Theoretical investigation of shock stand-offdistance for non-equilibrium flows over spheres

1.Introduction

When a supersonic/hypersonic flow over a blunt body like a sphere,a detached bow shock forms around the body,and the level of the non-equilibrium of the flow is measured by the following dimensionless reaction rate parameter,1where α is the dissociation fraction,D the diameter of the sphere,u the velocity;and the subscripts ‘∞” and‘s”means the corresponding quantities at freestream and immediately behind the shock,respectively.Depending on the value of Ω,the flow can be categorized into nearly frozen flow(Ω ≪ 1),nearly equilibrium flow(Ω ≫1),and nonequilibrium flow(otherwise).The distance between the bow shock and the stagnation point of the nose was referred to as the Shock Stand-off Distance(SSD).The SSD is much smaller than the size of the tested model,and hence experimental measurement admits large errors.Generally speaking,if there is no significant dissociation in the free stream,a larger free stream kinetic energy leads a smaller SSD,due to a higher level of vibrational excitation and chemical dissociation.But an increased SSD is observed in high enthalpy shock tunnels under the same free stream velocity and this phenomenon is attributed to the inevitable free stream dissociation in such facilities.2,3In order to understand the physics behind,it is crucial to explore the effects of the important flow parameters through theoretical analysis.Olivier et al.2first gave an estimation of the effect of free stream dissociation on SSD,but no quantitative solution was provided.

For frozen flows,Lobb4performed extensive experiments on the SSD for spheres of various diameters using a schlieren photography technique and derived the following correlation

where Δ is the SSD,ρ density,L a constant with a value of 0.41 for spheres.For dissociating flows,the accuracy of Lobb’s correlation is significantly degraded.5,6

Wen and Hornung5proposed an analytic correlation between generalized dimensionlessand the generalized reaction rate parameterwhich comprises two branches,namely afrozen-side and an equilibrium-side.The frozen-side solution is given by

which implies the SSD is independent of all parameters other than L.Meanwhile,the equilibrium-side solution is given by

Table 2 compares the values of the dimensionless SSD,obtained by Eqs.(12),Wen and Hornung’s theory5and Oliver’s theory,11respectively,given various values of the density ratio ρs/ρbranging from 0.4 to 0.9 provided with ρs/ρ∞=6.0.It is observed that a good agreement exists between the three sets of results.

Wen and Hornung5used a piecewise linear function to approximate the density pro file.They pointed out that the use of a piecewise linear function to approximate the density pro file between the shock and the body results in an overestimation of the average density,and hence an underestimation of the SSD.This error can be reduced by replacing the piecewise linear function with the following exponential function:

whereis the dimensionless tangential velocity gradient and the subscript ‘b” represents corresponding values at the stagnation point(body).For frozen air flows,i.e.,ρs/ρb=1 and ρs/ρ∞=6,the Olivier’s analytic solution has a value of~Δ=0.4,and is thus in good agreement with the solution obtained from Lobb’s correlation.significantly,Olivier’s model shows that the parameter L is not constant but depends on the gas properties.Nevertheless,since non-equilibrium processes increase the complexity of the conservation equations to such an extentthateven fora quasi-one-dimensional approach,analytic solutions cannot be obtained for the whole non-equilibrium flow regimes.12

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In view of the discussions above,the present study has two aims:(A)to derive a comprehensive analytic solution for the whole non-equilibrium flow regime without using the semiempirical parameter L;(B)to investigate the effect of two fundamental flow parameters,namely the frees tream kinetic energy,and the freestream dissociating level,on the SSD using a simple Ideal Dissociating Gas(IDG)model.13,14

2.Analytic solution for shock stand-off distance

Consider the control volume ΔV in the stagnation region between the shock and the body,as shown in Fig.1.The rate at which mass enters the control volume from the left-hand side is equal to ρ∞u∞b or ρ∞u∞b2,depending on whether the flow is two-dimensional or axisymmetric,respectively.Meanwhile,the rate at which mass leaves the control volume through the right-hand side is equal to

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where uτis the tangential velocity(i.e.,the component of velocity normal to the ray from the center of curvature),R is the radius of the sphere and dr is the differential element of the radius.Consequently,the mass balance is given as

and

for two-dimensional and axisymmetric flows,respectively.The integral terms in Eqs.(1)and(2)can be approximated using the average value,i.e.,

and

Furthermore,let only the flow region very close to the stagnation streamline be considered.Therefore,the following approximations can be applied:

As described above,the scaled SSD is dependent on ρs/ρ∞and ρs/ρe.For an ideal dissociating gas with no frees tream dissociation,ρs/ρ∞is equal to 7.For CO2withis equal to 7.67.Fig.3 plotsandversusfor different values of ρs/ρ∞.It is seen that whilehas a very weak dependence on ρs/ρ∞,has a strong dependence on ρs/ρ∞.For a constant~Ω,when ρs/ρ∞increases,decrease significantly,but~Δ almost remains the same.In other words,is a more universal dimensionless parameter thanin estimating the SSD.

Fig.1 Schematic of control volume and associated notations.

and

with solutions

and

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respectively,where

Sinceis O(1)(this point can be further veri fied later)and ρ/ρ∞is very large for hypersonic flows,it follows that

Substituting Eq.(11)into Eq.(9)yields the following simple solution for SSD in axisymmetric flow:

Obviously,the parameter χ is the measurement of the product of density and the tangential velocity gradient.Eqs.(8)and(12)imply that the dimensionless SSD is inversely proportional to χ.The above derivations using integral analyses are obviously more succinct than Olivier’s correlation derived from the differential analyses.Comparing Eqs.(8)and(12),the SSD for a cylinder exhibits the same qualitative behavior as that for a sphere.However,the tangential velocity gradient for a cylinder is smaller than that for a sphere,12and thus the SSD is more than twice that of a sphere.The following derivation will be focused on the SSD for spheres.

To determine the SSD for spheres using Eq.(9)or(12),the tangential velocity gradient must be solved.At the point immediately behind the shock,the velocity gradient can be determined from the conserved tangential velocity component across the shock,i.e.,

Meanwhile,from the momentum equation in the tangential direction at the body,we have15

where p is the pressure.Utilizing the approximation of velocity in Eq.(5)and assuming a Newtonian pressure distribution over the surface,i.e.,,Eq.(14)can be written as

From Eqs.(5)and(15),we can get the solution of tangential velocity gradient as

A comprehensive analytical solution has been derived to calculate the SSD and to correlate the SSD of hypervelocity nonequilibrium flows with the average density between the shock and the body without the need for any specific gas model or empirical parameters.Furthermore,using an exponential function to approach the density distribution between the shock and the body,the scaled SSD~Δ has been correlated with the reaction rate parameterIn general,the results have shown that:

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whereρavgis the average density along the stagnant line which is equal to ρsand ρefor frozen flows and fully equilibrium flows,respectively.For hypervelocity frozen air flows,ρs/ρb=1 and ρs/ρ∞=6.Hence,Eqs.(9)and (12)yield SSDs of~Δ=0.38 and 0.40,respectively.Both solutions are in good agreement with there sultsobtained from Lob b’s approximation and Olivier’s model,i.e.,0.41 and0.40,respectively.TheSSDof~Δ=0.38derived by the morerigorous Eq.(9)is slightly less than Lobb’s approximation and Olivier’s model.Nevertheless,it is interesting to note that for the frozen nitrogen flows,Hornung1derived a value for SSD of~Δ=0.39 which is also slightly less than that given by Lobb’s approximation.Moreover,~Δ calculated from Eqs.(9)and(12)has only a weak dependence on ρs/ρ∞for hypersonic frozen flows which is consistent with that first reported by Olivier.11When free stream Mach number Ma∞→∞,the value of ρs/ρ∞depends on the value of γ(adiabatic index).In order to compare the present model with Oliver’s model11,the dimensionless SSDs for different gases are listed in Table1.It is observed that the present model is not as sensiti veto ρs/ρ∞as Olivier’s model.For large value of ρs/ρ∞,the present theory agrees well with Olivier’s theory.But for the monoa to mic gas flow(γ=5/3,ρs/ρ∞=4.0),the difference between the two theories is more obvious.

The values of ρ/ρ∞for non-equilibrium and fully equilibrium flows are larger than that for frozen flows,and the solutions obtained from Eqs.(9)and(12),respectively,tend to converge.Therefore,only the concise correlation Eq.(12)is employed in the following calculations.Eqs.(12)and(17)show that the density ratio ρs/ρbplays an important role in determining the SSD in non-equilibrium dissociating flows,which is consistent with the observations of Wen and Hornung5and Olivier,11respectively.

Table 1Dimensionless SSDof frozen flows for gases with different values of ρs/ρ∞.

Model Dimensionless SSD(~Δ)CO2(ρs/ρ∞=7.67)Ideal dissociating gas(ρs/ρ∞=7.0)Present,Eq.(9)Monoatomic gas(ρs/ρ∞=4.0)0.38 0.38 0.38 Present,Eq.(12)0.40 0.41 0.40 Olivier110.38 0.44 0.39

Table 2Dimensionless SSD(~Δ)of fully equilibrium flows for gases with different values of ρs/ρbprovided with ρs/ρ∞=6.0.

Model Dimensionless SSD(~Δ)ρs/ρb=0.4 ρs/ρb=0.5 ρs/ρb=0.6 ρs/ρb=0.7 ρs/ρb=0.8 ρs/ρb=0.9 Present,Eq.(12) 0.162 0.201 0.241 0.280 0.320 0.361 Wen and Hornung50.164 0.205 0.246 0.287 0.328 0.369 Olivier110.164 0.203 0.241 0.280 0.319 0.359

which implies the importance of the density ratio ρs/ρe(note that the subscript ‘e” denotes the corresponding quantities at fully equilibrium states).This simple correlation is well validated by experiments,5,7CFD results8,9and a quasi-oned imensional model.10However,it relies on the semiempirical parameter L measured by experiments,and therefore cannot completely reveal the embedded physics.

3.Correlation between shock stand-off distance and reaction rate parameter

Eqs.(10)and(12)imply that if ρs/ρ∞is known,the SSD can be determined from the average value ofNote that,the tangential velocity gradient is already solved in the last section.On the other hand,the generalized reaction rate parameter,i.e.,can be rewritten as

where y denotes the horizontal direction.In other words,the reaction rate parameter is governed by the spatial gradient of the density immediately behind the shock.As a result,the SSD can be correlated with the generalized reaction rate parameter by means of the density pro file between the shock and the body.

3.1.A correlation using exponential density pro file

Based on a differential analysis of the governing conservation equations,Olivier11proposed the following analytic solution for the SSD in frozen and equilibrium flows:

where λ ranges from zero to infinity.As shown,Eq.(19)is a monotonic function for ρ with respect to λ and the density reduces to ρs(frozen flows)and ρe(fully equilibrium flows)when λ =0 and ∞,respectively.In other words,every flow regime within the range of the frozen flow to the fully equilibrium flow is represented by a specific value of λ between 0 and∞.

Using Eq.(19),the density ratio between the shock and the body and the product of density and tangential velocity gradient and can be given as

and

respectively.From Eq.(19),we can easily verify that

which represents the dimensionless density gradient right after the shock.Clearly,an explicit correlation is no longer possible.But the following uniform implicit correlation can be derived

3.2.Comparison and discussion

Eq.(23)shows that the correlation betweenanddepends on the values of ρs/ρ∞and ρs/ρe,respectively.Fig.2 shows the variation ofwith~Ω as a function of ρs/ρegiven a constant ρs/ρ∞=6.Notably,the physical significance ofis the ratio between the energy absorption rate by chemistry and the input rate of free stream kinetic energy.5For small,no chemical reaction occurs in the flow and thus the scaled SSD remains constant.However,asincreases,the amount of energy absorbed by vibrational excitations and chemical reactions also increases.As a result,the average density increases,while~Δ decreases.As expected for the non-equilibrium regime,using exponential density approach gives a higher value of SSD than Wen and Hornung’s correlation5using linear density approach.

Fig.2Scaled SSDas function of reaction rate parametergiven ρs/ρ∞=6.

Fig.3Effect of ρs/ρ∞on scaled SSDandfor different values of ρs/ρe.

As a result,the solution method is restricted to this area since only the stand-off distance at the stagnation point is of interest and Eqs.(1)and(2)can be re-written as

4.Analytic solution for stand-off distance of nitrogen flows using ideal dissociating gas model

4.1.Basic equations

The analytic solutions derived in the previous section are not restricted to any specific gas model,and show thatdetermined by both ρs/ρ∞and ρs/ρe.However,in experimental and simulation studies,the free stream condition is usually expressed in terms of free stream values of ρ∞,u∞,T∞and α∞(T∞is the free stream temperature).Wen and Hornung5qualitatively described the effect of free stream kinetic energy on the scaled SSD~Δ.However,no quantitative relation was derived.Thus,in the present study,the simple IDG model is used to quantify the effects of the main flow parameters on the scaled SSD analytically,for the illustrative case of nitrogen flows.The analysis is also suitable for other pure dissociating diatomic gases and can be extended to multi-component gases by using the approach proposed by Olivier and Gartz.16

The boundary conditions on the shock wave can be determined by enforcing the conservation of energy,momentum,mass and dissociation fraction across the shock,i.e.,

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where his the specific enthalpy.In general,the equation of state for a mixture of molecular and atomic nitrogen is given as

where Mis the molecular weight of N2,Tis the temperature and Ruis the universal gas constant.Meanwhile,the specific enthalpy for an IDG is given by

where θdis the characteristic dissociating temperature for nitrogen and has a value of 113200 K.The boundary condition for h at the shock is then expressed as follows:

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Utilizing the state equation and the definition of enthalpy,the temperature immediately behind the shock can be obtained from Eq.(27)with αs=α∞as

whereis the dimensionless kinetic energy parameter.For hypersonic flows,the approximationis widely adopted.Eqs.(25)and(28)thus yield the following expression for the density:

From equilibrium theory of Lighthill,14the equilibrium dissociation fraction αecan be determined as

Here,ρdis the characteristic dissociation density,and was reported by Lighthill14to have a value of 1.3×105kg/m3for nitrogen.

To solve αe,ρeand Tefrom Eq.(30),two more equations are required.The first equation can be derived by enforcing the conservation of the total enthalpy,i.e.,

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Meanwhile,the second equation can be derived directly from the state equation,i.e.,

From Eqs.(30)–(32), αe, ρeand Tecan all be solved.Although explicit solutions are impossible,they nevertheless demonstrate the roles of the dimensionless parameters T∞/θd,ρd/ρ∞,μ and α∞in determining the shock stand-off position.Notably,T∞and α∞can be very different from the real flight conditions in a free-piston shock tunnels.

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4.2.Effects of μ and α∞on SSD

In the following discussions,ρs/ρ∞and ρs/ρeare derived from(29)and(30)–(32),respectively.Then they are used as the inputs of the correlation ofand

Fig.4 shows the variation ofwithas a function of μ givenand α∞=0.It is seen that the scaled SSDdepends very weakly on μ on the frozen side(~Ω≪1).However,reduces significantly with increasing μ on the equilibrium side(~Ω≫1).When μ=0.15(u∞=3175 m/s),~Δ on the frozen side and equilibrium side are almost equal.It indicates that when the free stream velocity of nitrogen flow is smaller than 3175 m/s,the dissociating reactions in the flow can be neglected.Notably,although when the freestream velocity decreases to around 3.2 km/s,the dissociation is very weak,the vibrational excitation may decrease a few percentages of(see Houwing et al.17).When μ increases to 1 and beyond,the correspondingcurves are approximately superimposed.From the physics perspective,for nearly frozen flow,no chemical reaction occurs to increase the average density.As a consequence,~Δ is effectively independent of~Ω and remains almost constant.For non-equilibrium and nearly equilibrium flows,the amount of energy absorbed by chemical dissociation increases with increasing the freestream kinetic energy parameter μ.As a result,ρe/ρsincreases and~Δ decreases.For the particular case of μ=1.0,the freestream kinetic energy is equal to the specific dissociation energy of the gas and the amount of energy absorbed by chemical dissociation reaches to the upper limit.Consequently, ρe/ρsno longer increases even when μ increases,and~Δ reaches its minimum value.Overall,Fig.4 infers that the change in the scaled SSDis due primarily to the energy absorption caused by chemical reactions.

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Fig.4 Effect of freesteam kinetic parameter μ on scaled SSD

Fig.5 Effect of freestream dissociation level α∞on scaled SSD

The solution shown in Fig.4 is based on α∞=0 which is the case for the ballistic range experiment.18However,in the high enthalpy free-piston shock tunnel tests,19,20the freestream dissociation level is not zero anymore.Using the IDG model,we can quantitatively estimate the in fluence of freestream dissociation level on the SSD.Belouaggadia et al.12investigated the effect of the freestream dissociation level,α∞,on the shock stand-off distance for the cases of frozen flows and fully equilibrium flows.In the present study,the effect of α∞onis investigated over the entire non-equilibrium flow regime.As shown in Fig.5,α∞has only a weak effect onfor the case of nearly frozen flows,which is the case presented by Belouaggadia,et al..12In addition,it is seenincreases significantly with increasing α∞for moderate values of μ,but is insensitive to α∞at larger values of μ.When α∞=0.3 and μ =0.4,the SSD is even larger than that of α∞=0 and μ=0.3.It means the two opposite acting effects,decrease of the SSD by high freestream kinetic effects and increase of the SSD by free stream dissociation,may even cancel each other.2This finding is reasonable since in higher α∞flows,dissociating chemical reactions occur less readily due to the absence of educts,and hence the density change is less obvious than that in the case of flows with low α∞.When μ is sufficiently large(e.g.,μ=1),dissociation anyway takes place easily,for α∞ranging from 0 to 0.3,and hence no change ofcurve occurs.In general,the curves presented in Fig.5 imply that the effects of possible freestream dissociation in high-enthalpy wind tunnels must be considered,particularly for the case of moderate μ.

5.Conclusions

Following Olivier,11an assumption is made here that the tangential velocity gradient pro file varies linearly with distance between the body and shock wave.For frozen flows and fully equilibrium flows,the density in the stagnant region can be treated constant and the expression of χ can be simply written as

(1)the correlation curve is strongly dependent on ρs/ρe,but is only weakly dependent on ρs/ρ∞.

(2)is a more universal dimensionless parameter thanin estimating the SSD.The effects on the SSD of the freestream kinetic energy parameter,μ,and the freestream dissociation level,α∞,have been investigated using a simple ideal dissociating gas(IDG)model.The results imply an underlying physical mechanism which can be described as the reduction of the scaled SSDis due to the greater energy absorption caused by chemical reactions.The effects of possible freestream dissociation in high-enthalpy wind tunnels must be considered;particularly for the case of moderate μ.

Acknowledgements

This study was co-supported by the Research Grants Council of Hong Kong,China(No.C5010-14E)and the National Natural Science Foundation of China(No.11372265).

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4) 根据折臂式铁钻工的设计要求，规定铁钻工的回转速度为nH=1r/min，根据传动比可以计算出n1=5.4r/min,n2=3.1r/min 。

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where the velocity component normal to the shock is neglected in the shock layer.

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