1. Introduction
The characteristics of the unsteady flowfield over a hemispherecylinder model and a bulbous heat shield of a satellite launch vehicle at turbulent flow and at transonic speeds and a laminar flow over a conical spike attached to a forward facing blunt body at supersonic speed are numerically simulated by solving timedependent compressible axisymmetric NavierStokes equations.
Hsieh [1] has conducted windtunnel tests of a hemispherecylinder model at zero angle of attack and freestream Mach number M_{∞} = 0.7 − 1.3 to investigate viscousinviscid interaction. Hsieh [2] has solved full potential equation to analyze windtunnel results and found that the inviscid analysis is unable to predict the external flowfield satisfactory. This is due to the fact that the shock waveturbulent boundary layer interaction causes a separated flow between M_{∞} = 0.80 − 0.90 on the hemispherecylinder model in a high speed windtunnel testing. The numerical simulations analyze the unsteady flow caused by shock waveturbulent boundary layer at transonic Mach numbers.
A bulbous payload shroud is generally selected to accommodate an increased payload volume of the satellite in a launch vehicle. All launch vehicles require a heat shield to protect the satellite from aerodynamic loading, heating, aeroacoustic vibration, and other environmental conditions during the ascent phase of the flight and to provide aerodynamic forward surface. The windtunnel tests for Titan I − IV were conducted during 1955 to 1996 and summarized by Brower [3]. The estimation of flowfield characteristics around such a heat shield configuration is of great aerodynamic importance, as well as research interest. For the ascent flight, during the transonic speed range, their study is particular important because of such resulting phenomena as terminal shock wave movements, frequently coupled with substantial freestream dynamic pressure. Flow induced vibrations are important design requirement of aerospace launch vehicles. Awrejcewicz, and Krysko [4] have developed numerical simulation of a cylindrical panel within transonic idealgas flow stream and solved dynamics for all intervals of the frequency. These parameters directly depend on the intensity of the vorticity components of the turbulence, the strength of the shock wave, and the mechanism of their interaction, all of which are implicitly linked to the specific configuration of a bulbous heat shield of a satellite launch vehicle. The numerical flow simulation over a bulbous payload shroud at transonic Mach number range is very useful to decide the geometrical configuration for minimum buffeting load and minimum aerodynamic drag requirement. The terminal shock wave of sufficient strength to interact with the boundary layer can cause flow separation and flowfield may become unstable as observed in the high speed cinematography [5]. Therefore, it is desirable to determine the location of the terminal shock wave on the heat shield and the strength of the terminal shock wave as a function of transonic Mach numbers range. The strength of the terminal shock wave and the mechanism of their interaction are related to the specific configuration of the heat shield satellite launch vehicle. Fluctuations of pressure level in shock waves and in separated flow regions can cause flow instabilities and then leads to buffeting phenomenon [6] – [7]. The features of the transonic flowfield can be delineated through the windtunnel data such as schlieren photographs and oil flow patterns. It is characterized by a normal or a terminal shock wave, supersonic pocket on the cylindrical region of the heat shield, shock wave/turbulent boundary layer interaction, and a separation bubble may be caused by the shock wave/turbulent boundary layer interaction on the cylindrical section of the heat shield. The main features of transonic flowfield around a bulbous heat shield are illustrated in Fig. 1. The terminal shock waves are an essential feature of transonic flow [8]. As the freestream Mach number increases from subsonic values a shock wave system appears near the shoulder. The flow is called transonic if both subsonic (M < 1) and supersonic (M > 1) regions are present in the field.
The transonic range begins when the highest Mach number reaches unity (M = 1) on the heat shield. The general features of the flow are as present once the sonic velocity occurs at the shoulder and remains throughout the whole transonic range. There is a local supersonic region ahead of the main shoulder shock. The near normal shock wave grows and moves downstream as the freestream Mach number increases. The difficulties to analyze the flowfield are associated with the detail design and a quantitative theoretical prediction. For the former, a sufficient windtunnel test data is required; the latter is due to nonlinearity of the equation governing transonic flow requires Computational Fluid Dynamics approach. In the boattail region, a local separation results, due to sharp discontinuity in the longitudinal of the payload shroud. The regions of flow separation impose additional complexity to aerodynamic and structural design aspects [9] – [13]. The complex flowfield at the transonic speeds is also observed during the experimental investigation of the bulbous heat shield. Experimental studies [14] – [15] have been made to understand flow behavior at transonic Mach numbers. These experimental investigations were limited to the measurement of surface pressure distribution, oil flow patterns, shadowgraphs and schlieren pictures for various heat shield models at transonic Mach number range. Recently analyses of Ares launch vehicle are carried out in the transonic speed and reported in a series of papers by Pinier [16], Piatak et al. [17] and Sekula et al. [18]. The current work reveals the paramount importance of aerodynamics at transonic Mach range.
A highspeed flow over a blunt body generates a bow shock wave in front of it, which causes a rather high surface pressure, and as a result, high aerodynamic drag. The surface pressure on the blunt body can be reduced if a conical shock wave is generated by mounting a forward facing spike. The aerospike produces a region of recirculating separated flow that shields the bluntnosed body from the incoming flow as shown in Fig. 2. The literature review reveals that addresses the mechanism how the unstable flow is initiated and how it persists [19] – [20]. The combination of the numerical simulations with experimental investigations has found to be a powerful tool to analyze unsteady flow and first results of a renewed investigation of the aerospike problem. The aerospike has been known since the 1950’s to cause an unstable flow [21]. Wood [22] has distinguished five different types of flow regimes over spiked cones based on the semicone angle and flow characteristics which may correspond to the fluctuation and oscillation regimes.
Bogdonoff and Vas [23] were the first to identify the two modes of unstable axisymmetric separation, the fluctuation mode and the oscillation mode by varying flat faced and hemispherical blunt bodies. The flowfield problem associated with the bluntnosed spike bodies can be distinguished by a conical blunt body with a total angles of the conical faces varied from 30^{0} to 180^{0} [22] or a hemispherecylinder body [24]. Kabelitz [25] has observed two distinct unsteady flow modes, namely, oscillation and pulsation [26] in the spike attached to the bluntnosed (flatfaced) cylindrical body. Experimental studies have been focused on identifying the boundaries of the unsteady region. The flow just outside the separated shear layer approaching the body’s shoulder can be turned by an attached conical shock, and then the shock structure is stable because an equilibrium condition is reached between escaping and recirculating flows in the separated region. Kistler [27] was the first to make detailed fluctuating wall pressure measurements under the separated supersonic turbulent boundary layer upstream of a forward step.
For a range of spike lengths the flow can became unsteady with two modes of instability observed. The oscillation mode involves the motion of the foreshock due to the spike tip. The pulsation mode features a largescale motion of the bow shock associates with blunt body. Spike length to diameter (
Flowfield over a conical spike attached to a blunt body is analyzed to understand the periodic oscillations of flowfield. The laminar NavierStokes equations are solved using multistage RungeKutta time stepping method. If the turning angle of the flow is too large to be accomplished by an attached conical shock wave, a detached strong shock is generated, which pushes highpressure flow from the reattachment zone at the body’s face into the recirculating region of the separated shear layer. This highpressure flow that gets into the separated flow region inflates the separation bubble, and the shock structure is pumped upstream. This gives rise to selfexcited shock oscillations during which the conical foreshock wave and the accompanying shear layer oscillate laterally and their shape changes periodically from concave to convex. This type of flowfield is unsteady in nature. The separated shear layer with an inflection point in the velocity profile is inherently unstable [21], and when this hits the body at the reattachment point selective amplification of the disturbances takes place, and this would cause the surface pressure to fluctuate in the flow separation region. The point of reattachment could be shifting to and fro along the body surface because of these shock oscillations. Because of this unsteady oscillation of the separation bubble, pronounced variations in the locations of separation shock, conical shock wave ahead of the blunt cone, and the reattachment point on the blunt cone surface are observed in different models with identical freestream conditions. Panaras et al. [30] have numerically simulated unsteady flows at high speeds around spikedblunt bodies. The experimental studies are also carried out to know the effect of variations to the spike diameter to blunt body diameter ratio.
The main aim of the present Chapter to analyze the unsteady flow characteristics and wall pressure fluctuations and oscillations over the hemispherecylinder, the bulbous payload shroud of a typical satellite launch vehicle and the conical spike attached to the forward facing blunt body. The numerical simulations present glimpse of the instantaneous flowfield features over various models at high speeds.
2. Governing fluid dynamics equations
The NavierStokes equations describe the motion of a viscous, heat conducting compressible fluid. In the Cartesian tensor notation, let
where
Usually
where strain rate tensor
The heat flux component is
where
in which γ is the ratio of specific heats.
Turbulent flows can be simulated by the Reynolds equations, in which statistical average are taken of rapidly fluctuating Reynolds stress terms which cannot be determined from the mean values of the velocity and density. Represent
where the integration interval
The statistical theory needs the statistical properties of the fluctuations, such as frequency correlation. Estimates of the Reynolds stress terms must be provided by a turbulence model. The simplest turbulence models augment the molecular viscosity by an eddy viscosity,
3. Axisymmetric fluid dynamics equations
The one of the serious problems in transonic regime of the flight of a bulbous payload shroud is wall pressure fluctuations caused by shock waveturbulent boundary layer interaction. A terminal shock wave of sufficient strength interacting with a boundary layer may cause flow separation and boundary layer may become unstable. The strength of the terminal shock and the mechanism of its interaction with the boundary layer are linked to a specific configuration of heat shield of a satellite launch vehicle. The shock wave turbulent boundary layer interaction unsteadiness may produce large amplitude fluctuations of the loads acting on the heat shield. The frequency band of the acoustic loads is typically in the range of several hundred Hz to several kHz. The experimental results obtained from the windtunnel at zero angle of incidence depict that the flow pattern remains the identical with reference to the windtunnel configuration even when the model is rotated. The measurements are made at two diametrically opposite locations indicate that the flow is axisymmetric. Therefore, a numerical solution of the timedependent, compressible, turbulent, axisymmetric Reynoldsaveraged NavierStokes equation is attempted to analyze the flow at transonic speeds over the hemispherecylinder and the bulbous heat shield of a typical launch vehicle. Now, Equation (1) can be written as
where
Reynolds stresses and turbulent heat fluxes in the mean flow equations are modeled by introducing an isotropic eddy viscosity,
Temperature is related to pressure and density by the perfect gas Eq. (7). The coefficient of molecular viscosity is calculated according to Sutherland’s law. The value of the turbulent Prandtl number Pr_{t} is assumed to take a constant value of 0.90. The closure of the system of equations is achieved by introducing following the algebraic turbulence model of the BaldwinLomax [31]
where
in the outer region
The coefficient of
l_{max}F_{max}
0.25l_{max} [(u^{2}+v^{2})]^{0.5}/F_{max}
The scale length
The effective viscosity is given by
This algebraic model, which utilizes the vorticity distribution to determine the scale lengths, has been extensively used in conjunction with the Reynoldsaveraged NavierStokes equations [13, 32, 33].
4. Numerical scheme
4.1. Spatial discretization
To facilitate the spatial discretization in the numerical scheme, Equation (10) can be written in the integral form over a finite volume as
where Ω is the computational domain. Γ is the boundary domain. The contour integration around the boundary of the cell is taken in the anticlockwise sense.
Figure 3 depicts a typical stencil of the computing cell which has four edges
Where
4.2. Artificial dissipation
In cellcentered spatial discretization schemes, such as above which is nondissipative, therefore, artificial are added to Eq. (17). The approach of Jameson et al. [36] is adopted to construct the dissipative function
where
with
The adaptive coefficients
are switched on or off by use of the shock wave sensor ν, with
where
The spatial discretization can be summarized here which is employed in numerical simulations. The convective terms are nonlinear, hyperbolic and grid dependent. A structured nonoverlapping quadrilateral cell is used in the numerical simulations. The diffusive terms are quasilinear, elliptic, grid independent, cell centered use of dual control volume for evaluating the gradients at a given location. Thus, the discretized solution to the governing equations results in a set of volumeaveraged state variables of mass, momentum, and energy which are balance with their areaaveraged fluxes (inviscid and viscous) across the cell faces [34]. The finite volume code constructed in this manner reduces to a central difference scheme and is secondorder accurate provided that the mesh is smooth enough [34]. The cellcentered spatial discretization scheme is nondissipative; therefore, artificial dissipation terms are included as a blend of a Laplacian and biharmonic operator in a manner analogous to the second and fourth difference. The artificial dissipation term [36] was added explicitly to prevent numerical oscillations near the shock waves to damp highfrequency modes.
5. Multistage timestepping scheme
The spatial discretiztion described above reduces the governing flow equations to semidiscrete ordinary differential equations. The integration is performed employing an efficient multistage scheme [36]. The following threestage, timestepping scheme is used for the numerical simulation (for clarity, the subscripts
where
where
5.1. Initial and boundary conditions
Conditions corresponding to a freestream Mach number were given as initial conditions. On the surface, no slip condition is considered together with an adiabatic wall condition. The symmetric conditions were applied on the centerline. For the transonic flow simulations, nonreflecting far field boundary conditions are applied at the outer boundary of the computational cell. For supersonic flow, all the flow variables are extrapolated at the outflow from the vector of conservative vector,
5.2. Geometrical details of the Model
The dimension of the hemispherecylinder model is taken as
The maximum payload shroud diameter
The dimensions of the spikedblunt body are depicted in Fig. 5. The model is axisymmetric, the main body has a hemisphericalcylinder nose, and the diameter D is 7.62×10^{2} m. The aerospike consists of a conical part and a cylindrical part. The angle of the spike’s cone is 10^{0} and the diameter of the spike is 0.1D. The aerospike model has a simple stick configuration. The L/D ratios of the spike are 0.5, 1.0 and 2.0.
5.3. Computational grid
One of the controlling factors for the numerical simulation is the proper grid arrangement. The following procedure is adopted to generate grid in the computational domain of the model. The computational region is divided into a number of nonoverlapping zones. The mesh points are generated in each zone using finite element methods [38] in conjunction with the homotopy scheme [39]. The above models are defined by a number of mesh points in the cylindrical coordinate system. Using these surface points as the reference node, the normal coordinate is then described by the exponentially stretched grid points extending towards up to an outer computational boundary.
where subscripts
The outer boundary of the computational domain is varied from 5 to 8 times the cylinder diameter,
6. Results and discussion
6.1. Hemispherecylinder 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 hemispherecylinder at
Once the initial phase of the computation is over 16 axial location (
6.2. Spectral analysis
A spectral analysis is carried out on the computed pressuretime 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 hemispherecylinder model. The pressure values have been converted from Pascal to decibel (dB) of surface pressure levels.
The surface pressure levels are computed in terms of the pressure reference at
The function is nonperiodic, a period
where
where
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
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 vibrations 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 timedependent 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 boattail region, a local flow separation occurs, due to sharp discontinuity in the longitudinal curvature. The flow reattachment length (
Once the initial phase of the computation was completed, some unsteadiness in the flow characteristics was observed. The sixteen locations (
A set of histograms of
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
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:
The total time period
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 spikedblunt body
The flow is assumed to be laminar for the spikedblunt 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
Figure 17 shows the enlarged view of the computed density contour and velocity plots for semicone angle of spike
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
6.6. Flow characteristics for the spikedblunt body
Figure 19 show the enlarged view of the density contours and velocity vector plots for spike lengths of
Flowfield was analyzed for
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
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 [
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
Figure 24 represents the pressure amplitude versus frequency and phase plots for L/D = 0.5 and M_{∞} = 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.
6.7. Selfexcited oscillation for the spikedblunt body
The fluid dynamics of the selfsustained oscillatory flow is analyzed using springmass analogy as well as the nonlinear oscillatory model. The selfexcited 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 selfsustained oscillation of Van der Pol equation [51]. Figure 25 shows [
7. Conclusions
The numerical simulations are carried out over the hemispherecylinder model and the bulbous heat shield of a satellite launch vehicle at transonic Mach number. The timedependent compressible axisymmetric NavierStokes equations are solved employing multistage RungeKutta timestepping scheme with BaldwinLomax turbulence model. The pressure fluctuations are computed at different location on the model. The unsteady flowfield characteristics are analyzed using fast Fourier transform. The numerical analysis also includes the numerical flow visualization and comparison with the available experimental data.
The key concern of research into the area of protruding spikes ahead of blunt bodies is the unstable flow that has been observed to exist for particular families of blunt bodies in supersonic and hypersonic flow. The length of the spike had an impact on the frequency and mode of oscillations. In this study the spike length was the principle parameter of variation on both flat faced and hemispherical blunt bodies. To this point, the focus of numerical simulations had remained on the effects that variation to the spike length and blunt body profile had on the resulting flow. Research into spike tipped blunt bodies has typically focused on variations to two main design variables; the length of the spike relative to the diameter of the blunt body, and the geometric shape of the blunt body itself. This research has drawn conclusions about the different flow regimes and the relative spike lengths that this is observed to occur for specific flow conditions. It is the objective of this project to contribute to this understanding by analyzing the effect that variations to the spike diameter relative to the blunt body diameter have on the characteristics of the flow. The numerical analysis is extended to simulate the flow over spikedblunt body under laminar condition. The numerical simulations captured pressure oscillations in the separation region. A limit cycle is obtained that describes the selfsustained oscillation of Van der Pol equation.
Nomenclature























α = semicone angle 















' = turbulent fluctuation 
Acknowledgments
The author is indebted to his parents and Vikram Sarabhai Space Centre, Trivandrum, India for their valuable encouragement, support and contributions to build the research career.
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