4.1 Unperturbed flow

The velocity field around the SPP, in general a blunt body, in supersonic and hypersonic flows is characterized by the detached bow shock and the large subsonic region downstream of the shock wave. In figure 1(b) Mach number contours from case up_h (unperturbed, high Reynolds number) are shown. Close to the inflow they all collapse and show the location of the shock wave. The flow in the subsonic region downstream of the bow shock is accelerated to supersonic speed half-way along the rounded edge. It is completely contained in the uniformly refined part of the grid. At the outflow downstream of the shock wave Mach numbers of approximately 2.5 are reached. The numerically determined heat flux $\bar{q}$ at the stagnation point for both Reynolds numbers is tabulated in table 2. It is higher for the lower Reynolds number case up_l (unperturbed, low Reynolds number). For validation purposes the stagnation point heat flux of the unperturbed flow is compared to corresponding experimental and numerical data from Ali et al. (Reference Ali, Zárate Cárdenas, Radespiel, Schilden and Schröder2018). The numerical value for case up_l is $\bar{q}_{\star }=71\,000~\text{W}~\text{m}^{-2}$ and the experimental comparative value for case up_h is $\bar{q}_{\star }=56\,000~\text{W}~\text{m}^{-2}$ . The good agreement of the simulations for up_l is evident and also the comparison with the experimental data for case up_h is convincing since the heat flux sensor has a large mean heat flux deviation of 20 % (Kegerise & Rufer Reference Kegerise and Rufer2016). Note that the finite size of the heat flux transducer of $1\times 1~\text{mm}^{2}$ is negligible, i.e. integrated numerical data deviate less than 1 % from the stagnation point data.

Furthermore, in figure 2(a,b) velocity and temperature comparison of two simulations of case up_h with different grids is shown. The case no.  $c320$ indicates the solution using the grid utilized throughout this work for the lower Reynolds number flows, whereas for case no.  $c830$ a much finer grid having 830 cells between shock wave and stagnation point along the axis of symmetry is used. Both simulations agree well.

Figure 2.

Analysis of the flow field along the streamline between the shock wave and the stagnation point of case up_h: (a) axial velocity distribution for two different meshes, no.  $c320$ and no.  $c830$ indicate solutions using grids having 320 and 830 cells between shock wave and stagnation point along the axis of symmetry; (b) temperature distribution for two different meshes; (c) group velocities $v_{g\star }$ of the fast acoustic $u+a$ , a convective $u$ and the slow acoustic mode $u-a$ computed on grid no.  $c320$ .

Table 2.

Properties of the unperturbed flow: Reynolds number $Re_{D}$ , stagnation point heat flux $\bar{q}$ , numerical (num) and experimental (exp) stagnation point heat flux from Ali et al. (Reference Ali, Zárate Cárdenas, Radespiel, Schilden and Schröder2018), period of a slow acoustic ( $t_{sa}$ ), entropy ( $t_{en}$ ) and fast acoustic ( $t_{fa}$ ) wave propagating between the shock wave and SPP at the axis of symmetry, frequency $f_{ac}=1/(t_{sa}+t_{fa})$ , Strouhal number $Sr_{ac}=Df_{ac}/u_{\infty }$ .

In figure 2(c) the group velocities $v_{g}$ of the slow acoustic ( $u-a$ ), entropy ( $u$ ) and fast acoustic wave ( $u+a$ ) of case up_h are plotted versus the axis of symmetry. From these distributions an a priori estimate of the acoustic standoff frequency can be calculated from the duration $t_{sa}+t_{fa}$ of an acoustic wave travelling upstream from the stagnation point to the shock wave as a slow acoustic wave and then back to the stagnation point as a fast acoustic wave. The values of $t_{sa}$ and $t_{fa}$ are calculated applying

and the acoustic standoff frequency is $f_{ac}=1/(t_{sa}+t_{fa})$ . The values of $t_{sa}$ , $t_{fa}$ and $f_{ac}$ can be found in table 2 for both Reynolds numbers. The lower diameter probe from case up_l shows a higher standoff frequency $f_{ac}$ due to the smaller shock standoff distance. Non-dimensionalizing the standoff frequency to gain a standoff Strouhal number $Sr_{ac}$ leads to very similar values for both cases (table 2). The standoff frequency characterizes the superposition of waves downstream of the shock wave. Analogous to $t_{sa}$ and $t_{fa}$ , equation (4.1) can also be evaluated for a lower bound of $t_{en}$ , i.e. being an estimate for the duration of an entropy wave convecting at $v_{g}=u$ from the shock wave to the stagnation point, but the integration is truncated at the position of the maximum temperature. Close to the wall, the velocity tends to zero and diffusion, which is not accounted for in (4.1), is the mechanism of propagation. The values of $t_{en}$ are tabulated in table 2 and are one order of magnitude greater than the acoustic time scales owing to the low Mach number.

In the next step, to investigate the superposition of acoustic and entropy waves downstream of the shock wave, the impulse response of the SPP to acoustic and entropy disturbance modes is explored.

4.2 Impulse response

By inserting a single pulse of an incident slow acoustic, entropy and fast acoustic wave the modulation of the incident mode can be analysed without having to deal with the superposition of its reflections downstream of the shock wave. The incident single pulses are defined according to (3.1)–(3.6), where the argument $\unicode[STIX]{x1D719}=\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}-\unicode[STIX]{x1D714}t$ is processed by a filter $f(\unicode[STIX]{x1D719})$

with $\unicode[STIX]{x1D719}_{0}=\min (\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x})-\unicode[STIX]{x1D714}t_{0}$ to yield a sinusoidally shaped pulse that enters the computational domain at the most upstream position with respect to the pulse propagation at the time $t_{0}$ . The single pulse can be traced and, provided the time scales $t_{sa}$ , $t_{fa}$ and $t_{en}$ are known, the actual superposition can be elucidated. Time series of pressure and entropy fluctuations of cases sa_pulse_h (slow acoustic mode, incident pulse, high Reynolds number), en_pulse_h (entropy mode, incident pulse, high Reynolds number), and fa_pulse_h (fast acoustic mode, incident pulse, high Reynolds number) from table 1 at $x_{\star }=-1.5~\text{mm}$ , i.e. approximately in the centre between shock wave and SPP are shown in figures 3 and 4 where each panel represents one of the three cases. It is obvious that all incident modes lead to acoustic and entropy waves downstream of the shock wave. The first pressure and entropy pulses at $t_{\star }=4.7~\unicode[STIX]{x03BC}\text{s}$ in figure 3 and $t_{\star }=14.7~\unicode[STIX]{x03BC}\text{s}$ in figure 4 represent the modulated incident wave. They are called the initial diverging acoustic and entropy wave and are generated simultaneously during the interaction of incident pulse and shock wave. Due to the different group velocities of acoustic and entropy waves they reach $x_{\star }=-1.5~\text{mm}$ at different times. For all incident modes, the initial diverging acoustic and entropy wave possess opposite signs. Their postshock modulation differs mainly by the absolute value of the ratio of the initial diverging acoustic and entropy wave. For the pressure fluctuations in figure 3, the second pulse at $t_{\star }=12~\unicode[STIX]{x03BC}\text{s}$ is caused by the reflection of the initial diverging acoustic wave at the SPP. It is called the first body reflected wave. Since $x_{\star }=-1.5~\text{mm}$ is approximately in the centre between the shock wave and the SPP, the time between the first and second pulse is close to $(t_{sa}+t_{fa})/2$ . The third pulse at ${\approx}20~\unicode[STIX]{x03BC}\text{s}$ follows from the reflection of the body reflected wave at the shock wave. It is denoted the first shock reflected wave. Again, the time between the second and third pulse is $(t_{sa}+t_{fa})/2$ . This kind of propagation of the postshock acoustic wave continues, i.e. it is damped during its reflection at the shock wave and slightly amplified during the reflection at the SPP. The reflection at the shock wave leads to a change of sign of the pressure disturbance. A schematic diagram of the acoustic wave pattern is illustrated in the upper half of figure 5.

Figure 3.

Time traces of pressure fluctuations at $x_{\star }=-1.5~\text{mm}$ between the shock wave and the stagnation point for an incident pulse of the slow acoustic (sa_pulse_h), entropy (en_pulse_h) and fast acoustic (fa_pulse_h) mode.

Figure 4.

Time traces of entropy fluctuations at $x_{\star }=-1.5~\text{mm}$ between the shock wave and the stagnation point for an incident pulse of the slow acoustic (sa_pulse_h), entropy (en_pulse_h) and fast acoustic mode (fa_pulse_h).

Figure 5.

Acoustic and entropy mode wave pattern for any incident mode. All incident modes directly lead to initial diverging fast acoustic and entropy waves. The acoustic wave travels between the shock wave and the SPP alternating as fast and slow acoustic mode. The reflection of the acoustic wave at the shock wave generates a further entropy wave.

Table 3.

Postshock waves for all incident mode types at $x_{\star }=-1.5~\text{mm}$ ; pressure amplitude $\hat{p}_{i,\star }$ of the initial diverging acoustic wave at $t_{\star }=4.7~\unicode[STIX]{x03BC}\text{s}$ ; entropy amplitude ${\hat{s}}_{i,\star }$ of the initial diverging entropy wave at $t_{\star }=14.7~\unicode[STIX]{x03BC}\text{s}$ ; non-dimensional ratio of the initial diverging acoustic and entropy wave ${\hat{s}}_{i}/\hat{p}_{i}$ ; entropy amplitude of the first reflection generated entropy wave ${\hat{s}}_{g,\star }$ at $t_{\star }=30~\unicode[STIX]{x03BC}\text{s}$ .

The entropy disturbances in figure 4 show a different pattern. The second pulse at $t_{\star }=30~\unicode[STIX]{x03BC}\text{s}$ follows the initial diverging entropy wave with a lag of $t_{sa}+t_{fa}$ . It is generated during the reflection of the body reflected acoustic wave at the shock wave. It is called the first reflection generated entropy wave and the disturbance has the same sign as the body reflected acoustic wave. The schematic diagram of the entropy wave pattern is included in the lower half of figure 5.

The magnitudes of the initial diverging acoustic $\hat{p}_{i}$ and entropy ${\hat{s}}_{i}$ waves and their non-dimensional ratio are summarized in table 3. All incident modes lead to different ratios of the initial diverging acoustic and entropy waves $({\hat{s}}_{i}/\hat{p}_{i})(a_{0}^{2}\unicode[STIX]{x1D70C}_{0}/c_{p})$ and deviate by at least 36 %. This is a necessary condition for any modal decomposition that relies on the measurement of quantities caused by postshock acoustic and entropy waves such as pressure and heat flux fluctuations.

Time series of the stagnation point pressure and heat flux disturbances of cases sa_pulse_h, en_pulse_h and fa_pulse_h are shown in figures 6 and 7. In all panels, the vertical thin dotted lines represent multiples of $t_{sa}+t_{fa}$ , where the initial line is defined by the first detection of the initial diverging acoustic wave at $t_{\star }=6.1~\unicode[STIX]{x03BC}\text{s}$ . The vertical thin dashed lines in figures 6 and 7(a), and its first occurrence in figure 7(b) represent the lower bound $t_{en}$ . The following dashed lines in figure 7(b) represent further multiples of $t_{sa}+t_{fa}$ . The time between the first dashed line at $t_{\star }=48.3~\unicode[STIX]{x03BC}\text{s}$ and the first dotted line is $t_{en}-t_{fa}$ , thus giving a lower bound for the first appearance of the initial diverging entropy wave in any signal. As discussed before, the stagnation point pressure in figure 6 shows for all incident modes postshock acoustic mode related pulses with alternating sign. The acoustic wave diminishes to a very low level before the entropy wave can reach the SPP. The postshock entropy waves related pressure disturbance is negligible. The stagnation point heat flux in figure 7(a) also shows pulses caused by the postshock acoustic wave after the thin dotted lines. Since pressure and heat flux signals are not proportional, i.e. the heat flux pulses exhibit a visible undershoot after the main pulse, an additional mechanism leads to a more complex transfer function between postshock acoustic wave and heat flux fluctuation. This issue will be addressed later.

Figure 6.

Time traces of pressure fluctuations caused by postshock acoustic waves at the stagnation point for an incident pulse of the slow acoustic (sa_pulse_h), entropy (en_pulse_h) and fast acoustic (fa_pulse_h) mode.

Figure 7.

Time traces at the stagnation point heat flux fluctuations caused by postshock acoustic and entropy waves for an incident pulse of the slow acoustic (sa_pulse_h), entropy (en_pulse_h) and fast acoustic (fa_pulse_h) mode: (a) time trace beginning at $t_{\star }=0$ showing mainly the contribution of the postshock acoustic wave to the heat flux fluctuation; (b) detailed view of the entropy mode contribution to the heat flux fluctuation.

Increasing the scale of the heat flux for a subsequent time span in figure 7(b) emphasises the contribution of the initial diverging entropy wave. As in figure 7(a), the postshock acoustic wave related heat flux fluctuations appear as local extrema immediately after the thin dotted lines. For $t_{\star }>80~\unicode[STIX]{x03BC}\text{s}$ , they are hardly noticeable. The remaining heat flux deviations for $t>t_{en}$ are caused by the postshock entropy waves, i.e. the initial diverging entropy wave and the reflection generated entropy waves that follow in intervals of $t_{sa}+t_{fa}$ . Unlike the postshock acoustic wave related heat flux, distinct responses to singular postshock entropy waves appearing in intervals of $t_{sa}+t_{fa}$ cannot be observed. The transfer function between postshock entropy waves and heat flux fluctuations causes heat flux responses to the individual postshock entropy waves that sustain longer than the period of the inserted pulse and $t_{sa}+t_{fa}$ . Therefore, the responses to the individual postshock entropy waves merge and form the trend that is characterized by a local maximum around $73~\unicode[STIX]{x03BC}\text{s}$ . The peak heat flux fluctuation can be linked to the amplitude of the initial diverging entropy wave ${\hat{s}}_{i,\star }$ in table 3. At growing initial diverging entropy wave, the peak heat flux fluctuation increases. The following decay of the heat flux deviation is related to the first reflection generated entropy wave ${\hat{s}}_{g,\star }$ from table 3. In addition to the different behaviour of the transfer functions between postshock acoustic and entropy waves and heat flux fluctuations, the magnitude differs tremendously. The contribution of the postshock entropy waves to the heat flux fluctuations is lower than the contribution of the postshock acoustic wave.

As aforementioned, an additional mechanism causes the transfer function of acoustic waves to heat flux fluctuations to be not directly proportional. The necessity of this further mechanism can also be deduced from the isothermal boundary conditions enforcing $T^{\prime }=0$ and the isentropic relations demanding $T^{\prime }\propto p^{\prime }$ for acoustic waves. To determine the intermediate step between an acoustic wave and the heat flux, instantaneous pressure, entropy and temperature distributions on the axis of symmetry of case en_pulse_h are shown in figure 8 at two time instances $7.8~\unicode[STIX]{x03BC}\text{s}$ and $23~\unicode[STIX]{x03BC}\text{s}$ . The pressure disturbance in figure 8(a) exhibits its first and second temporal extremum, while the entropy wave in figure 8(b) did not reach the SPP, yet. Nonetheless, entropy fluctuations are evident close to the surface of the SPP. To fulfil the isothermal boundary condition in the presence of an acoustic wave reflecting at the wall as shown in figure 8(c), an entropy spot is generated close to the wall. When the acoustic wave travels upstream again, the entropy spot related temperature disturbance remains close to the wall and is balanced by diffusion, i.e. additional heat flux on the surface, causing the heat flux signal of figure 7(a) to show the aforementioned undershoot.

Figure 8.

Instantaneous flow field between the shock wave and the stagnation point for an incident pulse of the entropy mode (en_pulse_h); (a) pressure fluctuations caused by the postshock acoustic wave; (b) entropy fluctuations caused by the postshock entropy waves; (c) temperature fluctuations caused by the postshock acoustic wave and the postshock entropy waves.

It can be concluded that all types of incident disturbance modes lead to an initial diverging acoustic and entropy wave, whose ratio differs. However, for all incident modes the ratio has a negative sign, i.e. the waves possess a phase shift of $\unicode[STIX]{x03C0}$ . Moreover, the heat flux fluctuations show contributions of the postshock acoustic wave and the postshock entropy waves. However, the initial diverging entropy wave of all incident modes led to lower heat flux amplitudes than the initial diverging acoustic wave. Using the findings of §§ 4.1 and 4.2 a model equation is derived to investigate the superposition of single disturbance pulses onto continuous incident disturbance modes.

4.3 Model equation

Assuming weak continuous sinusoidal incident modes, the stagnation point pressure and heat flux fluctuations can be modelled as a linear superposition of the contributions of the postshock acoustic and entropy waves to each fluctuating quantity, e.g. see Morkovin (Reference Morkovin1960) for a similar model for pressure fluctuations. The model equations for pressure and heat flux fluctuations are formulated based on the time scales $t_{fa}+t_{sa}$ and $t_{en}$ deduced from the mean flow listed in table 2 and the postshock wave pattern extracted from the impulse response shown in figure 5. The amplitudes of the initial diverging acoustic wave $\hat{p}_{i}$ and the initial diverging entropy wave ${\hat{s}}_{i}$ also appear in the model equation and represent the input of the different incident waves, thus causing different pressure and heat flux fluctuations for various incident waves.

First, the model equation for pressure fluctuations is formulated. It reads

The first term for $j=0$ denotes the pressure fluctuation induced by the initial diverging acoustic wave. The terms for $j>0$ represent the shock reflected acoustic waves where the factor $c_{I}$ is the transfer function of an acoustic wave being reflected at the shock wave.

According to Chaudhry & Candler (Reference Chaudhry and Candler2016) $c_{I}$ depends on the Mach number and for $\unicode[STIX]{x1D6FE}=1.4$ , it is $-0.139<c_{I}<0$ . The argument $-\unicode[STIX]{x1D714}j(t_{sa}+t_{fa})$ of (4.3) introduces the phase shift of the impinging postshock acoustic waves at the stagnation point. It depends on the angular frequency of the incident disturbance and gives rise to interference. Furthermore, the factor 2 accounts for the contribution of the body reflected acoustic waves to the pressure fluctuations. At the wall, the velocity fluctuation is zero $u^{\prime }=0$ . Therefore, velocity fluctuations due to fast acoustic waves approaching the wall $u_{fa}^{\prime }$ and their reflections $u_{sa}^{\prime }$ , i.e. slow acoustic waves moving upstream away from the wall, need to cancel out $u_{fa}^{\prime }+u_{sa}^{\prime }=0$ . Using (3.2) to relate velocity and pressure fluctuations, we obtain identical pressure fluctuations of approaching and reflected wave, thus we account for all body reflected waves by introducing a factor of 2. Note that (4.3) assumes the amplitude of the postshock acoustic waves to be altered only by the interaction with the shock wave. Changes of the amplitude due to the non-uniform flow or the isothermal wall boundary condition are neglected. The postshock entropy waves do not contribute to pressure fluctuations.

The resonant behaviour of the pressure frequency response can be deduced from (4.3). The phase of the individual summands $\unicode[STIX]{x1D719}_{j}$ is

At multiples of the acoustic standoff frequency $f_{ac}$ the postshock acoustic waves interfere destructively since $\arg (c_{I})=\unicode[STIX]{x03C0}$ . Constructive interference occurs at frequencies $f=f_{ac}/2+nf_{ac}$ with $n\in \mathbb{N}$ . The Nyquist plot of (4.3) for $Ma=5.9$ is shown in figure 9. The outer locus denotes the hypersonic limit $Ma_{in}\rightarrow \infty$ of $c_{I}=-0.139$ . The rightmost intersections of the frequency response loci with the real axis resemble constructive interference, whereas the intersections on the left denote destructive superposition. Due to the intense damping of the body reflected acoustic wave, the phase difference of the final pressure signal compared to the initial diverging wave at $\arg (p^{\prime }/p_{j=0}^{\prime })$ is limited to small values. Furthermore, since in the case of constructive interference all summands have the same sign and for destructive interference the signs change alternately, the constructive interference can be expected to be more intense. This result is indicated in figure 9. The initial diverging acoustic wave $p_{j=0}^{\prime }$ at $1+0\text{i}$ does not coincide with the centre of the intersections of the frequency response locus on the real axis.

Figure 9.

Nyquist plot of the net pressure signal of (4.3); the inner locus shows the $\unicode[STIX]{x1D714}$ dependency at $Ma_{in}=5.9$ , the outer locus denotes the hypersonic limit $Ma_{in}\rightarrow \infty$ of $c_{I}=-0.139$ for $\unicode[STIX]{x1D6FE}=1.4$ .

To model the heat flux signal, the contribution of the combined pressure signal to the heat flux fluctuations can be written

where the factor $c_{\mathit{II}}$ is the transfer function between the pressure and pressure related heat flux fluctuations shown in figure 7(a). It contains the phase shift caused by the intermediate step of entropy generation and a potential decay with increasing frequency. The contribution of the postshock entropy waves, i.e. the initial diverging entropy wave and the reflection generated entropy waves, to the heat flux fluctuations is expressed as

In (4.6), $c_{\mathit{III}}$ is the transfer function linking entropy waves and their heat flux fluctuation contribution. The transfer function $c_{\mathit{IV}}$ relates body reflected acoustic waves and reflection generated entropy waves. According to McKenzie & Westphal (Reference McKenzie and Westphal1968) and figures 3 and 4  $c_{\mathit{IV}}$ is a positive real number. The arguments of both terms in (4.6) depend on the entropy wave time scale to consider the phase shift that results from the different group velocities of the postshock acoustic and entropy waves. Note that the reflection generated entropy waves follow from reflections of body reflected acoustic waves and are unlike the shock reflected acoustic waves not necessarily weaker than the initial diverging wave which is evident by the values of ${\hat{s}}_{i}$ and ${\hat{s}}_{g}$ in table 3. For the incident fast acoustic wave case fa_pulse_h, the absolute value of the reflection generated entropy wave ${\hat{s}}_{g}$ is higher than that of the initial diverging entropy wave ${\hat{s}}_{i}$ . Even though the amplitude ratio of the initial diverging entropy wave and the subsequent reflection generated entropy waves is not universal for all incident modes, their phase difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}_{en}$ is. Using $\arg ({\hat{s}}_{i})=\unicode[STIX]{x03C0}$ it can be written

In the case of destructive pressure interference $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}_{en}=\unicode[STIX]{x03C0}$ holds and thus also destructive entropy wave interference occurs for the postshock entropy wave related heat flux. For constructive pressure interference follows $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}_{en}=0$ resulting in constructive postshock entropy wave interference, too. Besides sheer destructive and constructive interference, the net phase of the postshock acoustic waves $\arg (p^{\prime })$ leads to a nonlinearity in $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}_{en}(\unicode[STIX]{x1D714})$ which increases at decreasing damping due to $c_{I}$ as can be seen in figure 9. Rearranging the terms of (4.5) and (4.6), the full heat flux signal $q^{\prime }=q_{ac}^{\prime }+q_{en}^{\prime }$ can be written

It is evident from (4.8) that besides the direct contribution of the postshock acoustic waves to the heat flux via $c_{\mathit{II}}$ also a certain part of the postshock entropy wave related heat flux, i.e. the third term due to reflection generated entropy waves, must be attributed to the postshock acoustic waves. Since in the postshock region various types of incident modes differ mainly by their amplitude ratio of the initial diverging pressure and entropy wave $\hat{p}_{i}/{\hat{s}}_{i}$ , the initial diverging entropy wave, i.e. the first term on the right-hand side of (4.8), needs to be measured by the heat flux transducer to differentiate between varying incident modes from their SPP pressure and heat flux signal. This makes the properties of the transfer function $c_{\mathit{III}}$ a crucial factor whenever a differentiation between various incident modes is required, i.e. for $c_{\mathit{III}}=0$ , a modal decomposition based on the differing ratios of the initial diverging acoustic and entropy waves cannot be posed since the initial diverging entropy wave is not captured in any signal.

In the following section, a further investigation of $c_{\mathit{III}}$ is pursued based on control theory. The step response is analysed to make a statement concerning the heat flux frequency response of an SPP to postshock entropy waves.

4.4 Step response

The step response of the heat flux sensor to postshock entropy waves is analysed to better understand the properties of the transfer function $c_{\mathit{III}}$ . Such a step response can be initiated by a sudden change of entropy, i.e. density at constant pressure, of the fluid surrounding the probe at the steady state. Considering the stagnation point heat flux, this change of state leads to a change of the local Reynolds number. An expansion of the Reynolds number, i.e. the total derivative, for small changes of entropy reads

The sudden change of the Reynolds number causes an imbalance of viscous and inviscid terms of the governing equations in the thermal boundary layer, i.e. the heat flux response shows the reestablishment to a new equilibrium. Besides the desired response of the thermal boundary layer, the sudden change of entropy at the inflow causes a Mach number deviation

which in turn results in a new steady state pressure field downstream of the shock wave. The acoustic wave that propagates downstream of the shock wave during the convergence of the pressure field leads to heat flux disturbances via $c_{\mathit{II}}$ and conceals the heat flux step response to the change of entropy. To avoid this, two different approaches are proposed.

First, for cases en_flood_l and en_flood_h the entropy wave inflow boundary condition from (3.5) is modified as

This approach is referred to as ‘flood’ since the computational domain is flooded with an entropy wave. Applying the entropy flooding, the Mach number deviation at the inflow still causes postshock acoustic wave related heat flux fluctuations during the pressure field convergence, but the entropy flood heat flux response is temporally shifted to times $t>t_{en}$ and the overlap of both heat flux contributions can be reduced. On the other hand, equation (4.11) does not resemble a discontinuous step function for the entropy and even if it were, the discontinuity would be smeared by diffusion during the convection from the inflow to the SPP.

Therefore, in the second approach, a rather non-physical numerical experiment is performed in cases step_l and step_h by restarting a converged simulation with a slightly altered Reynolds number. Similar to a sudden change of entropy, the equilibrium between viscous and inviscid terms of the governing equations is disturbed and the immediate heat flux response shows the desired reestablishment of a new equilibrium in the thermal boundary layer. Due to the temporal discontinuity introduced into the governing equations by the slight change of Reynolds number, this approach is referred to as ‘step’.

In figure 10(a,b), the response of the stagnation point pressure and heat flux signals to the aforementioned approaches of cases en_flood_h and step_h are shown. For the case en_flood_h the stagnation point pressure and heat flux converge to a new steady state. As soon as the incident entropy wave interacts with the shock, it generates the aforementioned acoustic wave and the pressure in figure 10(a) drops within the time scale of the fast acoustic wave $t_{fa}$ . Then, only the postshock acoustic waves can be seen as they diminish during the convergence. The heat flux step response in figure 10(b) shows for the en_flood_h case two time scales during its convergence. First, it rapidly changes due to the altered Mach number and postshock pressure. Then, starting at $t_{\star }\approx 60~\unicode[STIX]{x03BC}\text{s}$ the entropy flood leads to a slow increase of the heat flux. This second part of the convergence is related to the slight change of the local Reynolds number in the thermal boundary layer. For the case step_h, the pressure response is zero since the pressure at rest does not depend on the Reynolds number. The development of a new heat flux equilibrium shows the same behaviour as the entropy flood related part of the step response of the case en_flood_h. The asymptotic trend of the heat flux step response resembles the behaviour of a low pass filter. The transfer function of a low pass filter is

thus leading to a strong decrease of $|G|$ beyond $\unicode[STIX]{x1D714}>2\unicode[STIX]{x03C0}/t_{co}$ . The cutoff frequency $f_{co}=1/t_{co}$ can be determined from the step response via

The distribution of $t_{co}$ based on (4.13) from the data of figure 10(b) is shown in figure 10(c). For the case en_flood_h, the diminishing postshock acoustic wave related heat flux fluctuations render the application of (4.13) impossible. However, for times $t_{\star }>100~\unicode[STIX]{x03BC}\text{s}$ and the data of case step_h a value of approximately $20~\unicode[STIX]{x03BC}\text{s}$ can be obtained.

Figure 10.

Step response at the stagnation point for the cases of entropy mode flooding (en_flood_h) and a slight change of Reynolds number (step_h) at $Re=109\,375$ ; (a) pressure response; (b) heat flux response; (c) cutoff frequency evaluation by (4.13).

The analogous analysis for the lower Reynolds number cases en_flood_l and step_l is depicted in figure 11. For the smaller SPP, a value of approximately $12~\unicode[STIX]{x03BC}\text{s}$ can be extracted. All values for the cutoff time scale $t_{co}$ , cutoff frequency $f_{co}$ and cutoff Strouhal number $Sr_{D,co}$ are listed in table 4. The non-dimensional analysis shows a decreasing cutoff Strouhal number at decreasing Reynolds number. On the other hand, from the Strouhal number itself we can directly deduce that smaller SPPs lead to higher cutoff frequencies. The final cutoff frequency in table 4 shows the latter trend determined by the Strouhal number definition. The Reynolds number effect is benevolent but too weak to yield higher cutoff frequencies using larger SPP.

Figure 11.

Step response at the stagnation point for the cases of entropy mode flooding (en_flood_l) and a slight change of Reynolds number (step_l) at $Re=50\,000$ ; (a) pressure response; (b) heat flux response; (c) cutoff frequency evaluation by (4.13).

Table 4.

Properties of the heat flux response to entropy waves, i.e. the transfer function $c_{\mathit{III}}$ modelled as a low pass filter; Reynolds number $Re_{D}=\unicode[STIX]{x1D70C}_{\infty }u_{\infty }D/\unicode[STIX]{x1D702}_{\infty }$ , cutoff time scale $t_{co}$ , cutoff frequency $f_{co}$ and cutoff Strouhal number $Sr_{D,co}=f_{co}D/u_{\infty }$ .

Regarding the transient behaviour of the heat flux sensor, it can be concluded that the transfer function $c_{\mathit{III}}$ connecting postshock entropy waves and heat flux fluctuations can be expected to behave like a low pass filter. From the findings in § 4.3, i.e. the initial diverging entropy wave has to be measured by the heat flux sensor to differentiate between the incident mode types based on heat flux and pressure signal, we can expect the SPP frequency response to different incident mode types to be identical above the determined cutoff frequency. The analysis of the frequency response is given in the next section.

4.5 Frequency response

To determine the frequency response, continuous waves of the fast acoustic, slow acoustic and entropy mode at different frequencies are superimposed to the flow. The cases of this investigation are denoted continuous in table 1. The frequencies of the investigation are based on the expected resonant behaviour of the larger heat flux SPP from the case up_h. Five Strouhal numbers are chosen to cover the frequency range up to the second destructive interference of postshock acoustic waves. Since the response coefficients of the pressure and heat flux fluctuations are required at a certain unit Reynolds number and frequency, lower Strouhal numbers result for the smaller pressure SPP from the case up_l. The non-dimensional analysis of the unperturbed flow in § 4.1 showed very similar Strouhal numbers dictating the resonant behaviour. Therefore, the Strouhal numbers for the smaller pressure SPP cover a smaller non-dimensional range.

In the subsequent analysis of the frequency response, the response coefficients $r_{j,i}$ of the transfer matrix $\unicode[STIX]{x1D64E}$

in the ansatz $\boldsymbol{F}_{w}=\unicode[STIX]{x1D64E}\boldsymbol{F}_{\infty }$ for the modal decomposition are utilized. The complete non-dimensional definition of the modal decomposition using fluctuations reads

In this formulation, both incident disturbances in $\boldsymbol{F}_{\infty }$ are quantified by their density fluctuation to yield a common frame of reference for the response coefficients. The response coefficients $r_{j,i}$ depend on the Strouhal number and are complex numbers to account for phase differences between incident waves and stagnation point fluctuations, i.e. for a single Strouhal number (4.15) is correct in time. To apply the decomposition to experimental power spectral density (PSD) distributions or bandwidth filtered nRMS values, equation (4.15) is converted into an nRMS formulation assuming uncorrelated acoustic and entropy waves in the free stream

where angle brackets denote nRMS values. The absolute values of the complex response coefficients $|r_{j,i}|$ represent ratios of normalized amplitudes and are henceforth referred to as response coefficients. The analysis of the response coefficients can be pursued following the discussion of Chaudhry & Candler (Reference Chaudhry and Candler2016). They derived analytic PSD based pressure response coefficients for the low frequency limit of any incident mode type. For our response coefficient definition using normalized amplitudes, the low frequency limit is defined as

where $p_{t}$ is the total pressure given by the Rayleigh Pitot tube formula and $i$ denotes the incident mode type being sa, en or fa. Evaluating (4.17), the analytic low frequency limit reads

where the upper sign in (4.18) is valid for the fast acoustic mode and the lower sign denotes the slow acoustic mode. Chaudhry & Candler found the pressure response coefficients of all types of incident modes to collapse when scaled by their analytic low frequency limit value. In figure 12(a) the scaled pressure response coefficients of all incident mode types at $Re=109\,375$ are plotted against the Strouhal number. Since the frequencies of this investigation were chosen according to the resonant behaviour of the larger SPP, the trend shows local minima and maxima as expected. The peak value of approximately 2 is in good agreement with the results of Chaudhry & Candler who find 2.2 when scaled to the current definition from (4.14) for a cylindrical probe. Figure 12(b) shows the heat flux response coefficients of the larger SPP. The coefficients of the different incident modes are also scaled by their analytic pressure response coefficient. The collapse of these curves using the analytic low frequency limit for pressure fluctuations also evidences the contribution of the dominant postshock acoustic wave to the heat flux fluctuations.

Figure 12.

Frequency response to continuous incident slow acoustic (sa1_h to sa5_h), entropy (en1_h to en5_h) and fast acoustic (fa1_h to fa5_h) waves at $Re=109\,375$ ; (a) pressure response $r_{p}$ scaled by the analytic low frequency limit of the pressure response $\unicode[STIX]{x1D712}_{i}$ , $i$ being as, en or af; (b) heat flux response $r_{q}$ scaled by the analytic low frequency limit of the pressure response $\unicode[STIX]{x1D712}_{i}$ , $i$ being sa, en or fa.

Figure 13.

Frequency response to continuous incident slow acoustic (sa1_l to sa5_l), entropy (en1_l to en5_l) and fast acoustic (fa1_l to fa5_l) waves at $Re=50\,000$ ; (a) pressure response $r_{p}$ scaled by the analytic low frequency limit of the pressure response $\unicode[STIX]{x1D712}_{i}$ , $i$ being as, en or af; (b) heat flux response $r_{q}$ scaled by the analytic low frequency limit of the pressure response $\unicode[STIX]{x1D712}_{i}$ , $i$ being sa, en or fa.

Figure 13(a) shows the pressure response coefficients of the smaller SPP at $Re=50\,000$ . As aforementioned, the Strouhal number range is limited to lower Strouhal numbers and only the first peak and valley of the response coefficients are captured in this investigation. Figure 13(b) shows the corresponding heat flux response coefficients. Again, the scaling by the analytic low frequency limit causes the curves to collapse.

To analyse the decomposition of the pressure and heat flux fluctuation stagnation point signals into different incident modes, the ratio of the SPP heat flux and pressure response coefficients of all incident modes at $Re=109\,375$ is shown in figure 14(a). The ratios $r_{q}/r_{p}$ of all incident modes collapse starting at $Sr=0.655$ . Inserting the response coefficients of two types of incident modes into the transfer matrix $\unicode[STIX]{x1D64E}$ and calculating the condition number $\unicode[STIX]{x1D705}$ based on the Euclidean norm leads in figure 14(b) to high condition numbers starting at $Sr=0.655$ for all combinations of incident modes since the column vectors become more aligned at increasing Strouhal number. Recalling the findings of § 4.3, this can result from a diminishing initial diverging entropy wave related heat flux contribution to the heat flux response coefficient $r_{q}$ . Apart from the absolute value of the heat flux response coefficient $r_{q}$ , the diverging entropy wave related heat flux contribution could also result in a mere phase shift. Figure 14(c) shows the phase difference between the pressure and heat flux signal $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{p}-\unicode[STIX]{x1D719}_{q}$ of all incident modes versus the Strouhal number. At $Sr=0.655$ a difference between the initial modes is still visible, even though the ratio of the response coefficient collapse. The influence of the initial diverging entropy wave related heat flux contribution to the heat flux absolute value and phase vanishes between $Sr=0.655$ and $Sr=1.309$ . This agrees well with the a priori determined value of $Sr_{D,co}=1$ in § 4.4.

Figure 14.

Analysis of the decomposition of the pressure and heat flux fluctuation signals at $Re=109\,375$ ; (a) ratio of heat flux and pressure response $r_{q}/r_{p}$ versus the Strouhal number of the continuous incident waves; (b) condition number $\unicode[STIX]{x1D705}$ of the transfer matrix $\unicode[STIX]{x1D64E}$ using the Euclidean norm for all combinations of incident modes; (c) phase difference between the pressure and heat flux signal $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{p}-\unicode[STIX]{x1D719}_{q}$ .

An analogous analysis can be conducted for the smaller SPP at $Re=50\,000$ . The corresponding data are shown in figure 15. For the lower Reynolds number case, the diverging entropy wave related heat flux contribution disappears in the response coefficient ratio and phase difference between $Sr=0.6$ and $Sr=0.9$ . This again matches well with the a priori determined value of $Sr_{D,co}=0.7$ .

Figure 15.

Analysis of the decomposition of the pressure and heat flux fluctuation signals at $Re=50\,000$ ; (a) ratio of heat flux and pressure response $r_{q}/r_{p}$ versus the Strouhal number of the continuous incident waves; (b) condition number $\unicode[STIX]{x1D705}$ of the transfer matrix $\unicode[STIX]{x1D64E}$ using the Euclidean norm for all combinations of incident modes; (c) phase difference between the pressure and heat flux signal $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{p}-\unicode[STIX]{x1D719}_{q}$ .