Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2021.014690

ARTICLE

Laminar and Turbulent Characteristics of the Acoustic/Fluid Dynamics Interactions in a Slender Simulated Solid Rocket Motor Chamber

Abdelkarim Hegab*, Faisal Albatati and Mohammed Algarni

Department of Mechanical Engineering, Faculty of Engineering at Rabigh, King Abdulaziz University, Rabigh, 21911, Saudi Arabia
*Corresponding Author: Abdelkarim Hegab. Email: amhegab@kau.edu.sa
Received: 20 October 2020; Accepted: 07 January 2021

Nomenclature

1  Introduction

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 [1–3] while the second model is devoted to studying the cold model (non-reactive) that consider the unsteady fluid dynamics inside the chamber/nozzle geometry, as in [4–6]. The recent studies that account the combustion process for the propellant concluded that the proposed conditions at or near the side-wall injection of the cold models are found to be different from that that considered with the reactive ones.

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 [7–16] studied intensively the propagation of acoustic waves (irrotational field only) in closed or open-end tubes without consideration for the mass injection from the side-wall. These studies described the behaviour of finite amplitude waves at resonance frequencies and the effect of nonlinearity on the waveform. Erickson et al. [7] studied theoretically the behaviour of forced finite amplitude oscillations in cylindrical constant and variable area ducts whose ends are closed. Shock waves were observed in the form of saw-tooth-like waveforms in the case of constant area duct when the duct oscillated at a resonance frequency. Zapirov et al. [8] performed similar theoretical and experimental works. Merkli et al. [9] studied the thermo-acoustic effects experimentally in resonant closed tubes. Their results showed the steepness of the wave and the shock wave formation near or at the resonance frequencies.

Wang et al. [10] also studied the nonlinear oscillations in a resonance gas column. They studied both closed and open tubes. In the case of closed tubes, a shock wave appears with small limiting amplitude because of nonlinearity; in the case of an open-end, the shock wave appears intermittently. Jimeniz [11] also drew this conclusion. The effect of a convergent–divergent nozzle appended to a constant area duct on the behaviour of travelling acoustic waves was described theoretically by Sileem [12] and experimentally by Sileem et al. [13]. The wave steepness increased, and an N-type waveform was noticed for the duct that ended with a convergent-area portion.

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. [14] studied the acoustic wave damping from fully opened, partially opened, and fully closed gas columns. They described the wave pattern at resonance conditions. They found that for a fully opened tube, the wave damping comes from the energy radiation into the adjacent medium. In the case of closed-end tubes, no energy radiates outside of the tube and, in turn, a shock wave is formed. In the case of a partially opened end tube, part of the acoustic energy is radiated, which prevents shock wave formation. Disselhorst et al. [15] and Garcia et al. [16] have conducted similar experimental and theoretical studies.

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. [17], Rempe et al. [18] and Zhao et al. [19]. These studies oadopt the perturbation theory in order to include the unsteadiness of the mass addition from the injection surfaces of the simulated combustion chamber as a source of the complex wave pattern inside the chamber. These generated waves are travelled in low axial Mach number (M), high Reynolds number (Re) mean flow. They concluded that the interaction between the propagated acoustic waves and the mass injection adjacent to the wall causes transverse rotational flow field and transverse temperature gradients to be clearly seen at or adjacent to the side-wall. These vorticities and the heat transfer generations may penetrate toward the centerline of the chamber and finally fill the whole domain. Moreover, their results proved that the nondimensional acoustic axial velocity component is found to be O(1) and the temperature disturbance is proportional to the Mach number with an amplitude of O(M) for M≪1, while, the gradient of the transverse velocity component is inversely proportional to the Mach number with magnitude O(1/M) and the transverse temperature gradient is O(1).

Moreover, several investigators developed a competitive linear model, as in Flandro [20] and Xuan et al. [21]. The velocity and vorticity dynamics were emphasized. They considered irrotational (acoustic) velocity and pressure disturbances, to the exclusion of rotational flow (vorticity) transients. Moreover, these studies developed a competitive linear theory to demonstrate that accurate descriptions of internal flow dynamics frequently require the presence of coexisting rotational and irrotational velocity fields systematic nonlinear asymptotic modelling were developed by Staab et al. [17] and Zhao et al. [19].

Vuillot et al. [22] and Smith et al. [23] examined the presence of similar rotational flow fields using a computational solution to the Navier Stokes equation. The acoustic disturbances are generated by imposed pressure variations along an axial location. Tseng et al. [24] and Chu et al. [25] extended related computational studies with the reaction model to account for the generated real diffusion and premixed flames adjacent to the combustion surface. The temperature variations observed in these studies arise from flame zone effects, rather than the acoustic disturbance-injected fluid interaction mentioned previously. Experimental observation by Vetel et al. [26,27] supported the concepts of vorticity generation featured by Hegab et al. [28,29].

Many investigations to chamber flow turbulence modelling, based on κ–ε, κ–ω and full Reynolds stress methods, have been studied. Liou et al. [30] revealed that the turbulence models, like κ–ε and κ–ω, that used to predict the turbulence level, are not very successful in predicting the transverse location of the turbulence intensity peak as well as show over predicting turbulence level than the measured values. For example, Beddini [31] employed a full Reynolds stress turbulence model to analyze the flow in a porous channel. At a higher Reynolds numbers, the transition to a turbulent velocity profile is predicted. The comparison to the experimental data in cold-flow simulation by Yamada et al. [32] indicates agreement with the laminar description for the mean velocity profile. The results by Sabnis et al. [33] using the low Reynolds number κ–ε turbulence model show remarkably over predicting turbulence levels but indicates agreement with the experimental data for the axial mean velocity profiles. In general, despite the over predicting turbulence level and the unsuccessful in predicting the transverse location of the turbulence peak, the turbulence models proved satisfactory in the pre-transition region of the mean-flow.

Studies by Rempe et al. [18], Zhao et al. [19], and Flandro [20] have shown that inadequate treatment of boundary conditions may cause excessive numerical diffusion and, in turn, may result in an inaccurate solution or the complete elimination of small-scale phenomena.

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 [34]. The study also presents RANS equations for time-dependent, compressible turbulent flows along with the implemented turbulence model. In addition, the numerical procedure and associated boundary conditions are explored. Moreover, an experimental solution in our recent studies [35] is adapted to validate the current computational approach.

2  Mathematical Formulation for Laminar Flow

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:

∂Q∂t+∂E∂x+∂F∂y=0,(1)

where Q, E, and F are column vectors given by

Q={ρρuρvET}E={MρuMρu2+1γMp-43MReux-13MRevyMρuv-MRevxM[ET+(γ-1)p]u-γMRePrTx}.F={MρvMρuv-A2MReuyMρv2+A2γMp-13A2MReux-A23MRevyM[ET+(γ-1)p]v-γA2MRePrTy}.

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

P=ρT.(2)

The nondimensional properties can be written as

p=p′/po′,ρ=ρ′/ρo′,T=T′/To′,u=u′/Uzo′,v=v′/(Uzo′/δ) (3)

x=x′/L′,y=y′/H′,t=t′/ta′. (4)

2.1 Initial and Boundary Conditions

For t≤0, the axial variation of the vertical velocity at the side-wall when y=1 is specified as V|wall=-Vrws(x), as shown in the physical model in Fig. 1. For t > 0, an end-wall disturbance is initiated at the head end when x=0 by a moving piston u(t)=ε*sin(ω*t) for resonance and non-resonance solution, where ε is the amplitude of the generated acoustic waves. The pressure node is specified at the exit end of the chamber, while the symmetry conditions are imposed at the centerline of the chamber when y=0.

Figure 1: The physical model for the non-reactive solid rocket motor nozzle-less chamber

The boundary and initial conditions are written as follows:

x=0;u(t)=ε*sin(ω*t)andv=0. (5)

x=1;p=1,“pressurenode.” (6)

y=0;v=0,∂u∂y=∂T∂y=∂p∂y=∂ρ∂y=0. (7)

y=1;u=0,T=1,andv=-vrws(x). (8)

t=0;u=v=0,p=T=1. (9)

2.2 Analytical Approach

2.2.1 Steady-State Flow

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

(u,v)≈∑i=0Mi(vis,uis). (10)

(p,ρ,T)≈1+∑i=0Mi+2(pis,ρis,’Tis). (11)

The asymptotic theory by Zhao et al. [19] proves that the primary viscous stresses are in the transverse direction and, as a result, the axial transportation terms can be neglected in the full Navier–Stokes equations. Eqs. (10) and (11) can be used in Eq. (1) to obtain the following equations describing an incompressible, inviscid, rotational flow:

∂uos∂x+∂vos∂y=0. (12)

uos∂uos∂x+vos∂uos∂y=-1γ∂pos∂x. (13)

∂pos∂y=0, (14)

which are valid for the limit M → 0 and A≫one.

The boundary conditions are

x=0uos=0<opendquote>closed-end.” (15)

x=1,pos=0<opendquote>pressurenode.” (16)

y=0,vos=0, (17)

∂φ∂y=0,φ=(pos,ρos,uos),<opendquote>symmetricconditions.” (18)

y=1,vos=-Vw(x),uos=0,Tos=0. (19)

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 Eqs. (12)–(14) that satisfies the boundary conditions in Eqs. (15)–(19) can be written as

vos=-Vw(x)sin(π/2y), (20)

uos=(π2∫0xVw(τ)dτ)cos(π/2y), (21)

pos=γπ24∫x1[Vw(x)∫0xVw(τ)dτ]dx, (22)

where Vw(x) is an arbitrary time-independent sidewall injection distribution. The solution with constant sidewall injection velocity is

vos=-sin(π/2y). (23)

uos=π2xcos(π/2y). (24)

pos=γπ24(1-x2). (25)

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

|ϕi,jt+1-ϕi,jt|≈10-k,(26)

where ϕ=(u,v,p,T) and k is an arbitrary coefficient based on the solution precision.

2.2.2 Unsteady Flow for Irrotational Planar Component

In this study, let the initial conditions for the unsteady flow variables (û,v^,P^,ρ^,T^) be the converged steady-state solution; using the perturbation theory in the limit M→zero yields

u(x,y,t)≈uos(x,y)+∑n=0Mnûn(x,y,t).v(x,y,t)≈vos(x,y)+∑n=0Mnv^n(x,y,t). (27)

p(x,y,t)≈pos(x,y)+M∑n=0Mnp^n(x,y,t).T(x,y,t)≈p=Tos(x,y)+M∑n=0MnT^n(x,y,t). (28)

Inserting Eqs. (27) and (28) in Eq. (1), the resultant planar acoustic equation for the order of Mach number O(M) terms can be written as follows:

∂2u0~∂t2=∂2u0~∂x2,(29)

with the following initial and boundary conditions:

t=0,u0~=0,∂u0~∂t=0. (30)

x=0,u0~=εsinωt (31)

x=1,∂uo~∂x=0. (32)

Since the left boundary condition in Eq. (31) is nonhomogeneous, the following transformation equation u=u0~-εsinωt is used to generate homogeneous boundary conditions with a solvable nonhomogeneous wave equation as

∂2u∂t2=∂2u∂x2-εω2sinωt,(33)

With the new initial and boundary conditions

t=0,u=0,∂u∂t=-εω. (34)

x=0,u=0 (35)

x=1,∂u∂x=0. (36)

Assume the homogeneous solution for Eq. (33) is represented by u′(x,t) and keep the boundary conditions from Eqs. (34)–(36) for u′(x,t). Using the separation of variable technique, the derived general homogeneous solution can be written as

u′(x,t)=∑n=0n=∞{Cnsinλnt+Dncosλnt}sinλnx,(37)

where λn refer to the eigenfunctions of the chamber and are represented by λn=(n+12)π, n=0,1,2,3,…∞, Cn=2λn∫x=0x=1-εωsinλnxdx and Dn = 0. Once again assume the non-homogeneous solution for Eq. (35) is represented by û(x,t) and retain the boundary conditions in Eqs. (34)–(36) except ∂û∂t=0 in Eq. (33). The general solution for the nonhomogeneous part of Eq. (33) for the non-resonance case is derived and written as

û(x,t)=∑n=0∞{-2εω3λn2(λn2-ω2)sinλnt+2εω2λn(λn2-ω2)sinωt}sinλnx.(38)

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 Eqs. (37) and (38):

u0~(x,t)|n≠n*=εsinωt+∑n=0,n≠n*∞-2εωλn2sinλntsinλnx+∑n=0,n≠n*∞{-2εω3λn2(λn2-ω2)sinλnt+2εω2λn(λn2-ω2)sinωt}sinλnx.(39)

The resonance solution when ω=λn* is derived from the non-homogeneous part by assuming that the particular solution is proportional to the product of tcosωt. The final form is obtained and written as

u0~(x,t)|n=n*=εsinωt-({1λn*}sinλn*t+tcosλn*t)sinλn*x.(40)

Hegab [26] and Zhao et al. [19] described the irrotational acoustic system that was employed by Flandro [20] and showed that the rest of the planer irrotational functions ρ0~(x,t),T0~(x,t),andP0~(x,t) have mathematical relations to the velocity acoustic field u0~(x,t) as follows:

∂ρ0~∂t=-∂u0~∂x,∂u0~∂t=-1γ∂P0~∂x,∂T0~∂t=γ-1γ∂P0~∂t.(41)

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 Eqs. (39) and (41), considering the pressure node (P = 1) boundary conditions at the exit of the chamber. The final pressure equation for the non-resonance case is derived and written as

P0~(x,t)|n≠n*=γεωcos(ωt)(x-1)+∑n=0,n≠n*∞2γεωλn2-ω2(ω2cosωt-λn2cosλnt)cosλnx.(42)

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

P0~(x,t)|n=n*=γεωcos(ωt)(x-1)-γ(2cosλn*t-λn*tsinλn*t)cosλn*x.(43)

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

•   For non-resonance solution (n≠n*),

T0~(x,t)|n≠n*=γ-1γ{γεωcos(ωt)(x-1)+∑n=0,n≠n*∞2γεωλn2-ω2(ω2cosωt-λn2cosλnt)cosλnx}.(44)

•   For resonance solution (n = n*),

T0~(x,t)|n=n*=(γ-1){εωcos(ωt)(x-1)-(2cosλn*t-λn*tsinλn*t)cosλn*x}.(45)

Moreover, the perturbed density field is determined from

ρ0~(x,t)=P0~(x,t)-T0~(x,t).(46)

2.3 Computational Approach

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 [36–38]. This method applied recently in [39] and found to be phase accurate and therefore, suitable for describing many wave cycles and wave interaction problems.

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

∂ρ∂t+1C2[℘2+12(℘4+℘1)]+M∂ρv∂y=0. (47)

∂u∂t+12ρC[(℘4-℘1)]+Mv∂u∂y=χx/ρ. (48)

∂v∂t+℘3+Mv∂v∂y+A2γM1ρ∂p∂y=χy/ρ. (49)

∂p∂t+12[(℘4+℘1)]+Mv∂p∂y+γMp∂v∂y=χE. (50)

where Xx and Xy refer to the viscous terms and XE refers to the combination with the conduction terms and C=C′/C0′. The amplitudes of the left and the right-running waves (℘1,℘4) and the entropy waves (℘2,℘3) can be written as

℘1=λ1(1γ∂p∂x-MρC∂u∂x);λ1=uM-C. (51)

℘2=λ2(C2∂ρ∂x-1γ∂p∂x);λ2=Mu. (52)

℘3=λ3∂v∂x;λ3=Mu. (53)

℘4=λ4(1γ∂p∂x+MρC∂u∂x);λ4=uM+C. (54)

2.3.1 Exit Boundary Conditions

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

℘1=2χE-℘4,(55)

and the remaining conditions are

℘2=Mu(C2∂ρ∂x-1γ∂p∂x),℘3=Mu∂v∂x,and℘4=(uM+c)(1γ∂p∂x+MρC∂u∂x).(56)

The nondimensional equations can be written as follows:

∂ρ∂t+Mu∂ρ∂x-MuγC2∂p∂x+Mv∂ρ∂y+δ2M(γ-1)RePr1C2∂2T∂y2. (57)

∂u∂t+1ρC[(u+CM)(1γ∂p∂x+MρC∂u∂x)]+Mv∂u∂y-δ2(γ-1)RePr1ρC∂2T∂y2-Mδ2Re1ρ∂2u∂y2=0. (58)

∂v∂t+M(u∂v∂x+v∂v∂y)=0. (59)

dpdt=0. (60)

2.3.2 Head End Boundary Condition

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

u(0,y,t)=εsin(ωt);∂u∂y=0;∂u∂t=εωcos(ωt). (61)

λ1=εMsin(ωt)-C;λ2=Mεsin(ωt);λ3=Mεsin(ωt);λ4=εMsin(ωt)+C. (62)

The left-running wave {℘1} propagates out of the computational domain and can be computed from the interior points using the definitions Eq. (62). The right-running wave {℘4} points into the domain. However, information on the exterior points is unavailable. Thus, the derivative of the time-dependent axial velocity variation in Eq. (61) is used to infer the unknown amplitude of the reflected wave {℘4}. Eq. (47) suggests that the necessary condition is

℘4=2Cχx+℘1-2ρCεωcos(ωt)-2ρCMv∂u∂y,(63)

and the remaining amplitudes are

℘1=(εMsin(ωt)-C)(1γ∂p∂x-MρC∂u∂x). (64)

℘2=Mεsin(ωt)(C2∂ρ∂x-1γ∂p∂x). (65)

℘3=Mεsin(ωt)∂v∂x. (66)

The wave-mode forms of the Navier–Stokes equations are simplified by ignoring the axial transportation terms in (χx, χy, and χE) and the transverse pressure gradient as well. These reductions are justified by the asymptotic of Zhao et al. [19] and have been employed successfully by Kirkkopru et al. [36] and Hegab et al. [39].

The nondimensional equations can be written in the form:

∂ρ∂t+1C2[Mεsin(ωt)(C2∂ρ∂x-1γ∂p∂x)-1γdpdt]+Mv∂ρ∂y+δ2M(γ-1)RePr1C2∂2T∂y2=0. (67)

u(0,y,t)=εsin(ωt). (68)

∂v∂t+M(εsin(ωt)∂v∂x+v∂v∂y)=0. (69)

∂p∂t+∂p∂t+12[(2Cχx+(εMsin(ωt)-C)(1γ∂p∂x-MρC∂u∂x)-2ρCεωcos(ωt)-2ρCMv∂u∂y+(εMsin(ωt)-C)(1γ∂p∂x-MρC∂u∂x))]+Mv∂p∂y+γMp∂v∂y=χE.(70)

The nondimensional primitive variables (ρ, u, and v) can be computed using one-side finite-difference approximation.

More details about the NSCBC techniques are found in [28].

The accuracy of the current explicit numerical schemes has been justified by Courant–Friedrichs–Lewy (CFL) as in MacCormack [37]. They derived the following criterion to determine the size of the time step: (ΔtCFL)i,j=1[M(|ui,j|Δx+|vi,j|Δy)+Ci,j1Δx2+δ2Δy2+3MRe(1Δx2+δ2Δy2)], where Δt=min[ϑ(ΔtCFL)i,j] and the Courant number ϑ is less than one. Precisely, the numerical results from the current study show that the best results are found for 0.5≤ϑ≤0.65. In order to resolve the axial and transverse structure, the number of mesh sizes in the x and y direction is chosen to be 101×201 to capture the small amplitude wave cycles near the centerline with uniform grids.

3  Mathematical Formulation for Turbulent Flow Characteristics

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

∂ρ′∂t′+∂∂xi(ρ′ui)=0,(71)

Momentum Equation

∂(ρ′ui)∂t′+∂∂xj(ρ′uiuj)=-∂p′∂xi+∂∂xj[μ(∂ui∂xj+∂uj∂xi-23δij∂ul∂xl)]+∂∂xj(-ρ′ui′uj′¯),(72)

Energy Equation

∂(ρ′e′)∂t′+∂[uj(ρ′e′+p′)]∂xj=∂∂xj[(λ+Cpμt/Prt)∂T′∂xj-ui(ρ’ ui′uj′¯)],(73)

where ui denotes the mean velocities, ui′ and uj′ are the turbulent fluctuations. The total energy per unit volume is defined as ρ′e′=ρ′CvT′+0.5ρ’ ui2, where Cv is the specific heat at a constant volume and T′ is the temperature.

The momentum equations contain additional terms, known as Reynolds stresses, which represent the effects of turbulence. These Reynolds stresses, -ρ′ui′uj′¯, must be modelled to close the governing system of equations. This can be achieved through the appropriate selection of a turbulence model.

The Reynolds-averaged conservation Eqs. (71), (72), and (73) for unsteady compressible turbulent flow along with a turbulence model are integrated with the equation of state p′=ρ′RT′ to close the system of governing equations.

3.1 Turbulence Modeling

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 Eq. (72). The conventional method to relate the Reynolds stresses to the mean velocity gradients is based on the Boussinesq hypothesis [40]:

-ρ′ui′uj′=μt(∂ui∂xj+∂uj∂xi)-23(ρ′k+μt∂uk∂xk)δij,(74)

where δij is the Kronecker delta function (δij=1, if i=j, and δij=0, if i≠j), and k is the turbulent kinetic energy. One of the most challenging problems of turbulent flow is the estimation of the turbulent viscosity.

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 [41]:

∂(ρ′k)∂t′+∂∂xj(ρ′ujk)=∂∂xj[(μ+μtσk)∂k∂xj]+ρ′(Pr′-ε) (75)

∂(ρ′ε′)∂t′+∂∂xj(ρ′ujε′)=∂∂xj[(μ+μtσε)∂ε′∂xj]+ρ′(C1εPr′ε′k-C2εε′2k+C3εPr′2k) (76)

uj∂v′2¯∂xj=kf-6v′2¯ε′k+∂∂xj[(ν+νt)∂v′2¯∂xj] (77)

L2∂2f∂xj2-f=1T[(C1-6)v′2¯k-23(C1-1)]-C2Gk (78)

T=max[kε′,6νε′] (79)

L=CLmax[k3/2ε′,Cη(ν3ε′)1/4] (80)

where Pr′ and ε′ represent the production rate and dissipation rate of the turbulent kinetic energy, k, respectively. The production rate is related to the mean strain of the velocity field through the Boussinesq assumption:

Pr′=μtS2(81)

where S is defined as follows:

S=1ρ′(∂uj∂xi+∂ui∂xj-23δij∂uk∂xk)∂ui∂xj.(82)

The turbulent viscosity is given by the following:

μt=ρ′Cμv2T′.

The constants of the model are given in as follows:

Cμ=0.22,σk=1,σε=1.3,C1ε=1.4(1+0.05k/v2);C2ε=1.9,C1=1.4;C2=0.3;CL=0.23;Cη=70.(83)

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.

4  Experimental Setup

The experimental setup layout is shown in Fig. 2. It consists of the following parts:

Figure 2: The layout of the test rig [35]

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 (11). These transducers are connected to a computer through a built-in data acquisition system (13). The motor frequencies are changed using voltage Varies. The range of the generated frequencies from the available devices is small enough to catch the first natural frequencies (250 Hz). The pressure traces at x=0.125, ε=0.045, and ω=0.52 (f′=63.44 Hz) is recorded.

5  Results and Discussion

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. [17]:

u(x,y,t)=ur(x,y,t)+us(x,y)+up(x,t)(84)

where, us refers to the converged steady flow-field, which is used as an initial condition for the unsteady computations. The planar part of the flow-field, up, which describes weakly viscous acoustic effects, is created by the difference between the unsteady and the steady axial speed at y = 0, where the rotational part uv (0, t) = 0 is established from the boundary conditions. The unsteady rotational part uv is found from Eq. (84) as the difference between the total unsteady speed and the calculated us and up.

The total dimensionless unsteady vorticity is

Ω=-[∂uv∂y-1δ2∂vv∂x],(85)