particle and drop dynamics in the flow behind a shock wave

ISSN 0015-4628, Fluid Dynamics, 2007, Vol. 42, No. 3, pp. 433–441. © Pleiades Publishing, Ltd., 2007.Original Russian Text © V.M. Boiko, S.V. Poplavskii, 2007, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2007,Vol. 42, No. 3, pp. 110–120.

Particle and Drop Dynamics in the Flowbehind a Shock Wave

V. M. Boiko and S. V. [email protected], [email protected]

Received May 10, 2006

Abstract—The results of an investigation of the dynamics of hard particles and liquid drops in theflow behind a transmitted shock wave are presented. From the equation of motion of a particle in theshock wave, relations for the displacement, velocity and acceleration as functions of time and certainvelocity-relaxation parameters taking into account the properties of the gas and the aerodynamic dragof the particles are obtained for unsteady flow around the particles at an acceleration of 103–104 m/s2.It is shown that the velocity-relaxation parameters are universal. Approaches to finding the aerodynamicdrag of freely-accelerating bodies from the dynamics of their acceleration after being suddenly exposedto the flow are considered. It is established that under these conditions the drop dynamics observed canbe well described in terms of the same velocity-relaxation parameters with account for linear growth ofthe transverse drop size. All the kinematic functions obtained are confirmed experimentally.

DOI: 10.1134/S0015462807030118

Keywords: shock waves, multiphase media, velocity nonequilibrium, aerodynamic drag.

The velocity relaxation of the particles and gas is one of the basic problems in multiphase gas flowdynamics, since it significantly affects various processes in multiphase media, in particular, shock-wavepropagation in gas-dispersed mixtures [1]. One of parameters determining the interphase force interactionon the individual-particle scale is the aerodynamic drag coefficient CD. For spherical particles this parameterhas been studied in some detail (see, for example, [2, 3]) but most of the data have been obtained for steadyflows. However, there is good reason to believe that the drag of a particle suddenly exposed to a flow maybe significantly different from that in steady regimes. The drag of a particle accelerated in a shock wavemay be significantly affected by certain factors new as compared with steady flow, such as the accelerationitself [4] and the nonstationarity of the boundary layer and the aerodynamic wake [5].

Of great interest are the drop dynamics in flows, in particular, flows behind shock waves. This is a wideclass of problems of physical gas dynamics having numerous applications to power and aerospace engineer-ing [6, 7]. Depending on the fluid properties and flow parameters, numerous types of drop deformation andbreakup can be observed. These processes occur under conditions of dynamic interaction and may be mani-fested in different relaxation stages, which inevitably affects the acceleration of the drop as a variable-massbody with a variable midsection area. In this connection, the following questions arise. What is the aero-dynamic drag of the drops under these conditions? Is it correct to use the aerodynamic drag as a constantparameter for a body of variable shape and mass? If yes, then is it possible to use the same tools for dropsas for hard particles?

In the last ten years, due to the topicality of these problems, a number of publications has been devotedto the aerodynamic drag Cx of hard spherical particles in the flows behind a shock wave [8–11] and dropssuddenly exposed to a high-speed gas flow [7, 12]. In order to find Cx from the dynamics of the bodydisplacement within the flow, the approach proposed in [8] was used. First, the experimental “x–t” bodytrajectory is approximated (in [11], for example, a cubic polynomial is recommended); then, by twice differ-entiating the resulting displacement function, the body velocity and acceleration necessary for calculatingCx from the equation of motion can be found.

433

434 BOIKO, POPLAVSKII

This approach is certainly valid for setting up an experiment but where the processing of the data isconcerned several questions remain: (1) why is the trajectory approximated by a polynomial and not the“exact” displacement function that can be found from the equation of motion? and (2) if a polynomial, thenwhy a cubic one which, being twice differentiated, restricts the set of possible acceleration functions to alinear dependence and introduces an error into Cx?

In these circumstances, we need to return to the problem of a freely-accelerating body in the flow behinda shock wave in order to develop efficient algorithms for finding its aerodynamic drag and to describe thedynamics of hard and liquid particles in such systems.

1. AERODYNAMIC DRAG OF A FREELY-ACCELERATING BODYIN THE FLOW BEHIND A SHOCK WAVE

By definition, the aerodynamic drag coefficient of a body Cx (for a sphere CD) is the ratio of the aero-dynamic force Fa to the force that could be produced by a pressure equal to the dynamic head acting onthe midsection area s. If the body mass m is known, instead of the measured force it is possible to use theacceleration A of a free body in the flow:

Cx =Fa

S(0.5ρu2)=

2mASρu2 =

2mSρ

1λ

, λ =u2

A. (1.1)

This approach is possible due to the use of multiple-frame shadow recording with a laser stroboscopiclight source [13], which enables the “x–t particle trajectory” to be obtained with high spatio-temporal resolu-tion. Then, by double numerical differentiation of the initial data array we can formally find the accelerationA needed for the calculations in (1.1). However, a detailed analysis of this approach made in [14] showedthat the real error of the experiment makes it impossible to process the initial data in this way due to thesignificant error in the second derivative found. Therefore, in [14] this problem was solved using specialmathematical tools on the basis of smoothing regularizing splines adapted to the noise level.

However, a simpler approach based on finding the relaxation parameter λ in expression (1.1) can beproposed. In this method, the calculations are based on the possibility of approximating the experimentaldata on the particle position Si at moments ti by a function S(t), which can be obtained from the equation ofmotion in a certain approximation. This approach is also possible for the derived array of data on the bodyvelocity Vi = (Si+1 − Si)/(ti+1 − ti). This makes it possible to determine the quantity λ (and from thisquantity Cx) without finding the acceleration, although the main relation (1.1) is also used. For several of themost typical physical formulations, the analytical form of the displacement S(t) and velocity V (t) functionswill be considered below.

Early stage of velocity relaxation. For high Reynolds numbers the equation of motion of a free bodysuddenly exposed to a flow has the form:

mdVdt

= Cxsρ(u − V )2

2, V = 0, t = 0. (1.2)

Here, m and V are the body mass and velocity, and ρ and u are the gas density and velocity. On theassumption that Cx is constant in the early stage of gas relaxation, collecting all the constant parameters intoone parameter (in brackets λD for a sphere)

λ =2m

Cxsρ

(λD =

43

ρp

ρd

CD

)(1.3)

with the dimension of length, we can reduce the equation of motion to the form:

dVdt

=1λ

(u − V )2, V = 0, t = 0. (1.4)

FLUID DYNAMICS Vol. 42 No. 3 2007

PARTICLE AND DROP DYNAMICS 435

Equation (1.4) is satisfied by the particle velocity function with the relaxation parameter τ (in bracketsτD for a sphere)

V (t) = u

[1 −

(1 +

tτ

)−1], τ =

λu

,

(τD =

43

ρp

ρd

uCD

). (1.5)

For the dimensionless time θ = t/τ , integration of the velocity function (1.5) gives the displacementfunction

S(θ) = λ [θ − ln(1 + θ)], (1.6)

whereas differentiation yields the acceleration function

A(θ) =uτ

1(1 + θ)2 . (1.7)

These are functions used for approximating the experimental data on particle dynamics in shock waves.The quantities λ and τ can be found from a best approximation condition, for example, using the leastsquares method [15]. Then, from (1.3)

Cx =2m

λρS, CD =

43

ρp

ρdλ

. (1.8)

Definition (1.8) formally coincides with (1.1); however, when obtained together with motion integrals(1.5)–(1.7), it makes it possible to find λ not in terms of the body acceleration but as a dimensional coef-ficient in the displacement function (1.6). In accordance with the least squares method, λ is a root of theequation

∑i[Si − S(ti; u, λ )]

∂S(ti; u, λ )∂λ

= 0. (1.9)

In the presence of other unknown parameters (for example, u), we must solve a system of equations oftype (1.9) with partial derivatives with respect to the corresponding parameters (u). For a function S(t; u, λ )of form (1.6) this yields the system of equations

∑i

{Si −

[uti − λ ln

(1 +

uλ

ti

)]}[ti

uλ

(1 +

uλ

ti

)−1

− ln

(1 +

uλ

ti

)]= 0,

∑i

{Si −

[uti − λ ln

(1 +

uλ

ti

)]}ut2

i

λ + uti= 0.

(1.10)

We note that, as a rule, Eq. (1.9) or system of equations (1.10) has more than one solution. In the roughestapproximation, the “correct” solution can be chosen from the physical conditions of the experiment. How-ever, there is a more general, completely formalized approach that makes it possible to make a preliminaryestimate of the solution domain. This approach is described in the next section.

Approximate form of the displacement function. For quick engineering calculations and a preliminaryestimation of the solution domain for Eq. (1.9) or system (1.10), an approximate analytical form of theparameter λ was found. This makes it possible to calculate the coefficient Cx directly from the data arrays(Si, ti) without solving the transcendental equation (1.9) or a system of equations of type (1.10).

Since we are concerned with the early stage of velocity relaxation (θ = t/τ < 1), the expansion ln(1 +θ) ∼= θ − 1/2θ2 + 1/3θ3 − . . . is valid. Then the displacement function (1.6) can be replaced by itsapproximation

S(t; u, τ) ∼= 12

uτ

t2 − 13

uτ2 t3 +

14

uτ3 t4 − . . . . (1.11)

Introducing the new parameters x1 = u/2τ , x2 = u/3τ2, . . . and substituting in (1.9) the new function

S∗(t; x1, x2, . . . , xm) = x1t2 − x2t3 + · · ·± xmtm+1 (1.12)



(“+” for odd and “−” for even m; m < n is the number of points in the array), we obtain a system of m linearequations. This is the problem of finding the coefficients of polynomial (1.2) for which this polynomialapproximates the array of experimental data (Si, ti) with the least mean-square deviation. Since the generalform is cumbersome, we will consider only the case of two expansion terms

x1

n

∑i

t4i − x2

n

∑i

t5i =

n

∑i

sit2i ,

x1

n

∑i

t5i − x2

n

∑i

t6i =

n

∑i

sit3i ,

x1 =c + bx2

a, x2 =

a f − bcb2 − ae

,

a =n

∑i

t4i , b =

n

∑i

t5i , c =

n

∑i

sit2i , e =

n

∑i

t6i , f =

n

∑i

sit3i .

Then from (1.11) and (1.12)

τ∗ =23

x1

x2, λ ∗ =

89

x31

x22

. (1.13)

The approximate form of the displacement function (1.11) shows that approximating the experimentaldata on particle displacement in a shock wave by a polynomial of general form is physically incorrect.

2. PARTICLE DYNAMICS FOR VARIABLE AERODYNAMIC DRAG

We note that for high Reynolds numbers the effect of nonstationarity on the particle motion has beeninsufficiently studied. Only one case of an analytical solution of the equation of motion of a particle witha variable drag coefficient is known [15]. This is the problem of a particle decelerating in a stationary gaswith a drag coefficient in the form of a function of Re of the Klyachko formula type

mdVdt

= −CDSρV 2

2, CD =

24Re

(1 + a

3√

Re2)

with the initial condition V = V0 at t = 0. The solution of this equation V = V (t; Re0) contains the initialRe value, but it is known from experiment [13] that during velocity relaxation with transition through thetransonic flow regime a more important parameter affecting the particle drag is the relative Mach number.We will now consider the possibility of taking this parameter into account.

“Sonic” Reynolds number. In engineering calculations and theoretical models of two-phase flows it iscommon to assume all the particles to be spherical with a smooth surface. This is in part due to the fact that,although the sphere is a high-drag body, empirical data on its aerodynamic drag are available for a fairlywide range of physical parameters. For describing these data there are several empirical formulas, one ofwhich [13] is reproduced below:

CD =[

1 + exp

(− 0.43

M4.63

)](0.4 +

24Re

+4√Re

). (2.1)

After replacing Re by a quantity proportional to M

Re =u − V

CC dρ

μ= Mkre,

relation (2.1) becomes a function of a single variable, the relative Mach number M



Fig. 1. Aerodynamic drag coefficient of a sphere CD vs. relative Mach number M during velocity relaxation: CD calculatedfrom (2.1) (1–5) for fixed Re ∼ 104, 5×103, 3.333×103, 2.5×103, and 2×103 corresponding to the velocity relaxationtimes 0, τ , 2τ , etc.; CD calculated from (2.2) for the initial Mach number M = 1.8 (6) and Re = 104 (point I); II, III, IV,and V successive values of CD at moments τ , 2τ , etc. with decrease in the relative numbers M and Re during velocityrelaxation (7).

Si/λ

ti/τ

Fig. 2. Comparison between the experimental data and the displacement functions: initial number M ≈ 0.5–0.6 (1), (2);1.05–1.16 (3–6); 1.3–1.4 (7), (8); displacement function (6) for Cx = const (I); (15) for a linearly decreasing Cx (II);S(θ )/λ ∼= θ 2/2 − θ 3/3 (III); S(θ )/λ ∼= θ 2/2 − θ 3/3 + θ 4/4 (IV).

CD =[

1 + exp

(− 0.43

M4.63

)](0.4 +

24Mkre

+4√

Mkre

), (2.2)

where kre is a quantity constant for each specific problem and defined as the Reynolds number but in terms ofthe speed of sound: kre = cdρ/μ . Using the “sonic” Reynolds number kre, any representation of the drag as afunction of M and Re of type (2.1) can be transformed into a convenient form of type (2.2), in which a singlevariable, the Mach number, is present. Figure 1 graphically depicts the dependence of the aerodynamic dragcoefficient determined from (2.1) and (2.2) for a sphere on the Mach number range typical of shock waves.From the graph it can be seen that our previous simplifying assumption that the drag is constant may bereasonable for the supersonic acceleration segment, and not just for the early relaxation stage.

Linear fall in the aerodynamic drag. From Fig. 1, it follows that in the subsonic region, on the interval0.6 < M < 1, during the acceleration of a particle an almost linear fall in the drag can be observed. In orderto integrate the equation of motion over this range of M, it is convenient to reduce the drag function to alinear dependence on the particle velocity V : Cx = C0(1 −V/u), where C0 is the initial drag. Then, in termsof the dimensionless velocity Ψ = V/u and dimensionless time θ = t/τ , the equation of motion (1.2) canbe reduced to the form dΨ/dθ = (1 − Ψ)3, and its solution, the dimensionless function of the velocity

Ψ(θ) = 1 − 1√1 + 2θ

, (2.3)

yields, after integration and differentiation,



Fig. 3. Shadow image of drop deformation stages in the flow behind a shock wave: 50, 80, 110 μs (2), (3), (4); 410 μs (1).

S(θ) = λ (1 + θ −√

2θ + 1), (2.4)

A(θ) =uτ

1√(1 + 2θ)3

. (2.5)

Functions (2.3)–(2.5) can be used for approximating experimental data, as for Cx = const, but with certainrestrictions indicated below. The quantitative difference between the displacement functions (1.6) and (2.4)grows nonlinearly with time and at θ ≈ 0.5 is of the order of 10%. The λ value found from these relationscorresponds to the initial aerodynamic drag C0. However, experiments have shown that function (1.6) ismost suitable for approximating the displacement data in all regimes up to θ < 0.7. This follows from Fig. 2which gives, in dimensionless form, the time dependence of the particle displacement from 8 experiments.

It can be seen that, like function (2.4), polynomials of less than fifth degree can be used for approximatingthe data on particle displacement in a shock wave only for θ < 0.2. The relaxation parameters u, λ , andτ are universal and, as follows from (1.7) and (2.5), their combination u/τ = u2/λ is the initial particleacceleration A0. This makes it possible also to use trajectory measurements of particle dynamics for findingthe gas velocity, since it follows from the equation of motion that u =

√λA0 and for an arbitrary moment

u = V (t) +√

λA(t).

3. DROP ACCELERATION DYNAMICS IN THE FLOW BEHIND A SHOCK WAVE

Previous experiments have shown that the aerodynamic drag of a drop is more than twice that of a hardspherical particle of the same diameter d0. A more detailed study showed that the inhomogeneous pressuredistribution over the surface of a liquid drop in a gas flow leads to drop deformation with a significant (byseveral times) increase in the midsection area [16]. It seems reasonable to assume that this apparent sharpincrease in the drag results from the increase in the aerodynamic force due to the increase in the midsectionarea, whereas the aerodynamic drag coefficient itself as a characteristic of the drop shape differs only slightlyfrom that of a sphere, since the flow around the latter is also separated.



Fig. 4. Dynamics of drop displacement and development of ablation in accordance with the separation breakup mechanismfor an alcohol drop of diameter d ∼ 2 mm in a flow with velocity u = 157 m/s and We = 5×103: successive images of theprocess at 30 μs intervals (1–14).

In order to verify this assumption, we carried out experiments in which the acceleration dynamics anddrop deformation were investigated in detail in the early stage of interaction with the shock wave, whichprecedes the onset of ablation. It was established that over a wide range of Weber numbers 200 < We < 5000the midsection size increases up to a limiting value 2d0 by the moment of ablation onset (frame No. 3 inFig. 3). Then the windward side of the drop flattens and the flow separation direction changes to radial, asin the flow past a disk. As a result, the microdrop veil, following the separated streamlines in the radialdirection, is recorded by the shadow method as an increase in the transverse drop dimension. However,as can be seen from Fig. 3 (frame No. 13), the size of the whole drop core remains constant and equal toapproximately 2d0.

In view of these observations, we propose using the technique of quick engineering estimates of thevelocity-relaxation parameters based on the equation of motion and on the assumption of a certain givenmidsection increase rate, as previously proposed for hard particles.

Drop dynamics in the case of a linear increase in the transverse dimension. If we assume that in thisstage, in the first approximation, the transverse dimension of a drop grows almost linearly, then, using anempirical quantity t2d , the time in which the limiting deformation is reached, we can introduce the time-dependent transverse drop dimension θ2d = t/t2d by means of the formula d = d0(1 + θ2d).

In this case, the equation of motion of a particle in the flow behind the shock wave

πd30

6ρp

dVdt

=πd2

0

4(1 + θ2d)

2CDρ(u − V )2

2



Fig. 5. Drop velocity: calculated from (3.2) and (1.5) (1), (3); experiment shown in Fig. 4 (2).

has the exact solutionVu

= 1 −{

1 +13

t2d

τ[(1 + θ2d)

3 − 1]}−1

. (3.1)

Here, as in the case of a hard sphere, τ is a relaxation time constant calculated as in (1.5) for the initialdrop diameter d0.

Estimation of the aerodynamic drag of a drop. Another way of writing solution (3.1) makes it possibleto reduce this solution to the above-mentioned variable drag. We rewrite (3.1) in the form:

Vu

= 1 −{

1 + θ[

1 + θ2d +13(θ2d)2

]}−1

(3.2)

and compare the dependence obtained with the dimensionless velocity of a hard particle (1.5). It can be seenthat the drop velocity differs from the velocity of the hard particle with respect to a nonlinear multiplier fortime t or, what amounts to the same thing, a divisor for the relaxation time constant τ . Then, introducing thenew relaxation parameter τ∗

τ∗ =τ

1 + θ2d + 1/3θ22d

=43

ρP

ρd0

uCx

[1 + θ2d +

13

θ22d

]−1

=43

ρp

ρd0

uC∗ , (3.3)

in which the role of aerodynamic drag coefficient is played by the time-variable quantity

C∗ = Cx

[1 + θ2d +

13

θ22d

], (3.4)

we obtain the velocity function in the usual form:

V (t) = u

[1 −

(1 +

tτ∗

)−1]

.

From (3.4) it follows that by the moment t2d the parameter C∗ introduced as an effective drop drag hasreached the value C∗ ≈ 2.3Cx, which is, on the whole, in agreement with that obtained as a result of pro-cessing the experimental data in the present study and in [6, 17]. In Fig. 4 a typical series of shadow patternsof a drop interacting with a transmitted shock wave is presented, and in Fig. 5 the results of processing thedrop displacement data are shown.

In this experiment the separation mechanism of drop breakup was modeled at a high We = 5×103. Analcohol drop 2 mm in diameter was exposed to a transmitted shock wave (its front can be seen in frames 1and 2) which generated an air flow with a velocity u = 157 m/s and a density ρ = 2 kg/m3. The experimentalcurve 2 in Fig. 5, obtained by numerical differentiation of the drop displacement data, shows a sharp decreasein acceleration with the onset of intense ablation. The approximation of the data by a quadratic function (the



first term of expansion (1.11)) gives a coefficient C∗ = 2.33. The approximation was made from the dataobtained before intense ablation.

Thus, even in the simplest approximation, taking the increase in the transverse drop dimension intoaccount makes it possible to estimate the effective drag of a drop from (3.4) and satisfactorily to describe itsdynamics by means of (3.2) with account for deformation.

Summary. The determination of the aerodynamic drag Cx of a free body in terms of the instantaneous ac-celeration in the flow is introduced as the most appropriate measure of the aerodynamic force. A method offinding Cx from the body velocity-relaxation parameters behind the shock wave is proposed. An analyticalform of the kinematic functions is found in certain typical cases of flow interaction with a free body and fordrops with account for their deformation. The proposed displacement, velocity and acceleration functionsare confirmed by experiments with hard particles and drops in the flow behind a shock wave.

The work was supported financially by the Russian Foundation for Basic Research (projectNos. 98-01-00722, 01-01-00776, 04-01-00235, and 06-01-00299).

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particle and drop dynamics in the flow behind a shock wave

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