^{1,a)}, Robert D. Moser

^{2,b)}and Fady M. Najjar

^{3,c)}

### Abstract

Compressible turbulent flow in a periodic plane channel with mass injecting walls is studied as a simplified model for core flow in a solid-propellant rocket motor with homogeneous propellant and other injection-driven internal flows. In this model problem, the streamwise direction was asymptotically homogenized by assuming that at large distances from the closed end, both the mean and rms of turbulent fluctuations evolve slowly in the streamwise direction when compared to the turbulent fluctuations themselves. The Navier–Stokes equations were then modified to account for this slow growth. A direct numerical simulation of the homogenized compressible injection-driven turbulent flow was then conducted for conditions occurring at a streamwise location situated 40 channel half-widths from the closed off end and at an injection Reynolds number of approximately 190. The turbulence in this modelflow was found to be only weakly compressible, although significant compressibility existed in the mean flow. As in nontranspired channels, turbulence resulted in increased near-wall shear for the mean streamwise velocity. When normalized by the average rate of turbulence production, the magnitudes of near-wall velocity fluctuations were similar to those in the log region of nontranspired wall-bounded turbulence. However, the sharp peak in streamwise velocity fluctuations observed in nontranspired channels was absent. While streaks and inclined vortices were observed in the near-wall region, their structure was very similar to those observed in the log region of nontranspired channels. These differences are attributed to the absence of a viscous sublayer in the transpired case which in turn is the result of the fact that the no-slip condition for the transpired case is an inviscid boundary condition. That is, unlike nontranspired walls, with transpiration, zero tangential velocity boundary conditions can be imposed at the wall for the Euler (inviscid) equations. The results of this study have important implications on the ability of turbulencemodels to predict this flow.

This work was conducted at the Center for Simulation of Advanced Rockets at the University of Illinois and supported by the Department of Energy through the University of California under Subcontract No. B341494.

I. INTRODUCTION

A. Previous work

B. Present study

II. MATHEMATICAL MODEL

A. Flow configuration

B. Governing equations

C. Homogenization procedure

1. Multiple scale analysis

2. Modeling the slow derivatives

3. Validity of the homogenization procedure

III. SIMULATION DETAILS

A. Numerical method

1. Spatial discretization

2. Temporal discretization

3. Boundary and initial conditions

B. Simulation parameters

C. Adequacy of resolution and domain size

IV. NATURE OF INJECTION-DRIVEN TURBULENCE

A. Scaling

B. Mean statistics

1. Mean velocity profiles

2. Mean density and temperature profiles

3. Mean kinetic energy budget

C. Turbulence statistics

1. rms velocity fluctuation profiles

2. Reynolds stress and turbulent kinetic energy budgets

D. Structure of near-wall turbulence

E. Compressibility effects

V. DISCUSSION AND CONCLUSIONS

### Key Topics

- Turbulent flows
- 131.0
- Reynolds stress modeling
- 31.0
- Channel flows
- 29.0
- Vortex dynamics
- 29.0
- Turbulence simulations
- 28.0

## Figures

Schematic representation of a solid-propellant rocket showing casing, propellant, and nozzle (top) and an idealized injection-driven flow (bottom), which models a solid rocket by replacing the evolution of gas by combustion at the propellant surface with mass injection through the walls.

Schematic representation of a solid-propellant rocket showing casing, propellant, and nozzle (top) and an idealized injection-driven flow (bottom), which models a solid rocket by replacing the evolution of gas by combustion at the propellant surface with mass injection through the walls.

Schematic representation of the model solid rocket motor.

Schematic representation of the model solid rocket motor.

Actual (—) and modeled (-⋅-⋅-) values of (a) , (b) , and (c) near the center of the channel measured in the spatially developing injection-driven flow of Wasistho and Moser (Ref. 18). Other locations are similar or have better agreement.

Actual (—) and modeled (-⋅-⋅-) values of (a) , (b) , and (c) near the center of the channel measured in the spatially developing injection-driven flow of Wasistho and Moser (Ref. 18). Other locations are similar or have better agreement.

[(a) and (b)] One-dimensional velocity spectra and [(c) and (d)] premultiplied spectra, [(a) and (c)] near the wall and [(b) and (d)] near the channel center . Shown are the spectra for (—), (----), and (-⋅-⋅-). The quantity is defined in Sec. IV A.

[(a) and (b)] One-dimensional velocity spectra and [(c) and (d)] premultiplied spectra, [(a) and (c)] near the wall and [(b) and (d)] near the channel center . Shown are the spectra for (—), (----), and (-⋅-⋅-). The quantity is defined in Sec. IV A.

rms value of the residual normalized by the rms value of the time derivative, . The -spline knots are spaced according to , where is the knot spacing at the center line and is the knot spacing at a given location. 192 collocation points are used in the wall normal direction.

rms value of the residual normalized by the rms value of the time derivative, . The -spline knots are spaced according to , where is the knot spacing at the center line and is the knot spacing at a given location. 192 collocation points are used in the wall normal direction.

Mean streamwise and wall normal velocities over half the channel width for the turbulent (—) and laminar (-⋅-⋅-) cases. The center of the channel is at , and the streamwise and wall-normal velocities are symmetric and antisymmetric about the center, respectively.

Mean streamwise and wall normal velocities over half the channel width for the turbulent (—) and laminar (-⋅-⋅-) cases. The center of the channel is at , and the streamwise and wall-normal velocities are symmetric and antisymmetric about the center, respectively.

Terms in the mean streamwise momentum equation, normalized by : (a) Turbulent case and (b) laminar case. (—), (⋯), (----), (---), and (-⋅-⋅-).

Terms in the mean streamwise momentum equation, normalized by : (a) Turbulent case and (b) laminar case. (—), (⋯), (----), (---), and (-⋅-⋅-).

Mean density and temperature profiles: Turbulent case (—) and laminar case (-⋅-⋅-).

Mean density and temperature profiles: Turbulent case (—) and laminar case (-⋅-⋅-).

Terms in the mean kinetic energy equation, normalized by . Convection of mean kinetic energy (—), convection of Reynolds stress (----), production of turbulent kinetic energy (⋯), contribution from slow growth terms (-⋅-⋅-), work done by mean pressure gradient (–◇–), mean pressure diffusion (–⊖–), mean pressure dilatation (–◻–), mean viscous diffusion (×), and mean dissipation (∗).

Terms in the mean kinetic energy equation, normalized by . Convection of mean kinetic energy (—), convection of Reynolds stress (----), production of turbulent kinetic energy (⋯), contribution from slow growth terms (-⋅-⋅-), work done by mean pressure gradient (–◇–), mean pressure diffusion (–⊖–), mean pressure dilatation (–◻–), mean viscous diffusion (×), and mean dissipation (∗).

rms velocity fluctuation profiles normalized by (a) and (b) for the transpiration-driven channel and for a nontranspired turbulent channel at . Shown are (—) (----), and (-⋅-⋅). The curves for the nontranspired channel are indicated with the ● symbol.

rms velocity fluctuation profiles normalized by (a) and (b) for the transpiration-driven channel and for a nontranspired turbulent channel at . Shown are (—) (----), and (-⋅-⋅). The curves for the nontranspired channel are indicated with the ● symbol.

Rate of turbulent kinetic energy dissipation (—) and streamwise convection (slow growth term) (----) normalized by the volume-averaged rate of turbulent kinetic energy production in the transpired (current) and nontranspired channels (●). (b) is a zoomed version of (a) to show the central region of the channels.

Rate of turbulent kinetic energy dissipation (—) and streamwise convection (slow growth term) (----) normalized by the volume-averaged rate of turbulent kinetic energy production in the transpired (current) and nontranspired channels (●). (b) is a zoomed version of (a) to show the central region of the channels.

Rate of turbulent kinetic energy dissipation normalized by in the transpired (current) and nontranspired channels (●). Note that this quantity is unbounded at the wall because goes to zero.

Rate of turbulent kinetic energy dissipation normalized by in the transpired (current) and nontranspired channels (●). Note that this quantity is unbounded at the wall because goes to zero.

Normalized two-point correlations at the channel center line with separations in the (a) and (b) directions from the current transpiration-driven flow and the nontranspired channel (●). Shown are (—), , (----), and .(-⋅-⋅-)

Normalized two-point correlations at the channel center line with separations in the (a) and (b) directions from the current transpiration-driven flow and the nontranspired channel (●). Shown are (—), , (----), and .(-⋅-⋅-)

Terms in the Reynolds stress transport equation, normalized by the average production of turbulent kinetic energy. (a) , (b) , (c) , and (d) . Convection of Reynolds stress (—), production (⋯), turbulent diffusion (----), pressure diffusion (-⋅-⋅-), pressure strain (–⊖–), viscous diffusion (–◻–), dissipation (–◇–), contribution from slow growth terms (×), and compressibility terms (∗).

Terms in the Reynolds stress transport equation, normalized by the average production of turbulent kinetic energy. (a) , (b) , (c) , and (d) . Convection of Reynolds stress (—), production (⋯), turbulent diffusion (----), pressure diffusion (-⋅-⋅-), pressure strain (–⊖–), viscous diffusion (–◻–), dissipation (–◇–), contribution from slow growth terms (×), and compressibility terms (∗).

Terms in the Reynolds stress transport equation for nontranspired turbulent channel flow at , normalized by the average production of turbulent kinetic energy. (a) , (b) , (c) , and (d) . Production (⋯), turbulent diffusion (----), pressure diffusion (-⋅-⋅-), pressure strain (–⊖–), viscous diffusion (–◻–), and dissipation (–◇–).

Terms in the Reynolds stress transport equation for nontranspired turbulent channel flow at , normalized by the average production of turbulent kinetic energy. (a) , (b) , (c) , and (d) . Production (⋯), turbulent diffusion (----), pressure diffusion (-⋅-⋅-), pressure strain (–⊖–), viscous diffusion (–◻–), and dissipation (–◇–).

Terms in the turbulent kinetic energy equation for the (a) transpiration-driven (b) and non-transpired channels, normalized by the average production of turbulent kinetic energy. Convection of turbulent kinetic energy (—), production (⋯), turbulent diffusion (----), pressure diffusion (-⋅-⋅-), pressure dilatation (–⊖–), viscous diffusion (–◻–), dissipation (–◇–), contribution from slow growth terms (×), and compressibility terms (∗).

Terms in the turbulent kinetic energy equation for the (a) transpiration-driven (b) and non-transpired channels, normalized by the average production of turbulent kinetic energy. Convection of turbulent kinetic energy (—), production (⋯), turbulent diffusion (----), pressure diffusion (-⋅-⋅-), pressure dilatation (–⊖–), viscous diffusion (–◻–), dissipation (–◇–), contribution from slow growth terms (×), and compressibility terms (∗).

Streamwise vorticity fluctuations obtained for the transpiration-driven channel.

Streamwise vorticity fluctuations obtained for the transpiration-driven channel.

Two-point autocorrelation for velocities in the spanwise direction at . (—), (----), and (-⋅-⋅-).

Two-point autocorrelation for velocities in the spanwise direction at . (—), (----), and (-⋅-⋅-).

Contours of streamwise velocity fluctuations. (a) Present study: . Turbulent channel flow at : (b) and (c) . In all three plots, blue represents negative values of and red represents positive values of .

Contours of streamwise velocity fluctuations. (a) Present study: . Turbulent channel flow at : (b) and (c) . In all three plots, blue represents negative values of and red represents positive values of .

PDF of (a) and (b) in the transpiration-driven channel, where is the angle made by the vorticity vector with the positive vertical axis and is the angle made with the positive streamwise axis. Curves are for (—), (⋯), (----), and (-⋅-⋅-).

PDF of (a) and (b) in the transpiration-driven channel, where is the angle made by the vorticity vector with the positive vertical axis and is the angle made with the positive streamwise axis. Curves are for (—), (⋯), (----), and (-⋅-⋅-).

Joint PDF of and for the present study and for plane turbulent channel flow at . Present study: (a) and (b) . Turbulent channel flow at : (c) and (d) .

Joint PDF of and for the present study and for plane turbulent channel flow at . Present study: (a) and (b) . Turbulent channel flow at : (c) and (d) .

Conditionally averaged vorticity magnitude at and for the present study and for plane turbulent channel flow at . The magnitude of vorticity has been normalized by the square root of enstrophy at that particular wall normal location. Present study: (a) and (b) . Turbulent channel flow at : (c) and (d) .

Conditionally averaged vorticity magnitude at and for the present study and for plane turbulent channel flow at . The magnitude of vorticity has been normalized by the square root of enstrophy at that particular wall normal location. Present study: (a) and (b) . Turbulent channel flow at : (c) and (d) .

Contours of spanwise vorticity velocity fluctuations in an plane for (a) the present study and (b) the turbulent channel flow at . Blue represents negative values of and red represents positive values of .

Contours of spanwise vorticity velocity fluctuations in an plane for (a) the present study and (b) the turbulent channel flow at . Blue represents negative values of and red represents positive values of .

Spanwise vorticity contours in a spanwise plane from a LES of spatially developing solid rocket motor flow conducted by Wasistho and Moser (Ref. 18). The parameters and the geometry used in the simulation are the same as those used in the experiment of Traineau *et al.* (Ref. 5). The streamwise and wall normal coordinates in the figure are in meters.

Spanwise vorticity contours in a spanwise plane from a LES of spatially developing solid rocket motor flow conducted by Wasistho and Moser (Ref. 18). The parameters and the geometry used in the simulation are the same as those used in the experiment of Traineau *et al.* (Ref. 5). The streamwise and wall normal coordinates in the figure are in meters.

rms density and temperature fluctuations normalized by their corresponding mean values.

rms density and temperature fluctuations normalized by their corresponding mean values.

(a) Mean Mach number . (b) Turbulent Mach number (-⋅-⋅-) and rms Mach number (----).

(a) Mean Mach number . (b) Turbulent Mach number (-⋅-⋅-) and rms Mach number (----).

Ratio of (a) mean square dilatation to enstrophy and (b) pressure dilatation to homogeneous solenoidal dissipation .

Ratio of (a) mean square dilatation to enstrophy and (b) pressure dilatation to homogeneous solenoidal dissipation .

## Tables

Comparison of parameters used in numerical simulations of injection-driven flows including the current study. reported by Apte and Yang (Ref. 7) is evaluated at the head end. Liou *et al.* (Ref. 6) performed a two-dimensional simulation of the Euler equations.

Comparison of parameters used in numerical simulations of injection-driven flows including the current study. reported by Apte and Yang (Ref. 7) is evaluated at the head end. Liou *et al.* (Ref. 6) performed a two-dimensional simulation of the Euler equations.

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