**2. Governing fluid dynamics equations**

can be distinguished by a conical blunt body with a total angles of the conical faces varied from

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 fore-shock due to the spike tip. The pulsation mode features a large-scale motion of the bow shock associates with blunt body.

for oscillation modes and *L/D* ratio of *1.0*, half-cone angle of blunt body *90*<sup>0</sup>

Feszty et al. [29] have conducted a computational analysis of the pulsation mode using

Flowfield over a conical spike attached to a blunt body is analyzed to understand the periodic oscillations of flowfield. The laminar Navier-Stokes equations are solved using multi-stage Runge-Kutta time stepping method. If the turning angle of the flow is too large to be accom‐

for the pulsation modes are numerically investigated by Badcock et al. [28].

*, M*<sup>∞</sup> *= 2.21, Re*<sup>D</sup> *=*

*, M*<sup>∞</sup> *= 6,*

Spike length to diameter (*L/D*) ratio of 0.9, half-cone angle of blunt body *70*<sup>0</sup>

 to 1800 [22] or a hemisphere-cylinder body [24]. Kabelitz [25] has observed two distinct unsteady flow modes, namely, oscillation and pulsation [26] in the spike attached to the bluntnosed (flat-faced) 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

300

forward step.

86 Computational and Numerical Simulations

*0.12 × 10*<sup>6</sup>

*Re*<sup>D</sup> *= 0.13 × 10*<sup>6</sup>

computational fluid dynamics.

**Figure 2.** Schematic flowfield over spiked-blunt body.

The Navier-Stokes equations describe the motion of a viscous, heat conducting compressible fluid. In the Cartesian tensor notation, let *x*<sup>j</sup> , be the coordinates, *p*, *ρ*, *T* and *E* the pressure, density, temperature, and total energy, and *u*, the velocity components. The governing fluid dynamics equations can be written as

$$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_j} \left(\rho u\_j\right) = 0 \tag{1}$$

$$\frac{\partial}{\partial t} \left( \rho u\_i \right) + \frac{\partial}{\partial \mathfrak{x}\_j} \left( \rho u\_i u\_j + \mathcal{S}\_{ij} p - \mathfrak{x}\_{i\natural} \right) = 0 \tag{2}$$

$$\frac{\partial}{\partial t}(\rho E) + \frac{\partial}{\partial \mathbf{x}\_{\dot{j}}} \Big[ (\rho E + p)\boldsymbol{u}\_{\dot{j}} - \boldsymbol{u}\_{i}\boldsymbol{\tau}\_{\dot{i}\dot{j}} + \boldsymbol{q}\_{\dot{j}} \Big] = 0 \tag{3}$$

where *τij* is the stress tensor, which is proportional to the rate of strain tensor and the bulk dilatation

$$
\Delta \pi\_{ij} = \mu \left( \frac{\partial u\_i}{\partial \mathbf{x}\_j} + \frac{\partial u\_j}{\partial \mathbf{x}\_i} \right) + \lambda \mathcal{S}\_{ij} \left( \frac{\partial u\_k}{\partial \mathbf{x}\_k} \right) \tag{4}
$$

fluctuations are said to be statistically stationary. The root-mean-square value of *u*'

' '2

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, *μ*<sup>t</sup>

approximately represents the effects of turbulent mixing, and is estimated with some charac‐ teristic length scale such as boundary layer thickness. Baldwin-Lomax [31] proposed algebraic or zero-equation turbulence for the outer law, eliminating boundary layer thickness and momentum thickness, in favor of a certain maximum function occurring in the boundary layer. A typical transonic flow patter over a bulbous heat shield of satellite launch vehicle is illus‐ trated in Fig. 1. The effects of compressibility start to cause a radical change in the flow. This occurs when embedded pocket of supersonic flow appear, generally in the terminal shock

The one of the serious problems in transonic regime of the flight of a bulbous payload shroud is wall pressure fluctuations caused by shock wave-turbulent 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 wind-tunnel at zero angle of incidence depict that the flow pattern remains the identical with reference to the wind-tunnel configu‐ ration 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 time-dependent, compressible, turbulent, axisymmetric Reynolds-averaged Navier-Stokes equation is attempted to analyze the flow at transonic speeds over the hemisphere-cylinder and the bulbous heat shield of a typical launch vehicle. Now, Equation (1) can be written as

> U F1 H ( <sup>G</sup>) <sup>0</sup> *r t xr r r*

+ + +=

¶¶ ¶ (10)

¶ ¶ ¶

*rms u u* <sup>=</sup> (9)

Unsteady Flowfield Characteristics Over Blunt Bodies at High Speed

as

wave.

where

**3. Axisymmetric fluid dynamics equations**

is defined

89

http://dx.doi.org/10.5772/57050

that

Usually *λ = -2μ/3*, and velocity gradient tensor can be represented as

$$
\left(\frac{\partial \boldsymbol{u}\_{i}}{\partial \boldsymbol{\omega}\_{j}}\right) = \left[\boldsymbol{\mathcal{S}}\_{ij} + \boldsymbol{\Omega}\_{ij}\right] \tag{5}
$$

where strain rate tensor *S*ij and rotation rate tensor *Ω*ij can be written as

$$S\_{ij} = \frac{1}{2} \left[ \left( \frac{\partial \, u\_i}{\partial \, \boldsymbol{x}\_j} + \frac{\partial \, u\_j}{\partial \, \boldsymbol{x}\_i} \right) \right], \\ \Omega\_{ij} = \frac{1}{2} \left[ \left( \frac{\partial \, u\_i}{\partial \, \boldsymbol{x}\_j} - \frac{\partial \, u\_j}{\partial \, \boldsymbol{x}\_i} \right) \right]$$

The heat flux component is

$$q\_{j} = -k \frac{\partial T}{\partial \mathbf{x}\_{j}} \tag{6}$$

where *k* is the coefficient of heat conduction. The pressure is related to the density and energy by equation of state

$$p = \left(\gamma - 1\right)\rho \left(E - \frac{1}{2}\mu\_i\mu\_i\right) \tag{7}$$

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 *u(t)* at a particular location *(x, y, z)*. Then the time average and its mean square of *u* is defined as

$$
\overline{\mu} = \frac{1}{T} \int\_{t\_0}^{t\_0 + T} u dt, \quad \overline{u^2} = \frac{1}{T} \int\_{t\_0}^{t\_0 + T} u'^2 dt} u'^2 dt \tag{8}
$$

where the integration interval *T* is chosen to be large than any significant period of the fluctuation, *u*' . The integrals in the above equations are independent of starting time *t*0. The fluctuations are said to be statistically stationary. The root-mean-square value of *u*' is defined as

$$
\mu\_{rms}^{\cdot} = \sqrt{\mu^2} \tag{9}
$$

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, *μ*<sup>t</sup> that approximately represents the effects of turbulent mixing, and is estimated with some charac‐ teristic length scale such as boundary layer thickness. Baldwin-Lomax [31] proposed algebraic or zero-equation turbulence for the outer law, eliminating boundary layer thickness and momentum thickness, in favor of a certain maximum function occurring in the boundary layer. A typical transonic flow patter over a bulbous heat shield of satellite launch vehicle is illus‐ trated in Fig. 1. The effects of compressibility start to cause a radical change in the flow. This occurs when embedded pocket of supersonic flow appear, generally in the terminal shock wave.
