**2. Physical model, simulation methodology and validation of computational fluid dynamics code**

## **2.1 Physical model**

train, are maintained the position of the Mach wave train. The compressed air is adjusted in isolator to match with the condition that can enter into the combustor. When the pressure introduced in the reverse direction of the combustor zone obtained the variation in pressure from the isolator and combustor. The difference between isolator and combustor zone is adjusted by changing the shock wavelength [3]. While designing the isolator, need serious concern about the unstart phenomenon. The isolator may lead to severe effect due to high speed in the flight. The length of the isolator part in the scramjet engine maintains at a certain weight. The required shear and shock waves provided to avoid the communication of the instabilities that will arise and affect the inlet [4]. The system developed using the hypersonic inlet isolator under Mach 4 and Mach 5 flight conditions [5]. In [6], reported that the decrease in the pressure at the inlet of the domain is observed with an increase in the isolator length. The shock train in a fixed 2-D scramjet inlet with isolator showed some results by increasing wall and decreasing the total temperature [7]. In Mach 5 inlet-isolator model, the shock train jumping moments captured by separating flow at the head of the shock train and the contraction ratio of the local throat-like shape [8]. The scramjet isolator decreases the static pressure, and it becomes sharper. The experiment conducted on a constant-area scramjet isolator and observed that was relatively stable with time-resolved and low-frequency pressure [9]. In [10], the numerical simulation influences the movement of free stream characteristics leading to separation with an increase in adverse pressure. The dynamic model of the shock train is predicted on the shock wave layer. The dynamic model cannot suppress the pressure gradient as high as the other sustains [11]. In [12], the complex compression and expansion waves exist in the isolator, causing large stream-wise and transverse gradients upstream of the shock train. The adverse gradient pressure in stream-wise decreases with the duct curvature [13]. In [14], the experiments compared with the conventional approaches using boundary layer interaction large-eddy simulation of a hypersonic of Mach 8 flight vehicle. In [15, 16], they have conducted experiments on the multiple shock wave/turbulent boundary layer interactions in a rectangular duct using Mach numbers 2.45 and 1.6. Carroll et al. observed that the length of the communications and the tendency towards a repeated oblique was scaled directly with the level of confinement. The study of unstart and unstarted flows in an inlet/isolator model strongly associated with boundary-layer separation [17]. The numerical solutions of the Naiver-stokes equations for the interactions of a shock wave and turbulent boundary layer varying from 7.93 to 12.17, at a free-stream Mach number of 2.96 and Reynolds number 1.2 107. The free-stream predicts accurate results. When shock strength and overall rise pressure for the low viscosity pressure asymptotes. The large-eddy 3-D analysis in the area of uniform cross section with low aspect ratio rectangular duct

*Numerical and Experimental Studies on Combustion Engines and Vehicles*

In the open literature, by varying the adverse pressure gradient at the exit of the isolator the motion path and characteristics of the shock wave train are obtained. In this work, we intend to study the impact of combustion phenomena on shock wave train. Therefore, the significance of the angle of attack on the wall surface of the domain and the effect of the adverse pressure gradient on the movement of the shock wave train are analyzed. The present analysis focused on different divergent angles, i.e., 0, 0.5, 1, and 1.5° with constant pressure gradient of 90 kPa, and also with different negative pressures of 80 and 100 kPa with constant cross-sectional area of the isolator is discussed. All these effects are studied on similar computational domain with similar solver type parameters. The rest of the paper is as organized as follows, Physical model and simulation methodology is discussed in Section 2, the effects of back pressure and angle of attack are discussed in Section 3.

geometry is studied [18–22].

**4**

In this work, analysis has been carried out on the scramjet isolator of uniform cross section to analyze the movement of shock wave train to study the relation between shock train and the interaction of the boundary layer formation. As shown in **Figure 1** to study the impact of adverse pressure and the significance of divergent angles on the behavior of shock wave train are studied using ANSYS Fluent 16 [23]. The atmospheric air is injected at the entrance of the isolator with 220 mm length and 32 mm height. The hydrogen fuel is injected transversely from the either sides of the wall at a distance of 232.8 mm of the inlet of the computational domain.

#### **2.2 Simulation methodology**

The commercial software ANSYS Fluent 16 [23] was used to simplify twodimensional compressible fluid flow by considering the density-based solver with standard K-ε turbulence model, Reynolds-averaged Navier Stokes equation with finite volume method was considered. The species transport model with single step volumetric reaction mechanism is considered to simplify the combustion model (finite rate/eddy dissipation model) [24–27]. To maintain the proper mixing and optimizing the combustion phenomena in supersonic flow RANS approach is the most effective and faster method. The standard K-ε turbulence model is chosen due to its ability of simplifying the negative pressure gradient in the case of transverse injection flow field.

The appropriate governing Eqs. (1)–(5) describing the continuity equation, Navier Stokes equation and combustion model for fluid flow is written as [21, 22, 28, 29].

Continuity equation:

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

Conservation of momentum (Navier–Stokes equation)

$$\frac{\partial(\rho u\_i)}{\partial t} + \frac{\partial(\rho u\_i u\_j)}{\partial \mathbf{x}\_j} = -\frac{\partial \rho}{\partial \mathbf{x}\_j} + \frac{\partial}{\partial \mathbf{x}\_j} \left[ \mu\_{\epsilon \| \mathbf{f}} \left( \frac{\partial u\_i}{\partial \mathbf{x}\_j} + \frac{\partial u\_j}{\partial \mathbf{x}\_i} \right) \right] + \mathbf{S}\_{ui} \tag{2}$$

**Figure 1.** *Schematic diagram of the scramjet combustor.*

where the source term Sui includes Coriolis and centrifugal forces

$$\mathsf{S}\_{\mathsf{u}\mathsf{i}} = -\mathsf{2}\mathsf{Q} \times U - \mathsf{Q} \times (\mathsf{Q} \times r)$$

Conservation of energy equation:

$$\frac{\partial(\rho H)}{\partial t} - \frac{\partial \rho}{\partial t} + \frac{\partial(\rho u\_i H)}{\partial \mathbf{x}\_j} = \frac{\partial}{\partial \mathbf{x}\_j} \left( k \frac{\partial T}{\partial \mathbf{x}\_j} + \frac{\mu\_i}{\mathbf{Pr}\_i} \frac{\partial h}{\partial \mathbf{x}\_j} \right) + \mathbf{S}\_E \tag{3}$$

Turbulence transport equations:

K-ε turbulence model and turbulence eddy dissipation equation

$$\frac{\partial(\rho\mathbb{k})}{\partial t} + \frac{\partial(\rho k u\_j)}{\partial \mathbb{x}\_j} = \frac{\partial}{\partial \mathbb{x}\_j} \left( \left( \mu + \frac{\mu\_t}{\sigma\_{k3}} \right) \frac{\partial \mathbb{k}}{\partial \mathbb{x}\_j} \right) + \tau\_{ij} \frac{\partial u\_i}{\partial \mathbb{x}\_j} - \beta^\* \rho k w \tag{4}$$

$$\frac{\partial \rho \varepsilon}{\partial t} + \frac{\partial \left(\rho u\_j \varepsilon\right)}{\partial \mathbf{x}\_j} = \frac{\partial}{\partial \mathbf{x}\_j} \left(\Gamma\_k \frac{\partial \varepsilon}{\partial \mathbf{x}\_j}\right) + \frac{\varepsilon}{k} \left(\mathbf{C}\_{\varepsilon 1} \mathbf{P}\_k - \rho \mathbf{C}\_{\varepsilon 2} \varepsilon\right) \tag{5}$$

**2.5 Validation of numerical methods and grid independency**

*Wall pressure distribution at mid-plane and bottom wall of the domain.*

*DOI: http://dx.doi.org/10.5772/intechopen.92555*

and strength of the shock wave system [25].

*Mach number and pressure variation along the mid-plane.*

**3. Results and discussions**

**Figure 2.**

area of the isolator is discussed.

**Figure 3.**

**7**

To validate the accuracy of numerical results it is required to compare with the experimental results in order to validate the reliability of computational tool. Computational results predominantly depend on the quality of the mesh and size; therefore, it is also necessary to find out the ideal grid size. Initially, computational analysis is carried out to validate the commercial code and simultaneously find out the ideal grid size by considering 634,846 (CFD1, fine mesh) and 384,592 (CFD2, coarse mesh) [25]. The obtained simulated results (**Figure 2**) are then validated by with the experimental data [25] available in open literature and found to be in good qualitative agreement. It is observed that simulation which own fine grid and coarse grid has a good accuracy that the relative error is below 5%. Therefore, the CFD tool can be applied to capture the shock wave reasonably well in terms of both location

*Numerical Investigation of the Shock Train in a Scramjet with the Effects of Back-Pressure…*

In the present study, the significance of the angle of attack on the wall surface of the domain and the effect of the adverse pressure gradient on the movement of the shock wave train are analyzed. The present analysis focused on different divergent angles, i.e., 0, 0.5, 1 and 1.5° with a standard pressure gradient of 90 kPa, and also with different negative pressures of 80 and 100 kPa with constant cross-sectional

#### **2.3 Combustion modeling**

To simulate the combustion flow dynamics more attention is required as rapid turbulence creation and chemical reaction is required. The species transport model with single step volumetric reaction mechanism is considered to simplify the combustion model (finite rate/eddy dissipation model) which is mainly used in the present research work. The global one step chemical reaction of hydrogen combustion has been considered in this paper for its capability of predicting the overall performance parameters with considerably less computational cost for the scramjet combustor. In this global one step reaction mechanism the rate constants like preexponential factor (A) and activation temperature are considered as 9.87 � <sup>10</sup><sup>8</sup> . The one step volumetric reaction mechanism is defined as shown in the Eq. (6):

$$\text{H}\_2\text{H}\_2 + \text{O}\_2 = \text{H}\_2\text{O} \tag{6}$$

#### **2.4 Boundary condition**

An atmospheric air is injected at a velocity of 1200 m/s with a hydraulic diameter of 0.05 m and turbulence intensity of 5%. Hydrogen fuel is injected transversely through the either sides of the walls at x = 0.220 m at a velocity of 900 m/s with a turbulence intensity of 5%. No slip condition and constant heat flux is chosen along the solid surface with standard wall function. Interior combustor zone was chosen for the fluid domain (**Table 1**).


**Table 1.** *Inlet conditions for hydrogen and air jet.* *Numerical Investigation of the Shock Train in a Scramjet with the Effects of Back-Pressure… DOI: http://dx.doi.org/10.5772/intechopen.92555*

**Figure 2.**

where the source term Sui includes Coriolis and centrifugal forces

*Numerical and Experimental Studies on Combustion Engines and Vehicles*

*<sup>∂</sup>*ð Þ *<sup>ρ</sup>uiH ∂x <sup>j</sup>*

K-ε turbulence model and turbulence eddy dissipation equation

¼ *∂ ∂x <sup>j</sup>*

> ¼ *∂ ∂x <sup>j</sup>*

Conservation of energy equation:

*<sup>∂</sup>*ð Þ *<sup>ρ</sup><sup>H</sup> <sup>∂</sup><sup>t</sup>* � *<sup>∂</sup><sup>ρ</sup> ∂t* þ

Turbulence transport equations:

*∂ρε ∂t* þ

*∂ ρku <sup>j</sup> ∂x <sup>j</sup>*

> *∂ ρu <sup>j</sup>ε ∂x <sup>j</sup>*

*<sup>∂</sup>*ð Þ *<sup>ρ</sup><sup>k</sup> ∂t* þ

**2.3 Combustion modeling**

**2.4 Boundary condition**

for the fluid domain (**Table 1**).

*Inlet conditions for hydrogen and air jet.*

**Table 1.**

**6**

*Sui* ¼ �2Ω � *U* � Ω � ð Þ Ω � *r*

¼ *∂ ∂x <sup>j</sup>*

*<sup>μ</sup>* <sup>þ</sup> *<sup>μ</sup><sup>t</sup> σk*<sup>3</sup> *∂k*

To simulate the combustion flow dynamics more attention is required as rapid turbulence creation and chemical reaction is required. The species transport model with single step volumetric reaction mechanism is considered to simplify the combustion model (finite rate/eddy dissipation model) which is mainly used in the present research work. The global one step chemical reaction of hydrogen combustion has been considered in this paper for its capability of predicting the overall performance parameters with considerably less computational cost for the scramjet combustor. In this global one step reaction mechanism the rate constants like preexponential factor (A) and activation temperature are considered as 9.87 � <sup>10</sup><sup>8</sup>

one step volumetric reaction mechanism is defined as shown in the Eq. (6):

An atmospheric air is injected at a velocity of 1200 m/s with a hydraulic diameter of 0.05 m and turbulence intensity of 5%. Hydrogen fuel is injected transversely through the either sides of the walls at x = 0.220 m at a velocity of 900 m/s with a turbulence intensity of 5%. No slip condition and constant heat flux is chosen along the solid surface with standard wall function. Interior combustor zone was chosen

**Parameters Hydrogen Jet Free-stream jet** Mach number [M] 1.0 4.5 Temperature [K] 1000 1300 Pressure [Pa] 506,625 101,325 CH2 1.0 0 CO2 0 0.21 CH2O 0 0.032

Γ*k ∂ε ∂x <sup>j</sup>* 

*k ∂T ∂x <sup>j</sup>*

*∂x <sup>j</sup>*

þ *ε k* þ *μi* Pr*<sup>i</sup>*

> þ *τij ∂ui ∂x <sup>j</sup>*

2H2 þ O2 ¼ H2O (6)

*∂h ∂x <sup>j</sup>*

þ *SE* (3)

� *<sup>β</sup>* <sup>∗</sup> *<sup>ρ</sup>kw* (4)

. The

ð Þ *Cε*1*Pk* � *ρCε*2*ε* (5)

### **2.5 Validation of numerical methods and grid independency**

To validate the accuracy of numerical results it is required to compare with the experimental results in order to validate the reliability of computational tool. Computational results predominantly depend on the quality of the mesh and size; therefore, it is also necessary to find out the ideal grid size. Initially, computational analysis is carried out to validate the commercial code and simultaneously find out the ideal grid size by considering 634,846 (CFD1, fine mesh) and 384,592 (CFD2, coarse mesh) [25]. The obtained simulated results (**Figure 2**) are then validated by with the experimental data [25] available in open literature and found to be in good qualitative agreement. It is observed that simulation which own fine grid and coarse grid has a good accuracy that the relative error is below 5%. Therefore, the CFD tool can be applied to capture the shock wave reasonably well in terms of both location and strength of the shock wave system [25].

### **3. Results and discussions**

In the present study, the significance of the angle of attack on the wall surface of the domain and the effect of the adverse pressure gradient on the movement of the shock wave train are analyzed. The present analysis focused on different divergent angles, i.e., 0, 0.5, 1 and 1.5° with a standard pressure gradient of 90 kPa, and also with different negative pressures of 80 and 100 kPa with constant cross-sectional area of the isolator is discussed.

**Figure 3.** *Mach number and pressure variation along the mid-plane.*
