Head-on collisions of dense granular jets

Jake Ellowitz∗

The James Franck Institute and the Department of Physics, University of Chicago, Chicago, IL 60637(Dated: August 8, 2018)

When a dense stream of dry, non-cohesive grains hits a fixed target, a collimated sheet is ejectedfrom the impact region, very similar to what happens for a stream of water. In this study, as acontinuation of the investigation why such remarkably different incident fluids produce such similarejecta, we use discrete particle simulations to collide two unequal-width granular jets head-on intwo dimensions. In addition to the familiar coherent ejecta, we observe that the impact producesa far less familiar quasi-steady-state corresponding to a uniformly translating free surface and flowfield. Upon repeating such impacts with multiple continuum fluid simulations, we show that thistranslational speed is controlled only by the total energy dissipation rate to the power 1.5, and isindependent of the details of the jet composition. Our findings, together with those from impactsagainst fixed targets, challenge the principle of scattering in which material composition is inferredfrom observing the ejecta produced during impact.

I. INTRODUCTION

If we collide two ideal fluid streams head-on in two di-mensions, one larger than the other but both the samespeed, we would observe a steady state with two equallysized and inclined streams leaving the impact region.This is because the larger jet has more incident mo-mentum than the smaller, and these ejecta are how thesystem accommodates for the momentum excess. Thissteady-state is achieved because ideal fluids are time-reversible and is thus the reverse of two equal-width jetsincident at an angle. Due to symmetry, this reversedcase is in turn equivalent to a jet impinging onto an infi-nite wall, a situation that surely reaches a steady-state.Hence, head-on collisions of ideal-fluids reach a steady-state where the ejecta compensate for the nonzero inci-dent momentum.

Since this stationary steady-state impact owes its exis-tence to the fact that the flow field is entirely reversible,breaking reversibility by including dissipation, no matterhow small, should destroy the stationary solution. Intu-ition would then suggest that the head-on impact of dis-sipative jets should thus produce collective motion in thecentral impact region. Numerical simulations presentedin the current study verify this expectation by demon-strating that a wide variety of dissipational jet impactsproduce quasi-steady-state flows whose entire flow fieldand free surface translate at a constant drift speed. Thisdrift speed is shown to increase as power law relative tothe dissipation with exponent 1.5. More surprisingly, wefind that fluids with widely varying rheologies all collapseonto the same power law: the behavior of this bulk mo-tion depends only on the amount of dissipation in theimpact, and is independent of the dissipation mechanismor the constitutive material of the incident jets.

Our findings are at the intersection of two previouslystudied but disparate jet impact systems: that of directly

∗ ellowitz@uchicago.edu

incident turbulent Newtonian streams and that of jet im-pact against fixed targets. In the case of the impactof turbulent jets, previous experiments identified that infixed-length cavities, the impact stagnation point shiftstoward the nozzle with lower incident momentum flux,whose offset depends on the incident momentum fluxesand nozzle separation [1–4]. These studies were focusedon the stagnation point shifts, and reach steady stateflows in the lab frame due to the fixed nozzle separa-tion. Our study is different because we investigate theoffset mechanism and transient directly and are able tocompletely remove the nozzle effects.

Regarding impacts against fixed targets, previousworkers showed granular and high speed liquid jets as-toundingly both produce quite similar coherent ejectasheets [5–7]. Because the flow from a high speed liquidimpinging a fixed target is similar to a perfect fluid, thisled to a discussion of whether or not a granular jet im-pact against a target behaves like a perfect fluid [7–11].Despite the similar ejecta from the target impacts, it waslater shown that granular jet impact against a target pro-duces a large, stagnant dead-zone, and that furthermoreinserting large dead-zone-like structures into perfect flu-ids barely affected the ejecta [12]. This dead zone, whichis reminiscent of those found in dilute flows [13, 14], isa stark contrast to the stagnation point in perfect fluidimpacts, yet the ejecta remained generic in its presence,showing that the jet composition and internal structurehave little effect on the ejecta when the impact is coldand incompressible.

Directly colliding fluid streams furthermore parallelmultiple applications in manufacturing processes. Op-posed jet micro reactors have been utilized to facilitatecombustion and to produce highly controlled chemicalreactions for polymer processing, nanoparticles, and thecareful synthesis of organic compounds in the pharma-ceutical industry [2, 3, 15–17]. Furthermore, opposingjets are employed in drying particles and fluid absorp-tion [18, 19] and mineral extraction [20]. The presentfindings are relevant to the mechanics by which the im-pact center shifts between nozzles in these diverse indus-

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FIG. 1. (Color online) Two-dimensional head-on impact of two jets produces a steady-state flow with collimated ejecta and asteadily drifting flow configuration. (a) In two dimensions, A jet of width 2R, shown in blue, is incident against a jet of with2kR, shown in red. Both jets are initially at 81% packing fraction and traveling toward each other at the same speed U0. Here,R = 100RG, where RG is the grain width, and k = 0.465. (b) Snapshot at t = 2R/U0 after the two jets directly collide. Herewe indicate the coordinates x and y whose origin is located at the point of initial contact. (c) Snapshot at t = 10R/U0, wherethe central saddle point, shown with a dot, is drifting toward the smaller oncoming jet with drift speed Ud. Here, Ud = 0.052U0.Additionally, the impact produces 2 collimated ejecta streams with widths 2Rout and ejecta velocities with magnitude Uout

exiting at an angle Ψ0 from the horizontal. Note Ψ0 is not quite the same as the spatial angle formed between the ejecta jetand the horizontal due to the fact that the ejecta streams are drifting alongside the saddle point.

trial examples, in particular highlighting the effects ofdissipation which have not yet been considered deeply,and are explored in this paper.

II. METHODS

In this paper, we consider the two-dimensional head-on collision of two dissipative jets of different sizes: onejet of width 2R and a smaller jet of width 2kR, where0 ≤ k ≤ 1. The two jets are incident such that theircenters align. Their initial velocities are both U0 in mag-nitude, and are opposite in direction so that the twojets collide directly (Fig. 1 (a)). To simulate these im-pacts we used two separate numerical approaches: thefirst being a discrete particle simulation of dense gran-ular jets [11, 12, 21, 22], and the second being the con-tinuum dynamics of colliding jets using the multi-phase,incompressible fluid flow solver gerris [23–26].

The discrete particle simulations consist of simulat-ing the motion and collisions of hundreds of thousandsof rigid spheres with specified coefficients of restitutionand friction. We conduct our discrete particle simula-tions using rigid grains, in line with experiments whichused copper and glass beads with negligible deformationduring impact [7, 12]. The discrete particle simulationsutilize a hybrid timestep- and event-driven method de-signed to handle dense granular flow efficiently and accu-rately [21, 22]. This method has been used in many recentstudies of granular jet impact, and has also been quanti-tatively validated against experimental granular jet im-pact flow configurations [12].

We prepare our discrete particle jets at 81% packing

fraction, and because our simulations are in two dimen-sions, we use polydisperse grains to prevent spurious crys-tallization. The grains we use have radii uniformly dis-tributed from 0.8RG to 1.2RG, where RG is the averagegrain size. In this paper, we use a grain size RG = R/100.The jet-width-ratio k, restitution and friction coefficientsare varied and the effects on the dynamics of head-on col-lision are explored.

The continuum simulations of colliding streams areconducted for liquid jet impacts, where the shear stress isproportional to the shear rate, for frictional fluid modelsof granular jets, where the shear stress is proportional tothe pressure, and a shear thickening fluid whose effectiveviscosity increases with the shear rate.

Continuum solutions in gerris utilize an incompress-ible volume of fluid method to solve multi-phase flowwhile providing significant flexibility for specification ofthe deviatoric stress tensor [23–25]. Ideally, we are solv-ing Cauchy’s momentum equations with incompressibil-ity

ρDu

Dt= ∇ · σ, ∇ · u = 0 (1)

where D/Dt = ∂t +u ·∇ is the advective derivative, ρ isthe jet density, u is the fluid velocity field, and σ is thestress tensor. The initial conditions are

u =

{U0x x < 0−U0x x > 0

(2)

with an initial jet free surface corresponding to the twoimpinging jets right when they make first contact. Theboundary condition on the free surface corresponds tozero-stress with σ · n = 0, where n is the surface normal.

3

The boundary conditions within the jet are u = U0x forx→ −∞ and u = −U0x for x→∞.

Our stress tensor is in general given by

σ = −p1 + τ (3)

where 1 is the identity matrix, p is the pressure and τis the deviatoric stress. In the cases presented in thispaper, we consider three deviatoric stress tensors: thatof Newtonian liquid flow, where τLiq = ηγ, that of africtional fluid, where τFF = µpγ/|γ|, and that of a shearthickening fluid τST = κ|γ|1/2γ. Here, γ = ∇u + ∇uT

is the rate of strain tensor, and |γ| =√γ : γ/2.

For the Newtonian rheology, η is the dynamic viscosity,for the frictional fluid, µ is the dynamic frictional coef-ficient, and for the shear thickening fluid κ is the flowconsistency index. Speaking in particular to the fric-tional fluid, this frictional fluid model has been shown toreproduce simulations and experiments of dense granu-lar jet impact, and is therefore a natural choice for thisstudy [11, 12]. We additionally considered the Newto-nian and shear thickening fluids because their stress ten-sors are significantly different from dry granular flow.These differences are key in our results showing thatwidely varying materials produce similar head-on impactresponses.

Even though our idealized problem consists of free-surface jets of a single fluid type colliding in free-space,the volume of fluid method that gerris utilizes to solvethe equations based on the boundary and initial condi-tions above requires a surrounding fluid occupying thefree space outside the jets in our idealized problem. Inorder to minimize the effects of the surrounding fluid, weset its density to ρsurrounding = ρ/250, and we set its devi-atoric stress to be Newtonian with viscosity ηsurrounding =10−3RU0ρInter. We conducted our simulations with re-finement 26 grid points across the jet width.

We have extensively varied the grid refinement, thesurrounding fluid viscosity, and surrounding fluid densityto ensure that the above parameters are in a regime wherethe effects of the surrounding fluid and grid discretizationare minimal. Specifically, using a frictional fluid withµ = 0.3 as a representative impact, we found that therewas a variation in the drift speed of roughly 0.1%, and avariation of the energy dissipation rate of roughly 2%.

III. RESULTS

To briefly summarize our results, using discrete par-ticle simulations, we observe the resulting flow from thecollision of two unequal-width dense jets composed ofdry, rigid grains in two dimensions. We reveal that afteran initial transient, the impact produces a quasi-steadystate corresponding to a uniformly translating flow fieldand free surface at a fixed drift speed. This steady driftspeed increases as the size of the larger jet with respectto the smaller jet increases, and occurs alongside momen-tum being absorbed into the drifting impact center. We

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FIG. 2. (Color online) Following a short transient, the centralsaddle point drifts at a relatively steady speed Ud. Normalized(a) saddle position and (b) corresponding drift speed versustime for collision between two jets at k = 0.465 (red) andk = 0.25 (blue). The impact initially occurs at xsaddle = 0.Following a short transient of duration ∼ 4R/U0, the saddlepoint translates linearly in time. The direct measurement ofthe drift speed obtained by differentiating the saddle position.The drift speed starts high in the transient state and thenfluctuates about the mean drift speed after steady drift hasbeen reached. The drift speeds above are Ud/U0 = 0.052 fork = 0.465 and Ud/U0 = 0.100 for k = 0.25.

then consider multiple continuum jet impacts and findthat the drift behavior was extremely robust, to the ex-tent that the total energy dissipation controls the driftspeed regardless of the significantly different dissipationmechanisms present in our continuum and discrete parti-cle simulations. This dependence manifests itself via thedrift speed robustly scaling versus the dissipation ratewith a 1.5 exponent.

A. Steady saddle drift

To begin, we consider two unequal-width granular jetswith k = 0.465, and whose grains have coefficient of resti-tution 0.9 and coefficient of friction 0.2 (see Fig. 1(a)).Unless otherwise stated, these are the values we use inour representative simulation examples in our discreteparticle jet impacts. In Fig. 1 we show early snapshots(a) just before impact, (b) just after impact, and (c) afterthe impact flow configuration has been well established.

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t = 16R/U0t = 32R/U0 −1

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(b)

t = R/2U0t = 2R/U0t = 8R/U0

t = 16R/U0t = 32R/U0

FIG. 3. (Color online) The entire flow configuration arising from the head-on collision of unequal jets drifts steadily in time.(a) As viewed in the fixed lab reference frame, we show horizontal velocity ux/U0 versus position x/R along the line runningthrough the centers of the jets defined by y = 0. Velocities are shown at a variety of times, ranging from very soon afterimpact (blue) to long after impact (red). The velocity fields at these different times does not coincide when viewed from the labframe. (b) Horizontal velocity profile x/R along the center line as viewed in the comoving reference frame which is translatingsteadily at the drift speed Ud along with the drifting saddle. All of the profiles from (a) are included, where all of the later(t > 2R/U0) snapshots directly overlap in the comoving frame. Hence, excluding the profiles obtained in the initial transient,the flow configuration translates at a constant drift speed Ud, remaining preserved as it translates.

The collision produces some familiar behavior, namelythat two collimated ejecta streams are produced as inthe case of impact against a fixed target [7, 11, 12, 27].The collimated ejecta are traveling with velocity Uout atan angle Ψ0 relative to the horizontal.

However, we also find notably unfamiliar behavior inthis system: the steady collision of the two incidentstreams produces a steadily translating mode in whichthe impact center and ejecta streams drift at a constantspeed Ud. As a result of this drift velocity, the spatial an-gle traced by the ejecta stream relative to the horizontal(Fig. 1(c)) is slightly different from Ψ0, the ejecta veloc-ity angle. In contrast, the more studied case of impactagainst a fixed target, there is always a fixed steady statewhere the ejecta streamlines are equal to their pathlines.But, in general, this is not true for head-on collisions oftwo jets, since they obtain a quasi-steady-state where theflow field and free surface steadily translate.

Let us define the saddle position xsaddle as the centralpoint in the impact at the interface where the grains ofthe large jet meet the grains of the small jet. In Fig. 1(c),this is where the blue and red jets interface along the liney = 0 and moving toward the smaller incident jet. Thesaddle center is initially situated at x = 0 at the pointwhere the two jets first make contact (Fig. 1(b)). Becausethe granular jets are dense, and therefore cold [11, 12],there is negligible diffusion and mixing between the jetsand their interface is clearly defined. In Fig. 2 (a) wetrack the saddle position over time, and find that fol-lowing a short transient, the two jets produce a steadycentral motion where the saddle position moves linearlyin time.

We instantaneously measure the drift speed over time

by computing the time derivative of the saddle positionby fitting a slope to 10 measurements of xsaddle over atime frame R/U0. These drift speed measurements areshown in Fig. 2(b). In Fig. 2(a) ,we graph only one xsaddle

for 10 measurements conducted in from our simulationsso that each xsaddle corresponds to one instantaneousUd/U0 in Fig. 2(b). This ensures that each Ud is indepen-dently measured from all other Ud shown. In the discreteparticle jets, we see that there are slight fluctuations inthe rate of motion of the saddle point about the averagedrift speed on the order of U0/100, with an average driftspeed Ud/U0 = 0.052 to the right when k = 0.465 andUd/U0 = 0.100 when k = 0.25, where the large jet ispushing the small jet. Thus, smaller k produces a largerUd.

As we next show, not only does the saddle drift steadilyin time, but the entire flow field drifts along with thesaddle steadily in time. We measure the flow field alongthe jet centers by spatially binning grains into squareboxes of linear dimension 4.5RG along the center of thejet at y = 0 and averaging the particle speeds within eachbin. A typical bin contains 5 particles.

In Fig. 3(a), we show the instantaneous horizontal ve-locity field ux along the center line defined at variouspoints in time. We see the velocity in the large left jet(x . −4R) is U0, and the velocity at large x in the smallerjet (x & 4R) is −U0. The velocity fields at differentpoints in time as viewed from this fixed lab frame appeardistinct from one another.

We now enter the comoving frame whose origin is mov-ing to the right at the drift velocity Ud obtained fromtracking the saddle point motion as in Fig. 2. Here wefind a dramatically simpler picture than in the lab frame.

5

Let us denote coordinates in the comoving frame with abar above the variable, notation that will be continuedthrough the rest of this paper. The comoving frame co-ordinates in this notation are

x = x− Udt (4)

y = y (5)

ux = ux − Ud (6)

uy = Uy (7)

In the comoving frame, neglecting the short transient ofduration ∼ 4R/U0, the velocity fields are indistinguish-able. The clear conclusion here is that when two unequal-width granular jets collide, they produce a quasi-steady-state in which the entire flow field uniformly drifts in timeat a fixed drift speed Ud in the lab frame. In the comovingframe, the impact produces a steady-state, where partialtime derivatives of the velocity field or interface locations(∂t) vanish, and the streamlines match the pathlines, re-sulting in the spatial angle traced by the ejecta relativeto the horizontal to match the velocity ejecta angle.

Having established that a central drift emerges duringimpact of two head-on granular jets of unequal widths,we next explore the effects of jet width, and will profilethe flow configuration as k is varied.

B. Characterization of post-impact flowconfigurations

For the discrete particle jets with coefficients of resti-tution and friction of 0.9 and 0.2, we varied the jet-width-ratio k and observed the resulting impact configurationin the lab frame to provide a basic understanding of head-on jet impact response. This also allows us to verify thatthe simulations are producing sensible outcomes, for ex-ample, to verify that there is no central drift when theincident jets have equal widths.

The impact configuration is characterized by the driftspeed Ud, the ejecta speed Uout, and the ejecta angleΨ0. As later shown, the ejecta width Rout is determineduniquely from these parameters by mass conservation(Eq. (17)).

In Fig. 4(a) we see how the drift speed Ud varies as kis varied. When the jets are the same width, there is nodrift, as one would expect due to symmetry. When thesmall jet is 0.465 times the width of the large jet, as men-tioned earlier, there is a drift speed of Ud = 0.052U0, andwhen the small jet is 0.125 times the width of the largejet, the drift speed increases to Ud = 0.17U0. For anyk < 1 we found the drift emerge; there is not a thresh-old or yield k above which there is no drift and belowwhich it emerges. Rather the drift speed as a functionof k appears simply as an emergent result of unequal jetwidths.

In Fig. 4(b) we show the dependence of the ejecta speedUout on k. Here we see much less variation than we didfor the drift speed. Evidently, at fixed jet parameters, the

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FIG. 4. (Color online) As the jets approach equal widths,the drift speed continuously vanishes. (a) In the lab frame,we show the saddle point drift speed Ud/U0 versus the ratio ofthe incident jet widths k while keeping R = 100RG and therestitution and friction coefficients 0.9 and 0.2, respectivelyin the discrete particle simulation (red points). We further-more include the outcome from our control volume analysisfor comparison, where we see the analysis perform quite welldespite its simplicity. We see that Ud decreases monotonicallyas k increases, and eventually Ud vanishes when the jets areequal widths at k = 1. (b) Ejecta velocity in the lab (red)and comoving (blue) frames. Note that the ejecta velocityappears relatively constant for all k, especially in the driftingframe. For the grains used here, that constant is approxi-mately B = 0.73. (c) To complete the characterization of theflow configuration resulting from the collision we include theejecta angle in both the lab and comoving frames.

ejecta speed does not change much even though the jetsizes are changed dramatically. In the cases presented,the smaller jet varied in width by a factor of 8 while Uout

varies by just 13% in the lab frame.

Furthermore, we provide the ejecta angle Ψ0 as the fi-nal ingredient to characterize the impact flow configura-tion. We see that Ψ0 decreases as k decreases, indicatingthat when the larger jet is much wider than the smallerone, the ejecta are more acute and are less deflected thanthe directly vertical jets obtained at the maximum deflec-tion of 90◦ when k = 1.

6

C. Bulk momentum absorption

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FIG. 5. (Color online) Momentum balance incorporatesmomentum absorption into the drifting saddle. Here we showthe momentum absorbed by the drifting saddle Psaddle as afraction of the net incident momentum Pin as we vary thejet-width-ratio k. We see that the drifting saddle absorbs aconsiderable fraction of the net incident momentum. In thediscrete particle jets used here, at least 15% of the net incidentmomentum is captured by the drifting saddle.

We now show that the drifting mode absorbs a signifi-cant fraction of net incident momentum. This absorptioncorresponds to a new term in the momentum balance forhead-on jet collisions. Let us begin by defining the mo-mentum fluxes in the three parts of the collision: Pin isthe incident momentum flux, Pout is the momentum fluxof the ejecta, and Psaddle is their difference:

Pin − Pout = Psaddle (8)

Importantly, because of the y-symmetry of our impactsystem, P in all cases above refers only to the horizontalmomentum along the x-axis.

In the absence of drift, such as in ideal fluid impact,we know that the ejecta balance the momentum equa-tion and Pout = Pin. However, in the case of granularjets, or drifting jets in general, we observe Psaddle enter-ing the picture: by measuring the net incident momen-tum Pin and the net momentum in the ejecta Pout inour simulations, we find that their difference is nonzero.Namely, there is momentum being deposited into the sad-dle Psaddle.

In Fig. 5 we show the fraction of the initial momentumthat is absorbed by the saddle Psaddle/Pin as a functionof the jet width ratio k. We see that indeed a significantfraction of the new incident momentum is being absorbedby the saddle, in this case roughly 15% for the range ofk presented.

This absorption is due to the fact that the flow fieldis uniformly translating, which in the lab frame appearssimply as an elongation of the larger jet and a contractionof the smaller jet. Hence, in a unit time, the change inmomentum due to the large jet elongation is 2ρRU0Ud,and the change due to the contraction of the smaller jet

is 2ρkRU0Ud. Thus

Psaddle = 2ρRU0Ud(1 + k) (9)

This effect is not unique to the momentum. In fact, themass and energy also display this behavior, for example,the same approach yields a new term in mass balance dueto the saddle motion given by Min −Mout = Msaddle =2ρRUd(1− k).

D. Drift dependence on energy dissipation

Now we show that the main factor controlling the cen-tral drift is energy dissipation. For many jet-width-ratios,we vary the energy dissipation in the granular jet bychanging the coefficient of restitution and coefficient offriction of the grains. We furthermore are able to explorethe drift by varying dissipation in the impact of contin-uum fluids using the gerris flow solver, where frictional-fluid, Newtonian and shear thickening constitutive rela-tions are considered.

The energy dissipation rate is most easily computed inthe comoving reference frame. As mentioned, there arenet mass, momentum, and energy transfers to the driftingcenter in quasi-steady-state found in the lab frame, andfor simplicity we do our analysis in the comoving framewhere a steady state is reached to simplify the analysis.In the comoving reference frame, we denote UB and US,as the incident speeds of the large and small jets respec-tively, and Uout and Ψ0 as the ejecta speed and ejectaangle, respectively. These are related to their values inthe lab frame via

UB = U0 − Ud (10)

US = U0 + Ud (11)

Uout =

√(Uout cos Ψ0 − Ud)2 + U2

out sin2 Ψ0 (12)

Ψ0 = tan−1 Uout sin Ψ0

Uout cos Ψ0 − Ud(13)

The jet cross sections Rout, R, and kR are the same in thecomoving and lab reference frames. The transformationsof the ejecta speed and angle in the lab versus the co-moving frames can be seen in for representative impactsin Figs. 4(b) and (c).

In the comoving reference frame, when the impact pro-duces a steady-state flow configuration, the dissipatedenergy Ediss is equal to the energy advected in Ein sansthe energy advected out Eout. These energy fluxes aregiven

Ediss = Ein − Eout (14)

Beginning with the energy influx, Ein is simply the sumof the energy input from both the large and small jets

Ein = ρR(U3B + kU3

S) (15)

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ST k=0.314FF k=0.314Liq k=0.314

ST k=0.465FF k=0.465Liq k=0.465

ST k=0.688FF k=0.688Liq k=0.688

Control volumeanalysis

FIG. 6. (Color online) Drift speed reaches a fraction of its maximum based only on how much of the available incidentenergy is dissipated. This fraction is achieved entirely independently of all jet-width-ratios and dissipation mechanisms. Shownin (a) is the drift speed Ud/U0 versus the normalized energy dissipation (from Eqs. (18) and (15)) for the discrete particlesimulations and the continuum fluids. The discrete particle simulations (DPS) with coefficient of restitution (COR) 0.9 (black,triangles), 0.1 (dark grey, squares), and 0.05 (light gray, circles) are shown for k = 0.465. The continuum impacts are shownwith k = 0.314 (+×), k = 0.465 (+), and k = 0.688 (×) for the shear thickening fluid (ST, magenta), the frictional fluid (FF,red), and the Newtonian (Liq, blue) cases. At any given k, all of these cases produce similar drift speeds at a given energydissipation rate. For comparison, we additionally include our control volume analysis (green, dashed) at k = 0.465, whichqualitatively captures the increasing drift speed with increasing dissipation, but is completely unable to capture other features,most notably the concavity. (b) Drift speed Ud normalized by the maximum drift Umax

d versus the energy dissipation rateEdiss normalized by its maximum Emax

diss . Firstly, we find a dramatic collapse of the data, indicating that when the drift speedreaches a given fraction of its maximum dependent only on the dissipation rate and independent of the details of the highlyvarying impact materials and jet-widths-ratios. Secondly, we find that this fraction is robustly dependent on the normalizeddissipation rate to the power 1.5.

Next, the energy advected out by the two ejecta jets is

Eout = 2ρRoutU3out (16)

We can simplify the above slightly by noting that thereis no net mass flux, and that all of the mass entering isconsequently leaving. Denoting the mass fluxes by M ,we therefore have Min = Mout, giving

R(UB + kUs) = 2UoutRout (17)

because Min = 2R(UB + kUA) and Mout = 4ρRoutUout.Throughout the above we noted that the density ρ of thejets remains constant during the impact in dense granularjet collisions [11].

Combining Eqs. (14) and (17), we obtain

Ediss = ρR[U3

B + kU3S − (UB + kUS)U2

out

](18)

In Fig. 6 we show how Ud varies based on the frac-tion of energy dissipated during impact by comparingthe dissipated energy (18) to the input power Ein. Thisdependence is displayed for 4 different cases: the discreteparticle granular jets like those in our previous results,and the three continuum fluids computed in gerris. Thecontinuum fluids we used were a frictional fluid model ofdense granular jets [11, 12], a continuum Newtonian im-pact, and a continuum shear thickening fluid. The con-tinuum impacts were repeated for k = 0.314, k = 0.465

and k = 0.688 while the discrete case is only presentedfor k = 0.465.

We varied the dissipation rate in the discrete particlegranular jets by specifying the coefficient of restitutionat 0.05, 0.1, and 0.9, and for each coefficient of restitu-tion, we varied the friction coefficient from 0 to 1. Wevaried the dissipation rate in the frictional fluid modelby changing the dynamic friction coefficient µ from 0.05to 0.7. Furthermore, we varied the dissipation rate inthe Newtonian impact by varying the Reynolds number(Re) from 3 to 100 where the Reynolds number is definedRe = ρRU0/η. Lastly, dissipation was varied in the shearthickening fluid by varying the effective Reynolds num-

ber Reeff = ρR3/2U1/20 /κ from 6 to 100. For the frictional

fluid and Newtonian continuum simulations, we verifiedthat the dissipation rate showed reasonable behavior asµ and η were varied [28].

In Fig. 6(a) we see that increasing the energy dissi-pation rate relative to the input power in the comovingframe, or increasing Ediss/Ein, increases Ud. Also, in theasymptotic case of no dissipation, the data shows thatthe drift speed itself also vanishes entirely. Hence, in-creasing dissipation increases the drift speed, and havingno dissipation, such as in the head-on impact of idealfluids, there is no drift. Also, in Fig. 6(a) we see that,consistent with Fig. 4(a), reducing k increases Ud.

For all of these fluids, we found that the dependenceof Ud on Ediss/Ein all coincide and collapse together at a

8

given jet-width-ratio. The drift speed resulting from thehead-on impact of unequal-width jets is insensitive to theprecise type of the dissipation, rather, it only depends onthe amount of dissipation in the jets for a given k. Themechanisms by which a Newtonian fluid dissipates en-ergy are different from the generalized Newtonian shearthickening fluid, and in turn entirely different from thatin a discrete particle granular jet. It is remarkable howthe system produces such similar drift speeds for suchdifferent jet compositions hen controlled for the amountof dissipation.

E. Maximum drift speed and collapse

Here we briefly derive the maximum drift speed avail-able to the impact. Using the maximum drift speed as acharacteristic velocity scale, we will then show that thescaled drift speed versus the scaled dissipation rate col-lapses across an enormous range of simulations.

The drift speed Ud is clearly a strictly increasing func-tion in the dissipation rate, and so it is clear that themaximum drift speed is obtained when the dissipation ismaximized. Here, the maximum energy dissipated Emax

dissis simply bounded by the energy supplied to the system:

Emaxdiss = Ein (19)

Next, note that when drift is present, the momentum dif-ferential of the incident streams is lower in the comovingframe than in the lab frame. In the lab frame, the netincident momentum is

Pin = 2ρRU20 (1− k) (20)

whereas in the comoving frame the net incident momen-tum is

Pin = 2ρR((U0 − Ud)2 − k(U0 + Ud)2) < Pin (21)

The question at hand is how far can the dissipation pushthe comoving incident momentum differential; the an-swer is until the two jets contain equal momenta in thecomoving frame. This corresponds to directly verticalejecta streams with Ψ0 = π/2. At the maximum energydissipation, where the drift speed is maximized, Pin = 0and

(U0 − Umaxd )2 = k(U0 + Umax

d )2 (22)

yielding a maximum drift speed

Umaxd

U0=

1−√k

1 +√k

(23)

Using Emaxdiss as the characteristic energy scale and the

corresponding Umaxd as the characteristic velocity scale,

we see in Fig. 6(b) that our results do not only collapseacross dissipation mechanisms for a given k, but further

that all of our head-on impacts appear to be controlledby these characteristic velocity and energy scales. Hencewhen the dissipation is at a given fraction of the availableenergy Ediss/E

maxdiss , the drift speed is at a given fraction

of its maximum Ud/Umaxd as well, independent of the

jet-width-ratio k, and independent of the details of thedissipation mechanisms of the impact. Essentially, theemergent drift speed in the end appears to be a relativelysimple phenomenon. The effects of the jet-width-ratioare easily captured based on the total available energy tothe system to the extent where the drift speed is appar-ently determined by a self-similar equation

Ud = Umaxd f(Ediss/E

maxdiss ) (24)

where f(0) = 0, f(1) = 1, and f is strictly increasing.

F. Drift speed scaling relative to the dissipation

In Fig. 6(b), aside from the dramatic data collapse, wesee that the drift speed has a power law scaling relativeto the dissipation rate where Ud ∼ E1.5

diss. Though thisscaling is clearly and apparently present, we do not havean explanation for the scaling at this time. In particular,it appears surprisingly that Ud = Umax

d (Ediss/Emaxdiss )1.5

in Eq. (24).A particularly confounding aspect of the power law

are that straightforward analyses predict nothing like anexponent of 1.5. For example, in order to recover such alaw using dimensional analysis, our analysis would haveto yield, in some form

U2d ∼ E3

diss (25)

However, a physically relevant dimensional quantity de-pendent on the cube of the dissipation rate is highly ob-scure, and it is unclear to us where one might arise.

Furthermore, linear perturbations to the system, forexample to the similarity equation Eq. (24) to infer theform for f , would not yield a 1.5 or 3/2 power, but rathera linear dependence of f on the dissipation. For all ofthese reasons, the scaling identified in Fig. 6 is puzzling.

Below we attempt to understand the results exposedabove by presenting a very simple control volume analy-sis. As we will see, the analysis captures the broad natureof the effects of dissipation and unequal jet widths on thedrift speed from our simulation, but is far from capturingthe simple form for f as we empirically identified fromour simulations.

IV. CONTROL VOLUME ANALYSIS

Here we provide a simple approximation of the driftspeed resulting from the head on in pact of two unequal-width, dissipative jets as a qualitative exercise in showinghow dissipation and an unequal jet-size-ratio can natu-rally produce a drift speed. This approximation is largely

9

independent from measurements or observations from oursimulations, providing a from-scratch look at the effectsof dissipation and the jet-width-ratio.

We conduct our analysis in the drifting frame. Thecontrol volume analysis consists of two rough approxi-mations: one regarding the ejecta speed, and the otherapproximating the energy dissipation rate. These twoparts are related but distinct, since the ejecta speed de-pends not only on the dissipation rate, but also the driftspeed.

A. Approximating dissipation rate and ejectaspeed

In the following analysis, we presume that most of thedissipation occurs in the vicinity of the smaller jet nearthe impact saddle. This corresponds to a length scaleL ∼ CkR, a velocity scale U ∼ US, and, in two dimen-sions, a volume scale of V ∼ L2. Here, the constant Cis assumed to be independent of the jet width ratio andthe dissipation rate.

The control volume analysis begins with a very roughapproximation of the energy dissipated during impact.The total energy dissipation rate, shown in Eq. (18), re-lates the energy dissipation to the change in the incidentand outgoing energy fluxes. More plainly,

Ediss = −∮ρu2

2u · dS (26)

where the contour is along a control volume surroundingthe jet and extending far from the impact center in orderto enclose the entire flowing region with nonzero velocitygradients. This control volume is fixed in space sincethe flow in the comoving frame is in steady-state. Thisbecomes, noting fluid incompressibility,∮

ρu2

2u · dS =∫∇ ·

(ρu2

2u

)dV =

ρ

2

∫u ·∇(u2) dV (27)

Using dimensional analysis we can approximate Ediss

from Eq. (27) as

ρU∆(u)2

LV ∼ ρCkR(U2

out − U2S)US (28)

Above we considered an additional approximation where∆(u2) = U2

out − U2S , as this is the change in the velocity

squared from our final velocity state relative to our dom-inant velocity scale. This, together with Eq. (18), yields

U3B +kU3

S − (UB +kUS)U2out = Ck(U2

S − U2out)US (29)

For equal-width jets when k = 1, symmetry requiresUd = 0, and hence US = UB Ud = 0. In this case, Eq. (29)reduces to

2U3S − 2USU

2out = CUS(U2

S − U2out) (30)

which requires

C = 2 (31)

The final part of the approximation concerns the ejectaspeed. Our analysis simply approximates the ejectaspeed in the comoving frame as

Uout = BU0 (32)

In the above approximation, we allow B to vary whenfluid properties vary, such as changing the Reynolds num-ber in a Newtonian impact, or the friction coefficient in adiscrete-particle granular impact. But, we presume thatB does not change as k is varied. This approximationis motivated by Fig. 4(b), where we see that Uout is ap-proximately constant for the range of k tested. In thatexample with discrete particle jets whose coefficients ofrestitution and friction are respectively 0.9 and 0.2, weobtain B = 0.73.

Hence, with C determined by symmetry, the analysisbecomes dependent on the parameter B above which ac-counts for dissipation effects. Combining Eqs. (29) and(32), our analysis simply yields

U3B +kU3

S−(UB +kUS)B2U20 = 2k(U2

S−B2U20 )US (33)

The result of the control volume analysis, shown inEq. (33) is a cubic equation for the drift speed. Below weexplicitly show the cubic equation for Ud from expandingEq. (33) and recalling Eqs. (10) and (11):(

Ud

U0

)3

− 3

(Ud

U0

)21− k1 + k

+Ud

U0(3−B2)− (1−B2)

1− k1 + k

= 0 (34)

Because Eq. (34) has a negative discriminant for virtuallyall k and B, it in all practicality admits a unique realsolution for Ud, and the drift speed can be determinedrelative to U0 if provided k and B.

B. Analogy to equal dissipation in both incidentjets

Here we show that the energy dissipation approxima-tion in Eq. (29) is equivalent to the approximation thatan equal amount of energy is dissipated in fluid originat-ing in the two incident jets. That is, our energy dissipa-tion approximation in Eq. 29 analogous to approximatingthe energy drop across fluid originating in the large in-cident stream as equal to that in the smaller incidentstream.

Beginning with the small jet in the comoving frame, weknow that the incident mass flux for particles originatingin the small jet equals the outgoing mass flux for thosesame particles. Because the ejecta is at a speed Uout

ρkRUS = ρRS,outUout (35)

10

where RS,out is the width of the ejecta stream includingfluid originating from only the small incident jet. Asbefore, we calculate the energy dissipated in the smalljet, denoted ES,diss, via

ES,diss = ρkRU3S − ρRS,outU

3out

= ρkRUS(U2S − U2

out) (36)

Thus, if we set the energy dissipated in fluid from thelarge jet equal to that from the smaller jet EB,diss =ES,diss, we find that this approximation yields

Ediss = ES,diss + EB,diss = 2ES,diss (37)

Thus Eqs. (36) and (37) recover Eq. (29). Therefore ap-proximating the power dissipated in both jets as beingequal is equivalent to the rough dimensional approxima-tion done earlier.

C. Evaluation of the control volume analysis

We now discuss how drift emerges amid dissipation inour control volume analysis, as well as compare the anal-ysis to our discrete particle and continuum jet impactsimulations. In the comparisons against previous results,we first compare the control volume analysis to our dis-crete particle simulations with fixed particle properties(or fixed B) and varied k. We then test how the analysiscaptures the effect of dissipation by fixing k and varyingthe dissipation by changing B.

We first identify qualitative outcomes pertaining fromthe head-on collisions of two streams in the context ofthe control volume analysis from Eq. (33). Namely, wenaturally find the presence of dissipation produces a driftspeed. In the absence of dissipation, the jets behave asideal fluids, and thus B = 1 because the ejecta speedmust equal the incident speed. In that case, Eq. (33)reduces to

UB(UB − U20 ) = kUS(US − U2

0 ) (38)

whose only real solution is obtained when UB = US =U0, and no drift is present. Decreasing B from unitycorresponds to increasing dissipation, where UB < US,and a positive drift speed emerges.

Second, we check how the control volume analysis per-forms when fixing B and varying k. In particular, wecheck how the analysis performs in our granular jet sim-ulations with coefficients of restitution and friction of 0.9and 0.2 at various k. In Fig. 4(a) we show the drift speedversus k for the granular jets as well as for the control vol-ume analysis with B = 0.73, as obtained from the granu-lar jet noting that in the comoving frame, Uout = 0.73U0

(Fig. 4(b)). We see that the analysis captures the trendin Ud when B is fixed and k is varied.

The final check for our control volume analysis is whenthe jet-width-ratio is fixed at k = 0.465 and the dissipa-tion is varied by changing B from 0 to 1. The compar-ison of the analysis to our various simulations is found

in Fig. 6(a). The analysis captures the broadest qualita-tive nature found in the simulations, namely that there isno drift speed in the absence of dissipation and that in-creasing the dissipation rate monotonically increases thedrift speed. However, the analysis fails to capture theclear concavity found in the multiple collapsing simula-tion data sets. As such, the analysis fails to capture the1.5 exponent in the scaling identified in Fig. 6(b).

To its credit, the analysis does not have much tunabil-ity, and it nevertheless is in order the simulation results.However, we find that the analysis fails quantitatively. Itslargest shortcoming is when comparing the drift speedversus the dissipation rate, where the analysis predictsthe wrong concavity for Ud versus Ediss. and thereforedoes not capture the robust scaling that we identified inour simulations.

V. CONCLUSION

The impact of ideal fluid jets in two dimensions is aclassic pedagogical problem in fluid dynamics, both be-cause it is a conceptually simple generalization of theclassical mechanics particle collision problem to fluid dy-namics and because it is analytically tractable using clas-sical conformal mapping techniques [29–31]. While ithas long been appreciated that flow reversibility in ideal,dissipation-free jet impact dictates that the central im-pact region remains fixed in the laboratory frame, therehas been no study focused on how dissipation affects thisoutcome.

In this study, we investigated the effects of dissipa-tion in two dimensional head-on jet impacts, focusing onthe head-on impact of granular jets. We identified thatthe collision produces a steadily drifting flow configura-tion which absorbs a significant fraction of the incidentmomentum into the collectively drifting impact center.We further showed that energy dissipation is the mainingredient necessary to produce the emergent drift. Tothis point, we saw that the amount of dissipation dic-tates what fraction of the maximum drift speed is reachedacross a very wide variety of fluid dissipation mechanismsand jet-width-ratios, and identified that the drift speedrobustly scales relative to the energy dissipation rate witha 1.5 exponent. We lastly motivated a simple control vol-ume analysis of the collision which captured the generaleffect of dissipation on the drift speed, though this anal-ysis was short of explaining the scaling identified in oursimulations.

The findings presented above, together with ejecta in-sensitivity to internal structure in jet-target impacts be-gin to paint a picture of extreme ejecta insensitivity tofluid composition during dense jet impacts. In the caseconsidered in this study, multiple different types of flu-ids, from discrete particle to shear thickening, all pro-duce the same drift speeds from head-on collisions whencontrolled for total dissipation; the precise form of thedissipation evidently does not affect the outcome of the

11

impact, rather the only determining factor is the amountof dissipation. Corroborating this insensitivity in densejet impact in [12], when impinging granular and idealfluid jets against a fixed target, the internal structuresshowed very weak signatures in the ejecta. Thus, it ap-pears that the ejecta resulting from impact of dense jetsholds a very weak signature regarding the internal struc-ture or composition of the impact. This challenges thetraditional scattering approach in which the ejecta areused to infer material properties or internal structure,such as has been done to identify the double-helix struc-ture of DNA, the subatomic structure of Baryonic Matterin high energy physics, as two of several examples.

Furthermore, when investigating the impact betweentwo free streams or objects, it is natural to simplify thedynamics by fixing the first object and then collide thesecond against the first. Examples where this has hasbeen done when studying the off-center collision of gran-ular streams [27] or the formation of protoplanets [32–34]. However, because of the presence of central driftin dense, dissipative impacts, the outcome of collisionswhen fixing one object is fundamentally different thanthe free-collision counterpart. This consideration can im-pact the interpretation, and perhaps even the validity,of some fixed-target impact experiments mimicking thehigh-speed free collision of two dense objects. For exam-ple, since the decimation of a dust aggregate is an impor-tant consideration in protoplanetary formation, the totalforce felt during impact is a crucial experimental output.But, in this case, fixing one object produces larger im-pact forces than free impacts because a free impact wouldreduce the collision force via the central drift.

We would also like to highlight that the scaling behav-ior of the drift speed relative to the energy dissipationrate remains a mystery. Our simple approaches were notable to capture the exponent, and the similarity equationor control volume analysis presented cannot capture thescaling short of significant insight. In the control volume

analysis, we assumed that the dissipation occurred overa length scale dependent only on k and not the energydissipated. But, evidently the dissipation length scale ismore sophisticated, and our evidence suggests that therelevant impact length scale varies with the dissipationrate. This, taken together with the fact that the de-tails of the jet constitutive material are irrelevant, laythe groundwork for a more involved analysis with a morecareful accounting of the impact length scale. In particu-lar, this would manifest as a the coefficient C in Eq. (29)as no longer being constant, but varying based on thedissipation.

Lastly, we remark that though we believe drift shouldoccur in three-dimensional axisymmetric impacts, we areunsure if it too is insensitive to material type or results ina simple scaling behavior relative to the dissipation rate.The control volume analysis was clearly able to capturethe qualitative nature of the central drift speed, and thisanalysis can easily be extended to show the same result inaxisymmetric impacts. Whether or not the finding herethat the total dissipation is significantly more importantthan the material composition is unclear, as the analysiswas unable to entirely capture this effect, and a three-dimensional study with numerical diligence is requiredto determine if all of the two-dimensional findings in thispaper persist.

ACKNOWLEDGMENTS

The work was supported by the National Science Foun-dation through its Fluid Dynamics Program (CBET-1336489 to PI Wendy W. Zhang). The author wouldlike to particularly thank W. W. Zhang for discussions,insights, and careful readings of this paper, in additionto T. A. Witten, H. M. Jaeger, S. R. Nagel, I. Bischof-berger, N. Guttenberg and M. Z. Miskin for insights anddiscussions.

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