**6. Results and discussion**

### **6.1. Hemisphere-cylinder model**

A terminal shock wave of sufficient strength interacting with a boundary layer can cause flow separation and the process can become unsteady [40]. The numerical procedure described in the previous section is applied to compute the flowfield over the hemisphere-cylinder at *M<sup>∞</sup> = 0.90* and Reynolds number *5.1×106* . Figure 7 depicts the close-up view of velocity field. The strong attached flow near the hemisphere-cylinder junction and an expansion due to the hemisphere geometry makes way to a separation following a terminal shock wave accompa‐ nying with supersonic pocket. The separation and reattachment points are indicted by symbols *S* and *R*, respectively. The separation is confined to a short distance and flow reattach at *x/D* = 1.07 for M<sup>∞</sup> = 0.90. Figure 7 also shows the comparison between the numerical and the experimental results of Hsieh [1] – [2]. All the essential flowfield features of the transonic flow, such as supersonic pocket, terminal shock wave and expansion region are well captured and compare well with the shadowgraph picture.

**Figure 8.** Instantaneous vector plot over hemisphere-cylinder at M∞ = 0.90.

SPL value increases gradually in the local supersonic pocket.

fundamental frequency that is to be considered in the analysis

0

w

w w

0

*n n*

A spectral analysis is carried out on the computed pressure-time data for all possible modes of fluctuations employing fast Fourier transform [41], which converts the pressure history from time domain into frequency domain. Figure 9 shows the spectrum of sound pressure level SPL over the hemisphere-cylinder model. The pressure values have been converted from Pascal to

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99

( ) ( ) 20log *<sup>w</sup>*

The surface pressure levels are computed in terms of the pressure reference at *20×10-6* Pa. The frequencies for which assessment were carried out at the multiples of the fundamental frequency of 26 Hz for the hemisphere-cylinder model. A high value of SPL is found at 200 Hz at *x/D* = *1.7996*, *2.0660* and *2.2630*. A significant SPL of *168 dB* at about *26* Hz is found at *x/D = 2.611* which may be attributed to the separated flow associated with the terminal shock. The

The function is non-periodic, a period *T* is to be assumed. This is dictated by the lowest or

2

p

*T*

where *ω* is angular frequency. The period is divided into N equal intervals of *Δt* and the

*T*

=D=

 ww

=D =

*n*

 *= mΔt.* The Fast Fourier Transform FFT is a computer algorithm

*p t SPL db*

*ref*

(23)

(24)

*p* é ù <sup>=</sup> ê ú ê ú ë û

**6.2. Spectral analysis**

decibel (dB) of surface pressure levels.

function is sampled at time *tm*

**Figure 7.** Comparison of numerical with experimental results over hemisphere-cylinder model.

Once the initial phase of the computation is over 16 axial location (*x/D* = *0.16 - 2.50*) along the hemisphere-cylinder measured from the stagnation point of the hemisphere are selected to study the sensitivity of the unsteadiness in the flow. Figure 8 depicts instantaneous vector plot at *M<sup>∞</sup> = 0.90*. The calculated surface pressure data are analyzed for the time mean, root mean pressure and, fast Fourier transform FFT [41] – [42].

**Figure 8.** Instantaneous vector plot over hemisphere-cylinder at M∞ = 0.90.

### **6.2. Spectral analysis**

 *=*

**6. Results and discussion**

98 Computational and Numerical Simulations

**6.1. Hemisphere-cylinder model**

*0.90* and Reynolds number *5.1×106*

compare well with the shadowgraph picture.

= 1.07 for M<sup>∞</sup>

at *M<sup>∞</sup>*

A terminal shock wave of sufficient strength interacting with a boundary layer can cause flow separation and the process can become unsteady [40]. The numerical procedure described in the previous section is applied to compute the flowfield over the hemisphere-cylinder at *M<sup>∞</sup>*

strong attached flow near the hemisphere-cylinder junction and an expansion due to the hemisphere geometry makes way to a separation following a terminal shock wave accompa‐ nying with supersonic pocket. The separation and reattachment points are indicted by symbols *S* and *R*, respectively. The separation is confined to a short distance and flow reattach at *x/D*

experimental results of Hsieh [1] – [2]. All the essential flowfield features of the transonic flow, such as supersonic pocket, terminal shock wave and expansion region are well captured and

**Figure 7.** Comparison of numerical with experimental results over hemisphere-cylinder model.

pressure and, fast Fourier transform FFT [41] – [42].

Once the initial phase of the computation is over 16 axial location (*x/D* = *0.16 - 2.50*) along the hemisphere-cylinder measured from the stagnation point of the hemisphere are selected to study the sensitivity of the unsteadiness in the flow. Figure 8 depicts instantaneous vector plot

 *= 0.90*. The calculated surface pressure data are analyzed for the time mean, root mean

= 0.90. Figure 7 also shows the comparison between the numerical and the

. Figure 7 depicts the close-up view of velocity field. The

A spectral analysis is carried out on the computed pressure-time data for all possible modes of fluctuations employing fast Fourier transform [41], which converts the pressure history from time domain into frequency domain. Figure 9 shows the spectrum of sound pressure level SPL over the hemisphere-cylinder model. The pressure values have been converted from Pascal to decibel (dB) of surface pressure levels.

$$SPL(db) = 20\log\left[\frac{p\_w(t)}{p\_{ref}}\right] \tag{23}$$

The surface pressure levels are computed in terms of the pressure reference at *20×10-6* Pa. The frequencies for which assessment were carried out at the multiples of the fundamental frequency of 26 Hz for the hemisphere-cylinder model. A high value of SPL is found at 200 Hz at *x/D* = *1.7996*, *2.0660* and *2.2630*. A significant SPL of *168 dB* at about *26* Hz is found at *x/D = 2.611* which may be attributed to the separated flow associated with the terminal shock. The SPL value increases gradually in the local supersonic pocket.

The function is non-periodic, a period *T* is to be assumed. This is dictated by the lowest or fundamental frequency that is to be considered in the analysis

$$\begin{aligned} \alpha\_0 &= \alpha \Delta T = \frac{2\pi}{T} \\ \alpha \alpha \alpha\_0 &= \eta \Delta \alpha = \alpha\_n \end{aligned} \tag{24}$$

where *ω* is angular frequency. The period is divided into N equal intervals of *Δt* and the function is sampled at time *tm = mΔt.* The Fast Fourier Transform FFT is a computer algorithm for calculating Discrete Fourier Transform DFT. The FFT computes in the frequency domain can be written as

$$F\left(o\_n\right) = \frac{T}{N} \sum\_{m=0}^{N-1} F\left(t\_m\right) e^{i\left(-2\pi mm/N\right)}\tag{25}$$

pressure level

pocket, the terminal shock wave, and the expansion and compression regions, are well captured by present numerical simulations. It can be seen from the density contour plots of heat shield that the formation of the terminal shock and supersonic region over the payload shroud is a function of the geometrical parameters of the heat shield [43]. Envelope of Mach lines leads to formation of the terminal shock wave. The expansion regions of the axisymmteric supersonic flow are characterized by diverging Mach lines, whereas in the compression region they converge to form a shock wave. In the shock wave theory [8] in fact there is no "expansion shock". The flow separation on the payload shroud is caused by the terminal shock wave. The shock-induced separated flow on the cone-cylinder is not found for the heat shield with a cone angle of *15* deg. As freestream Mach number increases, the terminal shock moves downstream and the local supersonic zone increases rapidly. The terminal shock becomes so strong that, as a result of a shock wave-boundary layer interaction, boundary layer separation occurs and it is the function of shock strength, geometrical parameter of the heat shield and freestream Mach number. The density contour plots reveals that a shear layer is formed which accom‐ modates the recirculating flow for the transonic speeds. The downstream boundary layer is found to be thick, which is nearly the boat-tail height. It is worth to mention here that the main purpose to introduce the boat-tail is to increase payload volume of a satellite launch vehicle. The shock wave separated boundary layer and flow separation caused by boat tail geometry of heat shield generated high and low frequency pressure fluctuations. Flow induced vibra‐ tions are important issues to be taken into account the design requirement of the satellite. Shock wave separated boundary layer and flow separation caused by boat tail geometry of the heat shield generated high and low frequency pressure fluctuations. Analyses of the time-depend‐ ent flowfield feature are essential to design the bulbous heat shield of a satellite launch vehicle. Fluctuations of pressure level in shock waves and in separation areas induce flow instabilities

S R S R S S R R

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M∞=0.800 M∞=0.85

**Figure 10.** Velocity vector plots at transonic Mach number.

and then structural vibration leading to the buffeting phenomenon.

longitudinal curvature. The flow reattachment length (*Xr*

height *H*, where *Xc*

In the boat-tail region, a local flow separation occurs, due to sharp discontinuity in the

prediction of reattachment point is the point where the axial component of the velocity along the downstream wall changes from negative to positive. The non-dimensional separation

is the location of the boat tail shoulder and *Xr*

 *– Xc*

) is normalized by the boat-tail

is the reattachment point. The

where *N* is the number of data points, *Δt* is sampling interval and *T = NΔt* is record length. The FFT offers an enormous reduction of computer time as compared to DFT.

**Figure 9.** Pressure coefficient, histogram and SPL over the hemisphere-cylinder.

### **6.3. Bulbous heat shield of a satellite launch vehicle**

Figure 10 depicts the density contour plots for the freestream transonic Mach number of 0.80 and 0.90 for the bulbous heat shield and compare the density plots with schlieren pictures. The strength of the terminal shock wave initially increases with Mach number and later decreases. It can be observed from the figures that all of the essential flowfield features of the transonic flow, such as supersonic pocket, normal shock, and expansion and compression regions are very well captured and compare well with the schlieren pictures. The density contour plots reveal that the supersonic pocket increases with increasing freestream Mach number, and as a result, the terminal shock moves downstream with increasing freestream Mach number. It is important to mention here that the density increases ahead of the stagnation region of the heat shield moves close to the heat shield with the increasing transonic Mach number. It also depends on the cone angle of the heat shield as observed in the density contours plots of the flowfield.

The general flowfield along the payload shroud is shown in Fig. 11 for freestream Mach number of *0.80 – 1.00*. Figure 12 depicts comparison between density contours and schlieren pictures. All of the essential flow characteristics of the transonic flow, such as the supersonic

**Figure 10.** Velocity vector plots at transonic Mach number.

for calculating Discrete Fourier Transform DFT. The FFT computes in the frequency domain

*<sup>N</sup> i nm N*

p

<sup>=</sup> å (25)

pressure level

( ) ( ) ( ) <sup>1</sup> 2 / 0


where *N* is the number of data points, *Δt* is sampling interval and *T = NΔt* is record length.

(a) Pressure coefficient fluctuations (b) Histogram (c) Spectrum of sound

Figure 10 depicts the density contour plots for the freestream transonic Mach number of 0.80 and 0.90 for the bulbous heat shield and compare the density plots with schlieren pictures. The strength of the terminal shock wave initially increases with Mach number and later decreases. It can be observed from the figures that all of the essential flowfield features of the transonic flow, such as supersonic pocket, normal shock, and expansion and compression regions are very well captured and compare well with the schlieren pictures. The density contour plots reveal that the supersonic pocket increases with increasing freestream Mach number, and as a result, the terminal shock moves downstream with increasing freestream Mach number. It is important to mention here that the density increases ahead of the stagnation region of the heat shield moves close to the heat shield with the increasing transonic Mach number. It also depends on the cone angle of the heat shield as observed in the density contours plots of the

The general flowfield along the payload shroud is shown in Fig. 11 for freestream Mach number of *0.80 – 1.00*. Figure 12 depicts comparison between density contours and schlieren pictures. All of the essential flow characteristics of the transonic flow, such as the supersonic

*n m m <sup>T</sup> <sup>F</sup> Ft e N*

The FFT offers an enormous reduction of computer time as compared to DFT.

=

w

**Figure 9.** Pressure coefficient, histogram and SPL over the hemisphere-cylinder.

**6.3. Bulbous heat shield of a satellite launch vehicle**

can be written as

100 Computational and Numerical Simulations

flowfield.

pocket, the terminal shock wave, and the expansion and compression regions, are well captured by present numerical simulations. It can be seen from the density contour plots of heat shield that the formation of the terminal shock and supersonic region over the payload shroud is a function of the geometrical parameters of the heat shield [43]. Envelope of Mach lines leads to formation of the terminal shock wave. The expansion regions of the axisymmteric supersonic flow are characterized by diverging Mach lines, whereas in the compression region they converge to form a shock wave. In the shock wave theory [8] in fact there is no "expansion shock". The flow separation on the payload shroud is caused by the terminal shock wave. The shock-induced separated flow on the cone-cylinder is not found for the heat shield with a cone angle of *15* deg. As freestream Mach number increases, the terminal shock moves downstream and the local supersonic zone increases rapidly. The terminal shock becomes so strong that, as a result of a shock wave-boundary layer interaction, boundary layer separation occurs and it is the function of shock strength, geometrical parameter of the heat shield and freestream Mach number. The density contour plots reveals that a shear layer is formed which accom‐ modates the recirculating flow for the transonic speeds. The downstream boundary layer is found to be thick, which is nearly the boat-tail height. It is worth to mention here that the main purpose to introduce the boat-tail is to increase payload volume of a satellite launch vehicle.

The shock wave separated boundary layer and flow separation caused by boat tail geometry of heat shield generated high and low frequency pressure fluctuations. Flow induced vibra‐ tions are important issues to be taken into account the design requirement of the satellite. Shock wave separated boundary layer and flow separation caused by boat tail geometry of the heat shield generated high and low frequency pressure fluctuations. Analyses of the time-depend‐ ent flowfield feature are essential to design the bulbous heat shield of a satellite launch vehicle. Fluctuations of pressure level in shock waves and in separation areas induce flow instabilities and then structural vibration leading to the buffeting phenomenon.

In the boat-tail region, a local flow separation occurs, due to sharp discontinuity in the longitudinal curvature. The flow reattachment length (*Xr – Xc* ) is normalized by the boat-tail height *H*, where *Xc* is the location of the boat tail shoulder and *Xr* is the reattachment point. The prediction of reattachment point is the point where the axial component of the velocity along the downstream wall changes from negative to positive. The non-dimensional separation

shield. Before the analysis of the amplification factor and sound pressure levels are initiated, a statistical approach was employed to ensure that the computed data are free from transitional phase, i.e., the pressure values are representative of the numerical results, if the computation

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(a) Reattachment distance as a function of Mach number (b) Location of terminal shock on heat shield

A set of histograms of *Cp* is depicted in Fig. 15. The numerical data in the separation region and other stations exhibit a Gaussian distribution. A spectral analysis was carried out on the computed pressure data for all possible modes of oscillations using fast Fourier transform FFT of MATLAB [44]. A cyclic behaviour of the pressure coefficient is observed in the vicinity of the separation and reattachment points. From the spectrum analysis low frequency pressure

The wall pressure fluctuations may arise as an effect of unsteady pressure associated to the turbulent velocity field. The Mach variation on the flow physics is the change of the location of the intense shock wave which is originally generated on the fairing and moves towards the booster for Mach approaching unity. From the acoustic point of view, it is observed that the

wave is observed. These behaviors can be qualitatively seen in the schlieren picture. The visualizations suggest that a shock wave is generated in the fairing region and then it moves downstream for increasing Mach. The present chapter is focalized on the transonic range of flow conditions. The characterization of the pressure fluctuations is accomplished by statistical analysis, and flow visualizations are used to the physical interpretation of the results. The overall sound pressure level is mentioned in the density contour plots. The overall sound

 *= 0.80*, where a significant unsteadiness of the shock

fluctuations and sound pressure levels are found at freestream Mach number 0.95.

pressure level OSPL reaches the maximum amplitude at transonic conditions.

had continued for a very long period of time.

**Figure 13.** Overall flowfield features over the bulbous heat shield.

most critical situation correspond to *M<sup>∞</sup>*

**Figure 11.** Density contour plots over the bulbous heat shield at transonic Mach number.

**Figure 12.** Comparison between density contour plots and schlieren pictures.

length (*Xr – Xc* )/*H* is found *11.7* from the velocity vector plot as shown in Fig. 13(a). The location of the flow reattachment point is required to analyze the wall pressure fluctuations. The density contour plots of Fig. 11 reveal that the supersonic region increases with increasing freestream Mach number, and as a result, the terminal shock moves downstream with increasing freestream Mach number as shown in Fig. 13(b).

Once the initial phase of the computation was completed, some unsteadiness in the flow characteristics was observed. The sixteen locations (*x/D*) along the heat shield measured from the stagnation point are selected to study the unsteadiness of the flowfield. Figure 14 shows the instantaneous flow separation and pressure distribution at different location of the heat shield. Before the analysis of the amplification factor and sound pressure levels are initiated, a statistical approach was employed to ensure that the computed data are free from transitional phase, i.e., the pressure values are representative of the numerical results, if the computation had continued for a very long period of time.

(a) Reattachment distance as a function of Mach number (b) Location of terminal shock on heat shield

**Figure 13.** Overall flowfield features over the bulbous heat shield.

length (*Xr*

 *– Xc*

102 Computational and Numerical Simulations

freestream Mach number as shown in Fig. 13(b).

**Figure 12.** Comparison between density contour plots and schlieren pictures.

)/*H* is found *11.7* from the velocity vector plot as shown in Fig. 13(a). The location

of the flow reattachment point is required to analyze the wall pressure fluctuations. The density contour plots of Fig. 11 reveal that the supersonic region increases with increasing freestream Mach number, and as a result, the terminal shock moves downstream with increasing

**Figure 11.** Density contour plots over the bulbous heat shield at transonic Mach number.

Once the initial phase of the computation was completed, some unsteadiness in the flow characteristics was observed. The sixteen locations (*x/D*) along the heat shield measured from the stagnation point are selected to study the unsteadiness of the flowfield. Figure 14 shows the instantaneous flow separation and pressure distribution at different location of the heat A set of histograms of *Cp* is depicted in Fig. 15. The numerical data in the separation region and other stations exhibit a Gaussian distribution. A spectral analysis was carried out on the computed pressure data for all possible modes of oscillations using fast Fourier transform FFT of MATLAB [44]. A cyclic behaviour of the pressure coefficient is observed in the vicinity of the separation and reattachment points. From the spectrum analysis low frequency pressure fluctuations and sound pressure levels are found at freestream Mach number 0.95.

The wall pressure fluctuations may arise as an effect of unsteady pressure associated to the turbulent velocity field. The Mach variation on the flow physics is the change of the location of the intense shock wave which is originally generated on the fairing and moves towards the booster for Mach approaching unity. From the acoustic point of view, it is observed that the most critical situation correspond to *M<sup>∞</sup> = 0.80*, where a significant unsteadiness of the shock wave is observed. These behaviors can be qualitatively seen in the schlieren picture. The visualizations suggest that a shock wave is generated in the fairing region and then it moves downstream for increasing Mach. The present chapter is focalized on the transonic range of flow conditions. The characterization of the pressure fluctuations is accomplished by statistical analysis, and flow visualizations are used to the physical interpretation of the results. The overall sound pressure level is mentioned in the density contour plots. The overall sound pressure level OSPL reaches the maximum amplitude at transonic conditions.

**6.4. Statistics analysis**

A statistical approach was used in order to ensure that the data are free from transitional phase, i.e., the pressure values are representive of the data, if computational had continued for a long time. The computed surface pressure data along the shroud were analyzed for the time mean

1

*i*

( )

*Cp Cp n*


<sup>1</sup> 1 *<sup>n</sup> <sup>i</sup>*

is over. Higher order moments of pressure fluctuations, the skewness coefficient, and the

( )

*Cp Cp*

*i*

é ù ê ú ë û <sup>=</sup>

*Cp*

( )

*Cp Cp*

*i*

é ù ê ú ë û <sup>=</sup>

*Cp*

The skewness coefficient expresses the asymmetric of fluctuations around the mean value and the kurtosis coefficient expresses the symmetric property around the mean value. Figure 16

The flow is assumed to be laminar for the spiked-blunt body, which is consistent with the experimental study of Crawford [24] and Kenworthy [45] and the numerical simulation of Yamauchi et al. [46], Hankey and Shang [47] and Badcock et al. [28]. Therefore, the turbulent

Navier-Stokes equations are employed to analyze unsteady low over the spiked-blunt body.

is neglected in Eq. (10). The time-dependent axisymmetric compressible laminar

s

s

1 3

1 4

1 *<sup>n</sup>*

*i*

1 *<sup>n</sup>*

*i*

=

*n*

shows variation of mean pressure, rms, skewness, and Kurtosis coefficient.

The coefficient of molecular viscosity *μ* is calculated using Sutherland's law.

=

*n*

2

3

4

<sup>=</sup> å (26)

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<sup>=</sup> - <sup>å</sup> (27)

*s*) and is considered after 47 900 time step computation

<sup>å</sup> (28)

<sup>å</sup> (29)

1 *<sup>n</sup>*

*i Cp Cp n* <sup>=</sup>

and standard deviation values using the following relations:

*Cp*

s

*Cp*

*Cp*

s

s

The total time period *nΔt* (*Δt = 0.8 × 10-6*

**6.5. Flowfield over the spiked-blunt body**

viscosity *μ<sup>t</sup>*

kurtosis coefficient are expressed as

*i*

=

**Figure 14.** Velocity vector plot and fluctuations of pressure coefficient.

**Figure 15.** Histograms of pressure coefficient over the bulbous heat shield.

The numerical simulation is used to analysis the unsteady flowfield characteristics of the bulbous payload shroud.

### **6.4. Statistics analysis**

A statistical approach was used in order to ensure that the data are free from transitional phase, i.e., the pressure values are representive of the data, if computational had continued for a long time. The computed surface pressure data along the shroud were analyzed for the time mean and standard deviation values using the following relations:

$$\overline{Cp} = \frac{1}{n} \sum\_{i=1}^{n} \mathbb{C}p\_i \tag{26}$$

$$\sigma\_{Cp} = \sqrt{\sum\_{i=1}^{n} \frac{\left(Cp\_i - \overline{Cp}\right)^2}{n-1}} \tag{27}$$

The total time period *nΔt* (*Δt = 0.8 × 10-6 s*) and is considered after 47 900 time step computation is over. Higher order moments of pressure fluctuations, the skewness coefficient, and the kurtosis coefficient are expressed as

$$\sigma\_{\mathbb{C}p} = \frac{\left[\frac{1}{n} \sum\_{i=1}^{n} \left(\mathbb{C}p\_i - \overline{\mathbb{C}p}\right)^3\right]}{\sigma\_{\mathbb{C}p}^3} \tag{28}$$

$$\sigma\_{\text{Cp}} = \frac{\left[\frac{1}{n} \sum\_{i=1}^{n} \left(\text{Cp}\_{i} - \overline{\text{Cp}}\right)^{4}\right]}{\sigma\_{\text{Cp}}^{4}} \tag{29}$$

The skewness coefficient expresses the asymmetric of fluctuations around the mean value and the kurtosis coefficient expresses the symmetric property around the mean value. Figure 16 shows variation of mean pressure, rms, skewness, and Kurtosis coefficient.

### **6.5. Flowfield over the spiked-blunt body**

**Figure 15.** Histograms of pressure coefficient over the bulbous heat shield.

**Figure 14.** Velocity vector plot and fluctuations of pressure coefficient.

104 Computational and Numerical Simulations

bulbous payload shroud.

The numerical simulation is used to analysis the unsteady flowfield characteristics of the

(a) Velocity vector plot (b) Pressure coefficient fluctuations

The flow is assumed to be laminar for the spiked-blunt body, which is consistent with the experimental study of Crawford [24] and Kenworthy [45] and the numerical simulation of Yamauchi et al. [46], Hankey and Shang [47] and Badcock et al. [28]. Therefore, the turbulent viscosity *μ<sup>t</sup>* is neglected in Eq. (10). The time-dependent axisymmetric compressible laminar Navier-Stokes equations are employed to analyze unsteady low over the spiked-blunt body. The coefficient of molecular viscosity *μ* is calculated using Sutherland's law.

**Figure 16.** Variations of (a) mean pressure (b) rms (c) skewness (d) Kurtosis coefficients over the bulbous heat shield.

Figure 17 shows the enlarged view of the computed density contour and velocity plots for semi-cone angle of spike *α* = 10, 15, 20 and 30 deg at M<sup>∞</sup> = 6.0 and *L/D* = 0.5. Figure 18 shows the pressure variation (*p/pa*) along the surface of the spiked-blunt body for different semi-cone angle of the spike. The wall pressure is normalized by freestream pressure *pa*. The *x* = *0* location is the spike/nose-tip junction. The location of the maximum pressure on the surface of the spiked-blunt body is at a body angle of about 40 deg for all the semi-cone angle of the conical spike. This location corresponds to the reattachment point. A wavy pressure distribution is observed on the spike, which may be attributed the separated flowfield behavior. The maximum pressure level is occurred at the same location on the blunt body. The flowfield can be studied using the shock polar diagram in conjunction with the Computational Fluid Dynamics approach [48]. The computed conical shock wave angles are compared with Ref. [49] and found good agreement between them.

**Figure 18.** Pressure distributions along the spiked-blunt body.

**6.6. Flow characteristics for the spiked-blunt body**

**Figure 17.** Enlarged view of velocity vector plot over conical spiked-blunt body.

Figure 19 show the enlarged view of the density contours and velocity vector plots for spike

6.80. It can be seen from the figures that interaction between the conical shock emanating from the tip of the spike and the reattachment shock wave on the blunt body is observed. The reflected reattachment shock wave and shear layer from the interaction are observed behind the reattachment shock wave. A large separated flow region is visible in the vector plots. In the separation region, a number of vortices exist and the velocity magnitude is very low.

Flowfield was analyzed for *M<sup>∞</sup> = 2.01, 4.15, 6.80* for *L/D = 0.5* and for the *Re* = *0.14 × 10*<sup>6</sup>

on the diameter of the hemispherical body D. Computed density contour plots with schlieren pictures are compared in Fig. 20 for L/D = 0.5, 1.0, 2.0 for M∞ = 6.80. The computed flowfield shows agreement with the schlieren photographs taken in the experiment by Yamauchi et al.

= 2.01, 4.15 and

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based

lengths of *L/D = 0.5* and the semi-cone angle of the spike 10 deg (Fig. 5) at *M<sup>∞</sup>*

It can be observed from the figure that interaction between the conical oblique shock wave emanating from the tip of the spike and the reattachment shock wave on the blunt body is seen. The reflected reattachment shock wave and shear layer from the interaction are seen behind the reattachment shock wave. A large separated region is observed in front of the blunt body. Flow patterns are same as that for *L/D = 0.5*. When the spike is long, the angle of the conical shock wave emanating from the spike-tip decreases and flow separation occurs slightly downstream. Since the reattachment point moved backward and the spike is long, the length of the separated region extended.

**Figure 17.** Enlarged view of velocity vector plot over conical spiked-blunt body.

**Figure 18.** Pressure distributions along the spiked-blunt body.

Figure 17 shows the enlarged view of the computed density contour and velocity plots for

**Figure 16.** Variations of (a) mean pressure (b) rms (c) skewness (d) Kurtosis coefficients over the bulbous heat shield.

the pressure variation (*p/pa*) along the surface of the spiked-blunt body for different semi-cone angle of the spike. The wall pressure is normalized by freestream pressure *pa*. The *x* = *0* location is the spike/nose-tip junction. The location of the maximum pressure on the surface of the spiked-blunt body is at a body angle of about 40 deg for all the semi-cone angle of the conical spike. This location corresponds to the reattachment point. A wavy pressure distribution is observed on the spike, which may be attributed the separated flowfield behavior. The maximum pressure level is occurred at the same location on the blunt body. The flowfield can be studied using the shock polar diagram in conjunction with the Computational Fluid Dynamics approach [48]. The computed conical shock wave angles are compared with Ref.

It can be observed from the figure that interaction between the conical oblique shock wave emanating from the tip of the spike and the reattachment shock wave on the blunt body is seen. The reflected reattachment shock wave and shear layer from the interaction are seen behind the reattachment shock wave. A large separated region is observed in front of the blunt body. Flow patterns are same as that for *L/D = 0.5*. When the spike is long, the angle of the conical shock wave emanating from the spike-tip decreases and flow separation occurs slightly downstream. Since the reattachment point moved backward and the spike is long, the length

= 6.0 and *L/D* = 0.5. Figure 18 shows

semi-cone angle of spike *α* = 10, 15, 20 and 30 deg at M<sup>∞</sup>

106 Computational and Numerical Simulations

[49] and found good agreement between them.

of the separated region extended.

### **6.6. Flow characteristics for the spiked-blunt body**

Figure 19 show the enlarged view of the density contours and velocity vector plots for spike lengths of *L/D = 0.5* and the semi-cone angle of the spike 10 deg (Fig. 5) at *M<sup>∞</sup>* = 2.01, 4.15 and 6.80. It can be seen from the figures that interaction between the conical shock emanating from the tip of the spike and the reattachment shock wave on the blunt body is observed. The reflected reattachment shock wave and shear layer from the interaction are observed behind the reattachment shock wave. A large separated flow region is visible in the vector plots. In the separation region, a number of vortices exist and the velocity magnitude is very low.

Flowfield was analyzed for *M<sup>∞</sup> = 2.01, 4.15, 6.80* for *L/D = 0.5* and for the *Re* = *0.14 × 10*<sup>6</sup> based on the diameter of the hemispherical body D. Computed density contour plots with schlieren pictures are compared in Fig. 20 for L/D = 0.5, 1.0, 2.0 for M∞ = 6.80. The computed flowfield shows agreement with the schlieren photographs taken in the experiment by Yamauchi et al. [46] and Crawford [24]. Figure 21 reveals the effects of ratio of *L/D* and M<sup>∞</sup> over the flow over the spiked-blunt body.

**Figure 19.** Enlarged views of velocity vector plots.

**Figure 21.** Density vector plots over spiked-blunt body.

time steps *Δt* were 5.0 × 10-7

coefficient [*Cp = 2{(p/p<sup>∞</sup>*

*M<sup>∞</sup>*

Once the oscillatory motion is established in the flow, as can be visualized in the instantaneous velocity vector plots in Fig. 22, the periodic phenomenon is investigated by a spectral analysis to obtain information on the frequency and amplitude for various modes of oscillations. The

The periodic phenomenon is investigated by a spectral analysis to obtain information on the frequency and amplitude for various modes of oscillation. Figure 23 show the pressure

 *= 4.15* and *L/D = 0.5*. The interaction of the conical shock wave emanating from the spike tip with the separation vortices governs the pressure oscillations. The pressure oscillations are

are low in amplitude and depend on the location on the spike. This unsteady behavior of the

A spectral analysis is carried out on the computed pressure history employing FFT of MATLAB

computed using pressure oscillations data *p(t).* In the spectrum plots, there are pressure amplitude peaks of dominant frequency and multiples of the dominant frequency at various locations of the spike. The spectral analysis of the pressure reveals that the discrete frequencies

[44]. The pressure amplitude versus frequency and phase plots for *L/D = 0.5* and *M<sup>∞</sup>*

= 6.8 and *L/D = 0.5*.

] variation with respect to time on the spiked-blunt body at

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. These pressure oscillations

 *= 6.8* are

*s* taken for M<sup>∞</sup>

*2*

found more cyclic in nature and function of the *L/D* ratio and *M<sup>∞</sup>*

of higher modes of oscillations are multiples of the principal modes.

flowfield is caused by the separated region enclosed inside the reattachment.

*) – 1}/γM<sup>∞</sup>*

**Figure 20.** Comparison between density contours and schlieren photographs.

**Figure 21.** Density vector plots over spiked-blunt body.

[46] and Crawford [24]. Figure 21 reveals the effects of ratio of *L/D* and M<sup>∞</sup>

the spiked-blunt body.

108 Computational and Numerical Simulations

**Figure 19.** Enlarged views of velocity vector plots.

**Figure 20.** Comparison between density contours and schlieren photographs.

over the flow over

Once the oscillatory motion is established in the flow, as can be visualized in the instantaneous velocity vector plots in Fig. 22, the periodic phenomenon is investigated by a spectral analysis to obtain information on the frequency and amplitude for various modes of oscillations. The time steps *Δt* were 5.0 × 10-7 *s* taken for M<sup>∞</sup> = 6.8 and *L/D = 0.5*.

The periodic phenomenon is investigated by a spectral analysis to obtain information on the frequency and amplitude for various modes of oscillation. Figure 23 show the pressure coefficient [*Cp = 2{(p/p<sup>∞</sup> ) – 1}/γM<sup>∞</sup> 2* ] variation with respect to time on the spiked-blunt body at *M<sup>∞</sup> = 4.15* and *L/D = 0.5*. The interaction of the conical shock wave emanating from the spike tip with the separation vortices governs the pressure oscillations. The pressure oscillations are found more cyclic in nature and function of the *L/D* ratio and *M<sup>∞</sup>* . These pressure oscillations are low in amplitude and depend on the location on the spike. This unsteady behavior of the flowfield is caused by the separated region enclosed inside the reattachment.

A spectral analysis is carried out on the computed pressure history employing FFT of MATLAB [44]. The pressure amplitude versus frequency and phase plots for *L/D = 0.5* and *M<sup>∞</sup> = 6.8* are computed using pressure oscillations data *p(t).* In the spectrum plots, there are pressure amplitude peaks of dominant frequency and multiples of the dominant frequency at various locations of the spike. The spectral analysis of the pressure reveals that the discrete frequencies of higher modes of oscillations are multiples of the principal modes.

Figure 24 represents the pressure amplitude versus frequency and phase plots for L/D = 0.5

frequency and multiples of the dominant frequency at different stations of the spike. The spectral analysis of the pressure reveals that the discrete frequencies of higher mode of oscillation are multiples of the principal modes. The vortex pattern inside the separated region is different for different spike lengths. Therefore the second and third mode frequencies are

The fluid dynamics of the self-sustained oscillatory flow is analyzed using spring-mass analogy as well as the nonlinear oscillatory model. The self-excited oscillation is governed or autonomous and draws its energy from the external source by its own periodic motion. For small oscillations, energy is fed into the system and there is "negative damping" [50, 51]. For large flow oscillations, energy is taken from the system and therefore damped. The periodic pressure behavior is analogous with differential equation describing the self-sustained oscillation of Van der Pol equation [51]. Figure 25 shows [*d(Cp)/dt*] versus *Cp*. The phase plane portrait is analyzed to understand the characteristics of the oscillatory flow. The phase plane plots are computed using the time-dependent pressure data. The phase plots reveal the characteristics of the oscillatory pressure field. The motion tends to build up small oscillations and decrease for large oscillations, which indicates that the damping term is greater than zero, hence, after the initial transient, the motion becomes periodic, represented by a closed

= 6.80. In the spectrum plots, there are pr4essure amplitude peaks of dominant

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and M<sup>∞</sup>

different for different location.

**Figure 24.** Spectrum of sound pressure level and pressure oscillations.

**6.7. Self-excited oscillation for the spiked-blunt body**

trajectory which is also called a limit cycle.

**Figure 22.** Instantaneous vector plot, M∞ = 6.8 and L/D = 0.5.

**Figure 23.** Pressure oscillations, *M*∞ *= 4.15* and *L/D = 0.5*.

Figure 24 represents the pressure amplitude versus frequency and phase plots for L/D = 0.5 and M<sup>∞</sup> = 6.80. In the spectrum plots, there are pr4essure amplitude peaks of dominant frequency and multiples of the dominant frequency at different stations of the spike. The spectral analysis of the pressure reveals that the discrete frequencies of higher mode of oscillation are multiples of the principal modes. The vortex pattern inside the separated region is different for different spike lengths. Therefore the second and third mode frequencies are different for different location.

**Figure 24.** Spectrum of sound pressure level and pressure oscillations.

### **6.7. Self-excited oscillation for the spiked-blunt body**

**Figure 23.** Pressure oscillations, *M*∞ *= 4.15* and *L/D = 0.5*.

**Figure 22.** Instantaneous vector plot, M∞ = 6.8 and L/D = 0.5.

110 Computational and Numerical Simulations

The fluid dynamics of the self-sustained oscillatory flow is analyzed using spring-mass analogy as well as the nonlinear oscillatory model. The self-excited oscillation is governed or autonomous and draws its energy from the external source by its own periodic motion. For small oscillations, energy is fed into the system and there is "negative damping" [50, 51]. For large flow oscillations, energy is taken from the system and therefore damped. The periodic pressure behavior is analogous with differential equation describing the self-sustained oscillation of Van der Pol equation [51]. Figure 25 shows [*d(Cp)/dt*] versus *Cp*. The phase plane portrait is analyzed to understand the characteristics of the oscillatory flow. The phase plane plots are computed using the time-dependent pressure data. The phase plots reveal the characteristics of the oscillatory pressure field. The motion tends to build up small oscillations and decrease for large oscillations, which indicates that the damping term is greater than zero, hence, after the initial transient, the motion becomes periodic, represented by a closed trajectory which is also called a limit cycle.

**Nomenclature**

*Cp* = specific heat at constant pressure

*E* = total specific energy, *(e + 0.5(ui*

*F*, *G*, *H* = flux vectors *L* = length of spike *M* = Mach number

*qj*

*N* = number of data points *Pr* = Prandtl number *p* = static pressure

 = heat flux components *Re* = Reynolds number *SPL* = sound pressure level

 = velocity components *u, v* = axial and radial velocity

 = Cartesian coordinate *x, r* = axial and radial coordinate

β = angle of conical shock wave γ = ratio of specific heats δ = Kronecker delta μ = molecular viscosity

α = semi-cone angle

ρ = density

τ*ij* = stress tensor Δ = increment

σ*ij* = viscous stress tensor

*U* = conservative variables in vector form

*T* = temperature

*t* = time *ui*

*xj*

*ui ))*

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*Cp* = pressure coefficient

*D* = diameter *c* = sound velocity *d* = booster diameter *e* = internal specific energy

**Figure 25.** Phase trajectory, *M*∞ *= 6.8* and *L/D = 0.5*.
