Wei Wu，Hou-Qian Xu，Liang Wang，Rui Xue

(School of Power Engineering，Nanjing University of Science＆Technology，Nanjing 210094，China)

1 Introduction

The total drag is one of the most important parameters during the artillery projectile design，and the base drag component is a large part.The base bleed technology is an efficient tool to reduce the base drag.In the last two decades，many studies were performed experimentally and numerically. The influence of ambient temperature，pressure on the drag reduction rate was investigated by free stream wind tunnel experiments，cases without base bleed，with cold air bleed and with base combustion on different conditions were also examined［1-3］. With the developmentofcomputationalfluid dynamics，the simulation of base region flow field for shell configuration has been of great interest in military. Lu［4］employed 3-D Reynolds-averaged N-S equations with SA and RNG k-ε turbulent model separately to simulate the full flow of base bleed projectile.Luo［5］considered the influence of external combustion on the base drag reduction.The complex two-phase flow of base bleed projectile was investigated based on the simplified two-fluid model［6］.Chen［7］studied on the effect of the bleed gas energy on base drag using 3-D Euler equations.Petri［8］coupled with a thermodynamic model based on curve-fit for specific heat of the individual species to simulate the supersonic projectile with base bleed.The base flow with mass bleed was simulated using axisymmetric mass-averaged N-S equations with standard k-ω turbulent model by Lee［9］. Shin and Choi［10］predicted the base flow and base pressure successfully based on EDS methodology. Charles［11］used the ensemble-average N-S computational technology which included finite rate model to study on the base drag reduction.Choi［12］modeled the flow using two-equations k-ω SST turbulent closure and finite rate model successfully.

However， the mesh generation was timeconsuming in these studies， especially for the engineering problems with complex configuration.In last two decades，the gridless method used in solid mechanics broadly attracted the scholars’attention in CFD field.This method operated on the distribution of points in the flow field，and the cloud generation was more convenient，hence，it had great advantages for the flow fields involving complex geometrical configurations. Somesignificantachievementswere obtained in last 10 years.Several inviscid numerical flux scheme extended into gridless method，such as HLLC［13］，AUSM+-up［14］，AUFS［15］.The dynamic cloud methodology wasstudied forflow problems involving large displacement moving boundaries by Ma［16］and Zhou［13］.The hybrid Cartesian grid and gridless methods also were presented by Hong［17］and Cai［14］.The gridless method coupled with finite rate model was used to simulate the flows of shock-induced combustion［18］.

The numerical studies on the base bleed projectile referred above are almost based on traditional mesh method.Due to its flexibility and superiority，M864 external combustion flow fields are simulated using the gridless method in this paper，and the finite rate reaction model is employed to model the secondary combustion in the weak region.The computations are completed for Ma=1.5，2，and 3，the cases without base bleed(inert)and with hot air injection are simulated for comparison，and the numerical results are also discussed.

2 Governing Equations

The Euler equations have been used in the supersonic flow.The formulation written in vector conservation law form foraxisymmetric flow with chemical source in Cartesian coordinates is

where the state vector，U，the convective flux vectors，F(xiàn) and G，the chemical source term，W，and the axisymmetric source term，S，are defined as follows:

where ρiis the density of species i;ρ is the total density;u and v are the velocity components;ρE is the total energy per unit volume，and p is the pressure. The chemical source ωirepresents the production or destruction of species i through chemical reaction.

3 Numerical Method

3.1 Spatial Discretization

Spatial derivatives are calculated by the sum of the products of flow properties of the points in its cloud using a least squares approximation.A linear function is adopted as the basis，and the flow property f is expressed as:

Therefore，the central point and all the satellitic points satisfy Eq.(2).Point i is set as an example，and we can obtain:

Then a0，a1，a2can be calculated by least squares method as:

where A is the coefficient matrix of Eq.(4)，and then the spatial derivatives of point i can be expressed as:

where fijdenotes the value of flux term at mid-point between central point i and its satellitic point j，and the flux term of Euler equation can be expressed as:

3.2 Inviscid Flux

Several numerical schemes were applied successfully in foregone gridless methods，in this paper， the multi-componentHLLC scheme with simple-form and high-resolution is extended into gridless method，and the numerical flux term Wijof mid-point can be calculated as:

where the subscript K takes i or j according to Eq.(9). More details are in Ref.［18］.

3.3 Chemical Model

The source term is calculated using a finite rate chemistry model with the reaction expressed in the form as:

在進行房屋建設的時候，在電氣設備中最為常見的就是開關插座，在安裝開關插座的時候，并沒有將居民的需求放在首位進行位置的選擇，致使位置的安裝并不合理。除此之外，在進行電路連接的時候，經(jīng)常會發(fā)生誤接和錯接的現(xiàn)象，就非常容易發(fā)生短路的現(xiàn)象，并且電路會出現(xiàn)串聯(lián)的現(xiàn)象，極容易造成高層的建筑發(fā)生火災，為消防安全造成極大挑戰(zhàn)，使居民生命與財產(chǎn)的安全受到了威脅。除此之外，在進行著強弱電布電的時候，對二者之間在距離上的規(guī)劃并沒有進行好好的規(guī)劃，距離太近，就會使電信號受到非常強烈的干擾。

where ν'imandare the stoichiometric coefficients for ithspecies in mthreaction，and the parameter Kfmis the forward reaction rate calculated using the Arrhenius expression:

where Amis the pre-exponential factor;bmis the temperature exponent;Emis the activation energy.The source term of an individual species is calculated by summing the contribution of each reaction:

where Miis the molecular weight of ithspecies;Kbmis the backward reaction rate and can be calculated through equilibrium constant.

3.4 Temporal Discretization

An explicit four-stages Runge-Kutta scheme is employed to advance over a time step Δt as:

where R is the residual vector.φ represents 1/4，1/3，1/2，1 respectively.The time step Δt is calculated by local time technique for convergence.

3.5 Boundary Conditions

The free stream condition is imposed along the supersonic inlet boundary， and a non-reflecting boundary condition based on Riemann invariants is using at the outlet boundary，and all the surface boundaries are assumed as no-penetration.The base bleed boundary isdetermined by the bleed gas stagnation temperature T0jand mass injection rate I:

where˙m is the injection mass flow rate;A is the area.

With the mass injection rate specified，the bleed Mach number can be calculated as:

where the subscripts b，j and∞ refer to the conditions atshell base， exhaust port and free stream，respectively.The static temperature Tjis determined from isentropic relation，and then all otherflow variables can be obtained.

4 Numerical Results

The full flow fields of M864 base bleed projectile are simulated using the gridless method above.This method was applied to simulate of shock-induced combustion successfully in our previous work［18］.

The schematic of M864 base bleed projectile is shown in Fig.1.In this paper，the outflow is 8 calibers downstream from the base and 5 calibers from the axis of symmetry.As shown in Fig.2，the points are uniform from the base to the downstream outflow boundary and clustered to the shell surface in radial direction.It contains 20 points in the injection region，60 points at the base and 780 points along the shell surface.There are 131666 points totally in the flow field.

Fig.1 Schematic of M864 base bleed projectile(mm)

Fig.2 Points distribution near projectile base

The simulations are performed for the Mach number of 1.5，2，and 3，the cases with hot air injecting and without bleed are also computed.All cases are run for the atmospheric conditions of free flow temperature T∞=294 K，and the bleed stagnation temperature T0jis set to 1533 K，and the mass injection rate I is 0.0022. The combustion productions of propellant grain are consisted of H2，H2O，CO，CO2，N2，and the molar fractions are given in Table 1.The combustion mechanism involving 9 species and 11 reactions［19］is used in this paper.

Table 1 M864 propellant equilibrium species concentrations

The temperature contours for all cases are shown in Fig.3.With the additional H2-CO combustion in the weak region，the temperature increase can be seen to extend further into the recirculation zone than hot air injection for Ma=2.0，3.0 respectively.Due to the existence of vortex in the concave portion of projectile near base corner，burning happens strongly in this region，which results in high temperature. The maximum temperature is higher than 2000 K at Ma= 3.0.The temperature increase is not able on the case of Ma=1.5 and the high temperature appears only near the bleed hole.Additional evidence of base combustion can befound from thespecies mass fraction，and CO2and OH mass fraction contours at Ma=3.0 are shown in Fig.4，which implies the combustible mixture reacts with the entraining oxygen near the base corner and in the recirculation zone.

Fig.3 Temperature contours

Fig.4 Species mass fraction of CO2(top)and OH(bottom)，Ma=3.0

The Mach contours of Ma=1.5，2.0，and 3.0 are shown in Fig.5 compared with cases with hot air injection and inert case.It can be found that hot air injection and H2-CO combustion have a significant effect on the characters of recirculation zone，the base corner expansion and the recompression region.As a result of the base bleed，the recirculation zone enlarges and moves downstream obviously.According to axial velocity distribution(Fig.6)，the H2-CO combustion case isshown to shiftthe rearstagnation point downstream approximately 0.26 calibers at Ma=3.0. Axial velocity profiles at four longitudinal positions(all these stations lay inside the recirculation zone)are plotted in Fig.7.It can be described that the shear layer rises with H2-CO combustion comparing to the cases with hot air injection and without bleed.

A plot of the total drag force coefficient is shown in Fig.8.With the increase of Mach number，the drag force coefficient decreases for all cases，and H2-CO combustion case shows a relatively low value comparing to the inert case.The pressure contours are shown in Fig.9 for the H2-CO combustion case at Ma=3.0，and the area-averaged base pressure rises to 37542 Pa，which is about 53.4%higher than the inert case，and the total drag force coefficient decreases to 0.182 from 0.198. Theeffectmaybeexplained asfollows: combustion of H2-CO will move the recirculation zone downstream further，and this process will result in a downstream shift in the weak closure location，and reduce the expansion at the base corner.Hence the base pressure will increase，which leads to a decrease of base drag.Furthermore，the total drag force will reduce.

Fig.5 Mach contours

Fig.6 Axial velocity distribution along centerline，Ma=3.0

Fig.7 Axial velocity profiles in recirculation zone，Ma=3.0

Fig.8 Total drag force coefficient vs Mach number

Fig.9 Pressure contours，Ma=3.0 H2-CO combustion

The drag force coefficients obtained from the trajectory model and Navier-Stokes computation with combustion［11］are also shown in Fig.8. The comparison is shown to be in some agreement for the higher Mach numbers，but goes up for the lower values.It is considered that working in the areas of viscosity，turbulence and adaptive gridless algorithms are used to help resolving this problem.

5 Conclusions

The external combustion flow fields of M864 base bleed projectile are simulated using a least-square gridless method coupled with finite rate chemistry model，and a 9 species and 11 reactions mechanism is employed in this paper.The results show that with the H2-CO combustion，the recirculation zone moves downstream further and the shear layer rises comparing to the hot air injection and inert case.Due to the expansion at the base corner，the base pressure increases，and base drag and totaldrag force coefficient decrease correspondingly.The comparison with the trajectory model predictions is fair at the higher Mach number，but not ideal at the lower values，which implies that the influence of viscosity，turbulence should be considered for transonic and low supersonic flows.Besides，owing to its flexibility，gridless method has great advantage in the flows involving complex geometric structure，especially in three-dimensional flows.It is significant to develop the three-dimensional gridless method for applications in engineering problems.These will be the focus of our future work.

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