text{e}}}/{{\mu }_{\text{e}}}.$

The streamwise width of the slot is denoted by $d$. The cold gas,with the dimensionless velocity,the pressure,the temperature,and the density of $v_\mathrm s,p_\mathrm s,T_\mathrm s,\rho_\mathrm s$,is injected from the slot into the flow field. In fact,the gas velocity across the slot should not be uniform,however,the distribution of the velocity at the slot has very limited effects on the downstream profiles as long as the total mass flux keeps unchanged. For subsonic boundary layers,this has already been proved in Ref. ^[8],and we will show in Subsection 2.4 that it is also true in our case.

If the flow is assumed to be laminar and unmixed,then the flow field downstream of the slot would consist of two parts. The upper part comes from the oncoming high-speed flow,while the lower one comes from the injection slot. Along each wall-normal line,the speed of the oncoming flow will have its minimum value at the interface of the two streams. Correspondingly,the temperature will have the maximum value there. Since the velocity at the interface is obviously not zero,the highest temperature of the oncoming stream would certainly be smaller than its total temperature.

If there is no flow injected from the slot,then the minimum speed of the oncoming stream is obviously zero at the surface. Assume that the wall is adiabatic,then,the dimensionless temperature there would approximately be equal to the total temperature $T_0$,which can be calculated by

(1)

where the subscript ‘e’ refers to the quantity of the oncoming flow,whereas the superscript ‘*’ indicates a dimensional value,$c_p$ is the specific heat of constant pressure,and $\gamma=1.4$ is the ratio of the specific heats. For $M\!a\gg 1$,we have

(2)

Moreover,for the case with injection,the total temperature at the interface of the two streams,as calculated from the oncoming stream side,can be expressed approximately as

(3)

where $u_\mathrm i$,$T_\mathrm i$,and $(T_\mathrm i)_0$ are the dimensionless streamwise velocity,temperature,and total temperature at the interface,respectively. Since the velocity $u_\mathrm i$ is non-zero,the temperature $T_\mathrm i$ is always smaller than $T_0$. The larger $u_\mathrm i$ is,the smaller $T_\mathrm i$ will be. A simple estimation gives that if $u_\mathrm i\approx 0.5$,then $T_\mathrm i\approx 0.75T_0$. In fact,the temperature of the injected fluid is always much lower than $T_0$,therefore,the real temperature at the wall is always not larger than $T_\mathrm i$. This is exactly the working principle of the gas-cooling technique applied to high-speed flying vehicles,which apparently is very different from that applied to turbine blades.

2.2 Numerical configuration

Assume that the vehicle is flying at an altitude of 40 000 m,where $p_\mathrm e^\ast=287 \mathrm {Pa},T_\mathrm e^\ast=250.4 \mathrm {K},$ and $ \rho_\mathrm e^\ast=0.003 996 \mathrm {kg/m}^3$. Let the Mach number be $M\!a=10$,and correspondingly,the velocity of the oncoming flow be $u_\mathrm e^\ast=3 172$ m/s. The slot center is assumed to be located at $x_\mathrm s^\ast=0.42 \mathrm {m}$ downstream of the leading edge of the plate,where the displacement thickness of the boundary layer is found from the Blasius solution to be $\delta^\ast=2$ cm,and then it is calculated that the Reynolds number is $Re=15 800$. Three cases with different slot widths are considered in this section,with parameters displayed in Table 1. The mass fluxes for Cases 2 and 3 are 1.5 and 2.0 times of the first one,respectively. The Mach number of the injected gas is chosen to be subsonic,which is easier for the design of the slot nozzle.

The numerical method used in this section can be found in Refs. ^[9]-[11]. A Cartesian coordinate system $(x,y)$ is used with its origin located at the center of the slot. The computational domain is $-4\leq x\leq 88$ (or $-0.08 \mathrm {m}\leq x^\ast\leq 1.76 \mathrm {m}$) and $0\leq y\leq 14$. There are 1 159 grid points in the streamwise direction,which are refined in the vicinity of the slot with the finest grid spacing of 0.025. There are 151 grid points in the wall-normal direction,which are refined towards the wall with the finest grid spacing of 0.003.

The inflow and upper boundary conditions are given by the Blasius solution of the boundary-layer equations,and the extrapolation method is used as the outflow condition. No-slip,non-penetration,and adiabatic (which is the most dangerous case) conditions are used at the wall apart from the slot,while the velocity and the temperature at the slot are given as those shown in Table 1.

2.3 Resolution study

In order to verify if the resolution of our computation is sufficient,we decrease the grid-spacing either in the streamwise (Mesh 2) or in the wall-normal (Mesh 3) direction as shown in Table 2,and compare the results with those of Mesh 1. In order to save time,the computation is performed in a smaller domain. Figure 2 shows the comparison of the density and temperature at the wall. Good agreement is observed,implying that our grid spacing is sufficiently fine.

2.4 Tests for boundary condition at slot

In order to confirm that the distribution of the injected flow from the slot has no apparent effect on the downstream flow field in our computation,we also have tried another velocity distribution instead of being uniform. The velocity is of a parabolic distribution,i.e.,

(4)

with the total mass flux being the same,i.e.,

(5)

In this test case,the temperature and density still remain uniform.

Figure 3 compares the streamwise distribution of the wall temperature $T(x,0) $ and the wall-normal temperature profile $T(80,y)$ at $x=80$ as obtained under the two injection conditions. The difference of the numerical results is extremely small. Comparisons for the velocity and the density profiles show similar trends. It is also verified that non-uniform distributions of the density and the temperature at the slot nozzle have little influence on the downstream flow. The details are not shown here to save space.

2.5 Numerical results

Figure 4 shows the flood-type contours of the pressure,the streamwise velocity,the density,and the temperature for Case 1. The pressure field,as shown in Fig. 4(a),exhibits remarkable gradients,which represent a weak shock and a series of expansion waves. The velocity field,on the other hand,is not affected by the gas injection appreciably,as shown in Fig. 4(b). There exists a high-density zone (see Fig. 4(d)) near and downstream of the slot due to the injection of cold gas. The density decreases with $x$,and gradually approaches to that of the Blasius solution when $x>20$ (see Fig. 4(c)). Most importantly,the low-temperature zone extends further downstream. Even at the outlet of the computational domain,the temperature remains to be significantly lower than that upstream of the slot,implying that the gas film does work well in reducing the wall temperature.

Figure 5 shows more clearly the wall-normal distributions of the streamwise velocity and the temperature at several streamwise locations,in which the circle symbols indicate the locations of the interface between the two streams. The interface is obtained as the location below which the total streamwise mass flux is equal to that injected from the slot. Upstream of the slot,i.e.,at $x=-4$,the profile is the classical Blasius profile. At the location of the slot center ($x=0$),the injected gas produces a low-speed zone in the near-wall region (say,$y< 0.4$),in which the temperature is also quite low. Further downstream,the velocity profile approaches gradually to the adiabatic Blasius profile. However,even at $x=88$,the outlet of the computational domain,the temperature profile is still quite different from the adiabatic Blasius profile. The latter should be similar to the profile at $x=-4$ shown in the same figure.

One can also see from Fig. 5(a) that the velocities at the interface for different streamwise locations are all about 0.45. It can be predicted from Eq. (3) that the temperature should be about 0.8 times of the total temperature. However,Fig. 5(b) shows that the temperatures at $x=20,40$,and 88 are only 0.41,0.44,and 0.54 times of the total temperature,respectively,which are much lower than expected. The reason is that the gas injected from the slot has a very low total enthalpy,and its temperature would decrease further as its velocity increases from subsonic to supersonic. Therefore,it has the capacity to absorb a certain amount of heat from the oncoming stream above the interface,leading to a further reduction on the temperature at the interface.

Figure 6(a) shows the profiles of the streamwise velocity at different $x$ for the three cases shown in Table 1. Increasing the mass flux results in a thicker boundary layer,which can also be found in Fig. 6(b). Also,the thickness of the gas film increases with the injected mass flux. Figures 6(c) and 7 show the streamwise distribution of the wall temperature and the wall-normal temperature profiles,respectively. The temperatures both at the wall and in the near wall region decrease as the mass flux increases,implying that the cooling effect is more significant if the injected mass is larger. In addition,for all the cases,the wall temperature increases along the streamwise direction,whose reasons are twofold. First,the ability of the injected gas in absorbing heat from the oncoming flow at the interface decreases as propagating downstream,because its total capacity of absorbing heat is obviously limited. Second,the effectiveness of the cooling film depends on its relative thickness in comparison with the total boundary-layer thickness. As shown in Fig. 6(b),the increment of the boundary-layer thickness is faster than that of the cooling film,leading to a reduced effectiveness of the cooling film.

In the flying-vehicle design,the load is one of the most important factors to be considered. Therefore,whether a proposed technique is feasible depends on the load added due to the adaption of the technique. In our case,the weight of the total cooling gas is a part of the added loads. For a flat plate with a unit spanwise width,i.e.,1 m,the mass flux of the injected gas is readily calculated,according to Table 1,to be 0.005 36 kg/s for Case 1. If the vehicle is conducting a 30-minute flight,then 9.65 kg of gas will be needed. For Case 2 and Case 3,the corresponding loads should be 1.5 times and 2 times of that value,respectively.

2.6 By-product: drag reduction

If the gas film forms successfully,the wall temperature will be suppressed,which leads to a reduction on the viscosity $\mu_\mathrm w$. Simultaneously,as shown in Figs. 5(a) and 6(a),the velocity gradient at the wall is also reduced. Figure 8 compares the streamwise distributions of the viscous stress $\tau_\mathrm w=\mu_\mathrm w\frac {\partial u}{\partial y}|_\mathrm w$ for the three cases with that for the conventional Blasius boundary layers. The reduction of the viscous stress is found to be remarkable.

The total amount of the drag force for a plate with a unit spanwise width can be calculated by

(6)

where the lower and upper limits of the integral are $x_1=0$ and $x_2=88$,respectively. For the Blasius profile,it is calculated that $F=0.019 8$,while for the three cases studied in this paper,the total drags are 0.009 05,0.005 90,and 0.004 17,respectively. Correspondingly,the total drag reductions induced by the injected gas film are $54%$,$70%$,and $79%$,respectively.

3 Problem II: boundary layer with optical window on spherical cone 3.1 Physical model and numerical configuration

Optical windows often appear on certain high-speed vehicles,and they are vulnerable to the high temperature environment in two senses. First,the high temperature may cause damage to the structure of the window. Second,density fluctuations in high-temperature boundary layers over the optical windows may deteriorate the quality of the optical image. Naturally,people would ask if the cooling film technique proposed above can also be used in protecting the optical window in the high temperature environment. In this section,we will focus on this aspect.

The physical model to be studied is a spherical cone with an optical window,as shown in Fig. 9. The radius of the nose of the sphere cone is $r^\ast=5$ cm,and the semi-cone angle is $\theta=25 ^\circ$. Again,we assume that the vehicle is flying at an altitude of 40 000 m with the zero angle of attack. All the flow quantities are normalized by those of the oncoming flow,and the reference length is chosen to be $r^\ast$. In the figure,$r$ is the dimensionless radius of the nose. The Mach number of the oncoming flow is 8,and the Reynolds number is $Re=\rho_\mathrm e r^\ast u_\mathrm e/\mu_\mathrm e=31 700$.

A cylindrical coordinate system $(x,y,\phi)$ is used with its origin locating at the center of the nose. The body-fitted coordinate system $(\xi,\eta,\phi)$ is also shown in Fig. 9,which is used for analyzing the results. Since the problem is axisymmetric,only one meridian plane is taken into account,and hence the azimuthal angle $\phi$ is not shown in Fig. 9.

Assume that the upper surface of the optical window is a part of the cone surface,and its streamwise extent covers the region $x\in [8,13]$, or $\xi\in [10.4,15.9]$. The center of the slot for the gas injection is located at $x_\mathrm s=6.3$,or $\xi_\mathrm s=8.6$. For preliminary study,we assume that the length scale in the azimuthal direction is sufficiently large in comparison with the thickness of both the boundary layer and the gas-cooling film. Therefore,the base flow in the region apart from the ending points of the slot is considered as a 2D one.

In this problem,a strong bow shock will certainly be generated in front of the cone. The computational domain,as sketched in Fig. 9,is a part of the space surrounding the cone,which includes the bow shock. In order to capture the shock properly,the second-order nonoscillatory nonfree dissipative (NND) scheme^[12] is used for nonlinear terms. The second-order central difference scheme is used for the viscous terms,and the first-order implicit scheme is used for time advancing.

The quantities at the upper boundary are set as the oncoming flow,and the no-slip,non-penetration,and adiabatic conditions are again used at the wall,except those at the slot,where the cooling gas with the normalized velocity $v_\mathrm s=0.08$,the density $\rho_\mathrm s=25.8$,and the temperature $T_\mathrm s=0.7$ is injected. The Mach number of the injected flow is 0.9,which is again a subsonic jet. The streamwise width of the slot is 4 mm. At the inlet of the computational domain ($\xi=0$),the symmetrical boundary condition is used,and an extrapolation condition is used at the outlet.

3.2 Numerical results of base flow

Figure 10 shows the flood-type contours of the pressure and the temperature. We can observe from the pressure field that,a weak shock followed by a series of dilatation waves is generated due to the interaction of the oncoming flow with the injected gas,which later merges with the main shock. After that,the main shock layer is thickened. There is an apparent low-temperature layer downstream at the slot,which is the direct consequence of cooling effect of the gas film.

Figure 11 shows the upper surface of the cooling film and the shocks for both the cases with and without injection. It is found that the shock wave moves slightly outward if there is an injection from the slot.

Figures 12(a) and 12(b) compare the velocity profiles at both the beginning $\xi=10.4$ and the end $\xi=15.9$ of the optical window between the cases with and without injection. The boundary layer is thickened appreciably due to the injection. The interfaces of the two streams for the two profiles are located at $\eta=0.14$ and 0.18,respectively,where the streamwise velocities are 0.25 and 0.34 of the inviscid stream,respectively. Figures 12(c) and 12(d) show the temperature profiles at the same two locations as Figs. 12(a) and 12(b). For the case of no-injection,the temperature first undergoes a jump when crossing the shock in the inviscid region,and then increases further in the boundary layer. The injection of the cooling gas causes a slight increment on the thickness of the boundary layer,which consequently pushes the shock outward slightly. The main effect of the gas injection is the significant change of the distribution of the temperature in the near wall region,i.e.,the wall temperature is remarkably reduced.

Figure 13 compares the wall temperature between the cases with and without injection. For the latter case,the wall temperature,as expected,is close to the total temperature of the oncoming flow. However,when the cold gas is injected from the slot,the great suppression effect on the wall temperature is observed. For the case studied,the wall temperature in the region of the optical window ($x\in [8, 13]$) is kept below 300 K.

3.3 Linear stability analysis of boundary layer with gas injection

Although the cooling effect is favorable as shown above,there is one issue which has to be addressed before its practical application. That is whether the laminar-turbulent transition may occur within the region of the window. Because if the flow is turbulent,there will be significant density fluctuations,which may render the window optically unworkable.

It should be noticed that,a comprehensive analysis of this problem is not possible,because the laminar-turbulent transition remains to be one of the unsolved problems in fluid mechanics. However,the linear stability analysis,which although may not be 100% correct,could help engineers make the decision.

The flow downstream of the slot is essentially a mixing layer of the two streams,which supports the Kelvin-Helmholtz instability. This kind of instability is of inviscid nature,with much higher growth rates than those for the viscous Tollmien-Schlichting (T-S) waves in the boundary layer. Hence,it is necessary to see if the instability waves will be amplified too much to cause the laminar-turbulent transition.

For a 2D base flow,the primary instability waves could be both 2D and three-dimensional (3D),but the most amplified waves are usually 2D unstable waves,which is also verified in the present case. In the late laminar stage,when the primary instability waves have accumulated to finite values,the secondary instability modes,with 3D nature,will appear and increase,and eventually trigger transition to turbulence. Therefore,the amplitude-accumulation of the primary instability is a critical parameter in the transition prediction,and its evolution is investigated by the linear stability theory (LST).

We select a base-flow profile $[u,T,\rho](\eta;\xi_0) $ at a certain streamwise location $\xi_0$ and introduce the parallel-flow assumption. Then,the perturbation can be expressed as

(7)

where $\omega$ is real and represents the frequency,$\alpha=\alpha_\mathrm r+\mathrm i\alpha_\mathrm i$ is complex,with the real part $\alpha_\mathrm r$ representing the wave number and the imaginary part $-\alpha_\mathrm i$ representing the growth rate,$A$ is the amplitude,and $\widehat u,\widehat v,\widehat T,$ and $ \widehat \rho$ are the eigen-functions of the instability mode. Inserting Eq. (7) into the linearized Navier-Stokes (N-S) equation leads to the Orr-Sommerfeld (O-S) equation^[13].

Figure 14(a) shows the growth rate as a function of frequency for several streamwise locations. The maximum growth rate increases slightly with $x$,while the frequency of the most unstable wave decreases with $x$. The growth rate is so large that the most unstable wave can be amplified 2.7 times within just a distance of 5 cm ($r^\ast$). Figure 14(b) shows the phase speed of the unstable wave at the same locations. There is a slight discontinuity at $c_\mathrm r\approx 0.8$ for each curve,which is due to the intersection of two branches of the unstable modes. For the branch with lower frequencies than the intersection points,the phase speed can be even larger than the free stream,and the growth rate as shown in Fig. 14(a) is rather low. Therefore,this branch is not of our interests. For the other branch,whose frequencies are relatively high,the phase speed decreases with the frequency rapidly. The phase speed of the most unstable mode is very close to the velocity at the interface.

As pointed out by the latest version of $\mathrm e^N$ method for transition prediction^[14-15],if one of the unstable waves,starting from small initial amplitudes,accumulates to an amplitude of $O(1.5%)$,then the transition is likely to happen soon. Therefore,by assuming the initial amplitude of each unstable wave to be $A_0$,the downstream amplitude can be obtained by integrating the growth rate along $\xi$,i.e.,

(8)

in which the lower bound $\xi_0$ is chosen to be at the beginning of the optical window,i.e.,10.4,and $N$ is the amplification factor.

Figure 15 shows the value of $\mathrm e^N$ for the unstable waves with different frequencies. In the region of our interests,i.e.,$10.4\leq \xi\leq 15.9$,the unstable waves with frequencies $\omega\approx 2.0$ have the largest amplification factor,and the corresponding value $\mathrm e^N\approx200$. If one knows the amplitude of the initial disturbance,then he can predict whether the transition would happen or not by using this $\mathrm e^N$ value.

4 Concluding remarks and discussion

For aero-thermal heating of high-speed vehicles,the origin of the high temperature is fundamentally different from that of gas turbines. The high temperature is the consequence of converting the high kinetic energy of the oncoming air into the thermal energy as it decelerates in reaching the surface of the vehicle. Therefore,the working principle of the corresponding gas-film-cooling technique is totally different. The gas injected from the holes,slots,etc.,on the surface forms a film around the surface of the vehicle. The gas film prevents the oncoming air from reaching the surface of the vehicle,such that a large portion of its kinetic energy will not be converted to heat,leading to a reduction on the maximum temperature. Moreover,the injected gas can also take away a certain amount of heat,which makes the protection even more effective.

In order to verify this concept,two problems have been studied,i.e.,a boundary layer on a flat plate and a boundary layer with an optical window on a spherical cone. Results do confirm that the concept is correct. For the second problem,we also perform the linear stability analysis on the boundary layer with the gas film. According to the latest version of the $\mathrm e^N$ method,an argument whether the laminar-turbulent transition would occur is provided.

Admittedly,the investigation of this paper only shows that in principle,the gas-film-cooling technique could be effective for the aero-thermal protection of the high-speed flying vehicles. For its real application,several other issues,for example,whether the film can keep its spanwise uniformity or smoothness in the 3D case,and how the effectiveness of the technique would be affected in terms of the occurrence of the laminar-turbulent transition,need to be addressed.

Acknowledgements The author is grateful to Prof. Heng ZHOU of Tianjin University. It was him who pointed out to the author that the working principle of gas-film cooling technique applied to high-speed flying vehicles may be fundamentally different from that applied to gas turbine blades. Frequent discussion with him throughout this work was very helpful. The author would also thank Prof. Xuesong WU of Imperial College London for the valuable comments and suggestions.