a. l. kuhl- a baroclinic model of turbulent dusty flows

8/3/2019 A. L. Kuhl- A Baroclinic Model of Turbulent Dusty Flows

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UCRL-JC-111404PREPRINT

JUl 191893IA Baroclinic Model of Turbulent Dusty Flows

9

A. L. Kuhl

This paper was prepared for submittal to theDNA Numerical Methods SymposiumStanford Research Institute

April 28-30,1992Menlo Park, California

April 1992

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A BAROCLINIC MODEL OF TURBULENT DUSTY FLOWS*A.L. Kuhl

Lawrence Livermore National LaboratoryE1 Segundo, California

ABSTRACTThe problem considered here is the numerical simulation of the turbulent dusty flow induced byexplosions over soil surfaces. Some of the unresolved issues are: (i) how much dust is scoured fromsuch surfaces; (2) where does the dust go in the boundary layer; (3) what is the dusty boundary

,f layer height versus time; (4) what are the dusty boundary layer profiles; (v) how much of the dustmass becomes entrained into the dust stem; and (6) where does the dust go in the buoyant cloud?While others have used numerical models based on mean-flow equations and turbulence closuremodels, here we propose a Baroclinic Model for flows with large density variations that actuallycalculates the turbulent mixing and transport of dust on an adaptive grid. The model is based onthe following idealizations: (1) a loose dust bed; (2) an instantaneous shock fluidization of the dustlayer; (3) the dust and air are in local equilibrium (so air viscosity enforces the no-slip condition);(4) the dust-air mixture is treated as a continuum dense fluid with zero viscosity; and (5) theturbulent mixing is dominated by baroclinically-generated vorticity. These assumptions lead toan inviscid set of conservation laws for the mixture, which are solved by means of a high-orderGodunov algorithm for gasdynamics. Adaptive Mesh Refinement (AMR) is used to capture theturbulent mixing processes on the grid.One of the unique characteristics of these flows is that mixing occurs because vorticity is producedby an inviscid, baroclinic mechanism: 0) -- _p _p. For example, vorticity is created byshock interactions with strong density gradients such as the fluidized bed, the thermal layer,and the fireball. These shear layers are hydrodynamically unstable and roll up into turbulentmixing layers. The rotational structures in the boundary l,_yer entrain dust from the fluidizedbed; turbulent mixing and transport of the dust then occurs via the velocity field induced bythe vorticity distribution. Additional vorticity generation occurs because of large-scale pressuregradients (e.g., during the positive and negative phase of the blast wave), because of local pressure

- gradients (induced by the rotational structures), and because of buoyancy forces.A number of examples will be presented to illustrate these baroclinic effects including shock in-teractions with dense-gas layers and dust beds, and dusty wall jets of airblast precursors. The

conclusion of these studies is that dusty boundary layers grow because of mass entrainment fromthe fluidized bed (and not because of viscous wall drag) as proven by the Mass Integral Equation.

*Work performed under the auspices of the U.S. Dept. of Energy by the Lawrence LivermoreNational Laboratory under Contract number W-7405-ENG-48. Also sponsored by the DefenseNuclear Agency under DNA IACRO #92-824 and Work Unit 00001.


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1. INTRODUCTIONThe problem considered here is the numerical simulation of the turbulent dusty flowinduced by explosions over soil surfaces. Figure 1.1 presents a photograph of EventPRISCILLA, which provides an excellent example of the type of dusty flow to bemodeled here. Some of the unresolved technical issues associated with such flowsare: (i) How much dust is scoured from soil surfaces? (ii) Where does the dust goin the boundary layer? (iii) What is the dusty boundary layer height versus time?(iv) What are the dusty boundary layer profiles? (v) How much dust mass becomesentrained into the dust stem? (vi) Where does the dust go in the buoyant cloud?We have tried to answer such questions by means of direct numerical calculationsof the turbulent mixing in such flows, as illustrated in Figure 1.2.

!One of the unique characteristics of these flows is that mixing occurs because of

- inviscid, baroclinic effects. This is perhaps best illustrated by considering the Vor-ticity Transport Equation:0_w + (u. V)w = (w" V)u + w(V' u) - V(1/p) x rP+ V x (B/p) + v V2w

I II III IV Vwhich is derived by taking the curl of the momentum equation. The terms on theleft-hand side represent the time rate-of-change of the vorticity vector w and theconvection of vorticity by the velocity field u, The vorticity transport is balancedby a number of source terms (and a diffusion term) on the right-hand side:

I vorticity stretching along the streamlines (a three-dimensional effect),II vorticity dilatation (due to compressibility effects),4

III vorticity creation by baroclinic effects (pressure gradients interacting with" density gradients),

IV vorticity creation by buoyancy forces B (gravitational forces accelerating den-- sity gradients), and

V diffusion of vorticity (due to molecular transport effects).Baroclinic effects predominate during the blast wave phase of the explosion. Forexample, vorticity is created by shock interactions with strong density gradientssuch as the fluidized bed, the thermal layer, and the fireball. These shear layers are


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hydrodynamically unstable and roll up into turbulent mixing layers. The rotationalstructures in the boundary layer entrain dust from the fluidized bed; turbulentmixing and transport of the dust then occurs via the velocity field induced bythe vorticity distribution. Additional vorticity generation occurs because of large-scale pressure gradients (e.g., during the positive and negative phase of the blastwave) and because of local pressure gradients (induced by the rotational structures).Buoyancy effects dominate the cloud rise phase. Molecular diffusion effects arenegligible for large-scale explosions because the Reynolds numbers are so large:Re = 10s to 101 (and Term V scales as 1Re).

While others have used numerical models based on mean-flow equations and turbu-lence closure models, here we propose a Baroclinic Model for flows with large densityvariations that actually calculates the turbulent mixing and transport of dust onan adaptive grid. A number of examples will be presented to illustrate these baro-clinic effects--including shock interactions with dense-gas layers and dust beds, anddusty wall jets of airblast precursors. The conclusion of these studies is that dustyboundary layers grow because of mass entrainment from the fluidized bed (and notbecause of viscous wall drag) as will be proven by the Mass Integral Equation.

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Figure1.1.Photographof thePRISCILLAEvent (37-KTburstat 700 ftoverFrenchmansFlat drylakebed).


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2. BAROCLINIC MODELThe BaIoclinic Model is based on the following idealizations: (1) a loose dust bed;(2) an instantaneous shock fluidization of the dust layer; (3) the dust and air arein local equilibrium (so air viscosity enforces the no-slip condition); (4) the dust-airmixture is treated as a continuum dense fluid with zero viscosity; (5) the turbulentmixing is dominated by baroclinically-generated vorticity. These assumptions leadto an inviscid set of conservation laws for the mixture,0

p -I- V" (pu) -- 0 (1)0ONpu + V(puu) = - V P (2)0ONpE + V" (pEu) - - V .(pu) (3)

where u denotes the velocity and E represents the total energy: E - e + 0.5 u. u.The pressure p was related to the density p and internal energy e by the perfect gasequation of state"

p-- (")'-)pe (4)where V represents the effective V of the mixture. For simplicity a value of 7 - 1.4was used in the calculations. This value is an adequate model for diatomic gasessuch as air. For the dust-air mixture, it also takes into account the kinetic pressurecreated particle-particle interactions of the dust which occur in the fluidized bed. Ifone employs a multi-fluid formulation, more complete equations of state are possible,however, this was considered to be beyond the scope of the present investigations.In the above equations, p actually represents the mixture density. The dust densityPd may be calculated from the relation:

pd= pc (5)This requires an additional transport equation for the dust concentration C, namely:

00-7c + (u. v)c = 0 (6)The above equations were integrated numerically by means of a high-order Godunovscheme. 15 Adaptive Mesh Refinement (AMR) is used to capture the turbulent mix-ing processes on the grid.In addition, we have a two-phase model to study nonequilibrium effects in dustyflows. The equations are presented in Table 2.1. One can see that there are separateconservation laws for the gas and particulate phase, which are coupled by drag andheat transfer effects.


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I> i _ i I + _, +I II II i-- II II ,-_ _ ,i_II

++ I I _ .._ + + I II II _ I 7 II ._E _ _ _ "_ _ E _ II II _' =I,, =_ f_ II II II II _ _ _ II II II II II _ r_ II II II II%

_- _ II IIII _ _ -_, ._

_ _ ,,_ ,e _ + _- ""_,. I::> + + I _a

II II II I II II II II _ _ II II II '_ "_, _. _. II II II II

,._ I_ i

.:" _ r.,,,) I I '_._, ++ E "_ r_

o _" _ _" ,,.t '_ _" " "-"., ,.-- _ ..-.- I:> i::> _ ''_I I III II II II II II ,._ II II II II _ II II II II II II

....

III

o _, or_ M r._ _ r._

....

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3. PLANAR SHOCK INTERACTIONS WITH ADENSE-GAS WALL LAYER

Shock interactions with density discontinuities provide a rich source of fluid-dynamicphenomena. For example, normal shock interactions with planar density interfaceslead to instabilities and turbulent mixing at the interface, as studied by Richtmyer(1960) and Meshkov (1960). Planar shock interactions with inclined density in-. terraces lead to a spectrum of shock refraction effects as shown in the shock tubeexperiments of Fattah et al. (1976), and to the rollup of the interface as demon-strated by the calculations of Zabusky et al. (1991). Planar shock interactions with

low density layers located along the wall create shock precursor effects. This leadsto a turbulent wall jet as shown in the shock tube experiments of Reichenbach and

. Kuhl (1987). In these examples, the shock interaction with the interface createsvorticity by a baroclinic mechanism: Ap A(1/p). This shear layer is hydrody-- namically unstable, and rapidly evolves into a turbulent mixing region.Considered here is a new class of such problems, which can be used to study turbu-lent mixing in boundary layers that are dominated by density effects- in contrastwith classical boundary layers that are dominated by viscous effects. In this respect,the current problem can be considered as a boundary layer version of the classicfree shear layer experiments by Brown and Roshko (1974). The problem consists ofthe interaction of a planar shock wave with a Freon layer located along the floor ofthe test section of a shock tube. Shock interactions with the density interface createa wall shear layer that rapidly rolls up into a turbulent boundary layer. Describedhere are shock tube experiments and direct numerical simulations of this problem.3.1 Problem DescriptionExperiments were performed in the Ernst-Mach-Institut shock tube. Figure 3.1shows a schematic of the test section which had an effective rectangular cross-sectional area of 7.5 cm by 4 cm. The wall layer was created by means of a fixture- that was inserted into the shock tube. It contained a plenum for the injected gas,covered by a 65 cre-long porous ceramic plate (Filtrokelit) that spanned the 4 cmwidth of the shock tube. The plenum was filled with Freon-12 at a pre.csure ofApF _ 0.5 bars and run for about one minute -- to achieve essentially 100 per-cent Freon concentration in the plenum, and to cleanse the pores of the ceramicplate of residual air. Next, the Freon valve VR was closed and the shock tube was

. purged of contaminated gases. Then the Freon valve was opened and the plenumwas pressurized to /kpF -- 1.5 bars for a time AtF (typically 2 s) and the shocktube diaphragm was broken mechanically. To increase the reproducibility of theexperiments, all filling procedures were computer controlled.


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This procedure created a thin layer (,,_ 0.3 cm) of pure Freon-12 on the floor of thetest section. Above this, the Freon concentration decreased gradually in a thicker(,,_ 1.5 cre) diffusion layer. This was caused by residual air pollution in the pores ofthe ceramic plate, and by bi-molecular diffusion processes. The resulting layer wasa 2 cm-thick by 65 cre-long laminar layer of Freon whose concentration profile wasindependent of x. The layer could be controlled by the Freon plenum pressure _PFand duration AtF; best results were achieved with ApF -- 1.5 bars and AtF -- 2 s.High-speed photography was used as the primary flowfield diagnostic -- to makevisible the turbulent mixing processes occurring in the wall layer. It consisted of

24 sequential frames of shadow-schlieren photographs that were recorded by EMI'sCranz-Schardin camera. In addition, Mach-Zhender interferometry was used to. evaluate the preshock density profile of the Freon layer.

Figure 3.2 shows a schematic of the computational domain that was used for numer-ical simulations of the experiments. A rectangular x-y cartesian grid of 150 cells by600 cells was employed. The mesh spacing was uniform with/kx = Ay = 0.05 cre.The grid was initialized with a quiescent air atmosphere: pl = 1.223 10-3 g/cre 3,Pl = 1 atm, ul = vi = 0. The Freon layer was then modeled by a Tanh(y) pro-file which approached the Freon density on the wall PF = 4ps. The left boundaryof the grid was fed with constant conditions corresponding to the state behind aMx = 1.38 shock wave: p2 -" 1.65 Pl,/92 -_ 2.06 Pl, U2 = 1.86 104 cm/s.Outflow boundary conditions were used on the right boundary. The roof and floorwere treated as inviscid (slip) walls. Both the air and the Freon were modeled asan ideal gas with -_- 1.4.

3.2 Experiments

Figure 3.3 presents a sequence of photographs showing the interaction of a MI -1.38 air shock with a Freon layer. The preshock structure of the layer is visiblein the first frame. The white band along the floor consists of pure Freon, about0.3 cm thick. The black band above it is the aforementioned diffusion layer, wherethe Freon concentration gradually decreases to zero. The complete layer is about

" 2 cm thick in this experiment.

9

m


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The sound speed in the layer was smaller than that in the air above it, hence theshock propagation was retarded in the layer. This caused the shock front to becurved near the floor. The oblique shock compressed the layer and reflected fromthe floor as a regular-reflection shock structure. The reflected shock reflected off thelayer interface as a rarefaction wave which stopped the compression of the layer.The rarefaction wave reflected back and forth within the layer, causing periodicaccretions and depletions of density. These effects are clearer in the numerical

. simulation presented in the next section.By about 10 cm behind the shock, the wave dynamics effects had damped out,

and mixing processes began to dominate. Figure 3.3 shows that the flow was veryunstable, and for At >_ 0.35 ms it rapidly evolved into a turbulent mixing layer

even in these small-scale experiments. Although the flow was no doubt three-dimensional, one can occasionally identify large-scale structures (e.g.,/kt = 0.85 msand 0.95 ms) that entrain Freon from the wall layer. These structures createddensity striations that point up and to the right at an angle of about 30 deg fromthe floor. Similar striation effects are found in the numerical simulations. One canalso observe fine-scale structures, especially near the top of the layer.3.3 Numerical SimulationsA numerical simulation of the preceding experiment was also performed, using theBaroclinic Model. Figure 3.4 presents contour plots that show the initial shockinteraction with the wall layer. The pressure contours show the refraction of theshock front in the layer, the reflection at the floor, and the formation of periodicexpansion and compression waves in the layer. The internal energy contours showthe periodic compaction (z = 61 cm and 71 cm) and expansion (x = 66 cre) of thesurface. The vorticity contours show that vorticity is generated at the point wherethe shock interacts with the layer interface. The Freon concentration contours showthat the interface begins to rollup almost immediately behind the shock front. Atdistances greater than about 10 cm behind the shock (x < 61 cre) the rotational. structures pair and merge into larger-scale structures.Figure 3.5 depicts flowfield contours at /kt - 0.5 ms after shock passage at x --70 cm. They show an intense mixing with a spectrum of length scales. As is char-acteristic of variable density flows, the vorticity contours show that counter-signvorticity (solid lines) are baroclinically generated in the braid region between rota-. tional structures. Peak values are about -0.5 times the peak value of the vorticityof the main flow. The counter-sign vorticity, of course, increases the complexity ofthe mixing. The tendency in these two-dimensional calculations is to form multiplemerging of vortices. At later times, this tendency leads to predominantly large-

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scale structures. In three-dimensional flow, however, vortex stretching and tanglingwill maintain the fine-scales within the large structures m as is evident in the pho-tographs of Figure 3.3. Future calculations should include the three-dimensionaleffects.

The flowfield was sampled at station x - 70 cm, and time-averaged profiles in thewall layer were evaluated. The mean flow profiles are presented in Figure 3.6. Flow variables were nond':mensionalized by the freestream values, denoted by subscript

oo. The profiles are qualitatively similar to our previous simulations of the dustyboundary layer behind a planar shock. The streamwise velocity profile indicates that

the layer remained about 2 cm thick during this time period -- thus eliminatingthe need for self-number, similarly stretching the grid to account for growth of thelayer. Density effects cause the velocity to decrease to _ - 0.2 Uoo at the floor; ifviscous effects were included in the laminar sublayer, then the velocity would go to- zero at y = 0. In accordance with the negative displacement effect of shock-inducedboundary layers, the mean transverse velocities are negative in the layer and reacha peak value of _ = -0.035 Uoo. The mean density reaches a peak value of about

= 3.8 pco because of the Freon near the floor; hence, this mixing layer is stronglyinfluenced by density effects. The mean pressure is essentially constant throughoutthe layer.Figure 3.7 presents the corresponding fluctuating flow profiles. Again, these profilesare qualitatively similar to our dusty boundary layer simulations. The streamwisevelocity fluctuations peak at a value of u_= 0.31 Uoo, which is typical of boundarylayers. The transverse velocity fluctuations peak at a value of v _ - 0.26 Uoo: this isprobably too large by a factor of two m due to the two-dimensional flow approxima-tion. The Reynolds stress u_v_ is positive, indicating that mixing is feeding fluctu-ating kinetic energy back into the mean flow; peak values reach u_v_= 0.0075 U2.Density fluctuations are of order 100 percent, due to the Freon. Fluctuations indynamic pressure and stagnation pressure are also large (e.g., 50 percent) near thefloor.3.4 SummaryShock interactions with a dense-gas layer create a shear layer on the wall by aninviscid (i.e., baroclinic) mechanism. The wall shear layer is unstable, and rapidlyrolls up into a three-dimensional, turbulent mixing layer with a variety of mixing. scales. One of the most interesting features of this problem is that it provides amethod for studying high-Reynolds number, turbulent wall layers that are domi-nated by baroclinically-generated vorticity and density effects _ in contrast withclassical boundary layers that are dominated by viscous effects. This problem also

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provides a gasdynamic simulation of turbulent dusty boundary layers if the dustparticle diameters are very small. Measurements of both the mean and fluctuatingflow profiles of this mixing layer will be performed to check the present numericalpredictions.3.5 References

Brown, G.L. and Roshko, A. (1974) "On density effects and large structures inturbulent mixing layers," J. Fluid Mech., 64:775-816.

Abd-el-Fattah, A.M., Henderson, L.F., and Lozzi, A. (1976) "Precursor shock wave--at a slow-fast gas interface," J. Fluid Mech. 76(1):157-176.Meshkov, E.E. (1960) "Instability of the interface of two gases accelerated by a- shock v.:e.ve_"[zv, AN SSSR Mekhanika Zhidkosti i Gaza 4(5):151-157.Richtmyer, R.D. (1960) "Taylor instability in shock acceleration of compressiblefluids," Comm. Pure Appl. Math, 13:297-319.Reichenbach, H. and Kuhl, A.L. (1987) "Techniques for creating precursors in shocktubes, 16rh Int. Syrup. on Shock Tubes and Waves (ed. H. Gronig), VCH press,Weinheim, Germany, 847-853.Zabusky, N. et al. (1991) "Vorticity deposition, evolution and mixing for shocked,density stratified interfaces and bubbles," 18rh Int. Syrup. on Shock Waves (inpress).


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0cm 45cm 65 cm\\,_'k\\\\\\\_\\\\\\\\\\\\\\\\\\\ \\\\\\\\_ \\\\\\ \\\\\\\ ___

IMI Opticalwindow A A /_ ///A /// 7.5 cm' "_\\'_\\\'_\\\\\\\\\\'c\\\\\\\\\\lli'_ "\"u'\\I,,_.._,, Porousplate I_

-"-'"\ (Filtrokclit) If. Plenum _],_v_

FREON-12

Figure 3.1.Schematic of the test scction.

' I I Z _O_l, :=\N ___\\\\\\\\\\\\\\\\\\'__\\\\\\\\\\\\_ ,'_,\\\\\\5

Figure3.2.Schematicfthecomputationalomain.

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MAt=O

t

-O,35rn.,I

,s

O,_5msi

O,55ms,

i

Figure 3.3, Shadow-schlieren photography of the interaction of a MI = 1.38shock wave with a Fr_on wall layer (PI = I atm).i4


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4_

"e

0,75ms ,.! -. : .

O,BSms

Figure 3.3. Concluded.


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0.075 , ,_,

"o'o.o_! \_2-...,, _:_Ill]

0.O75F (c)COME ZONEI 0.050)y (m) I0.025I C =00

0.075 / (d)VORTICrrY0.050

1

y (m) 0.02500.51 0.56 0.61 0.66 0.71 0.76x(m)

Figure3.4.Contourplotsshowingtheinitialhockinteractioniththewa.II_ayer:(a)internalnergy;(b)pttssur_;c)Fmon concentration;d)vo_.city.0.075o.o5oIc- - : . oo: ](.)__._oY\J ...._... i """ 'Y(m) .- , " "_'"'" ' ,, x _ "0.025 _ :"-" 2'-_-..''-_'_'_:"- ',//_.-(

0.075 Co)DE/qSrl_0.050y (m) .._..0.025 :' , --'o_,i,o:I I=o

0.075,(d)VORTIClTY0.050y (m) 0.025

00.60 0.65 0.70 0.75 0.80 0.85x (m)

Figure 3.5. Contour plots showing intermediate-dme mixing in the wall layer(At = 0.5 ms): (a) inramal energy; Co)density; (c) entropy; (d) vorticity.

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Figure 3.6. Mean flow profiles in the wall layer (x = 70 cre): (a) srreamwise veloci_;Co)transverse velocity; (c) density; (d) pressure; (e) dynamic pressure;(f) stagnation point.

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Figure3.7. Fluctuating-flow profiles in thewalllayer (x - 70 cna):(a)streamwisevelocitCo)transversevelocity; (c)Reynoldsstress;(d)density; (e) dynamicpressure(f) stagnation pressrre.18


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4. TURBULENT DUSTY BOUNDARY LAYERBEHIND A SHOCK

Dusty boundary layers axe a common feature of explosions over soil or groundsurfaces. Direct theoretical predictions of such flow fields are extremely difficultbecause of the two-phase, turbulent nature of the flow. In addition, wall boundaryconditions (e.g., dust scouring) are particularly intractable. Hence, many of the. initial studies (e.g., Ausherman, 1973) were empirically based. Hartenbaum (1974)measured the mean boundary layer profiles for steady flow over a dust bed, anddeduced an empirical dust scouring rate for flow velocities of 30 to 120 m/s. More

- recently, Batt et al. (1988) have measured the velocity and density profiles in theturbulent boundary layer behind a shock propagating along a dust bed. However,such nonsteady experiments can give only instantaneous point values at variousdistances behind the shock; mean-flow profiles and R.,k_.S. fluctuations cannot be" obtained from such data.Mirels (1984) has published analytic solutions for this problem, but he assumedthe boundary layer profiles. Others have performed finite difference simulations;however, they used an empirical mass injection boundary condition on the wall anda tc-e model of the turbulence (Denison, 1988; Wolf and Strawn, 1982). Since thesemodels were developed for clean flow, and since the dust dramatically modifies theboundary layer, the adequacy of this approach is questionable.Described here is a numerical calculation based on the Baroclinic Model for theturbulent boundary layer formed by a planar shock propagating along a dusty wall.It differs from previous studies in two respects: first, it follows the dynamic evo-lution of the unstable flow without resorting to turbulence modeling; second, theentrainment of dust into the turbulent boundary layer is included as part of thecomputational flow field, thus eliminating the need for a "dust scouring" model.The formulation of the calculation is described in the next section; this is followedby the results and a discussion that offers suggestions for accurate simulations of such dusty, turbulent flows.4.1 Formulation

m Figure 4.1 depicts a planar shock propagating with constant velocity Ws along adusty wall. In stationary coordinates (Fig. 4.la), a boundary layer is formed behind. the shock because of the no-slip condition on the wall. In addition, air percolatesinto the dust layer, forming a fluidized mixture of dust and air. For simplicity,the mixture was treated as equilibrium dense fluid of 50 mg/cc, as inferred fromexperiments (Batt et al. 1988).The specific case considered was that of an M8 -- 1.7

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shock. As in previous analytic studies (Mirels, 1956), the analysis was performed inshock-fixed coordinates, (Fig. 4.1b). In this coordinate system, the velocities varyfrom the value at the edge of the boundary layer, ue = Ws - u2, to a maximumvalue on the wall, Uw= Ws.

The computational grid consisted of 500 fine cells in the x direction and 60 finecells in the y direction; a few coarse zones were used above and to the right of the, fine mesh to remove any effects of the computational boundaries. The grid was

initialized with the appropriate states from Table 1 and the profiles depicted inFigure 4.1c. The shear layer on the wall was approximated by a Tanh(y) function- with proper asymptotes of u/al = 1.7 at y = 0 and u/ai = 0.77 at y = c. The

density profile consisted of a three-cell-thick fluidized bed (53 mg/cc) with shockedair above it. Note that the inflection point (IP) of the shear layer was located inthe air, three cells above the fluidized bed (FB). The left-hand boundary of the grid" was then driven by these same profiles (Fig. 4.1c) with sinusoidal perturbationson the velocity field. Their frequencies corresponded to the frequency of maximumamplification rate from linear stability analysis and its first nine subharmonics. Themaximum perturbation amplitude was 1 percent. A more complete description ofthis calculational approach may be found in Kuhl et al. (1988).The dynamic evolution of the flow field was calculated by means of the BaroclinicModel. The calculation was run for 6,000 cycles (6 hours) on a Cray-XMP to createa sufficient database for statistical analysis of the flow.4.2 ResultsFigure 4.2 depicts the location of the material interface of the fluidized bed. Fig-ure 4.3 presents the density (p), internal energy (e), vorticity (w), and pressure(p) contours near the end of the fine-zoned grid. These figures are useful for ftowvisualization in the boundary layer region. The shear layer in the air rolled up intolarge rotational structures that entrained material from the fluidized bed. Dense material from the fluidized bed formed long striations because of the flow field ofthe rotational structures of the turbulent boundary layer. Thus, the dust scouringoccurred naturally in the calculation without any modeling.

o Figure 4.4 depicts the boundary layer thickness 6 (i.e., the height of the 99rh per-centile point of the mean velocity profile) as a function of distance behind the

o shock. The dusty boundary layer grows linearly with distance (6 - .024x) as aresult of entrainment and merging of large-scale vortices. Note that this is in con-trast to the clean boundary layer on a flat plate, which grows as the 0.8 powerof distance from the leading edge (6 a x'S). The instantaneous flow field at the

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top of the fluidized bed (y = yo) is presented in Figure 4.5. The flow is clearlyvery chaotic; hence, useful representations of the flow require some type of averag-ing. To that end, the calculated environment was sampled for all y-cells at stationsx = 400, 600, 800, and 950 for each computational cycle. The mean-flow environ-ment was then calculated from these station histories by integrating over the last5,000 cycles.

. The calculated mean density profiles are depicted in Figure 4.6a. The profiles be-come thicker with increasing distance from the shock because of turbulent convec-tion. Near the wall all profiles converge to a density of about fi/poo = 6(,,_ 17mg/cc).

" This convergence point defines the calculated height of the fluidized bed: yo = 3.5. cells. Only the flow field above yo was used in the boundary layer profiles; the

solution below yo was then considered to be part of the computational modeling ofthe dust scouring. Note that these calculated profiles are qualitatively similar tothe measured dust density profiles shown in Figure 4.6b.The mean-flow velocity (fi/U_), specific volume (_/Ac), and dynamic pressure(4/Q_) profiles are depicted in Figure 4.7 (subscript c_ denotes the freestreamconditions above the boundary layer). Note that by using the boundary layer scalingr/BL "-- (y- yo)/_, the calculated profiles collapse to a similarity profile, independentof distance behind the shock. Note also that the calculated profiles are consistentwith the LDV and X-ray measurements (shaded regions in Fig. 4.7) over a soil bedof 10-pm-diameter loose dust (Batt et al. 1988). Near the wM1, densities approachfive to six times the freestream value. This causes the velocities in the wall regionto be very small (fi/U_ < 0.1). Hence, the velocity profile in the wall region seemsto be dominated by inertia effects of the dust, and not by fluid viscosity. Notethat this situation differs considerably from that of a turbulent boundary layer on aflat plate, where fi/Uc _- 0.5 in the law-of-the-wall region, and viscous effects playan important role. Using a control volume analysis of the fine-grid region (witha bottom boundary at yo) and the mean-flow profiles, the nondimensional massscouring rate was evaluated as: rho/pooU_ = 0.036.The root-mean-square (R.M.S.) fluctuations about the mean were also evaluated.These fluctuating-flow profiles are presented in Figure 4.8. The streamwise (u'/Uc_)

. and transverse (v'/Uoo) velocity fluctuations peak at about 30 and 13 percent, re-spectively, and extend many boundary layer thicknesses from the wall. Thus, dustwill be transported well above the mean boundary layer height. The turbulent

- Reynolds stress (u-_v_/U_) peaked at about -0.003, similar to other boundary lay-ers. Density fluctuations (p_/p_) peaked at about 8ix times the freestream valuebecause of turbulent entrainment of dense

21


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material from the fluidized bed. The static (p'/Po,,) and dynamic (q'/Qoo) pressurefluctuations peaked at about 17 and 42 percent, respectively.The local fluctuation-intensity profiles are presented in Figure 4.9. In contrast withclean boundary layers, turbulent velocity fluctuation_ were much larger than theirmean values (e.g., u'/fi .'_:>1 and v'/9 >> 10) because fi is so small in the wall region.Hence, the fluctuations are the most important component of the flow. This seems

to pose severe challenges for both experimentalists and turbulence modelers.A high-order Godunov scheme has also been developed for nonequilibrium gas-dust- mixtures. This solves the conservation of mass, momentum and energy for eachphase, with drag and heat transfer interactions between phases. A nonequilibriumcalculation of the dusty boundary layer behind a shock was run assuming veryfine (0.1#rn diameter) dust particles. The nonequilibrium mixture results (NE) aredepicted by dashed curves in Figures 4.7 and 4.8. Both the mean and the R.M.S.profiles are essentially identical to the dense gas results. Surprisingly then, the densegas approximation is quite an accurate model for this two-phase flow problem.4.3 SummaryDust lofting behind a shock can be viewed as a two-step process--the formation ofa fluidized bed, followed by entrainment of the dense material from the bed by therotational structures of the turbulent boundary layer. The principal effect of thedust is to change the velocity of the flow. In the wall region, mean-flow velocitiesare small because dust densities are large. Also, turbulent velocity fluctuations inthe wall region are much larger than the mean values due to this same effect.The present calculations focus on the two most important physical processes in theturbulent boundary layer, namely, the nonsteady velocity field of the large rotational" structures, and the inertia effects of the dust. An accurate numerical simulation ofthese processes allows one to capture the first-order physical effects. For example,- the velocity and density profiles, the dust scouring rate, and the boundary layergrowth agree with the available data. 3 By inference then, nonequilibrium effects,fluid viscosity and three-dimensionality must have a secondary effect on the meanflow. Nevertheless, experimental data are needed to check the accuracy of thecalculated R.M.S. fluctuations.

. This calculation vividly demonstrates the advantage of working in shock-fixed coor-dinates where the smooth or laminar solution is steady. By performing a nonsteadycalculation in these coordinates, one can not only capture the rotational structuresof the turbulent flow, but also record time histories at fixed distances behind the

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shock. These can be used to determine both the mean and fluctuating flow pro-files without resorting to turbulence modeling--the main limitation being the 2-Dflow approximation. This approach represents a significant advance over previousapproaches that must assume turbulence properties which are, of course, Unknownfor this dusty boundary layer flow.4.4 ReferencesAusherman, D., (1973) Initial Dust Lofting: Shock Tube Experiments, DNA 3162F.

Haxtenbaum, B., (!974) Lofting of Particles by a High-Speed Wind, DNA 2737.Batt, R.G., Kulkarny, V.A., Behrens, H.W., Rungaldier, H., (1988) "Shock-inducedBoundary Layer Dust Lofting," Shock Tubes and Waves, ed., H. Gronig, VCH,Weinheim, Germany, pp. 209-215.Mirels, H., (1984) Blowing Model for Turbulent Boundary-Layer Dust Ingestion,AIAA, 22(11), pp. 1582-1589.Denison, M.R., A Two-Layer Model of Dust Lofting (in press).Wolf, C.J., Strawn, R.C., Boundary Layers in Surface Burst Flowfields (in press).Mirels, H., (1956) Boundary Layer Behind a Shock or Thin Expansion Wave Movinginto a Stationary Fluid NACA TN 3712.Kuhl, A.L., Chien, K.-Y., Ferguson, R.E., Glaz, H.M., and Colella, P., (1988) Invis-cid Dynamics of Unstable Shear Layers, RDA-TR-161604-006, R & D Associates,Marina del Rey, CA.

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(b)W= O Ws-U(9 ., __gbiLz " " ,.......

,,,,,,, FLUIDIZED BED (FBI FB_' Ii, lC) ld)" 20 olyl C) I--Ue-- INFLOW B.C. OUTFLOW B.C.w

o _" lP UolY), Poly) 10- _. Tanh(y)- P(Y)3 - U ,-"i ....... "o ---rl w 5' lo' "" 5o'1oo _/i_/.4Z/////I/I/I/i/I/A_,,,,......,,,,_/HH/'/////III///m. p(mg/cc) \ WALL SLIP WALL B.C.

Figure 4.1. Dusty bounda_ layer behind a shock: (a) stationary coordinates,(b) shock-fixed coordinates, (c) inflow profiles, (d) grid.40, 40

i 1 (aiY 20 r FB Y 20t f0_) iii i -- 01O0 200 800 900 1000X X40_ 1Y 20_oL _ (FB_- i _ 4o b) _ -_200 300 4..0 Y 20_,,__ _',!_-__'_.

X 0

40 f , i _]t 800 900X 1000Y 20

0 _FB

400 500x 600 40 | tc} ' _,jLHL__40 - Y 20L "-'.-_ .....

0 "_'_:_"_" " "Y 20 800 900 10000 x 600 700 800X

40 .... 40t (di - -t. y 20 _ .,_ ..

0"--' -- --": -'-_.i..-'_.,_._, . --"ii,_- , 0 , _,_i .. ..-800 900 1000 800 900 1000X X

Figure 4.2. Material interface plots at Figure 4.3. Flowfield contours att = 8202 showing entrainment t = 8202: (a) density,of dust from the fluidized bed (b) internal energy,(FB). (c) vorticity,(d) overpressure.24


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_ 1 1 'Jl 0.6 ......... I ........ lb. t@oG .,/ I 0.4 ]

6 20-- _ 0%% /

p/S._.._ " i " !. -0.4L_ 1 =1' I I ' ", 0 200 400 600 800 _000 -0.6_- 4'l -.!

X , , , i , l i "Figure 4.4. Boundary layerthicincss 8, momentum o 500 looothicirJlCSSO, arid c_vs distance behind shock.50 L , 0.60 i. - t (ai : :- , :i , : ! (c)_

" :l ] } '

_.o- .,I ,! ii: ; . .t: , i_l ! I ;i i'- ,. I , 1 l-,ig !. 3o_ ', t ,f II, ,;i ia z ._

i' i t, o.o II l+ , '* i ' ii I lUl " l_ . ', t I , .,10- .... !'_ ,, , t t, I :': i S_ II i!l / i i i ! , ' I " ,'1,; :L/_ im _L_ :,;"_.,


28/58

I i I I i ! I I i [ I ! !

0.8 _ -:.!:_':

0.68 _'__0.4 - {3X = 400_ O 600

0.2 - A 8000 950. 0 ., I C'---- = a _ _ _ , i0.01 O.1 0.2 0.4 0.6 0.8

17BL1 i i J _ _ = J i j ,

(bl0.8

0.68 ,.'i'_S'':'' -0.4 {3 X = 400 -:_'_-_'_;*'=.:."o oo ;":;":':_:"'0.2 0 950 . F.

0 I I I 1 I f I I I I I I t I I I =0.01 0.1 0.2 0.4 0.6 0.8

T/BL

1 - (C)

0.8

' 0.6-8oIO"0.4 - {3X = 400 O 600

Z_ 8000.2- 0 950

0 t 1 l i I s 111 i i i I i I I I0.01 0.1 0.2 0.4 0.6 0.8 1_BL

Figure 4.7. Mean-flow boundary layer: (a) velocity, (b) specific volume,(c) dynamicpressure.ShadedregionsdenotedatabandsofBatt ct al. (1988). Dashed curve denotes NE mixture resultsat x = 950.

26fir


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0% la) O.2 i ' (e)O.3!_4_: B X = 400

o__r_ _ _oo _-__ _o"_ "a.0.1

00.2 Ib)

v

0.3 ,' _.

oo 0 "

0 1.0 2.0 3.0 4.0I:; I,o. ,i d:u 100-6 , i i" v'/;ld)

6-V'/_10

U'/O8 4

2 1 o'lBm

0. 0 1.0 2.0 3.0 4.0'qBL 0.1

Figure 4.8. R.M.S. fluctuating-flow profiles: o.ol o. 1 , _.o(a) streamwise, (b) transverse, T/BL(C) Reynolds stress, (cDdensity, At't(e) pressure, (f) dynamic pressure. Figure .,.:,. Loca] fluctuating-intensityDashed curve shows NE mixture profiles at x = 950.results at x = 950.


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5. DUSTY WALL JET IN A PRECURSORAirblast precursors are a shock refraction effect accompanying explosions with alarge thermal radiation output. Thermal radiation from the fireball is absorbed bythe ground and creates a heated layer of gasma thermal layer (TL)mahead of theblast wave. The shock front refracts into this high-sound-speed layer, forming theprecursor shock structure characteristic of such explosions (Glasstone, 1962). The

. major fluid-mechanic feature of this flow is a supersonic wall jet consisting of a freeshear layer (FSL) and a wall boundary layer (BL). The jet is unstable and rapidlybecomes turbulent.

P

Airblast precursor flows have been studied experimentally under a variety of con-ditions. These include small-scale laboratory experiments (Reichenbach and Kuhl,1988), 1000-1b HE tests of the DIAMOND ARC series (Reisler et al., 1988), andthe ANFO surface-burst field test (Lutton, 1988). Precursor experiments have alsobeen conducted in the AFWL 6-ft shock tube (Newell, 1987). The main drawbackof all these nonsteady experiments is that the data are always taken in laboratory-fixed coordinates, and thus there is no possibility of time-averaging the nonsteadydata to produce accurate mean-flow profiles of the turbulent wall jet and boundarylayer.A number of researchers have performed numerical simulations of airblast pre-cursors. Typically, these numerical calculations were based on the time-averagedNavier-Stokes equations, turbulent transport was then modeled; e.g.,

A mixing length model (Rosenblatt et al., 1989), based on an analytic pre-cursor solution (Mirels, 1988)

A two-equation turbulence model, such as the k- _ model (Barthel, 1985)or the k -w model (Traci and Su, 1988) A six-equation turbulence model (Donaldson et al., 1982)

Such simulations attempt to calculate the evolution of the mean flow. As mentioned. above, however, the available data are all instantaneous point measurements of theturbulent flow and cannot be used to directly check the mean-flow calculations.

In addition, these turbulence models were all developed for steady, incompressibleo

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clean flow, whereas the airblast precursor flow field is inherently a nonsteady, com-pressible dusty flow. Hence, the accuracy of such turbulence models is questionablefor airblast precursor flows.In the past we used an alternative approach; we performed a direct calculation ofthe turbulent mixing by following the dynamic evolution of the large-scale turbu-lent eddies on the computational grid. This approach was used to parametrically

. investigate precursed flows (Glowacki et al., 1986), to design precursor simulationexperiments in large shock tubes (Kuhl et al., 1985) and on large HE tests (Kuhlet al., 1987), and to simulate the turbulent, dusty boundary layer behind a shock

' (Kuhl et al., 1989).This section describes an extension of such direct calculations of the turbulentmixing. The problem considered is the interaction of a planar, constant-velocity" shock with a constant-property thermal layer over a loose soil bed. The calculatedturbulent environment was stored along similarity lines; then the solution was time-averaged to produce the mean and r.m.s, profiles of the turbulent wall jet and dustyboundary layer flow.5.1 FormulationThe problem considered here is the precursor flow field created by a planar, constant-velocity shock interacting with a constant-property thermal layer, TL (see Fig. 5.1).Dust scouring effects were incorporated in the flow by inclusion of a loose fluidizedbed, FB, along the wall below the thermal layer.A 2-D Cartesian grid was used for the computational mesh. This consisted of aslowly growing fine-mesh region (100 < i


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and a ten-cell thick; constant-property thermal layer (subscript TL):

aTc/a_ = 3.18; UTL = 0 0.3m < y < 1.3mNote that the thermal layer somld speed is about three times the ambient value.The left boundary of the mesh (x = 0, y,t) was driven by the conditions (state 2)

behind an MI = 1.7 shock:p2/Pl = 3.2; p2/pl = 2.2; e2 el = 1.46

a2/al = 1.21; u2/al = 0.94; v2/al = 0Wall drag was neglected at the bottom of the fluidized bed; hence, an inviscidsnp boundary condition (v = 0, Ou/Oy = O, Op/Oy = 0) was used at the bottom. boundary. An outflow condition was used at the upper boundary of the mesh.The calculation was run for 6,000 computational cycles to accumulate enough datafor a good statistical analysis of the turbulent flow. This required about ten CPUhours on a Cray XMP computer. The results are described in the next section.5.2 Precursor Flow VisualizationFigure 5.1 presents internal energy contours at various times near the beginningof the calculation (t = 100 to 300 ms). They show the initial formation of theprecursor shock structure (shocks P-T-I and triple point TP) and the wall jet. Thewall jet is very unstable and becomes turbulent almost immediately. At a timeof about 300 ms, a large rotational structure breaks off the rear of the jet. Thisstructure was the result of large-scale vortex merging at x __ 133 m and t = 220 ms.Figure 5.2 presents a series of snapshots of the flow that shows the evolution of theturbulent wall jet from t = 400 ms to 1000 ms. From t = 300 ms to 800 ms the wall, jet seems to propagate in a quasi-steady mode, with all features growing linearlywith time. Then, at about t = 800 ms the turbulent mixing again penetrates all theway to the wall; large-scale merging occurs and this piece of the flow is shed fromthe rear of the jet at about t = 900 ms. This is accompanied by an almost total

" collapse of the jet. At the time of about t = 1000 ms, a new precursor, P', formsand re-energizes the jet. The process then repeats itself with a period of about500 ms. This quasi-periodicity has been confirmed by a separate calculation using" a one-zone-thick thermal layer where the calculated quasi-period was 50 ms.Figure 5.3 gives a more detailed view of the flow field in the quasi-steady regime(t = 600 ms). The wall jet is supersonic; turbulent mixing on the free shear layer

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(FSL) creates weak shocklets above and below the shear layer, as depicted in theinternal energy and pressure contours of Figures 5.3a, b. The velocity field from thevortex structures on the free shear layer trips the wall shear layer (WSL) and causesit to roll up (Fig. 5.3c). Pairing of the boundary-layer vortex structures with thefree-shear layer vortex structures causes the boundary layer to separate locally fromthe wall at many locations along the wall jet. This phenomenon is similar to thestrong interactions found in low-speed wall jets (Bajura and Catalano, 1975). This effect is visible in the entropy contours (Fig. 5.3d), which show that material fromabove the free shear layer mixes all the way to the wall, and dust from the fluidizedbed mixes to the top of the free shear layer. Note that conventional turbulence

" raodels are not designed to simulate such strong interactions between two shearlayers containing vorticity of opposite signs.5.3 Wall Jet Profiles

,i

Figure 5.4 presents the time-averaged mean-flow profiles of the wall jet during thequasi-steady phase of propagation (300 ms to 700 ms). The profiles are nondimen-sionalized by the peak values and scaled by the wall jet similarity variables:

= - xsp(t)]/[xj(t)- xsp(t)],j = (y- yo)/(y,- yo)where r/j equals 1 at the centerline of the free shear layer and 0 at the top of thefluidized bed. The wall jet scaling seems to collapse the _/Um velocity profilesquite well, especially in the free shear layer region (0.5 < r/j < 1.5). At the tipof the wall jet (_ -- 1) the boundary layer and free shear layer are decoupled bya constant-velocity plateau. Near the front vortex (_ _ 0.8), one can observe aseparating boundary layer profile with an inflection point. The inflection point iscloser to the wall in the stagnation point region (_ = 0). Near the wall, the meanvertical velocities are positive, because the turbulent fluctuations scour (entrain)dust from the fluidized bed; the mean density values are quite large (ft/pe "" 10)

, for the same reason. The mean pressures are constant in the wall jet (except near= 0), indicating an isobaric mixing process, similar to other turbulent flows.

Figure 5.5 presents the root-mean-square (r.m.s.) fluctuating-flow profiles of the wall jet during the quasi-steady phase of the propagation (300 ms to 700 ms). The

wall jet scaling also seems to collapse these profiles. The streamwise velocity fluc-tuations peak in the boundary layer (r/g " 0.2) and near the center of the free shear

layer (r/j = 1.0). The transverse velocity fluctuations peak near r/j = 0.6, and arenon-zero at the bottom of the boundary layer (r/j = 0) due to turbulent fluctu-ations from the fluidized bed. The Reynolds stress peaks near the wall and nearr/j = 1. The density fluctuations are very large (p'/pe " 6) near the wall because

31


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of turbulent scouring of dust from the fluidized bed. The pressure fluctuations arerelatively constant in the jet.5.4 Discussion

A shear layer is formed when the precursor shock interacts with the thermal layer.A shear layer is also created at the top of the fluidized dust bed on the wall. The. free shear layer is unstable and rolls up into large rotational structures on top of the

wall jet. The wall shear layer is also unstable; the velocity field from the turbulentstructures on the free shear layer trips the wall shear layer, which then also rolls up." There is a strong interaction between the two; for example, gas from above the free

shear layer mixes all the way to the wall, and dust from the fluidized bed mixes tothe top of the free shear layer. Boundary layer separation occurs at many locationsalong the wall jet. Turbulence models are not formulated to model such nonlinear

. interactions between shear layers; hence the turbulent mixing in the precursor walljet should be calculated directly--as was done here.The precursor shock structure and wall jet did not continue to grow indefinitelywith time. Turbulent mixing eventually destroyed the coherence of the wall jet andlimited its propagation to about 500 thermal layer thicknesses. The flow was quasi-periodic with a duration of about 500 ms in the present calculation. During thattime, a quasi-steady phase existed where the precursor shock structure and wall jetfeatures grew self-similarly.Similarity coordinates _ and 77were used to sample the flow field during the quasi-steady phase of propagation. Time-averaging in these coordinates was used tofind both the mean and r.m.s, profiles of the wall jet and boundary layer flowwithout resorting to turbulence modeling. The calculations showed that there is nosingle boundary layer velocity profiles in the precursor wall jet; instead, the profiles" depend on _ because of gradients in the mean flow and because of increasing dustmass effects at increasing distances behind the precursor. Behind the wall jet region, the profiles are similar to the DG3 calculation of a dusty boundary layer behind aplanar shock (Kuhl et al., 1989).In closing, we recommend a direct calculation of the turbulent mixing for suchproblems--because the dust mass modifies the turbulence, and because of the com-plexity of the nonlinear interactions between the free shear layer and the wall bound-

. ary layer. In other words, one should time-average the solution, not the equations.In the future, this approach should be used to calculate the turbulent mixing innonsteady blast reflection problems.

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5.5 References

Bajura, R.A. and Catalano, M.R. (1975), Transition in a Two-dimensional PlaneWall Jet, J. Fluid Mech., 70, pp. 773-799.Baxthel, J. (1985), $-D Hydrocode Computations Using a k- _ Turbulence Model:

" Model Description and Test Calculations, SSS-R-85-7115, S-Cubed, La Jolla,California." Colella, P. and Glaz, H.M. (1985), "Efficient Solution Algorithms for the RiemannProblem for Real Gases," J. Comp. Phys., 59(2), pp. 264-289.

Donaldson, C. et al. (1982), Second-Order Closure Model: Comparison with a Num-" ber of Complex Turbulent Flows, ARAP 469, Aeronautical Research Associates

of Princeton, Princeton, New Jersey.Gilmore. F. (1955), Equilibrium Composition and Thermodynamic Properties of

Air to 2_,000 K, RM-1543, Rand Corp., Santa Monica, California.Glasstone, S. (1962), The Effects of Nuclear Weapons, U.S. Atomic Energy Com-

mission. Glowacki, W.J., Kuhl, A.L., Glaz, H.M., and Ferguson, R.E. (1986),"Shock Wave Interaction with High Sound Speed Layers," Proc. 15rh Int. Sym.on Shock Waves and Shock Tubes, eds. D. Bershader and R. Hanson, StanfordUniversity Press, Stanford, California, pp. 187-194.

Kuhl, A.L., Glowacki, W.J., Glaz, H.M., and Colella, P. (1985), "Simulation ofAirblast Precursors in Large Shock Tube," Proc. 9rh Int. Sym. on MilitaryApplications of Blast Simulation, Oxford, England.

Kuhl, A.L., Glowacki, W.J., Glaz, H.M., Ferguson, R.E., Collins, P., and ColeUa, P.' (1987), "Simulation of Airblast Precursors on Surface Burst HE Tests," Proc.

l Oth Int. Sym. on Military Applications of Blast Simulation, Bad Reichenhall,Germany.

o

Kuhl, A.L., Chien, K.-Y., Ferguson, R.E., Collins, J.P., Glaz, H.M., and Colella, P.(1989), "Simulation of a Turbulent, Dusty Boundary Layer Behind a Shock,"

(in preparation).Lutton, T. (ed.) (1988), Proc. MISTY PICTURE Symposium, POR-7187-5,Defense Nuclear Agency, Kirtland Air Force Base, New Mexico.

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Mirels, H. (1988), "Interaction of Moving Shock with Thin Thermal Layer," Proc.16rh Int. Sym. on Shock Tubes and Waves, cd. H. Gr6nig, VCH, Weinheim,Germany, pp. 177-183.

Newell, R. (1987), "Nonideal Airblast Simulation in Shock Tubes," Proc. lOth Int.Sym. on Military Applications of Blas_ Simulation, Bad Reichenhall, Germany,pp. 302-321.

Reichenbach, H. and Kuhl, A.L. (1988), "Simulation of Precursors in Shock Tubes,"Proc. 16rh Int. Sym. on Shock Tubes and Waves, cd. H. GrSnig, VCH,

Weirtheim, Germany, pp. 847-853.Reisler, R., et al. (1988), DIAMOND ARC87-Blast Phenomenology Results fromHOB: HE Tests with a Helium Layer, DNA-TR-87-99-V2-AP-A-G, Defense

Nuclear Agency, Washington, D.C.Rosenblatt, M. et al. (1989), Nonideal-Surface Airblast and Vehicle Loads Predic-

tions in Support of HML. 1. CRT-3770F, California Research & Technology,Chatsworth, California.

Sedov, L.I. (1959), Similarity and Dimensional Methods in Mechanics, AcademicPress, New York.Traci, R.M. and Su, F.Y. (1988), Turbulent, Two-Phase Flow Modeling for DustyBoundary Layer Analyses, FP1-R88-06-05, Fluid Physics Ind, Encinitas,California.

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40. I I i i I I I I I

sz. t = I00 ms -xs = 86.30m

16. TP p

o. (_7_o o " " TLJ

20. ilO. _,, aO. |00. |_0.

45. I I I I I I I I I

FigureS.1. Internal energy contours showing the formationof a precursor wall jet, t = i00 to 300 ms.

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M. ' I ' l ' I ' I '

4s. - I t = 220 msxs = 179.09 m

E

:m. T P

I t = 26(Ims49.XS l 207.09 m

37. TPE

zs. pT

13.%

O, cuelot. 128. 154. lot. 207. 234.x Cm)

Figure 5.1. (Concluded).36


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r_ _ t = 400 msXs 295.10 m.. 4, TPE>,

'8

O. ._,tsl. lO0. _. 2m. 267. _F/.

W. ' I , I ' I , i ,

t - 500 msxs - 358.38

"' TP_ P

19. ._

Oo

' I ' ' I ' ' i s I J

t 600 msr_. -- i I Xs 426,46 m

Q

a.

"_s 1 ( TP P>_ 411.T

2_,

O.2'30. 216. 321. 35"/. 3B3. q2B.x (m)

FigureS.2. Evolution of the turbulent wall jet from t = 400 to i037


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iTp t : 700 ms -

xs = 495.73mr

T24.

38


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(a)_) t : 600 ms

-'E4:. I T XsP: 426.46 mv>,,22.

WALLJETO.

2'JO. 286. 321. 3S'/. 393. J,28.

-.. c_>_,__\_,_ -:1. _- \ P -22.

O, 4SP

_ l ' ' ' ' ' ' ' ' ' "' '

I jt:)

E 22. FSL>_ BL WSL

O.x (m)

43. ' i ' I , l " , l m i(d)

O. -0.5 0 0.2 0.4 0.6 0.8 1.0 1.2

Figure5.3. Contours showing the flow-field details att = 600 ms: (a) internal energy, (b) pressure,(c) vorticSty, (dl entropy,

3


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I. IIm -m 'gO= (a)= 40 == _=Ov ,i

0.8 .2 ta .84 _ '6 *

2 .8 301.0 #1.2 m _ , -.2o.6 o

:=E l,=.. I_ BL Y* 20 _0

0.4 t

I0 ' FBlr"

0 2 FSL ,_ _I,.

FB m 0 --I "e-ro.]2 ]. 2-"'lumnumnumnimmmnnmmnmnmmmnini._/_.Cb) amffiffiimmffiffimffiffi

0.08 ,,= 1.0.FB == ,,

0.04 "== 0.8-t GI

j_ o o.6. .z o-0.04 O.4.

-0. 0.2-

-0.12 _@ 0 , ......... !-0.2 0 0.4 0.8 1.2 1.6 2.0 10.2 0 4 .8Tj I 2 l 6Tj

Figuro5.4. Mean-flow profiles of the wall jet (0 _ 1 2):,_, =_=_,,w==e velocity, tD) transversevelocity, (c) density, (d) pressure,

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I

" 0.I 16-" _=Ov

FB "% .2 u ,4ii I0 12 .6 8 "I.30- _ 1.0:(b) -= _ 12 mA

; l0.24 p, '_ FB

I _1 ql/"ip

' 4i.'p,0.18 .&INi=E _ _ 0- o

0.12 0.8i I"" _ FB

eB ii

" 0.40.06II. ,, laFB " ,"

IIt I

0 IliliiII 010.2 0 0.4 0.8 1.2 1.6 2.0 10.2 0 0.4 0.8 1.2 1.6nj njFigure5.5. R.MoS. fluctuating-flow profiles of the wall jet(0 _ _ 1.2): (a) streamwise velocity, (b) transver

velocity, (c) shear stress, (d) density, (e) pressu

++ 41i


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6. DISCUSSIONThis section explores the mechanisms of the growth of the wall layer in the contextof boundary layer theory. A complete derivation of the equations can be found inthe Appendix. We start by defining the mass thickness 6,,, which is related to theboundary layer thickness 6 according to:

. =z.,6 (6.1)Here, Im represents the integral of the mass and mass-flux profiles taken over the

" boundary layer: +']0' /0'-- (h - 1)dr/+ (1 - hf)drl (6.2)U2 where u2 denotes the gas velocity behind the shock. If the density and velocityprofiles in the wall layer are self-similar (i.e., if they are independent of x so that

h = _/p_ = h(r/) and f -- fL/U_ = f(r/), respectively), then the mass integralIm equals a constant; such was the case for the normal shock calculation whereIm _ 1.85.Next, consider the Mass Integral Equation

d [In 6/R,] = dno - voolu2 (6.3)6"= d---(which may be derived from a control volume analysis of the mass flux in the bound-ary layer. In the above, rho represents the rate that mass is being entrained intothe bottom of the boundary layer at 77- 0 due to turbulent mixing:

rho = PoVo/p2u2 (6.4)" Assuming that the velocity and density profiles are self-similar, then 6" = Ira6' and

Equation (6.3) becomes: 6'=(mo-vlu )lS., (6.5) Thus the Mass Integral Equation states that the fundamental reason that the dustyboundary grows is because of turbulent mass entrainment from the fluidized bed

(i.e., because of rho). This is true regardless of momentum considerations.In a similar manner, one can define a momentum thickness 60 of the boundary layer:

60= Io 6 (6.6)


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where= _ h(1 - f)do - hf(1 - f)drl (6.7)IL2

Again, using a control volume analysis of the momentum flux in the boundary layer,one can derive the Momentum Integral Equation (for the case where there are nopressure gradients in the freestream):

d [Io 5/R_I - rho + cii2 (6.8). where cI denotes the local wall drag. This states that the momentum thicknessgrows because of mass entrainment and wall drag. Typically, rho >> cl/2 (e.g.,

rho "_ 0.04 and cii2 < 0.001), so again one finds that the boundary layer growth iscaused by mass entrainment.Solving Equation (6.5) for the mass entrainment rate, one finds

rho = I,-, 6' + vo_/u2 (6.9)The boundary layer slope 8' may be evaluated from the empirical relation (6.13):6'= #a_ a-1

= 0.022_ -2/5 (for a planar shock) (6.10)and eliminated from Equation (6.9), yielding:

,:no= I,n _a_ _-1 + voo/u2 (6.11)Evaluating the constants in the above for the planar-shock calculation case gives:

rho = 0.041 _-2/5 _ 0.01 (6.12)This relation shows that the mass entrainment rate decays as _-2/5 of the distancebehind the shock front. The mass entrainment rate for a dusty boundary layer inli an ANFO surface burst is presented in Figures 6.1 and 6.2.The dusty boundary layer growth for a variety of problems is presented in Table 6.1.We found that the boundary layer grew as a power function

5/R, = a__ (6.13). (6.14)of the distance behind the shock Rs. The exponent was case dependent (3/5 _


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Table 6.1. Boundary Layer GrowthCASE BOUNDARY LAYER GROWTH /_

Clean Flat Plate is 6/)(. = 0.37Re_ 1/55 _-_ X 4/5 4/

SQUARE WAVE SHOCK REFLECTIONS 22Normal Shock- Case 1 5/Rs = 0.037 _a/5 (0.1 < _ < 0.7) 3/(MI = 1.7, _,_ = 0)RR- Ca_e 2 5/Rs = 0.0157_ 3/5 (0.1 < _ < 0.6) 3/(Mt = 2, Ow= 60 )SMR- Case 3 5/R, = 0.0147_ a/5 (0.1 < _ < 0.3) 3/(Mi = 2, Ow = 27 )DMR- Case 4 5/R_ = f_(_)(MI - 10, _w = 30) __ 0.0213_ a/5 (0.1 < _ < 0.25) 3/Precursor Case '7 5/Rj = f2(_)(MI = 1.7, PTL/Pl = 0.1, PFB/Pl = 50) __ 0.0325_ 5/6 5/Normal Shock, infinitely-long fluidized bed 13(MI = 1.7,PFB/Pl = 50) 5Rs = 0.024_ 1

SHOCK TUBE EXPERIMENTSNormal shockoverloosesoilbed9(MI = 1.7) 6,= 0.0325(Ax)5/6 5[x]

5,= tangentslopethicknessNormal shockalonga cleanwall9(Mz = 1.7) 5,= 0.00983(Ax)'9z 0.

DECAYING BLAST WAVESANFO SurfaceBurstoverloosedustbed(20 < Apz(psi) < 80, PFB/P, = 50) 5Rs = 0.0256_ 5/6 5Point Explosion Surface Burst over loosedust bed(1000


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' I , I ' I '

jc////0.03 0 ///

0.02 _

I

0.01 _ _

//0 I / ! i I I ! I0.6 0.7 0.8 0.9 1.0X

Fi_e 6.1. No_cnsio_ mass en_r_nt _ MOv_s_ _s_ce xan ANFO sm'face burstexplosion (Mo = xP-_o/P2u2).i i0._ , , , ,

O0.4 -

. 0.3- _mo

m

0.2 - O\x .\ !\

0.1- \ _\ \- \-0 x i I I i I t0.6 0.7 0.8 0.9 1.0

XFi___ 6.2. _ mass sco_ng .ra_ rha versus die.ce x m an

ANTO sm'fcc burst (mo = p-_olP,j.],).45


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7. SUMMARY AND CONCLUSIONSThe preceding illustrations have shown that turbulent dusty flows are dominatedby baroclinic effects. For example, vorticity is created by pressure interactions withdensity gradients. The dust scouring is a function of the local vorticity. Also, turbu-lent mixing is driven by the velocity field associated with the vorticity distribution.

* The Baroclinic Model demonstrates that dusty boundary layers grow by entrain-ment of mass from the fluidized bed -- as proven by the Mass and MomentumIntegral Equations.

Wall drag effects are unimportant for turbulent boundary layers with large dustloadings. The role of air viscosity is reduced to keeping the dust-air mixture inlocal equilibrium, while the mixture viscosity is essentially zero.The remaining issues are: (1) convergence of the numerical solution; (2) three-dimensional effects on the turbulent mixing; (3) improved constitutive relationsfor the dust-air mixture; (4) improved zoning in the boundary layer (via implicitGodunov schemes; and (5) nonequilibrium effects. These issues will be addressedin future calculations.

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APPENDIX AMASS AND MOMENTUM INTEGRAL EQUATIONS

FOR DUSTY BOUNDARY LAYERS

Described here is an analysis of the turbulent dusty boundary layer created by ashock wave that is propagating along a loose dust bed. The problem is depicted" in Figure A-1. In stationary coordinates (Fig. A-la), the shock wave S propagates

with a constant velocity Wj; states ahead and behind the shock are denoted by. subscript 1 and 2, respectively. Flow interactions with the fluidized bed FB, create

a velocity deficit (shown as the shaded regions D_ and Db) in the mean streamwisevelocity profiles. Densities increase near the wall due to entrainment of dust fromthe fluidized bed. The boundary layer grows because of turbulent entrainment ofdust.

To analyze the flow, we define the following similarity coordinates. First, assumethat the shock front propagates as a linear function of time:

R,(t) = W,t (Al)and that the shock-induced flow field above the boundary layer is constant alonglines of

x = r/R, = 1 - _ (A2)y=z/R, (A3)

These similarity lines propagate with a wave velocity' w =wo (A4)

The streamwise velocity in these similarity coordinates becomes:' =W - u =xW0- u (A5)

This transformation modifies the velocity profiles as depicted in Figure A-lb. Inthis case, the velocity begins with the freestream value fioo= xW, - uoo at the edgeof the boundary layer, and increases to a maximum value of fiw = xW_ on the wall.

A-1


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FigureA-I. Schematicftheturbulentustyboundarylayernducedbyashock(S)propagatinglongaloosedustyed(FB):(a)stationaryoordinates;o)'imilarity(xmlinatcs_.y).

A-2


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Described here is a control volume analysis of the mass and momentum balancefor such turbulent boundary layers, assuming that the mean velocity and densityprofiles, f -- u/uoo and h- p/poo, are known.Mass Integral EquationLet Fi represent the mass flux across surface i of the control volume in the similaritycoordinates of Figure Al-b. Then the streamwise fluxes across surfaces a and b aregiven by:

_0y _01F',.= p,.fia dy = (sh/ns) pafiadr/ + Poofioo(Yc - 6a/Rs) (A6)/o /o1b "" pbfibdy --- (6b/Rs) Pbfibdr + pccfioo(y_ - 6b/Rs) (A7)Similarly, the mass fluxes through the bottom and top of the control volume are:

Po= poloa_ (AS)Since the flow is steady in these similarity coordinates, then the conservation ofmass requires that the sum of the fluxes is equal to zero:

ior _o- _ +,_o- _ = 0 (A_0)Solving the above equation for the streamwise flux yields:

(_, - Fb)/Az = -poVo+ poovoo (All)iTaking the limit as Ax approaches zero and using d_ -- -dx, we find the massconservation law: d

,, d-'-__'' = povo - poovoo (Al2)where

= (6/n_,) (pfi - poofic)dr/ (Al3)A-3

,'.


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The latter represents the surplus mass flux (relative to the freestream values) cre-ated by the wall boundary layer: The mass conservation law (Eq. Al2) can benondimensionalized by the mass flux p2u2, yielding:

d [HF 6",Rs] = rho - poovoo/p2u2 (14), where

rho = poVo/p2u2 (Al5)" H(x) = Poo/P2 (Al6)" F(x)=uoo/u2 (Al7) In the above, rho represents the nondimensional mass entrainment rate, and H(x)

and F(x) denote the nondimensional flow field above the boundary layer which canbe a function of x. In addition, 6., represents the mass thickness of the boundarylayer:

6., = 6 (hi- ]co)&7 (ALS)where

h(x, rl) = p/po _ (Al9)/(x,_7) = fi/uoo = xWs/uoo - f (A20)

, /_(x)=xW_/uoo-foo (A21)f(x,71) = u/u_ (A22)

. Next, we convert the above integrand to lab-fixed velocity profiles (f = u/uoo):, h/-/oo = h[xW,/u_ - f]- xW_,/uoo + 1

_ W, x (h -1) + (1- hf) (A23)u2 F(z) The expression for the boundary layer mass thickness then simplifies to:

5.,=I.,5 (A24)Qwhere /o /o'2 F(x) (h - 1)drl + (1 - hf)drl (A25)

A-4


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Using the above relations, the Mass Integral Equation becomes:d

d_ [HF I,n 51R8] = rho - HF vooluoo (A26)This equation may be integrated to determine the growth of the mass thickness orboundary layer thickness as a function of _:

" 5m(_)/R_ = I.., 5(_)/R, = (rho - HF voo/uoo) d_IHF (A27)Thus, mass conservation in the boundary proves that the fundamental cause of

' dusty boundary layer growth is mass entrainment from the fluidized bed. Note that- this is true independent of momentum considerations (e.g., for zero wall drag).o If the freestream conditions are independent of x (e.g., in the normal shock case),then the above relations reduce to a particularly simple form:

dd--_In 5/n,] = rho- vooluoo (A2S)

and L'm(_)/R,= sm_(_)/R, = (,ho- _/u_)a_ (A29)Momentum Integral EquationNow let Pi represent the momentum flux across surface i of the control volume insimilarity coordinates. Then the streamwise fluxes across surfaces a and b are givenby:

.. Z/o /o'b = pba_ dy + pbdyI' -2 (yoo - 5,1Rs) + pbyoo (A31)(SblR_) Pbfi_dT1+ PooUooSimilarly, the momentum fluxes through the bottom and top of the control volumeI are:

Po = (poVo_W + ro)Ax (A32)-_oo= po_v_fiooAx (A33)A-5


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Since the flow is steady in these similarity coordinates, then conservation of mo-mentum requires that the sum of the fluxes equals zero:

orF_ - Fb+ Fo - Foo= 0 (A34)

Solving the above equation for the streamwise flux yields:lP

(F. -/_'b)/Ax -- --PoVofi_ - ro + p=voofioo - y_(p. - ph)lAx (A35)Taking the limit as Az approaches zero and using d_ = -dz, we find the momentum conservation law:

dd _ = poVo5W + 7"o- p_vooftoo - Yoo_ P (A36)_

where

1

= (6/n_) (o_,2 "_p_Uoo) d_7 (A37)But from the mass conservation law (Eq. A12) we recall that

dpoovoo-poVo-"_.ff'm

@This can be used to eliminate p_v_ from Equation A36, yielding

d d.--_ _'o = poVoUoo+ ro- yoo --;-;p (A38)1, ta agwhere

= (6/n.) pa(a- fioo)d_ (A39)A-6


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The latter represents the surplus momentum flux (relative to the freestream values)created by the boundary layer. The momentum conservation law (Eq. A38) can benondimensionalized by the momentum flux p2u_, yielding:

d-'_[HF26/Rsl = Fgno + HF 2 Cf/2- P2 y_ -_ G (A40)where

" Cf=ro/(O.hpccu 2) (A41)G(x) -'P_/P2 (A42), v_/p_ = (_-1)/9

In the above, C! represents the local wall drag coefficient which the fluidized bedexerts on the boundary layer, and G(x) denotes the nondimensional pressure aboveo the boundary layer which can be a function of x. In addition, 6o represents themomentum thickness of the boundary layer:

60=6 hi(i- ]_) d,7 (A43)Next, we convert the above integrand to lab-fixed velocity profiles (f = u/u_)"

hf(f - f_)=h (xW,/uoo- f) (l-f)u2 F(x) h(1 f)- hf(1 - f) (A44)

The expression for the boundary layer momentum thickness then simplifies to:,5o= Io,5 (A45)

whereu2 F(x) h(1 - f)drl .- hf(1 - f)drl (A46)

Using the above relations, the Momentum Integral Equation becomes:|d"-_[HF2 Io ,5/R_,] = Fdno + HF 2 Cf/2 - _ --_

This equation may be formally integrated to determine the growth in the momentumthickness as a function of _:

_o(_)/a, = Xo6/R," lo'(F,ho+ HF2C_/2)d(/HF_

- P_ fo_(6/R,) da d_ (A4S)2u_ HF 2 "_A-7


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Thus, momentum conservation in the boundary layer demonstrates that the mo-mentum thickness grows because of three effects: mass entrainment, wall drag andexterior pressure gradients.If the freestream flow is independent of _ such as in the normal shock case (whereH - F -- G - 1), then the above relations reduce to a particularly simple form:

, d [Xo6/R,] = 'no + C/12 (A49)d_, and

6o(_)/R,- zo6/R, = (,ho+ c_/2)a_ (A50)

A-8


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a. l. kuhl- a baroclinic model of turbulent dusty flows

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