'The structure of a shock wave close to a wall

THB FWW field ·at the foot of a shock wave, travelling along a wan, is investigated. Results of density measurements are compared with numerical data which were calculated by means of the direct Monte Carlo simulation method. Oose to the wall Jarae gradients in the flow variables develop; the influence of the parameters, shock strength, the intermolecular force Jaw and ac commodation coefficient will be diSCUSied. Finally, the first numerical results for a binary gas mixture of He-Ar show a different separation drect isl the shock wave than in the shear layer close to the wan.

'The structure of a shock wave close to a wall F. SElLER, M. WORNER and B. SCHMIDT (KARLSRUHE) THB FWW field • at the foot of a shock wave, travelling along a wan, is investigated.Results of density measurements are compared with numerical data which were calculated by means of the direct Monte Carlo simulation method.Oose to the wall Jarae gradients in the flow variables develop; the influence of the parameters, shock strength, the intermolecular force Jaw and accommodation coefficient will be diSCUSied.Finally, the first numerical results for a binary gas mixture of He-Ar show a different separation drect isl the shock wave than in the shear layer close to the wan.

Numerical treatment of tJW problem
The direct simulation Monte Carlo method, as developed by BIRD [1], has been used to obtain numerical data for comparison with the experimental results.Figure 2 shows the model used for the two-dimensional simulation.In a two-dimensional channel a piston is suddenly set in motion and moves into the stagnant gas.The gas in front of the piston is simulated by a limited number of model molecules (8000 to 15 000 molecules).The simulated flow field, being bound by the piston, the centre or symmetry line, a right border far enough from the piston and the solid wall, is subdivided into a limited number of cells.The size of a cell is such that the change of the molecular quantities is small over a cell.For the solid wall a simple accommodation model has been chosen.At the three other field boundaries the molecules are reflected specularly.For the simulation a nondimensional model-time is advanced in discrete steps.The molecules move according to their individual velocities and collision pairs are chosen randomly in a cell according to their relative speed.For the collision process an intermolecular force law is chosen.The molecular movement and the collision process are decoupled.After a certain number of discrete time steps the two-dimensional flow pattern with a shock wave moving along the solid wall evolves.The macroscopic quantities (density, pressure, temperature, mean velocity) are extracted by sampling and averaging in each cell over the different moments of the cell distribution function.Since the model number density per cell is small,.the macroscopic quantities fluctuate about an unknown mean value.The fluctuations can be reduced by taking more molecules per cell or by repeating the simulation process several times and averaging the macroscopic quantities.Limits of these procedures are set by the demand in computer storage capacity and in computing time.
___ .. _______ ___ ___ ____ ____ _ _______ .,:..__ To get information about the influence of the side walls on the flow, a three-dimensional simulation of the real flow field in the test chamber is set up.Here the possibilities are much more limited than in the two-dimensional case because the demand in storage capacity and computing time soon become excessive.
The only theoretical treatment of the problem known to the authors is a continuum approach by SICHEL [2].Sichel used modified Navier-Stokes equations to get an analytical solution for the two-dimensional problem without piston and extending to infinity perpendicular to the wall.Due to the fact that the main influence of the wall is limited to' a region very close to the wall, a comparison of Sichel's results with numerical results should allow one to draw some conclusions.The continuum approach limits Sichel's results to weak shock waves (Ms ~ 1.3).

Experiment
A simple low density shock tube of a 150 mm inner diameter was used to generate the shock waves.At the end of the driven section a 90 by 90 mm test section was attached and a square cookie cutter in front of the test section cut out a square piece of the shock wave.For the measurements close to a wall a flat plate with a sharp leading edge was inserted into the test section with the flat upper surface parallel to the top wall and with H = 77 mm below it (Fig. 4).The shock wave •adapts very fast to the new surfaces as can be deduced from the experimental results.A fully-developed flow can be expected at the location of the density measurements.The local gas density has been estimated from the measured local density gradients of a multibeam laser differential interferometer.Four interferometers with a beam spacing of 3.3 mm were arranged perpendicular .to the flat plate (see Fig. 4).The effective beam diameter in the test section was 0.19 mm.This arrangement, shown in Fig. 5 for one of these interferometers, provided a sufficient resolution in time and space if p 1 ahead of the shock wave was below ~ 13.33 N/m 2 (0.1 Torr).As Fig. 6 shows, the density profiles calculated with the use of Sichel's continuum ap-.proach and those of the Monte Carlo simulation are similar.The agreement may improve if the continuum approach is not extended to its limits (M.~ 1.3) and a ~nore appropriate intermolecular force law is used for the simulation.Further, the simulation gives averages for the density over the ceU height LJy whereas the continuum approach gives the density profile at exact y = const.

Results
In Figs. 7 to 10 experimental results for shock Mach numbers between 2.13 and 9.21 are compared with n~erica11y calculated• ones.The most interesting d~sity profile is the one closest to the wall yf,\ 1 == 0.28.For the numerical results a pure repulsive intermolecular force law, F ~ ,-•, "== 10 and full accommodation at the wall (a:= 1.0) is used.On the other hand the calculated density profiles for y I l 1 = 0.28 grow steadily to higher density values with increasing shock strength and for the same distance behind the shock lneasured ones reach a maximum at about M.   http://rcin.org.pl at the wall with increasing shock strength.Full accommodation (ex.= 1.0) seems to be appropriate for weak shocks only.
hard and a weaker repulsive force law with v < I 0 should be used or a force law with an attractive and a repulsive part (Lennard-}ones potential).
The influence of the wall decreases very fast with the distance from the wall.Two or three mean free path distances from the wall the density profiles do not go much above  For the shock strength used here (M, =-3.55), the Mott-Smitt profile is known to be almost identical to experimental results (3}.A • three-dimensional simulation (see Fig. 3) is started to obtain information•• about the distortion of the interferometer signal due to the influence of the shear layers at the windows, _ especially at the corners .close to the ftat plate.Fm. 14. Qange in the separation of a_ binary pa mJxtule in tbo resion close to tbe wall.
http://rcin.org.pl between the two-dimensional simulation and the experimental tesults.Qualitatively com-• parable experimental density profiles should fail above two-dimensional calculated ~nes.
Figure 13 shows constant density and constant temperature shock front profiles for M 11 = 3.55, calculated with the simulation method.The density profile develops a forewatd facing foot close to the wall.This foot becomes more pronounced with increasing shock strength.The temperature shock fropt does not develop such a foot.As indicated by the .: :~ insert in Fig. 14, the density measurements show this foot, too: The temperature could not be measured, but shock front arrival measurements, done with thin film gauges [4], do not show such a foot.Thin film gauges .respond to the gas temperature and density .and the temperature effect seems to override the density effect A gaskinetic explanation for the foot may be that fas~ molecules coming from the rear part of the shock wave penet• rate the slower ones at the front and are retarded by contact with the wall.This results in an accumulation of molecules in front of the shock wave close to the wall. 5. Binary gas mixture Figure 14 shows the results for a simulation • calculation for a binary gas mixture of SO% heavy gas (argon) and SO% light gas (helium).The mass ratio is 10: I.The separation of the two components in the shock wave and in the shear layer at the wail is clearly visible.In the shock wave the light gas compresses first whereas in the shear layer and close to the wall the heavy component compresses more.At more than one mean free path distance from the wall the compression of the heavy component is always behind the compression of the light component.Figure 15  Experiments on gas fuixtures are in preparation.It is planned to do density measurements by using the electron •beam luminiscence method.

CoocJusioDs .
The large gradients observed in a shock wave ~lose to a wall make this region particularly interesting for the investigation of effects connected with large deviations from local thermal equilibrium.Close to the wall the gradients are much larger than in an undisturbed normal shock wave.The comparison of calculated and experimen~l results allows the separation of the influence of the intermolecular force law and the accommodation at the wall.Full accommodation does not hold for strong shock waves and the pure repulsive intermolecular • force law ,-, with v = 10 does not give density profiles similar to the measured profiles for all shock strengths (2.13 ;S M:s ;S 9.25).The density profile clo~est to the wall is the most sensitive one to these parameters.The calculated density profiles for a binary gas mixture of SO% He -SO% At show some unexpected results.In the shock wave the heavy component lags the light one in compression whereas very close to the wall the density of the heavy component rises above that of the light one.This is not observed outside one to two mean free path distances from the wall.The large amount of numerical work has been done on the UNIV AC 1108 and the Burroughs B 7700 c;>f the university computing center at Karlsruhe and the IBM 370/168 0f the university computing center at Heidelberg.The support of this work by the Deutsche Forschungsgemeinschaft is appreciated.

FIG. 3 .
FIG. 3. Model for the three dimensional direct simulation Monte Carlo method.

F
FIG.4.Test section close to the end of the driven section of the shock tube.

FIO. 12 .
An influence of the wall is visible up to the center of the channel.Compare Mott-Smith profile and profile for the center closest boxes yfA 1 = 49.2. the value (e-(!t)/((!2-(!t) = I (see Fig. 12).Nevertheless the influence of the wall is visible up to the center of the channel.The profile closest to the centerline (Fig. 12, yf A. 1 = = 49.2) deviates markedly from the Mott-Smith profile for an undisturbed no' rmal shock.