INFLUENCE OF SPATIAL EFFECTS ON COHERENT STRUCTURES FORMATION IN A SUPERSONIC FLOW NEAR A PLATE

A large number of experimental works investigating vortex structures in threedimensional (3D) unsteady §ows of gas were published [16]. Numerical modeling of processes of instability in mixing layers accompanied by vortex structures formation was carried out in [712]. The mechanisms leading to the cross-§ow formation near the plate of ¦nite width under strong falling shock wave were described in [10, 11]. It was shown that interaction of longitudinal vortices with the transversal vortices may lead to the structures formation similar to the ̃-structures investigated in subsonic §ows [5]. In [8], receptivity of the §ow to the entropy disturbances was investigated. As was shown in [12], width and length of the plate play signi¦cant role in pressure pulsations level increase. The purpose of the present article is the cross and longitudinal structures investigation which are formed due to shock wave falling on a plate along with harmonic waves in§uence. Disturbances ampli¦cation coe©cients were determined as functions of wavelength and plate width ratio. In [9,10], investigation was conducted for Mach numbers 6 and for high values of stagnation pressure. Still unresolved was a problem to describe the §ows with smaller speeds and for smaller values of Reynolds numbers. For this purpose, in this paper, the §ow regime corresponding to Mach number 2 is investigated. Progress in Flight Physics 9 (2017) 423-434 DOI: 10.1051/eucass/201709423


INTRODUCTION
A large number of experimental works investigating vortex structures in threedimensional (3D) unsteady §ows of gas were published [16].Numerical modeling of processes of instability in mixing layers accompanied by vortex structures formation was carried out in [712].The mechanisms leading to the cross- §ow formation near the plate of ¦nite width under strong falling shock wave were described in [10,11].It was shown that interaction of longitudinal vortices with the transversal vortices may lead to the structures formation similar to the ˜-structures investigated in subsonic §ows [5].In [8], receptivity of the §ow to the entropy disturbances was investigated.As was shown in [12], width and length of the plate play signi¦cant role in pressure pulsations level increase.
The purpose of the present article is the cross and longitudinal structures investigation which are formed due to shock wave falling on a plate along with harmonic waves in §uence.Disturbances ampli¦cation coe©cients were determined as functions of wavelength and plate width ratio.
In [9,10], investigation was conducted for Mach numbers 6 and for high values of stagnation pressure.Still unresolved was a problem to describe the §ows with smaller speeds and for smaller values of Reynolds numbers.For this purpose, in this paper, the §ow regime corresponding to Mach number 2 is investigated.

PROBLEM FORMULATION
The §ow schemes are presented in Fig. 1 where Ábcd corresponds to the supersonic §ow near plate with a re §ected strong shock wave and supersonic §ow near the plate in §uenced by travelling harmonic wave.
In the ¦rst case, as a result of shock wave re §ection, boundary layer separation takes place with the unsteady vortex in the separated region [7,9,10].
In the second case, harmonic waves lead to the TolmienShlichting instability and separated zones formation having lengthscale comparable with the wavelength.
Wave re §ection from side edges of a plate transforms from originally twodimensional (2D) §ow to three-dimensional (3D) §ow.In both cases, the §ow in the wake contains transversal vortices with the parameters depending on the plate width, wavelength, and Mach number.Symmetric §ow is supposed so that only the §ow near the lower plate surface is considered.The numerical results were obtained by means of direct numerical solution of NavierStokes equations [9,10].The accuracy of calculations was estimated by the value of the boundary layer thickness.Data obtained coincide with the theoretical results for central longitudinal section.For separated §ow region, the calculated results coincided with the experimental results for cross structures thickness [1] and for pressure pulsations frequencies [2] as well.
To get nondimensional results, all geometrical parameters were divided to the plate length, §ow parameters were divided to undisturbed pressure and density, and velocity components were divided by undisturbed speed of sound.
The corresponding Reynolds number takes then 1.21.5 million length and is equal (1.21.5) • 10 6 .The number of grid points is equal to one million.

RESULTS OF CALCULATIONS OF THE SHOCK WAVE REFLECTION
Let us compare the results of calculations corresponding to di¨erent Mach numbers (M = 6 [10] and 2).The pictures of vortex structures for 4 time values (from top to down) in the form of lines of constant density are given in Fig. 2. For M = 6, the time interval of ¦elds of density is equal to dt = 4 µs, and for M = 2 ¡ 20 µs.It may be concluded that for Mach number 2 (see Fig. 2b), for smaller values of Re number, the size of vortices increases.Larger vortices promote to form new vortices arising in viscous sublayer (2).In Fig. 2, §ow Vorticity ¦elds along with the skin friction distributions are depicted in Fig. 3.It may be concluded that for M = 6 (see Fig. 3, left column) downstream from the shock wave, streaky structures arize and interact with cross vortices.This process leads to the formation of cross structures consisting of two longitudinal vortices having opposite rotation.In the region of vortices interaction, viscous sublayer arizes leading to the skin friction increase almost on two orders (Fig. 3b, left column).
In the §ow near plate for M = 2, §ow picture for vorticity ¦eld and skin friction distribution di¨er from the data obtained for larger Mach number.Here, one can see stronger viscosity in §uence along with the change of the shock wave amplitude.
Thus, comparison of the results obtained for two sources of disturbances leads to the conclusion about similarity of §ow ¦elds with the §ow velocity as a main factor in §uencing the vortices parameters.

RESULTS OF CALCULATIONS FOR THE TRAVELLING WAVE INFLUENCE
Let us consider a travelling of sound wave and in §uencing supersonic §ow near §at plate with M = 2.This problem contains additional signi¦cant parameter ¡ wavelength.It was found that nondimensional parameter proportional to the wavelength to the plate width ratio strongly in §uences the wavelength amplitude: it can diminish or grow.It was investigated the §ow near the plate having the length of 2.53 cm with the wavelength 12.5 mm.Frequency of waves changed in the range of 25100 kHz.The calculations with the amplitude of the incident wave ≈ 1% of the pressure in the unperturbed gas P 0 are presented.In Fig. 4, change of a 2D initial pro¦le of a wave near the plate having unity length is presented.From distribution of pressure on a plate (see Fig. 4, at y = 0), it is visible that on the plate border (the light line), there is a re §ection of disturbances which leads to a wave bend in the cross direction and forms a number of secondary waves.As a result, the §ow in the boundary layer becomes 3D.As a result, the amplitude of pulsations nearby the trailing edge considerably diminishes.The change of an initial pro¦le in longitudinal central section is shown in Fig. 4, TollminShlichting waves formation can be seen.The results of travelling wave in §uence on the wake §ow downstream from the plate are presented in Figs.57.In particular, for waves with short wavelength λ 1 ≈ 1 mm, the pressure distribution on a plate (at y = 0) is depicted in Fig. 5a.It may be concluded that on a narrow plate already at distance z ≈ 0.2, originally, 2D wave §ow turns to be 3D.Vortices along the plate are visible.Downstream from the plate, the strong expansion wave is formed providing gas acceleration coming from the boundary layer region.Pressure and density ¦elds  To show in §uence of the wavelength on the §ow, the calculations were made for larger wavelength λ 2 ≈ 2.3 mm.From the results presented in Fig. 6, it may be concluded that the travelling wave on the plate is deformed as well but does not separate into series new waves, that is, for such plate when its width of an order of length of the travelling wave, ¦nal destruction of an initial wave does not occur.Therefore, in cross sections behind a plate (z = 1.1 and 1.4), longitudinal vortices are not visible because their intensity is insigni¦cant.Such di¨erence in §ow structures for §ows with di¨erent wavelengths in §uences the §ow parameters.Especially, it is con¦rmed by the pressure distributions on the plate in longitudinal central section presented in Fig. 6.Here, it is shown that amplitude of waves with a small length (curve 1) nearby plate leading edge slightly grows up and then, after waves destruction (z ≈ 0.2), considerably decreases.Amplitude of waves with the doubled length (curve 2) increases more in comparison with initial amplitude.Wavelength λ ≈ 2δ is also equal in this option to the plate width h, having boundary layer thickness δ = 0.9 mm.In Fig. 7a where pressure ¦eld is presented for the wide plate and for the travelling wave with the wavelength λ 2 , it is visible that the wave does not break up to separate waves.Dependence of number of vortex structures on long and short waves is presented in Figs.7b and 7c for z = 1.5.It is visible that number of vortex structures is in inverse proportion to the wavelength which follows from the pressure distribution in the wake (Fig. 7d ).Dependence of amplitude of pulsations on a wide plate from wavelength is shown in Fig. 7e.Just as on a narrow plate for short waves (3) at the leading edge of plate, there is a destruction of a wave and amplitude of pulsations falls.For doubling wavelength λ 2 = 2λ 1 , there is a sharp growth of amplitude of a wave (1).However, at further increase in wavelength twice (λ 3 = 2λ 2 ), its amplitude falls almost twice (2), that is, there is a certain wavelength providing the maximum strengthening of disturbances.Comparison of pressure distributions on the narrow (Fig. 6d ) and on the wide plates (Fig. 7e) shows that strengthening of amplitude of waves is twice more on a narrow plate.It means that strengthening of disturbances on a plate depends both on the plate width and on the wavelength of the travelling wave.

THE ANALYSIS OF DEPENDENCE OF AMPLIFICATION OF PRESSURE PULSATIONS ON THE WAVELENGTH
In Fig. 8, the ampli¦cation coe©cients -P max /-P 0 are presented that depend on the dimensionless parameter proportional to the wavelength ratio to the boundary layer thickness δ.It may be concluded that the maximum strengthening takes place at wavelength equal to 2 boundary layer thickness.For this case, sig-ni¦cant increase in amplitude of travelling wave takes place.In §uence of width of a plate on ampli¦cation coe©cient for two waves with di¨erent wavelengths is given in Fig. 8a where curves 1 and 2 show essential distinction in ampli¦cation coe©cient for narrow and wide plates, that is, for narrow plates when interaction of side vortices is essential.It is possible to explain e¨ect of strengthening of amplitude by means of waves interaction with the boundary layer §ow.When the wave moves along a mixing layer, there is its refraction because of action of a positive and negative phase of a wave.The bend of a mixing layer leads to strengthening of pressure in a positive phase and to reduction in the negative.When wavelength is less than δ, these changes are compensated in the boundary layer, that is, the gain and reduction of pressure extinguish themselves without amplifying, without reaching a plate.In a case when wavelength is of the order of 2δ, compression and expansion waves reach a wall and amplify in amplitude.In the process of movement of the wave along the mixing layer, there is an accumulation of these disturbances that leads to the further growth (reduction) of amplitude of a wave and formation of cross vortices.In fact, a typical KelvinHelmgolz instability takes place.As in the problem with a shock re §ection on side borders, in the cross direction sound waves exist and the scenario of development of instability is, probably, the same.But here, waves interact in the middle of a plate and are re §ected from each other.When wavelength of the waves re- §ected from the side borders several times is less than a half of width of a plate, collision of phases of compression and expansion occurs not on the plate border but in its central part.In this place, the weak vortex arize.But there is the main longitudinal speed and, as a result of this interaction, the longitudinal vortex is formed.As the length of a cross wave is less than h/2 for several times, some longitudinal vortices of weak intensity which are visible in Figs. 4 and 5 are formed.In the case when the length of a cross wave is of the order of h, in internal area on a plate, the longitudinal vortices are not formed (see Figs. 6 and 7a).It should be noted that the analysis of laminarturbulent transition [8] in 2D formulation is not correct and should take into account the 3D e¨ects.

CONCLUDING REMARKS
Generalizing results of experimental and numerical modeling, the known fact is justi¦ed.This fact is associated with the secondary §ow instability arising due to 3D e¨ects.It means that any 3D surface disturbance may lead to the longitudinal vortices and, eventually, to the secondary instability.In particular, in the problem of re §ected shock wave weak longitudinal vortices interacting with the lateral vortices can arise and, as a result, cross structures can appear.Gas emission from the boundary layer provides transition to the turbulence.In the second problem associated with the travelling waves in §uencing the §ow near the plate, there is an additional parameter, the wavelength.Depending on the ratio of the wavelength to the plate width, di¨erent solutions are realized in the wake of the plate and the intensity of the wave on the plate becomes stronger or weaker.In the case when wavelength is less than the width of the plate, as a result of interaction of the travelling 2D wave and vortices arising on lateral parts of the plate, the travelling wave becomes 3D.When wavelength is comparable or more than the width of the plate, vertical structure does not arise.The maximum ampli¦cation of travelling wave corresponds to narrow plates and waves having wavelength larger than the width of the plate.

Figure 1
Figure 1 Flow shemes nearby the §at plate: 1 ¡ strong shock wave falling on the plate; and 2 ¡ harmonic wave

Figure 4
Figure 4 Pressure ¦eld of harmonic wave in the §ow on a plate (Á) and in longitudinal central cross section (b)

Figure 5 Figure 6
Figure 5 Flow parameter distributions in the case of §ow past for the ¦nite-length plate: (a) pressure on the plate; (b) and (c) pressure and density in the central longitudinal section of the computational domain; (d ) and (e) vortex structures in the transverse sections z = 1.17 and 1.4, respectively

Figure
Figure Vortex structure on the wide plate: (a) wave structure of the §ow on the plate; (b) and (c) vortex structures in the transverse sections of the wake (z = 1.5) for long and short waves; (d ) comparison of the wake pressures (y = 0 and z = 1.5) for the long (1) and short waves (2); and (e) pressure §uctuation growth on the plate for the long (1 and 2) and short waves (3)