CCD reportNumerical simulation of energy deposition in a

supersonic flow past a hemisphereReport No. 2013-3

December 18, 2013

Mahsa Mortazavi and Doyle KnightCenter for Computational Design

Dept Mechanical and Aerospace EngineeringRutgers University

98 Brett RoadPiscataway, NJ 08854

mahsa.mortazavi89@gmail.com, doyleknight@gmail.com

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Introduction

In this project the effect of energy deposition in a supersonic flow past a hemisphereis studied. The energy deposition disturbs the flow after the blunt body shock andchanges the pressure distribution through time. The drag coefficient changes dueto the change of the pressure on the hemisphere instantaneously. In this projectthis phenomenon is simulated using the Euler equations.

Not all the energy that is deposited to the flow will be transformed to heating thegas. A large portion of the energy goes to vibrational energy, dissociation, etc.The efficiency of the energy deposition (based upon the assumption of an inviscid,perfect gas) is calculated using the experimental data [1].

Description of the Problem

In this study the experiment of Adelgren et al. [1] is simulated assuming an inviscid,perfect gas. The Mach 3.45 flow forms a blunt body shock in front of the hemisphere.The laser discharge is modeled as an instantaneous heated region added to theprevious steady state solution on the centerline.

It is assumed that the initial temperature distribution is spherically symmetric.The form of the initial temperature within the discharge region is therefore

(1) T =

{T∞ + ∆T f(r/ro) r ≤ roT∞ r > ro

where r is the spherical radius measured from the focal point of the laser dischargeand ro is the initial radius of the instantaneously heated region. The dimensionlessfunction f(r/ro) describes the shape of the initial temperature distribution withinthe heated region. The focal point is a distance L from the surface of the hemisphereon the centerline (Fig. 1) and is upstream of the blunt body shock. The initialdensity ρ and velocity u in the discharge region are set equal to their freestreamvalues since the discharge is assumed to be instantaneous. The initial pressure isobtained from the Ideal Gas equation.

The change in energy of the gas within the energy deposition volume V is

(2) ∆E =

∫V

ρ(cvT + 12u.u)

∣∣∣∣after

dV −∫V

ρ(cvT + 12u.u)

∣∣∣∣before

dV

Since the density and velocity do not change due to the instantaneous energy de-position,

(3) ∆E =

∫V

ρcvT

∣∣∣∣after

dV −∫V

ρcvT

∣∣∣∣before

dV

and therefore

(4) ∆E = ρ∞cv∆TV α

where α is a dimensionless parameter determined by the assumed shape of theinitial temperature profile

(5) α =1

V

∫V

f(r/ro) dV

3

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ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppp pppppppppppppppp pppppppp pppppppp

pppppppppppppppp

shock wave

M∞

L D

laser discharge

Figure 1. Schematic

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

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pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppp BA

C

D

shock wave

M∞i

j

laser discharge

Figure2. Computationaldomain

The application of dimensional analysis to the inviscid, perfect gas simulation im-plies that the peak pressure on the centerline ppeak (due to the interaction of theblast wave on the hemisphere centerline) normalized by the stagnation pressure onthe centerline po2 (in the absence of the laser discharge) is

(6)ppeakpo2

= g(M∞, ε, γ; ro/D,α)

where M∞ is the freestream Mach number, and ε is the dimensionless energy de-position parameter

(7) ε =∆E

p∞L3

The semicolon is introduced in (6) as it is expected that the dimensionless peak pres-sure ppeak/po2 is relatively insensitive to both ro/D and α for ro/D � 1, providedthat the dimensionless energy deposition parameter ε is based on the distance L ofthe discharge from the hemisphere surface. This non-dimensionalization embodiesthe fact that the blast wave strength (for a fixed ∆E) decreases with increasingdistance from the discharge region.

The dimensionless energy deposition parameter for the experiment is

(8) εe =Q

p∞L3

The dimensionless energy deposition parameter for the simulation is

(9) εs =4

3

π

(γ−1)

∆T

T∞

(roL

)3α

The thermal efficiency is defined as

(10) η =∆E

Q

where ∆E is the energy required to achieve agreement with the experimentalppeak/po2 . For a simulation using dimensional variables there is no requirement

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that the freestream static pressure and static temperature be specified in accor-dance with the values in the experiment provided that the freestream Mach numberM∞, ratio of specific heats γ match the experiment (and, of course, the dischargeposition is upstream of the blunt body shock). In this case, the equivalent energydeposition of the simulation corresponding to the experimental conditions is

(11) ∆Es =

[p∞L

3

]e

[4

3

π

(γ−1)

∆T

T∞

(roL

)3α

]s

where e and s correspond to the experiment and simulation, respectively.

Method of Solution

The fluid is assumed to be inviscid, perfect gas. Therefore, the Euler equationswith the Ideal Gas Equation apply for this flow.

∂ρ/∂t+∇.(ρu) = 0

∂(ρu)/∂t+∇.(u⊗ (ρu)) +∇p = 0

∂e/∂t+∇.(u(e+ p)) = 0

p = ρRT(12)

where e is the total energy per unit volume. Slip boundary conditions are appliedto the hemisphere-cylinder surface BC (Fig. 2). Symmetry boundary conditionsare applied on the axis AB. Freestream boundary conditions are applied on theouter boundary AD, and zero normal gradient boundary conditions applied on thedownstream boundary CD.

The simulation is performed using the commercial software GASPex[5]. The secondorder upwind-biased Van Leer method [2] in all i, j and k directions is used for theflux algorithm with the Modified ENO limiter. Prior to the energy deposition,the steady state condition is obtained using Gauss-Seidel[3] relaxation with a fixedCFL number of one. For the unsteady simulation the explicit temporal integrationis performed using a second order Runge-Kutta [4] scheme.

Three sequences of mesh are considered for the numerical simulation: coarse, baseand fine. The grid spacing in each sequence is one half of the previous one. Detailsof the mesh are presented in the Table 1. The accuracy of the simulations is assessedusing the coarse, base and fine grids to extrapolate the exact pressure ppeak/po2 onthe centerline vs time for the duration of the interaction. The average error for thebase and fine grids is 0.402% and 0.372%, respectively.

Table 1. Details of Grid

Grid ∆min,i/D ∆min,j/D No. of nodes

coarse 6.540 · 10−4 8.333 · 10−4 0.270 · 106

base 3.270 · 10−4 4.167 · 10−4 1.080 · 106

fine 1.645 · 10−4 2.083 · 10−4 4.320 · 106

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Initial Condition for Laser Discharge

The laser discharge is added as an in-stantaneous change to the steady stateflowfield. The Top Hat function is usedin the heated region; therefore the f inEquation 1 is one and the r0/D is cho-sen to be 0.05; which is close to the non-dimensional radius of the heated region inthe experiment. With the Top Hat func-tion that has been assumed for the heatedregion α in the Equation 5 becomes one.The schematic description of the initialcondition for the laser discharge is shownin the Figure 3. Figure 3.

Interaction of Laser Discharge with Hemisphere Cylinder

The pressure drag coefficient is defined as

(13) cd =F

12ρ∞U

2∞A

where F is the force in the streamwisedirection on the hemisphere and A =πD2/4 is the frontal area. Fig. 4 showsthe frontal pressure drag coefficient vsnon-dimensional time τ = tU∞/D forε = 1.178 ·10−2 (∆T/T∞ = 9 and ro/L =0.05). The drag coefficient is observedto initially increase and then drop dra-matically before recoverying to the steadystate value. The overall duration of theinteraction is approximately three timesthe characteristic timeD/U∞ which is thesame as L/U∞ for this simulation. Fourpoints (labeled A through D) are identi-fied in Fig. 4 and Figs. 5 to 8 display theMach contours and instantaneous stream-lines for the four points. Figure 4. cd vs τ

At point A (Fig. 5) the drag coefficient is halfway between the steady state value andthe first peak. The blast wave has already intersected the blunt body shock and thetransmitted shock wave has impacted the hemisphere surface causing an increasein drag. The heated region, however, has just reached the blunt body shock. Atpoint B (Fig. 6), the heated region has interacted strongly with the blunt bodyshock causing a lensing forward (upstream) of the blunt body shock. As the result

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of the interaction a vortex is generated inside the shock layer due to the Richtmyer-Meshkov instability. A stagnation point has formed on the axis upstream of thehemisphere surface due to the decreased stagnation pressure in the heated region1.At point C (Fig. 7) the heated region has completed passage through the bluntbody shock and the lensing effect has reached its maximum extent. The interactiongenerates a rarefaction wave which reduces the static pressure on the hemisphere.The drag coefficient achieves its minimum value. At point D (Fig. 8) the bluntbody shock has begun to collapse back towards its steady state position and theheated region has convected past the hemisphere.

Figure 5. Point A Figure 6. Point B

Figure 7. Point C Figure 8. Point D

1The static pressure within the heated region rapidly relaxes to the ambient pressure p∞ given

sufficient distance of the focal point from the hemisphere cylinder. Thus, at the moment the heatedregion reaches the blunt body shock, the static pressure within the heated region is essentially the

ambient pressure p∞ and the velocity is approximately U∞. However, the Mach number is lower

since the temperature is higher than T∞. Hence, the stagnation pressure in the heated region atthe moment of interaction with the blunt body shock is below the freestream stagnation pressure

pt∞ .

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Thermal Efficiency

In order to determine the thermal effi-ciency η, it was decided to match the ex-perimental peak pressure on the center-line due to the impingement of the blastwave. The experimental centerline pres-sure vs time is displayed in Fig. 9. Thepeak pressure due to the blast wave is anincreasing function of the laser dischargeenergy. Note, however, that the mini-mum pressure is insensitive to the laserdischarge energy and therefore cannot beused to determine the thermal efficiency.Multiple simulations with different valuesof ∆T were performed for each of thethree experimental cases in order to de-termine the value of ∆T that achievedagreement between the computed and ex-perimental ppeak/po2 .

Figure 9. Centerlinepressure vs t[1]

The value of η was determined using (10) where ∆E was calculated using (4) or (11)as appropriate. The computed and experimental dimensionless peak pressures werematched to within 1%. Matching of the computed and experimental peak pressureis illustrated in Fig. 10(a) to 10(c). The peak pressure is accurately matched towithin 1%. A second peak pressure is observed in the simulation for correspondingto the impact of the second shock on the hemisphere; however, this second peakis not observed in the experiment. The reason is that viscous effects would tendto mitigate the rapid pressure rise associated with the second shock, and thermaldiffusion would tend to diffuse the heated region and the viscous effects are notconsidered in this simulation which assumes inviscid, perfect gas.

The calculated efficiencies are listed in Table 2. The computed value of η is multi-plied by 100 to obtain the efficiency in percent.

Table 2. Thermal Efficiency

Q 13 mJ 127 mJ 258 mJ

η 1.081% 0.580% 0.436%

As it is shown in the Table 2, the thermal efficiencies of the energy deposition areso small and only a small portion of the energy deposited goes to heating the gasand increasing its translational-rotational energy.

8

(a) 13 mJ (b) 127 mJ

(c) 258 mJ

Figure 10. Computed and experimental p/po2 vs τ

Dependency on the Freestream Condition

In this project the dependency of the flow parameters on the dimensional freestreamcondition is studied. Two separate freestream conditions were chosen for two sepa-rate numerical simulations. The freestream conditions and the efficiencies and net

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energy deposited to the flow due to the energy deposition in each numerical simu-lation are presented in the Table 3. The net energy deposited to the heated regionin the numerical simulations and the efficiencies are calculated using the Equations11 and 10, respectively.

Table 3. Details of Simulations

p∞ T∞ ∆T/T∞ L/D r0/D ∆Es η(Pa) (deg K) (mJ) (%)

Simulation No. 1 101325 288 4 1 0.05 1.125 0.436Simulation No. 2 13100 77.8 4 1 0.05 1.125 0.436

Figure 11. Comparison between two simulations and the experiment

As shown in the Figure 11, the two numerical simulations match perfectly. Theresults show that the non-dimensional parameters in the flow (e.g., p/p02) onlydepend on the parameters.

Conclusion

The interaction of a laser-generated plasma with a hemisphere at Mach 3.45 issimulated. The instantaneous laser discharge is assumed to create a heated regionwith higher pressure and temperature but the same density as the freestream con-dition. The peak pressure on the hemisphere due to the impact of the blast waveis matched in each simulation to determine the thermal efficiency of the laser dis-charge. The time history of the centerline pressure shows significant disagreementwith experiment for all simulations. The results indicate that the perfect gas Euler

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simulations with the assumed initial condition are incapable of accurately predict-ing the surface pressure due to the interaction of the laser-generated plasma withthe hemisphere and hence the net drag reduction.

References

[1] Adelgren, R. G., Elliott, G. S., Knight, D. D., Zheltovodov, A. A., and Beutner, T. J., Energy

Deposition in Supersonic Flows, AIAA Paper 2001-0885, Jan. 2001.[2] Leer, B. V., Flux-Vector Splitting for the Euler Equations, In Lecture Notes in Physics, Vol.

170, 1982, pp. 507-512.

[3] Knight, Doyle. Elements of Numerical Methods for Compressible Flows. Cambridge: Cam-bridge University Press, 2006.

[4] Jameson, A., Schmidt, W., and Turkel, E., Numerical Solution of the Euler Equations by

Finite-Volume Methods Using Runge-Kutta Time Stepping Schemes, AIAA Paper 81-1259,1981.

[5] Aerosoft Inc. GASPex. Blacksburg, Virginia, 2009. http://www.aerosoftinc.com/