les modeling of scramjet combustion

American Institute of Aeronautics and Astronautics

1

LES Modeling of Scramjet Combustion

A Ingenito*, M.G. De Flora.† and C. Bruno‡ University of Rome “La Sapienza”, Rome, Italy.

E. Giacomazzi§ ENEA, S. Maria di Galeria (Rome), Italy

and

J. Steelant¶ ESA-ESTEC, Noordwijk, The Netherlands

The physics of supersonic combustion is analyzed in order to derive a new subgrid scale model for Large Eddy Simulation. Anisotropy associated to the directional Mach number typical of supersonic flows (i. e., M > 1 in only one direction) is explicitly considered by means of non-dimensionalized Navier-Stokes equations. The study shows that high Mach number flows experience mainly streamwise vorticity and consequently maximum helicity. Both affect mixing and may alter the turbulent kinetic energy decay by decreasing its rate, i.e., decreasing its characteristic spectral slope below that predicted by Kolmogorov scaling. Furthermore, it is analytically predicted that transversal pressure gradients increase vorticity, thus plausibly explaining the improved mixing realized by certain injectors. The supersonic regime is also found to affect the combustion regime: analysis of characteristic acoustic and convective times shows that while subsonic combustion takes place at approximately constant pressure, supersonic combustion takes place at approximately constant volume. Furthermore, collisional frequency is shown to increase due to local dilatation, resulting in faster kinetics and shorter ignition delay times. This effect could explain flame anchoring observed in some SCRJ combustor experiments. All the physical features and aspects of supersonic combustion found are used as ingredients to build a new subgrid scale model. In particular, micro-scale physics has been included by means of a subgrid kinetic energy equation that is algebraically modeled to provide the velocity fluctuation needed by the eddy viscosity SGS closure. Numerical simulations of a supersonic combustion NASA – Langley test case provide qualitative validation of the proposed model.

I. Introduction CRJ engines are promis ing candidates for future TSTO launchers. To design them, understanding combustion is mandatory: in this regard numerical simulations may help, and even become a fundamental tool in designing SCRJ combustors. In particular, by focusing on the unsteadiness of the flow, Large Eddy Simulation (LES) can help in understanding how to improve mixing, flame anchoring and combustion efficiency in supersonic reacting flows. LES sub-grid scale models developed for subsonic

combustion (such as the Smagorinsky-Lilly 1 and Fractal Model2) and adapted to supersonic flows are insufficient to reproduce well experimental results3,4, and in particular flame anchoring. In fact, past experimental results with hydrogen injected at Mach 2.5 in Mach 2 airstream showed combustion taking place in about 2 ft .

* Ph.D., Department of Mechanics and Aeronautics, Via Eudossiana 18, 00184 † Ing., Department of Mechanics and Aeronautics, Via Eudossiana 18, 00184, AIAA Member ‡ Professor, Department of Mechanics and Aeronautics, Via Eudossiana 18, 00184, AIAA Associate Fellow § Ph.D., Sec. ENE-IMP, C.R. Casaccia, Via Anguillarese 301, 00060, AIAA Member ¶ Project Manager, Sec. TEC-MPA, Keplerlaan 1, 2200 AG

S


2

Despite studies5 on supersonic combustion dating back to forty years, still not much is known on this topic. The goal of this work is to analyze the influence of Mach number on mixing and combustion, and to develop a new subgrid scale model.

The following analysis emphasizes three points: a. mixing is not driven only by vortex stretching, but also by compressibility; b. compressibility tends to favour combustion by increasing kinetic rates; c. supersonic combustion occurs at about constant volume. All these effects have been considered in the new model described here (ISCM 6 model). This new model has

been validated so far with two sets of experimental data: - cross flow injection of hydrogen at Mach 1 into an airstream at Mach 2 in a combustor similar to the one built

at the University of Tokyo by Takahashi et al7; - 30°-oblique injection of hydrogen at Mach 2.5 into an air stream at Mach 2 in a combustor similar to the one

built at NASA Langley by Drummond et al8. Numerical simulations obtained with the ISCM subgrid model show better agreement with experimental data

than with Smagorinsky-Lilly and FM models.

II. Effects of Mach number on turbulence In order to understand the influence of the Mach number on turbulence, a preliminary non-dimensional analysis

of the Navier Stokes equations has been conducted. By assuming dynamic pressure (p), total enthalpy (h) and inlet variables as reference quantities, the non-dimensional equation for vorticity ω is:

( ) ( ) ( ) ( ) ( ){ }2 22 2

1 1 1 1 2Re Re

pu u u u u Etω ρω ω ω ν ω ρ σ µ µ

ρ ρ ρ ∂ ∇ ×∇ + ⋅∇ = ⋅∇ − ∇⋅ + + ∇ + − ∇ × ∇⋅ + ∇ × ∇ +∇∇⋅ + ∇× ∇ ∂

Eq. (1)

where µ is the dynamic viscosity, u the velocity vector with components (u, v, w), ρ the density, σ the stress

tensor, E the strain rate and Re uLρµ

= the Reynolds number.

Eq. (1) points out that in supersonic flows the dilatational term has the same order of magnitude of vortex stretching, ( )u ω⋅∇ , at all scales. Therefore, vorticity transport is not exclusively driven by vortex stretching;

compressibility and the baroclinic terms play also important roles. Since supersonic flows are characterized by a high Mach number in the streamwise direction, a non-dimensional

analysis based assuming that the flowfield is characterized by two different velocity scales, that of the supersonic streamwise u component, and that of the transversal components v and w, where u a v w> >> ≈ , has been carried on, to see the influence of the flow anisotropy.

If k s is the ratio between the “transversal” (subsonic) Mach ( )sub vMa a= and the (supersonic) streamwise Mach

number ( )sup uMa a= ,

uv

MaMak

sub

s == sup Eq. (2)

in supersonic flows is a small parameter, i.e. ks<<1.

Then since ks<<1, the non-dimensional continuity equation

0=

∂∂+

∂∂+

∂∂+

∂∂+

∂∂+

∂∂+

∂∂

zw

yvw

zv

ytk

xuu

x s ρρρρρρ Eq. (3)

states that the variation of u along x is balanced by a density variation and not by local generation of transversal velocity gradients. Neglecting for the time being viscous effects, the scalar non-dimensional momentum equations in Cartesian coordinates are:


3

∂∂

−=

∂∂

+∂∂

+∂∂

+∂∂

∂∂

−=

∂∂

+∂∂

+∂∂

+∂∂

∂∂

−=

∂∂

+∂∂

+∂∂

+∂∂

zp

kzw

wyw

vtw

kxw

u

yp

kzv

wyv

vtv

kxv

u

xp

zu

wyu

vtu

kxu

u

ss

ss

s

ρ

ρ

ρ

11

11

1

Eq. (4)

pointing out that in a ducted supersonic flow, even moderate gradients of pressure along y e z can generate high x-

gradients of v e w, i.e., xv

∂∂ and

xw

∂∂ . Then, from the vorticity definition, the presence of

xv

∂∂ and

xw

∂∂

produces

vorticity in the y and z directions favouring better mixing that can enhance combustion: in fact, experimental results by Yang et al.9 show that flame anchoring is favoured by increasing the pressure ratio between fuel and oxidizer. Pressure gradients may also be produced by combustion itself, or by shaping combustor geometry and injectors. Strong variations of pressure are visible close to shock waves and they are known to increase mixing and combustion. The non-dimensional vorticity equation along x, written in Cartesian coordinates and neglecting viscous terms, is

∂∂

∂∂

−∂∂

∂∂

+∂∂

+∂∂

=∂

∂yp

zzp

ykxw

xv

xu

szy

x ρρρ

ωωω

2

11 Eq. (5)

Since 1/k s>>1, Eq. (5) shows that pressure and density variations along y and z influence the vorticity transport

along x. Also the transverse vorticity components ωy and ωz affect streamwise vorticity. To understand how vorticity develops and to clarify the role of different scales in mixing, it is convenient to

introduce the helicity, H. Its definition, H uω= ⋅ , shows that when vorticity and velocity are aligned, helicity is

maximum and the term ( )uω∇× × , that includes the vortex stretching term, is nearly equal to 0. This means that

where helicity is maximum the vortex stretching term responsible in subsonic flows for the cascade process is not intense in supersonic flows: mixing due to vortex stretching is less efficient. In fact, this conclusion may be derived from the non-dimensional helicity equation:

∂∂−

∂∂+

∂∂−

∂∂+= v

xww

xvkw

yuv

zuuH sxω Eq. (6)

showing that, since in the supersonic stream considered in the present analysis zu

∂∂

and yu

∂∂

are appro ximately nil,

and v e w are small with respect to u, vorticity is mainly streamwise and helicity is maximized. The implication is that in supersonic flows vortex stretching is lower than in subsonic case. Thus, in zones where dilatational and baroclinic phenomena are negligible (e.g. far from shocks), the contribution by stretching to kinetic energy transfer along the inertial cascade is low and characterized by fewer scales; if contributions by baroclinic and dilatation effects were missing, the slope of the kinetic energy vs. wave-number curve would be at most -5/3, i.e. lower than that in the classical Kolmogorov subsonic case. At the same time, in a flow dominated by streamwise vorticity, large scale mixing is much more efficient than where vorticity has no preferential direction; so, even though vortex stretching is lower than in the subsonic regime, the efficient mixing observed in some experiments must be driven by other effects, such as baroclinic and dilatational: if combustor geometry and boundary conditions allow the supersonic flow to become helicoidal, mixing may be more efficient and combustion distance short10.

Recent DNS simulations tend to confirm these theoretical results. In particular, Kida and Orszag11 noted that near shocks the baroclinic term plays an important role in vortex production and that compressibility amplifies vorticity, while vortex stretching becomes again the main mixing driver far from shocks. Finally, DNS simulations


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of supersonic flows show turbulence is mainly streamwise, with velocity and vorticity generally angled either at 0° or 180°12.

III. Effects of the Mach number on reaction rate To see whether there are effects of compressibility also on combustion, a new parameter, S, the ratio between

temperature increase due to kinetic energy, ∆Ts, and that due to combustion, ∆Tc

=∆∆

=c

s

TT

S ( )cT

TMa

∆− 02

21 1γ Eq. (7)

has been defined by Libby and Williams 13. When S<<1, the increase of temperature due to combustion is larger than that of kinetic energy, and vice versa for S>>1. They stated that in subsonic regime and low turbulence intensity, for

Zel’dovich numbers Z >1

∆=

00 TT

TT

Z ac , combustion occurs in the reaction sheets regime. This is still true for

supersonic flows only if S<1, but it is no longer so above a critical Mach number such that S>1, where compressibility effects cannot be neglected.

Since generally ∆Tc/T0 ~ 1-10, S>1 for 2Ma ≥ : the coupling between compressibility and chemical kinetics becomes important above a Mach number that depends on ∆Tc/T0 , thus that depends also on the fuel/air thermo -kinetics and on initial conditions. That means that, to define a flame regime for supersonic flows, the well known Da-Re plot in turbulent subsonic combustion must become 3-D, that is, extended in a third direction having Ma or S as parameter.

A second effect due to compressibility associated to high Mach numbers is on collisional frequency per unit volume, zV, and thus on reaction rates. In fact, in the simple Arrhenius theory the two are related by14:

[ ][ ]218

A AE ET T

VA A

Tz e e X Y

N N Mπ

ω σµ

− −ℜ ℜℜ

= =& Eq. (8)

where ℜ is the universal gas constant, µ the dynamic is viscosity, M the molecular weight, NA the Avogadro number, s the collisional cross-section, [X] and [Y] the molecular concentrations, EA the activation energy.

The Arrhenius pre-exponential A is the expression 22AM

π σℜ= . Detailed studies on the effect of

compressibility on chemistry have been conducted15,19 but this (nonequilibrium) effect is still not well characterized. The collision frequency per unit volume given by the kinetic theory of gases (and used in the Arrhenius theory):

2 8X A Y A

V X YX Y

n N n N Tz

m m Mπσ ρ ρ

µ ℜ

=

3

1seccm

Eq. (9)

corresponds to the zeroth order solution of the Boltzmann equation using the Chapman-Enskog method, the zero th order being associated to the assumption of zero bulk velocity. The 1st order Chapman-Enskog solution, instead, accounts for finite flow velocity, and then the effect of compressibility appears. Several models have been developed16,17,18 to understand how this effect influences collisional frequency; some of these models predict a correction to the exponential term that, however, does not satisfy the second law of thermo dynamics.

Eq. (9) shows that the reaction rate depends on temperature and density (or concentration). This dependence is obtained from gas kinetic theory for thermo dynamic statistical equilibrium. Strictly speaking, in supersonic flows where the acoustic time is longer than convective time it is not possible to define thermodynamic equilibrium, i.e., close to, or inside, a shock wave it is not possible to define a thermodynamic temperature19. At the microscopic level, indeed compressibility favours a shortening of the mean free path and, as a consequence, an increase of collisional frequency not predicted by Eq. (9). In order to rigorously evaluate this effect on reaction rate in high speed flows, the collisional frequency should be calculated with the help of the Boltzmann equation. Here the cruder but more practical alternative chosen is to analyze the effect of a small variation from equilibrium. A 1st order Taylor expansion of the function zV (t,ρ,T, M) of Eq. (9) around the equilibrium instant t0 yields:


5

( ) ( )0

0 02 1 1 2, , , 1 . .

2VV

t t

D DT Dnz t T M z t t hotDt T Dt nDtρρ

ρ=

= + + + − +

Eq. (10)

Eq. (10) tells that collisional frequency changes due to the combined effects of variations of density, temperature

and number of moles. The effect associated to dilatation is that due to D

Dt

ρ.

By neglecting (for the time being) the other two effects , Eq. (10) simplifies to:

( ) ( )0

0 0, , , 1 2VV t t

z t T M z u t tρ=

= + ∇⋅ − Eq. (11)

Eq. (11) shows that while in subsonic flows ( 0)u∇⋅ ≅

r r the influence of compressibility on collisional frequency

is negligible, it becomes important in supersonic flows. Non-dimensional analysis of the continuity equation shows that the divergence of velocity scales with Ma2, and that Eq. (11) may be crudely approximated as:

( ) 20, , , 1 2

VVz t T M z Maρ γ + ∼ Eq. (12)

i.e. in compressible flows the collisional frequency varies also with the local Mach number, and so the reaction rate:

( ) 20, , , 1 2t T M Maω ρ ω γ = + Eq. (13)

is raised by positive dilatation effects.

This conclusion might explain the prompt flame anchoring shown by experimental results and still hard to reproduce by numerical simulations. E.g., supersonic combustion tests at NASA Dryden in 1961-62, with nominal air stream conditions Mach = 3.5, T0 = 2390 K, P0 = 0.29 atm were carried on with hydrogen injected through a row of holes on a flat plate. The fuel jets were injected at 90° with respect to the air stream. Shadowgraphs and IR fast-camera pictures shown in Figure 1 indicate ignition was fast. If, as it appears, at Ma>1 compressibility affects the reaction rate ω& , this effect must be taken into account in modeling chemical specie sources.

IV. Effects of Mach number on combustion regime To assess the type of supersonic combustion regime, a new parameter has been defined:

1chem

conv

acus

conv

chem

acus DaMa

ττττ

ττ

= =

By saying that the heat release time (chemical time) is shorter than the acoustic time means that the increase of molecular kinetic energy due to breaking inter-atomic bonds is higher than chaotic molecular kinetic energy. In that case molecules have no time to distribute and to space out over a distance equal to the equilibrium mean free path

2πσpkTl = , so combustion must occur at constant volume. Since molecular velocity is increased by heat release, local

pressure increases (in turn, increasing, typically, reaction rates). The condition for constant volume combustion is

then 11chem

acus DaMa= <

ττ

, or 1DaMa

> . For constant pressure combustion, instead, 1 1chem

acus DaMaττ

= > or 1

DaMa

< .

Figure 1 Supersonic combustion flame anchoring on a flat plate. The picture on the right was taken with a IR filter and shows water (McDonnel Douglas ASD-TDR-63-236 report, circa 1962). Flow is from left to right.


6

In flows at Ma<1 both kinds of combustion regime are possible, e.g. at constant pressure, as in Diesel engines and at constant volume, as in Otto engines.

In supersonic flows combustion meeting the Da>1 condition implies 1DaMa

> . This means that in supersonic

flows combustion may occur only at (approximately) constant volume. This result is confirmed by experiments by Bellenoue et al.20. They observed pressure increases within a

supersonic constant area combustor; this is less evident with a divergent channel, where pressure is deliberately made to decrease or stay constant to prevent thermal choking. These effects can also be deduced by using gasdynamic relations21 relating the effect of the heat release on pressure (or density) to the Mach number.

As a consequence of the nearly constant volume combustion and of the low turbulence intensity at small scale, supersonic combustion can be conjectured to be in the reaction sheets regime, just as at subsonic Mach numbers.

V. ISCM subgrid supersonic combustion model The previous considerations point to some basic features about coupling between turbulence and combustion at

Ma>1: 1. reaction rates depend on the local Mach number; 2. vorticity is mainly streamwise, and its creation and transport are driven also by baroclinic and

dilatational terms; 3. combustion occurs at roughly constant volume.

These conclusions have been the first step towards development of the ISCM subgrid scale model accounting for

the physics of supersonic combustion. Turbulent closure of the Navier Stokes equations with ISCM is based on the Prandtl eddy viscosity concept

t ml uµ ρ ′= . The characteristic turbulent length scale lm is the grid dimension; the

velocity fluctuation u’ is calculated as SGSk , kSGS being the subgrid kinetic energy. To each computational cell are

associated the kinetics energy obtained by the filtered equations,k , and the kinetic energy k ∗ of the subgrid fine structures. These structures, that simultaneously react and dissipate, are assumed22 to occupy only a fraction γ* of the cell volume. Kinetic energy equation specialized at the fine structures level (superscript *) is :

( ) ( )* ** * *

* * * * * * * * * * * * * * * * ** * * * *

1 1 1 1

1 1 1N N N Nj i

p p i i i u i j i i ii i j ii ij

IIIIII IV V VI

Xk p Tu k C Cu T T hYV T V V Y f u Vt t t wD

αλ

ρ ρ ρ= = = =

∂ ∂ ∂ + ⋅∇ = − − ⋅∇ + ∇⋅ ∇ −∇⋅ − ∇⋅ ℜ − + ⋅ + ∂ ∂ ∂ ∑ ∑∑1424314243123 14243 1442443 14444444244444443 {

** * * *

* *1

1 N

i ii

VIIIVII XIX

u hτ ωρ ρ =

Φ+ + ⋅∇ +∑ ∑&144424443 1424314243

Eq. (14)

obtained from the total energy equation (mechanical plus thermal). Eq. (14) includes all effects that the -dimensional analysis has shown important in supersonic flows: solving this equation would involve a DNS-type approach and grid, therefore within the LES constrains, this equation has been algebraically modeled term by term (not reported here for brevity).

To account for the contribution of subgrid reactor kinetic energy k ∗ to the filtered kinetic energy k , a

“corrected” filtered kinetic energy Nk is introduced and defined as the weighted average of k ∗ and k , with γ* as

weight:

* * *(1 )Nk k kγ γ= + − Eq. (15)

The difference between the filtered Navier Stokes and th is weighted or “corrected” filtered kinetic

energy is assumed to be the subgrid kinetic energy, k SGS, to be used in the model:

( )* *SGS Nk k k k kγ= − = −

All thermodynamics variables within the subgrid reactor (superscript *) have been calculated assuming a zero-

dimensional perfectly mixed reactor at constant volume 23.


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Subgrid reaction rate has been modelled by means of Eq. (13); coupling between turbulence and combustion is implicit in the assumption that reactions occur only within the fine structures volume, thus the filtered reaction rate is:

* 20 1 2 Maω γ ω γ = +

The main differences between this SGS model and others are that ISCM takes into account the effect of

compressibility on mixing by considering the [complete] equation for kinetic energy, by assuming the fine scales reactor operates at constant volume, and by introducing the effect of the Mach number on reaction rates.

VI. ISCM model validations The ISCM model has been validated with two test cases:

1. cross-flow injection of hydrogen at Mach 1 into a Mach 2 airstream (experiments by Takahashi et al.7); 2. oblique (30°) injection of hydrogen at Mach 2.5 into a Mach 2 airstream in a combustor similar to that

built at the NASA Langley by Drummond et al.8 . For brevity, here only the results and comparisons with the second test case are reported. The 3-D LES

simulations were performed with the FLUENT 6.7 ™ (double precision24), both with the Smagorinsky-Lilly SGS model and the ISCM model (the ISCM model has required a user-defined function). For simplicity only, hydrogen chemical kinetics has been modeled by means of a global 1-step reaction and 3 reacting species25:

H2 + ½ O2 = H2O

kglobal = 1.8×1013 exp(-17614 K/T) [H2]1.0[O2]0.5 The LES solver is coupled upwind explicit,

second-order accurate both in space and time. The geometry of the SCRJ combustor

configuration is shown in Figure 2a. The combustor is divided into two sections: the main section (shown in Figure 2b) is made of copper. It is fastened to a 3° divergent in carbon steel. The hydrogen injector is located at 43.8 mm from the step and inclined of 30° with respect to the air stream.

This SCRJ combustor geometry has been mapped by a structured 3-D grid; the characteristic cell scale in the constant cross section is 1 mm, except near the hydrogen injection where is 0.2 mm. In the divergent channel the grid is stretched along the main flow direction (the stretching factor is 1.01). In the hydrogen duct the spacing is 0.5 mm. The number of hexahedral elements is 303792, for a total of 324864 nodes; the time step is 10-8s.

The inlet BCs are in the following table; these data correspond to a flight Mach about 7. Temperatures are sufficiently high to produce self-ignition of the air-hydrogen mixture.

The combustor walls are non adiabatic: measured heat flux varies from 0.7 MW/m2 to 1.8 MW/m2,

corresponding to about 10% of combustion heat release.

Figure 2 Test model: (a) nozzle, copper and carbon steel duct sections, (b) detail of fuel injector and pilot.


8

Figure 3 shows experimental results, predicting a flame anchoring before cross section 6 (at 77 cm from the inlet

combustor plane, about 60 cm from the H2 injector), where it is already well developed. Maximum temperatures are close to the wall and are somewhat lower than 2200 K.

VII. Simulation Results Simulations performed by the ISCM subgrid model in the non reactive case show the H2 jet penetrating up to 0.5

diameters, and afterwards bending down towards the bottom wall. A bow shock is located immediately in front of the injector; intense shock reflections between upper and bottom walls are predicted. Two counter-rotating vortices move along the stream direction, straining the H2 jet.

In the reactive case, simulations conducted with the Smagorinsky-Lilly model predict weaker and “more diffused” shocks than those predicted by ISCM.

Flow at Air Intake GH2 Injection

Mach 2.0 Mach 2.5

Temperature 1204 K Temperature 133 K

Pressure 101325 Pa Pressure 202650 Pa

Table 1 inlet conditions

Figure 3 3D view of the average temperature field


9

In fact, the turbulent viscosity predicted by the Smagorinsky-Lilly model (∼0.06 m2/s to 0.69 m2/s) is higher than that using ISCM (10-4 m2/s – 10-2 m2/s) and tends to smear out and dampen convective mixing and fluctuations.

The ISCM model predicts sharper vortex structures: two counter-rotating vortices are shown on all planes. These

vortices promote mixing thereby favouring flame anchoring. Figure 6 shows that the Smagorinky-Lilly model does not predict combustion or flame anchoring occurring

within the combustor, whilst it is evident in ISCM model at about 25 cm from the injector. Experimental results don’t show exactly where the flame anchors, but this occurs between 25 cm (section 5) and 70 cm from injector (section 7) where it is well developed.

Figure 4 Averaged Pressure field; median XY plane

Figure 5 Averaged YH2 mass fraction field and streamlines on the ZY plane at X =

0.2 m, X = 0.35 m e X = 0.7 m

Smagorinky-Lilly model

ISCM MODEL


ISCM MODEL


10

Maximum temperatures predicted by the ISCM model are about 2200 K, in very good agreement with the

experimental data. Figure 6 shows that the flame develops close to the upper wall and is absent close to the bottom; the flame

length, in good agreement with the experimental results, reaches the end of combustor and continues out. Further, this figure shows vorticity is mainly streamwise. The numerical results point out that velocity fluctuations, defined

here as SGSk , are less than 100 m/s, thus turbulence intensity is not high, ∼ 2% with rare peaks at 10%. Figure 7

shows that ISCM predicted turbulent viscosity is ~ 10-4 to 10-1, with high viscosity spots of predicted close to the H2 jet/air interface; their spatial distribution is intermittent.

Numerical simulations show that the ISCM subgrid model is in better agreement with experimental data than the

Smagorinky-Lilly model in terms of flame anchoring and temperatures. However, experimental results show a flame reaching also the bottom wall, still not predicted with the ISCM model, possibly a consequence of a still too rough grid or likely a too short computation (10-3 s corresponding to slightly more than a single residence time, simulated in ~ 10 days with a time step of 10-8 s ).

Simulations of the Takahashi test case, not reported here for brevity, have also confirmed that combustion occurs at nearly constant volume and that vorticity is mainly streamwise.

VIII. CONCLUSIONS The effects of high Mach number on turbulence and combustion have been theoretically analyzed. The study

shows that high Mach number flows experience mainly streamwise vorticity and consequently maximum helicity. Both affect mixing and improve it at large scales. The turbulent kinetic energy decay is affected by the baroclinic and dilatational effects and reduced with respect to the Kolmogorov scaling, due to the higher helicity. It is also

Figure 7 Instantaneous turbulent viscosity with ISCM model in the middle XY plane

Figure 6 Averaged Temperature field and streamlines in the middle XY plane


ISCM MODEL

ISCM MODEL


11

found that mixing can be improved by forcing transversal pressure gradients, e.g., by means of particular geometries (generating shocks) and devices (injectors). Furthermore, it shows that supersonic combustion takes place at approximately constant volume, and that collisional frequency increases due to local dilatation, resulting in faster kinetics. This effect could explain flame anchoring observed in certain SCRJ combustor exp eriments.

All these theoretical features have been used to build a new subgrid scale model. In particular, micro-scale physics has been included by means of a subgrid kinetic energy equation that is algebraically modeled to provide the velocity fluctuation to be used in an eddy viscosity SGS closure. Numerical simulations of a test case provide qualitative validation of the proposed model, showing better agreement with experimental data than the Smagorinky-Lilly SGS model. Future developments include grid refinement and further validation with the test case by Kanda et al.26 at JAXA-Kakuda Research Center.

Acknowledgments The financial contribution of EU within the Sixth Framework Program, Priority 1.4, Aeronautic and Space,

Contract No. 12282 (LAPCAT) to the investigation of the supersonic and hypersonic airbreathing vehicles, and the discussions with all pan-European LAPCAT consortium partners are gratefully acknowledged.

References

1 Piomelli, U., Modeling Requirements for Turbulent Flows, Lecture Series 1998-05, Advances in Turbulence Modeling, Von Karman Institute for Fluid Dynamics, 23-27 March 1998 2 Giacomazzi, E., Bruno, C., Favini, B., “Fractal Modeling of Turbulent Combustion”, Combustion Theory and Modeling, Vol. 4, 2000, pp. 391-412; 3 Giacomazzi, E., Del Rossi, A., Bruno, C., “Numerical Simulation of a Scramjet Combustion Chamber”, 54th International Astronautical Congress, IAC-03-S.5.05, Bremen, Sept. 2003; 4 Del Rossi, A., Giacomazzi, E., Ingenito, A., Parisi, V., Bruno, C., “`LES of Supersonic Combustion in Cross Flow H2/Vitiated Air SCRAMJET Configuration'', 40th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, AIAA 2004-3877, Fort Lauderdale, Florida, 11-14 July 2004; 5 Curran, E. T., “Scramjet Engines: The First Forty Years”, Journal of Propulsion and Power, Vol. 17, No 6, November –December 2001, pp. 1138-1148; 6 Ingenito, A., “Supersonic Combustion Modeling”, Ph.D. Dissertation, Mechanics and Aeronautics Department, University of Rome “La Sapienza”, Rome, October 2005, (in Italian); 7 Takahashi, S., Demise, S., Oshita, M., Ikeda, T., "Correlation between Heat Flux Distribution and Combustion Mode in a Scramjet Combustor", paper ISABE 2001-1191, Bangalore, India, 2001; 8 Drummond, J. P., Diskin, G. S., Cutler, A. D., “Fuel-Air Mixing and Combustion in Scramjets”, Technologies for Propelled Hypersonic Flight, NATO Research and Technology Organization, Working Group AVT 10, Final Report, January 2001; 9 Choi, J. Y., Yang, V., “Flow and Flame Oscillations in Hydrogen-Fueled Scramjet Combustors”, 2003 International Colloquium on Hypersonic Propulsion, Institute of Mechanics, Chinese Academy of Sciences, Beijing, September 2003; 10 Scheel, F., Ciezki, H.K., Haidn, O., “Connected Tube Experiments: Description of Boundary Conditions and Compilation of Test Data”, European Union LAPCAT Project Report, Progress Meeting 2, Capua, Italy, 21-23 November 2005; 11 Kida, S., Orszag, S. A., “Entropy budget in decaying compressible turbulence”, J. Comput. Sci. 5,1, 1990;


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