In this paper, analytical, computational, and experimental studies are integrated to examine unsteady acoustic/vorticity transport phenomena in a solid rocket motor chamber with end-wall disturbance and side-wall injection. Acoustic-fluid dynamic interactions across the chamber may generate intense unsteady vorticity with associated shear stresses. These stresses may cause scouring and, in turn, enhance the heat rate and erosional burning of solid propellant in a real rocket chamber. In this modelling, the unsteady propellant gasification is mimicked by steady-state flow disturbed by end-wall oscillations. The analytical approach is formulated using an asymptotic technique to reduce the full governing equations. The equations that arise from the analysis possess wave properties are solved in an initial-boundary value sense. The numerical study is performed by solving the parabolized Navier–Stokes equations for the DNS simulation and unsteady Reynolds-averaged Navier–Stokes equations along with the energy equation using the control volume approach based on a staggered grid system with the turbulence modelling. The v2-f turbulence model has been implemented. The results show that an unexpectedly large amplitude of unsteady vorticity is generated at the injection side-wall of the chamber and is then penetrated downstream by the bulk motion of the internal flow. These stresses may cause a scouring effect and large transient heat transfer on the combustion surface. A comparison between the analytical, computational, and experimental results is performed.

The designers of the Solid Rocket Motor (SRM) are facing several technological problems requiring expertise in diverse-sub-disciplines in order to understand the nature of the fluid flow dynamics along with the associated complex wave patterns inside the chamber/nozzle geometry of the SRM. These problems include the burning of the solid propellant and its solid mechanical properties, the chamber case, and the fluid dynamics inside the cavity, shock physics and mechanical properties of the Ammonium Perchlorate oxidizer (AP)/Hydroxyl Terminated Poly Butadiene fuel (HTPB). Moreover, these complexities are characterized by very high energy densities, extremely diverse length and time scales, complex interface, and reactive, turbulent, and multiphase flows.

In order to have a stable rocket engine with desirable performance, there are two different essential models for the reacting and non-reacting systems must be examined first. The first model is directed to study the combustion of real microscale propellant that accounts the Ammonium Perchlorate oxidizer (AP)/Hydroxyl Terminated Poly Butadiene fuel (HTPB), as in [

As a result, the current study is conducted to develop more accurate numerical techniques combined with the Navier–Stokes Characteristic Boundary Condition (NSCBC), that used to solve the parabolized, unsteady, compressible Navier–Stokes equations at the boundaries, to examine the unsteadiness of the fluid flow dynamics in the cavity of the SRM chamber and the associated complex wave patterns and their impact on the burning of real solid propellant surfaces. NSCBC is used to specify the numerically reflected boundary conditions, in order to eliminate numerically generated reflected waves and to minimize numerical dissipation near the aft end. We try in this study to assign constant velocity injection to mimic the real variation of composite propellants with end wall disturbance that generate longitudinal acoustic waves similar to the pattern that may exist in the rocket chamber. To describe the nature of a low Mach number and weakly viscous flow field generated in a long, narrow chamber with steady side-wall mass injection and end-wall disturbances, analytical, computational, and experimental studies were conducted. The inviscid rotational steady injected flow interacts with the irrotational acoustic field to generate unsteady vorticity at the side-wall and penetrates toward the centerline by the steady transverse velocity component. The axial distribution of the vertical velocity on the side-wall is prescribed.

The experimental and theoretical approaches in [

Wang et al. [

Furthermore, for the duct that ended with a convergent–divergent area, the wave amplitude near the end of the constant portion of the duct was greater than that of the convergent part only. Seymour et al. [

The interaction between the propagation of acoustic waves in closed or open-end tubes with mass injection from the side-wall has been studied by Staab et al. [

Moreover, several investigators developed a competitive linear model, as in Flandro [

Vuillot et al. [

Many investigations to chamber flow turbulence modelling, based on

Studies by Rempe et al. [

Therefore, in this study, careful attention has been paid to the employment of modern algorithms based on higher-order accuracy techniques to solve the parabolized, unsteady, compressible Navier–Stokes equations at the boundaries. A boundary condition treatment method, namely, Navier Stokes Characteristics Boundary Condition (NSCBC), is used to specify the numerically reflected boundary conditions, in order to eliminate numerically generated reflected waves and to minimize numerical dissipation near the aft end. This technique shows that low dispersion errors and in turn, significant improvement over the old methods, e.g., extrapolation and partial use of Riemann invariant. The current study is conducted to develop more accurate numerical techniques combined with NSCBC to examine the unsteady flow field dynamics inside the solid rocket motor chamber and the associated complex generated wave and vorticity patterns and their impact on the burning of real solid propellant surfaces. Here, the analytical solutions for the steady and unsteady flow regimes are derived using the asymptotic techniques.

Moreover, in the present study, the turbulence characteristics of the induced flow inside the chamber are studied numerically. The numerical model was specially developed for this study in FORTRAN environment using our programmed code. The developed model was validated using a wide range of computational fluid dynamics (CFD) problems in different aerodynamics applications with reacting and non-reacting flows [

The nondimensional conservation form of the unsteady, two dimensional, complete Navier–Stokes equations describing both fluid dynamics and acoustics in a perfect gas within a rectangular chamber can be written as follows:

where

With the aid of the equation of state for a perfect gas

The nondimensional properties can be written as

For

The boundary and initial conditions are written as follows:

The steady-state flow is generated by constant side-wall injection speed and is used as an initial condition for the unsteady flow computations. The analytical velocity profiles for incompressible, inviscid, rotational flow in a long narrow duct are used as starting profiles for the steady, compressible, viscous flow computations. The steady-state variables for the incompressible, inviscid, rotational flow can be described using the asymptotic expansions as

The asymptotic theory by Zhao et al. [

which are valid for the

The boundary conditions are

Isothermal wall with no-slip boundary conditions and side-wall injection are specified at the side-wall.

The solution to the first-order calculations in

where

To ensure that the solution converges with the steady state, the mass conservation is examined through

where

In this study, let the initial conditions for the unsteady flow variables

Inserting

with the following initial and boundary conditions:

Since the left boundary condition in

With the new initial and boundary conditions

Assume the homogeneous solution for

where _{n}

Then, the final non-resonance solution for the homogeneous and non-homogeneous wave equation that describes the acoustic velocity fields is derived from the summation of

The resonance solution when

Hegab [

In this study, the experimental measurement is constructed to record only the pressure acoustic fields at several locations along the chamber. For comparison considerations, the pressure equation can be derived from

Consequently, the final pressure equation for the resonance acoustic equation is derived as

Also, the thermal, acoustic field for the non-resonance and resonance solution is derived and written as follows:

For non-resonance solution

For resonance solution (^{*}),

Moreover, the perturbed density field is determined from

Multidimensional, rotational, transient flow-dynamics in a simulated SRM chamber model are obtained from the solutions to the mass, momentum, and energy conservation equations. Numerical simulations are used to identify the acoustic fluid dynamics interactions as well as to evaluate the vorticity distribution as a function of time and space. The Two-Four explicit scheme, combined with the Navier–Stokes Characteristic Boundary Condition (NSCBC) and higher-order accuracy technique at the symmetry line, is developed to obtain a solution to the unsteady, compressible Navier–Stokes equations in the SRM chamber model as in [

Using the NSCBC techniques, the fluid-dynamics equations can be written in nondimensional wave modes form as follows:

where

In this case, the amplitude of the reflected wave based on the prescribed pressure node at the exit boundary should be

and the remaining conditions are

The nondimensional equations can be written as follows:

The velocity component normal to the boundary (u (0, y, t)) is non-zero since the head end is the moving boundary. Thus,

The left-running wave

and the remaining amplitudes are

The wave-mode forms of the Navier–Stokes equations are simplified by ignoring the axial transportation terms in (

The nondimensional equations can be written in the form:

The nondimensional primitive variables (

More details about the NSCBC techniques are found in [

The accuracy of the current explicit numerical schemes has been justified by Courant–Friedrichs–Lewy (CFL) as in MacCormack [

The features of the flow field are essentially transient, weakly viscous and contain vorticity distribution that has a length scale small compared to the overall geometry. These flow characteristics have similar behaviour of the turbulent flow. In this sense, the objective of this section is utilized to examine the relationship between results of the traditional studies of turbulence in injection driven channel and those found in the DNS as in Section 2.

The governing conservation equations of the flow can be written for time-dependent, compressible, and turbulent flow in the dimensional form as follows:

Continuity Equation

Momentum Equation

Energy Equation

where _{i}_{v}

The momentum equations contain additional terms, known as Reynolds stresses, which represent the effects of turbulence. These Reynolds stresses,

The Reynolds-averaged conservation

In industrial CFD applications, RANS modelling remains one of the main approaches when dealing with turbulent flows. The Reynolds-averaged approach to turbulence modelling requires appropriate modelling of the Reynolds stresses in

where

For the calculation of the turbulent viscosity, the v2-f well known turbulence model is applied. The distinguishing feature of the v2-f model is its use of the velocity scale, v2, instead of the turbulent kinetic energy, k, for evaluating the eddy viscosity. The v2, which is the velocity fluctuation normal to the streamlines, provides the proper scaling in representing the damping of turbulent transport close to the wall.

This model requires the solution to three transport equations, where the transport equation of the turbulent kinetic energy and its dissipation rate can be written as follows [

where P

where S is defined as follows:

The turbulent viscosity is given by the following:

The constants of the model are given in as follows:

The importance of the turbulence studies with the aspect ratio of twenty, which is not large enough to enable a fully turbulent mean flow to develop in the chamber, suggests that the DNS may provide a better understanding for the impact of the transverse movement of the vorticity peak on the behaviour of the erosive burning.

The experimental setup layout is shown in

The piston oscillates at the head end of the tube. The acoustic pressure fields are recorded at several locations along the tube by capacitive pressure transducers

In this study, the asymptotic methodologies are combined with numerical and experimental approaches to give precise fundamental insights for the unsteady flow dynamics in a chamber with end-wall disturbances and steady side-wall injection. The complete axial flow speed may be divided into three parts, as by Staab et al. [

where, _{s}_{p}_{v}_{v}_{s}_{p}

The total dimensionless unsteady vorticity is

where the vorticity is non-dimensionalized as follows:

where primed quantities are dimensional. The transverse speed component _{v}_{s}

The parameters used to define the internal flow field include the characteristic flow Mach number (_{i} = M/

Once a converged steady-state solution is established at given values of ^{5}, and

The current results pertain to the pure planner irrotational acoustic waves and the rotational fluid flow as described in the following paragraphs.

The first category of the results deals with the acoustic fields in the chamber without injection from the side-wall using the experimental, computational, and analytical approaches. The analytical results are obtained to describe acoustic and viscous flow dynamics in an impermeable duct when an end-wall disturbance is used to derive axial waves. The results are used to show that resonance between forced and propagating acoustic wave modes may be a source of large acoustic instability as well as to assert how nonlinear effects alter the energy in acoustic eigenfunction. The experimental results are compared with computational results to validate the numerical technique.

The first set of the results shows a comparison between experimental and computational pressure traces at

The result illustrates the wave oscillations (quasi-steady wave motion) after many wave cycles. A stable or bounded solution is observed at this frequency. A reasonable comparison between the experimental and computational approaches is noticed. The deviations are directed to the linearity and complexity of the experimental test rig design. As a result, undesirable harmonics are seen with the experimental work due to some leaks it the connection between the circular portion of the piston and the duct shape.

In the current study, the computational and asymptotic approaches are extended to cover a wide range of frequencies away from the first fundamental mode (

Moreover, the pressure-time history at the mid-length of the chamber for

It is noted that the frequencies increase as time increases for both pressure and velocity fields when the forced frequency (

Coupling effects between the burning process and the flow characteristics inside the SRM chamber are associated with small-amplitude pressure oscillations, which can be described mathematically in terms of appropriate eigenfunctions. The co-existence of forced acoustic modes and propagating modes has a significant role in the energy exchange mechanism and acoustic instability inside the chamber of an SRM. One of the primary objectives of this research is to illustrate how acoustic wave patterns in a duct are generated by a simple time-dependent boundary condition at the head end in terms of an initial-boundary value formulation. The numerical solution to the full parabolized nonlinear, unsteady, compressible Navier–Stokes equations is integrated with the analytical solutions using the perturbation theory to the weakly nonlinear analysis of the acoustic waves to examine the complex acoustic/fluid dynamics mechanism and their impact on the combustion of real solid fuel. Moreover, both approaches are used to show that the resonance between forced and propagating acoustics wave modes may be the source of large acoustic instability.

Lower and higher-order calculations are formulated to find a weakly nonlinear solution within the core flow, which defines the eigenfunctions’ acoustic damping time. The lowest order acoustic equations, valid for the limit

Our previous study [

Moreover, the higher-order calculations revealed that the quadratic nonlinear convective terms could not be a source of intermodal energy transfer at the O(M2) approximation level. As a result, energy in eigenfunction modes cannot be damped by quadratic convection effects when “open box” eigenfunctions are the basis set for the Fourier series. At the same time, the O(M3) cubic nonlinearities are responsible for nonlinearization in the cavity of the SRM chamber. It follows that wave-wave interactions (non-harmonic behaviour of eigenfunction amplitudes) will occur over a time scale long compared to the acoustic time

Finally, we may conclude that

The second category of the results is devoted to examining the unsteadiness of the rotational fluid flow that arises from the interaction between the longitudinal axial pressure waves and the steady injection from the side-wall. The first case of the second set is implemented for the rotational velocity and vorticity when

The results show that the rotational fluid flow is generated at the side-wall due to the interaction process and is then convected vertically toward the centerlines as time increases. It is found that the amplitude of the vorticity wave oscillations near the surface is generally larger than that away from the wall for both the non-resonance (

The effect of the injected Mach number on the generation and evolution of the unsteady vorticity is presented in

It is noticed that the amplitudes of the axial rotational velocity and the generated unsteady vorticity increase as the injected Mach number increases. Moreover, the vorticity front for

The above derived front location in

The comparison between the analytical result for the vorticity front in

The vorticity topography for

Time | Analytical unsteady vorticity front |
Computational vorticity front | Deviation | Percentage (%) |
---|---|---|---|---|

1.0 | 1.0 | 0 | 0 | |

0.902 | 0.880 | 0.02 | 2.21 | |

0.835 | 0.805 | 0.03 | 3.59 | |

0.679 | 0.720 | −0.041 | −5.2 | |

0.60 | 0.621 | −0.021 | −3.33 |

Here again, it is noted that the vorticity is generated at the sidewall and then penetrated and fill the entire chamber as time increase. ^{−2}). This verification reveals that the numerical techniques with the treatment of the boundary conditions using NSCBC are suitable for solving the whole Navier–Stokes equations in similar chambers to the solid rocket.

To verify results for Mach number of 0.02, the transverse variations of rotational axial velocity (_{v}

The results imply that the rotational field extends to cover nearly the entire chamber by _{e}^{5},

From the results, it is noted that the absolute amplitude of the unsteady vorticity increases with increasing axial distance downstream. Moreover, the vorticity front moves almost parallel to the injection surface, and monotonic variation is noticed.

The last case study of the second set is for the generation and evolution of the temperature disturbance that accompanies the vorticity generation in the first case study.

It is clearly noted that the fluctuation of the transverse temperature gradient at the injection surface is larger than that away from the injection surface. Here again, the variation can be attributed to a combination of oscillations at the forcing frequency and eigenfunction oscillations that exist for ^{−2}).

Time | Analytical unsteady vorticity front |
Computational vorticity front | Deviation | Percentage (%) |
---|---|---|---|---|

1.0 | 1.0 | 0 | 0 | |

0.9 | 0.880 | 0.02 | 2.22 | |

0.8 | 0.77 | 0.03 | 3.75 | |

0.59 | 0.565 | 0.025 | 3.85 | |

0.43 | 0.408 | 0.022 | 5.1 | |

0.315 | 0.295 | 0.02 | 6.3 |

An investigation to chamber flow turbulence modelling in a rectangular channel with a side-wall mass injection has been studied and validated in our previous study in [

In the present paper, we only explain some important features of this flow configuration. Due to the continuous side-wall mass addition, the axial flow velocity increases towards the duct exit, and the flow is subjected to a strong streamline curvature, as shown in

This velocity gradient and streamlined curvature results in an increase in turbulent kinetic energy production. As a result, the turbulent kinetic energy increases towards the duct exit, as shown in

Therefore, the possibility of a laminar-to-turbulent transition exists. One way to identify the transition from laminar to turbulent is to examine the wall friction coefficient.

where

The results shown reveal that the flow is laminar near the head end and transitions to turbulence at a certain axial distance. Dunlap et al. [

Computational, analytical, and experimental approaches are integrated to study the complex wave patterns generated in a chamber with end-wall disturbances and the interactions with steady side-wall mass addition.

Results for the acoustic field show a bounded solution for the non-resonance frequency, while beats phenomena are observed near resonance frequency. A reasonable comparison between the experimental and computational approaches is seen.

The computational results for the rotational fields show that severe shear stresses may be generated beside the injection side-wall and transferred via convection over time to fill the entire chamber. The analytical and computational results show an excellent agreement for the propagation of the unsteady vorticity and its edges. It is seen that the amplitude of the generated shear stresses and the wavelength are many times larger beside the injection surface than near the centerline. These important phenomena reveal that the grid generation techniques must cluster near the centerline to capture the shorter wave cycles around the chamber centerline. By comparing the weakly nonlinear analytical solution and the computational solution for the vorticity front locations at different times and Mach numbers, the deviations between the two results were found to be O (10^{−2}). This verification reveals that the numerical techniques with the treatment of the boundary conditions using NSCBC is suitable for solving the whole Navier–Stokes equations in chambers similar to the solid rocket chamber.

The effect of including turbulence models in predicting the internal flow characteristics is explained by a comparison between the laminar and turbulent simulation. The comparison showed coupling between the generated unsteady vorticity arising from acoustics-fluid dynamics interaction and the turbulence intensity that exists strongly downstream of the rocket chamber. Moreover, the importance of the turbulence studies with the aspect ratio of twenty suggests that the DNS may provide a better understanding for the impact of the transverse movement of the vorticity peak on the behaviour of the erosive burning.

In the context of propellant combustion, the intensive unsteady vorticity generation and the associated shear stresses can be anticipated to have a direct impact on the stability of burning real solid rocket propellant. Moreover, more intensive computations are required to account for the existence of the convergent–divergent nozzle along with the end-wall disturbances.

Dimensionless axial length x

Dimensionless vertical length y

Aspect ratio L

Dimensionless Temperature T

Dimensionless velocity in x-Dir. u

_{i}

Mean velocity

Turbulence fluctuations

_{m}

Local mean streamwise velocity

_{f}

Friction coefficient

Dimensionless velocity in y-Dir. v

Dimensionless pressure p

Dimensionless time t

Solid Rocket Motor

Navier Stokes Characteristic Boundary Condition

Dimensionless vorticity,

Amplitude of end wall disturbances

Dissipation rate

Chamber natural frequency

Dimensionless mass density

Dimensionless forcing frequency

Dynamic viscosity, Pa.s

Viscous and conduction terms

Wave amplitude

Aspect ratio

Steady State

Wall conditions

Eigenvalues

First fundamentals

Reference values

Acoustic

Rotational component

Planar component

Converged steady state

Resonance conditions

Perturbed values

Dimensional values

_{e}

Characteristics Mach number M ^{−1})

Prandtl number

Production rate

The authors wish to record their gratitude to DSR, King Abdulaziz University, for their technical and financial support. This research was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia, Grant No. 829-722-D1435.