Jinghui GUO, Guiping LIN, Xueqin BU, Hao LI

Laboratory of Fundamental Science on Ergonomics and Environmental Control, School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, China

KEYWORDS

Abstract Implementation of an opposing jet in design of a hypersonic blunt body significantly modifies the external flowfield and yields a considerable reduction in the aerodynamic drag. This study aims to investigate the effects of flowfield modeling parameters of injection and freestream on the flow structure and aerodynamics of a blunt body with an opposing jet in hypersonic flow.Reynolds-Averaged Navier-Stokes (RANS) equations with a Shear Stress Transport (SST)turbulence model are employed to simulate the intricate jet flow interaction. Through utilizing a Non-Intrusive Polynomial Chaos (NIPC) method to construct surrogates, a functional relation is established between input modeling parameters and output flowfield and aerodynamic quantities in concern.Sobol indices in sensitivity analysis are introduced to represent the relative contribution of each parameter. It is found that variations in modeling parameters produce large variations in the flow structure and aerodynamics. The jet-to-freestream total-pressure ratio, jet Mach number,and freestream Mach number are the major contributors to variation in surface pressure, demonstrating an evident location-dependent behavior.The penetration length of injection, reattachment angle of the shear layer, and aerodynamic drag are also most sensitive to the three crucial parameters above.In comparison,the contributions of freestream temperature,freestream density,and jet total temperature are nearly negligible.

1. Introduction

A strong shock wave generated in front of the blunt nose of a high-speed vehicle causes a rather high surface pressure,leading to a substantial aerodynamic drag.In terms of a hypersonic cruising vehicle, the shock-wave drag may account for two thirds of the total drag and is associated with severe aerothermal loads.1To overcome the large drag, the vehicle demands a suited thrust with massive fuel consumption, and meanwhile, its available flying speed range as well as payload carrying capacity is possibly limited to some extent.2Nevertheless,since using a slender body shape reduces the wave drag yet gives a rise in surface heating, a hypersonic vehicle generally has a blunt nose to alleviate surface heating, which consequently raises the wave drag.3Thereby,the issue of drag reduction of a hypersonic vehicle during its ascent or cruising phase has received much attention and efforts from researchers, and has been extensively investigated over the years.4–7

Among the variety of drag relieving techniques, an opposing jet is suitably applied in both drag and heat reductions for blunt nose shapes at hypersonic speeds, where the total drag of a vehicle is dominated by the wave drag.1,8Prior investigations have found through experiments and numerical simulations that the structure and stability of jet flow change with the total-pressure ratio of jet and freestream.9–11Accordingly,two modes of jet interaction with incoming freestream are indicated, namely the Long Penetration Mode (LPM) and the Short Penetration Mode (SPM).12,13

As the total-pressure ratio rises,the unstable multi-cell LPM emerges with oscillating motions.9,13The long upstream length of jet penetration into bow shock considerably enlarges the shock stand-off distance and attenuates the shock strength(by dispersing the shock), yielding decreased surface pressure and wave drag.At the threshold point,a minimum total drag is realized.13,14Further increasing the total-pressure ratio beyond the threshold value, a nearly stable single-cell SPM flowfield is formed, in which the jet does not penetrate the bow shock,and the shock stand-off distance distinctly diminishes when compared with that of the LPM.9,15However, although both the pressure and wave drag undergo a sharp jump immediately after the threshold point, they maintain a continuous drop as the total-pressure ratio mounts up.10,13The total drag falls with a small slope and is relatively low despite a little higher than the lowest value at the threshold point.13As confirmed,both modes have the ability to achieve a drag reduction.13,16Considering the fact that the LPM jet is less feasible to obtain for its existence under a narrow range of flow conditions16,17and unstable with undesirable fluctuations of flow properties incurring flight disturbances and control complexity,10,13the stable SPM jet mode is chosen for this investigation.

To better understand the underlying flow physics and attain the appropriate characteristics of an opposing jet,many previous studies have focused on exploring and revealing the mechanism of key modeling parameters and consequent flow structure and aerodynamics. Finley9carried out systematic experiments and established an analytical model to investigate opposing jets from various blunt shapes with sonic and Mach 2.6 jet nozzles of different sizes. Three flow regimes successively appeared and were found to be governed by the increasing total-pressure ratio of jet to freestream. The variation of the critical total-pressure ratio was fitted linearly relative to the body dimension, which provided a sufficient condition for judging the occurrence of steady flow. Further, the flowfield and aerodynamic features of steady flow were found to be dictated primarily by the jet flow-force coefficient (or the jet-to-freestream total-pressure ratio)and the jet Mach number at the nozzle exit.With an increase of the jet flow-force coefficient, the blunt-body surface pressure dropped, and the reattachment point moved downstream, while the positions of the bow shock,Mach disk,and interface shifted forward.Likewise, Romeo and Sterrett18,19developed a theoretical method based on their experiment to calculate the locations and sizes of the jet shock and interface by using the known total-pressure ratio of jet to freestream and freestream Mach number.Hence,a counter-jet flowfield could be primarily constructed. McGhee20experimentally studied supersonic jets issuing from a spherically blunt cone at freestream Mach numbers of 3.0, 4.5, and 6.0. He found that in steady regimes, the positions of the bow shock, jet shocks, and flow interface as well as separation and reattachment pressures were all mainly functions of the nozzle-thrust coefficient.Later,Zhou and Ji10recovered and extended Finley’s experimental findings with a set of total-pressure ratios, jet nozzle size ratios, and angles of attack by numerically solving three-dimensional unsteady RANS equations. Apart from gaining results consistent with Finley’s, they demonstrated that in a stable flow, fluctuation of the drag coefficient was largely subdued,and the drag coefficient mainly diminished with an increasing total-pressure ratio at an angle of attack up to 10°.

Besides, Shang et al.13,21gave an insight into the shock bifurcation of a jet-spike from a blunt body in hypersonic stream, and revealed its driving mechanism through ample experimental data with attendant numerical simulation by differing jet total pressures and freestream total pressures. For different freestream total pressures, they discovered similar variation trends of the total drag relative to growing totalpressure ratios covering two flow modes. The role of the total-pressure ratio as a key modeling parameter was also reaffirmed by Chen,11Venukumar,22and Kulkarni23et al.Rong24attempted to represent the jet intensity by introducing a new parameterRPAas the product of total-pressure ratio and jet flux.The sameRPAbrought the same shock position and drag.AsRPAincreased, the shock wave moved far from the surface while the drag degraded. Zhang et al.25applied a pulsed opposing jet to mitigate aerodynamic drag and heating. Periodic variations and hysteresis phenomenon of the shock wave stand-off distance and drag force were observed,and the effect of the total-pressure ratio was examined.

It can be concluded from the aforementioned literature survey that there exist a variety of modeling parameters together to dictate the essential flow features and aerodynamic characteristics of a vehicle with an opposing jet. The examination of the effects of each parameter on the flowfield and aerodynamics can be assisted by utilizing a high-fidelity CFD approach to establish surrogate models and performing a sensitivity analysis. Ahmed and Qin26,27constructed surrogate models of hypersonic spiked bodies based on a generic representation of forebody and aerospike geometric dimensions. The Latin Hypercube Sampling (LHS)technique was used to select samples from a design space. A quadratic response surface model and three Kriging surrogates were tested and compared. The significance of a design parameter in determining the design performance was roughly inferred by a parameter θ while accurately computed by Sobol indices. Sun et al.28conducted multi-objective design optimization to improve the combinational novel cavity and opposing jet concept in minimization of both the drag and heat load. The sampling method was the LHS technique, and a Kriging surrogate was validated to have sufficient accuracy in modeling the complex problem.Ju et al.29performed a sensitivity analysis to show the influences of geometric parameters of the heatshield on the aerothermodynamic environment of a Mars entry vehicle. A Non-Intrusive Polynomial Chaos (NIPC) response surface method coupled with the LHS method was used to generate the functional relationship between the geometry design and aerothermodynamics.The dependence of aerothermodynamics on geometric parameters was indicated by Sobol indices. Ju et al.30also exploited a variance analysis method to analyze the effects of energy addition parameters upon the scramjet nozzle performance, and later, LHS experimental design was employed to construct a surrogate model to link energy deposition parameters and aerodynamic coefficients.31

In addition, Huang et al.32–34performed a systematic analysis on the sensitivity of component performances of a scramjet engine to layout parameters based on a variance analysis method and orthogonal experimental design. Ou et al.35also used a variance analysis to show the importance of design variables to the drag and heat reductions of a combined spike and opposing jet thermal protection system. Based on the sensitivity analysis, a Kriging model33,34or a quadratic response surface model35was applied in multi-objective design optimization.In terms of sensitivity analysis(i.e.,an indication of the role of each design variable in dictating the design performance),variance analysis of orthogonal test results is on the basis of sampling data, and suggests, in a manner, a type of qualitative statistical significance of a parameter. In contrast,Sobol indices are computed based on the coefficients of surrogate, and signify the relative contribution of a design parameter to the design performance in a quantitative way.The design of experiment is responsible for spreading a limited number of training samples in the whole design space.The LHS technique is stratified sampling which ensures space-filling and structure similarity of a sample and a population, thus improving the accuracy of evaluation. The orthogonal test method can largely reduce the number of sample points especially for many design variables, thus improving the efficiency of evaluation.In regard to surrogate models,Kriging interpolates all training samples36while a response surface model is polynomial regression in nature. A response surface model is capable of providing an explicit functional relation of sampled data and quantities in concern with a small set of sample points.35,37

Based on the research work mentioned above, a construction of the association between the flowfield modeling parameters and the flowfield and aerodynamic characteristics of a vehicle facilitates the examination of the effects of the modeling parameters on the flow structure and aerodynamics, and can further be used to find a proper way to improve the aerodynamic performance. In this study, numerical simulation of an opposing jet issued from a hypersonic blunt body is performed by using the CFD approach to illustrate basic flow features and aerodynamic characteristics. The flowfield modeling parameters of injection and freestream are designed here, and LHS38,39is utilized to select samples in the design space. The response surface surrogate model between the flowfield modeling parameters and the flowfield and aerodynamic quantities of interest is constructed by the point-collocation NIPC method,40–44which needs no modification of the original CFD approach and enables to propagate the variations of input modeling parameters to output quantities in concern.The relative contribution of each modeling parameter on the flow characteristics and aerodynamics is evaluated and expressed by Sobol indices obtained with sensitivity analysis.45,46This investigation is aimed to assess the variations in flowfield and aerodynamic quantities caused by the variations in flowfield modeling parameters and to identify the crucial modeling parameters that show importance to the flow structure and aerodynamic performance of a hypersonic blunt body with an opposing jet.

2. CFD approach

2.1. Governing equation and turbulence model

The flowfields and aerodynamics of freestream-jet interactive flows are calculated by numerically solving three-dimensional compressible Navier-Stokes equations with a two-equationk-ω SST (Shear Stress Transport) turbulence model. The governing equation in a differential form is written as47,48

whereQis the vector of conservative variables,fc,gc, andhcare the vectors of convective fluxes, andfv,gv, andhvare the vectors of viscous fluxes, which are calculated by

in whicheinis the internal energy,γ=1.4 is the ratio of specific heats,μ is the dynamic viscosity coefficient,η is the thermal conductivity coefficient,Tis the temperature, and δijis the Kronecker delta function.

In hypersonic flows, Favre- and Reynolds-averaging are utilized in the Navier-Stokes equations to take account of turbulent fluctuations.47,48Thek-ω SST model developed by Menter49is a hybrid of the originalk-ω model of Wilcox and the standardk-ε model, which gives rise to improvement in predicting flows involving adverse pressure gradients and has been widely employed in simulating efforts of drag and heat reduction by means of replacing a strong bow shock with oblique shocks as well as forming flow separation and recirculation in front of a vehicle.25,28,50,51The SST model is implemented in the form of49

wherekis the turbulent kinetic energy, and ω is the specific dissipation rate of turbulence.PkandPωare the production terms ofkand ω, respectively. μLis the laminar viscosity calculated by Sutherland formula. μTis the turbulent viscosity given by48

where Ω is the magnitude of vorticity.The total viscosity coefficient μ and the total thermal conductivity coefficient η are respectively defined by47

whereCpis the specific heat at constant pressure, andPrL=0.72 andPrT=0.9 are the laminar and turbulent Prandtl numbers, respectively. Besides, σk, σω, σω2, β, β*,anda1are model constants, andf1andf2are blending functions.47–49To account for compressible effects due to high Mach numbers, the compressibility term devised by Wilcox52,53is applied to modify the corresponding values of β and β*, aiming to obtain reasonably accurate predictions for wall-bounded and free shear-layer flows.54,55

2.2. Boundary conditions

For all the cases in this study,no-slip,zero normal gradient of pressure,and isothermal conditions are prescribed on the solid wall. On the symmetry plane, the normal velocity is zero, as well as the normal gradients of scalar quantities and that of the tangent velocity.47,48The opposing injection is determined by the jet Mach number,the jet total temperature,and the jetto-freestream total-pressure ratio PR. PR is defined as

in whichp0jis the jet total pressure, andp0∞is the freestream total pressure. The outer boundary is imposed with a farfield condition, which assigns the inflow boundary with the freestream Mach number, density, and static temperature, and then calculates variables on the outflow boundary using extrapolation from internal cells because of supersonic flow.For the SST model, the values ofkand ω on the solid wall are given by47

where μ1is the laminar viscosity,ρ1is the density,andd1is the distance to the solid wall, which are all calculated at the cell centers of the first mesh layer from the wall.On inflow boundaries, the freestream levels ofk, ω, and μTare specified as47

and on outflow boundaries, the values ofkand ω are extrapolated from interior domains. Herein,ais the speed of sound.The subscript ∞denotes the variable at freestream.

2.3. Discretization

The governing equations with boundary conditions above are numerically solved by a multi-block finite-volume approach.47,48Inviscid fluxes are computed using a fluxdifference splitting scheme of Roe with a second-order spatial accuracy accomplished by MUSCL reconstruction coupled with a minmod limiter. A central difference method is used to discretize viscous fluxes with a second-order accuracy. For time marching, the implicit Lower-Upper Symmetric Gauss-Seidel(LU-SGS)procedure is utilized,and local time stepping is adopted to obtain a faster convergence of calculation.

2.4. Numerical validation

Validation of the CFD approach described previously is realized with a numerical simulation of Finley’s experiment of a hemispherical model with an opposing jet.9The jet species is air, and a total-pressure ratio of 6.54 is employed confirming a steady-state of the jet structure. The freestream Reynolds number is 2.76×107/m,thereby the flowfield can be simulated by solving the three-dimensional Navier-Stokes equations with a turbulence model.Experimental flow conditions are given in Table 1.

The jet flowfield structure of numerical simulation and experiment is compared in Fig.1(a). As shown, the CFD approach is capable of resolving and capturing the flow details of freestream and jet interaction, and the calculated densitygradient contour is in decent accordance with the experimentalSchlieren image. Calculated results of the surface pressure for cases of jet and no jet are plotted in Fig.1(b), compared with experimental data (Exp.) and predicted results from Ref. 10,where the surface pressurepwis normalized by the freestream Pitot pressurep0f. The calculated surface pressure of the jet case nearly coincides with experimental and referencepredicted values except for being slightly lower in the recirculation portion ahead of the reattachment point.The calculated surface pressure of the no-jet case agrees with the experimental and predicted results of reference very well.It is suggested that the present CFD approach has high accuracy and reliability,and can be applied for simulations of opposing jet flows.

Table 1 Flow conditions of experiment.

3. Sensitivity analysis method

Sensitivity analysis is aimed to examine the significance of flowfield modeling parameters to the flow features and aerodynamic performance of a hypersonic blunt body with opposing injection,and to determine the variations of the flow structure and aerodynamics due to these parameters.The freestream and jet exit parameters are designated as the input flowfield modeling parameters, while the flowfield features, surface pressure distributions,and drag coefficient are the output performances of concern. In this study, sensitivity analysis is performed by using Sobol indices.45,46A surrogate model is created via a stochastic response surface by implementing the pointcollocation NIPC method,40–44in order to establish the relation between output quantities and modeling parameters as well as to provide necessary coefficients to obtain Sobol indices.

Fig.1 Numerical validation results.

As a widely used tool for global nonlinear sensitivity analysis,Sobol indices are capable of intuitively demonstrating the relative contribution of each input parameter to the total variation of output quantities of interest,based on the corresponding weight in the variance of model response to the total variance.39Sobol indices (Si1，...，is) are defined as

whereDi1，...，isis the partial variances,andDtotalis the total variance. Calculations of the partial variances and the total variance utilize the Polynomial Chaos Expansion (PCE)coefficients in a way that

and the relation between partial and total variances can be illustrated via Sobol decomposition, given as56

in which α is the deterministic component, Ψ is the random variable basis functions of multi-dimensional Legendre polynomials with its bounded nature,57ξ is then-dimensional standard random variable vector, andnis the number of input variables.

The PCE coefficients (deterministic components) are acquired via construction of a surrogate model using the NIPC method. Among several methods developed for NIPC, the point-collocation NIPC method is employed in this paper.Within this approach, a stochastic response function α* (such as the drag coefficient,a flowfield quantity,or surface pressure at a given location) can be expanded into separable deterministic and stochastic components in a polynomial chaos series40as follows:

wherexis the deterministic vector involving spatial coordinates and other deterministic factors of the problem, and α*is assumed to be a function ofxand ξ.Intrinsically, the series of expansion is infinite. However,for application of PCE, it is truncated to the output modes of a finite number(P+1),and thus can be replaced by a discrete sum.43Accordingly, a total number ofNssample pointes are selected in a random space by employing the LHS technique, where the order of PCEp′, the number of random variablesn, and the oversampling rationpare used to dictateNs40,43as follows:

In order to compute the PCE coefficients αi,the deterministic CFD approach is evaluated at theNsselected sample points,thus obtaining the corresponding response values of α*.In this way, a linear system ofNsequations is formulated as40,43

The combined effect of each input parameter on an output quantity is the summation of its individual and mixed contributions, i.e., including all the partial Sobol indices containing this parameter, given as41,43

In this paper,the extreme values of surrogate models represent the varying ranges of output quantities of interest. To do this, a number of 105samples are selected in a random space by Monte Carlo sampling, and the response values to these sample points are computed via the already well-constructed PCE, which is much more computationally efficient than via deterministic CFD calculations. By identifying the maximum and minimum of the response surface, the varying ranges of output quantities can be determined.41–43

4. Computational details

4.1. Geometric model

As illustrated in Fig.2, the geometric model employed in this study is a hemisphere-cylindrical blunt body, which has been introduced in Ref. 58. In this design, the hemispherical part in front and the cylindrical part behind are referred to as the forebody and main body of the model,respectively.The hemisphere and the cylinder have the same diameter ofD=500 mm, and the diameter of the jet orifice isd=40 mm.The overall length of the model isL=D,measuring from the initial apex of the hemisphere.

4.2. Flow conditions

The flowfield modeling parameters are consistent with the freestream and jet boundary conditions mentioned previously,namely the freestream Mach numberMa∞, the freestream density ρ∞, and the freestream static temperatureT∞as well as the jet Mach numberMaj, the jet total temperatureT0j,and the jet-to-freestream total-pressure ratio PR, which are independent flow variables combined to fully determine the flow conditions.

The standard values of flowfield modeling parameters are tabulated in Table 2,which are derived from Ref.58.The standard freestream values correspond to a flight altitude of 30 km at a 0°angle of attack.The air is used as the jet species issuing from the nose tip of the hemispherical forebody against the hypersonic incoming stream. The standard jet total pressure isp0j=375 kPa, and a constant wall temperature ofTw=295 K is prescribed. As calculated, the freestream Reynolds number is as high as 2.26×106/m, and thus turbulent effects should be considered.

The baseline case is investigated employing the standard flow conditions,based on which the upper and lower variation limits of each flowfield modeling parameter are specified, as summarized in Table 2. The variation ranges of modeling parameters are consistent with those in Ref. 58. The modeling parameters are assumed with a symmetrically-varied deviation from the corresponding standard values, respectively. As stated, steady jet patterns substantially subdue flow oscillations caused by unsteady jet patterns and are favorable for drag reduction as well. Therefore, selected ranges of modeling parameters are required to construct stable flowfields of jet and freestream interaction.A total pressure ratio PRfis implemented here to judge the stability of flow,which is defined as9

wherep0fis the freestream Pitot pressure (i.e., the total pressure behind the normal shock). The criterion based onPRfis9,58

where PRf,critis the critical total pressure ratio, indicating the threshold point of mode transition from an unstable multi-cell structure to a stable single-cell structure.If Eq.(14)is satisfied,a stable structure of the opposing jet is formed.

To estimate the variation range of PRfin line with Eq.(13),p0jis calculated according to Eq. (4) and the isentropic relation59as

Fig.2 Schematic diagram of jet geometric model.

Table 2 Variation ranges of flowfield modeling parameters.58

wherep∞is the freestream pressure, andp0fis approximated by the Rayleigh Pitot tube formula59as

Therefore, the estimation of PRfcan be deduced and simplified as

in whichH(Ma∞) is a monotonically-increasing function forMa∞in [5.4, 6.6], and thus PRfis positively related to PR andMa∞.Accordingly,using the known upper and lower limits of PR andMa∞,the variation range of PRfis determined as[2.61, 14.02]. According to Finley’s experimental study,9PRf,critis positively associated with the ratio of blunt body diameter to jet orifice diameter as

For the geometric model in this study, PRf,crit is approximately 2.32. Consequently, the variation of PRfsatisfies the criterion of Eq. (14), and the selected ranges of modeling parameters are proven to be able to generate stable jet flowfields. This criterion for the determination of jet flow stability has also been implemented in Refs. 35,58.

Based on the aforementioned sensitivity analysis method,fifty-six samples (i.e., fifty-six ξ vectors) are selected by the LHS technique to construct surrogate models for output quantities of interest using the point-collocation NIPC method.Therefore,the effects of input modeling parameters on the output flowfield properties and aerodynamics of a hypersonic blunt body with an opposing jet can be examined.

4.3. Computational mesh and convergence

A three-dimensional two-block structured mesh is generated for the blunt model.The mesh of the wall,jet exit,and symmetry plane is depicted in Fig.3.The coordinate origin is located at the initial apex of the hemispherical forebody,and thex-axis coincides with the axis of the hemispherical cylinder.The freestream direction is parallel to the positivexdirection,while the jet stream flows along the negativexdirection.Since the model is geometrically symmetric and flies without a yaw or slip angle, only a half-body mesh is employed, and the symmetry plane is on thexOyplane.

The cell heights of the first mesh layer at the wall are strictly governed by the cell Reynolds numberRec,55namely

where Δhis the height of the first mesh layer near the wall,andu∞and μ∞are the freestream velocity and viscosity coefficients, respectively. A regulation ofRec≈10 is guaranteed to achieve convergent and reliable predictions of aerodynamics.54Moreover, the mesh in the region close to and behind the bow shock is carefully refined to capture the flow details of freestream-jet interaction.58

Under standard flow conditions, a mesh independence study is conducted with three levels of mesh size, that is,coarse, medium, and fine,58as tabulated in Table 3. The drag coefficientCDis defined as

whereDwis the drag force(i.e.,a streamwise component of the integration of pressure and skin friction over the entire body),andAref=πD2/8 is the reference area.The centerline distributions of surface pressure obtained for the three levels of mesh are compared in Fig.4, and theCDvalues of the three mesh levels are listed in Table 3. It is found that taking the finemesh results as reference, the pressure predictions of medium mesh and fine mesh show a slight difference with a maximum discrepancy of only 0.2%, and the difference ofCDbetween medium mesh and fine mesh is within 0.4%.Therefore,a mesh size of fine level is selected for all cases to ensure numerical precision.

A solution is considered to be converged in the case that the residuals of continuity, velocity, and energy decline to below 10-4respectively, as well as the change of the drag coefficient spanning 5000 iteration steps does not exceed 0.1%. The convergence history of the baseline case is plotted as an example in Fig.5. As shown, the residuals are smaller than 10-4and still tend to drop when the calculation stops at the iteration step of 50,000, and the change of the drag coefficient for iteration steps between 45,000 and 50,000 is less than 0.78%,indicating that the solution can be judged to be converged.

Fig.3 Computational mesh (fine level).

Table 3 Mesh information and drag coefficient.

4.4. Assessment of sensitivity model

The fidelity of the sensitivity analysis method (i.e., the accuracy of Sobol indices) depends highly on the accuracy of the NIPC surrogate.60To validate the surrogate model, a set of 30 test sample points is randomly selected in the design space,and their CFD results are obtained for comparison with the surrogate-predicted responses. The surrogate accuracy is evaluated using the coefficient of determination (R2), which addresses the consistency of the surrogate-predicted responses with the actual CFD results.28The definition ofR2is

whereyiis the CFD value of a sample point, ^yiis the corresponding response value of the surrogate,y-iis the mean of sampling values, andNtis the number of test sample points.The closer to unityR2is, the higher accuracy the surrogate achieves. TheR2contour of surface pressure is shown in Fig.6.R2exceeds the default threshold value of 0.9 in most of the region of surface indicating a high modeling accuracy,and for the locally banded part,R2is higher than 0.75 implying a moderately acceptable accuracy.28Therefore, it is suggested that the modeling accuracy of the NIPC surrogate is adequate for variation propagation and sensitivity analysis.

Fig.4 Centerline surface pressure comparison between three mesh levels.

Fig.5 Convergence history of baseline case.

5. Results and discussion

5.1. Baseline results

Before constructing surrogate models and performing sensitivity analysis, a baseline solution is obtained upon the blunt model under standard flow conditions(see Table 2),to present the base flow features and aerodynamics of injection opposing hypersonic airstream.A no-jet case under standard freestream conditions is also shown for comparison.

Fig.6 Coefficient of determination (R2) of surface pressure for 30 test samples predicted by NIPC surrogate model.

The baseline flow structure of the opposing jet ejected at the nose of the blunt model is illustrated in Fig.7, featured by density-gradient in the upper half and pressure in the lower half, respectively. As the jet emanates from the orifice and moves contrary to hypersonic mainstream, the bow shock is pushed away from the body surface and bulged out to admit the jet protrusion, therefore the shock stand-off distance is greatly enlarged compared to that of the no-jet case. The upstream jet flow forms a single-cell structure and undergoes a Mach reflection. The barrel shock inside the jet column is generated by the coalescence of compression waves reflected from the free jet boundary, and the jet column is finalized by a normal shock (i.e., the Mach disk) across which the jet total pressurep0jdrops to the freestream Pitot pressurep0f. A reflected shock originates at the intersection of the barrel shock and the Mach disk, thus forming a triple point.

The jet layer meets with the incoming mainstream at the interface,and both the jet and the mainstream rest at the Free Stagnation Point (FSP) where the pressure isp0f. The flow of the jet layer then deflects laterally and reverses its direction towards the blunt body. Thereby, a recirculation region that screens a large portion of the forebody surface is generated surrounding the jet column,and a conical free shear layer covering the recirculation region is formed. The impingement of the shear layer on the reattachment ring leads to compression waves in the vicinity of body surface, and a reattachment shock is subsequently created. The reattachment shock interferes with the bow shock, creating a triple point. The reattached jet layer flows downstream along the body surface.To this, a steady-state flowfield dominated by the interaction of the freestream and the opposing jet is well-constructed,and its flow features are clearly illustrated.

The baseline Mach number (Ma), pressure, and temperature distributions along the stagnation line are plotted in Fig.8, in which the locations of the bow shock, the FSP,and the Mach disk can be distinctly recognized.The upstream jet and the incoming mainstream are decelerated by the Mach disk and the bow shock, respectively, and then both cease at the FSP,where the pressure(i.e.,the freestream Pitot pressure)is about 55.4 kPa. As shown, the high-pressure region is pushed away from the surface and confined between the bow shock and the Mach disk. The cool jet stream effectively reduces the temperature of the hot mainstream behind the bow shock.

Fig.7 Baseline flow pattern of opposing jet around blunt model.

Fig.8 Baseline Mach number, pressure, and temperature distributions along stagnation line.

The centerline surface pressure and skin friction (τw) of both baseline jet and no-jet cases are exhibited in Fig.9. Also shown are the model surface contour,the location of the Baseline Reattachment Point (BRP), and the dividing line between the forebody and the main body for better description.On the surface in the vicinity of the recirculation region, the pressure and friction are remarkably alleviated compared to those of the no-jet case.Specifically,the pressure rises as moving closer to the reattachment point, and attains its peak shortly downstream from the reattachment point due to the formation of a reattachment shock. As the flow approaches the main body,the pressure drops and tends to the values of no injection.The friction caused by the recirculating flow is negative,and at the reattachment point, the friction accordingly vanishes, while it becomes positive as the flow moving downstream. Nevertheless, for the reason that the pressure is roughly two orders of magnitude over the friction, the construction of a surrogate model and analysis of sensitivity will not directly aim at the friction, but the contribution of the friction to the aerodynamic drag is still reckoned in.

Fig.9 Centerline surface pressure and skin friction for baseline and no-jet cases.

In order to examine the flowfield and aerodynamic characteristics triggered by the opposing jet, four representative quantities, namely, the penetration length of injection ΔP, the reattachment angle of the shear layer θR, the drag forceDw,and the jet thrustDTare presented in Table 4. Herein, ΔPis defined by the distance from the initial apex of the hemispherical forebody to the FSP, and θRis the angle between the reattached-corresponding radial vector from the center of the hemisphere and the injection orientation, both of which are denoted in Fig.7.DTis calculated as

5.2. Variation quantification

Table 4 Comparison of representative quantities between baseline and no-jet cases.

wherep∞is varied with different samples.

Fig.10 Surface pressure distribution.

The baseline surface pressure distribution is exhibited in Fig.10(a). At each surface mesh point, a surrogate model is constructed for the pressure. Thus, the variation results for surface pressure can be presented by a contour on the whole body surface, such as the mean pressure in Fig.10(b) and the variation interval width of pressure in Fig.10(c).The variation interval width denotes the varying extent of a quantity,computed by a subtraction of the maximum limit with the minimum limit of a surrogate model. For clarity, the centerline maximum limit, minimum limit, and variation interval width of pressure are plotted with the baseline and mean pressure in Fig.11. As a benchmark, the position of the BRP is sketched in Figs. 10 and 11.

The baseline surface pressure demonstrates an almost axisymmetric distribution. The pressure on the hemispherical forebody is appreciably higher than that on the cylindrical main body, and the peak pressure is located immediately behind the BRP with a value of 12.7. The distribution and exact values of mean pressure nearly coincide with those of the baseline. The distributions of the maximum and minimum limits of pressure variation also exhibit similar trends with those of the baseline.The peak of the pressure maximum limit lies at a short distance downstream from the BRP, and its value is 144%of the baseline peak pressure.The varying trend of the pressure variation interval width generally resembles that of the baseline, while in contrast, it mainly changes with its peak occurring ahead of the BRP, indicating that the surface pressure experiences the most intense variation adjacent to the recirculation region. This is possibly explained from the varying recirculating flow pressures affected by different jet-freestream conditions with fierce interaction. Downstream from the BRP, the pressure variation interval shrinks, where the flow gradually becomes relatively mild. As observed, the variation in pressure owing to the variable input modeling parameters is more prominent on the forebody than that on the main body. The variation interval width of pressure is approximately axisymmetrically distributed, and at the same location,its value is in the same order of magnitude as the corresponding baseline pressure. The peak value of pressure variation is as high as 13.8, which is 109% of the baseline peak pressure, whereas on the main body, the pressure variation range is merely below 1.8.

The influences of variations in flowfield modeling parameters on variations in ΔP,θR,andCDare also considered.Since drag reduction is an important application of an opposing jet,CDis chosen as one quantity of concern. ΔPand θRare also selected as output quantities of interest for reasons in the following twofold aspects.On one hand,the size of the recirculation region is vital for determination of pressure distribution,and a decrease in the surface pressure benefits from a larger recirculation region. ΔPand θR, in a manner, can be used to measure the size of the recirculation region.A larger ΔPimplies a longer shock stand-off distance, yielding a lower forebody pressure. As the reattachment point moves toward the forebody shoulder (a larger θR), the reattachment shock becomes weaker because the angle required to turn the flow outside the shear layer parallel to the downstream surface is declined,which mitigates the pressure elevation.On the other hand,the drag for such a blunt-body vehicle is expected to be dominated by the pressure drag,which is proportional topwcos θ,where θ is the angle between the radial vector from the center of hemisphere and the injection orientation. Distinctly, larger ΔPand θRcontribute to a lower level of drag.

The variation intervals of ΔP,θR,andCDare presented with their respective baseline values in Table 5. The mean ΔPis slightly larger than the baseline ΔP, while the variation range of ΔPis about 86% of that of the baseline with 59% for the maximum limit and 27% for the minimum limit. The mean θRis close to the baseline θRwith tiny augmentation.θRvaries by as much as 39% of that of the baseline with 25% for the maximum limit and 14% for the minimum limit. The meanCDis also slightly higher than the baselineCD. The variation inCDis approximately 118% of that of the baseline with 66% for the maximum limit and 52% for the minimum limit.

Thereby, it can be seen that significant variations of flowfield and aerodynamic quantities have been induced by variations in flowfield modeling parameters. With respect to ΔP,θR, andCD, an interest of note is that all the mean values approach to the corresponding baseline values only with minor increases, and relative to each baseline value, the changing degrees of maximum limits obviously surpass those of minimum limits. Accordingly, the variation intervals of ΔP, θR,andCDare found to be asymmetric from their respective baseline values in spite of the variations in all of the input modeling parameters assigned with symmetric varying bounds from the corresponding standard values.

5.3. Sensitivity analysis

Fig.11 Centerline surface pressure variation.

Table 5 Variations of output flowfield and aerodynamic quantities.

Fig.12 Centerline Sobol indices for surface pressure.

The top three contributors to the variation in surface pressure areMa∞, PR, andMaj, which together contribute more than 92% to the pressure variation throughout the entire surface. Accordingly, contributions fromT∞, ρ∞, andT0jare much smaller. PR andMajare relatively significant on the front part of the forebody surface, which are responsible for most of the enhanced variation in surface pressure there. It is inferred that in the surface portion close to recirculation regions, PR andMajare the major sources of variation to affect the surface pressure. Downstream, PR andMajgradually degrade in their importance to pressure variation. Over the entire body,PR contributes evidently more thanMaj,possibly associated with the widest variation range assigned for PR out of all the input parameters. The contribution ofMa∞towards surface pressure variation is lower than that ofMajon the front part of the forebody surface. However,Ma∞gradually grows in its significance after the BRP, and becomes predominant on the main body surface where the effects of PR andMajare decreased. In respect toT∞, ρ∞,andT0jthat have weak influences on the whole body,one common feature observed is that they make almost little contribution to pressure variation on the front part of the forebody,and downstream, their contributions to pressure variation merely rise slightly. Thus in a sense, the contributions of PR andMajas well as that ofMa∞stand for the dependence of surface pressure variation on the injection modeling parameters and the freestream modeling parameters, respectively.

Fig.13 Sobol index contours of top three contributors for surface pressure.

The Sobol indices ofMa∞, PR, andMajfor surface pressure are depicted by contours on the blunt body in Fig.13.The Sobol indices show nearly axisymmetric or slightly three-dimensional distributions,which may be related to some degree of asymmetry in a steady single-cell jet flow structure.61As seen, the influences of PR andMajon pressure variation are greater on the forebody than those on the main body,whileMa∞shows a higher significance to pressure variation on the main body than that on the forebody, which means that in general, from the jet orifice to the downstream surface, the importance of PR andMajtakes on a descending trend while the effect ofMa∞becomes more and more pronounced.This is a clear reflection for the location-dependent behavior of the sensitivity of surface pressure to modeling parameters.

Fig.14 Sobol indices for output flowfield and aerodynamic quantities.

The Sobol indices of modeling parameters for ΔP, θR, andCDare conveyed via pie charts in Fig.14. The variations inMa∞, PR, andMajrepresent more than 99% of the variation in ΔP. It is clear that the variations in these three contributing parameters produce an amount of variation in ΔP. Similarly,θRis also highly dependent on PR,Ma∞,andMaj,which combine to account for over 99% of the variation in θR. The contributions fromMa∞and PR to the variation in ΔPare comparable, and comparable contributions fromMa∞and PR are also found to the variation in θR, all of which exceed 41%. The fractional contributions ofMajto variations of ΔPand θRare 14% and 15%, respectively. The impacts ofT∞,ρ∞,andT0jon variations in both ΔPand θRare quite smaller.Since the drag arises primarily from surface pressure which is affected by the formation of a recirculation region,the importance of modeling parameters to theCDvariation is in a qualitative agreement with that to variations in ΔPand θR.CDis most sensitive to PR andMa∞, which contribute about 44%and 43%to theCDvariation separately.Majis again the third contributor representing about 13% to theCDvariation. The contributions fromT∞, ρ∞, andT0jto theCDvariation are almost negligible.

6. Conclusions

The objective of this study is to quantify variations and perform a sensitivity analysis of the flow structure and aerodynamic characteristics of a hypersonic blunt body with an opposing jet due to variations in freestream and jet modeling parameters. Variations of flowfield and aerodynamic quantities(i.e.,surface pressure,penetration length of injection,reattachment angle of the shear layer, and drag coefficient) are calculated through surrogate models constructed by the point-collocation NIPC method.The significance of each modeling parameter to the variations in output quantities of interest is evaluated by the corresponding Sobol indices derived from sensitivity analysis. Main findings are concluded as follows:

(1) The baseline surface pressure and aerodynamic drag are effectively reduced due to the formation of a slender equivalent body and a low-pressure recirculation region compared to those of a no-jet case. Flowfield and aerodynamic quantities possess an amount of variation due to variations of flowfield modeling parameters. The most intense variation in surface pressure emerges on the forebody portion ahead of the BRP, and the peak pressure variation is 109%of the baseline peak pressure.The pressure experiences more remarkable variations on the forebody than on the main body. The drag coefficient primarily as a result of the pressure drag and affected by the formation of a recirculation region varies by 118% of the baseline.

(2) The jet-to-freestream total-pressure ratio and jet Mach number are the main contributors to the large variation in pressure on the front part of the forebody, while the freestream Mach number becomes prominent downstream and on the main body. The three key factors above represent more than 92% of the pressure variation for the entire body and show a high dependence on the surface location. In regard to the penetration length of injection and the reattachment angle which dictate the size of the recirculation region, the totalpressure ratio, freestream Mach number, and jet Mach number contribute significantly to over 99% of the resulting variations.Those three major sources of variation for the surface pressure, penetration length, and reattachment angle are also found important for the variation of the drag coefficient, with comparable contributions from the total-pressure ratio and freestream Mach number along with a less effect of the jet Mach number. The flowfield and aerodynamic quantities are relatively insensitive to the freestream temperature,freestream density, and jet total temperature.

Furthermore, an extension of this investigation is expected to incorporate geometric modeling parameters for a wider design space, giving a comprehensive study on different geometric shapes involving realistic vehicles. On the basis of surrogate construction and sensitivity analysis,aerothermodynamic design optimization is ready to realize better drag and heat reduction efficiency aided by an opposing jet.