Experimental investigation of the effect of transpiration cooling on second mode instabilities in a hypersonic boundary layer

The experimental results presented in this paper were obtained in the High Enthalpy Shock Tunnel Göttingen (HEG) which is a free-piston driven reflected shock tunnel. It provides a pulse of gas to a hypersonic nozzle at stagnation pressures of up to 200 MPa and stagnation enthalpies of up to 25 MJ/kg (Eitelberg et al. 1992; Eitelberg 1994). Its range of operating conditions was successively extended to allow investigations of the flow past hypersonic flight configurations from low altitude Mach 6 up to Mach 10 at approximately 33 km altitude (DLR 2018). In the present study HEG was operated at Mach 7.4 using a nozzle exit diameter of 0.59 m and an expansion ratio of 218.

The test conditions used are listed in Table 1. The stagnation values in the table correspond to the average properties of all corresponding runs considered for this work, with their respective standard deviations in parenthesis. The values at the nozzle exit are obtained through a calibration procedure which is described in detail in DLR (2018). The gas at the nozzle exit is found to be in thermal and chemical equilibrium. Test times are typically around 3 ms.

The investigated model is a \(7^\circ\) half-angle cone at \(0^\circ\) angle-of-attack with a \(44 \times 82\) mm (streamwise \(\times\) spanwise) porous C/C insert with constant 10 mm thickness designed for transpiration cooling, starting 658 mm from the cone tip, as shown in Fig. 2. An exchangeable tip allowed investigating three different nose radii, namely 5 mm, 2.5 mm and 0.1 mm. The cone was mounted in the test section such that line-of-sight schlieren visualization around the region of transpiration and further downstream was facilitated. At this position the cone partly reached into the contoured nozzle.

By varying both the free-stream unit Reynolds number and the model nose radius, a wide range of boundary layer transition locations, with respect to the transpiration insert were realized. Figure 3 shows representative normalized surface heat flux distributions obtained in the present study without gas transpiration. Free-stream conditions are represented as colors, and nose radii as symbols. The porous insert location is also shown for reference. The correlation between transition onset location, unit Reynolds number and the cone tip radius is depicted. Thus, the variable transition locations on the cone allowed investigating the effects of local mass addition on boundary layers at different transitional states.

Lines of Medtherm coaxial Type E thermocouples distributed along two generatrices of the cone were used to monitor boundary-layer transition by means of surface heat flux measurements. On the opposite side of the porous patch, a line of approximately uniformly spaced thermocouples provided repeatable reference measurements in each run. On the porous patch side, thermocouples were installed along a streamline crossing its center, with increased spatial resolution downstream of the transpiration location. Two shorter lines of thermocouples were positioned downstream of the porous insert on the left and the right of the center line, to monitor potential effects caused by the finite width of the injection patch. The locations of these thermocouple lines are illustrated in Fig. 2.

Four pairs of GE NPP-301 and Kulite XCEL-100 pressure transducers were installed in radially opposing positions in different sections of the model to monitor the angle-of-attack and yaw angle of the model. Two additional GE NPP-301 pressure transducers in the plenum chamber below the porous surface monitored the plenum pressure during the tests. The mass flow rate of nitrogen into the plenum was monitored using a MKS Instruments GmbH digital mass flow monitor of type 579A.

Three Atomic Layer Thermopiles transducers of type FOR-ALTP 2.5/1/100DYN-CAL were installed 23, 48 and 78 mm downstream of the porous insert offering high-frequency heat flux information. The transducers have an active area of 2.5 \(\times\) 1 mm (width × length), a response time of less than 1 \(\mu s\) and a bandwidth of 17 Hz to 1 MHz with \(-3\) dB. Due to the large uncertainty of the nominal ALTP sensitivity of around 25%, an in-situ calibration of the ALTP DC output against Medtherm thermocouples readings was conducted yielding sensitivities in the range of 35–50 \(\mu V/(W/cm^2)\).

All transducer signals were recorded using Amotronic transient recorders, with sampling rates of 1 MHz for the thermocouples and up to 10 MHz for the ALTPs, depending on the test conditions. If present, sharp peaks due to electrical noise were filtered out from the ALTPs timeseries using second-order IIR bandstop filters.

Heat flux spectral information were obtained as amplitude spectral densities (ASD) using Welch’s average estimator. The time series recorded by the ALTPs during steady-state time were processed using Hamming windowed segments of 2048 samples each and 40% overlap.

The Z-type high-speed schlieren used a Phantom v2012 camera, driven by a Cavilux laser source. Frames were recorded at 170 kHz with a resolution of either 1024 \(\times\) 112 or \(1280 \times 96\) pixels. This yielded a spatial resolution of 0.14 mm/pixel, while the undisturbed second modes were expected to have a wavelength of around 6.5 mm. The camera was tilted by 7 degrees to match the inclination of the cone surface. The knife edge was oriented horizontally. The images were post-processed following Petervari (2014) and Benjamin (2017) to extract information on second modes such as convection speed and frequency.

The DLR TAU code is used for all base flow computations. The code is a three-dimensional parallel hybrid multi-grid code and has been validated for hypersonic flows (see e.g. Mack and Hannemann 2002, Schwamborn et al. 2006 or Reimann and Hannemann 2010). The calculations, including the expansion process through the nozzle and the mass injection on the cone, are conducted using non-equilibrium gas modelling with 5 species for air: N\(_2\), O\(_2\), NO, N, O in combination with thermal equilibrium and chemical non-equilibrium (one temperature model), which was shown to be a suitable approach for the test conditions investigated previously (Wartemann et al. 2019c; DLR 2018). The transport coefficients are expressed in terms of collision integrals (Bottin 1999). The mixing rules are based on constant Schmidt numbers. Further, lookup tables are used, see Esser (1991). Axisymmetric grids are applied. Previous analyses have shown that the base flow simulations should include the nozzle, test chamber, and cone model itself (Wartemann et al. 2019c), which is consequently applied in this paper. For all grids, grid refinement towards the walls as well as around the shock is applied. A grid study can be found in Wartemann et al. (2019c). Based on the nozzle calibration of Wagner (2014), the nozzle boundary-layer is set turbulent, which is modelled with the Spalart–Allmaras turbulence model. The model wall temperature for all tests is assumed to be isothermal at 293 K. This temperature is also applied for the gas injected into the boundary layer. Air is used as transpired gas after a cross-check for selected cases with nitrogen yielded good agreement, thus, complementing previous studies presented in Wartemann et al. (2019a). Instead of a complete simulation of the mass flow through the porous ceramic, a mass flow wall condition is applied. A detailed description of this wall condition for the effusion section can be found in Hannemann (2006).

For the stability analyses the NOLOT code (NOnLocal Transition analysis, Hein et al. 1994) is used. NOLOT was developed in cooperation between DLR and the Swedish Defence Research Agency (FOI) and can be used for local as well as non-local spatial analyses. The equations are derived from the conservation equations of mass, momentum and energy, which govern the flow of a viscous, compressible, ideal gas. All flow and material quantities are decomposed into a steady laminar base flow \({\bar{q}}\) and an unsteady disturbance flow \({\tilde{q}}\):The laminar base-flow \({\bar{q}}\) is calculated by the DLR TAU code (see previous Sect. 3.1). The disturbance \({\tilde{q}}\) of Eq. (1) is represented as a harmonic wave:with the complex-valued amplitude function \({\hat{q}}\). The three coordinate directions of the equations 1 and 2 are denoted as \(x_c\), \(y_c\) and \(z_c\) and describe a curvilinear surface-oriented orthogonal coordinate system. Previous investigations of the same geometry with similar free-stream conditions, including different mass fluxes, show that the differences between Linear Stability Theory (LST) and Parabolized Stability Equations (PSE) are negligible (Wartemann et al. 2019a). Consequently, the local-parallel LST approach is applied for the investigations of this paper. Currently, the NOLOT stability code is limited to perfect-gas assumptions (calorically or/and thermally perfect). For low-enthalpy cases such as those investigated in this paper, it is possible to neglect real-gas effects for the stability calculation, as long as they are included in the base flow calculations (Wartemann et al. 2019c).

In the present work, mass addition into the boundary layer is achieved through a C/C porous patch. The dependency between the velocity of a fluid through a porous medium and the pressure gradient across the medium can be described using the Darcy–Forchheimer law, Eq. (3). In this equation, \(k_D\) and \(k_F\) are permeability coefficients which account for the viscous and inertial contributions, P is the pressure, v the fluid velocity, \(\mu\) its viscosity and \(\rho\) its density. It will be shown that for the range of mass flow rates investigated here, the inertial term cannot be neglected.This equation may be worked into a more practical form with a few considerations.

The transpired gas is assumed to be an ideal gas at constant temperature. The pressure gradient across the porous material is assumed to be large enough to consider the external pressure to be negligible. The latter assumption holds since the pressures needed to realize the present mass flow rates are at least one order of magnitude higher than the model surface pressure during a test.

The porous patch was machined from a C/C block such that its outer shape fits the cone contour. However, the main orientation of the pores between the parallel C/C layers was uniform, as shown in Fig. 4. Hence in this study the flow through the porous patch is approximated as planar, rather than radial as in Schmidt et al. (2016). Mass flux in the direction perpendicular to the orientation of the pores is neglected, since the material is anisotropic and permeability is considerably larger in the direction of the pores (Dittert et al. 2015).

Finally, due to the parallel fiber orientation and the uniform pressure distribution provided in the plenum chamber below the patch, the mass transpiration is assumed to be uniform along the dimensions of the insert, as seen in Dittert et al. (2015).

With the above assumptions, Eq. (3) yields:where the subscript p refers to values in the plenum, \({\dot{m}}\) is the mass flow rate, T the temperature of the injected gas, R the gas constant, A the surface area of the porous patch and t its thickness.

The permeability of the porous insert is determined experimentally by applying constant pressure in the plenum below the porous patch and measuring the mass flow rate simultaneously. The best fit of the coefficients \(k_D\) and \(k_F\) are calculated using a least square fit. Figure 5 shows the observed dependency between plenum pressure and mass flow rate. The resulting fit is plotted as solid line and yields permeability values of \(k_D=5.547 \cdot 10^{-13}~m^2\) and \(k_F=2.104\cdot 10^{-7}~m\). The dotted lines show a 95% confidence interval of the fit.

Figure 5 also shows as a dash-dot line the result of applying Darcy’s law considering only the \(k_D\) contribution in Eq. (4) as often proposed in the literature. The differences are small for low mass flow rates, but increase with increasing plenum pressure. This evidences the relevance of the second term in the present case.

Potential roughness effects caused by the porous patch were analyzed comparing the measured heat flux distributions on both sides of the cone, without mass transpiration. No significant differences on the boundary layer transition onset were observed. Thus, roughness effects due to the porous patch are neglected in the present study.

To allow a controlled and predictable mass addition into the boundary layer, the model was designed with a plenum below the porous insert, as seen in Fig. 4. Due to the short test times the plenum was pressurized with nitrogen already before the test in HEG to ensure steady-state injection conditions. This was particularly important for low mass flow rates.

Both the mass flow rate and the pressure readings in the plenum were recorded for each run. Figure 6 shows examples of the recorded values for three runs with condition B and nose radius 5.0 mm. They are representative for low, medium and high mass flow rates in the context of the present study. The local surface pressures obtained by a transducer in the vicinity of the patch are shown as well. All three cases show similar features, with flow arrival at around 1 ms and steady-state times from roughly 3 to 6 ms. No significant changes of the mass flow rate and the plenum pressure are observed in the test time. This observation supports the assumption made for Eq. (4), i.e. assuming the external pressure to be negligible. Furthermore, the plenum pressure was verified not to show a significant increase for cases with no transpiration.

All mass flow rates investigated here and the corresponding test conditions are provided in Table 2.

In the course of the analyses presented in the results section of this paper, it will be necessary to define the location of boundary layer transition. This will be assessed by means of heat flux distribution on the cone surface, measured by the line of thermocouples crossing the center of the porous insert. Transition onset is assumed to happen at the location where the heat flux distribution stops following the laminar level and starts increasing towards the turbulent level.

Extra considerations are necessary in the cases analyzed in this work, since the cold gas transpiration also affects the heat flux measurements due to cooling effects. To prevent this from biasing the results, CFD quantities obtained for a laminar boundary layer without transpiration are used as an auxiliary parameter. The transition onset location is estimated based on the intersection between the numerical laminar prediction and a linear fit through the transitional portion of the measured data. Figure 7 shows an example of this approach, with the resulting estimated transition onset location highlighted.

Fully turbulent conditions at the location of the porous surface are obtained with conditions C and D, and 0.1 mm nose tip. Figures 8 and 9 show the averaged Stanton numbers for different mass transpiration rates for these conditions, obtained from the thermocouples along the transpiration side of the cone. In these figures, the position of the three ALTP sensors and the location of the porous surface are shown, for reference. The numerical laminar and turbulent Stanton number distributions are depicted as dashed and dashed-dot lines, respectively.

Figures 8 and 9 show that for both sets of cases there is a clear cooling effect resulting from the transpiration of nitrogen, with decreasing surface heat flux as the mass flow rates increase. Immediately following transpiration, a reduction in Stanton number values of up to 40% is achieved. As the cooling effect diminishes, the Stanton number reduction is still as high as 18% on the most downstream measurement on the cone compared to the test case without transpiration. On one hand, this represents an overall heat flux reduction of 30% on the cone surface downstream of the transpiration location. On the other hand, however, the heat fluxes are still higher compared to a fully laminar case, meaning an action towards preventing boundary layer transition is still more advantageous. Furthermore, the transpiration mass flow rates necessary for observing such cooling effect on the turbulent boundary layer are an order of magnitude larger than the flow rates used in the present study to modify the second mode development.

With a 5 mm nose radius, the boundary layer is entirely laminar along the cone length for a unit Reynolds number of \(1.4 \cdot 10^6 \, {\text {m}}^{-1}\) (condition A), recall Fig. 3. Indications of boundary layer transition onset can be observed in the figure as the unit Reynolds number was increased to \(2.3 \cdot 10^6 \, {\text {m}}^{-1}\) (condition B), evidenced by the rising trend of the heat flux towards the end of the model. In the laminar boundary layer cases, the frequency spectra measured with the ALTP sensors show distinct peaks. The convection speed of the corresponding flow structures is estimated from the cross-correlation between the ALTP time signals. For the cases without gas transpiration, the spectral peaks are centered around frequencies corresponding to wavelengths between 2 and 3 times the boundary layer height, estimated by means of CFD computations. LST calculations confirmed the observed peaks to correspond to the most amplified second mode frequencies.

With amplified second modes confirmed to exist downstream of the porous patch, different mass flow rates were applied to assess the effect on the instabilities.