Experimental investigation of heating augmentation by particle kinetic energy conversion in dust-laden supersonic flows

Abstract

The presence of particles in supersonic flows can cause significant increases in stagnation point heat fluxes (Dunbar et al. in AIAA J 13:908–912, 1975). This effect is commonly named ‘particle-induced heat flux augmentation’ or just ‘heating augmentation.’ Heating augmentation can be described as the sum of the conversion of kinetic energy of the particles into thermal energy, characterized by the energy conversion efficiency, also called accommodation coefficient, and the increase of convective heat flux (Polezhaev et al. in High Temp 30:1147–1153, 1992; Vasilevskii and Osiptsov in Experimental and numerical study of heat transfer on a blunt body in dusty hypersonic flow 33rd thermophysics conference, American Institute of Aeronautics and Astronautics, 1999). Although the accommodation coefficient is fundamental for heating augmentation characterization, there is only a small number of experimental datasets for it. This work focusses on the experimental determination of the accommodation coefficient in flow regimes at Mach number 2.1, Reynolds number, based on the probe nose diameter, from approx. 6e5 to 1.8e6, and nominal particle sizes of approx. 20 µm. The decrease of particle velocity and kinetic energy flux in the shock layer is measured with highly resolved shadowgraphy for individual particles. The particle kinetic energy flux is decreased by 29% on average by particle deceleration in the shock layer. Negligible kinetic energy fluxes of rebounded particles were measured. The accommodation coefficient is approx. 0.36 for Al₂O₃ and SiO₂ particles, while it is approx. 0.09 for MgO particles. Hence, it is significantly smaller than the widely used value of 0.7, based on the study of (Fleener and Watson in Convective heating in dust-laden hypersonic flows 8th thermophysics conference, 1973), but in good agreement with values given in (Hove and Shih in Reentry vehicle stagnation region heat transfer in particle environments 15th aerospace sciences meeting, 1977) and (Molleson and Stasenko in High Temp 55:87–94, 2017. https://doi.org/10.1134/S0018151X1701014X ). No difference between erosive and elastic particle reflection mode was detected on the conversion efficiency. The data from a simplification of the modeling approach of the conversion efficiency for elastic particle reflection by Molleson and Stasenko (2017) are in poor agreement with experimental data.

1 Introduction

In the early 70’s, stagnation point heat fluxes in dust-laden supersonic and hypersonic flows were measured first to be far in excess of heat fluxes in dust-free flows (Dunbar et al. 1975). This heat flux increase is commonly named ‘particle-induced heat flux augmentation’ or ‘heating augmentation.’ A wide range of physical effects were identified to explain heating augmentation, among them were the conversion of kinetic energy of incident particles into thermal energy (Fleener and Watson 1973; Polezhaev et al. 1992), rebounded particles which were crossing the bow shock and generating additional shear layers or ring vortices (Alkhimov et al. 1982; Holden et al. 1976; Kudin et al. 2013; Levin 1993), turbulence increase of the boundary layer (Bakum and Komarova 1971), the change of the bow shock structure by particle accumulations in front of the probe (Kudin et al. 2013), general bow shock disturbances by particles (Dunbar et al. 1975; Fleener and Watson 1973), and a change of surface roughness and erosion (Dunbar et al. 1975; Osipov et al. 2001).

A common approach to describe heating augmentation is the differentiation of conversion of kinetic energy of the particles into thermal energy and the increase of convective heat flux (Polezhaev et al. 1992; Vasilevskii et al. 2002):

$$\dot{q}_{{\Sigma }} = \dot{q}_{0} + {\Delta }\dot{q} + \dot{q}_{{\text{a}}}$$

(2.1)

The measurable stagnation point heat flux in a dust-laden flow ${\dot{q}}_{\Sigma }$ is the sum of the stagnation point heat flux in a dust-free flow ${\dot{q}}_{0}$, the increase of convective heat flux due to the presence of particles $\Delta \dot{q}$, and the direct energy transfer from particles into the probe ${\dot{q}}_{\text{a}}$.

The direct particle energy transfer during impact consists of two parts: the conversion of kinetic energy into thermal energy and heat conduction due to differences between particle temperature T_p and probe surface temperature T during particle impact:

$$\dot{q}_{a} = a_{{{\text{kin}}}} *\left( {\dot{e}_{{\text{kin inc}}} - \dot{e}_{{\text{kin reb}}} } \right) + a_{{{\text{heat}}}} *G_{{\text{p}}} * c_{{\text{p}}} *\left( {T_{{\text{p}}} - T} \right)$$

(2.2)

This formulation assumes that all particles have the same particle temperature during impact. The parameter a_kin is the energy conversion efficiency, also called accommodation coefficient or a_kin coefficient, describing the “effectiveness” of particle kinetic energy to be converted into probe heat. The values ${\dot{e}}_{\text{kin inc}}$ and ${\dot{e}}_{\text{kin reb}}$ are the kinetic energy flux of incident and rebounded particles. The factor a_heat and G_p are the thermal energy conversion efficiency coefficient and the particle mass flux of incident particles, respectively. The heat capacity of particles is described with the factor c_p.

The formulation of Eq. (2.2) deviates from those given in literature (Golubkina et al. 2011; Vasilevskii and Osiptsov 1999). This is caused by the fact, that those studies assumed uniform particle sizes as well as uniform particle velocities at impact, as it is also done in (Polezhaev et al. 1992). The kinetic energy flux of rebounded particles ${\dot{e}}_{\text{kin reb}}$ is commonly neglected (Fleener and Watson 1973; Polezhaev et al. 1992), as well as the heat conduction aspect, namely the second term on the right-hand side of Eq. (2.2) (Fleener and Watson 1973; Kudin et al. 2013; Polezhaev et al. 1992). This can be reasoned by the short collision time of the particles with the probe surface (Molleson and Stasenko 2014; Polezhaev et al. 1992).

The accommodation coefficient a_kin depends on the reflection modus between particles and probe surface (Hove and Shih 1977; Levin 1993; Molleson and Stasenko 2017). The reflection can be described as elastic or inelastic/erosive. An elastic reflection is described for example in (Lashkov 1991) and strongly depends on the material properties of the probe and the particle, as well as on the impact velocity and the impact angle related to the surface. If the particle velocity exceeds a certain value at impact, no rebound from the probe surface occurs and the reflection can be termed erosive; otherwise, it is elastic or partially elastic (Molleson and Stasenko 2017). This critical value is called limiting impact velocity V_p* in the following and is defined in (Molleson and Stasenko 2017; Stasenko 2007):

$$V_{{{\text{p}}^{*} }} = \left[ {\frac{3}{{\rho_{{\text{p }}} }}\left( {\frac{1}{{ \sigma_{{\text{p}}} }} + \frac{1}{{ \sigma_{{\text{s}}} }}} \right)^{ - 1} } \right]^{1/2}$$

(2.3)

The parameters ρ_p, σ_p and σ_s are the particle density and the elastic limit of the particle and sample material, respectively. It must be noted that V_p* is not equivalent to the so-called ‘critical velocity’ of cold spray applications, at which particles start to deposit on the probe surface. Although it is stated that there is no critical velocity for brittle materials like ceramics (Schmidt et al. 2009), the authors have made a comparison between both velocities for classification purposes. Taking the formulation given in (Schmidt et al. 2006) into account and assuming identical particle and probe material, V_p* is approx. 2–4 times lower than the critical velocity for several materials.

The authors of (Molleson and Stasenko 2017) have deduced a theoretical model describing the energy flux from particles to the solid for elastic reflection. This model was only based on the experimental data of (Kudin et al. 2013). It includes normal and tangential velocity components of incident, rebounded, and chaotized particles. Chaotized particles are incident particles interacting with rebounded particles. The authors of (Molleson and Stasenko 2017) mentioned that the concentration of chaotized particles is low in flows with initial particle mass concentrations up to 30%. Since in this work only particle concentrations lower than 1% were investigated, it is assumed that chaotized particles are of minor importance in the following analyses. Considering only the normal velocity components and neglecting chaotized particles, the parameter a_kin can be modeled with the following equations:

$$a_{{{\text{kin}}}} = \frac{{\dot{q}_{{{\text{SP}}}} }}{{\dot{e}_{{\text{kin inc}}} - \dot{e}_{{\text{kin reb}}} }} = \left[ {1 + \frac{{\sigma_{{\text{s}}} }}{{\sigma_{{\text{p}}} }}\left( {\frac{{b_{{\text{s}}} }}{{b_{{\text{p}}} }}} \right)^{3} } \right]^{ - 1}$$

(2.4)

with:

$$b_{{\text{j}}} = \left( {\frac{{E_{{\text{j}}} }}{{\rho_{{\text{j}}} }}} \right)^{0.5} {\text{of j}} - {\text{material}}$$

(2.5)

The parameters ${\dot{q}}_{\text{SP}}$, E_j and ρ_j are the measured stagnation point heat flux, the Young’s modulus and the density of the j-material, respectively. The heat conduction aspect is neglected in Eq. (2.4).

If particles reflect erosively/inelastic, no a_kin formulations but distinct values are given in literature. All literature-listed a_kin values for both reflection modes, elastic and erosive, are summarized in Table 1.

Table 1 Summary of a_kin coefficients from literature

(Fleener and Watson 1973) provided the largest dataset of the a_kin coefficient, containing 20 measured values in the range of 0.5–0.9. They have made heating augmentation tests in a hypersonic Mach number 6.1–9.5 air flow regime and 100–200 µm large SiC and MgO particles. The achieved particle mass concentrations were in the range of 0.001–10% (Dunbar et al. 1974, 1975; Fleener and Watson 1973). The stagnation point heat flux was measured with backface thermocouples. The a_kin coefficient was derived from the calculated heat flux at that time at which the probe surface reached the gas stagnation temperature in a dust-laden flow. The authors of (Kudin et al. 2013) measured an a_kin coefficient of 0.17 in a Mach number 2.3 combustion gas and nitrogen flow regime and 27–111 µm large SiO2 particles. The particle mass concentration was up to 25%. Both studies approximated the reduction of particle impact velocity in the shock layer and measured a time-averaged particle mass flux to calculate the incident particle kinetic energy flux. A detailed summary of selected experimental studies is given in Table 12 in the Appendix. This selection considers only studies in which explicit heating augmentation data were presented.

There is only a limited number of experimentally determined a_kin coefficients for a limited number of materials (see Table 1). Many particle-induced heating augmentation studies, among them (Golubkina et al. 2011; Hove and Shih 1977; Hove and Taylor 1976; Ishii et al. 1990; Levin 1993; Polezhaev et al. 1992), are based only on the results of (Dunbar et al. 1975). An extension of experimentally determined a_kin coefficients in supersonic dust-laden flows is therefore of high importance. This work presents new experiments concerning the determination of the a_kin coefficient. While the heat conduction aspect of Eq. (2–2) is neglected, both particle kinetic energy fluxes, namely ${\dot{e}}_{\text{kin inc}}$ and ${\dot{e}}_{\text{kin reb}}$, are derived from individual particle sizes and velocities, which are measured with highly resolved shadowgraphy. This spatially and temporally resolved measurement technique was tested and validated in previous investigations (Allofs et al. 2022). It allows analyzing both, incident and rebounding particles. To the best knowledge of the authors, this is the first time that temporally resolved individual particle sizes and velocities are accounted for the analysis of the accommodation coefficient a_kin.

The purposes of this work are summarized as follows:

• Measurement of velocity and kinetic energy reduction of individual particles within the shock layer

• Determination of particle reflection mode by considering V_p*

• Check, if the common neglection of the kinetic energy of rebounded particles for a_kin coefficient determination is appropriate

• a_kin coefficient determination for 20–60 µm sized Al₂O₃, MgO, and SiO₂ particles in a Mach number 2.1 air flow regime

• Comparison of the a_kin coefficient formulation by (Molleson and Stasenko 2017) to experiments

2 Methods

2.1 Test facility setup

Tests were performed in the multi-phase flow facility (GBK), which is a blow down facility, using high-pressurized air from reservoir tanks. The maximum design pressure of the GBK facility is 5.4 MPa, while the maximum design temperature is 800 K. Two different air flows can be controlled automatically: A heatable pure air flow, named ‘main’ flow, and an unheated dust-laden flow, named ‘bypass’ flow in the following. Particles were seeded with an in-house developed seeding device. A detailed description of the GBK can be found in (Allofs et al. 2022). The GBK measurement section is illustrated in Fig. 1.

Fig. 1

Sectional side view of the GBK measurement section, dimensions given in mm (adapted from (Allofs et al. 2022))

The heatable main flow was mixed with the cold two-phase bypass flow in the injection adapter. Behind the injection adapter, a stagnation chamber with a diameter of 70.3 mm and an ideal-contoured Ma = 2.1 de-Laval nozzle with a nozzle exit d_nozzle of 30 mm were placed. The nozzle flow was directed into a diffusor, which was located at the end of a test chamber. The flow stagnation temperature T₀ was measured with a 1.1 mm diameter type K thermocouple in front of the nozzle. The stagnation pressure p₀ was reconstructed by means of the wall pressure close to the T₀ sensor.

2.2 Shadowgraphy

Particles were analyzed with a highly resolved shadowgraphy system. This system was one of two implemented optical systems, namely PTV and shadowgraphy, whose setup were the same as described in (Allofs et al. 2022). An overview of both optical setups is sketched in Fig. 2. In this study, only data of the shadowgraphy system are considered.

Fig. 2

Front view on the PTV and shadowgraphy setup, dimensions given in mm (adapted from Allofs et al. 2022)

The illumination source was a ‘SpitLight DPSS 250 PIV’ laser system of InnoLas Laser GmbH, generating two light pulses with a nominal time separation (Δt) of 400 ns at a repetition rate of 100 Hz and a wavelength of 532 nm. The 532 nm laser light was used for the PTV measurements and for feeding the shadow diffusor of Dantec Dynamics GmbH. The shadow diffusor generated a background illumination with a wavelength range of 532 nm to approx. 564 nm. Several wavelength filters were used to separate PTV and shadowgraphy measurements properly. Only light in the wavelength range of 550 nm to approx. 564 nm was used for shadowgraphy measurements by implementing 550 nm long-pass filters behind the shadow diffusor and in front of the shadowgraphy cameras. The 532 nm laser light is marked green and the shadowgraphy background illumination is marked yellow in Fig. 2. The maximum illumination area of the shadow diffusor was 112 mm in diameter. With a set of half-wave plates and beam splitters, the energy for shadowgraphy and PTV illumination was controlled. A maximum pulse energy of 30 mJ was used to feed the shadow diffusor. An additional photodiode was used to control and to correct the pulse time separation Δt. The timing of laser and cameras as well as the camera data acquisition was controlled by a PTU-X timing unit of LaVision and the LaVision DaVis ParticleMaster software V10.1.

The high-magnification shadowgraphy system consisted of two LaVision Imager sCMOS cameras (named C1 and C2 in Fig. 2), having a pixel size of 6.5 µm. A long-distance microscope K2 Distamax of Infinity Photo-Optical Company was equipped with a CF-1b lens, a ‘Zoom Module’, and an optical beam splitter so that both cameras used the same optics. To avoid double exposure by the 100 Hz laser system, only a central sensor area of 1060 × 2560 px² for C1 and C2 was used, ensuring a double-image rate of 50 Hz for each camera. The cameras were recording one after the other, resulting in a shadowgraphy double-image recording rate of 100 Hz. The optical magnification was increased to 327.5 px/mm by increasing the optical amplification level of the ‘Zoom Module.’ The aperture control of the long-distance microscope was set to the middle position, resulting in an aperture opening of approx. 17 mm in diameter. C1 and C2 were equipped with 550 nm long-pass filter. The working distance between lens and focus plane was 361 mm. The resulting shadowgraphy field-of-view (FOV) was 3.2 × 7.8 mm². An example of the entire shadowgraphy FOV is given in Fig. 3. Shadowgraphy was capable to see a part of the incident flow, the bow shock, the shock layer, as well as a part of the probe tip. Incident particles were analyzed in an area in front of the shock, called ‘freestream’ analysis area and marked with a purple rectangle, while rebounded particles were analyzed in the shock layer, called ‘shock layer’ analysis area and marked with an orange rectangle. The distance between shock and probe called the shock stand-off distance (Δ) was measured for each test. On average, the incident particles analysis area was approx. 1.4 × 4.8 mm², while the rebounded particles analysis area was approx. 1.8 × 4.8 mm². The freestream analysis area for the detection of incident particles was kept similar to those described in (Allofs et al. 2022) to comply with the validated results of particle mass flux. The effect of the analysis area selection on the resulting incident particle mass flux and incident particle’s kinetic energy is covered in Sect. 3.1.

Fig. 3

Sketch of shadowgraphy’s FOV: ‘freestream’ analysis area (purple) in front of the bow shock, while the ‘shock layer’ analysis area contains the entire shock layer (orange)

DaVis ParticleMaster software measured the individual particle velocity with an implemented particle tracking algorithm. Particle shadow displacements were between 40 and 80 px. Particle size and particle mass flux were measured with LaVision DaVis ParticleMaster V10.1 software and an additional particle size and measurement volume correction. The analysis procedure is described in detail in (Allofs et al. 2022). The smallest detectable particle size was 6 µm. The depth-of-field (DOF) was less than 6 mm for particles smaller 60 µm.

Velocity filters were applied to distinguish between incident and rebounded particles. These filters are listed in Table 2 and Table 3, respectively. The freestream gas velocity was calculated with one-dimensional isentropic relations and is named ‘theoretical velocity’ in the following.

Table 2 Overview of applied filters to detect incident particles

Table 3 Overview of applied filters to detect rebounded particles

Individual particle characteristics were used to calculate the particle kinetic energy flux ${\dot{e}}_{\text{kin}}$. Assuming spherical particles and a constant particle density ρ_p for all particles, the particle kinetic energy flux was calculated as follows:

$$\dot{e}_{kin} \left( {t,X,Y} \right) = \frac{4}{3}*\pi *\frac{{\rho_{{\text{p}}} }}{{{\text{Lx}}*{\text{Ly}}}}*\mathop \sum \limits_{i = 1}^{{n_{{\text{p}}} \left( {t,X,Y} \right)}} \frac{{\left( {\frac{{d_{{{\text{p}} i}} }}{2}} \right)^{3} *V_{{{\text{p}} i}}^{3} }}{{d_{ i} }}, \left[ {\dot{e}_{{{\text{kin}}}} } \right] = \frac{W}{{m^{2} }}$$

(3.1)

The Volume-Of-Interest (VOI) has the dimensions Ly and Lx with the thickness d. It is located at the position X and Y. All particles in VOI at time t are summarized with n_p (t, X, Y). Shadowgraphy’s measurement volume thickness d is a function of the particle diameter d_p, and hence, it must be placed within the sigma sign in Eq. (3–1). The parameter V_p is the magnitude of the particle velocity.

The particle kinetic energy flux was calculated for each shadowgraphy image. It was additionally smoothed with a rolling-average filter with a length of 25 samples. This filter was used to reduce strong oscillations in particle kinetic energy flux and the resulting a_kin coefficient data. Those oscillations were caused by several “empty” shadowgraphy images in which no particles were detected.

The resulting particle size uncertainty for all particle sizes is 1.25 µm (Allofs et al. 2022). The overall particle velocity uncertainty is 0.2 px. It was found that the overall particle mass flux measurement uncertainty is 30% and is dominated by particle mass flux variations due to changes in particle detection settings (Allofs et al. 2022). The authors assume that a measurement uncertainty of the same order of magnitude can be applied to the measured particle kinetic energy fluxes.

2.3 Probe installation

The stagnation heat flux was derived from temperature data of a type-E coaxial thermocouple. This thermocouple was manufactured by the Supersonic and Hypersonic Flow Technologies Department of DLR. It had a thread-less 1.9 mm diameter tip and a metric M2.5 thread for mounting. Its total length was 18 mm. The coaxial thermocouple was mounted into a hemispherical-shaped axisymmetric probe head made of stainless steel (1.4539) and fixed with a nut. The probe head was insulated to decrease the coaxial-thermocouple signal noise. The probe head and the insulator were clamped into the head mount with clamping screws. Alignment edges were used for accurate re-positioning of the probe head. The head mount including the probe head and the insulator had a total length of 60 mm. The probe nose diameter d_Probe was 12 mm. A sketch of the coaxial-thermocouple installation as well as the probe head is illustrated in Fig. 4. The distance between probe tip and nozzle exit varied between 4 and 6 mm, which was caused by thermal expansion of the nozzle.

Fig. 4

Sketch of probe setup, dimensions given in mm

The probe head as well as the coaxial thermocouple was polished with a 2000-grit sandpaper before each test run. Subsequently, an activation of the thermocouple was done. The seeded particles slightly eroded the probe head, resulting in a change of the probe head shape from hemispherical to parabolic. The probe head shape was scanned once with a Keyence VR-5000 optical microscope after the tests made with Al₂O₃ and before tests made with SiO₂ and MgO. The probe shape profiles of a vertical (z = 0 mm) and of a horizontal (y = 0 mm) plane are depicted in Fig. 5. A hemispherical profile is shown for reference purposes. The probe hat shape change is underlined by the decrease of the shock stand-off distance Δ from 1.93 to 1.68 mm during the entire test series (see Table 8, tests are not sorted in chronological order).

Fig. 5

Probe shape during the test series, horizontal (y = 0 mm) and vertical (z = 0 mm) profile

The following equations are based on the Cook–Felderman equations (Cook and Felderman 1966) and were used for stagnation point heat flux $\dot{q}_{{{\text{SP}}}}$ calculation (like in (Thiele et al. 2018)):

$$\dot{q}_{{{\text{SP}}}} \left( {t_{{\text{n}}} } \right) = \frac{{2\sqrt {\rho {\text{ck}}} }}{\sqrt \pi } \mathop \sum \limits_{i = 1}^{{\text{n}}} \frac{{\Theta \left( {t_{{\text{i}}} } \right) - {\Theta }\left( {t_{i - 1} } \right)}}{{\left( {t_{{\text{n}}} - t_{i} } \right)^{0.5} + \left( {t_{{\text{n}}} - t_{i - 1} } \right)^{0.5} }}$$

(3.2)

with:

$${\Theta }\left( {t_{i} } \right) = T\left( {t_{i} } \right) - T\left( {t_{0} } \right)$$

(3.3)

The factor ρck is the coaxial-thermocouple material constant, and θ_ti is the difference between stagnation point surface temperature T at time t_i and the initial stagnation point surface temperature at time t₀.

The Eq. (3.2) is based on the theory of one-dimensional heat conduction into a semi-infinity body with constant material properties. The authors assume that lateral heat fluxes can be neglected in this analysis. The given equation is only valid as long as the temperature at the coaxial thermocouple’s back end T_back is constant. T_back was measured with a second thermocouple placed at the coaxial thermocouple’s back end. The authors assumed that Eq. (3.2) can be applied as long as follows:

$$\Theta_{{{\text{back}}}} \left( {t_{i} } \right) = T_{{{\text{back}}}} \left( {t_{i} } \right) - T_{{{\text{back}}}} \left( {t_{0} } \right) < 0.1 K$$

(3.4)

The thermocouple’s raw signals were recorded with 50,000 Hz. In a first step, the raw signal was smoothed with a 500-sample-length rolling-average filter and down-sampled to the shadowgraphy’s recording rate of 100 Hz. This signal was additionally smoothed with a second rolling-average filter with a length of 25 samples to agree with shadowgraphy data smoothing. The resulting temperature signal was used for heat flux computation.

The measurement uncertainty of the heat flux is determined as follows: The material constant uncertainty is assumed to be 5%. The measurement uncertainty for each of the down-sampled temperature signal was assumed to be the standard deviation of all considered 500 temperature samples. The heat flux measurement uncertainty was calculated by applying linear error propagation theory.

2.4 Particles and material constants

Three different particle materials were used for seeding, namely alumina (Al₂O₃), magnesium oxide (MgO) and silica (SiO₂). Volume-based diameters are given in Table 4 for a rough size characterization. As an example, 90% of the entire particle volume is cumulated on all particles smaller than d_V90; 10% of the entire particle volume is cumulated on all particles smaller than d_V10. The shown values are taken from shadowgraphy measurements at the reference flow condition (see Sect. 2.5). Additional detailed particle size distributions can be found in (Allofs et al. 2022). A pycnometer was used to measure particle density ρ_p. While Al₂O₃ and SiO₂ densities were close to values from literature, MgO density was determined to be significantly smaller. This was explained by a partial chemical reaction of MgO with humidity to Mg(OH)₂, which has a density of 2380 kg/m³ (Allofs et al. 2022). Considering the measured particle density of 3134 kg/m³, the nominal MgO material could be a MgO/Mg(OH)₂ mix with a mass mixing ratio of 63% MgO–37% Mg(OH)₂. Since MgO is dominating, the authors assume that the mechanical properties of that mix can be approximated with those of MgO. Furthermore, this material is named ‘MgO’ in the following. The parameter σ is the Yield strength or the elastic limit of the material and were taken from (Molleson and Stasenko 2017) and (Azom 2023). Table 4 lists all relevant particle properties. Characteristic probe and coaxial-thermocouple material properties are summarized in Table 5. These values were required for the modelling processes.

Table 4 Particle properties

Table 5 Probe and coaxial-thermocouple material properties

The limiting impact velocity V_p* is calculated with Eq. (2.3) and summarized for each particle/probe material combination in Table 6. Material properties of chromel were taken into consideration for following analyses, because the major cross section of the coaxial-thermocouple is made of it and because lateral heat conduction from the stainless-steel probe head into the coaxial-thermocouple is neglected in the formulation given in Eq. (3.2). The resulting V_p*, calculated with chromel material properties, is plotted in Sect. 3.1 and is marked bold in Table 6.

Table 6 V_p* for all particle/probe material combinations

2.5 Test conditions

A total of 16 tests were performed. While the nozzle contour and the resulting Mach number remained constant, T₀, p₀, and particle material were varied. The Reynolds number Re_dProbe, based on the probe nose diameter d_Probe and freestream conditions, ranged from 6e5 to 1.8e6. Specific flow properties are assumed to be constant, namely the gas constant R, the specific heat ratio γ, the specific heat capacity c, as well as the Mach number and the Prandtl number Pr. Their values can be found in Table 7.

Table 7 General flow constants

An overview of all tests is given in Table 8. Tests are sorted by particle material, T₀, and p₀. The temporal mean of the particle mass concentration is expressed by c_m. The reference flow condition was T₀ = 373 K and p₀ = 0.96 MPa. Four different p₀ levels, namely 1:0.6 MPa, 2:0.96 MPa, 3:1.3 MPa, and 4:1.7 MPa, and three different T₀ levels, namely 1:303 K, 2:338 K, and 3:373 K, were tested with Al₂O₃ particles. For the other materials, only the variation in p₀ was performed. For each run, a synonym was defined in the form: material–temperature level/pressure level. For example, the synonym ‘A-32’ stands for the run with Al₂O₃ particles, on the third temperature level of 373 K, on the second pressure level of 0.96 MPa. The active seeding time was set to 10 s. Table 8 also lists the air mass flow ${\dot{m}}_{\text{air}}$, the Reynolds number based on the probe nose diameter d_Probe, and the shock stand-off distance Δ.

Table 8 Test matrix, sorted by particle material, T₀, and p₀

The injection time of the probe was set to 185 s. The idea was to heat the probe in a dust-free flow up to a quasi-stationary temperature level, so that convective heating effects can be excluded. This procedure is similar to those described in (Dunbar et al. 1974, 1975; Fleener and Watson 1973). The probe was heated up in a dust-free flow for 180 s, which was sufficient to achieve a constant temperature at the thermocouple’s back end. The particle injection was activated for 10 s, and the probe was removed from the flow 5 s after the particle injection start.

The time curves of the probe’s temperatures are given in Fig. 6 for the reference flow condition. The measurement period is in between the two red dashed lines. The particle injection started at t = 180 s, marked with a blue dotted line. The particles’ impact increased the measured stagnation point surface temperature T. The authors assumed that the application of the Cook–Felderman method (Cook and Felderman 1966) (see Eq. (3.2)) is valid for the period of particle injection. This was reasoned by constant temperatures at both ends of the coaxial-thermocouple prior particle injection. As a consequence, the resulting stagnation point heat flux corresponds only to the direct energy transfer from particles into the probe ${\dot{q}}_{\text{a}}$. It was observed that a lower quasi-stationary surface temperature at the stagnation point T was measured, compared to the stagnation temperature T₀. The measured quasi-stationary surface temperature at the stagnation point T is in fair agreement to the recovery temperature.

Fig. 6

Time curves of the coaxial-thermocouple (purple: front, red: back) and the stagnation temperature T₀ of test run A-32. The measurement period is in between the two vertical red dashed lines. The particle injection started at t = 180 s, marked with a blue dotted line

3 Results

3.1 Energy flux of incident particles

Particles are generally decelerated when passing the shock layer. The effect of particle deceleration on the resulting kinetic energy flux of all incident particles at reference flow condition for three particle materials is illustrated in Figs. 7, 8, and 9. The orange error bars represent the velocity IQR of all particles in the freestream (‘fs’, see Sect. 2.2), while the blue error bars represent the velocity IQR of all particles in the shock layer (‘sl’). The red dotted lines represent the limiting impact velocity V_p* (see Table 6). The total numbers of all considered particles are given in the figure legends.

Fig. 7

Relation between particle velocity, size, and the kinetic energy of incident particles in the freestream and in the shock layer, run ID A-32

Fig. 8

Relation between particle velocity, size, and the kinetic energy of incident particles in the freestream and in the shock layer, run ID M-32

Fig. 9

Relation between particle velocity, size, and the kinetic energy of incident particles in the freestream and in the shock layer, run ID S-32

The kinetic energy flux of incident particles with sizes up to the given value on the x-axis is called ‘cumulated’ kinetic energy flux (${\text{cum}. \dot{e}}_{\text{kin}}$) in the following. The parameter ${\text{cum}. \dot{e}}_{\text{kin}}$ is considered for both analysis areas, the freestream and the shock layer. Both of the cumulated kinetic energy fluxes are scaled to ${\dot{e}}_{\text{kin fs}},$ the kinetic energy flux of all incident particles in the freestream. These ratios are shown as light bars. As an example, approximately 50% of the kinetic energy flux in the freestream is distributed on particles smaller than 22.5 µm (Al₂O₃), 17.5 µm (MgO), or 40 µm (SiO₂) (light orange bars). The darker parts of the bars represent the ratio of the cumulated erosive kinetic energy flux $\text{cum}. {\dot{e}}_{\text{kin erosive}}$ and ${\dot{e}}_{\text{kin fs}}$. The cumulated erosive kinetic energy flux considers only particles faster than V_p* (see Sect. 2.4).

The absolute velocity difference of particles in the freestream and in the shock layer increases with decreasing particle size. The velocity IQR range of particles in the shock layer is larger compared to those of particles in the freestream, especially for smaller particle sizes. This is caused by the continuous deceleration of particles in the shock layer and the summation of particles just entering the shock layer and particles already traveling through the shock layer.

The kinetic energy flux of incident particles in the shock layer is approx. 8–29% smaller than those of particles in the freestream. The detected percentage of the erosive kinetic energy flux is reduced significantly for Al₂O₃ and SiO₂ particles in the shock layer. These energy flux reductions are mainly caused by the particle deceleration. An overview of detected kinetic energy flux reduction in the shock layer is given in Table 9.

Table 9 Kinetic energy flux reduction due to particle deceleration within the shock layer

The distribution of kinetic energy flux on different particle sizes is a complex function depending on particle size, particle material, and flow conditions (see Figs. 7, 8, and 9)). The kinetic energy of incident particles in the shock layer ${\dot{e}}_{\text{kin inc sl}}$ is considered in following analyses.

3.2 Energy flux of rebounded particles

The scaled difference between incident particle kinetic energy flux ${\dot{e}}_{\text{kin inc}}$ and rebounded particle kinetic energy flux ${\dot{e}}_{\text{kin reb}}$ is compared in Fig. 10 to estimate the influence of the rebounded particle kinetic energy flux on the determination of the a_kin coefficient (see Eq. (2.4)). The bars indicate the mean and the interquartile range (IQR) of each test. The scaled difference of the kinetic energies ${\dot{e}}_{\text{kin reb}}$ and ${\dot{e}}_{\text{kin inc}}$ is slightly increasing with increasing ${\dot{e}}_{\text{kin inc}}$ for Al₂O₃ and SiO₂ tests, but it is less than 2.5%. Almost no rebounded MgO particle was detected. The authors assume that the neglection of the kinetic energy flux of rebounded particles ${\dot{e}}_{\text{kin reb}}$ is reasonable for the following analysis.

Fig. 10

Significance of rebounded particle kinetic energy flux, compared to incident particle kinetic energy flux for all tests

3.3 a _kin Coefficient

The a_kin coefficient is estimated with the formulation in Eq. (2.4) to compare with experimental results. This formulation is only valid for elastic particle reflection, meaning that the percentage of erosive kinetic energy flux is zero. Calculated values are given in Table 10. The influence of the selected probe material on the estimated a_kin coefficient is not significant.

Table 10 Estimated a_kin coefficient for all particle/probe material combinations. The right term of Eq. (2.4) was used for computation

The temporal changes of the measured relative stagnation point surface temperature θ as well as ${\dot{e}}_{\text{kin inc}}$ are plotted in Figs. 11 and 12, respectively. Data of all tests with Al₂O₃ particles were considered for these plots. The time on the x-axis is set to 0 at the start of the particle injection, which is approx. 180 s after probe injection (see Fig. 6). Significant unexpected variations of the ${\dot{e}}_{\text{kin inc}}$ were observed. While for most of the tests ${\dot{e}}_{\text{kin inc}}$ is increasing with time, some tests show maximum ${\dot{e}}_{\text{kin inc}}$ right after seeding start with a decrease with increasing time. The relative stagnation point surface temperature behaves similar to ${\dot{e}}_{\text{kin inc}}$. Data before the maximum achieved stagnation point surface temperature are marked with a solid and a dashed line; the data beyond this point are marked with a dotted line. The difference between the solid line and a dashed line is explained below.

Fig. 11

Temporal change of θ for all tests with Al₂O₃ particles

Fig. 12

Temporal change of ${\dot{e}}_{\text{kin inc}}$ for all tests with Al₂O₃ particles

Considering the results of Sect. 3.2 and neglecting the heat conduction aspect, the a_kin coefficient is determined experimentally with the following equation:

$$a_{{{\text{kin}}}} = \frac{{\dot{q}_{{{\text{SP}}}} }}{{\dot{e}_{{\text{kin inc}}} - \dot{e}_{{\text{kin reb}}} }}\sim \frac{{\dot{q}_{{{\text{SP}}}} }}{{\dot{e}_{{\text{kin inc}}} }}$$

(4.1)

This ratio of all Al₂O₃ tests is shown in Fig. 13. Like in Figs. 11 and 12, data before the maximum achieved stagnation point surface temperature are marked with a solid and a dashed line, the data beyond this point are marked with a dotted line. The a_kin coefficient is decreasing when the maximum stagnation point surface temperature is reached. Furthermore, it becomes smaller than zero when the stagnation point surface temperature is decreasing for a longer time period (see A-12). The larger the temperature difference θ, the larger the potential impact of convection effects. To minimize the effect of convection cooling on the determination of the a_kin coefficient, only data are considered in which θ is smaller than 5 K. These data are marked in Figs. 11, 12, and 13 with a solid line. Peaks of the a_kin coefficient larger than unity can be explained by short time periods, in which only few particles were detected. Figure 13 does not contain measurement uncertainties for better visibility of the measured mean values.

Fig. 13

a_kin Coefficient versus relative stagnation point temperature θ

The ratio of erosive kinetic energy flux ${\dot{e}}_{\text{kin inc erosive}}$ and the kinetic energy flux of all incident particles ${\dot{e}}_{\text{kin inc}}$ was considered to distinguish between an elastic and an erosive particle reflection environment. Material properties of chromel were taken into consideration for the calculation of V_p*. If stainless steel is considered as probe material, V_p* is reduced and hence, the percentages of erosive kinetic energy flux would increase.

The measured a_kin coefficients of all tests are summarized in Table 11. The experimentally determined a_kin coefficient is in the range of 0.26–0.50, 0.36–0.85, and 0.06–0.12 for Al₂O₃, SiO₂, and MgO particles, respectively. The respective mean values are 0.37, 0.52, and 0.09. All MgO particle tests provided a complete erosive particle reflection environment. Generally, it seems that the determined a_kin coefficient of test run S-32 has an outlier character. If this outlier is excluded, the determined mean a_kin coefficient of SiO₂ particle tests is 0.36, very close to those of Al₂O₃ particle tests.

Table 11 Measured a_kin coefficients

The measurement uncertainty of the mean a_kin coefficient as well as the IQR of the considered temporally resolved a_kin coefficients for each test is also listed in Table 11. The mean a_kin coefficient values as well as the measurement uncertainty were rounded to two decimal figures. The estimated measurement uncertainties are smaller than the temporal a_kin coefficient variations, which are shown in Fig. 13 and which are considered in the IQR.

Since there is a link between particle-induced heating augmentation and Re_dProbe (Fleener and Watson 1973), this study analyzed the dependency of the a_kin coefficient on Re_dProbe and plotted it in Fig. 14. A monotonically decreasing a_kin coefficient was observed only in tests with MgO particles. The IQR bars of the a_kin coefficient of Al₂O₃ tests are too large for an appropriate relation interpretation.

Fig. 14

Measured a_kin coefficient versus Re_dProbe for all tests

4 Discussion

The implementation of highly resolved shadowgraphy measurements allowed to quantify the degree of kinetic energy reduction of particles passing a bow shock. Test data showed that the kinetic energy flux of incident particles is reduced by approx. 8–29% due to the deceleration of particles within the shock layer. The distribution of the kinetic energy flux on different particle sizes depends on the particle size distribution and the flow condition. In this context, the consideration of individual particle characteristics for kinetic energy flux determination within the shock layer seems to be essential.

The kinetic energy flux of rebounded particles within the entire shock layer is less than 2.5% of the incident particle kinetic energy flux. Hence, the kinetic energy flux of rebounded particles was neglected for the measurement of the a_kin coefficient in this study. Since almost no rebounded MgO particles were found, the question arises whether these particles broke up during impact into pieces which are smaller than 6 µm (the resolution limit of the implemented shadowgraphy system) or whether they stuck on the probe surface. Future investigations of the probe surface could give answers to this. Generally, a larger kinetic energy flux of rebounded particles would increase the a_kin coefficient, if the formulation of Eq. (2.4) is considered.

The a_kin coefficient is assumed to be the ratio between measured heat flux and kinetic energy flux of incident particles in the shock layer. The time curves of the temperature and kinetic energy flux indicate that the particle seeding was not homogeneous during these tests. It seems that the uncertainty of determination of the a_kin coefficient itself depends on the temporal change of the kinetic energy flux of incident particles. It is shown that convective cooling phases of the probe reduced the a_kin coefficient significantly, even to negative values. On the one hand, extending the range of considered θ would possibly smooth the measured a_kin coefficient by increasing the number of data points, but on the other hand, potential convection cooling effects become stronger. Therefore, limiting the θ range of up to 5 K seems to be a good trade-off to exclude convection cooling effects from the author’s point of view, although even within this small temperature difference range some cooling phases were observed. Future studies should focus on the improvement of homogeneous particle seeding, since a temporal-homogeneous seeding should simplify the data analysis.

The experimentally determined a_kin coefficients are between 0.26 to 0.50, 0.36 to 0.85, and 0.06 to 0.12 for Al₂O₃, SiO₂, and MgO particles, respectively. The respective means are 0.37, 0.36, and 0.09, respectively, considering one outlier exclusion for the SiO2 tests. The a_kin coefficients determined for Al₂O₃ and SiO₂ particles are in good agreement to the erosive a_kin coefficient value of 0.3, mentioned in the study of (Hove and Shih 1977). Moreover, the experimentally determined MgO a_kin coefficient of 0.09 is in good agreement with the suggestion of (Molleson and Stasenko 2017) for the a_kin coefficient in erosive environments.

The decrease of the MgO-based a_kin coefficient with increasing Re_dProbe can be explained with stronger cooling or ablation effects in erosive environments. Previous particle-induced heating augmentation studies state a decrease of the a_kin coefficient from elastic to erosive environments. This behavior cannot be reproduced with the experiments presented in this study, based on the characterization of the particle reflection behavior with V_p* (Stasenko 2007). If stainless-steel material properties were considered in the calculation of V_p*, instead of chromel properties, V_p* would decrease and hence, the percentage of erosive incident particle kinetic energy flux would be higher.

The a_kin coefficient for elastic particle reflection, suggested by (Molleson and Stasenko 2017), was simplified and compared to experimental data. It seems that the simplified formulation overestimates the a_kin coefficient by a factor of approx. 2 for Al₂O₃ and SiO₂ particles. This might be caused by a higher ${\dot{e}}_{\text{kin}}$ detection rate of shadowgraphy, compared to the measurement techniques used in (Kudin et al. 2013). The difference of up to 29% of the kinetic energy flux of particles in the shock layer and in the freestream seems to be too small to be an explanation. Because MgO tests were completely erosive, a direct comparison is not feasible.

5 Conclusion

The key findings of this work can be summarized as follows:

• The deceleration of incident particles in the shock layer reduced the kinetic energy flux of incident particles up to 29%, depending on particle material and flow conditions.

• the measured kinetic energy flux of rebounded particles in the entire shock layer was less than 2.5% of the kinetic energy flux of incident particles.

• The experimentally determined a_kin coefficient is approx. 0.36 for Al₂O₃ and SiO₂ particles, while it is approx. 0.09 for MgO particles. No difference between erosive and elastic particle reflection mode was detected on the a_kin coefficient.

• The a_kin coefficient decreases with increasing Re_dProbe for MgO particles.

Appendix

1.1 Overview of selected experimental particle-induced heating augmentation analyses

See Table 12.

Table 12 Experimental setups for supersonic two-phase flows